WYDZIAŁY POLITECHNICZNE KRAKÓW

BIBLIOTEKA GŁÓWNA

L. inw.

Biblioteka Politechniki Krak

owskiej

«TATE OF RHODE ISLAND'S VICTORY BOOK CAMPAIGN FOR THE ARMED FOKCLS RHODE ISLAND STATE LIBRARY EXTENSION SERVICE

-b-'Vf

Jr &

THE

ELASTICITY AND RESISTANCE

OF THE

MATERIALS OF ENGINEERING.

BY

WM. H. BURR, C.E.,

FORMERLY WILLIAM HOWARD HART PROFESSOR OF RATIONAL AND TECHNICAL MECHANICS AT RENSSELAER POLYTECHNIC INSTITUTE.

SECOND EDITION.

REVISED AND ENLARGED.

NEW YORK:

JOHN WILEY & SONS. 1888.

X>s

i g:

W

Copyright, 1883,

By WM. H. BURR.

BIIU8ÏÏÜA Pülü'Büälsm

IŁ & A K, Ów

13385

PRESS OF J. J. LITTLE 4 CO.,

NOS. 10 TO 20 ASTOR PLACE, NEW YORK.

PREFACE.

THIS work has been the outgrowth of lectures on the elasticity and resistance of materials, given by the author to succeeding classes of students in the department of civil engineering at the Rensselaer Polytechnic Institute. Although those lectures, as given, form the basis of the work, they have, of course, been considerably elaborated and extended, so as to cover many details of the subject which it would be impossible to include in any ordinary technical course of study, but which, at the same time, are necessary to a complete and philosophical treatment.

The first, or “ Rational,” part of this work, is intended to furnish an analytical or rational basis for the “ Technical” or practical development contained in Part II. It will undoubtedly impress a great number, and perhaps all engineers in active practice, that it is unnecessary to the proper treatment of such a subject. Indeed, a very considerably extended experience in iron and steel constructions places the author himself in position to fully appreciate the weight of such a criticism at the first glance. But it may be contended, and he thinks must be admitted, that the present advanced state of engineering as a profession implies the existence of something that may be called the “ natural philosophy ” of engineering. In other words, the engineer of the present time must meet the increased and increasing demands upon him in some one or more specialty, not only by the aid of sound common sense and a well-trained judgment, but also by a systematic knowledge of so much of natural philosophy as is involved in practical engineering operations. The ideal simplicity of stresses and strains in a perfectly isotropic body, and the clearness of action of “ external forces ” applied at any “ point ” or distributed over any “ surface ” according to some known and well-defined law, are not, it is evident, the things the technical student will encounter in his practice as an engineer. He will find few, if any, of the ideal conditions realizêd, and the difficulties constantly confronting him will be those involving modifications of the analytical or mathematical results based upon ideal quantities and conditions. Nevertheless, it is certainly true that in engineering practice he deals with precisely the same quantities as in the natural philosophy of engineering, but in different amounts and with far different and vastly more complicated conditions. And it is equally true that a correct knowledge of the consequent modifications, both in kind and amount must be based not only upon a correct recognition of the actual circumstances into which the ideal conditions transmute themselves in engineering works, i.e., upon sound practical knowledge, but also upon a thorough comprehension of the things involved, in the abstract, and the laws governing their actions and relations. In other words, but in essentials the same, an engineer’s preparation for active practice must consist both of that philosophical training in what is largely ideal, and which he acquires in the technical school, and of the purely practical training of the first few years of his professional life.

The first, or “ Rational,” part of this work is, then, designed for few others than technical students, although there are engineers whose tastes induce or circumstances require investigations in connection with the elasticity and resistance of materials. The writer would esteem himself fortunate if the mathematical portion of the book should find favor with such individuals and be useful to them.

In Part II. the mathematical results obtained in Part I. are subjected to the test of experiment. By the aid of experimental results in a great variety of material, empirical coefficients are established which involve the varied and complicated circumstances of material in actual use. The formulae, which otherwise express ideal conditions only, are thus rendered of the greatest practical value ; in fact they constitute the only reliable practical formulae in use by engineers.

All the experimental results are, of course, compilations only, but they have been taken in all cases from what are believed to be trustworthy sources, and it has been the intention to give credit to the experimenter in every case. It may appear that too great a profusion of experimental results has been introduced. But it has been the aim of the author, even at the risk of being tedious, to represent truly and completely the great variety of both quantitative and qualitative phenomena exhibited by material under test ; to show not only the variation in .products of different mills but the variation in different products of the same mill ; to exhibit the variations due to difference in size, shape, relative dimensions and condition of specimens ; to show that specimens apparently identically the same may even give considerable diversity in results and to prove the difference between the finished member and its component parts, as well as to indicate the direction in which further investigations may most profitably be prosecuted. A few groups of tests are not sufficient to the attainment of such a series of results.

In the course of the preparation of the MSS. the author found it necessary to reduce a very great amount of experimental quantities from the crude shape of a mere record of tests to a useful condition, and to change many others from one unit to another. These numerical operations involved much labor, and although they were performed with great care and repeated in almost every instance, it is very probable that errors have crept in, though it is believed that there are few,

fb

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ÂMES i

if any, of importance. The writer will feel indebted to any one who will discover them. In all cases, unless otherwise specifically stated, the ultimate resistance, elastic limit and coefficient of elasticity are expressed in pounds per square inch of original area of section.

In a few of the tables of Art. 32 the “ strains,” i.e., amounts of stretch, are given as decimal fractions (hundredths) of original length, while the otherwise uniform method of expression is by means of whole numbers giving per cents, of original dimensions. This diversity is unintentional and due to the fact that a part of the MSS. was a portion of that used in lectures.

The distinction between “ stress ” and “ strain ” conflicts, so far as the latter word is concerned, with ordinary usage. But some distinction is absolutely necessary, and that used has had a long existence, and is at least consistent with the etymology of the words. There certainly can be no way of filling the hiatus caused by the absence of a word to concisely express changes of shape or dimensions, without some inconvenience, and that followed will probably cause as little as any.

W. H. B.

Rensselaer Polytechnic Institute, 1883.

PREFACE TO SECOND EDITION.

The present edition of this work is the result of a careful revision and extension of the first edition. Since the issue of the latter, very considerable developments have been made in constructions of iron and steel, particularly in those of steel ; civil engineers are also pushing their investigations in timber, cement, cement mortar, building stones, bricks, etc., with energy and corresponding success. Abstract results in pure engineering science are constantly finding their applications in the practical operations of the engineer ; while experimental results with members built for actual use in structures are continually furnishing bases for new inductions of the greatest practical or technical value.¹ It has been the design to bring the present volume into £uch a condition as to be quite abreast of these material advances. Considerable old matter has been canceled and new matter supplied, and Addenda to many Articles have been written. For convenience of reference it is believed well to state that new mattef and Addenda will be found in or added to Arts. 20, 21, 24, 32, 34, 42, 45, 46, 51, 57, 65, 66, 67, 70, 73, 76, 78, 84, 85, 86, 87, 89, 90, 91 and Addenda at the end of the book. These additions are entirely in the domain of engineering practice and contain valuable practical data.

W. H. B.

Phœnixville, Pa., Sept., 1887.

CONTENTS.

PART I.—RATIONAL.

CHAPTER I.

GENERAL THEORY OF ELASTICITY IN AMORPHOUS SOLID

BODIES.

ART.

1. —General Statements.....

2. —Coefficients of Elasticity

3. —Lateral Strains.........

4. —Relation between the Coefficients of Elasticity for Shearing and Direct

Stress in a Homogeneous Body.................................

5. —Expressions for Tangential and Direct Stresses in Terms of the Rates of

Strains at any Point of a Homogeneous Body....................

6. —General Equations of Internal Motion and Equilibrium.................

7. —Equations of Motion and Equilibrium in Semi-Polar Co-ordinates........

8. —Equations of Motion and Equilibrium in Polar Co-ordinates.............

PACK

14

20

27

CHAPTER II.

THICK, HOLLOW CYLINDERS AND SPHERES, AND TORSION.

9.—Thick, Hollow Cylinders. 10.—Torsion in Equilibrium ..

36

43

Equations of condition in rectangular co-ordinates

Solutions of Equations (13) and (21)............

Elliptical section about its centre................

Equilateral triangle about its centre of gravity ....

50

53

54

Rectangular section about an axis passing through its centre of

gravity..... ......................................

Circular section about its centre .    .........................

General observations......................................

xi.—Torsional Oscillations of Circular Cylinders..........................

12.—Thick, Hollow Spheres............................................

59

75

77

78

84

CHAPTER III.

THE ENERGY OF ELASTICITY.

13. —Work Expended in Producing Strains......................

14. —Resilience....... . ...................................

15. —Suddenly Applied External Forces or Loads. ................ ..

16. —Longitudinal Oscillations of a Straight Bar of Uniform Section

O O 00 O' O' O'

lun

CHAPTER IV.

THEORY OF FLEXURE.

17. —General Formulae.................................................

18. —The Common Theory of Flexure....................................

19. —Deflection by the Common Theory of Flexure........................

20. —External Bending Moments and Shears in General ...................

21. —Moments and Shears in Special Cases................................

Case I................................................'.

Case II.................................................

Case III................................................

22. —Recapitulation of the General Formulae of the Common Theory of

Flexure..........................................

23. —The Theorem of Three Moments.........................

23a.—Reactions under Continuous Beam of any Number of Spans

Special method for ordinary use.................

Special case of equal spans......................

24. —The Neutral Curve for Special Cases .....................

Case I..

Case II.

Case III

106

W M W

WWW

CO CO CO co

139

142

143

146

157

165

169

171

172

174

25. —The Flexure of Long Columns......................................

26. —Graphical Determination of the Resistance of a Beam.................

27. —The Common Theory of Flexure with Unequal Values of Coefficients of

Elasticity...........,.......................................

28. —Greatest Stresses at any Point in a Beam.............................

190 b

196

199

202

PART II.—TECHNICAL.

CHAPTER V.

TENSION.

2g.—General Observations.—Limit of Elasticity

30. —Ultimate Resistance....................

31. —Ductility.—Permanent Set...............

208

210

211

32.—Wrought Iron

212

Coefficient of elasticity............

Ultimate resistance and elastic limit

212

223

33.—Cast Iron

Wrought-iron boiler plate.

Effect of annealing......

Effect of hardening on the tensile resistance of iron and steel.. . 247 Variation of tensile resistance with increase of temperature... . 247

Effect of low temperatures on wrought iron.................

Iron wire..............................................

Tensile resistance of shape iron............. ..............

English wrought iron....................................

Fracture of wrotight iron ..................................

Crystallization of wrought iron............................

Elevation of ultimate resistance and elastic limit . ............

Bauschinger's experiments on the change of elastic limit and

coefficient...........................................

Resistance of bar iron to suddenly applied stress..............

Reduction of resistance between the ultimate and breaking points. 269

270 272 276 276 279 282

Effects of chemical constitution Kirkaldÿs conclusions........

Coefficient of elasticity and elastic limit

Ultimate resistance..................

Effect of rcmelting..................

242

245

251

254

257

Ln cn CO CO

259

260

262

269

ART.

34.—Steel

Effect of continued fusion . Effect of repetition of stress Effect of high temperatures.

CO CO

286

286

Coefficient of elasticity......................

Ultimate resistance and elastic limit..........

Boiler plate..............................

Effects of hardening and tempering steel plates

Rivet steel.................................

Effect of reduction of sectional area, in connection with hammering and rolling...................................

Effects of annealing steel.................................

Steel wire...............................................

Shape steel..............................................

Steel gun wire...........................................

Effect 0f low and high temperatures on steel.................

Effect of manipulations common to constructive processes ; also

punched, drilled and reamed holes......................

Bauschinger’s experiments o?i the change of clastic limit and

coefficient of elasticity.................................

Fracture of steel ........................................

Effect of chemical composition.............................

35.—Copper, Tin, Zinc and their Alloys.—Phosphor Bronze................

Coefficients of elasticity....................................

Ultimate resistance and elastic limit........................

286

294

306

314

315

315

319

319

321

322

323

325

333

334

334

336

336

338

Gun metal..............................................

General results..........................................

Phosphor bronze, and brass and copper wire..................

Experiments on rolled copper by the '''Franklin Institute Committee ”............................................

Variation of ultimate resistance and stretch at high temperatures ...............................................

341

342

344

345

345

Bauschinger's experiments with copper and red brass

348

36.—Various Metals and Glass

351

Coefficients of elasticity............................

Ultimate resistance and elastic limit..................

37-—Cement, Cement Mortars, etc.—Brick........................

Experiments and conclusions of JVm. IV. Mac lay, C. E

Artificial stones....................................

Bricks...........................................

Adhesion between bricks and cement mortar............

351

352

in in W

363

364

364

38.—Timber

365

CHAPTER VI.

COMPRESSION.

ART.

39. —Preliminary........................................

40. —Wrought Iron.....................................

41. —Cast Iron .......................................

42. —Steel..............................................

43. —Copper, Tin, Zinc and their Alloys...................

44. —Glass.............................................

45. —Cement.—Cement Mortar.—Concrete.—Artificial Stones

46. —Brick and Brick Piers...............................

47. —Natural Building Stones............................

CHAPTER VII.

COMPRESSION.—LONG COLUMNS.

PACE

w e* <0 o o o

N N N 00 OO CO cn cn cn n n n

49. —Preliminary Matter................................................

50. —Gordon’s Formula.................................................

51. —Experiments on Phoenix Columns, Latticed Channel Columns and Chan

nels ......................................................

409

430

Latticed channel columns and channels................

52.—Euler’s and Tredgold’s Forms of Long Column Formulae........

53-—Hodgkinson’s Formulae......................................

54.—Graphical Representation of Results of Long Column Experiments 55-—Limit of Applicability of Euler’s Formula......................

56. —Reduction of Columns at Ends...............................

57. —Timber Columns.............................................

449

455

463

469

473

477

479

480

Formula of C. Shaler Smith, C. E

435

CHAPTER VIII.

SHEARING AND TORSION.

58. —Coefficient of Elasticity .

59. —Ultimate Resistance. .. .

Wrought iron Cast iron.....

487

490

491

ART,

Steel... Copper. Timber

60.—Torsion

Coefficients of elasticity...............................

Ultimate resistance and elastic limit...................

Wrought iron.......................................

Cast iron............................................

Steel...............................................

Copper, tin, zinc arid their alloys.......................

Timber............................................

Relation between ultimate resistances to tension and torsion

vO vO vO vO vO vO vO CO co 00 co O' cn co

UX U\

o s

4- O

S' ?

m m

510

CHAPTER IX.

BENDING, OR FLEXURE.

61. —Coefficient of Elasticity............................................

62. —Formulas for Rupture .............................................

63. —Solid Rectangular and Circular Beams...............................

Wrought iron......................... .................

Cast iron...............................................

Steel...................................................

Combined steel and iron...................................

Copper, tin, zinc and their alloys...........................

Timber beams...........................................

Timber beams of natural and prepared wood.................

Cement, mortar and concrete ..............................

Stone beams............................................

Practical formułce for solid beams..........................

Comparison of modulus of rupture for bending with ultimate resistances...........................................

64. —Flanged Beams with Unequal Flanges...............................

Equal coefficients of elasticity..............................

Unequal coefficients of elasticity...........................

Cast-iron flanged beams..................................

Deflection of cast-iron flanged beams........................

Wrought-iron T beams...................................

Deflection of wrought-iron T beams........................

65. —Flanged Beams with Equal Flanges........................ ........

Experiments of U. S. Test Board..........................

512

512

514

515

518

520

523

524

CO to Os O'

537

543

543

545

546

547

550

554

558

559

562

564

ART.

66. —Built Flanged Beams with Equal Flanges.—Cover Plates ...

67. —Built Flanged Beams with Equal Flanges.—No Cover Plates

68. —Box Beams........................................

69. —Exact Formulae for Built Beams..........................

70. —Examples of Built Beams Broken by Centre Weight.........

Example I.— Wrought-iron beam...............

Example II.—Steel beam...........•............

71. —Loss of Metal at Rivet Holes............................

72. —Thickness of Web Plate.................................

H CO

o

< ««

ui cn

vC CO Lrx O

59°

598

598

600

6ox

601

CHAPTER X.

CONNECTIONS.

73.—Riveted Joints

606

74. —Welded Joints .

75. —Pin Connection

Kinds of joints...................

Distribution of stress in riveted joints

Effect of punching ................

Wrought-iron lap joints, and butt joints with single butt strap. 616 Steel lap joints, and butt joints with one cover,

Wrought-iron butt joints with double covers .

Steel butt joints with double cover plates......

Efficiencies...............................

Riveted truss joints........................

Diagonal joints.......................

Friction of riveted joints...................

Hand and machine riveting...............

606

607

615

627

632

635

638

640

642

642

643

644 644

76.—Iron, Steel and Hemp Cables or Ropes._—Wrought-iron Chain Cables... 648 Wrought-iron chain cables

652

CHAPTER XI.

MISCELLANEOUS PROBLEMS.

77. —Resistance of Flues to Collapse.........................

78. —Approximate Treatment of Solid Metallic Rollers..........

79. —Resistance to Driving and Drawing Spikes.................

80. —Shearing Resistance of Timber behind Bolt or Mortise Holes

81. —Bulging of Plates......................................

655

659

663

664

ART.

82.—Special Cases of Flexure

674

Flexure by oblique forces.................

General flexure by continuous, normal load.

674

679

CHAPTER XII.

WORKING STRESSES AND SAFETY FACTORS.

83.—Definitions

681

84. —Specifications for the Cincinnati and Covington Bridge, 1887..........

85. —Specifications for the Blair Crossing Bridge, 1886.....................

86. —General Specifications for Iron Railroad Bridges and Viaducts, by Theo

dore Cooper, C. E., 1887....................................

87. —Standard Specifications for Iron and Steel Railway Structures by the

Phoenix Bridge Co., 1887.....................................

88. —Niagara Suspension Bridge.......................................

89. —Specifications for Boiler and Fire Box Steel, Penn. RR. Co., 1883......

90. —The St. John Cantilever, St. John, N. B., 1885.....................

91. —Penn. RR. Specifications, 1887 ...................................

682

688

691

697

702

702

702 a 702 b

92. —Specifications for Steel Cable Wire for the East River Suspension Bridge. 703

93. —Specifications for Steel Wire Ropes for the Over-Floor Stays and Storm

Cables of the East River Suspension Bridge...................

94. —Specifications for Steel Suspenders, Connecting Rods, Stirrups and Pins,

for the East River Suspension Bridge..........................

95. —Specifications for Certain Steel Work .    .    . East River Bridge,

705

705

CHAPTER XIII.

THE FATIGUE OF METALS.

96. —Wohler’s Law.....................

97. —Experimental Results...............

98. —Formulae of Launhardt and Weyrauch

99. —Influence of Time on Strains........

CO O' xr> O' O O *-h w

CHAPTER XIV.

THE FLOW OF SOLIDS.

100. —General Statements. .

101. —Tresca’s Hypotheses

ART.

PAGE

102. —The Variable Meridian Section of the Primitive Central Cylinder..... 727

103. —Positions in the Jet of Horizontal Sections of the Primitive Central

Cylinder

104. —Final Radius of a Horizontal Section of the Primitive Central Cylinder. 731

105. —Path of any Molecule

729

731

ADDENDA.

To Art. 34. To Art. 65, To Art. 70 To Art. 73

734

736

740

740 b

Elasticity and Resistance of Materials.

PART I.—RATIONAL.

CHAPTER I.

General Theory of Elasticity in Amorphous Solid

Bodies.

Art. I.—General Statements.

The molecules of all solid bodies known in nature are more or less free to move toward, or from, or among each other. Resistances are offered to such motions, which vary according to the circumstances under which they take place, and the nature of the body. This property of resistance is termed the “ elasticity ” of the body.

The summation of the displacements of the molecules of a body, for a given point, is called the “ distortion ” or “ strain ” at the point considered. The force by which the molecules of a body resist a strain, at any point, is called the “stress ” at that point. This distinction between stress and strain is fundamental and important.

Stresses are developed, and strains caused, by the application of force to the exterior surface of the material. These stresses and strains vary in character according to the method of application of the external forces. Each stress, however, is accompanied by its own characteristic strain and no other. Thus, there are shearing stresses and shearing strains, tensile stresses and tensile strains, compressive stresses and compressive strains. Usually a number of different stresses with their corresponding strains are coexistent at any point in a body subjected to the action of external forces.

It is a matter of experience that strains always vary continuously and in the same direction with the corresponding stresses. Consequently the stresses are continuously increasing functions of the strains, and any stress may be represented by a series composed of the ascending powers (commencing with the first) of the strains multiplied by proper coefficients. When, as is usually the case, the displacements are very small, the terms of the series whose indices are greater than unity are exceedingly small compared with the first term, whose index is unity. Those terms may consequently be omitted without essentially changing the value of the expression. Hence follows what is ordinarily termed Hooke’s law :

The ratio between stresses and corrcspondmg strains, for a given material, is constant.

This law is susceptible of very simple algebraic representation. As the generality of the equation will not be affected, intensities of stresses and distortions or strains per linear unit, only, will be considered.

Let p' represent the intensity of any stress, and /' the strain per unit of length, or, in other words, the rate of strain. If E’ is a constant coefficient, Hooke’s law will be given by the following equation :

(O

/ = ET.

If the intensity of stress varies from point to point of a body, Hooke’s law may be expressed by the following differential equation :

=E

dV

If p' and V are rectangular co-ordinates, Eqs. (1) and (2) are evidently the equations of a straight line passing through the origin of co-ordinates. It will hereafter be seen that the line under consideration is essentially straight for very small strains only.

Art. 2.—Coefficients of Elasticity.

In general, the coefficient E’ in Eq. (1) of the preceding Art., is called the “ coefficient of elasticity,” or, sometimes, “ modulus of elasticity.” The coefficient of elasticity varies both with the kind of material and kind of stress. It simply expresses the ratio between stress and strain.

The characteristic strain of a tensile stress is evidently an increase of the linear dimensions of the body in the direction of action of the external forces.

Let this increase per unit of length be represented by /, while p and E represent, respectively, the corresponding intensity and coefficient. Eq. (1) of the preceding Art. then becomes :    ,

p = El, or, E =

E is then the coefficient of elasticity for tension.

The characteristic strain for a compressive stress is evidently a decrease in the linear dimensions of the body in the direction of action of the external forces. Let /, represent this decrease per unit of length, p₁ the intensity of compressive stress, and E₁ the corresponding coefficient. Hence:

to

Pi — EJii or, Ii i —

E1} consequently, is the coefficient of elasticity for compres

sion.

The characteristic strain for a shearing stress may be determined by considering the effect which it produces on the layers of the body parallel to its plane of action.

17

The

Fig.1

Representing, therefore, the intensity of shear, coefficient and strain by S, G and ą>, respectively, Eq. (1) of Art. 1 be

In Fig. i let ABCD represent one face of a cube, another of whose faces is fixed along AD. If a shear acts in the face BC, whose plane is normal to the plane of the b paper, all layers of the cube parallel to the plane of the shearing stress, i.e., BC, will slide E over each other, so that the faces AB and DC will take the positions AE and DF. amount of distortion or strain per unit of length ° will be represented by the angle EAB — cp. If F the strain is small there may be written (p, sin or tan cp indifferently.

comes :

II

'QlOo

CO

It will be seen hereafter that there are certain limits of stress within which Eqs. (i), (2) and (3) are essentially true, but beyond which they do not hold ; this limit is called the “ limit of elasticity,” and is not in general a well defined point.

Art. 3.—Lateral Strains.

If a body, like that shown in Fig. 1, be subjected to tension, all of its oblique cross sections, such as FE and GH, will sustain shearing stresses in consequence of the components of the tension tangential to those oblique sections. These tangential stresses will cause the oblique sections, in both directions, to slide over each other. Consequently the normal cross sections of the body zvill be decreased ; and if the normal

Fig.1

cross sections of the body are made less, its capacity of resistance to the external forces acting on AB and CD will be correspondingly diminished.

If the body is subjected to compression, oblique sections of the body will be subjected to shears, but in directions opposite to those existing in the previous case. The effect of such shears will be an increase of the lateral dimensions of the body and a corresponding increase in its capacity of resistance.

These changes in the lateral dimensions of the body are termed “lateral strains”; they always accompany direct strains of tension and compression.

It is to be observed that lateral strains decrease a body’s resistance to tension, but increase its resistance to compression. Also, that if they are prevented, both kinds of resistance are increased.

Consider a cube, each of whose edges is a, in a body subjected to tension. Let r represent the ratio between the lateral and direct strains, and let it be supposed to be the same in all directions. If /, as in Art. 2, represents the direct strain, the edges of the cube will become, by the tension : a{ 1 + /), a(i — rl) and a{i — rl). Consequently the volume of the resulting parallelopiped will be :

a\i + /) (1 — rl)² — a³[i -f /( 1 — 2r)\    .    .    .    (1)

if powers of / higher than the first be omitted. tween o and x/₂, there will be an increase of volume, but not otherwise.

With x be-

If the body is subjected to compression, the edges of the cube become : a(i — /,), a( i + rf) and a( i + rf) ; while the volume of the parallelopiped takes the value :

aKl ~ 4) (i + rjtf = ^³[i + lt(2rx - i)] . . • (2)

As before, the higher powers of /, are omitted. If the volume of the cube is decreased, r₁ must be found between o and y₂.

Art. 4.—Relation between the Coefficients of Elasticity for Shearing and Direct Stress in a Homogeneous Body.

A body is said to be homogeneous when its elasticity, of a given kind, is the same in all directions.

Let Fig. i represent a body subjected to tension parallel to CD. That oblique section on which the shear has the greatest

intensity will make an angle of 45⁰ with either of those faces whose traces are CD or BD ; for if a is the angle which any oblique section makes with BD, o P the total tension on BD, and A' the area of the latter surface, the total shear on any section whose area is A' sec a, will be P sin a. Hence the intensity of shear is :

hV'h

F'

.JF-

Nn

G>TXi

Fig.1

P sin a

= —r. sin a cos a A sec a A

(O

The second member of Eq. (1) evidently has its greatest

Hence, if the tensile intensity on BD is represented by ~ = p, the greatest intensity of shear will be :

value for a — 45⁰.

to

Then by Eq. (3) of Art. 2 :

(3)

In Fig. i EK and KG are perpendicular to each other, while they make angles of 45⁰ with either AB or CD. After stress, the cube EKGH is distorted to the oblique parallelopiped E'KG'ET. Consequently EKGH and E'KG'H' correspond to AB CD and AEFD, respectively, of Fig. 1, Art. 2. The angular difference EKG — E'KG' is then equal to cp ; and EKE

= GKG' - Also EKE' = 45⁰ - ^

2

Using, then, the notation of the preceding Arts., there will result, nearly :

tan f 45⁰ —

= i - /(i + r) ; . . (4)

remembering that EK — FK(i -f- /) ; and that EF = FK{ i - rl).

From a trigonometrical formula, there is obtained, very nearly :

tan

(«•-î)

o    <P

tan 45 — tan —

<P

I--—

a

tan 45 -f tan —

I+f

From Eqs. (4) and (5) :

(6)

Substituting from Eq. (3), as well as from Eq. (1) of Art. 2 :

G=-j^—

2(1 + r)

It has already been seen in the preceding Art. that r must be found between o and y₂, consequently the coefficient of elasticity for shearing lies between the values of y and of that of the coefficient of elasticity for tension.

This result is approximately verified by experiment.

Since precisely the same form of result is obtained by treating compressive stress, instead of tensile, there will be found, by equating the two values of G :

i + rt

or,

_ i + rz E I+r* ’

• • (8)

It is clear, from the conditions assumed and operations involved, that the relations shown by Eqs. (7) and (8) can only be approximate.

Art. 5.—Expressions for Tangential and Direct Stresses in Terms of the Rates of Strains at any point of a Homogeneous Body.

Let any portion of material, perfectly homogeneous, be subjected to any state of stress whatever. At any point as O, Fig. i, let there be assumed any three rectangular co-ordinate planes ; then consider any small rectangular parallelopiped whose faces are parallel to those planes. Finally let the stresses on the three faces nearest the origin be resolved into

components normal and parallel to their planes of action, whose directions are parallel to the co-ordinate axis.

6—7C

d.

X.

Fig.1

The intensities of these tangential and normal components v/ill be represented in the usual manner, i.e., pxy signifies a tangential intensity on a plane normal to the axis of X (plane ZY), whose direction is parallel to the axis of Y, while pxx signifies the intensity of a normal stress on a plane normal to the axis of X (plane ZY) and in the direction of the axis of Two unlike sub- o scripts, therefore, indicate a tangential stress, while two of the same kind signify a normal stress.

From Eq. (3) of Art. 2 and Eq. (7) of Art. 4, there is at once deduced :

2(1 + r) ⁹

= Gcp

(0

Now when the material is subjected to stress the lines bounding the faces of the parallelopiped will no longer be at right angles to each other. It has already been shown in Art. 2 that the angular changes of the lines, from right angles, are the characteristic shearing strains, which, multiplied by G, give the shearing intensities.

Let be the change of angle of the boundary lines parallel to X and Y.

Let cp₂ be the change of angle of the boundary lines parallel to Y and Z.

Let (p₃ be the change of angle of the boundary lines parallel to Z and X.

Eq. (1) will then give the following three equations :

Pxy

2(1 + r)

Ex

(2)

Py*

2(1 + r) (p3

• • (3)

pzx —

2(1 + r)

• - (4)

In Fig. i let the rectangle agfh represent the right projection of the indefinitely small parallelopiped dx dy dz. If u, v and w are the strains, parallel to the axis of x, y and z, of

the original point h, the rates of variation of strain —

du dv dw

dx' dy’ dz'

etc., may be considered constant throughout this parallelopiped ; consequently the rectangular faces will change to oblique parallelograms. The oblique parallelogram dhck, whose diagonals may or may not coincide with those of agfh, therefore, may represent the strained condition of the latter figure.

Then, by Art. 2, the difference between dhc and the right angle at h will represent the strain q>v But, from Fig. 1, <pt has the following value :

(5)

cpx — dhe + bhc

But the limiting values of the angles in the second member are coincident with their tangents ; hence :

dc be ' n ~ dy + 5

But, again, de is the distortion parallel to OX found by moving parallel to O Y, only ; hence it is a partial differential of u, or, it has the value :

du

de — ~r dy

dy

(7)

In precisely the same manner be is the partial differential of v in respect to x, or :

be —

dv

dx

dx.

By the aid of these considerations, Eq. (6) takes the form :

_ du    dv

1¹ dy    dx

(8)

If XV be changed to XZ, and then to ZX, there may be at once written by the aid of Eq. (8) :

dv dw dz dy

(9)

dw du

V' ⁼ dï⁺7z.....

Eqs. (2), (3) and (4) now take the following form :

(10)

„ /du , dv\ ^ = G\dy ⁺ aw)

(”)

. (12)

pzx — G

'dw , du'

4x

dz)

• • (13)

The direct stresses are next to be given in terms of the displacements u, v and w. Again, let the rectangular parallelo-piped dx dy dz be considered. Eq. (1), of Art. 1, shows that the strain per unit of length is found by dividing the intensity of stress by the coefficient of elasticity, if a single stress only exists. But in the present instance, any state of stress whatever is supposed. Consequently the strain caused by pxx, for example, acting alone must be combined with the lateral strains induced by pyy and pzz. Denoting the actual rates of strain along the axes of X, Y and Z by /„ l₂ and lv therefore, the following equations may be at once written by the aid of the principles given in Art. 3 :

fjr “ 4 + {j’yy + A* ) -jß • •

1 — TT— T>/ir and — Tÿÿ»

ydy\-,&Ai

+ 3Ą«» +

+ =°

• (7)

= (^f)• or-('-*)•

(29)

In general terms, i.e.y for any degree of strain a

suits are somewhat irregular, the lowest ultimate

49. It is necessary to know correctly the exact co

TABLE VIII.—Continued.

Mr. Hill draws the following conclusions :

Table I. contains the results of the tests made by

The series of alloys presents some interesting res

o

p

TABLE Via.

1882), show that the full resistance of the column

The elastic limit varied from 46 to 58 per cent, o

loads about half the breaking weight, not only the

• • (15)

Fig. 1.

k — 0.71 V(J> — d)d + 0.5^. . . . (11)

TABLE X.

ly • • • <6)-

L/zy

(I)

fv.-'izmmi

. . (14)

fjr “ 4 + {j’yy + A* ) -jß • •

Py.

f--4 + (jxx + pzz ^ -jß •    •

• * OS)

. . (16)

— 4 + (pyy + Pxx ) ^ • •

Eliminating between these three equations :

XX

Pyy =

/,+ — i — 2 r

i* + j _ 2r (/t + 4 + 4) • • • (l8) 4 + v _ 2r(4 + 4 + 4) • • • (:9)

• • (17)

But if u, v and w are the actual strains at the point where

dx(\ -f- l^)dy(\ -f- la)dz(l -(- 4) — dy dz{\ + 4 + 4 + 4)>

if powers of 4, 4 and 4 above the first be omitted. The quantity (4 + 4 + 4) ^s> then, the rate of variation of volume, or the amount of variation of volume for a cubic unit. If there be put

2(1 + r)'

these stresses exist, the rates of strain /₂ and l₃ will evi-

respectively. The volume

dently be equal to ^ and of the parallelopiped will be changed by those strains to

dw

dz ’

0 __ du ^ dv dw dx dy'dz'

Eqs. (17), (18) and (19) will take the forms :

2 Gr

du

2re + 2GTx ■ ■

.    .    (2O)

■ • (21)

■ ■ (22)

The form in which Eqs. (14), (15) and (16) are written, shows that if pxx, pyy or pzz, is positive, the stress is tension, and compression if it is negative. Consequently a positive value for any of the intensities in Eqs. (20), (21) or (22) will indicate a tensile stress, while a negative value will show the stress to be compressive.

The Eqs. (14) to (19), together with the elimination involved, also show that the coefficients of elasticity for tension and compression have been taken equal to each other, and that the ratio r is the same for tensile and compressive strains.

Further, in Eqs. (11), (12) and (13), it has been assumed that G is the same for all planes.

Hence Eqs. (11), (12), (13), (20), (21) and (22) apply only to bodies perfectly homogeneous in all directions.

It is to be observed that the co-ordinate axes have been taken perfectly arbitrarily.

Art. 6.—General Equations of Internal Motion and Equilibrium.

In establishing the general equations of motion and equilibrium, the principles of dynamics and statics are to be applied to the forces which act upon the parallelopiped represented in Fig. I, the edges of which are dx, dy and dz. The notation to be used for the intensities of the stresses acting on the different faces will be the same as that used in the preceding Article.

Let the stresses which act on the faces nearest the origin be considered negative, while those which act on the other three faces are taken as positive.

The stresses which act in the direction of the axis of X are the following :

0 ; —Pxx dy dz.

(a* + ~fXd*)dy dz'

O ; — pzxdy dx.

On the face normal to X, nearest to

“    “    “ farthest from O;

“ dy dx nearest to

“ farthest from

i^Pzx +

dpz

dz

-dz^jdy dx.

“ dz dx nearest to

“ farthest from

O', —pyxdzdx.

(*+’%*)

<9;

dz dx.

The differential coefficients of the intensities are the rates of variation of those intensities for each unit of the variable, which, multiplied by the differentials of the variables, give the amounts of variation for the different edges of the parallelopiped.

Let X₀ be the external force acting in the direction of A on a unit of volume at the point considered ; then X₀ dx dy dz will be the

amount of external force „

z

acting on the parallelopiped. ^

These constitute all the forces acting on the parallelopiped inthe direction of the axis of X, and their sum, if un-

(Z²n

balanced, must be equal to m ——dx dy dz ; in which m is the

mass or inertia of a unit of volume, and dt the differential of the time. Forming such an equation, therefore, and dropping the common factor dx dy dz, there will result :

dy

dx

.>•

dy

d²u

dz

dx

Y_

Fig-1

dp xx , dpyx dp%x . j _    ~ ^

dx    ⁺ dz ⁺    d!*‘    ■

■ ■ (O

Changing x to y, y to z, and z to x, Eq. (1) will become :

dpzy

dpxy i dpXy .__

dx dy dz

. . (2)

Again, in Eq. (1), changing x to z, z to y, and y to x :

dpXz i dpyz . dpzz „ ^_

dx dy dz ⁰

• • (3)

The line of action of the resultant of all the forces which act on the indefinitely small parallelopiped, at its limit, passes through its centre of gravity, consequently it is subjected to the action of no imbalanced moment. The parallelopiped, therefore, can have no rotation about an axis passing through its centre of gravity, whether it be in motion or equilibrium. Hence, let an axis passing through its centre of gravity and parallel to the axis of X, be considered. The only stresses, which, from their direction can possibly have moments about that axis, are those with the subscripts (yd), (zy), (yy), or (zz). But those with the last two subscripts act directly through the centre of the parallelopiped, consequently their moments are zero. The stresses

7.    7.

dz . dx dy are two of six forces whose

resultant is directly opposed to the resultant of those three forces which represent the increase of the intensities of the normal, or direct, stresses on three of the faces of the parallelopiped ; these, therefore, have no moments about the assumed axis. The only stresses remaining are those whose intensities are pzy and pyz. The resultant moment, which must be equal to zero, then, has the following value :

dpyz

dy

dy . dx dz and

dpzy

dz

pyzdx dz . dy -j- pzydx dy . dz — o

• • (4)

• * pyx    pzy

(5)

Hence the two intensities are equal to each other.

The negative sign in Eq. (5) simply indicates that their moments have opposite signs or directions ; consequently, that the shears themselves, on adjacent faces, act toward or from the edge between those faces. In Eqs. (1), (2) and (3), the

tangential stresses, or shears, are all to be affected by the same sign, since direct, or normal, stresses only can have different signs.

The Eq. (5) is perfectly general, hence there may be written :

pxy — P yxi and pzx — pxz

(6)

Adopting the notation of Lamé, there may be written :

Pxx — N„ pyy - N„ ptt — Nv Pxy — Tv    pxz — T„ pxy = Tv

by which Eqs. (1), (2) and (3) take the following forms :

dNT . dT„ . dT₂ . „ d²u Hx + ~dy + W ⁺    = ’"S’ •    •

• • (7)

dT% d/V₂ d7\ v _ dx^ dy ^ dz ^    ⁹

d²v

. . (8)

dy

dz

d²iv

dt² '    '

• • (9)

The equations (11), (12), (13), (20), (21) and (22) of the preceding Art. are really kinematical in nature ; in order that the principles of dynamics may hold, they must satisfy Eqs. (7), (8) and (9). As the latter stand, by themselves, they are applicable to rigid bodies as well as elastic ones ; but when the values of N and T, in terms of the strains u, v and w, have been inserted they are restricted, in their use, to elastic bodies only. With those values so inserted, they form the equations on which are based the mathematical theory of sound and light vibrations, as well as those of elastic rods, membranes, etc.

In general, they are the equations of motion which the different parts of the body can have in reference to each other, in consequence of the elastic nature of the material of which the body is composed.

v d²w

- and will each be equal to zero. Hence, Eqs. (7),

If all parts of the body are in equilibrium under the action of the internal stresses, the rates of variation of the strains d*u d²v dt² ’ dt

(8) and (9) will take the forms :

■ ■ (10)

dNx dT₃ dT _ lx + ~dy + HI + x° - °

dT* + -T + -p + k = o .... (to

dx

dy

dT* ₊    4. dJh ₊ z₀ = o .

dx

dy

dz

.    .    (12)

These are the general equations of equilibrium. As they stand, they apply to a rigid body. For an elastic body, the values of N and T from the preceding Art., in terms of the strains u, v and w, must satisfy these equations.

The Eqs. (10), (11) and (12) express the three conditions of equilibrium that the sums of the forces acting on the small parallelopiped, taken in three rectangular co-ordinate directions, must each be equal to zero. The other three conditions, indicating that the three component moments about the same co-ordinate axes must each be equal to zero, are fulfilled by Eqs. (5) and (6). The latter conditions really eliminate three of the nine unknown stresses. The remaining six consequently appear in both the equations of motion and equilibrium.

The equations (7) to (12), inclusive, belong to the interior of the body. At the exterior surface, only a portion of the small parallelopiped will exist, and that portion will be a tetrahedron, the base of which forms a part of the exterior surface of the body, and is acted upon by external forces. Let

— be the area of the base of this tetrahedron, and let p, q and

axes

r be the angles which a normal to it forms with the three of X, F and Z, respectively. Then will

da cosp — dy dz, da cos q — dz dx, and da cos r = dx dy.

Let P be the known intensity of the external force acting on da, and let n, y and p be the angles which its direction makes with the co-ordinate axes. Then there will result :

X₀ — P da . cos 7t, Y₀ — Pda . cos y and Z₀ = P da . cos p.

con-

The origin is now supposed to be so taken that the apex of the tetrahedron is located between it and the base ; hence that part of the parallelopiped in which acted the stresses involving the derivatives, or differential coefficients, is wanting ; sequently those stresses are also wanting.

The sums of the forces, then, which act on the tetrahedron, in the co-ordinate directions, are the following :

— (Aj dy dz -f- T₃ dz dx    7\dy dx) + Pda cos n — o, — (T₃dz dy N₃ dz dx -f- T, dy dx) + Pda cos y = o, — (T₂dz dy -f- Tz dz dx + N₃ dy dx) -f- Pda cos p = o.

Substituting from above :

T₃ cos p -f- N₂ cos q -f- 7i cos r = P cos x •    •    (14)

T₂ cos p + Txcos q + N₃ cos r — P cos p . .    (15)

These equations must always be satisfied at the exterior surface of the body ; and since the external forces must always be known, in order that a problem may be determinate, they will serve to determine constants which arise from the integration of the general equations of motion and equilibrium.

Art. 7.—Equations of Motion and Equilibrium in Semi-Polar Co-ordinates.

For many purposes it is convenient to have the conditions of motion and equilibrium expressed in either semi-polar or polar co-ordinates ; the first form of such expression will be given in this Article.

The general analytical method of transformation of coordinates may be applied to the equations of the preceding Article, but the direct treatment of an indefinitely small portion of the material, limited by co-ordinate surfaces, possesses many advantages. In Fig. 1 are shown both the small portion of material and the co-ordinates, semi-polar as well as rectangular. The angle made by a plane normal to ZY, and containing OX, with the plane XY is represented by cp ; the distance of any point from OX, measured parallel to ZY, is called r ; the third co-ordinate, normal to r and p, is the co-ordinate x, as before. It is important to observe that the co-ordinates x, r and p, at any point, are rectangular.

The indefinitely small portion of material to be considered will, as shown in Fig. 1, be limited by the edges dx, dr and r dp. The faces dx dr are inclined to each other at the angle

dp.

The intensities of the normal stresses in the directions of X and r will be indicated by Nx and R, respectively. The remainder of the notation will be of the same general character as that in the preceding Article ; i.e.y Txr will represent a shear on the face dr. r dip in the direction of r, while is a normal stress, in the direction of q>, on the face dx dr.

The strains or displacements, in the directions of x, r and <p, will be represented by u, p and w ; consequently the unbalanced forces in those directions, per unit of mass, will be :

dx

h-.

q.

7c

Fig.i

d² u    d²p . d²w

m —and m —y— dt² dt²    dt²

.    .    (I)

Those forces acting on the faces hf, fe, and he, will be considered negative ; those acting on the other faces, positive.

Forces acting in the direction of r.

— R . r dq) dx, and ;

-f- Rr dq) dx -f-

(ARr) \ dr

dr =

r dr -f- R dr'jdq) dx.

— T$rdr dx, and ;

+ T$rdrdx +

dT

dcp

<l'r dcp. dr dx.

— Txr . r dcp dr, and ;

+ Txr. r dcp dr + ~~j^rdx . r dcp dr.

On the face dr dx, nearest to ZOX, there acts the normal

stress ('N^dr dx +    dcp. dr dx^j

- - AT'. Now N’ has a component acting parallel to the face fe and toward OX, equal to N’ sin (dcp) — N,r-^y- = N’dcp. But the second term of this

product will hold (dcp)², hence it will disappear, at the limit, in the first derivative of N'dcp N'dcp = N^dcp dr dx. Since this force must be taken as acting toward OX, it acts with the normal forces on hf, and, consequently, must be given the negative sign.

If R₀ is the external force acting on a unit of volume, another force (external) acting along r will be R₀ . r dcp dr dx.

The sum of all these forces will be equal to

m. r dcp dr dx.-

dap

dt²'

Forces acting in the direction of cp.

— Nw dr dx, and ;

-j- Nąą dr dx +

dN*

dcp

dcp .dr dx.

— Tr<t> . r dcp dx, and ;

Art. 7.] EQUATIONS IN SEMI-POLAR CO-ORDINATES. 23

+ Tr$ . r dcp dx -f-    dr — r C~-^~dr + rr^dr\dcp dx.

— Tx$ . r dq) dr, and ;

+ Tx$ . r dq) dr + ^x . r dq) dr.

As in the case of N^, in connection with the forces along r, so the force T$rdr dx has a component along (normal to fe) equal to T$r dr dx . sin (dqj) = r d(p dr dx. It will have a positive sign, because it acts from OX.

The external force is, <Z>₀ . r dq) dr dx.

Forces acting in the direction of x.

— N₁ . r dq) dr, and ;

-j- iVj r dq) dr d--'

dN.

dx

dx . r dq) dr.

— Trx . dx r dcp, and

+ Trx . dx r dq) +

Trx)

dr

dTrx

dr

dr + Trx dr ^ dx dq).

— Tą,x dx dr, and ;

+ T+gdx dr +

dT$x

dq>

dq). dx dr.

The external force is, X₀ . r dq) dx dr.

Putting each of these three sums equal to the proper rates

of variation of momentum, and dropping the common factor, r dcp dx dr :

dNx

dx

dTrx dT\x , _TV dr ' r dcp r

■ t>    d*u

+ X° = mHr ⁽²)

dTxr dR dx    dr

dZ*

r dcp

+ R° = m ^ (3)

dTx<!) dTrih

sr⁺ -^- +

dr

dN’’(fjcf}

r dcp

Trfy -f- Tr<j,

d²w , N

SF ⁽⁴⁾

These are the general equations of motion (vibration) in terms of semi-polar co-ordinates ; if the second members are made equal to zero, they become equations of equilibrium. Eqs. (2), (3), and (4) are not dependent upon the nature of the body.

Since x, r, and cp are rectangular, it at once follows that :

• • (5)

Trx — Txr, Th = 7;r, and Tx<t, = T$x. .

In order that Eqs. (2), (3), and (4) may be restricted to elastic bodies, it is necessary to express the six intensities of stresses involved, in terms of the rates of variation of the strains in the rectangular co-ordinate directions of r, and cp. Since these co-ordinates are rectangular, the Eqs. (11), (12), (13), (20), (21), and (22) of Article 5, may be made applicable to the present case by some very simple changes dependent upon the nature of semi-polar co-ordinates.

For the present purpose the strains in the co-ordinate directions of x, y, and z will be represented by u', v', and Since the axis of x remains the same in the two systems, evidently :

du* _ du dx dx'

From Fig. 1 it is clear that the axis of y corresponds exactly to the co-ordinate direction r ; hence :

dv

dp

dy dr

From the same Fig. it is seen that the axis of £ corresponds to cp, or rą>. But the total differential, dw', must be considered as made up of two parts ; consequently the rate of variation dW'

—— will consist of two parts also. If there is no distortion in

dz

the direction of r, or if the distance of a molecule from the origin remains the same, one part will be

dw

div

If,

d(rcp) ~ rdcp'

however, a unit’s length of material be removed from the distance r to r + p from the centre O, Fig. i, while cp remains

(■+')

constant, its length will be changed from i to i .

which p may be implicitly positive or negative. Consequently there will result :

, in

dw' _ dw dz rdcp

For the reasons already given, there follow :

du!

du

dy

dr

and

dd

dp

dx dx'

In Fig. 2 let dc be the side of a distorted small portion of

the material, the original position of

which was d'e. Od is the distance r °

from the origin, ad — dr and ac = dw,

while dd' — w. The angular change

.    ...    . . . etc dw ,

in position of dc is = —; but an ad dr

a'

Jw

»___

i*

Fig.2

cv

ab

w

— — is due to the movement of r, and is r

not a movement of dc relatively to the material immediately adjacent to d.

Hence :

amount equal to

ad

dw' dw

dp

dy

dr

W . dv

y 3.1SO —    j .

dz r dcp

There only remain the following two, which may be at once written :

and

dw' _ dw dx dx

dii _ du dz r dą>

The rate of variation of volume takes the following form in terms of the new co-ordinates :

a    du■ , dv' , dw

u — —— -4------

dx dy dz

8 I.H

's

^ I'fc

• (6)

Accenting the intensities which belong to the rectangular system x, y, z, the Eqs. (ii), (12), (13), (20), (21) and (22), of Art. 5, take the following form :

7Vr

= N’ =

2Gx a , r^dll

- V -f- 2C7T

— 2r    dx

• (7)

R — N₃’ =

2Gx

dp

—- 6 -j- 2G-j-2x    dr

(8)

- N' =

2Gv

i — 2r

6+2g(

dw

r dqj

■s l'a

• • (9)

dll

dp'

r~= Ti = C(£+*

(10)

T,* = t; = g(

dp

dw

r dcp dr

• - (II)

(dw du ⁷** = T* = GKd~x ⁺

r dcp.

(12)

If these values are introduced in Eqs. (2), (3) and (4), those equations will be restricted in application to bodies of homogeneous elasticity only.

The notation r is used to indicate that the r involved is the ratio of lateral to direct strain, and that it has no relation whatever to the co-ordinate r.

The limiting equations of condition, (13), (14) and (15) of Art. 6, remain the same, except for the changes of notation, shown in Eqs. (7) to (12), for the intensities N and T.

Art. 8.—Equations of Motion and Equilibrium in Polar Co-ordinates.

The relation, in space, existing between the polar and rectangular systems of co-ordinates is shown in Fig. 1. The angle cp is measured in the plane ZY and from that of XV; while ip is measured normal to ZY in a plane which contains OX. The analytical relation existing between the two systems is, then, the following :

x = r sin ip, y = r cos ip cos cp, and z = r cos ip sin cp.

The indefinitely small portion of material to be considered is a/ic d. It is limited by the co-ordinate planes located by

(p and ip, and concentric spherical surfaces with radii r and r + dr. The directions r, <p and ip, at any point, are rectangular ; hence, the sums of the forces acting on the small portion of the material, taken in these directions, must be found and put equal to

d²p

dt²

d²rf

dt²’

, d²oo and in—;—, dH*

in which expressions, p, 77 and go represent the strains in the direction of r, cp and ip respectively.

(0

w.

z.

V-

Vi

•T

T^-jL V

'V

Fig.1

Ss

'À''

Those forces which act on the faces ah, bd and cd will be considered negative, and those which act on the other faces positive.

The notation will remain the same as in the preceding Articles, except that the three normal stresses will be indicated by Nrt and

Forces acting along r.

— Nr . r dtp r cosip dq\

+ Nr . r¹ cos

tp dip dcp + (~

cos tp dip dcp.

dr —

dNr r* ——■

dr

dr + 2r Nr dr'j

— T4>r. r dtp dr.

+ T+r, r dip dr

dT

dcp

— dcp . r dip dr.

— T^r . r cos tp dcp dr.

+ T^r. r

cos tp dcp dr +    dip = cos tp dip

— T^r sin tp dip^ r dcp dr.

— N$ . r dip dr . sin aOc — —    . r d>p dr . cos tp dcp ;

on face ce.

— Ną, . r cos tp dq> dr , sin aOb — —    . r cos tp dq) dr . dip)

on face be.

Forces acting along cp.

— Tr<t>. r cos tp dqi r dip.

+ T > r* cos ip dcp dtp -f

cos tp dip dcp.

dr—r*

dr

dr-\-2r T^drJ

—    . r dtp dr.

+    . r dtp dr +

dN.

dcp

dcp r dip dr.

— Zij4 . r cos ip dcp dr.

+ Th cos ip .

r dcp dr +    gp — Cos p

— sin ip d//^Jr dcp dr.

dtp

dp

-f- Tr dp dr . cos p dcp ; on face ce.

r dp dr (sin akc

r cos p dcp

r cot p on face cc.

— — r dp dr. sin p dcp ;

The lines ak and ck are drawn normal to Oc and Oa.

Forces acting along p.

— Tr^j. r cos p dcp . r 'dp.

+ Trip r² cos p dcp dp +

cos p dp dcp.

dr — r²

dT

dr

~ ar + 2r    dr)

—    . r dp dr.

+ r dp dr + dcp . r dp dr.

dcp

— Ny . r cosp dcp dr.

+    . r cos

tp dcp dr +

COS >p)

dN

dtp

dtp — cos tp ,, dtp

±dtp

-N*

+ T^r. r cos tp dtp dr . dtp ; on face be.

+    . r dp dr . sin akc — + r dtp dr . sin tp dtp ; on face ce.

The volume of the indefinitely small portion of the material is (omitting second powers of indefinitely small quantities) :

r cos tp dcp . r dtp . dr — AV ;

and its mass is m multiplied by this small volume. The latter may be made a common factor in each of the three sums to be taken.

The external forces acting in the directions R, op and tp will be represented by :

R₀AV, <P₀AV and V₀AV,

respectively.

Taking each of the three sums, already mentioned, and dropping the common factor A V, there will result :

dNr

dT^r

---_l_ -

dr r cos tp . dcp

dT

\pr

r dtp

2Nr — A⁷^ — 'Ny — Ttan tp

(0

d²p

in -Ą-

dTr<j)

dN.

dr

r cos tp . dcp

r dtp

2^+ T$r— 7)4 tan tp — 7)4 tan tp

dirj

Since r, qj and ip are rectangular at any point :

1    — TT— T>/ir and    — Tÿÿ»

Hence :

2^ -f- TĄ,r — tan i]'(74^ -(- T'^P) _ 3    — 2 tan ?/; .

2Tr^ + T^r — tamp{N^ — N+) __ 3Tri, — tan j'(N^ — N$)

These relations somewhat simplify the first members of Eqs. (2) and (3).

Eqs. (1), (2) and (3) are entirely independent of the nature of the material ; also, they apply to the case of equilibrium, if the second members are made equal to zero.

The rectangular rates of strain, at any point, in terms of r, cp and ip are next to be found. As in the preceding Art., the rates of strain in the rectangular directions of r, cp and ip will be indicated by :

dv dw' du' dv du!

dy" dz' ’ dx' dx" dy" CtC‘

dw'

dr

dT^

dN+

r cos ip d(p r dip

2 Tr^ + T^r— tan ip + tan ip

+ W₀ = vi

d²oo

dt²

(3)

Remembering the reasoning in connection with the value of in the preceding Art., and attentively considering Fig. 1, there may at once be written :

In Fig. I, if ac — I and ab — go, while ak = r cot. p (ak is perpendicular to aO), the difference in length between ac and bit will be :

GO

go tan p

r cot p

This expression is negative because a decrease in length takes place in consequence of a movement in the positive direction of rtp.

Again, a consideration of Fig. i, and the reasoning connected with the equation above, will give :

SI*

dw’

dr)

go tan p

dz

r cos tli dcp

Without explanation there may at once be written :

dp

dv'

dÿ dr '

Fig. i of this, and Fig. 2 of the preceding Art. give :

du

doo

dÿ dr

—, and

dv' _ dp dx' r dp '

These are to be used in the expression for T^r. Precisely the same Figs, and method give :

dp

and

dv'

dz r cos p dcp ’

dw' dp p dÿ dr r

which are to be used in finding T$r.

dw' dx'

parts. In Fig. 2, ab is the original position of r dtp, and after the strain 77 exists it takes the position ec. Consequently ac (equal and parallel to bd and perpendicular to ak) represents the strain 77, while ed represents dr}. Since, also, fc is perpendicular to ck, the strains of the kind 77 change the right angle fck to the angle fee ; or the angle eck is equal to

= ecd -f- dek —

dc ak

_ df _i_ V

r dtp r cot tp

The expression for

will be composed of the sum of two

dw'

es

a.

dx'

f.-------c'y

Fig-2

In Fig. 2, the points a, b and k are identical with the points similarly lettered in Fig. 1. The expres-

du'

sion for — may be at once written from Fig. 1. There may, dz

then, finally be written :

dw' _ drj , 77 tan tp dx r dtp

, and.

du!

dco

dz' r cos tp dq)

These equations will give the expression for T^. The value of

du'

dr'

dw'

0 —    4- _ J----_

dx dy' dz

now

takes the following form :

dp

dt}

dr

doo 2P ⁰⁰ tan tp r cos tp dtp r dtp r

. . (4)

The last two terms are characteristic of the spherical coordinates.

The equations (20), (21), (22), (11), (12) and (13), of Art. (5), take the forms :

Nr =

2 Gx

dp

i — 2r

dr

(5)

2 Gx

■■

dp

i — 2r

r cos ip dą>

go tan ^ {G')

(7)

2Gx

i — 2r

dp

dao

p tan ip'

r d'p

r cos ip dcp

. . (8)

= g(

dco

dr

---u Ae.)

r ' r dp)

<0

(9)

dp

dr -

Tri = G(

dp

r cos <p dcp

If these values are inserted in Eqs. (i), (2) and (3), the resulting equations will be applicable to isotropic material only.

As in the preceding Art., r is used to express the ratio between direct and lateral strains, and has no relation what

ever to the co-ordinate r.

It is interesting and important to observe that the equations of motion and equilibrium for elastic bodies, are only special cases of equations which are entirely independent of the nature of the material, of equations, in fact, which express the most general conditions of motion or equilibrium.

CHAPTER II.

Thick, Hollow Cylinders and Spheres, and Torsion.

Art. 9.—Thick, Hollow Cylinders.

In Fig. i is represented a section, taken normal to its axis, of a circular cylinder whose walls are of the appreciable thickness t. Let p and p, represent the interior and exterior intensities of pressures, respectively. The material will not be stressed with uniform intensity throughout the thickness t. Yet

if that thickness, comparatively speaking, is small, the variation will also be small ; or, in other words, the intensity of stress throughout the thickness t may be considered constant. This approximate case will first be considered.

Fig.1

The interior intensity p will be considered greater than the exterior consequently the tendency will be toward rupture along a diametral plane. If, at the same time, the ends of the cylinder are taken as closed, as will be done, a tendency to rupture through the section shown in the Fig. will exist.

The force tending to produce rupture of the latter kind will be :

If N₁ represents the intensity of stress developed by this

force,

n{r² - r'a)

₌ pr'²~ PS*

r² - r²

. . (2)

If the exterior pressure is zero, and if r' is nearly equal to r. + r' m

• • (3)

pp _ pr'

²(ri~r )    2t

In this same approximate case, the tendency to split the cylinder along a diametral plane, for unit of length, will be :

F' = pr - psrv

If N' is the intensity of stress developed by F' :

(4)

N' is thus seen to be twice as great as Nx when px = 0. If, therefore, the material has the same ultimate resistance in both directions the cylinder will fail longitudinally when the interior intensity is only half great enough to produce transverse rupture ; the thickness being assumed to be very small and the exterior pressure zero.

Nx and N' are tensile stresses, because the interior pressure was assumed to be large compared with the exterior. If the opposite assumption were made, they would be found to be compression, while the general forms would remain exactly the same.

The preceding formulas are too loosely approximate for many cases. The exact treatment requires the use of the general equations of equilibrium, and the forms which they take in Art. 7 are particularly convenient. As in that Art., the axis of .r will be taken as the axis of the cylinder.

Since all external pressure is uniform in intensity and normal in direction, no shearing stresses will exist in the material of the cylinder. This condition is expressed in the notation of Art. 7 by putting :

T$x — Trx — Tr>, — o.

Again the cylinder will be considered closed at the ends, and the force F, Eq. (1), will be assumed to develop a stress of uniform intensity throughout the transverse section shown in Fig. I. This condition, in fact, is involved in that of making all the tangential stresses equal to zero.

Since this case is that of equilibrium, the equations (2), (3) and (4) of Art. 7 take the following form, after neglecting X₀, R₀ and &₀ :

(5)

(6)

(7)

dNt

dx

dR R — dr H “

dN^ rd<p

These equations are next to be expressed in terms of the strains u, p and w.

In consequence of the manner of application of the external forces, all movements of indefinitely small portions of the material will be along the radii and axis of the cylinder. Hence :

u will be independent of r and cp;

p “    “

w — o.

The rate of change, therefore, of volume will be (Eq. (6) of Art. 7) :

(8)

n du . dp . e = dx + dr +

dd

d²u

; hence if the value of

As p is independent of x, — =

dx

be taken from Eq. (7) of Art. 7 and put in Eq. (5) of this Art.:

dx²

dNx

dx

2 Cr d²u i — 2r dx²

d²u

~dx² - °*

d²u    j    .    ,

— O, and & = au: + a. dx³

But the transverse section in which the origin is located may be considered fixed. Consequently if x = o, u — o and thus a! — o. The expression for u is then : u = ax.

The ratio u ■— x is the /' of Eq. (1), Art. 1 ; while the p' of the same equation is simply Nx of Eq. (2), given above. Hence :

• . (9)

Aj _ pr'² - ps* E(rz² — r²)

Again, Eq. (8), of Art. 7, in connection with Eqs. (8) and (6) of this, gives:

c\ or:

O.

hi >3

2 Gx fd²p    dp

I — 2r \dr* r dr

to

'li'ö

o.

••

n.l'ö

dl —

d²P , __

dr²    dr

= o.

r dp -j- p dr — d{pr) = cr dr.

cr²

Pr = — +

»I*

HH

This value of p in Eqs. (8) and (9) of Art. 7 will give :

• . (ii)

R = 2G

t(a + c)

I — 2X

I ‘ I i i — 2r 2 r²

• • (I²)

At the interior surface R must be equal to the internal pressure, and at the exterior surface to the external pressure. Or, since negative signs indicate compression ;

If r = r'

R — —p.

If r = rt

R = — pi-

Either of these equations is the simple result of applying Eqs. (13), (14) and (15) to the present case, for which,

cos p = cos r — cos n — cos p = o,

-A-

cos q — cos x = I, and P — — / or

Applying Eq. (n) to the two surfaces:

U i. #

2

— p — 2G

r(^ -f- o')

i — 2r

■ • (13)

-A = 26

I r(^ + <0

— 2r

• • (14)

Subtracting (14) from (13):

2Gb =

(Pi - p) r* r’²

Inserting this value in Eq. (13):

*(a + c) , £.

i — 2r ‘    2

}-

Pi*i - Pr'²

r" - r² *

The general expressions of R and freed from the arbitrary constants of integration, can now be easily written by inserting these last two values in Eqs. (11) and (12). By making the insertions there will result :

R ₌    - Pr'²

r² - r²

(A - P) r?r'² l_

‘ r² *

r² - r²

• - (IS)

. . (16)

ar _ Piri - Pr" i (A - P) rir'² 1

The stress N,^ is a tension directed around the cylinder, and

has been called “ hoop tension.” Eq. (16) shows that the hoop tension will be greatest at the interior of the cylinder. An expression for the thickness, /, of the annulus in terms of the greatest hoop tension (which will be called h) can easily be obtained from Eq. (16). *

If r = r' in that equation :

2Ą?T - P ir'² -r A²)

AI A

h -T p

V,

2/1 —p +h.

.*. rx — r' — t — r'

h +p

2/r- P + K

• • 07)

Eq. (17) will enable the thickness to be so determined that the hoop tension shall not exceed any assigned limit h. If px is so small in comparison with p that it may be neglected, t will become :

t — r'

{m- « j-

• • (is)

If px is greater than /, becomes compression, but the equations are in no manner changed.

The values of the constants b and c may easily be found from the two equations immediately preceding Eq. (15).

It is interesting to notice that the rate of change of volume, 6, is equal to {a + f) and, therefore, constant for all points.

Art. io.—Torsion in Equilibrium.

The formulas to be deduced in this Article are those first given by Saint-Venaut, but, with one or two exceptions, established in a different manner.

It will in all cases, except that of the final result for a rectangular cross section, be convenient to use those equations of Art. 7 which are given in terms of semi-polar co-ordinates.

Let Fig. i represent a cylindrical piece of material, with any cross section, fixed in the plane ZY, and let the origin of co-ordinates be taken at 0. Let it be twisted, also, by a couple

P. ab = PI,

the plane of which is parallel to p ZY. The material will thus be subjected to no bending, but to pure torsion.

The axis of the piece is supposed to be parallel to the axis of Aas well as the axis of the couple.

Fig.1

Normal sections of the piece, orig- z/ inally parallel to ZOY, will not re- / main plane after torsion takes place. But the tendency to twist any elementary portion of the piece about an axis passing through its centre and parallel to the axis of X will be very small compared with the tendency to twist it about either the axis of r or cp ; consequently the first will be neglected. In the notation of Art. 7, this condition is equivalent to making Tr4> = o.

As the piece is acted upon by a couple only, all normal stresses will be zero.

2fft    • • (3)•

— 2r

Eqs. (7), (8), (9) and (11), of Art. 7, then become : 2Gx n , sdu

0+2Gfk

* =r

— 2r

(O

2Gx

i — 2r

0 ₊

dr

(2)

^ I SL

łh

(4)

After introducing the values of Trx and Z\x, from Eqs. (io) and (12) of Art. 7, in Eqs. (2), (3) and (4) of the same Article, at the same time making the external forces and second members of those equations equal to zero, and bearing in mind the conditions given above, there will result :

dTrx dT^x dr r dcp

Trx _ r( d²u +    G\~d? +

d²p

d²w

d²u

dr dx r dcp dx

T^dcp*

(5)

du dp r dr r dx

dTrx

d²u

d²p

dx

dr dx

dx²

. . (6)

dx

'ïï = g(~~ +

•    \dx²

d²u

r dcp dx

Also by Eq. (6) of Art. 7 :

The cylindrical piece of material is supposed to be of such length, that the portion to which these equations apply is not affected by the manner of application of the couple. This portion is, therefore, twisted uniformly from end to end ; consequently the strain u will not vary with any change in x. Hence :

(9)

II

a LH

■I Ił

^ I'fc

00

du

dx

Eq. (i) then shows that 8 — o. This was to be anticipated, since a pure shear cannot change the volume or density. Because 8 = 0, Eqs. (2) and (3) at once give :

dp

dw p dr    r dcp    r

(10)

As the torsion is uniform throughout the portion considered :

dp

dp

dx

r dx

(n)

Eq. (11) in connection with Eq. (10), gives:

(12)

d²w

r dx dcp

Eqs. (11) and (12), in connection with Eq. (10), reduce'Eq. (5) to the following form :

Both terms of the second member of Eq. (6) reduce to zero by Eqs. (9) and (11), and give no new condition. The second term of the second member of Eq. (7) is zero by Eq. (9) ; the remaining term therefore gives :

(H)

('S

dr

• (13)

d²w

dx²

As the stress is all shearing, p will not vary with cp. Hence :

dp

r dp °

05)

(4) to :

Eqs. (10), (11) and (15) show that p = o, and reduce Eq.

dzu

dr

(16)

Eq. (10) now becomes

dw

= o, and shows that w does not

r dcp

contain cp ; while Eq. (14) shows that w does not contain x² or any higher power of x. The strain zu, in connection with these conditions, is to be so determined as to satisfy Eq. (16). If a is a constant, the following form fulfills all conditions :

zu = arx

(i/)

Eq. (17) shows that the strain w, in the direction of p, i.e., the angular strain at any point, varies directly as the distałice

from the axis of X, and, as the distance from the origin measured along that axis. This is a direct consequence of making Tr^ = o.

The quantity a is evidently the angle of torsion, or the angle through which one end of a unit of fibre, situated at unit’s distance from the axis, is twisted ; for if ;

r — x — i, w — a.

An equation of condition relative to the exterior surface of the twisted piece yet remains to be determined ; and that is to be based on the supposition that no external force whatever acts on the outer surface of the piece. In Eqs. (13), (14) and (15) of Art. 6, consequently, P = o. The conditions of the problem also make all the stresses except :

and T₂ — T^x

equal to zero, while the cylindrical character of the piece makes :

p = 90°    .•. cos p — o.

If cos t be written for cos r :

cos t = sin q.

Eq. (13), just cited, then gives :

. . (18)

Txr cos q + T$x sin q = o .

But since p = o and zv = arx :

Txr — G

du

• dr

<49)

and

Tx<t> — g(^

du

r dcp

+ ar

. . (20)

Eq. (18) now becomes:

du

du

dr

= — tan q — —

dr₀

r dcp

-f- ar

r₀d(p '

• • (21)

in which r₀ is the value of r for the perimeter of any normal section.

Eqs. (13) and (21) are all that are necessary and all that exist, for the determination of the strain u. Eq. (13) must be fulfilled at all points in the interior of the twisted piece, while Eq. (21) must, at the same time, hold true at all points of the exterior surface.

After u is determined, Txr and Tx$ at once result from Eqs. (19) and (20). The resisting moment of torsion then becomes :

Tx$ r² dcp . dr = GJJ^ . r dr dcp -j- Gal^ .    (22)

r² . r dcp dr is the polar moment of

In this equation Ip =

inertia of the normal section of the piece about the axis of X, and the double integral is to be extended over the whole section.

According to the old, or common, theory of torsion :

M — Galp

The third member of Eq. (22), shows, however, that such an expression is not correct unless u is equal to zero, i.e., unless all normal sections remain plane while the piece is subjected to torsion. It will be seen that this is true for a circular section only.

It may sometimes be convenient to put Eq. (22) in the following form :

M — G r dr .

dep + Godp — Gju . r dr + Gal}

(²3)

In this equation u is to be considered as :

'Jdu

dtp ;

dep

while the remaining integration in r is to be so made that the whole section shall be covered.

It is very important to observe that the equations of condition for the determination of u, and consequently the general values of Txr and 7^, are wholly independent of any considerations regarding the position of the axis of torsion, or the axis of X. It follows from this, that the resistance of pure torsion is precisely the same wherever may be the axis about which the piece is twisted. It is to be borne in mind, however, that, if the axis of the twisting is not the axis of the cylindrical piece, the latter will be subjected to combined bending and torsion ; the bending being produced by a force sufficient to cause the piece to take the helical position necessitated by the torsion. The cylindrical axis is the straight line locus of the centres of gravity of all the normal sections.

If, as in Fig. 2, there are n cylinders whose centres c are all at the same distance Ce — l from the centre C of twisting, or motion ; and if M is the total moment of torsion of the system,

/ ...

ci

~o

Fig.2

while m is the moment of torsion of each cylinder about its own axis or centre cy then will M = nm ; and each cylinder will be subject to a bending moment whose amount can be determined from the condition that the diameter of each piece lying along Cc before torsion, must pass through C after, and during, torsion, also.

Since Txr and Tx$ act at right angles to each other, the resultant intensity of shear at any point in an originally normal section of the twisted piece will be :

(²4)

According to the ordinary methods of the calculus, the coordinates of the point at which T has its greatest value must satisfy the equations :

dT _ dT dcp dr

= o;

(25)

d' 7

d²T y _ d²T d²T/ dcp dr ) dcp² ' dr²

After the solution of Eqs. (25), it will usually be necessary only to inspect the resulting value of T, in order to determine whether it is a maximum or minimum, without applying the tests which follow those equations.

Equations of Condition in Rectangular Co-ordinates.

In the case of a rectangular normal section, the analysis is somewhat simplified by taking some of the quantities used in terms of rectangular co-ordinates.

In the notation of Art. 6, all stresses will be zero except

T₃ and T₂. Hence Eqs. (10), (11) and (12) of that Article reduce to :

dT₃ , dT2 -7 A H—7-^ = o.

dy

dz

d_T₃

dx

= o.

= o.

dT₂

dx

The strains in the directions of x, y and z are, respectively, u, v and w. Introducing the values of T₃ and T₂ in the equations above, in terms of these strains, from Eqs. (11) and (13) of Art. 5 ; and then doing the same in reference to the conditions

Nt = N₂ = N₃ = T1 = o:

the following equations will result :

d²u d²u

dy ^

dz²

.    . (26)

dv

dw

dz

dy

.    .    (27)

The operations by which these results are reached are identical with those used above in connection with semi-polar coordinates, and need not be repeated.

Eq. (27) is satisfied by taking :

axz ; w = — axy ;

in which a is the angle of torsion, as before.

The element of a normal section is dz dy. Hence the moment of torsion is

j j(r₃z - T₂y) dy dz.

'IIS'

y) dydz+Gal, .    (30)

.*. M = Gj(zu dz — yu dy) -f- Galp ....    (31)

Ip — ||(^ + y²) dy dz

i

is the polar moment of inertia of any section about the axis of X.

The integrals are to be extended over the whole section ; hence, in Eq. (31), zu dz is to be taken as :

z dz

Eqs. (ii) and (13) of Art. 5 then give:

s ■ <i, + a) - <

) • • • (28)

.    . (29)

f+y° du j

and 9^ dy as :

j (⁺*°du j

ydy\-,&Ai

in which expressions, y₀ and z₀ are general co-ordinates of the perimeter of the normal section.

Eq. (26) is identical with Eq. (13), and can be derived from it, through a change in the independent variables, by the aid of the relations:

z — r cos <p\ and y = r sin cp.

Solutions of Eqs. (13) and (21).

It has been shown that the function u, which represents the strain parallel to the axis of the piece, must satisfy Eq. (13) [or Eq. (26)] for all points of any normal section, and Eq. (21) (or a corresponding one in rectangular co-ordinates) at all points of the perimeter ; and those two are the only conditions to be satisfied.

It is shown by the ordinary operations of the calculus that an indefinite number of functions u, of r and <p, will satisfy Eq.

(13); and, of these, that some are algebraic and some transcendental.

It is further shown that the various functions u which satisfy both Eqs. (13) and (21) differ only by constants.

If u is first supposed to be algebraic in character, and if cv c₂, c₃ etc., represent constant coefficients, the following general function will satisfy Eq. (13) :

j    Cj r sin cp -\- c₂r² sin 2cp -f- c₃ r³ sin 3<p -f- .

( -j- c\ r cos cp c'₂r² cos 2 q) -j- c'₃r³ cos 3 cp -|- .

and the following equation, which is supposed to belong to the perimeter of a normal section only, will be found to satisfy Eq. (21) :

r²

— -f- cz r cos cp + c₂ r² cos 2 cp -j- c₃ r³ cos 3    .

— c\r sin cp — c₂r¹ sin 2 <p — c₃ r³ sin ^cp —    — C (33)

C is a constant which changes only with the form of sec

tion.

du

If — and

du

dr₀

be found from Eq. (32), while

be taken

rdcp

from Eq. (33), and if these quantities be then introduced in Eq. (21), it will be found that that equation is satisfied.

The. only form of transcendental function needed, among those to which the integration of Eq. (13) or Eq. (26) leads, will be given in connection with the consideration of pieces with rectangular section, where it will be used.

dr

rQ dtp

Elliptical Section about its Centre.

Let a cylindrical piece of material with elliptical normal section be taken, and let a be the semi-major and b the semiminor axis, while the angle cp is measured from a with the centre of the ellipse as the origin of co-ordinates, since the cylinder will be twisted about its own axis. The polar equation of the elliptical perimeter may take the following shape :

r¹ r¹ b¹ — a¹ 2 2 * a¹ + b¹

By a comparison of Eqs. (33) and (34), it is seen that :

COS 2cp =

a¹b¹

a¹ + b¹ '

• • (34)

; and C =

a¹b¹

a¹ + b¹ '

and that all the other constants are zero. Hence Eq. (32) gives :

• (35)

b¹ — cB .    01

a    ^ r¹ sin 2<p — — fr¹ sin 2(p .

The quantity represented by f is evident.

By Eqs. (19) and (20) :

Since

Txr

r₀. r₀ dtp

„ b² — a²

= r 2(p

(36)

* « (37)

— dA, A being the area of the ellipse, or

nab, the second member of Eq. (22), by the aid of Eq. (37), may take the form :

= Ga\Mlh^

- a²

r³ cos 2 tp -f- r³^ dr.

Then using Eq. (34) :

M = Ga

a²b²

a² -j- b².

dA — Ga

na³b³

a² + b² '    '

• (38)

If Ip is the polar moment of inertia of the ellipse (i.e., about an axis normal to its plane and passing through its centre), so that

nab [a² + b²)

then:

M = Ga

A*

4n²Ip

(39)

Using f in the manner shown in Eq. (35), the resultant shear at any point becomes, by Eq. (24) :

T — Gar Vf² + 2f cos 2cp    I.

= o,

dT

dcp

gives :

sin 2cp — o, or cp — 90° or o°.

Since f is negative, T will evidently take its maximum when cp has such a value that 2/ cos 2cp is positive ; or, cp must be 90°.

Hence the greatest intensity of shear will be found somewhere along the minor axis. But the preceding expression shows that T varies directly as the distance from the centre. Hence, the greatest intensity of shear is found at the extremities of the minor axis.

Making cp — 90° and r — h in the value of T:

T — Tm — Gab{i -/) = Ga

2 a²b

• (40)

Taking G a from Eq. (40) and inserting it in Eq. (38) :

(41)

M= T„^ = 2rjf ;

2    b

in which :

nab³

or the moment of inertia of the section about the major axis.

Equilateral Triangle about its Centre of Gravity.

Wç>

This case is that of a cylindrical piece whose normal cross section is an equilateral triangle, and the torsion will be supposed about an axis passing through the centres of gravity of the different nor- p mal sections. The cross section is represented in Fig. 3, G being the centre of gravity as well as the origin of co- H~ ordinates.

Fig.3

Let GH = y₂ GD — a. Then from the known properties of such a triangle • y

FD — DB — BF = 2a V3.

Hence, the equation for DB is ; r sin <p —

— r cos q>

-=—— = o.

V3

— o.

Hence, the equation for BF is ; r cos cp -J- a

Hence, the equation for FD is ; r sin tp -f-

2a — r cos cp

V3

= o.

Taking the product of these three equations, and reducing, there will result for the equation to the perimeter :

r*

2aa

— — cos 3 cp —- ,

6a    3

Comparing this equation with Eq. (33) :

C~

Hence:

r³ sin 2,(p

6a

• (43)

And by Eqs. (19) and (20) :

Txr — — G a

r² sin 3<p

(44)

Tx$ —

Gal r

r² cos 3 ql

• • (45)

Eq. (22) then gives :

M — Gal a — G a

’ fr⁴ cos 3 <p

dr dcp.

=. Galp — G a

r⁴ sin yp

6a

- dr.

= G a (jp — — a⁴V 3 ^ = 0.6 Galp — 1.8 G a a⁴V 3 ; (46)

since Ip — polar moment of inertia = 3a⁴V3. By Eq. (24) :

Gay r²

r³ cos 3 <p ------H

r⁴

4a³

• • (47)

= 0, gives sin 3 cp — o ;

dT

dcp

or

cp — o°, 6o⁰,. 120°, 180⁰, 240°, 300° or 360°. The values I20°, 240° and 360° make:

cos 3<p = + I ;

hence, for a given value of r, these make T a minimum. The values 6o°, i8o° and 300° make: hence, for a given value of r these make T a maximum. Putting cos 3<p = — I in Eq. (47) :

(48)

Ga(r + ï)

This value will be the greatest possible when r is the greatest. But = 6o°, 180⁰ and 300°, correspond to the normal a dropped on each of the three sides of the triangle from G. Hence r — a, in Eq. (48), gives the greatest intensity of shear Tm, or :

(49)

Tm — — Gota

Or, the greatest intensity of shear exists at the middle point of each side. Those points are the nearest of all, in the perimeter, to the axis of torsion.

The value of Ga, from Eq. (49), inserted in Eq. (46), gives :

M - 0.4-pTm = a

PTmm

20 ’

• • (50)

in which l = side of section = 2a a/J.

Rectangidar Section about an Axis passing through its Centre

of Gravity.

In this case it will be necessary to consider one of the transcendental forms to which the integration of Eq. (13) [or

(26)] leads ; for if the polar equation to the perimeter be formed, as was done in the preceding case, it will be found to contain r⁴, to which no term in Eq. (33) corresponds.

If c is the base of the Napierian system of logarithms (numerically, e = 2.71828, nearly), and A any constant whatever, it is known that the general integral of the partial differential equation (13) may be expressed as follows:

. . (51)

it — A Cnr cos ^ Cnr stn ^ i

when 11² -f- n² = 0. For :

d²u

dr²

d²u

du

r² d(fj

r dr

— A(n² -j- U²) Cnr cos $ (>n'rsin<!?'

But the second member of this equation is evidently equal to zero if

(n² -j- n'a) = o,

or n' = V — u².

These relations make it necessary that either n or u! shall be imaginary.

It will hereafter be convenient to use the following notation for hyperbolic sines, cosines and tangents :

sih t =

d — e-*

; co h t

d -f e~ł

; and, tah t =

d — e~ł

By the use of Euler’s exponential formula, as is well known, and remembering that n² — — n², Eq. (51) may be put in the following form :

2cnr c°sï [An sin (nr sin cp) + A'n cos (nr sin <p)] ;

in which the sign of summation is to be extended to all possible values of An and A'„. At the centre of any section for which r is zero, u must be zero also, for the axis of the piece is not shortened. This condition requires that A'„ — o ; u then becomes :

u = 2enrcot* An sin {nr sin cp).

The subsequent analysis will be simplified by introducing the form of the hyperbolic sine, and this may be done by adding and subtracting the same quantity to that already under the sign of summation, in such a manner that :

u = 2[An sin {nr sin cp) . sih {nr cos cp) y₂ A n sin {nr sin cp) e~

nr cos

(52)

Now if the product :

sin {nr sin cp) e ~ nr cos $

be developed in a series and multiplied by An, one term will consist of the quantity :

— r² sin cp cos cp

multiplied by a constant, and if :

2An sin {nr sin cp) e~nr cos^

be replaced by simply :

— (XT² sin cp cos cp

all the conditions of the problem will be found to be satisfied. This is equivalent to putting:

— ar³ sin cp cos cp

for a general function of r sin cp and r cos give the following form to u, first used by Saint-Venant :

This change will

u = 2A„ sin (nr sin cp) . sih (nr cos cp) — ar² sin cp cos cp .    (53)

Fig. 4 represents the cross section with C as the origin of co-ordinates, and axis. The angle cp is measured positively

Oi

•x-4----3-

\ oa

Li

Fig.4

from CN toward CH. At the points N, H, K and Z, in the equation to the perimeter, dr₀ will be zero. Hence, at those points, by Eq. (21) :

du

An sin (nr sin cp) . n cos qj . coh (nr cos cp) + An . n sin cp . cos (nr sin cp) . sih (nr cos cp)]

— 2ar sin cp cos cp — o.

At the points under consideration cp has the values o°, 90°, 180⁰, 270° and 360°. At the points Vand K, cp — o° or 180⁰ ;

dzi

hence, sin cp =0, and both terms of the second member of —

dr

reduce to zero, whatever may be the value of 71. But at H and A = 90° and 270° ; hence, sin cp = + 1 or — 1 and cos cp = o. dti

In order then, that = o at H and L. these must obtain : dr

— cos (— nr) — o.

cos nr

If HL — c ; and, KN = Æ ; then :

ni

cos — = cos

ne

• • (54)

If the signification of n be now somewhat changed so as to represent all possible whole numbers between o and oo, Eq. (54) will be satisfied by writing :

7t

2n — I

for 11, in that equation. Eq. (53) will then become :

u — 2An sin

211 —

. s ill

'2 n — i

nr cos (p

• • (55)

— on² sin cos cp .

The quantity An yet remains to be determined by the aid of Eq. (21), which expresses the condition existing at the perimeter of any section.

Now, for the portion BN oi the perimeter :

r cos cp = —

and

drQ

rQ dep

will be the tangent of (— <p) ; or,

drQ    , s

— r dep ~ — ^an (— <^>' ~ ^an

Hence, Eq. (21) becomes :

or:

dll

dr

dn

r dcp

+ ar

du

= tan cp

• (56)

ar sin cp — — cos cp —

du

r dcp

sin cp.

Substituting from Eq. (55), then making:

r cos cp —

r sin cp = 2A„ .

2 ac

- 7T . coh

. sin

(2n — i    \

—-— nr sm cp).

If r sin cp be represented by the rectangular co-ordinate y, and another quantity by H, the above equation may be written :

y = H₁ sin — + H₂ sin + H₃ sin — c    c    c

2n — i

n ) y 4- ...

If both sides of this equation be multiplied by

sin

2n — I

ny

) . dy,

1 • •    c

and if the integral then be taken between the limits o and -, it

is known from the integral calculus that all terms except the nth will disappear, and that :

„ = j y . sin (y-y—- ny ^ • dy

) • dy.

7ty

f* , [2tl — I S171² ( -

Jo \ C

Completing these simple integrations :

n.= (

(2n — 1) 7T

³(- i)*-**.!.

Hence :

Ah

(2« — i)² n²

■'tl 's»

2ac

(2« — i) 7T

coh

If this value of be put in Eq. (55), and if rectangular coordinates :

y — r sin cp,

and z — r cos cp,

be introduced, that equation will become :

u — — azy +

. ać*2

m (- l)M *sin(*ny-l ny ^ . sih (^y^ n* )

(2« — i)³ coh (y~yc—~ nb ^

. (57)

This value of u placed in Eq. (31) will enable the moment of torsion to be at once written.

c    c

The limits -j-y₀ and — y₀ are +- and — and the limits

Hence :

,    , b .    b

+ z₀ and — z₀ are -1— and — -

2 sih

2n — i

TCZ )

a be

-T +

00

2

¹ (2 n — i)³ coh ^

2n — i

2c

nb

— Q, for brevity.

oo(—I )n~¹.2sih

2n — I

nb ) . sin

2n — i

7ty

(2n — i)³ coh (^n2c—~ nb

For the next integration :

2b c

f+\

J 'fizdz-

cube

r _ by_

L ¹³

“(2»—l)7T

. coh

272 —

2£

- xb-r-£---. sik (MJ nb)

, \2n— l)²7T²    \ 2f /

(2n-~ i)³ coh (²”-i nb^j

I * Ry dy — abc ^

».

ĄC²

sih

- nb

4°, (2n — i)² n2 ¹ (27i — if coh (^n'2c ~~ nb

2C

)_

Thus the integrations indicated in Eq. (31) are completed. Hence :

= GI Qz dz + jfiy dy + alp j- .

Remembering that :

n/r _ r P^³ I l6^3    1

M— Ga\ -~r -f- -— 2-j-r-

L6 ti* I (2n — i)⁴

64^ -

7T5 t

taJi

(~ ")1

iyn — if

• * (58)

But it is known that :

co

_v_ _ _ _

! (2n — i)⁴    1.2.3* 2s ‘

Hence Eq. (58) becomes :

and remembering that :

295.1215 *

and since:

64 ₌

w5 0.209137,

tah n

M ~ Gäbe³

tah

00

— nb

(2« — i)⁵

• (59)

Since :

I — tah n

(7 + P + j₅ + •

, I — tah in ₍ i — tah 5n

r    _r    ■ ~T --

3⁵

5s

-)

tah in

3⁵

tah 5 n

5s

)⁵=i+p+p+---=0-i)

Eq. (59) becomes :

+ O.209137 T

Gäbe³ p-

nb

I — tah -—

2c

k)

00

OJ

I — tah

3 nb

3⁵

(60)

Eq. (60) gives the value of the moment of torsion of a rectangular bar of material.

If z had been taken parallel to b, and y parallel to c, a moment of equal value would have been found, which can be at once written from Eq. (60) by writing b for c and c for b. That moment will be :

Gacb³ Tj —

to

CO

dl ¹

+ 0.209137

— tah—.

i — tah

3nc

2b

3⁵

• (61)

Eq. (60) should be used when b is greater than c, and Eq. (61) when c is greater than b, because the series in the parentheses are then very rapidly converging, and not diverging. It will never be necessary to take more than three or four terms and one, only, will ordinarily be sufficient. The following are the values of,

(' -tahT)

for a few values of n :

- »ah

= 0.083 : 0.00373 : 0.000162 : 0.000007 •

Square Section.

-tah f)]

If c = b either Eq. (60) or Eq. (61) gives : M = Gab⁴ J — 0.2I0I + 0.209(^1

in which vl is the area ( = b²) and h is the polar moment of inertia

Rectangle in which b — 2c.

If b = 2c, Eq. (60) gives :

M — Ga . 2<4

M — 0.1406 Gab⁴ = G a

A*

427 h ’

.    .    (62;

Q- — 0.105 + 0.1046 (1 — tah 7r)^

.’. M = O.457 Gac⁴ = G a

A*

42 Ip

(63)

in which A is the area (= 2c²') and = polar moment of inertia

_ be³ + b³c _ 5(4

12

Rectangle in which b = 4c.

If b = 4c, Eq. (60) then gives :

M — Gabe³

(j - 0.0525)

= 1.123 Gac⁴

M — Ga

A*

40.2 It ’ in which ^ = area = 4<4 and / = polar moment of inertia

be³ -j- b³c _ 17 c⁴

12

If b is greater than 2c, it will be sufficiently near for all ordinary purposes to write :

Pi

I CO

HH

VO

OJ

• • (65)

Greatest Intensity of Shear.

There yet remains to be determined the greatest intensity of shear at any point in a section, and in searching for this quantity it will be convenient to use Eqs. (28) and (29).

It will also be well to observe that by changing z to y, y to — z, c to b and b to c, in Eq. (57), there may be at once written :

Il

azy

7T5¹

f 2n — I ' Sth \ ~b

TTJ/

(:2îi — i)³ coh

2n

ne

(66)

This amounts to turning the co-ordinate axes 90°. Since the resultant shear at any point is:

t=Vt; + Ti,

it will be necessary to seek the maximum of

dn

du

—r az

dy

⁺ \dz~ ay

T²

" G² ‘

The two following equations will then give the points de

sired :

T²'

d²u

dy

dy²

du

dz-ay

) (py _ a\

/ \dz dy )

• - (67)

'(£)

dz

s+“) (

d²u

dz dy

+• «

d²u

dz²

. . (68)

It is unnecessary to reproduce the complete substitutions in these two equations, but such operations show that the points of maximum values of T are at the middle points of the sides of the rectangular sections ; omitting the evident fact that T — o at the centre. It will also be found that the greatest intensity of shear will exist at the middle points of the greater sides.

This result may be reached independent of any analytical test, by bearing in mind that an elongated ellipse closely approximates a rectangular section, and it has already been shown that the greatest intensity in an elliptical section is found at the extremities of the smaller axis.

By the aid of Eqs. (28), (29), (57) and (66), it will also be found that T₃ = o at the extremities of the diameter c, and T₂ = o at the extremities of the diameter b. The maximum value of T will then be :

Tm=- Ta=-G

du

■ ■ (69)

y=i

By the use of Eq. (57):

00

du

Tz~“y

(— i)*⁻¹ . — . sin

v J c

=-2^+d)'

271 — I

Tty

). coh (2Lr ¹

7tZ

(2 n — i)² coh {~~c—~    ^

c .

Putting z — o and — — in this equation, there will result :

Tm — Gac

¹ (2n — i)² coh Ç

- 7tb

• (70)

If ^ is greater than ^ the series appearing in this equation is very rapidly convergent, and it will never be necessary to use more than two or three terms if the section is square, and if b is four or five times c there may be written :

(71)

Tm = Gac

Square Section.

Making b — c in Eq. (70) and making n — 1, 2 and 3 (i.e., taking three terms of the series) there will result :

Tm — 0.676 Gac .*. Ga — 1.48—.

Inserting this value in Eq. (62) :

M = 0.21 b³Tm —

1-2 6ITm

• • (72)

/. Tm = 0.8 -j- a

= 5 TT

• • (73)

in which :

re

Q.

Rectangular Section ; b — 2c.

Making b — 2c in Eq. (70) and making n — 1, only, there will result :

Tm — 0.93 Gac .*. G a — 1.08 —- .

Inserting this value in Eq. (63) :

IT

x ^ m

M = 0.49 czTm = 1.47 —

a

• • (74)

• • (75)

Tm = 0.68 a

in which :

Circular Section about its Centre.

The torsion of a circular cylinder furnishes the simplest example of all.

If r₀ is the radius of the circular section, the polar equation of that section is :

~ — C, (constant).

Comparing this equation with Eq. (33), it is seen that :

= o.

Rectangular Section ; b — 4c.

Making b — 4c in Eq. (70) and making n— 1, only:

Tm — 0.997 Gac . '. G a —

⁰⁰³ e- ’

Inserting this value in Eq. (64) :

M= 1.126 c*Tm = 1.69

. . (76)

7^ = 0.6 -j a = 0.9 — ; .

• » (77)

in which :

^ M l-l

Cj —- c₂ C^ i . .    Cj — c₂ • •

By Eq. (32) this gives u — o. Hence, all sections remain plane during torsion.

Eqs. (19) and (20) then give :

• • (78)

Txr — o ; and, Tx<t> = Gar .    .

Eq. (23) gives for the moment of torsion :

(79)

M = Gal

or :

M — 0.5 nr* . Ga =

A*G

(80)

In which equation, A is the area of the section and

nr*

The greatest intensity of shear in the section will be obtained by making r = r₀ in Eq. (78) ; or :

(so

Tm = Gar₀ Ga — —

Eq. (80) then becomes :

M = 0.5 nr% Tm — 2

IT

± ± m

r₀

(82)

• - (83)

Tm — 0.64    — 0.5 j r₀ ;    .

in which

=T

It is thus seen that the circular section is the only one treated which remains plane during torsion.

General Observations.

The preceding examples will sufficiently exemplify the method to be followed in any case. Some general conclusions, however, may be drawn from a consideration of Eq. (33).

If the perimeter is symmetrical about the line from which ą> is measured, then r must be the same for -j- <p and — cp ; hence :

Cj — Ca — Cj — ...

= o.

If the perimeter is symmetrical about a line at right angles to the zero position of r, then r must be the same for :

<p = 90° -j- (p' and 90° — cp' ;

hence :

— Ca — CA — Cÿ — ...

= o.

In connection with the first of these sets of results, Eq. (32) shows that every axis of symmetry of sections represented by Eq. (33) will not be moved from its original position by torsion.

If the section has two axes of symmetry passing through the origin of co-ordinates, then will all the above constants be zero, and its equation will become :

— -\- eg¹ cos 2cp -f eg⁴ cos 4<p + eg⁶ cos 6cp + ...

= K.

Art. ii.—Torsional Oscillations of Circular Cylinders.

Two cases of torsional oscillations will be considered, in the first of which the cylindrical body twisted is supposed to be the only one in motion. In the second case, however, the mass of the twisted body will be neglected, and the motion of a heavy body, attached to its free end, will be considered. In both cases the section of the cylinder will be considered circular.

Since these cases are those of motion, the internal stresses are not, in general, in equilibrium ; hence, equations of motion must be used, and those of Art. 7 are most convenient. Of these last, the investigations of the preceding Art. show that Eq. (4) is the only one which gives any conditions of motion in the problem under consideration.

Putting the value of :

dw

T- T+x-G

in Eq. (4) of Art. 7, that equation may take the form :

d²w

d²w _ G d²w . dt² m dx² ’    °r’ dl²

d²w

dx²

• • (o

For brevity, b² is written for — .

That dimension of the cross section of the body which lies in the direction of the radius will be assumed so small that w may be considered a function of x and t only. The results will then apply to small solid cylinders and all hollow ones with thin walls.

The general integral of Eq. (1), on the assumption just made, is (Books’ “ Differential Equations,” Chap. XV., Ex. 1) :

— f(x + bt) + F(x - bt) ;

in which f and F signify any arbitrary functions whatever. Now it is evident that all oscillations are of a periodic character, i.e., at the end of certain equal intervals of time, w will have the same value. Hence since f and A are arbitrary forms, and since circular functions are periodic, there may be written :

w = A„\sin iocnx + ajbt) -f sin {anx — anbt)}

— Bn\cos (aHx -j- anbt) — cos (anx — anbt)\ ; .

• (2)

in which an, An and Bn are coefficients to be determined.

Substituting for the sines and cosines of sums and differences of angles :

w = 2 sin anx(Ancos ajbt -j- Bn sin a„bt) ...    (3)

Let the origin of co-ordinates be taken at the fixed end of the piece ; w must then be equal to zero, as is shown by Eq. (3). But there may be other points at which w is always equal to zero, whatever value the time t may have. These points, called nodes y found by putting w = o ; or :

sin ax = o

(4)

This equation is satisfied by taking :

cl*

nn #

and x — a ; in which a is the length of the piece. Hence, at the distances :

ft

to I a

from the fixed end of the piece, there will exist sections which are never distorted or moved from their positions of rest. These are called nodes, and one is assumed at the free end, although such an assumption is not necessary, since a is really the distance from the fixed end to the farthest node and not necessarily to the free end.

If, as is permissible, An and Bn be written for twice those quantities, the general value of w now becomes :

sin —(a

a \

J cos

+ sin

™(a

a \

a COS

COS

— + B, sin ^-) a    a J

~nbt , „    . 2nbt\

--1- B₂ sin-)

a    a J

Æ ₊ B3sin¥ÉÎ)

a    ³ a J

rnzbt

+ sin ^-(am a \

nnbt , ^

cos--b Bn sin

a

The coefficients A and B are to be determined by the ordinary procedure for such cases, Let :

wt = cp(x)

be the expression for the initial or known strain at any point, for which the time t is zero. Then if An is any one of the coefficients A :

<n !

P , x . nnx j J cp(x) sin —— ax .    .

• • (6)

The velocity at any point, or at any time, will be given by :

dw

dt

. 7tx ( .    . 7xbt    „ nbt

= — sin — Ar sm--B, cos-

a \    a    a

nb

• - (7)

In the initial condition, when the time is zero, or t — o, it has the given, or known, value :

dt

= 4>(x) = — (bj sin — + 2B₃ sin — a \    a    a.

¹

{a¹ + b¹)

+ 3Ą«»    +

Then, as before :

3.=

2 fa ,, v .    ,

^-1    -S772 ——

(8)

Thus the most general value of w is completely deter

mined.

The intensity of shear at any place or time is given by:

dw

T= G~;

dx

w being taken from Eq. (5).

The second case to be treated is that of the torsion pendulum, in which the mass of the twisted body is so inconsiderable in comparison with that of the heavy body, or bob, attached to its free end that it may be neglected.

Let M represent the mass of the pendulum bob, and k, its radius of gyration in reference to tJic axis about which it is to vibrate ; then will Mh² be its moment of inertia about the same axis.

The unbalanced moment of torsion, with the angle of torsion a, is, by Eq. (9) of Art. 10:

G ah.

The elementary quantity of work performed by this unbalanced couple, if ß is the general expression for the angular velocity of the vibrating body, is :

Galp . ß dt.

This quantity of energy is equal in amount but opposite in sign to the indefinitely small variation of actual energy in the bob ; hence :

= — MJdß dß.

Gafß dt = - d

But if a is the length of the piece twisted :

d(ad)

, and dß clt

_ d²(ad) dt

-• (^>)

= - Mh²

d\aa)

dt²

ting:

Multiplying this equation by 2d (ad), and for brevity put

GI

= //; (Mh²) = K\

then integrating and dropping the common factor a² :

Ha* = - K

When a = at, the value of the angle of torsion at the ex-

doc

tremity of an oscillation, the bob will come to rest and will be zero. Hence :

C = Ha*,

and

da

«.it.

. a

• sin~¹ — = t

<*x

C — o because a and t can be put equal to zero together. At the opposite extremities of a complete oscillation a will have the values :

(+ <*t) and (- a,).

Putting these values in the expression :

K . a --- . sin~* —

«1

(10)

and taking the difference between the results thus obtained,

the following interval of time for a complete oscillation will be found :

M/Pa

“SÇ"- •

• . (II)

The time required for an oscillation is thus seen to vary directly as the square root of the moment of inertia of the bob and the lengt/i of the piece, and inversely as the square root of the coefficient of elasticity for shearing arid the polar moment of inertia of the normal section of the piece twisted.

The number of complete oscillations per second is i. If

this number is the observed quantity, the following equation will give G :

² 7t²MJfa

The formulas for this case should only be used when the mass of the cylindrical piece twisted is exceedingly small in comparison with M.

Art. 12.—Thick, Hollow Spheres.

In order to investigate the conditions of equilibrium of stress at any point within the material which forms a thick hollow sphere, it will be most convenient to use the equations of Art. 8. As in the case of a thick, hollow cylinder, the interior and exterior surfaces of the sphere are supposed to be subjected to fluid pressure.

Let r' and rx be the interior and exterior radii, respect

ively.

Let — p and — px be the interior and exterior intensities, respectively.

Since each surface is subjected to normal pressure of uniform intensity no tangential internal stress can exist, but normal stresses in three rectangular co-ordinate directions may and do exist. Consequently, in the notation of Art. 8,

T$r — 7    — o.

With a given value of r, also, a uniform state of stress will exist. Neither nor can, then, vary with cp or ip. By the aid of these considerations, and after omitting 7?₀, <h₀, W₀, and the second members, the Eqs. (1), (2) and (3) of Art. 8 reduce

to :

dNr

2Nr — N,), — N^

dr

.    .    (i)

(2)

— N^ -j- N* — o

By Eq. (2) :

N+=Nt.

Eq. (1) then becomes:

• - (3)

On account of the existing condition of stress, which has just been indicated, it at once results that :

rj = go — o,

and that p is a function of r only.

Eqs. (4) to (10), of Art. 8, then reduce to :

(4)

dp 2p dr r

Nr =

2Gx

dp

i — 2r

6Jr2Gdr ‘    *

. . (5)

2Gx

i — 2r

0 2G —

(6)

After substitution of these quantities, Eq. (3) becomes :

2 Gx f d²p I — 2r\ dr²

2rdp-2pdr\ 2G^₊₄gIL

r² dr J — dr*    r dr

-4 g£

r²

— o.

or :

d²p

dr3 +

(d²E)

= o.

dr

One integration gives :

dp

_

dr r

(7)

Hence 6, the rate of variation of volume, is a constant quantity. Eq. (7) may take the form :

r dp + 2p dr = cr dr.

As it stands, this equation is not integrable, but, by inspecting its form, it is seen that r is an integrating factor. Multiplying both sides of the equation, then, by r\

r² dp -j- 2 rp dr = d (r²p) = cr² dr.

Vs

Q.

er

r² '

(8)

Substituting from Eqs. (7) and (8) in Eq. (5) :

Nr =

2G1    . 2Gc

-c 4----

i — 2r 3

4 bG

r³

• • (9)

r³

It is obvious what A represents.

When r' and rl are put for r, Nr becomes — p and — pv Hence :

= -A;

4bG _

r³

and :

r,³

= “A-

These equations express the conditions involved in Eqs. (13), (14) and (15), of Art. 6.

The last equations give:

46Ć =

. a - AA³ - Pr'*

r'³ — r³

These quantities make it possible to express Nr and independently of the constants of integration, c and b, for those intensities become :

Nr =

r'³ — r³

_ pxr³ — pr'3 r'³ — r³

(Pi - P) r³r’³

Nt=N* =

_ Ari³ ~ P^

r'i — r3

Thus it is seen that Afy = Afy has its greatest value for the interior surface ; that intensity will be called h.

It is now required to find rx — r' — t in terms of h, p

and pv

If r — r' in Eq. (i i) :

2Ji(r'³— rp) — 2>Pirł — p(2r'³ + rp).

Dividing this equation by r'³ and solving :

Lh —    2(h + A .

r'³ 2h — p + 3pi '

rx — r' = t — r'

³/ 2(h + p)

2h - p + 3A

If the intensities / and px are given for any case, Eq. (12) will give such a thickness that the greatest tension h (suppos-ing y>x considerably less than p) shall not exceed any assigned value. If the external pressure is very small compared with the internal, A may be omitted.

The values of A and 4Gb allow the expressions for c and b to be at once written.

If pt is greater than p, nothing is changed except that becomes negative, or compression.

CHAPTER III.

The Energy of Elasticity.

Art. 13.—Work Expended in Producing Strains.

The general expressions, in rectangular co-ordinates, for the unbalanced forces which act in the three co-ordinate directions upon any indefinitely small parallelopiped of material subjected to any state of stress whatever, are given by multiplying each of Eqs. (7), (8) and (9) of Art. 6 by {dx dy dz). If an indefinitely small change in the state of stress takes place, that indefinitely small parallelopiped will suffer a displacement whose rectangular components are du, dv, dw ; and the amount of work performed in moving it will be found by multiplying each of the three unbalanced forces, determined as above, by each of the three small strains belonging to the same direction with the force (as in Art. 6, u, v and w are strains in the directions of x, y and z). This differential quantity of work, integrated throughout the extent of the body, will give the elementary quantity of work required for the small deformation and motion of the whole body.

The resulting equations form the foundation of investigations in elastic vibrations and resilience ; they also furnish the means of reaching some general conclusions in reference to suddenly applied loads.

Let dW represent the elementary quantity of work required for the motion only, then the operations which have just been indicated will give the following expression :

~(dN1 _ \ dx

dT.

dx dy dz Ą—-j— dx dy dz +

dT₂

~ dx dy dzj du + ((~p³ dx dy dz    dx dy dz + dx dy dzj dv

(—J'² dx dy dz +

+ (xo du + Y₀ dv + Z₀ dw ^ dx dy dz

dy

dß

dT,

dy

dx dy dz +    ³ dx dy dz\dw

II m (du

d²u .    .

+ dV

d²v

di

- + dw

d²w

dt²

^ dx dy dz = dW .    (1)

This equation, however, can be put in a much simpler form, and, caused to take a shape which will show at a glance the true character of each part ; dx, dy and dz are differentials of independent variables, hence they are arbitrary and independent. Integrating by parts, therefore :

' [dN,

dx

jj(AY du! - Nt" du") dy dz

\WN'd(\ii)dx dy dz'

dx . dy dz . du =

in which the primes indicate the values of N₁ and u at one point of the exterior surface of the body, and the seconds those values for another point of the exterior surface ; these points being taken at opposite extremities of a bar of the material whose normal section is (dy dz) and which extends entirely through

body in the direction of x. Maintaining the same notation and proceeding :

I dy . dx dz . du —

dy

- Tf du") dx dz

\\\T'dXIy) dy dX dZ ■

f.dT₂

J J dz

dz . dx dy .du —

( Tar dii! — Tf du') dx dy

T₂ d dz dx dy.

But by referring to the equations which immediately precede (13), (14) and (15) of Art. 6, it will be seen that the sum of these three double integrals will represent the amount of work performed on the body by the external forces acting in the direction of the axis of x. Precisely the same general results are obtained for the directions of y and z by treating in the same manner the remaining derivatives of the internal intensities in Eq. (1). The preceding operations are typical, therefore they need not be repeated.

Again, by reference to the notation and demonstrations of Art. 5 :

= dcp₁ ;

= dcpy,

© +0

du

.dx,

— dlx\ d

'dv

,dy

— dl₂ »

'©)

= dlt.

Finally :

d²u

i \ ldu\²

dt²

,dt

; dv

d²v

dt²

dt

dw

d²w

HH

dt²

fell

Introducing these reductions and quantities, Eq. (i) becomes :

jp'da' (cos ri du’ -f- cos x' dv' -f- cos p' dw)

— jp"da" (cos n" du" + cos x" dv" -f- cos p" dw")

— J11(Ntdlt + N₃dl₃ -f- N₃d!₃ + T₃d<px -f- Tzd<p₃ -f- Tyicp₃) dx dy dz + HJ(X₀ du + Y₀ dv + Z₀ dw) dx dy dz

dx dy dz — dW. (2)

©’+ ©j

Eq. (2) shows clearly the distribution of the different portions of work expended. The first two (single) integrals evidently represent the total amount of work performed by forces acting on the exterior surface of the body ; it will be indicated by dWv If the forces Pr and P" are of the same kind (i.e., both pulls or both pushes), the algebraic sum of any two terms of these integrals will be a numerical sum if they involve cosines of the same letter but of opposite signs.

The correct application of Eq. (2) depends largely upon the proper observance of the signs which should affect P', P" and the cosines.

The first triple integral in the first member of Eq. (2), in which each intensity of stress is multiplied by the differential of its characteristic strain, and which will be indicated by dW₂, is evidently the amount of work required for the small distortion alone, of the body. The quantity within the parentheses is called the potential energy of the elasticity of a cubic unit of material, since, if it be multiplied by (dx dy dz), the product will express the amount of work that small portion of material can perform in returning to its original condition.

This potential energy for a cubic unit is easily integrated by the aid of Eqs. (11), (12), (13), (17), (18) and (19) of Art. 5. Making the substitutions from those equations and integrating :

Nx dlx -J- N₂ dl₂ 4- N₃ dl₃-\- T₃ dcpx -f- T₂ dą>₃ -(- Tt depPj

f + ip + 4a

zGr (4 + 4 + If

H is the potential energy of a cubic unit of material for a change of state extending from the limit o to the strains /„ l₂, etc.

The last triple integral in the first member of Eq. (2) expresses the work done by external forces which take hold of the mass of the body. Let it be represented by dWy This triple integral added to the first two single integrals, which belong to the surface of the body, will give the total work done by external forces.

The second member of the equation is the small variation of actual energy, which usually exists in consequence of vibrations.

Let V be the resultant velocity of the parallelopiped, then

will :

- dV² = VdV = - d

l du Y _\dt J

dv\²

'dw

2~|

dt

By transferring dlV₂, the first two members of Eq. (2) may take the form :

dWt + dW₃ = dWa + IjjmVdVdx dy dz.

wt + W₃ = W₂ + H mVdVdxdydz . . (3)

Or, the total external work performed is equal to the work done in distorting the body added to the change of actual energy.

This result expresses the law of the conservation of energy for the elastic bodies considered.

If the external work is nothing, the first member of Eq. (3) is zero. The actual energy will then exist in consequence of a state of vibration. Let its variable value be represented by U. Since dx, dy, and dz are arbitrary :

U-

11m ~2 dy dz — C \

C representing a constant of integration. Under the circumstances assumed, then :

• (4)

W₂ + U = C

Hence, the total energy of the vibrating body (i.e., the sum of the actual and potential') will be constant.

Art. 14.—Resilience.

The term resilience is applied to the quantity of work which is required to be expended in order to produce a given state of strain in a body. The analytical expression for this amount of work is obtained directly from Eq. (2) of the preceding Art.

Let the simple case of a single straight bar be considered ; and let all the external forces act parallel to the axis of the bar while they take hold of the end surfaces, which are normal sections. These external forces will be considered equal to the internal stresses developed ; consequently no vibrations will exist. The action of the external forces X₀, Y₀ and Z₀ will also be omitted.

Now, if the axis of ^ be taken parallel to the axis of the bar, and if that end of the bar to which P" is applied be fixed, there will result from the preceding conditions :

COS 7t' — COS 7t" — I,

cos y = cos f — cos x" = cos ft" = du!’ — o,

Nt = N₃ = Tt = Ta = T₃ = o.

Eq. (2) will then become :

P'da du' ~ J J J A/j dlx dx dy dz . . . . (1)

But if the intensity P' is uniform and A the area of normal section, Eq. (1) becomes :

(2)

P'A du' = AN\ x■ dlx

in which xT is the length of the bar. From Eq. (1) of Art 1 :

Nx = Elx,

hence :

rli

l?

P'A du! — Resilience = Axx E /, dlx = Axx E — . (3)

The quantity :

A?

is called the “ Modulus of Resilience.” This term is usually applied when Nx is the greatest intensity allowed in the bar.

If one end of a bar, placed in a vertical position, is fixed, while a falling body whose weight is w, acts upon the other end, the height of fall may be sufficient to produce rupture. Let h be the height of fall required and Nx =/ the ultimate resistance of the material of the bar. In order that rupture may take place :

A p¹

. h = xx — .    . .

2 w E

• . (4)

Eq. (4) shows that the height of fall varies directly as the length of the piece. It is virtually assumed, however, that the extension or compression is uniform throughout the length of the bar, to the instant of rupture. This, in reality, is not true, and h will not vary as rapidly as xv The principle established in Eq. (4) is equally true for torsion and bending.

Art. 15.—Suddenly Applied External Forces or Loads.

A very important deduction can be reached by an attentive consideration of Eq. (2) of Art. 13, if it be assumed that the external forces P' and P" are simple and direct functions of the external strains u, v and w. In such a case the following relations will hold, in which a, b and c are constants :

P' cos 7t' — au ;

P' cos x' — bv' ;

P' cos pi = cw

P" cos 7t" - au" ; P" cos x" = bv" ; P" cos p" = cw" .

Consequently the external work performed, omitting X₀, Y0 and Z₀, in changing the body from a state of no stress to that indicated by the strains it', n', 10', tt", u", tu", will be :

dWx =

da’ (a'S₊fS \ 2 2-

to'²

+

= W;

in which equations the integrals are to be made to cover the whole extent of the surface.

If, instead of being variable, the forces P' and P" are constant and equal to the fi?ial values of the preceding case {i.e., equal to au’, bo', rto', an", etc.), the external work performed in bringing the body to the final state n', o', etc., will be :

dWT = da! (an'² -f- bn'² + <ou ²)

I da" (an"² + fo"² + ^tu"²) = 2 W.

This last case is that of “ suddenly ” applied external forces or loads, while the former is that of gradual application, in which the external forces, at each instant, are equal to the internal resistances. In the case of sudden application it is seen that the amount of work expended is twice as great as in the other case ; consequently when the body arrives at the state of strain indicated by u', n', etc., there remains to be expended just as much work as has already been performed, and at the instant in question it exists in the body in the shape of actual energy.

But if an amount of energy equal to W will produce the strains it', n', etc., and if, while the force acts which performed the work, an additional amount of energy equal to W be expended on the body, additional strains equal to it', n', etc., will be produced in the body.

When the body comes to rest, therefore, the external strains will be 2it', 211', 2m', etc. There is then no actual energy, all is potential.

Since the external strains are 21t', 20', etc., the external work which has been performed up to this instant will be found by putting those quantities in the place of it', n', etc., in the expression for W, above. That expression will then become 4 W.

For gradually applied loads Eq. (2) of Art. 13 becomes simply :

I ! H dx dy dz ;

in which H is the potential energy per cubic unit for the state

of strain corresponding to n', 1)', to', etc. But, if the loads be suddenly applied, in accordance with what has been given, the Eq. (2) of Art. 13 becomes :

4 W = HI4H dx dy dz.

Now the expression for H, given in Art. 13, shows that multiplying H by 4 is the same thing as doubling the strains :

ln I21 ty <Pn ând *Py

But by doubling the strains the intensities of stresses are doubled. Hence, if the same loads are first applied gradually and then suddenly, the strains and stresses in the latter case will be double those in the former. This is a very important principle in engineering practice, for it covers all cases of tension, compression, torsion and bending. It also finds many important extensions in special cases of such structures as iron and steel bridges, particularly suspension bridges. For the considerations involved in this Art. show that in all cases of sudden applications of loads, actual energy will be stored and restored during different intervals of time and, consequently, that vibrations will be initiated.

Eq. (2) of Art. 13 furnishes a most convenient and elegant point of departure for investigations in such special cases, as will be exemplified in the next Art.

Art. 16.—Longitudinal Oscillations of a Straight Bar of Uniform

Section.

The complete solution of this problem will not be given, though it may be reached.

Let the bar be fixed at one end in a vertical position and let a heavy weight, W, act on the other. Also, let the axis of x be taken parallel to the axis of the bar, whose uniform normal section will be represented by A.

On account of the circumstances of application of the external forces and position of bar, the following equations of condition will exist :

cos x' = cos P' = cos x" — cos P" — du" — N₂ = N₃

= Tx — T₂ = T₃ = o = Y₀ = Z₀.

dv _ dw dt dt

du

hence they will be

will be very small compared with — ,

omitted. P' is the heavy weight attached to the free end of the bar divided by A ; consequently:

cos 7t' = I.

Eq. (2) of Art. 13, now reduces to :

jp'da' du

— Elx dlx dx dy dz -j- j X₀ du dx dy dz

(O

= jm i d    dx dy dz......

The integrals are to be extended throughout the whole of the bar. Since strains and stresses are uniform for any one cross section of the bar, and because X₀ = w — weight of a unit of volume of the bar (the force of gravity is the only external force which acts on the mass of the bar), Eq. 1 becomes :

Wdu' - AEly~dx + Awudx = ^ A(^f) dx + Cdx • (²)

This equation (C being a constant of integration) involves the complete problem of longitudinal oscillations. Two special cases, only, however, will be treated, in which the weight of the bar is so small compared with W that it may be neglected. This condition involves the omission of :

1H

Awu dx and — A

'du\²

At

dx,

in Eq. (2), and makes /, constant throughout the length of the bar.

Since the equation must be homogeneous, C will represent a quantity of actual energy ; in fact, a part of that quantity stored, at any instant, in W.

If xx represents the length of the bar, C may be put equal to :

W Iduf 2gxl \dt J

Also, because L is constant for the whole bar :

Introducing all these changes in Eq. (2) and integrating :

wu' - AE— = — (ęy

2X\    2g \dt J

If W is suddenly applied to the bar while in a state of equilibrium or rest, for which :

dii

u' = — = O,

dt

C will be zero, as the equation shows by such a substitution. For this case Eq. (3) becomes, after omitting the primes :

dt =

Wx,

dii

A Eg

\j EA

2 Wx.

u' — li •

Wx,

A Eg

ver sin~

j AEu

Wxt * *

• - (4)

The limits of the amplitude are discovered by putting :

du

^ (the velocity) = o,

in Eq. (3), remembering that C is also equal to zero. That operation will give :

u — o and u

_ 2 Wx1 ~ ~AE~'

Putting these values in Eq. (4), successively, and taking the difference of the results, the time occupied by one oscillation will be :

Wx,

A Eg

ver sin~l 2 — n

• • (5)

in which equation :

Wxt

U' ⁼ AË

is the strain in the bar caused by a gradual application of W.

In the second case to be treated the bar is first supposed to take a vertical position, with the weight attached to its free end, in a state of equilibrium. An external force then depresses the free end a distance u₀, measured from its position of equilibrium. If the force F is now removed, the weight will make excursions on each side of its position of rest.

Let ux represent the value of u! corresponding to the weight W alone, as in the previous case ; then let :

u' = ux + u,

u being measured from the position of equilibrium of the weight W.

Eq. (3) will then take the form :

W [ux + u) —

AE

2Xx

y+c- ®

When u — u₀ the body comes to rest. Hence :

W(ux + u₀) -

AE

2Xx

Oi + O’ = C. . . . (7)

Subtracting Eq. (7) from Eq. (6) :

W{u - u₀) -    [2ut (u - u₀) + u³ - uf = ^ (^) . (8)

since :

d (ut -f- u) — du.

Remembering that :

W (u — u₀)

_ AEux

0 - o,

Eq. (8) may take the form :

dt —

Wxx

du

A Eg V«₀² —    ’

' • (9)

u, . u — sin~¹ —

u₀

(io)

Eq. (9) shows that the amplitude of a vibration is found by putting :

u = + u₀ or — u₀.

Putting these values in Eq. (10) and taking the difference of the results, the time of a single oscillation is found to be :

T= it,

Uy

cr

(II)

Eq. (11) is seen to be identical with Eq. (5). In this case the amplitude is 2u₀, and the body oscillates through its position of rest. Both oscillations are completely isochronous for the same weight W.    *

If n is the observed number of oscillations per second, either Eq. (5) or (11) gives :

7iaWxt

= n²

7T² WXy

Ag

• • (12)

from which E may be computed, if W is very great compared with the weight of the bar or wire.

CHAPTER IV.

Theory of Flexure.

Art. 17.—General Formulae.

If a prismatic portion of material is either supported at both ends, or fixed at one or both ends, and subjected to the action of external forces whose directions are normal to, and ait, the axis of the prismatic piece, that piece is said to be subjected to “ flexure.” If these external forces have lines of action which are oblique to the axis of the piece, it is subjected to combined flexure and direct stress.

Again, if the piece of material is acted upon by a couple having the same axis with itself, it will be subjected to “ torsion.”

The most general case possible is that which combines these three, and some general equations relating to it will first be established.

The co-ordinate axis of X will be taken to coincide with the axis of the prism, and it will be assumed that all external forces act upon its ends only. Since no external forces act upon its lateral surface, there will be taken :

Tx = Na = N₃ = o;

retaining tne notation of Art. 6. These conditions are not strictly true for the general case, but the errors are, at most, excessively small for the cases of direct stress or flexure, or

for a combination of the two. By the use of Eqs. (12), (21) and (22) of Art. 5, the conditions just given become :

h-4

r!

I — 2r

(du dv dw\ \dx dy dz )

dw

+ u~° •

dv , dw

a+^=°

dy

(3)

Eqs. (1) and (2) then give :

(4)

dw

dz

In consequence of Eq. (4), Eqs. (1) and (2) give :

dv dw

du

dy

dz    T dx

(5)

By the aid of Eq. (5) and the use of Eqs. (11), (13) and (20) of Art. 5, in Eqs. (io), (11) and (12) of Art. 6 (in this case X₀ — Y₀ = Zg = o), there will result :

(6)

d²u . d²u , d²u

2 —t— + —;--—— — o

dx² dy² dz²

d²u , d²v

dx dy

+ =°

d²U

d²w

dx dz

dx²

(8)

The Eqs. (3), (5), (6), (7) and (8) are five equations of condition by which the strains u, v and w are to be determined. Let Eq. (6) be differentiated in respect to ^ :

dh( d³u

²2? ⁺

d³u

dy² dx

dz² dx

= o.

From this equation let there be subtracted the sum of the results obtained by differentiating Eq. (7) in respect to y and (8) in respect to z :

d³u d³v

²

dzw

dx² dy dx² dz

= o.

In this equation substitute the results obtained by differentiating Eq. (5) twice in respect to x, there will result :

'du

(9)

dhi dx³

This result, in the equation immediately preceding Eq. (9) by the aid of Eq. (5), will give :

d³v

= o.

dx² dy

After differentiating Eq. (7) in respect to y, and substituting the value immediately above :

d(-)

(ic)

d³u _ \dxj _ dy² dx    dy²

Eqs. (9) and (10) enable the second equation preceding Eq. (9), to give :

(ii)

d³u

dz² dx

<i)

dz²

Let the results obtained by differentiating Eq. (7) in respect to z and (8) in respect toy, be added :

d³v

dx² dz

d³u

2-

dx dy dz

d³w _ dx² dy °*

The sum of the second and third terms of the first member of this equation is zero, as is shown by twice differentiating Eq. (3) in respect to x. Hence :

d³u

dy dz dx dy dz

<g).

.    .    (12)

The Eqs. (9), (10), (11) and (12) are sufficient for the de-

dii

termination of the form of the function if it be assumed to be algebraic, for :

Eq. (9) shows that .ar² does not appear in it ; “    (10)    “    “ y² “    “

“    (u)    “

“    (12)    “

“ £2 “ yz

The products xz and xy may, however, be found in the function. Hence if av a₂, b, bz, and b₂ are constants, there may be written :

du

dx

= a + aTz + aj + x (b + bxz + b₂y) .    .    (13)

Eq. (5) then gives :

dv __ dw dy dz

- r \a + a,z + a₂y + x [b + bxz + b₂y)\ (14)

Substituting from Eq. (13) in Eqs. (7) and (8) :

(IS)

d²v

dx²

d²w

dx²

— — al — b^x

(16)

The method of treatment of the various partial derivatives in the search for Eqs. (13) and (14) is identical with that given by Clebsch in his “ Theorie der Elasticität Fester Körper.”

It is to be noticed that the preceding treatment has been entirely independent of the form of cross section or direction of external forces.

It is evident from Eqs. (13) and (14), that the constant a depends upon that component of the external force which acts parallel to the axis of the piece and produces tension or compression only. For, by Arts. 2 and 3, it is known that if a piece of material be subjected to direct stress only :

du

— a and

fril-

ra

the negative sign showing that ra is opposite in kind to a, both being constant.

Again, if z and y are each equal to zero, Eq. (13) shows

that :

du

—y = a + bx.

dx

Hence bx is a part of the rate of strain in the direction of x which is uniform over the whole of any normal section of the piece of material, and it varies directly with x. But such a portion of the rate of strain can only be produced by external force, acting parallel to the axis of X, and whose intensity varies directly as x. But, in the present case such a force does not exist. Hence b must equal zero.

The Eqs. (13), (14), (15) and (16), show that ax, bx and a₂, b2 are symmetrical, so to speak, in reference to the co-ordinates z and y, while Eqs. (13) and (14) show that the normal intensity Nj is dependent on those, and no other, constants in pure flexure, in which a — o. It follows, therefore, that those two pairs of constants belong to the two cases of flexure about the two axes of iT and Y.

No direct stress N₁ can exist in torsion, which is simply a twisting or turning about the axis of X.

Since the generality of the deductions will be in no manner affected, pure flexure about the axis of Y will be considered. For this case :

a — a₂ — b₂ = o = b.

Making these changes in (13) and (14) :

du

= a,z + b,xz

(17)

2 Gr a . s^du

Nx =

--6 + 2 G-r- .

— 2 r    dx

• (²°)

Also :

&II

du

— r — — — r (azz + bzxz) .

.    (i8)

Nz = 2G {r + i) (tfj + l\x)z — E(az + bzx)z

since :

2 Gir + i) = E.

Taking the first derivative of Nz :

dNt

dz = E X + I'X)

(21)

This important equation gives the law of variation of the intensity of stress acting parallel to the axis of a bent beam, in the case of pure flexure produced by forces exerted at its extremity. That equation proves, that in a given normal section of the beam, whatever may be the form of the section, the rate of variation of the normal intensity of stress is constant ; the rate being taken along the direction of the external forces.

It follows from this, that Nz must vary directly as the distance from some particular line in the normal section considered in which its value is zero. Since the external forces A are normal to the axis of the beam and direction of Nv and because it is necessary for equilibrium that the sum of all the forces Nx dy dz, for a given section, must be equal to zero, it follows that on one side of this line tension must exist and on the other, compression.

Let N represent the normal intensity of stress at the dis-

Q-4

Fig.l

tance unity from the line, b the variable width of the section parallel toy, and let a = b dz. The sum of all the tensile stress in the section will be :

r*'

z'

Nz A

The total compressive stress will be :

J~*i

The integrals are taken between the limits o and the greatest value of z in each direction, so as to extend over the entire section. In order that equilibrium may exist therefore :

IL^a + L/a}=°-

tz'

J~zl

(22)

Eq. (22) shows that the line of no stress must pass through the centre of gravity of the normal section.

This line of no stress is called thz neutral axis of the section. Regarding the whole beam, there will be a surface which will contain all the neutral axes of the different sections, and it is called the neutral surface of the bent beam. The neutral axis of any section, therefore, is the line of intersection of the plane of section and neutral surface.

Hereafter the axis of X will be so taken as to traverse the

centres of gravity of the different normal sections before flexure. The origin of co-ordinates will then be taken at the centre of gravity of the fixed end of the beam, as shown in Fig. i.

4- e, -

The value of the expression (a₁ + btx), in terms of the external bending moment, is yet to be determined. Consider any normal section of the beam located at the distance x from O, Fig. I, and let OA = l. Also let Fig. 2 represent the section considered, in which BC is the neutral axis and d' and dx the distances of the most remote fibres from BC. Let moments j of all the forces acting upon the portion {l—x) ¹ of the beam be taken about the neutral axis

Fig.2

BC. If, again, b is the variable width of the beam, the internal resisting moment will be :

rd-

J-dx

Aj bz dz —E (a₁ -f- bzx)

rd'

-dl

z². b dz.

But the integral expression in this equation is the moment of inertia of the normal section about the neutral axis, which will hereafter be represented by I. The moment of the external force, or forces, F, will be F(l — x) and it will be equal, but opposite in sign, to the internal resisting moment. Hence :

• • (23)

•••    - (*, + Ær*) =

A/ * *

• • (24)

Substituting this quantity in Eq. (16) :

(25)

d²w _ dx² El

It will hereafter be seen that Eq. (25) is one of the most important equations in the whole subject of the “ Resistance of Materials.”

An equation exactly similar to (25) may, of course, be written from Eq. (16) ; but in such an expression M will represent the external bending moment about an axis parallel to the axis of Z.

No attempt has hitherto been made to determine the complete values of u, v, and w, for the mathematical operations involved are very extended. If, however, a beam be considered whose width, parallel to the axis of Y, is indefinitely small u and w may be determined without difficulty. The conclusions reached in this manner will be applicable to any long rectangular beam without essential error.

If y is indefinitely small all terms involving it as a factor will disappear in u and w ; or, the expressions for the strains u and w will be functions of z and x ortly. But making u and w functions of z and x only is equivalent to a restriction of lateral strains to the direction of z only, or, to the reduction of the direct strains one half, since direct strains and lateral strains in two directions accompany each other in the unrestricted case. Now as the lateral strain in one direction is supposed to retain the same amount as before, while the direct strain is considered only half as great, the value of their ratio for the present case will be twice as great as that used in Arts. 3 and 4. Hence 2r must be written for r, in order that that letter may represent the ratio for the unrestricted case, and this will be done in the following equations.

Since w and u are independent of y :

dw _ du dy dy

dv

= 0, and T,= G₇

But by Eq. (14) :

- 2r (ax + bxx)zy + f(x, z).

By Eq. (3), since :

dw

dy °;

dv

dz

= — 2r (a, + bjx)y + ^/(^, z) = o.

This equation, however, involves a contradiction, for it makes f{x, z) equal to a function which involves^, which is impossible. Hence :

f{x, z) = o.

Consequently :

dv

dz

= — 2 r (az + bxx)y ;

which is indefinitely small compared with :

dv

dy

= — 2r{ax + l\x)z,

and is to be considered zero.

Because f(x, z) — o :

dv

~r == — 2tb,zy. dx

This quantity is indefinitely small ; hence :

Tz — — 2 Grbysy

is of the same magnitude.

Under the assumption made in reference to y, there may be written from Eqs. (17) and (18) :

x¹

u — ayez + bï z + f’(z)

(26)

— r^z* + b.xz¹) + fix) ....    (27)

Using Eq. (26) in connection with Eq. (6) :

dz*

By two integrations :

• • (28)

m = - b-f - *+<r . .

Using Eq. (27) in connection with Eq. (8) :

dx¹

— bxx — at.

By two integrations :

f(x) = - bt ~ - ~~ + CxX + Cu.

The functions u and w now become :

u = axxz -f- b₁ — z

b^z*

—— —c'z 4- c” .

3

.    . (29)

- ra,z² - rbtxz² - b/^ -    + c,x + cu .    (30)

The constants of integration c, c", etc., depend upon the values of u and w, and their derivatives, for certain reference values of the co-ordinates x and z, and, also, upon the manner of application of the external forces, F, at the end of the beam, Fig. I. The last condition is involved in the application of Eqs. (13), (14) and (15) of Art. 6.

In Fig. i let the beam be fixed at 0. There will then result for x = o and z — b :

du

dz

X — o

z =0

(u = o, and w — o)_o.

In virtue of the last condition :

c" — cn = o.

In consequence of the first :

ć — o.

After inserting these values in Eqs. (29) and (30) :

du

x²

7;= £

tb = a* + i.T-ts,

-j-2

^ = - rb^-bĄ - as + c,;

(Ê⁺È) = -G^' ⁺ ^ + G<' ■ ö'*)

The surface of the end of the beam, on which F is applied, is at the distance / from the origin 0 and parallel to the plane ZY. Also the force F has a direction parallel to the axis of Z. Using the notation of Eqs. (13), (14) and (15) of Art. 6, these conditions give :

cos p — I

cos q — o,

cos r — o,

cos 7t — o,

cos x = o,

cos p — i.

Since for x = l :

M — F {I — x) = o,

Eqs. (24) and (20) give N₁= o for all points of the end surface. Eq. (15) is, then, the only one of those equations which is available for the determination of cv That equation becomes simply :

Tu = P.

For a given value of z, therefore, any value maybe assumed for Tv For the upper and lower surfaces of the beam let the intensity of shear be zero ; or for z = ± d let Tt = o. Hence, by Eq. (31) :

Ci = bt (l + r)d² ;

Ta= Gb, < I + r) (d² — z²) ;

(32)

Eb

T, =    (d² — z²)

The constants a, and b, still remain to be found. The only-forces acting upon the portion (/— x) of the beam, are A'and the sum of all the shears T₂ which act in the section x. Let A y be the indefinitely small width of the beam, which, since z is finite, is thus really made constant. The principles of equilibrium require that :

f+<*    C+rf

T₂ . Ay . dz — Gbz (1 -J- r) (d². Ay. dz — z². Ay. dz)>— F. J — d    J — d

The first part of the integral will be 2 Ayd³ and the second part will be the moment of inertia of the cross section (made rectangular by taking Ay constant) about the neutral axis. Hence :

2Gb, (i + r)I = F\ or b, —

26(1 +r)I El

(33)

(34)

If x = o in Eq. (24) :

a,~ -

FI

El *    *

• • (35)

Thus the two conditions of equilibrium are involved in the determination of a, and b,. The complete values of the strains u and w are, finally :

;r²

u — -=- i z —

El \ 2

• • (36)

W — -=r-

EI

Irz² — rxz² —

X* lx²\

■6 + Tj

Fd²x

• (37)

These results are strictly true for rectangular beams of indefinitely small width, but they may be applied to any rectangular beam fixed at one end and loaded at the other, with sufficient accuracy for the ordinary purposes of the civil engineer. It is to be remembered that the load at the end is supposed to be applied according to the law given by Eq. (34) ; a condition which is never realized. Hence these formulae are better applicable to long than short beams.

The greatest value of TM in Eq. (34), is found at the neutral axis by making 2 = 0; for which it becomes :

to

to

II

8.1^

•    •    (38)

is the mean intensity of shear in the cross section ; hence,

the greatest intensity of shear is once and a half as great as the mean.

In Eq. (36) if z — o, u — O. Hence no point of the neutral surface suffers longitudinal displacement.

In Eq. (37) the last term of the second member is that part of the vertical deflection due to the shear at the neutral surface, as is shown by Eq. (38). The first term of the second member, being independent of x, is that part of the deflection which arises wholly from the deformation of the normal cross section.

The usual modification of the preceding treatment, designed to supply formulae for the ordinary experience of the engineer, ’ will be given in the succeeding Arts.

Art. 18.—The Common Theory of Flexure.

The “ common theory ” of flexure is completely expressed by Eq. (25) of Art. 17. That equation involves the condition that no external force acts upon the exterior surface of the bar or beam. In reality this condition is never fulfilled. External loads are applied in any manner whatever, causing normal compressive stresses to exist at any or all points of the exterior surface. It is assumed in the common theory of flexure that the equation :

(O

d²w _ M di* ~ m

holds true, for pure bending; whatever may be the number or manner of application of the external forces or loads.

By “ pure bending ” is meant the action of external forces whose directions are normal to, and cut, the axis of the beam.

As has already been seen in Art. 17, w, strictly speaking, is a function of x, y and z.

It is further assumed in the common theory of flexure that w is a function of x only.

This is equivalent to an assumption that the lateral dimensions of the piece are so small that they can have no influence on the value of w, and consequently that they will not appear in it. In other words the common theory of flexure is the theory of the flexure of pieces, one or two of whose cross dimensions are indefinitely small in comparison with their length. The neglect of this fact has led to some erroneous applications and deductions in connection with long column formulae.

Eq. (1), taken in connection with these two important assumptions, constitutes the “ Common Theory of Flexure,” which is always used in engineering practice.

Since the intensity of external loading is almost invariably very small compared with the internal stress Nx, the first of the above assumptions involves very little error in all ordinary cases.

The second assumption, as was stated above, is equivalent to taking the bar or beam so small that the strain or “ deflection ” w is essentially the same at all points of a given cross section. With such small strains and large ratios of length to lateral dimensions as almost always occur, this assumption, also, involves no considerable error.

It is well known that if the curvature is very small, the reciprocal of the radius of curvature, in the plane zxy is repre-

Hence Eq. (1) may

sented with no essential error by take the form :

d²w dx³ '

El

(2)

in which p is the radius of curvature.

Let M' and Mt represent two bending moments which will produce the two radii of curvature p! and pv Eq. (2) will then give the following :

El

=-r- = M' . .

P'

• • (3)

(4)

• • (5)

El

Pi

Hence *.

El(- - 1) 3=

\Px P /

The second member shows that a bending moment :

— M' ~ M,

applied to a curved beam whose radius of curvature at any section is p', will produce a change of curvature expressed by :

.Pi P

In other words : the common theory of flexure is applicable to curved beams of slight curvature.

In such a case Eq. (2), expresses the variation (increase

or decrease) of curvature caused by the moment M. It is to be distinctly borne in mind, however, that Eq. (2) itself is made approximately true only by considering the curvature very small.

The limits within which the common theory is applicable to

curved beams, and the degree of approximation of the application, will be shown by the following investigations, in which the longitudinal compression and extension, due to the external forces, will be neglected.

In the figure let a portion of any curved beam, whose lateral dimensions are small

ft -—

"a

vc

Fig-1

compared with its length, be represented. Let AB represent an indefinitely short length, ds, of the neutral surface. C is the centre of curvature of ds before flexure, and C the same point after flexure. Since the

lateral dimensions are small compared with the length, if the strains are not great, any normal cross section may, without essential error, be taken as plane after flexure, and such planes passing through A and B will then contain the radii of curvature at the points A and B. Let :

AC' — p’ and AC = p

also :

Aa — Ab — Be — Bd — unity.

Aa and Bd are the positions taken by Ab and Be after flexure. The angle, before flexure, between two radii AC and BC, in-

ds

definitely near to each other, is — ; after flexure, as the figure

shows, the same angle becomes

ds

Hence the change in

curvature (or change of angle between consecutive radii) caused by flexure is :

P’ ’

HH

*>1

In the figure, the amount of strain for

Now let the amount of shortening or lengthening of a unit of length of fibres, parallel to the neutral surface and situated at unit’s distance from it, be represented by u\ concisely stated, u is the rate of strain for any point at unit’s distance from the neutral surface.

AB — ds is :

ab -|- cd — u ds.

But the difference between the angles aC'd and bCc is :

(ab + cd) -f- Ab = ab -f- cd — u ds.

But this difference is the change of curvature ; hence :

U = —r

¹ i_ ¹

P' P

(6)

This relation is purely kinematical ; a value for u must next be determined in terms of the bending moment M.

Under the circumstances of the case it has been seen that the longitudinal strains parallel to the neutral surface vary essentially directly as their distances from it (this law is the assumption that plane normal sections before flexure are also plane afterwards). The strain at any distance z from the neutral surface will then be uz. But it was shown in Art. 17 that the intensity of longitudinal stress Nt varies directly as z ; hence there may be written :

A⁷; = Euz.

If b is the variable width of cross section, taken parallel to the neutral surface, the internal resisting moment of the section will be :

M = Nx b dz . z =

Euj

bz² dz.

(7)

El

(8)

The integration is to be extended over the whole section. Then, if the “ neutral axis ” is the line of intersection of the neutral surface with the normal section, I is the moment of inertia of the normal section about the neutral axis.

Eqs. (6) and (8) then give :

El

P' P

(9)

This equation is true, under the assumption made, for any degree of curvature whatever in the original beam.

If w and ;r are rectangular co-ordinates in the plane of the beam, x being the independent variable, the expressions for the reciprocals of the radius of curvature before and after flexure, are :

_3

.    . (10)

d²w₁ /    dw²\ 2

~ dx² V    dx²)

P'

d²w’ dx²

dw'²

dx²

-A

• • (ii)

By the binomial formula :

. dw²

1 + nï

-3.

ts)

Ml"

M I M

2-

and an exactly similar expression for —

After introducing

these in Eqs. (io) and (n), and supposing the deflections to be small, there may be written :

d²w

dx²

d²wI

dx²

dw² dw'²\ d²w' dx²    dx² ) dx²

15 /dw,⁴    dw'*\ d²w₁

8 \ dxA    dx⁴ / dx² 6 C*

l -, )

P'” ‘J::.;

If, in addition to small deflections, the values of :

dw' dx’

and

dw₁

dx

are not great, the equation just written shows that with a considerable degree of approximation :

d²w'

dx²

d²w1 dx²

• • (12)

The smaller the curvature the more nearly accurate is Eq. (12). If, as before, w is the deflection or strain normal to x :

w = w’ — w\ .*. d²w — d²w' — d²w₁ ;

hence, from Eqs. (9) and (12) :

(*3)

M _ d²w EI~

Eq. (13) is exactly the same as Eq. (1) for straight beams.

These investigations show that the common theory of flexure is not strictly applicable to the general case of curved beams. In order to obtain Eq. (12) it was necessary to assume the same law for stresses and strains, in any normal section, both for curved and straight beams, which is not exactly true» It was also necessary to assume small values of

dzv j

dx

and

dw'

dx

for a close approximation. Yet the application of the common theory of flexure to curved beams, even within these restricted limits, is of the highest importance.    r

In Art. 22 a generalization of the common theory of flexure is given, in which the differential of the centre line of the beam is used instead of dx. The resulting formulae are accurately applicable to curved beams of any curvature. The only assumption involved, in addition to those of the common theory, is the identity of the law of variation of stresses and strains in curved and straight beams ; and that causes very little error.

One of the most important forms of Eq. (7) yet remains to be established.

Let d₁ represent the distance from the neutral axis of any normal section of the beam to that point of the section farthest from it. Let K represent the intensity of tensile or compressive stress (as the case may be) existing at this same point ; K will be the greatest intensity in the section. Because the intensity of stress varies directly as the distance from the neutral axis, the intensity at distance unity from that axis will be :

But by Art. 2, this intensity also has the value Eu. Consequently Eq. (7) becomes :

KI

M — -E-

d*

(14)

If the external moment is sufficient to break the beam, and if Eq. (14) is applied to the section at which failure begins, K is called the “Modulus of Rupture” for flexure. It is an empirical quantity.

Art. 19.—Deflection by the Common Theory of Flexure.

*

The common theory of flexure, as developed in the pre, ceding Art., leads to very simple and, in nearly all ordinary cases, very closely approximate formulae.

Let x₀ be the co-ordinate of some point at which the tangent of the inclination of the neutral surface to the axis of ^ is known ; then, from Eq. (1) of Art. 18 :

dw

dx

. Et dx

xo

(0

dw

—— will be at once recognized as the general value of the tangent of the inclination just mentioned, or, in the case of curved beams, as approximately the difference between the tangent, before and after flexure.

Again, let x₁ represent the co-ordinate of a point at which the deflection w is known, then, from Eq. (1) :

* rx M

----- dx²

*1 Jxo

El

(2)

The points of greatest or least deflection and greatest or least inclination of neutral surface are easily found by the aid of Eqs. (1) and (2).

The point of greatest or least deflection is evidently found by putting :

dw

dx

(3)

dw

and solving for x. Since is the value of the tangent of the

inclination of the neutral surface, it follows that a point of greatest or least deflection is found where the beam is horizontal.

Again, the point at which the inclination will be greatest or least is found by the equation :

dm.

dx

d²w

dx²

(4)

is the reciprocal of the radius of

But, approximately,

d²w dx²

curvature ; hence, the greatest inclination will be found at that point at which the radius of curvature becomes infinitely great, or, at that point at which the curvature changes from positive to negative or vice versa. These points are called points of “ contra-flexure.” Since :

__    „T d²iv

M=EIdE'

there is no bending at a point of contra-flexure.

The moment of the external forces, M, will always "be expressed in terms of x. After the insertion of such values, Eqs.

(1) and (2) may at once be integrated and (3) and (4) solved.

The coefficient of elasticity, E, is always considered a constant quantity ; hence it may always be taken outside the integral signs. In all ordinary cases, also, / is constant throughout the entire beam. In such cases, then, there will only need to be integrated the expressions :

fX

M dx and

Jx0

r r

Jx₀

M dx².

Before applying these formulae to particular cases it will be necessary to consider some other matters.

Art. 20.—External Bending Moments and Shears in General.

Beams subjected to combined bending and direct stress will not be treated. Such cases are of little or no real value to the engineer, and approximate solutions, even, are only to be reached by the higher processes of analysis. In all beams, therefore, pure bending only is to be treated. A beam is said to be non-continuous if its extremities simply rest at each end of the span and suffer no constraint whatever.

A beam is said to be continuous if its length is equal to two or more spans, or if its ends, in case of one span (or more) suffer constraint.

A cantilever is a beam which overhangs its span ; one end of which is in no manner supported. Each of the overhanging portions of an open swing bridge is a cantilever truss.

Let any beam be horizontal, and suppose it to be subjected to vertical loads. The results will evidently be applicable to any beam acted upon by loads normal to its axis. Let P be any single vertical load, and let x be any horizontal co-ordinate measured from any point as an origin. Let xx represent the co-ordinate, measured from the same origin, of the point of application of any load P. Finally, let it be required to determine the external bending moment M at any section, x, of the beam. The lever arm of any load P is evidently (x — xz).

Hence, for any number of forces :

00

M = 2P(x - xt)

The summation sign 2 refers only to xx and is to cover that portion of the beam on one side of the section xy as is evident from the manner of forming the equation.

If the origin of x is in the section considered :

(²)

M = - 2Pxz

From Eq. (1) :

dJI

dx

- = 2P= S

(3)

Now 2P = S is the algebraic sum of all the forces on one side of the section considered, it is consequently the total force acting in the section tending to move one portion of the beam past the other; it is therefore called the “ shear ” in the section. This quantity (the shear) is a most important one in the subject of the resistance of materials.

The reactions, or supporting forces, applied to the beam, are to be included both in the sum 2P, and in the moment :

2P (x — Xj).

Eq. (3) shows that the shear at any section is equal to the first differential coefficient of the bending moment considered as a function of x.

The sum of all the loads on the other side of the section x would give the same numerical shear, but it would evidently have an opposite direction.

As is well known, the analytical condition for a maximum or minimum bending moment in a beam is :

dM

dx

• (4)

From Eqs. (3) and (4) is to be deduced the following im-portant principle : The greatest or least bending moment in any beam is to be found in that section for which the shear is zero.

The importance of this principle lies in the fact that in the greater portion of cases of loaded beams which come within the experience of the civil engineer, the section subjected to the greatest bending moment can thus be determined by a simple inspection of the loading.

These principles can be well illustrated by the following simple example.

Fig. i represents a non-continuous beam with the span /,

u\

4-

iTTHfr

—_j

Fig. i.

supporting two equal weights P, P. These two weights or loads are to be kept at a constant distance apart denoted by a.

It is required to find that position of the two loads which will cause the greatest bending moment to exist in the beam, and the value of that moment. The reaction R is to be found by the simple principle of the lever. Its value will therefore be :

(*+1)

.2 P    . .

• - (5)

Since the reaction R can never be equal to 2jP, 2P, or the shear, it must be equal to zero at the point of application of one of the loads P. In searching for the greatest moment, then, it will only be necessary to find the moment about the point of application of one of the forces P. It will be most convenient to take that one nearest R.

The moment desired will be :

M — Rx

ax

2/

• . (6)

dM

dx

a\

2P ( T ⁻ 2?)

This value in Eq. (6) gives :

¹

by — —

•    (7)

Since :

d²M dx²

it appears that M₁ is a maximum.

If the load is uniformly continuous and of the intensity/,

in Eqs. (i), (2) and (3)pdx\ is to be put fori⁵, and the sign j for

2. Hence :

pj(x - xt) dxv — pjxi dxt.

But since dx and dxx are perfectly arbitrary, they may be taken equal to each other, hence :

d²M dx²

Or, the second differential coefficient of the moment, considered as a function of x, is equal to the intensity of the continuous load.

A very important problem arises in connection with the principles discussed in this Art. It is the following :

A continuous train of any given varying or uniform density advances along a simple beam of span l. It is required to detcr-mine what position of loading will give the greatest shear at any specified section.

In Fig. 2, CD is the span /, and A is any section for which

Fig.2

dM

dx

p\

dx\.

it is required to find the Tp position of the load for ^ the greatest transverse shear. The load is supposed to advance continuously from C to any point Bt Let R be the reaction at D, and 2P the load between A and B. The shear S' at A will be :

(8)

Let R' be that part of R which is due to ISP, and R" that part due to the load on CA ; evidently R' is less than 2P. Then :

R' + R" - 2P= S'.

If AB carries no load, R' and 2P disappear in the value of 6“. Hence : is the shear for the head of the train at A. S is greater than S' because 2P is greater than R. But no load can be taken from AC without decreasing R". Hence: The greatest shear at any section will exist when the load extends from the end of the span to that section, whatever be the density of the load.

In general, the section will divide the span into two unequal segments. The load also may approach from either direction. The greater or smaller segment, then, may be covered, and, according to the principle just established, either one of these conditions will give a maximum shear. A consideration of these conditions of loading in connection with Fig. 2, however, will show that these greatest shears will act in opposite directions.

When the load covers the greater segment the shear is called a main shear ; when it covers the smaller, it is called a counter shear.

Addendum to Art. 20.

The position of the moving load for the greatest bending moment at any section of a non-continuous beam may be very simply determined. In Fig. 3, let F G represent any such beam of the span /, and let any moving load whatever, as W₁ . . . W„> . . . Wn advance from F toward G. Let C be the section at which it is desired to determine the

,R

i-

£l

W,

JX

4-

k-z—4

d>

IF

maximum bending moment, and let 7Ï loads rest to the left of C, while 7i is the total number of loads on the span. Finally-let x' represent the distance of Wn> from C and to the left of that point, while x is the distance of Wn to the left of F. If a is the distance between Wx and W« ; b the distance between IV₃ and W₃ ; c the distance between Ws and etc., etc., the reaction R at G will be :

Wx

• • (9)

- +w4

The bending moment M about C will then take the value :

Wx (a + b + c +

+

+ x')

+ *')

I + W„> x'.

Or, after inserting the value of R from above :

l-j [IV, a + (W, + W,) b + (TV, + W, + W.)e +

+ W„)x]~ W₁a-(JV₁+ W₂) b .... -(JVt + Wt+Wt+......

(IV, + IV, + W, + ....

(io)

-(TV, + IV, + W,)c-+ WJ) x' ...    .

If the moving load advances by the amount ax, the moment becomes, since ax = ax' :

Hence, for a maximum, the following value must never be negative :

I-Ç (Wi + w₂ + w:; +

+ W„) - (Wx

M'=M+ ^ (Wx + W₂ + Wa +

+ Wn') A X

₊ W„)ax-(W₁ + W%

(II)

M'-M= ax

+ W₂ +

nf

(12)

Or, the desired condition for a maximum takes the form :

wx+w*+w% +

• • (13)

wx + w₂ +

+ W„>

+ Wn

It will seldom or never occur that this ratio will exactly exist if Wn' is supposed to be a whole weight ; hence W„> will usually be that part of a whole weight at C which is necessary to be taken in order that the equality (13) may hold.

It is to be observed that if the moving load is very irregular, so that there is a great and arbitrary diversity among the weights W, there may be a number of positions of the moving load which will fulfil Eq. (13), some one of which will give a value greater than any other; this is the absolute maximum desired.

From what has preceded, it follows that Wm may always be taken at the point C in question; hence, x' in Eq. (10) may always be taken equal to zero when that equation expresses the greatest value of the moment. The latter then becomes:

y [Wxa + (Wt + Wt)b +

+ Wn) x ] — Wxa- (Wx 4- W₃)b -+ Wn'-1) (?).....

+ (Wx + Wa+ ......

- {Wx + Wt

(14)

In this equation x, of course, corresponds to the position of

maximum bending, while the sign (?) represents the distance between the concentrations W1 and W„,.

+ W„) between G and C ; the

It has already been shown in this Art. that for any given condition of loading the greatest bending moment in the beam will occur at that section for which the shear is zero. But if the shear is zero, the reaction R must be equal to the sum of the weights (W₁ + Wa + latter now being the section at which the greatest moment in the span exists.

Hence, for that section, Eq. (13) will take the form :

+ W„ *

l - w,+ w₂+ w₃ +

or, the centre of the gravity of the load is at the same distance from one end of the beam as the section or point of greatest bending is from the other. In other words, the distance between the point of greatest bending for any given system of loading and the centre of gravity of the latter is bisected by the centre of span.

If the load is uniform, therefore, it must cover the whole

span.

It is to be observed that Eq. (14) is composed of the sums Wu Wx -I- W₂, etc., multiplied by the distances a, b, c, etc. Hence tabulations of these quantities for any given system of loading will expedite and simplify computations of actual moments.

With a given system of concentrated loads it sometimes becomes necessary to determine at what particular length of span n weights W cease to give the maximum bending moment, and (n + 1) weights begin to be employed, for a special

/'

and constant fraction, —, of the span. Eq. (14) gives the solu

tion of this question at once. Let M be the moment for n weights and distance x, while hf is the moment for (n + 1) weights corresponding to the distance xx. Also let k be the

distance between the weights or loads IV„ and W, there results :

« +1*

Then

= Ç-[(k-x)(W₁ + wa + .....

-(W* + wa +

+ W„) + (Wt

+ Wa +......+ Wu+l)xt

+ WJ) ?!

The last term of this equation will not exist if, as is frequently the case, the maximum moment continues at the same load Wn>. Hence either :

xx — (x — k)

Wx + W» +

+ wn

wx + w₂ +

+ wH

. . (15)

Or,

_ l (Wx+Wa+.. + Wn,)?x

Wx +1^2+... + Wn+1 V W\ +    + .. + Wn +1

Since xx cannot be negative, Eq. (15) shows that x — k for the condition to which it belongs. Eq. (16) gives xx, when n loads cease to be used and (n + 1) begin, if the point of maximum bending at the same time changes from Wn>-1 to Wm.

xx —(x—k)

Wx+Wa+... + W»'

.(16)

Art. 21.—Moments and Shears in Special Cases.

Certain special cases of beams are of such common occurrence, and consequently of such importance, that a somewhat more detailed treatment than that already given may be deemed desirable. The following cases are of this character.

Case I.

Let a non-continuous beam, supporting a single weight P

at any point, be considered, and let such a beam be represented in Fig. I. If the r' span RR is represented by

Fig.1

I = a + b = RP+ R'P,

the reactions R and R' will be :

cd

CL

Consequently, if x represents the distance of any section in RP ixom. R, while x represents the distance of any section of R'P from R', the general values of thebending moments for the two segments a and b of the beam will be :

• • (²)

M — Rx, and M' — R'x . .

These two moments become equal to each other and represent the greatest bending moment in the beam when

x — a and x' — b,

or, when the section is taken at the point of application of the load P.

Eq. (2) shows that the moments vary directly as the distances from the ends of the beam. Hence, if AP (normal to RR!) is taken by any convenient scale to represent the greatest

ah

moment, P, and if RAR is drawn, any intercept parallel to

AP and lying between RAR and RR will represent the bending moment for the section at its foot, by the same scale. In this manner CD is the bending moment at D.

The shear is uniform for each single segment ; it is evidently equal to R for RP and R for RP. It becomes zero at P, where is found the greatest bending moment.

Case II.

Again, let Fig. 2 represent the same beam shown in Fig. 1, but let the load be one of uniform intensity, /, extending from end to end of the beam. Let C be placed at the centre of the span, and let R and R, as before, represent the two reactions. Since the load is symmetrical in reference to C,

R = R.

Fig.2

For the same reason the moments and shears in one half of the beam will be exactly like those in the other ; consequently, reference will be made to one half of the beam only. Let x and xx then be measured from R toward C. The forces acting upon the beam are R and p, the latter being uniformly continuous. Applying the formulae of the preceding Art., the bending moment at any section ;tr will be :

p [ (x — Xt) dxx .

" O

(3)

/. M — Rx — P—

If / is the span, at C, M becomes :

m' = T

Rl    pi²

(4)

But because the load is uniform :

Hence :

(5)

if W⁷ is put for the total load. Placing : in Eq. (3) :

• » (6)

M — —^{lx — x³)

The moments M, therefore, are proportional to the abscissae of a parabola whose vertex is over C, and which passes through the origin of co-ordinates R. Let A C, then, normal to RR', be taken equal to Mif and let the parabola RÂR be drawn. Intercepts, as FH, parallel to AC, will represent bending moments in the sections, as H, at their feet.

The shear at any section is :

dM

S=-j- = R-px =

P^2-X) '

■ ■ (7)

or, it is equal to the load covering that portion of the beam between the section in question and the centre.

Eq. (7) shows that the shear at the centre is zero ; it also shows that 5 = R at the ends of the beam. It further demonstrates that the shear varies directly as the distance from the centre. Hence, take RB to represent R and draw BC. The shear at any section, as H, will then be represented by the vertical intercept, as HG, included between BC and RC.

The shear being zero at the centre, the greatest bending moment will also be found at that point. This is also evident from inspection of the loading.

Eq. (2) of Case /., shows that if a beam of span / carries a W

weight — at its centre, the moment M at the same point will be :

Sloo

• • (8)

The third member of Eq. (8) is identical with the third member of Eq. (5). It is shown, therefore, that a load, concentrated at the centre of a non-continuous beam, will cause the same moment, at that centre, as double the same load uniformly distributed over the span.

Eqs. (5) and (8) are much used in connection with the bending of ordinary non-continuous beams, whether solid or flanged : and such beams are frequently found.

Case III.

The third case to be taken, is a cantilever uniformly loaded ;

Fig.3

it is shown in Fig. 3. Let x and xl be measured from the free end A, and let the uniform intensity of the load be A represented by p. The entire loading ²³ is uniformly continuous. Hence the principles and formulae of Art. 20 give, for the moment about any section :

= - P [ (x— XI ) dxt = -

J o

px²

. . (9)

If AB = /, the moment at B is :

Mt= -

pi'

(10)

The negative sign is used to indicate that the lower side of the beam is subjected to compression. In the two preceding cases, evidently, the upper side is in compression.

The shear at any section is :

dM

dx

= - Px

(ii)

Hence, the shear at any section is the load between the free end and that section.

Eq. (9) shows that the moments vary as the square of the distance from the free end ; consequently, the moment curve is a parabola with the vertex at A, and with a vertical axis. Let BC, then, represent by any convenient scale, and draw

the parabola CD A. Any vertical intercept as Z>Awill represent the moment at the section, as F, at its foot.

Again, let BG represent the shear//, at B, then draw the straight line AG. Any vertical intercept, as HF, will then represent the shear at the corresponding section F.

Art. 22.—Recapitulation of the General Formulae of the Common Theory of Flexure.

It is convenient for many purposes to arrange the formulae of the Common Theory of Flexure in the most general and concise form. In this Art. the preceding general formulae for shears, strains, resisting moments and deflections will be recapitulated and so arranged. In order to complete the generalization, the summation sign 2 will be used instead of the sign of integration.

,B

Fig.l

---X----> pT-jp--

:

In Fig. I, let ABC represent the centre line of any bent beam ; AF, a vertical line through A ; CF, a horizontal line through C, while A is the section of the beam at which the deflection (vertical or horizontal) in reference to C, the bending moment, the shearing stress, etc., are to be determined. As shown in figure, let x be the horizontal co-ordinate measured from A, andj the vertical one measured from the same point ; then let z be the horizontal distance from the same point to the point of application of any external vertical force P. To complete the notation, let D be the deflection desired ; Mv the moment of the external forces about A ; S, the shear at A ; P\ the strain (extension or compression) per unit of length of a fibre parallel to the neutral surface and situated at a normal distance of unity from it ; I, the general expression of the moment of inertia of a normal cross section of the beam, taken in reference to the neutral axis of that section ; E, the coefficient of elasticity for the material of the beam ; and M the moment of the external forces for any section, as B.

Again, let A be an indefinitely small portion of any normal cross section of the beam, and let ÿ be an ordinate normal to the neutral axis of the same section. By the “ common theory ” of flexure, the intensity of stress at the distance ÿ from the neutral surface is (y'P'E). Consequently the stress developed in the portion A, of the section, is EP'y'A, and the resisting moment of that stress is EP'y'²A.

The resisting moment of the whole section will therefore be found by taking the sum of all such moments for its whole area.

Hence :

M = EP'2y'*A = EP'I.

Hence, also .

El’

If n represents an indefinitely short portion of the neutral surface, the strain for such a length of fibre at unit’s distance from that surface will be nP’.

If the beam were originally straight and horizontal, n would be equal to dx.

P' being supposed small, the effect of the strain nP¹ at any

section, B, is to cause the end K of the tangent BK, to move vertically through the distance nP'x.

If BK and BR (taken equal) are the positions of the tangents before and after flexure, nP'x will be the vertical distance between K and R.

By precisely the same kinematical principle, the expression iiP'y will be the horizontal movement of A in reference to B.

Let 2nP'x and 2nP'y represent summations extending from A to C, then will those expressions be the vertical and horizontal deflections, respectively, of A in reference to C. It is evident that these operations are perfectly general, and that x and y may be taken in any direction whatever.

The following general, but strictly approximate equations, relating to the subject of flexure, may now be written :

(O

(²)

= 2Pz

P' =

- El

(3)

2nP' = 2n 4L

El

(4)

D = 2 nP'x = 2

nMx

El

• • (5)

Dh — 2nP'y = 2

nMy

El

(6)

Dk represents horizontal deflection.

The summation ~£Pz must extend from A to a point of no bending ; or from A to a point at which the bending moment is M-l. In the latter case :

(7)

Mz' may be positive or negative.

Art. 23.—The Theorem of Three Moments.

The object of this theorem is the determination of the relation existing between the bending moments which are found in any continuous beam at any three adjacent points of support. In the most general case to which the theorem applies the section of the beam is supposed to be variable, the points of support are not supposed to be in the same level, and at any point, or all points, of support there may be constraint applied to the beam external to the load which it is to carry ; or, what is equivalent to the last condition, the beam may not be straight at any point of support before flexure takes place.

Before establishing the theorem itself, some preliminary matters must receive attention.

If a beam is simply supported at each end, the reactions are found by dividing the applied loads according to the simple principle of the lever. If, however, either or both ends are not simply supported, the reaction, in general, is greater at one end and less at the other than would be found by the law of the lever ; a portion of the reaction at one end is, as it were, transferred to the other. The transference can only be accomplished by the application of a couple to the beam, the forces of the couple being applied at the two adjacent points of support ; the span, consequently, will be the lever arm of the couple. The existence of equilibrium requires the application to the beam of an equal and opposite couple. It is only-necessary, however, to consider, in connection with the span AB, the one shown in Fig. 1. Further, from what has immediately preceded, it appears that the force of this couple is

yR

NR'

Fig=1

equal to the difference between the actual reaction at either point of support and that found by the law of the lever. The bending caused by this couple will evidently be of an opposite kind to that existing in a beam simply supported at each end.

These results are represented graphically in Fig. 1. A and B are points of support, and AB is the beam ; AR and BR are the reactions according to the law of the lever ; RF = R'F is the force of the applied couple ; consequently :

AF = AR + RF and BF = BR - (R'F = RF)

are the reactions after the couple is applied. As is well known, lines parallel to CK, drawn in the triangle ACB, represent the bending moments at the various sections of the beam, when the reactions are AR and BR’. Finally, vertical lines parallel to AG, in the triangle QHG, will represent the bending moments caused by the force RF.

In the general case there may also be applied to the beam two equal and opposite couples, having axes passing through A and B respectively. The effect of such couples will be nothing so far as the reactions are concerned, but they will cause uniform bending between A and B. This uniform or constant moment may be represented by vertical lines drawn parallel to AH or LN (equal to each other) between the lines AB and HQ. The resultant moments to which the various sections of the beam are subjected will then be represented by the algebraic sum of the three vertical ordinates included between the lines A CB and GQ. Let that resultant be called M.

Let the moment GA be called Ma, and the moment :

BQ = LN — HA = My

Also designate the moment caused by the load P, shown by lines parallel to CK in A CB, by Mx. Then let x be any horizontal distance measured from A toward B ; l the horizontal distance AB ; and z the distance of the point of application, K, of the force P from A. With this notation there can be at once written :

Eq. (i) is simply the general form of Eq. (2).

It is to be noticed that Fig. 1 does not show all the moments Ma, Mb and Mx to be of the same sign, but, for convenience, they are so written in Eq. (1).

The formula which represents the theorem of three moments can now be written without difficulty. The method to be followed involves the improvements added by Prof. H. T. Eddy, and is the same as that given by him in the “American Journal of Mathematics,” Vol. I., No. 1.

Fig. 2 shows a portion of a continuous beam, including two spans and three points of support. The deflections will be supposed measured from the horizontal line NQ. The spans

'Ma

11s-

■cc

•Ja

*•

~v

-*ï

IC.

7\

? s*

crq

tO-33-.

:cn

are represented by la and lc ; the vertical distances of NQ from the points of support by ca, cb and cc ; the moments at the same points by Ma, Mb and Mc, while the letters .S' and R represent shears and reactions respectively.

In order to make the case general, it will be supposed that the beam is curved in a vertical plane, and has an elbow at b, before flexure, and that, at that point of support, the tangent of its inclination to a horizontal line, toward the span la is t, while t' represents the tangent on the other side of the same point of support ; also let d and d' be the vertical distances, before bending takes place, of the points a and c, respectively, below the tangents at the point b.

A portion of the difference between ca and cb is due to the original inclination, whose tangent is t, and the original lack of straightness, and is not caused by the bending ; that portion which is due to the bending, however, is, remembering Eq. (5), Art. 22 :

D — ca - cb - Ij — d - 2b

a Mxn

El *

. (2)

+ Mb

In this equation, it is to be remembered, both and z (involved in Mj) are measured from support a toward support b. Now let a similar equation be written for the span l„ in which the variables x and z will be measured from c toward b. There will then result :

E(cc -cb- // - d')

xn

By the aid of Eq. (1) this equation may be written :

E («ca — cb — lat — d)

• • (3)

When the general sign of summation is displaced by the integral sign, n becomes the differential of the axis of the beam, or ds. But ds may be represented by u dx, u being such a function of as becomes unity if the axis of the beam is originally straight and parallel to the axis of x. The Eqs. (2) and (3) may then be reduced to simpler forms by the following methods :

In Eq. (2) put :

Kl^-)

xn

_ I fa U (4 - x)x dx

_ xa[au (4 — x) dx J b

Also :

xjL^u{lg -x)dx ₌ Ÿf*u(/a _x)dx .    .

lg

• (5)

Also :

IgXg

[ «(4

J b

_ tgXgU-a

lg

[ (4 — x) dx — J i

IgX gU J a    çq

In the same manner :

a X ²1l

*bT7

= ir

üb

ux² dx

‘hV

a ux dx

• « (7)

Also :

a ux dx

igXg [a

L J*

ux dx .    .

■ ■ (8)

And,

tgXgj*

lg

ux dx —

i'gX'gUg j“

lg

x dx

_ IgX gM-gig

.    .    (9)

Again, in the same manner :

tlaula2Mtx Ax .    . . . (10)

« Mtxn

*b~~r~ -

Using Eqs. (4) to (10), Eq. (2) may be written :

£ {cc - cb - lat - d) =    (MauJaxa + MbujX)

Proceeding in precisely the same manner with the span lc, Eq. (3) becomes :

E (cc — cb — 1/ — d') — ~ (M'îtj'X, + Mbu/cX'c)

.    .    (12)

+ Uxchc^b Mxx Ax .    .

The quantities xa and xc are to be determined by applying Eq. (4) to the span indicated by the subscript ; while ua, ia, uc and ic are to be determined by using Eqs. (5) and (6) in the same way. Similar observations apply to u'a, ia, x’a, u'c, i'c and x'c, taken in connection with Eqs. (7), (8) and (9).

If / is not a continuous function of x, the various integrations of Eqs. (4), (5), (7) and (8) must give place to summations

(2) taken between the proper limits.

Dividing Eqs. (11) and (12) by la and lc, respectively, and adding the results :

— Cb

Cc - Cb

lc

la

2b Mtx Ax -f

lc

'A 2b M,x Ax

4- Yz(Mauaiaxa + Mbuai'axa + Mcucicxc + Mbu'cl'cx'c) .    (13)

in which F = t + t'.

i

Eq. (13) is the most general form of the theorem of three moments if E, the coefficient of elasticity, is a constant quantity. Indeed, that equation expresses, as it stands, the “ theorem ” for a variable coefficient of elasticity if (le) be written instead of i; e representing a quantity determined in a mannef exactly similar to that used in connection with the quantity i.

In the ordinary case of an engineer’s experience T = o, d = d ' = o, I — constant, u — ua — uc— etc.y = ć = secant of the inclinatio7i for which t = — t' is the tangent ; consequently :

T'a —

H.

II

IS

From Eq. (4) :

xa -

xc

- Ik

From Eq. (7) :

■*4

_ 44

_ 44

The summation 2Mtx Ax can be readily made by referring to Fig. I.

The moment represented by CK in that figure is : consequently the moment at any point between A and K, due to P, is :

Mt

Between K and B :

= (^f)• or-('-*)•

Using these quantities for the span la :

Mxx Ax — I Myc dx + f M^x dx = }4P(42 — £²)^.

J O    J z

For the span lc, the subscript a is to be changed to c. Introducing all these quantities Eq. (13) becomes, after providing for any number of weights, P:

6EI

( Cq - Cb ,

V 4    ^

— Mala + 2 Mii (4 + 4) + Mfc

"a

Eq. (14), with c equal to unity, is the form in which the theorem of three moments is usually given ; with ć equal to unity or not, it applies only to a beam which is straight before flexure, since :

T=tft' = 0 — d=d'.

If such a beam rests on the supports a, b, and e, before bending takes place,

and the first member of Eq. (14) becomes zero.

If, in the general case to which Eq. (13) applies, the deflections ca,, 4, and cc belong to the beam in a position of no bending, the first member of that equation disappears, since it is the sum of the deflections due to bending only, for the spans la, and 4, divided by those spans, and each of those quantities is zero by the equation immediately preceding, Eq. (2). Also, if the beam or truss belonging to each span is straight between the points of support (such points being supposed in the same level or not), ua — u'a = ula = constant, and uc — u'c = ulc — another constant. If, finally, / be again taken as constant, xa and xc, as well as 2MjX Ax, will have the values found above.

From these considerations it at once follows that the second member of Eq. (14), put equal to zero, expresses the theorem of three moments for a beam or truss straight between points of support, when those points are not in the same level, but when they belong to a configuration of no bending in the beam. Such an equation, however, does not belong to a beam not straight between points of support.

The shear at either end of any span, as la, is the next to be found, and it can be at once written by referring to the observations made in connection with Fig. 1. It was there seen that the reaction found by the simple law of the lever is to be increased or decreased for the continuous beam, by an amount found by dividing the difference of the moments at the extremities of any span by the span itself. Referring therefore, to Fig. 2, for the shears S, there may at once be written :

S. = 2P

Ma - Mb

• • OS)

sb = ip 4- +

Ma — Mb

• • (16)

sb =

Me-Mb

• • (17)

St = 2P

Mc - Mb

. . (18)

The negative sign is put before the fraction,

Ma - Mb

in Eq. (15), because in Fig. 1 the moments Ma and Mb are represented opposite in sign to that caused by P, while in Eq. (1) the three moments are given the same sign, as has already been noticed.

Eqs. (15) to (18) are so written as to make an upward reaction positive, and they may, perhaps, be more simply found by taking moments about either end of a span. For example, taking moments about the right end of la :

SJa - 2P(la - s) + Ma = Mb.

From this, Eq. (15) at once results. Again, moments about the left end of the same span give :

Sbla - 2P* + Mb = Ma.

This equation gives Eq. (16), and the same process will give the others.

If the loading over the different spans is of uniform intensity, then, in general, P = w dz\ w being the intensity. Consequently :

12P(J² — za)z —

\l    I'*

w (/a — z²) z dz = w — .

J ©    4

In all equations, therefore, for

— 2P{/: ~ P) z

there is to be placed the term wa -

to

; and for

le

4³

the term wc The letters « and ^ mean, of course, that

reference is made to the spans la and lc.

From Fig. 2, there may at once be written :

R =S'a + Sa

(19)

R' =S’b + Sb

(20)

R" = 5; + Se

(21)

etc. = etc. + etc.

Art. 23a.—Reactions under Continuous Beam of any Number of Spans.

The general value of the reactions at the points of support under any continuous beam have been given in Eqs. (19), (20), (21), etc., of the preceding Art. Before those equations, however, can be applied to any particular case, the values of the bending moments, which appear in the expressions Sa, Sb, Sb, etc., for the shears, must be determined. In the application of the theorem of three moments, it is invariably virtually assumed that the continuous beam before flexure is straight between the points of support, and that the latter belong to a configuration of no bending. The moment of inertia I and the coefficient of elasticity E are also assumed to be constant. This is frequently not strictly true, yet it will be assumed in

what follows, since the method to be used in finding the moments is entirely independent of the assumption, and remains precisely the same whatever form for the theorem of three moments may be chosen.

Agreeably to the assumption made, Eq. (14) of the preceding Art. takes the following form, which is almost, or quite, invariably used in engineering practice :

Mala -j- 2Mb(Ja -j- /c) Mclc —    7    (l² z²)z

(I)

~ kp{i? - z*)z

Let Fig. i represent a continuous beam of n spans, equal or unequal in length. At the points of support, o, 1, 2, 3, 4, 5,

la

O_ll

Fig.l

etc., let the bending moments be represented by M₀, Ma, My etc. The moment M₀ is always known ; it is ordinarily zero, and that will be considered its value.

An examination of Fig. 1 shows that, by repeated applications of Eq. (1), the number of resulting equations of condition will be one less than the number of spans. But if the two end moments are known (here assumed to be zero), the number of unknown moments will also be one less than the number of

Hence the number of equations will always be sufifi-

spans.

cient for the determination of the unknown moments.

For the sake of brevity let the following notation be adopted :

«1 — 2(4 + 4) ;

b₂ — 2(4 + 4) ;

4 = 4.

«1 =

u₂= -

«3 =

-1 iv>(/₃²

- ~    - *■)* - I kpy? - ty.

etc. =

etc. —

etc.

4s — ²(4 + 4) ;

= 4;

dĄ — 2(4 -(- 4) ;

/₄ = 4 •

pi — 4 ;

q{ — 2(4 + 4 +1) ;

It is

z denoting any number of the series I, 2, 3, 4, thus seen that, in general,

— ²{Pi    Si) »

also that a₂ = bt, c₂ = bv d₃ = c₄, etc. These relations can be used to simplify the final result.

By repeated applications of Eq. (1) the following n equations of condition, involving the notation given above, will result :

alMt b₁M₂

= «I

— 2/₂

-f £₂Æf₂ + c₂TT₃

+ b₃Ma -f- c₃M₃ + d₃M^

+ cąM₃ + d₄M₄ + f₄Ms

= U4

(2)

+ d₅M₄ + fsMs + gsM₆

= un

The moment M.

will also be equal to zero. In consequence of this last condition it is seen that the coefficients of the Ms occupy precisely the places of the elements of a determinant of the 72th degree. Of the array indicating the determinant, however, there exists only the leading diagonal and one diagonal on each side of it. The determinant for n equations, or {n + 1) spans, has, then, the value :

’ a„ b„ o, o, o, o......

(3)

.    . o, pn, (Jn _

Also let Dt represent the value of the determinant D when

the column indicated by the zth letter of the series a, b, c, d, f, etc., is replaced by the column ut, u₂, uy u₄, etc. If, for example, i — 3, the zth letter is c. Hence :

^1, ^i>    0,0,0,. •

&₂> ^2f ^2, 0,0,0,. .

O , by Uy dy O , O

• - (4)

Ą= ■

0,0, Uy dy f y O, .    .    .

0,0 ,Uy dy fy gy .    .    .

0,0, ZZM, 0,0,0,.    • o, pn, çn

Then, in general :

(5)

Eq. (5) will give the value of the bending moment at any point of support, whatever may be the number of spans or the law of loading on any or all the spans.

Precisely the same formulae are to be used if M₀ and Mn are not zero, but have definite values and are known. In such a case, however, ux and un would be replaced by :

u\ — ux — a₀Af₀.

u'„ - un - rnM,

The same equations also hold true whatever form of the theorem of three moments may be chosen. It is only to be

remembered that the values of the quantities a, b, c, etc., uv u₂, uv etc., will depend upon the choice.

If all the moments are desired, it will be most convenient to put the vertical column uiy u₂, uv . . . uH in place of the vertical column a„ a₂, o, o, . . . o, in Eq. (4), and then find the resulting determinant Dx. Eq. (5) will then give the value of Mv which, placed in the first of Eqs. (2), will enable M₂ to be at once found. M₃ will then result from the second of Eqs. (2), M₄ from the third, etc., etc.

So far as the general treatment of the question is concerned, there yet remains to be considered the expansion of the determinants D and Dt.

The expansion of the determinant D is very simple, and leads to the following results :

For two spans :

(6)

(7)

. - (8)

D — ax

For three spans :

D — axb₂ — a₂bx

For four spans :

D — axb₂cg — axb₂c₂ — a₂bxcz .    .

For five spans :

D = axb₂czd₄ — axbj₂d₄ — a₂bxc^₄ — a₁b₂cAdi + a₂bxc₄d₃ .    (9)

For six spans :

D — ajb^dJs - axbj₂d₄fs - a₂btc₃d₄f₅ — axb₂c₄d₃fs + a₂bxc₄d₃fs - axb₂c₃dsf₄ + aĄcĄf₄ + a₂bxc₃dj₄ .    .    . (10)

By the observance of two or three simple rules, the determinant for (n + 1) spans, or n points of support, may easily be written.

A series of numbers such as 1, 2, 3, 4, 5, 6, etc., is said to be written in its natural order. Let any permutation of this series, 2, 1, 3, 6, 5, 4, be written, in which 2 is placed before 1, 6 before 5 and 4, and 5 before 4. In this permutation, therefore, there are said to be (1 -fi- 2 + 1) — 4 inversions.

Let (A,) represent any letter of the series a, b, c, d, etc., which has the subscript n ; also, let (AM)„ and (AM)M _, represent the nth and (n — i)th letters of the same series which have the subscripts 11. In general, the letter inside the parenthesis represents the subscript figure in the determinant, and that outside, the place of the letter in the series a, b, c, d, f etc.

The nth determinant for (n -f- 1) spans, or n points of support, will then be :

Dn — Dn _ j (Am)„ + Dn _ ₂ (A_ j (Aw _ j )n.

Now, with the notation taken, if the letters in each term of the determinant are written in their natural order, as abcdfg, etc., the number of inversions in the subscript figures of any term will determine the sign of that term, i.e., if the number of inversions is odd, the sign is minus, but if the number is even the sign is plus.

Since n is the greatest subscript in any term, and since (A„)M occupies the most advanced place in the series of letters, no inversions are introduced in multiplying Dn_t by (A„)„. Hence, all terms of Dn_t (A*)* will have the same signs as the corresponding terms of Dn_₁.

Similarly, since 11 is greater than (n — 1), the product (A*\*-i(A*-i\, involves one inversion. Hence, all terms of

will have signs contrary to those of the corresponding terms of Dn _ 2 •

The number of terms in Dn will evidently be the sum of the numbers of terms in Dn_x and Dn_₂.

An examination of the notation will at once show that :

('!«)« — 2(4, -f- ln + j) ,    — In > and (^»-i)» — 4»*

Hence there will result :

Dn = 2Dn_₁(l„ + 4 + I) - Dn_fn .... (ii)

The minus sign before the last term of the second member is on account of the inversion introduced, as already explained.

The general value of the determinant Dt (shown in Eq. (4) when i — 3) can be most easily expanded by considering it the sum of two determinants ; and in order to illustrate this method let it be supposed that M₃ is desired. It will then be necessary to expand the determinant Dy given in Eq. (4). As is known from the theory of determinants, D₃ may be written as follows :

0,0,. .

&u 0,0,0,0, . . . a2, b„ u₂, o o > by n y dy 0,0,. . . 0,0, uv dv /₄, o , . . . o , o , o, dv fy g₅, . . .

tt'vt 0,0,0, . . a₃, b„ o_, o , o, o , . . o, by o , dy 0,0, . . o , 0,0 , dv fy o , . . 0,0, uv ds, fy g₅, . .

O, O, O, O, O, 0,0,.. pny Qu

O, O, Un, O, 0,0,.. pni Qn

(13)

A = A + A"

Eq. (12) shows at a glance what D₃ and Z>₃" represent.

D₃ is precisely the same in form as D, and is given at once by the Eqs. (6) to (11) after writing u₂, u₃ and u₄ for c₂, c3 and c₄.

In general, D/ is found by simply writing Ui _ „ and ut for    and (Ai + I) in the determinant D.

As a general method, that of alternate numbers is probably as simple as any for the expansion of the determinant D-’. For example :

Z^3 —    ~i~ y^’2 + ^1^3) (<^2^1 ~h ^2^2)    “f"    * • •

.    .    (H)

or :

. . . (tlnC₃ -f~ PrTn - id⁻ Çn^n) > •    •

in which ^₂, ^₃, etc., are the units of the alternate numbers.

The circumstances of any particular case will frequently either furnish a more expeditious method than that of alternate numbers, or allow the expansion of D-¹ to be written at once from an inspection of the array given in Eq. (12).

In any case the method of alternate numbers may be used as a check.

Special Method for Ordinary Use.

If the number of spans is large, the expansion of the determinant Di will, at best, be found somewhat tedious. Special methods may be employed which involve only the determinant D, given in Eqs. (6) to (11) ; and it has already been seen that that determinant admits of a very simple expansion.

Let any one span carry any load whatever, while all other spans carry no load. In such a case, P will be zero for every

span but one, and, in consequence of the notation employed, all but two quantities in the series ux, uv us, etc., will also become equal to zero.

If /,• (the zth span) carries the load, there will result :

Ui-1

Ui =--T 2P (/t² — z²)z.....(16)

All other us reduce to zero. Although Eqs. (15) and (16) have the same form, they are not identical except in special cases, since z is not measured from the same end of the span in both expressions.

Now let ui_₁ and u{ take the place of those letters in that column of D formed with the zth letter of the series a, b, c, d, etc., which have the subscripts i and i — I ; ui + x is equal to zero. Or in the notation already employed, let u{ _ , and ut take the place of (/!,-_ j),- and while zero takes the place of (A,- + I)t-. The resulting determinant, A, will then be precisely the same as D in general form. The expansion of Dt can then be at once made by simply putting in D the substitutions above indicated. There will then result :

(17)

In order to find    Math the same loading on the same

Then :

span, Ui_x and 11 { must take the place of (/\,• _ x),- _ t and (Aj)i-lt respectively, while (A^₂),‘_ t becomes equal to zero. Making these substitutions in the determinant D, there will result the determinant D

(18)

The values of and Mi_x> thus obtained, placed in the zth and {i — i)th of the Eqs. (2) will at once give :

i + I-

Mi _ ₂ and M.

Similar substitutions in the other equations will give all the moments. Thus the solution is complete, for the span and loading taken, with the use of the expanded determinant D only.

Each span may be treated in the same manner and the same expansion of D will be the only one necessary.

This method is equivalent to splitting the elements u₂, vv uĄ, etc., of the general determinant D{.

In order to determine the bending moment at any point of support, for loading which covers more than one span, or portions of more than one span, it is only necessary to take the algebraic sum of the separate moments (as above determined), at the point of support in question, found for the loading in each single span. The result will be the moment due to the combined action of all the loading.

It is thus seen that the solution of the most general case is made to depend on the one expansion of the determinant D.

Example.

Let there be a continuous beam of six spans, and let any loading rest upon the fourth ; it is required to find the expansions of the determinants D{ and

The expansion of D is given in Eq. (10), and need not be repeated here.

Using the preceding notation :

= 4-

= d₄.

2-i = 3-

ui

— u₄.

Ob' + i)* — dy

Ui_x

= UV

In Eq. (10) then, d₄ and d₃ are to be displaced by u₄ and u₃, while zero is to take the place of d₅. Hence :

Di — axb₂c₃u₄f₃ axb₃c₂uj%    4f^    cixb₂c₄u₃f₃

+ a₂bxc₄u₃f₃

(19)

Again :

Then, in Eq. (10), placing u₃ and u₄ for c₃ and c₄, and placing

(^t - 2)* — x ' C₂ — O,

there will result :

Di _ i — ciJj ₂u₃d₄j'Ci    a₂bxu₃d₄f₃    uxb₂u ₄d₃_f₃    t,

axb₂u₃d₃f₄ -f* a₂bxu₃dj₄.....

(20)

These values placed in Eqs. (17) and (18) will give M4 and M₃.

The lengths of span may be any whatever ; if they are equal, the results will be simplified.

Special Case of Equal Spans.

If all the spans are of equal length, each may be represented by l. There will then result :

(21)

a2 — ^3 — ^4 — ^5 — • • • —Pi — ^1    ^2 — ^3 —J4 — ... Si — /

aI = b₂ = c₃ = d₄= f₃ — . . . — Çi — 41.

These values of a, b, e, etc., placed in Eqs. (6) to (10) give :

For two spans :

D — 4/.

For three spans :

D = 15l\

For four spans :

D = 56/3.

For five spans :

# D — 209/⁴.

For six spans :

Others may be easily and rapidly written by the aid of Eq. (11), which now becomes :

.    . (22)

Dn = 4lDn _ j - l²Dn _ ₂ .

If the determinant for seven {i.e., n + 1) spans is desired :

Dn_x — 780/⁵ and At-2 = 209/⁴.

Hence :

Dn = D₆ — 3120/⁶ — 209/⁶ = 2911/⁶.

Similarly for eight spans :

D — 4 x 291 il⁷ — 780/⁷ = 10864/⁷.

For nine spans :

D — 4 X 10864/⁸ — 2911/⁸ = 40545/⁸.

For ten spans :

D — 4 X 40545/⁹ — 10864/⁹ = 151316/⁹.

The values given in Eq. (21) will correspondingly simplify the expansion of the determinant A, either in its general form as exemplified in Eq. (4) or as given in the special method. As an illustration, Eqs. (19) and (20) become, respectively :

= 224 l*u₄ — 6o/⁴uy

Di_x = 225/^3 — 6o/⁴zq.

These values then give :

A 56^ - 15^3

195^

-A* -1_ 15 ^3    4^4

52/

Then by Eqs. (2) :

m₄

Ma= Uf -(aMs + M₄). Ms=^-(M₃ + 4M<) = - ₂

m₂

Mx = -

Thus all the moments are known for this example, i.e, with six spans and loading on the fourth span only.

Reactions.

After the moments are found either by the general or special method, for any condition of loading, the reactions will at once result from the substitution of the values thus found in the Eqs. (15) to (21) of the preceding Art., which it is not necessary to reproduce here.

Art. 24.—The Neutral Curve for Special Cases.

The curved intersection of the neutral surface with a vertical plane passing through the axis of a loaded, and originally straight, beam may be called the “ neutral curve.” The neutral curve is the locus of the extremities of the ordinates w of Art. 19 ; it therefore gives the deflection at any point of the beam.

The method of finding the neutral curve for any particular case of beam or loading can be well illustrated by the operations in the following three cases.

Case I.

This case is shown in the accompanying figure, which

'Ś

represents a cantilever carrying a uniform load with a sin-“*j gle weight W at its free end. =jB As usual, the intensity of the uniform loading will be represented by p.

Fig.1

Measuring x and w from B, as shown, the general value

of the bending moment is :

„rd²w M = El

dx²

= Wx

2

(O

Integrating between x and /, remembering that :

dw dx

for x — l :

0\|S-

CO

Hence :

ESI

0\|S-

CO

The greatest deflection, occurs for x — l. Hence :

El

(^+4) • •

• - (4)

The greatest moment, Mv exists at A, and its value is :

(5)

Mx = Wl + P—

These equations are made applicable to a cantilever with a uniform load by simply making W — o. They then become :

M = El

d²w _ px²

dx²

(6)

E[dm _t ,xi_ ^

dx

(7)

. . (8)

(9)

w - -É-1 6EI

//⁴

SEI (10)

Again, for a cantilever with a single weight only at its free end, / is to be made equal to zero in the first set of equations. Those equations then become :

M — El

d²w

dx²

= Wx . .

• • (u)

EI~

dw ₌ K - /’)

dx

(12)

IV

X³

W ” 2El \₃

• • (Iß)

w, = —

Wl³

3 El

(H)

05)

The general expressions for the shear and the intensity of loading are :

S= El

d³w

dx³

= W + px

(16)

(17)

EIdV=P

dx* r

Case IT.

This case, shown in the figure, is that of a non-continuous beam, supported at each end, and carrying both a uniform load

■k----x---->

Fig.2

(whose intensity is p) and a single weight W at its middle point. The reaction R, at either end, will then be :

The general value of the moment will then be :

M = El

d*w

dx²

= Rx-**-

The origin of x and w is taken at A. Remembering that :

II

to I *Ni

CL

era

rt

2.

aq

to

o.

ta

§ LH

to I >3

Ov|S-

001

VO

Again integrating :

û'

""N

OV|S-

°°l^

to

The greatest deflection wI occurs at the centre of the span, for which :

Hence :

WI = —

/3

4SEI

//!■ .... (21)

001 <-n

. . (22)

These formulae are made applicable to a non-continuous beam carrying a uniform load only, by putting W = o. They then become :

(²3)

dx²

The greatest moment, also, is found by putting :

It has the value :

to

to

12

• - (24)

iv —

24 El

(2 xH — x* — l*x) ....    (25)

w₁ = —

5//⁴

384^/

co

KJ

VO

^i =

Pi²

(2;)

The formulae for a beam of the same kind carrying a single weight at the centre, are obtained by putting/ = o in the first

set of equations. Those for the greatest deflection and greatest moment, only, however, will be given. They are :

w₁ —

Wl*

48 El

(28)

Wl

(²9)

The general values of the shear and intensity of loading

are :

dM

S = —-,— — R — px dx

(30)

d²M

dx³

= -P

(31)

Case III.

The general treatment of continuous beams requires the use of the theorem of three moments. The particular case to

p'

Fig. 3

be treated is shown in Fig. 3. The beam covers the three spans, BA, AB and BC, and is continuous over the two points of support A and B.

Let DA = If

“ AB — l₂ - Let l₂ = tz/, = tz7₃.

Let the intensity of the uniform load on AB be represented by p and let the two single forces P and P' only, act in the spans DA and BC respectively. Also let the two distances :

DR — zx = alx and CF — a'l₃

be given. It is required to find the magnitudes of the forces P and P\ if the beam is horizontal at A and B.

Since the beam is horizontal at A and B, the bending moments over those two points of support will be equal to each other, for the load on AB is both uniform and symmetrical. Let this bending moment, common to A and B, be represented by M₂. As the ends of the beam simply rest at D and C, the moments at those two points reduce to zero.

Because the four points D, A, B and C are in the same level, the first member of Eq. (14), of Art. 23, becomes equal to zero.

If that equation be applied to the three points D, A and B, the conditions of the present problem produce the following results :

Ma = o, Mb = Mc — M₂

and

lc    4

Hence the equation itself will become :

-^4(24 + 34) H    y- (4² - •s'i²)'sri + / ~

• (32)

4

M₂= -

4^(4² - ^²K + /44³ . 44(24 + 34)

4P (1 — a²) a + pnHx

4(2 + 3«)

• • (33)

Reaction at D — R₂ =

P¹!—-Il + Ę²

•    • (34)

As the origin of ^ is at D, x will be measured from the same point.

Separate expressions for moments must be obtained for the two portions, DE and EA, of lv because the law of loading in that span is not continuous.

Taking moments about any point of EA :

El

(35)

Remembering that :

dw dx

for x = /„ and integrating between the limits x and 4 :

dw

dx

f C*-a - 4a) - T (^ - 4²) + P*i (* ~ 4) (36)

Again, remembering that w = o for x = 4, and integrating between the limits x and 4 :

+ p*' (t _ ^ ⁺ t)

(37)

(38)

Taking moments about any point in ZTA :

El ** = R,*

dw

(39)

Making x — z₁ in Eqs. (36) and (39), then subtracting :

dw

A¹/ -T—

dx

y (** - A²) - — Oi² - A²) + -ffel (*i - A) (40)

Remembering that w — o for x — o, and integrating between the limits x and O :

EIw — — (—--ł²x) — — (zf — lf)x + Pzt (%i — Ay* (4¹)

2 \ 3    /    2

Making x = zt in Eqs. (37) and (41), then subtracting :

Rtf P

- J (Ił - tf) + P-' w

zł') — o . . (42)

Putting the value of M₂ from Eq. (33) in Eq. (34), then inserting the value of RI} thus obtained, in Eq. (42), after making z₂ — al\ :

a{ i — a²)

[<■

- (I - a*) + i a(I - a‘) J

_ pffili

~ 4(2 + 3«) *

pn²lx

pnl₂

6a{ i — a²)

60(1 —ci²)

• (43)

This is the desired value of P, which will cause the beam to be horizontal over the two points of support A and B when the span AB carries a uniform load of the intensity p.

By the aid of Eq. (43), Eq. (33) now gives :

' <«>

It is to be noticed that M₃ is entirely independent of/t or /₃. Eq. (43) also gives :

6a( I — a²)

11²/,

: (45)

Hence :

.    . (46)

Ma = - ^ (I - a²)a .    .

Thus any of the preceding equations may be expressed in terms of p or P.

Rx, also, becomes :

6a( I + a)

pnl,

12

(47)

or :

*,=

P(i - a) [1

— a( I +    •    •    •    (48)

It is clear that there cannot be a point of no bending in DE. Hence, the point of contra-flexure must lie between E and A, Fig. 3. In order to locate this point, according to the principles already established, the second member of Eq. (35) must be put equal to zero. Doing so and solving for :

P — Ri

(49)

Since P is always greater than Rlt there will always be a point of contra-flexure.

All these equations will be made applicable to the span EC, by simply writing a! for a, /₃ for and n' for n.

As an example, let :

a — — and n = 1.

Eqs. (43), (44) and (47) then give :

-EL = -Æ

Ma —

12

16 ⁵

/j —    /₂    —    /₃    —    /.

In general, the span /, is called “ a beam fixed at one end, simply supported at the other and loaded at any point with the single weight, P.”

Let it, again, be required to find an intensity, °fa uniform load, resting on the span which will cause the beam to be horizontal at the points A and B.

Since the load is continuous, only one set of equations will be required for the span. The equation of moments will be :

(50)

>0

VO I to

1-0

CO

0\

5J

after writing :

- , d²w

El

Integrating between the limits x and lx :

§ LH

II

OM^

l^ł

Integrating between the limits x and o :

hi

*?

to

But, also, w = o when x = /,. Hence :

*,=

oo|o>

Nà-

co

This equation gives the value RT when p' is known. Making x — lx in Eq. (50) and using the value of Rx from Eq. (53) :

M

-vT

col 00.

S-

00

• • (54)

Adapting Eq. (32) to the present case :

Ma(2l, + 34) + 7 (P' 4³ + X³) — o. 4

Ma = -

(/ -f /k³)/,²

4(2 + 3«)

(55)

Equating these two values of M₂ :

(56)

²    2

— fin²

3

Thus is found the desired value of p'. In this case the span /j is called “ a beam fixed at one end, simply supported at the other and uniformly loaded/’

The points of contra-flexure are found by putting the second member of Eq. (50) equal to zero and solving for x, after introducing the value of Rt from Eq. (53). Hence :

or :

col Tj-

x — o and ^ sa

Between the simply supported end and point of contra-

flexure the beam is evidently convex downward, and convex upward in the other portion of the spans lx and ly whether the load is single or continuous. Moments of different signs will, then, be found in these two portions, and there will be a maximum for each sign. The location of the sections in which these greatest moments act may be made in the ordinary manner by the use of the differential calculus ; but the negative maximum is evidently M₂, given by Eqs. (44) and (55). On the other hand the positive maximum is clearly found at the point of application of P in the case of a single load, and at the point

CO loo

in the case of a continuous load. These conclusions will at once be evident if it be remembered that the portion of the beam between the supported end and point of contra-flexure is, in reality, a beam simply supported at each end. These moments will have the values :

Mt = Plx (i — d)a — 4

ąP{i — a²)a² -f- partią

4(2 + 3 n)

• (57)

(58)

In case of a single load if P is given, and not p, Eq. (45) shows :

Mx = Plx (1 — a)a i — i a(i -j- a) .

The points of greatest deflection are found by putting the

second members of Eqs. (36), (40) and (51) each equal to zero, and then solving for x. They are not points of great importance, and the solutions will not be made.

The following are the general values of the shears for a single load on lx :

In AE ;

dHo

S ~ EI dx* ~ R* ~ P> [from ET (35)]-

[from Eq. (38)].

In ED ; St = ElÇj = K ;

The shear in /, for the uniform load p' is :

S' = EI

d^w

dxz

= Rz - p'x\

[from Eq. (50)].

Also :

d⁴w

Intensity of load — EI

As has already been observed, all the equations relating to the span /, may be made applicable to the span /₃ by changing a to a and n to n.

The span /₂ remains to be considered.

Since the bending moments at A and B are equal to each other, and since the loading is uniformly continuous, half of it (the load p/₂) will be supported at A and the other half at B. In other words, the vertical shear at an indefinitely short distance to the right of A, also to the left of B, will be equal to pi

—Let x be measured to the right and from A. The bending moment at any section x will be :

EI

d²w

pi

dx²

M₂-\-EELx

or :

d²w El ——

dx²

to iNâ-

On

Integrating between the limits x and o :

dw

dx

(Lf

?)• •

. (60)

Again integrating between the same limits :

EIw —

Since :

dw akr

for /₂, Eq. (60) will give Af₂ independently of preceding equations. Following this method, therefore ;

M₂ = -

pi²

12

This is the same value which has already been obtained. Introducing the value of M₂ :

d²w

El —T—

4²

• • (62)

The points of contra-flexure are found by putting the second member of Eq. (62) equal to zero. Hence :

4²

X — l₂

0.7894

0.2114

The moment at the centre of the span is found by putting,

in Eq. (62) :

24

Ej. dw _ p fl₂x²

X³

dx

• • (63)

EIw

X²

4²

. . (64)

Mt

This is the greatest positive moment. The general value of the shear is :

c    d³w

o = El = p dx³

And the intensity of load :

d*zv

EIH*=-p-

The span l₂ is generally called “ a beam fixed at both ends and uniformly loaded.”

It is sometimes convenient to consider a single load at the centre of the span /₂ while the beam remains horizontal at A and B ; in other words, to consider “ a beam fixed at each end and supporting a weight at the centre.”

Let W represent this weight : then a half of it will be the shear at an indefinitely short distance to the right of A and left of B. As before, let be measured from A, and positive to the right. The moment at any point will be :

El

d²zi>

Wx

dx²

= Ma- — ²    2

(65)

Integrating between .v and o :

If;r =

hence :

El

then will

dzv

dx

= M₂x —

Wx²

. . (66)

dzv

dx °⁵

Ma ==

Wla

The general value of the moment then becomes :

M-El

dszv Wl> Wx

- -3--— -»» -.

dx³

• • (67)

j in this equation, the bending moment at the centre (where W is applied) has the value :

Ux =

Wl₂

Centre moment — —

Hence, the bending moments at the centre and ends are each equal to the product of the load by one eighth the span, but have opposite signs.

A second integration between x and o gives :

El

Wlye*

Wx*

16

12

. . (68)

Hence, the deflection at the centre has the value :

Centre deflection —

Wl*

192El '

By placing M — o, the points of contra-flexure are found at the distance from each end :

*?

II

•vT|

Addendum to Art. 24.

The formulae of this Art. furnish the solutions of many practical questions of maxima deflections and moments. The latter for several ordinary cases are given in the following tabulation.

Pis the weight in pounds at end of beam or centre of span, p is the load in pounds per lin. ft. of beam.

BEAM.

MAX. MOMENT.

MAX. DEFLECTION.

POINT OF CONTRAFLEXURE.

PI at A.

II

pcruntP^

kpP at A.

of length.

Ill

\Pl at centre.

PI³

576 p j- at A.

PI³

36 ~pj at centre.

IV

P per unit of length.

ipi³ at centre.

pi⁴

22.5 ^ryat centre.

,A

'P

— -&PI at A.

A PI at centre.

l6 l6 dill at 0.447/ ¹⁰⁻¹⁰ El from B.

VI

— I pi'¹ at A.

jP per unit of length.

Pll at 0.4215/

+ tIï pl³ at f/ from B. 9’35 El from B.

ft/ from B.

Reaction at B — -pgP.

It from B.

Reaction at B = \pl.

VII

— \Pl at A.

\Pl at centre.

PI³

9 -£j at centre.

\l from each end.

iA

VIII

*P per unit of length.

— ~bipl³ sit A. fpl‘ at centre.

PI'

4.5 yryat centre.

0.211/from each end.

I is the length of beam or of span in feet.

E is the coefficient of elasticity in pounds per sq. inch.

/ is the moment of inertia of the normal section of the beam with all dimensions of section in inches.

The “Max. Moments” will be in foot pounds; and the “ Max. Deflections ” will be in inches.

In the use of Eq. (2), Art. 62, in its many practical applications, it is best to have the moment M in inch pounds, which will result from simply multiplying the “ Max. Moments ” of the preceding Table by 12.

Case I results from Eqs. (14) and (15) ; Case II, from Eqs. (9) and (10) ; Case III, from Eqs. (28) and (29) ; Case IV, from Eqs. (26) and (27). In Case V the reaction is found by putting - a = ^ in Eq. (48) ; the point of “ Max. Deflection ” is found by

dw

equal

dx

placing zx = in Eq. (40), and the resulting value of

to zero and solving for x, which latter value in Eq. (41) will give “ Max Deflection.” Case VI results from treating Eqs. (53), (51) and (52) in precisely the same manner. Case VII results directly from the formulae on pages 189 and 190. Case VIII results directly from the equations on pages 187 and 188.

The preceding cases are those which commonly occur with constant values of E and I. Other cases, such as a single load at any point, or partial uniform load over any part of span, are to be treated by the same general principles.

Art. 25.—The Flexure of Long Columns.

A “ long column ” is a piece of material whose length is a number of times its breadth or width, and which is subjected to a compressive force exerted in the direction of its length. Such a piece of material will not be strained, or compressed, directly back into itself, but will yield laterally as a whole, thus causing flexure. If the length of a long column is many times the width or breadth, the failure in consequence of flexure will take place while the pure compression is very small.

As with beams, so with columns, the ends may be “ fixed,” so that the end surfaces do not change their position however great the compression or flexure. Such a column is frequently, perhaps usually, said to have “ flat ” ends. If the ends of the column are free to turn in any direction, being simply supported, as flexure takes place, the column is said to have “ round ” ends. It is clear that if the column has freedom in one or several directions, only, it will be a “ round ” end column in that one direction, or those several directions, only. It is also evident that a column may have one end “ round ” and one end “ flat ” or “ fixed.”

’///////,

tumiiiwunl/d

Fig.l

In Fig. i let there be represented a column with flat ends, vertical and originally straight. After external pressure is imposed at A, the column will take a shape similar to that represented. Consequently the load P, at A, will act with a lever arm at any section equal to the deflection of that section from its original position. Let y be the general value of that deflection, and at B let y=yv Let x be measured from A, as an origin, along the original axis of B._ the column. In accordance with principles already established, the condition of fixedness at each of the ends A and C is secured by the application of o' a negative moment — M. Now it is known from the general condition of the column that the curve of its axis will be convex toward the axis of x at and near A, while it will be concave at and near B (the middle point of the column). Hence, since y is positive toward the left, and since the ordinate and its second derivative must have the same sign when the curve is convex toward the axis of the abscissas, the general equation of moments must be written as follows :

- Ei =

dx²

— M + Py • •

Multiplying by — 2dy :

2 M dy — P 2y dy.

EI 2dy d2y = dx²

EI

(£) =*M?-Pf + (c = o). .

• (2)

c — o because the column has flat ends, and,

dy

-dr- — o

dx

when y = o. Also :

dy

-d- = o

dx

when y — yv

_ Py,

• • (3)

Eq. (2) now becomes :

EI

dy

— dx.

;r

IE

• • (4)

7t,

El

(5)

In this equation / is the length of the column. From Eq. (5) there may be deduced :

p- 4n2RI

l²

(6)

It is to be observed that Pis wholly independent of the deflection, i. e., it remains the same, whatever may be the amount of deflection, after the column begins to bend. Consequently, if the elasticity of the material were perfect, the weight P would hold the column in any position in which it might be placed, after bending begins.

Eq. (6) forms the basis of “ Hodgkinson’s Formula ” for the resistance of long columns, of which more will be given hereafter. It was first established by Euler.

Some very important results flow from the consideration of Fig. i in connection with the preceding equations.

The bending moment at the centre, B, of the column is obtained by placing y = y₁ in Eq. (1) ; its value is, consequently:

• • (7)

M' = - M + Py, = M . .

Hence the bending at the centre of the column is exactly the same (but of opposite sign) as that at either end. Between A and B, then, there must be a point of contra-flexure.

Putting the second member of Eq. (1) equal to zero, and introducing the value of M from Eq. (3) :

Introducing this value of y in Eq. (4), and bearing in mind Eq. (5) :

. . (8)

'El _ l P ~ 4

The points of contra-flexure, then, are at H and D, — / and — / from A.

Hence, the middle half of the column (HD) is actually a column with round ends, and it is equal in resistance to a fixed-end column of double its length.

Hence writing /' for ~ and putting 2/' for lin Eq. (6) :

(9)

71* El

Eq. (9) gives the value of P for a round-end column.

Again, either the upper three quarters (AD) or the lower three quarters (CH) of the column is very nearly equivalent to a column with one end flat and one end round, and its resistance is equal to that of a fixed-end column whose length is ^ its own. Putting, therefore :

•vT

|uj

and introducing :

-vT

in Eq. (6) :

P = 2.25

n²EI

/x²

.    . (IO)

The last case is not quite accurate, because the ends of the columns HC and AD are not exactly in a vertical line.

In reality, the column under compression may be composed of any number of such parts as HD, with the portions HA and CD at the ends, thus taking a serpentine shape, so far as pure equilibrium is concerned. In such a condition the column would be subjected to considerably less bending than in that shown in the figure. In ordinary experience, however, the serpentine shape is impossible, because the slightest jar or tremor would cause the column to take the shape shown in Fig. I. Hence, the latter case only has been considered.

If r is the radius of gyration and 5 the area of normal section of the column, Eqs. (6) and (9) will take the forms :

ĄPEr²

l²

and

n²Er²

Eq. (io) will, of course, take a corresponding form.

These equations evidently become inapplicable when -^r approacheś C, the ultimate compressive resistance of the material in short blocks. The corresponding values of at

the limit, are :

2 7t

-ç ; and

.    . (ii)

for fixed and round ends respectively ; other conditions of ends will be included between those two.

If, for wrought iron :

E — 28,000,000 and C = 60,000,

the above values become 136 and 68, nearly.

Euler’s formula, therefore, is strictly applicable only to wrought-iron columns, with ends fixed or rounded, for which l -T- r exceeds 136 and 68, respectively.

If, for cast iron :

E = 14,000,000 and C — 100,000,

Eqs. (11) give :

CL

Co

Euler’s formula evidently becomes inapplicable considerably above the limits indicated, since columns in which has

those values will not nearly sustain the intensity C.

The analytical basis of “ Gordon’s Formula ” for the resistance of long columns is so closely associated with the empirical, that both will be treated together, hereafter.

Art. 26.—Graphical Determination of the Resistance of a Beam.

The graphical method is well adapted to the treatment of beams whose normal sections are limited either wholly or in part by irregular curves. In Fig. 1 is represented the normal section of such a beam, the centre of gravity of the section being situated at C. The lines HL, AB and DF are parallel. As is known by the common theory of flexure, the neutral axis will pass through C.

Let aa be any line on either side of AB, then draw the lines ad normal to AB, having made MN and HL equidistant from

AB. From the points a', thus determined, draw straight lines to C. These last lines will include intercepts, bb, on the original lines aa. Let every linear element parallel to AB, on each

H---TT,

a'

-L

JL\

—Df-t-

¥

M a'

sr

a/ N

Fig. 1

side of C, be similarly treated. All the intercepts found in this manner will compose the shaded figure.

This operation, in reality, and only, determines an amount of stress with a uniform intensity identical with that developed in the layer of fibres farthest from the neutral axis, and equal to the total bending stress existing in the section ; this latter stress, of course, having a variable intensity. HL represents the layer of fibres farthest from the neutral surface, consequently MN was taken at the same distance from AB. Any other distance might have been taken, but the intensity of the uniform stress would then have had a value equal to that which exists at that distance from the neutral axis. Again, a different intensity might have been chosen for the stress on each side of AB. It is most convenient, however, to use the greatest intensity in the section for the stress on both sides of the neutral axis ; this intensity, which is the modulus of rupture by bending, will be represented, as heretofore, by K.

Let c and c be the centres of gravity of the two shaded figures. These centres can readily and accurately be found by cutting the figures out of stiff manilla paper and then balancing on a knife edge. Let s represent the area of the shaded surface below AB, and s' the area of that above AB.

Because this is a case of pure bending, the stresses of tension must be equal to those of compression. Hence :

Ks = Ks' ; or, s = s' .    .

The moment of the compression stresses about AB will be :

Ks X c'C.

The moment of the tensile stresses about the same line will be :

Ks x cC.

Consequently the resisting moment of the whole section will be :

M — Ks{c'C + cC) — KsY.cc. .    .    .    (2)

Thus, the total resisting moment is completely determined. In some cases of irregular section the method becomes absolutely necessary.

It is to be observed that the centre of gravity, c or c, is at the same normal distance from AB as the centre of the actual stress on the same side of AB with c or ć.

Art. 27.—The Common Theory of Flexure with Unequal Values of Coefficients of Elasticity.

In all cases it has hitherto been assumed that the coefficient of elasticity for tension is equal to the same quantity for compression. In reality, this is exactly true for probably no material whatever, though the error, fortunately, is not serious for the greater portion of the material used by the engineer. By the aid of the assumptions used in the common theory of flexure, formulae involving this difference of coefficients may be deduced. As these are of little real value, however, a few general results, only, will be obtained.

Let E represent the coefficient of elasticity for tension.

Let E' represent the coefficient of elasticity for compres

sion.

As has before been assumed, the normal sections of the beam, which are plane before flexure, will be taken as plane and normal to the neutral surface after flexure. Also, as before (Art. 18), let u represent the rate of strain (strain for unit of length of fibre) at unit’s distance from the neutral surface-; let the variable width of the section be represented by b, while y represents the variable normal distance of the element bdy from the neutral axis of the section. The element of the tensile stress in the section will be :

Euy . b dy.

The elementary moment of the same will be :

Euy² b dy.

In precisely the same manner, the elementary compressive moment will be :

E 'uy*b dy.

Consequently, the total resisting moment will have the

value :

a| ¹ y²b dy -)- E' y²b dyJ .

The ordinates y and are those belonging to the extreme fibres of the section, while K and K' represent stress intensities in those fibres. The general value of y is also affected with the negative sign on the compression side of the beam.

It has been shown in Art. 18 that : also, in the case of straight beams, that :

d²w dx* ’

w being the deflection and ^ the abscissa measured along the axis of the beam. For the sake of brevity, let the quantity in the brackets in the second member of Eq. (i) be represented by EJ, in which, consequently, E ' will be displaced by nE, n being the ratio between E and E'. Eq. (i) may then take the form :

M=EJ

d²w

dx²

• . (2)

or :

EJ

d²w

dx²

• (3)

If M and J are expressed in terms of x, w may at once be found. If, as is usual, the section is uniform, then will J be constant and Monly, will be a function of x.

If the section is rectangular, b will be constant and J will take the following value :

J=bl

by'³

CJt- _L. -ft — ,

3    3

(4)

- Ox³ + ny'³)

Because the internal tensile stress in any section must equal the internal compressive stress in the same section :

• - (5)

tyj    rÿ

En by dy — E 'u by dy .    .

Eq. (5) will enable the neutral axis of any section to be located. If the section is symmetrical, the neutral axis will evidently be situated on that side of the centre of gravity of the section on which is found the greatest coefficient of elasticity.

Art. 28.—Greatest Stresses at any Point in a Beam.

If the approximate conditions on which are based the formulae found in the latter part of Art. 17 are assumed, some interesting and important results may be very easily obtained.

The Eqs. (13), (14) and (15) of Art. 6 are those which lead to the ellipsoid of stress, and hence to all of its special cases and consequences. The equation representing the ellipsoid of stress might first be found, and then the special form relating to the case considered. It will be more simple and direct, however, to use those equations immediately.

If, as in Art. 17, a rectangular beam carrying a load at its end be assumed, in which :

Tx= T₃ = N₂ = N₃ = o,

Eqs. (13), (14) and (15) of Art. 6 reduce to :

iV, cos p + T₂ cos r — P cos n ;

T₂ cos p

— P cos p.

But since all stress is assumed to be found in planes parallel to ZX :

cos r = sin p, and cos p — sin n.

Hence :

.    .    (I)

A⁷! cos p + T₂ sin p — P cos tc .    .    .

T₂ cos p

— P sin 7t . .

.    .    (2)

in which P is the intensity of the resultant stress on any plane

at any point ; p the angle which the normal to that plane makes with the axis of X (the axis of the beam) ; and n the angle which the direction of P makes with the same axis.

Let it first be required to find the plane, at any point, on which the normal or direct stress is the greatest.

It is known from the theory of internal stress that this greatest normal stress will be the resultant and, hence, a principal stress. Hence the relation : 7t — p ; or:    •

(3)

A, -j- T₂ tan p — P

T₂

= P tan p .    .

•    • (4)

If F is the weight carried by the beam at its end; / the moment of inertia of the beam’s cross section ; and d its half depth, or greatest value of z, it has been shown in Arts. 17 and 18 that :

Fx

A, — z, and T₂ ~

2/

(d‘ - O ■ ■ • (5)

Inserting the value of P from Eq. (4) in Eq. (3) :

T₂ — T₂ tan¹ p — Ny tan p.

Nx

tan¹ p -j- tanP — l-

T₂

Solving this quadratic equation and then inserting the values of T₂ and Ar from Eq. (5) :

tan p — —

xz

d¹ — z¹

V (d¹ — z²)² + x²z2 ±    d² - z²

. . (6)

This value of tan p put in Eq. (3), or Eq. (4), will give the greatest value of the direct or normal stress (also resultant) at any point in the beam.

At the exterior surface, d — z ; hence :

co.

tan p — o or

Since for this point T₂ = o, the first value gives, by Eq. (3), P— Nv The second value, by Eq. (4), gives, P = o. These results might have been anticipated.

At the neutral surface, 2 — o; hence :

tan p = ± i = tan ± 45 °.

Hence, at the neutral surface there are two planes on which the stress is wholly normal, and these planes make angles of 45⁰ with the neutral surface, or 90° with each other (i.e., they are principal planes).

Since N₁ = o at the neutral surface, either of the Eqs. (3) or (4) gives :

Fd²

P=±T^±-2T

(7)

Hence each of these normal or principal stresses equals in intensity that of the transverse or longitudinal shear at the neutral surface ; also, one of these principal stresses is a tension and the other a compression.

xz

P — T₂ cot p = —

(d² - z²)²

=F V(d² - s²)² + x²s²

is the equation of the locus of the point of constant greatest

normal intensity of stress, if P be taken constant and equal to any possible value.

Let it next be required to find the plane of greatest shear at any point i?i the beam, and the value of that shear.

The shear on any plane will be :

(8)

P sin (n — p) = T

Multiplying Eq. (1) by (— sinp) and Eq. (2) by cosp, then adding :

— Nx cos p sin p T₂ {cos² p — sin²p) — P {sin 7r cos p — cos 7i sin p) = P sin {n — p') — T.

• • (9)

sin 2p -j- T₂ cos 2p

It is now required to find what value of p will make the general value of T [given by Eq. (9)] a maximum. Hence :

dP

dp

— — Nx cos 2p — 2 T₂ sin 2p — o.

xz

2 T₂

• • (10)

d² - z²

± — ... ........ —.

p x²z² + {d² — z²)²

Eqs. (10) give the value of/ which is to be placed in Eq. (9), in order to obtain the greatest value of T at any point of the beam.

±    + (d² — z³)²    ....    (11)

At the exterior surfaces of the beam :

z = ± d.

Hence :

±    ±K.

For this case, also :

cos 2p — o, or p — 45⁰.

Hence, at the exterior surfaces of the beam the planes of greatest shear make angles of 45⁰ with the axis of the beam, and the intensity of the shear is half that of the direct stress at the same place.

At the neutral surface : z = o. Hence :

± i.

From Eq. (9) :

T — T₂ cos

2p\-

2T₂

Fd²

± —zr = T₂ ; and cos 2p = 21

Hence, 2p — o or 180⁰ ; or / = o or 90° ; i.e., the planes of greatest shear are the transverse and longitudinal planes, and the greatest shear itself is, consequently, the transverse or longitudinal shear.

If T is given any possible value and considered constant, Eq. (11) will give the locus of the point of constant greatest shear.

The result expressed in Eq. (7) is of great value in determining the thickness of the web of flanged beams, as will be seen hereafter.

PART II.—TECHNICAL.

CHAPTER V.

Tension.

Art. 29.—General Observations.—Limit of Elasticity.

HITHERTO, certain conditions affecting the nature of elastic bodies and the mode of applying external forces to them, have been assumed as the basis of mathematical operations, and from these last have been deduced the formulae to be adapted to the use of the engineer. These conditions, in nature, are never realized, but they are approached so closely, that, by the introduction of empirical quantities, the formulae give results of sufficient accuracy for all engineering purposes ; at any rate, they are the only ones available in the study of the resistance of materials.

In determining the quantity called the “coefficient of elasticity,” it is supposed that the body is perfectly elastic, i.e., that it will return to its original form and volume when relieved of the action of external forces, also, that this “ coefficient ” is constant. There is reason to believe that no body known to the engineer is either perfectly elastic or, possesses a perfectly constant coefficient of elasticity. Yet, within certain not well defined limits the deviations from these assumptions are not sufficiently great to vitiate their great practical usefulness.

The “ not well defined ” limit for any one given material is called its “ limit of elasticity,” or “ elastic limit.” The u limit of elasticity,” then, may be defined as that degree of stress within which the coefficient of elasticity is essentially constant and equal to the stress divided by the strain.

In some materials, like many grades of wrought iron and steel, the limit of elasticity approximates, to a greater or less degree, to the condition of a well defined point. If a piece of such a material is subjected to stress in a testing machine, at the elastic limit, the amount of strain caused by a given increment of stress will be observed, comparatively speaking, to rapidly increase. This increase may be uniform for a considerable range of stress, but it finally becomes irregular, after which failure takes place.

In other materials, there seems to be no simple relation between stress and strain for any condition of stress whatever. For such a material it obviously is impossible to assign either any definite elastic limit or coefficient of elasticity.

Between these limits, of course, all grades of material are

found.

It should be stated that some authorities have given arbitrary definitions of the elastic limit, and that these definitions have been very much used. Wertheim and others have considered the elastic limit to be that force which produces a permanent elongation of 0.00005 of the length of a bar. Again, Styffe defines, as the limit of elasticity, a much more complicated quantity. He considers the external load to be gradually increased by increments, which may be constant, and that each load, thus attained, is allowed to act during a number of minutes given by taking 100 times the quotient of the increment divided by the load. Then the “ limit of elasticity ” is “ that load by which, when it has been operating by successive small increments as above described, there is produced an increase in the permanent elongation which bears a ratio to the length of the bar equal to 0.01 (or approximates most nearly

to 0.01) of the ratio which the increment of weight bears to the total load.” (Iron and Steel, p. 30.)

The most natural value, however, seems to be that stress which exists at the point where the ratio between stress and strain ceases to be essentially constant, though the assignment of the precise point be difficult in many cases and impossible in some ; and in that sense it is here used, though seldom in ordinary testing.

Again, in the common theory of flexure, modes of application of external forces and a constitution of material are assumed, which are never realized ; yet the resulting formulae are of inestimable value to the engineer.

Finally, it will be shown in the first section of Art. 32 that it is in general impossible to produce a uniform intensity of stress in a normal cross section of a body subjected to pure tension, and, consequently, that the ultimate resistance, as experimentally determined, is a mean intensity which may be, and usually is, considerably less than the maximum sustained by the test piece.

These general observations are to be carefully borne in mind in connection with all that follows.

Art. 30.—Ultimate Resistance.

After a piece of material, subjected to stress, has passed its elastic limit, the strains increase until failure takes place. If the piece is subjected to tensile stress, there will be some degree of strain, either at the instant of rupture or somewhat before, accompanied by an intensity of stress greater than that existing in the piece in any other condition. This greatest intensity of internal resistance is called the “ Ultimate Resistance.

In very ductile materials this point of greatest resistance is found considerably before rupture ; the strains beyond it increasing very rapidly while the resistance decreases until separation takes place.

The ultimate resistances of different materials used in engineering constructions can only be determined by actual tests, and have been the objects of many experiments.

It has been observed in these experiments that many influences affect the ultimate resistance of any given material, such as mode of manufacture, condition (annealed or unannealed, etc.), size of normal cross section, form of normal cross section, relative dimensions of test piece, shape of test piece, etc. In making new experiments or drawing deductions from those already made, these and similar circumstances should all be carefully considered.

Art. 3i.—Ductility.—Permanent Set.

One of the most important and valuable characteristics of any solid material is its “ ductility,” or that property by which it is enabled to change its form, beyond the limit of elasticity, before failure takes place. It is measured by the permanent “ set,” or stretch, in the case of a tensile stress, which the test piece possesses after fracture ; also, by the decrease of cross section which the piece suffers at the place of fracture.

In general terms, i.e.y for any degree of strain at which it occurs, “ permanent set ” is the strain which remains in the piece when the external forces cease their action. It will be seen hereafter that in many cases, and perhaps all, permanent set decreases during a period of time immediately subsequent to the removal of stress. Indeed, in some cases of small strains it is observed to disappear entirely.

Some experimenters, with the aid of very delicate measuring apparatus, have observed permanent set even within what is ordinarily termed the limit of elasticity, and have been led to believe that a very small permanent set exists with any de-gree of stress whatever. In such cases, however, it is probable that the greater part or all of the permanent set disappears after the lapse of a few hours.

Art. 32.—Wrought Iron.—Coefficient of Elasticity.

Before considering the experimental results which are to follow, it will be interesting as well as important to examine some of the circumstances which attend the experimental determination of the coefficient of elasticity.

If tensile stress is uniformly distributed over each end of a test piece, it will not be so distributed over any other normal section. For since lateral contraction takes place, the exterior molecules of the piece must move towards the centre. But if this motion takes place, the molecules in the vicinity of the centre must be drawn farther apart, or suffer greater strains, than those near the surface.

Hence the stress will no longer be uniformly distributed, but the greatest intensity will exist at the centre and the least at the surface of the piece. These effects will evidently increase, for a given kind of cross section, with its area. But the stretch, or strain, from which the coefficient of elasticity is computed, is measured on the surface of the piece, and corresponds, as has just been shown, to an intensity of stress less than the mean, while the latter is actually used in the computation. In the notation of Eq. (1), Art. 2,p is too great and / too small ; hence E will be too large.

As these effects increase with the area of the cross section, while other things are the same, larger bars should give greater coefficients of elasticity than smaller ones.

These effects will evidently be intensified, also, if the external force is applied with its greatest intensity near, or at, the centre of the bar, as is the case in testing eye-bars.

Again, on the other hand, if the ends of the test piece are

gripped on the surface, or skin, as is usually the case with small pieces, these effects will be very much modified, and possibly entirely counteracted, so that the greatest intensity will exist at the surface. In the latter case, the resulting coefficient would be too small.

Between these extreme cases, all grades will be found.

From these considerations, it is clear that the manner of gripping the test piece, length, character and area of cross section all affect the value of the coefficient of elasticity, and should be given in connection with the latter.

These conclusions apply to any other material, as well as to wrought iron.

Table I. gives the results of some experiments made by the Phoenix Iron Co., of Phcenixville, Penn., on some flats and rounds of the dimensions shown in the column headed “ Size.”

TABLE I.

NO. OF BARS

SIZE.

LENGTH.

STRETCH.

/•

E.

12

24

24

23

24

24

Inches.

4 x I«

4 x ifV 32 x if 3Î x ii

3 x ^

3 x f .

2X1

36

68

120

48

72

48

000000

Ft. In. 35 O 27 6 35 o 35 o 35 o 35 o

24 9żl

ii 9 ii ii

ii

ii 9 ii 9

ii

Inches. 0.2692 0.2033 0.2500 0.2617 0.2587 0.2633 o. 1948 0.0953 0.0998 0.0947 0.0955 0.0940 0.1008

Pounds.

20,000.00

Pounds.

31,203,000.00 32,464,700.00

33,600,000. OO

32,098, 32,470,000.00 31,902,000.00 30,544,000.00 29,380,000.00 28,056,000.00 29,567,000.00 29,319,000.00 29,787,000.00

27,777,777-00

000.00

The column “p ” is the intensity per square inch which caused the stretches shown in the column headed “ Stretch.” From Eq. (1) of Art. 2 :

E = t.

In this case, for any individual bar :

^ _ Stretch Length ’

remembering that the stretch and length must be reduced to the same unit.

Let the above formulae be applied to the twenty-four bars 3 X ^ inches X (35 ft. = 420 ins.) long.

20000 X 420

0.2633

= 31,902,000.00 pounds.

The other values are found in precisely the same way. The quantities in the column E are the averages of the number of experiments given in the extreme left hand column. The fact that the results are the averages of a great number of experiments gives the table peculiar value. This table is taken from “ Useful Information for Architects and Engineers,” published by the Phoenix Iron Co. The following reference to the table is taken from the same source : “ The annexed table gives the results attained in testing with the proof load of 20,000 pounds per square inch, a number of bars for the International Bridge over the Niagara River, near Buffalo, N. Y. The recovery of each bar, after the removal of the load, was perfect, no permanent set occurring at less than 25,000 pounds. It will be observed that the stretch per foot of the flat bars is less than that of the rounds, giving them higher moduli of elasticity.” It is interesting and important to observe this last point.

It is to be observed, finally, that these coefficients of elasticity are determined for one intensity of stress, only, i.e., 20,000.00 pounds per square inch. It is probable that values a little different might be given by other intensities.

Table II. contains coefficients of elasticity for tension in

TABLE II.

NUMBER.

h cs co Tf mvo r^co O'

O w CS cO *ovO Nco O w d co tJ" MMWMMMhiWMWCSCSCSC^CSOIOl

SIZE OF BAR.

INCHES.

OOOOOOOONNNMNNOoSoSonMWNNN

xxxxxxxxxxxxxxxxxxxxxxxxxx

cqncqcqcqncqcqmTj-cnN cn to vmn vn m vn >nco co    vO

OOOOOOOOOOOOOOOOOOOOOOOOOO cqcqchcothihchcoininvDnininchcocqcotocoinun/nnmin

GAUGED

LENGTHS.

INCHES.

aocooocococccocooococococococooococococococococococo

STRETCH IN

GAUGED LENGTH.

E IN POUNDS

INCHES.

.029 .0279 .0272 .0238 .0281 .0215 .0273 .0260 .0275 .029 .0278 .026 .0256 .0277 .028 .0284 .0279

OOOOOOOOO

PER SQ. IN.

27,586,240

28.673.760

29.411.760 33,613,440

28.469.760 37,209,300 29,304,000 30,769,200 29,090,880 27,586,240

28.776.960 30,769,200 31,250,000 28,880,860 28,571,430

28.168.960

28.673.760 28,070,160 25,000,000 25,396,800 32,000,000 28,571,430 31,250,000 30,769,200

29.411.760 29,629,600

pounds per square inch, computed from data given in “ Report of Tests of Metals ” for 1881, made on the government testing machine at Watertown, Mass. These results, like those in the preceding table derive additional interest and importance from the fact that they belong to full size bars and such as are ordinarily used in engineering practice.

Those bars whose numbers are preceded by an “ S ” are of single rolled material, while those preceded by a “ D ” are double rolled. A portion of the bars given in Table II. are those on which the subsequent tests shown in Table XIII#. were made.

The values of “ E,” the coefficient of elasticity, were computed by Eq. (1), l representing the stretch in 80 inches divided by 80. The stretch in every case was measured at a stress of 10,000 pounds per square inch, which is the limit quite generally specified as a maximum in railway bridges of ordinary length. It is seen that the values thus determined are not on the whole very different from those shown in table I. for double this intensity of stress.

Bar No. 9 was slightly warped and No. 25 not originally straight, but the coefficients do not seem to be appreciably affected.

A comparison between the results for single and double rolled iron shows that there is no appreciable difference between them either in uniformity or magnitude. In the aggregate, the values run from 25,000,000 to 37,209,300 pounds; giving a variation of fifty per cent, of the lowest amount. This fact has a most important bearing on those theories of continuous girders which assume E to be constant.

Double rolling, which materially increases the cost of the metal, is thus seen to give it no elastic advantage.

Prof. Woodward, in “ The Saint Louis Bridge,” gives the results of 67 experiments on specimens varying from 6 to 18 inches long and from 0.45 inch to 1.13 inches in diameter, from 17 different producers. In these results the range of variation was very great ; in fact the coefficient of tensile elasticity varied from 9,500,000 lbs. per sq. in. to 65,500,000, and some of the widest variations were in specimens of the same brand.

Table III. gives the results of the experiments of Mr. Eaton

TABLE III.

Tensile Experiments on two Annealed “Best” Wrought Iron Bars ten feet long and one inch square.

BAR NO. I.

BAR NO. 2.

LI.

Sets.

E.

LI.

Sets.

E.

2,668

5.335

8,

*°,

003

70

t8

*3*33

16,005

18,673

21,340

24,008

26,676

29,343

32,01¹

Inches.

.00986

.02227

.03407

.04556

.05705

,.06854

.07993

.09193

.10485

.12163

.15458 .26744 .28271 in 5 minutes

Inches.

.OOO3O5 .OOO4O7 .OOO5O9 .OO061O .OOO813 .OOI525 .OO3966 .OO9966 .O3I424

.13566

32.457,

28,198,

28.18

OOO

OOO

0,000 ^,.01,000 28,056,000 28,020,000 ²8,o33,ooo 27,855,000 27,475, 26,308, 22,702 M,36i

20.1

28.1

,000

000

000

,000

1,262

2,524

3,786

5,047

6,309

7.571

8.⁸33

10,095

”,

12,

J3,

357

019

880

34,678

37,346

Repeated

40,013

.5148

1.095

*.*949

i .220

in 5 minutes

.36864 1.01695 I.02966

1.093

8,083,000

4,077,000

3,924,ooo

Repented d left on

i .4*1

after i hours

>5,142

16, (04 17,666 18,928 20,190

21,452

22,713

23,795

25,237

26,499

27,761

1.424 after 2 h

*•433 after 3 h

*.434 after 4 h

1.436

after 5 hours

*.437 after 6 h>

1.443 after 7 h

*.443 after 8 h

1.443

after 9 h

*.443 after 10 hours

- if!

ours

ours

■ 'Mr-,

29,023

30,285

3⁰,²85

30,285

30,285

Repeated

31,546

Repeated

Inches.

.00520

.01150

.01690

.02240

.02772

.03298

.0379°

.04300

.04854

•0537°

.05950

.06480 .06980 .07530 .08170 .08740 .09310 .09920 I .10570 .11250 .i2040 .12880 .14500 .1991 .2007

after 5 min. .2018

after 10 min.

.2054

after 15 min.

.2080

after 20 min.

.2oq6

after 1 hour

.2366

after 17 hours

Inches.

.00050

.00060

.00050

.00045

.00050

.00050

.00070

.00130

.00270

.00410

.00680

4 0120

0736

.0774

.0796

.0814

. 1082

29,125,OCX) 26,347,000 26,851,000 26,990,000 27,312,000 27,551,000 27,953,000 28,198,000 28,077,000 28,199,000 27,984,000 28,041,000 28,186 28,153,000 27,794,000 27,734 27,644,000 27,474,000 27,213,000 26,919,000 26,420,000 872,

86,

,coo

,000

25,

000

000

23,9 18,244,000 18,030,000

42,681

2.148

in 5 minutes

1.983

2,384,000

32,808

Repeated

.242

after 5 min. .2449

after 5 min.

.5506

.1083

.1111

.4141

15,617,000

7,132,000

Repeated

. ²*339

in 5 mini

utes

.7024

after 5 min.

.70966

after 10 min.

.5635

.6558

* TABLE III.—Continued.

BAR NO. 2.

BAR NO. I.

LI.

Sets.

E.

LI.

Sets.

£.

Repeated

45,34⁸

Repeated

«6

48,016

Repeated

it

50,684

Repeated

cc

53,35i

Inches.

2.383

in 10 minutes 2.428

after 46 hours

2.580

after 5 min.

2.605

fter i hour

2.606

after 2 hours

2.606

after 19 hours

2.975 after 5 min.

3.019 after 1 hour

3.029

fter u hours

. 4.19.5

in 10 minutes

4.226

• A²²⁷

in 7 hours

4.227 in 12 hours

Broke

Inches.

2.212

Repeated

Inches.

‘•⁰¹4. after 15 min.

Inches.

.866

2.237

2.377

2.403

2.733

3.941

34,070

1.346

after 1 min.

2,839,000

2,109,000

1,936,000

1,448,000

34,070

34,070

Repeated

((

35,332

Repeated

36,594

37.856

1.400 after 2 min.

1.600

t.65

after i min.

1.786 after i hour

2.04

fter 5 min. 2.18

after 5 min.

2.254

2.54

after 6 min. 2.894

1.44

I.628

1.874

2,078,000

2.01

2.08

1,743,000

1,571,000

Hodgkinson on the tensile elasticity and permanent set of two wrought iron bars. The coefficients of elasticity E have been computed from the data contained in the first three columns as given by Mr. B. B. Stoney in his “ Theory of Strains in Girders and similar Structures.” The following is the notation used :

= pounds per square inch ;

LI

= total elongation, or strain, for the bar ;

“ Sets ” = permanent set ;

= coefficient of tensile elasticity = p x L -f- LI.

These experiments show some very interesting results.

In the first place permanent sets were observed with the low intensities of stress of 8,003 and 3,786 pounds, and it becomes a question whether permanent sets would not have been observed with lower intensities and more delicate apparatus, at least for a short time after the material is subjected to stress.

In both bars the largest value of E is found for the smallest intensity of stress. In bar No. 1, the values of E decrease, with one exception, regularly from the greatest. In bar No. 2, however, greater irregularity is observed ; there are two maxima, one for the intensity 1,262 pounds, and the other for about 12,000 with nearly regular gradations from these values.

Considering the whole range in both bars, E may be considered nearly constant until an intensity of about 24,000 pounds per square inch is reached in each case; it then begins to fall off very rapidly. 24,000 pounds per square inch, then, may be considered about the limit of elasticity for both bars.

It is very important to observe the increase of strain with the lapse of time after the limit of elasticity has been considerably passed.

Values of the coefficient of elasticity, therefore, mean little after that limit is exceeded.

The results of the experiments on bar No. 1 are shown graphically in Fig. 1. The values of “p " are laid off vertically through O to a scale of 20,000 pounds to the inch ; the tensile strains are the horizontal co-ordinates of the curve laid down at full size. The essentially straight portion of the curve between O and a is within what is ordinarily known as the “ elastic limit.”

The equation for this portion of the line is :

P = El;

E being assumed constant if Oa is considered straight.

50000

_l__.1 I

M__LLU

11 rm ' 11—Hi

EE

+FFT

eeeiiee

cJ

40000

TIT

INI

__k

3000(7

TT i i III!

JJ

:=r

20000

'Ll

yöoö

inch

inches

F'ig.1

inches

inches

The point a is at a vertical distance above O indicating about 24,000 pounds per square inch, i.e., about the elastic limit. Above this point the curvature of the line is very sharp, indicating a rapid fall in the value of E and a rapid rise in the values ' of the strains / or LI. For “p ” = 27,000 (nearly) the table shows E — 23,000,000 (nearly) and LI = 0.12 inch ; while for up" = 37,000, E = 4,100,000 and LI — 1.095 inches (nearly). These phenomena are always characteristic of the limit of elasticity.

Above the point b the curvature is slight, indicating (what the table shows) a comparatively slow change in the values of E.

The table shows that bar No. 2 would exhibit a curve of precisely the same character but with a more rapid decrease to E above the elastic limit. The tests of this bar were not carried to failure on account of the breaking of one of the holding details.

Within the elastic limit, the mean values of E may be taken about as follows :•

For bar No. i :

E — 28,000,000 pounds.

For bar No. 2 :

E — 27,500,000 pounds.

The next, Table IV., contains values of the coefficient of tensile elasticity (E) determined by Knut Styffe (“ The Elasticity, Extensibility and Tensile Strength of Iron and Steel,” translated from the Swedish by Christer P. Sandberg).

TABLE IV.

KIND OF IRON.

Hammered Bessemer Iron

Puddled, from Low

“ Dudley

“ Motała, Sweden

Moor

(square)

(round)

PER CENT

AREA OF

OF

CARBON.

SECTION.

SET.

E.

From Surahammar

u    u

Swedish Rolled Iron from Aryd

(square)

0.1

0.15

0.20

0.09

0.09

0.05

0.20

0.14

0.20

Sq. inch.

o.1003 o.1107

0.19

0.18

07

61

0.2 O.I942 O.I229 O.2176 O.I269

O.2087

~ N M ko 00    r-^co    00    ci    r-~.

I § 8 8 8    &■ 8    5    8    ?

1—000' o"    o' o"    o’    0’    o'

Pounds.

32,320,020

34,²4I,3⁸°

31,976,920

28,408,680

,448,000

30,261,420

29075,220

31,084,860

30,467,280

27

u Hallstahammer (square).

0.18

0.07

0.07

0.2279 o.1891 0.1965

0.003

0.013

o.ooi

26,761,800

27,791,000

28,957,64°

30,810,380

The “ Set’ is the permanent elongation which “ the bar had just before the modulus (E) was taken.” In the per cent, of carbon no distinction is here made between “ in the bar tested ” and “ in the bars of the same kind,” the two quantities given by Styffe.

As a result of his experiments in regard to the effect of a change of temperature on the coefficient of tensile elasticity, he states (page 112 of the work above cited) :

“ That the modulus (coefficient) of elasticity in both iron and steel is increased on reduction of temperature and diminished on elevation of temperature ; but that these variations never exceed .05 per cent, for a change of temperature of i.8° Fahr., and therefore such variations, at least for ordinary purposes,, are of no special importance.”

In his “ Physique Mécanique,” page 58 of the “ Premier Mémoire,” M. G. Wertheim gives three coefficients of tensile elasticity for wrought iron, each having about the value of 29,680,000 pounds per square inch, and one for iron wire of about 26,474,000 pounds per square inch.

Redtenbacher (Resultate für den Maschinenbau, Zweite Auflage, page 36) gives as the limits of the values of the coefficient of elasticity, expressed in pounds per sq. in., about 21,330,000 and 35,550,000.

Reviewing the preceding values, therefore, it would appear that the coefficient of tensile elasticity for good wrought iron may be ordinarily taken to lie between 25,000,000 to 30,000,000 pounds per square inch, with extreme values arising from variation of mode of manufacture, chemical constitution, size of bar, etc., lying some distance either side of those limits.

P    J

Since E — E-, if / = 1, / = — will be the elongation or i    E

tensile strain for each unit of stress ; hence, the coefficient of elasticity is the reciprocal of the strain for a unit of stress. For an intensity of stress of 20,000 pounds, for example, then :

to

20,000

25,000,000

20,000

30,000,000

1250

to

1500 ’

or a bar of wrought iron will be stretched :

1250

th to —— th 1500

of its length.

The coefficient of elasticity is thus seen to be a measure of the stiffness of the material.

Ultimate Resistance and Elastic Limit.

It has been found by experiment that bars of wrought iron which are apparently precisely alike, in every respect, except in area of normal section, do not give the same ultimate tensile resistance per square inch. Other things being the same, bars of the smallest cross section give the greatest intensity of ultimate tensile resistance.

Aside from the absence of uniform distribution of stress in the interior of the bar, as was shown in the section “coefficient of elasticity,” and the intensified effects of the processes of production on pieces with comparatively small cross sections, this result is to be expected from the circumstances which attend fracture. When a piece of material is subjected to tension to the point of rupture, not only a tensile strain of essentially uniform character, from end to end, takes place, but also a very considerable local transverse strain, or contraction, at the place of fracture. This latter manifests itself only shortly before rupture as a short “ neck ” in the piece. Now a given percentage of “ local ” contraction in the case of a large section involves a much larger absolute lateral movement of the molecules than in the case of a small section. But it is evident that this absolute lateral movement will exert a much more potent influence toward severing the molecules sufficiently for rupture, than the percentage of contraction. Hence the degree of local and lateral movement, required by rupture, will be reached with a less mean intensity of stress in the cases of large section than in those of small ones. But this is equivalent to a greater intensity of ultimate resistance for the small sections, and, as has been indicated, this conclusion is verified by experiment.

The same considerations result in the additional conclusion that, other things being equal, the smaller sections will give the greater final contraction. But a greater intensity of ultimate resistance with greater final contraction involves a greater final stretch, for the same length of piece.

These last two conclusions will also be found to be hereafter verified by experiment.

Again, it is found independently of the effects of the processes of production, as might be anticipated, that the length in terms of the lateral dimensions of the test piece, within certain limits, affects very perceptibly the ultimate resistance.

If a specimen of the shape shown in Fig. 2 be broken by a tensile stress, it will, of course, fail in the reduced section MN. But before failure takes place, the reduced portion will be con-A    B    cd siderably elongated and the

Fig.2

normal section correspondingly reduced, in consequence of the shearing E strains in the oblique planes shown by the dotted lines. (See Arts. 3 and 4.) When the reduced portion in the vicinity of MN is very short in comparison with its lateral dimensions, it includes the whole of very few of these oblique planes, if any at all, consequently very little movement of these oblique layers over each other can take place ; in other words little or no reduction of section can take place before rupture. In this latter case, then, a greater area of metal section will offer its resistance to the external tensile force, at the instant of failure, than in the former, and a correspondingly greater in-

tensity of ultimate resistance will be found. Thus the shape and dimensions of the test piece will considerably influence the ultimate resistance and strains, as will soon be shown by experimental results.

All the preceding conclusions, though given in connection with wrought iron, are independent of the nature of the material, and apply equally to steel and cast iron.

Since the reduction of area of the fractured section and the elongation of the bar are true measures of the ductility of the iron, these are or should be always measured with care.

Table V. exhibits in a very plain manner the decrease of ultimate tensile resistance with the increase of sectional area of round bars; it is taken from the “Report of the Committees of the U. S. Board appointed to test Iron, Steel and other Metals, etc.,” by Commander L. A. Beardslee, U.S.N.

This decrease is probably partly due to the effect produced upon the iron by the rolls as it passes through them ; the bars of smaller sections being more “ drawn,” and at a lower temperature in consequence of the lesser mass cooling more quickly.

The notation of the table is the following :

“ Dia." = diameter of the round bar in inches ;

“ rr

= ultimate tensile resistance ;

“ E. L.” = elastic-limit.

It will be observed that the ultimate resistance per square inch varies between widely separated limits, in some cases, for the same diameter of bar. This is due to the fact that the different bars, even of the same diameter, were from a number of different mills, and consequently involved different treatment in manufacture, chemical constitution, etc. A general view of the table, however, shows in a marked and satisfactory manner the decrease of T with the increase of the diameter or area of normal section. The last fourteen bars of the table are of the same manufacture, and show a decrease in T as nearly uniform as could be expected.

TABLE V.

Ultimate Resistance and Elastic Limit in Pounds per square inch of Original Normal Section.

Dia.

T.

E. L.

Dia.

T.

E. L.

Dia.

T.

E. L.

59>g85

54>°9°

62,700

59,000

57,7°°

55.400 52,275

55,45° 52,050 57,660 51,546 50,630 61,727 57,3⁶3 57.S07 56,790

51,921

52,819

51.400 60,458 57,470

57,49⁸

55,927

54,644

53,9oo 53,035 52,267 61

40,980

39,126

35,933

33,931

37,415

39,230

36,885

300

32,267

34,6oo

37,344

31,900

4T>3¹¹

37,250

34,6q5

26,787

31,

¹fr

i Va

53,⁰¹⁶

51,296

50,594

57,052

56,505

55,131

54,540

55,415

54,354

54j544

35,379

3i,992

34,94°

38,417

32,496

33,77i

32,869

34,6i7

33,027

53.

52,

512

8iq

59,4

57,8

97

55,782

56,334

55,253

53,895

53,247

53,752

52,970

53,022

50,040

58,926

58,021

54,949

54,277

52,733

5",557

52,537

34,410

32,019

36,501

32,469

35,596

33,921

34,784

32,712

32,520

32,075

30,730

37,548

32,152

31,030

33,622

34,606

33,650

34,469

52,73

52,700

52,155

5i,994

51,456

51,047

56,344

57,402

56,227

54,334

53,339

53,6i4

52,675

52,314

52,401

51,205

50,970

56,595

34,840 34,901 35,880 27,708

32,054

34.591

35,889

35,7oi

33,207

32,163

33,54°

3O.664

33,745

29,364 34,012 33,318 33,625 38,310

54,

114

789

34,160

57,7 57,874

54,410    3i,354

53,846    36,573

018    34,283

55,

53,264

53,i54

51,509

50,395

50,

49,

547

816

50,129

56,577

35,323

29,404

36,254

35,954

31,214

32,271

a    w fororoco't

tli I If 111 11 SIÎ i II IIISÎII mili ilittSlfll

In the words of the Report, as given by Wm. Kent, C.E., in

the abridgment, “ The elastic limit as given is not from perfectly accurate data ; it is simply the amount of stress which produced the first perceptible change of form, divided by the bar’s area.”

TABLE Va.

Rectangular Bars.

PER CENT. OF

STRESS IN LBS. PER SQ. IN.

NO.

KIND OF IRON.

SIZE OF BAR.

Elastic

Limit.

Ultimate.

Final elongat’n Final contrac-in 80 inches.    tion.

Single Refined Double Single Double Single Double Single Double

Inches. 3x1 3x1 5 X 5 x

3X1

3X1

5 x ii 5 x ii

•“ht ’—N*

W M

29,000

31,000

27.330

27.170

28.330

29.170 24,830

27.170

52,470

53,550

50,410

50,920

48,700

5i,37o

49,240

51,0x0

18.0

16.0

16.6 19.0

131

22.2

16.0

19.7

31.0

27.7

24.1

25-7

27.1 35-6

18.1

29-5

Table Va. shows the results of some tests in the U. S. Govt, machine during 1881, at Watertown, Mass. Nos. 1 and 2 are means of four tests ; the others are means of three. Nos. I, 2, 3 and 4 are for bars from the Elmira Iron and Steel Rolling Mill Co.; Nos. 5, 6, 7 and 8 are from the Passaic Rolling Mill Co. As a rule the large bars give the least elastic limit and ultimate resistance.

It is also important to observe that the double refined iron, with two exceptions, gives the highest results of all kinds.

It appears from an examination of the tables that the elastic limit varies, approximately, from a half to two-thirds the ultimate resistance.

The ultimate resistance, it is to be particularly observed, is given in pounds per square inch of original sectional area. On account of the reduction of the fractured section, the ultimate resistance should be specifically referred either to its own section (to be noticed hereafter) or to the original section. The customary reference is to the latter, though it is frequently interesting and important to make an accompanying reference to the former.

The influence of the reduction of the piles between the rolls was next examined by the same committee. It was found that the additional working involved in the increased reduction of the pile, as it passes through the successive rolls, in the process of manufacture, considerably increases both the ultimate resistance and elastic limit. Tables VI. and VII., condensed from those containing the results of the committee’s experiments, show this effect in a very satisfactory manner. The notation is as follows :

D = diameter of bar in inches ;

A — area of normal section of original pile in square inches ;

Per cents. = area of bar in per cent, of area of pile ;

T = ultimate tensile resistance in pounds per square inch of entire bar ;

T' = ultimate tensile resistance in pounds per square inch of core of bar ;

E. L. — elastic limit in pounds per square inch of entire bar ;

E'. L'. = elastic limit in pounds per square inch of core of bar.

As is to be anticipated in such cases, some irregularities are exhibited in the tables, but they are very few, while the general result is unmistakable. On the whole, a considerable increase in the values of T is observed in connection with

to

VO

a decrease in the values of “ Per cents' Values of the elastic limit show greater irregularities.

TABLE VI.

Comparison of the Reductions by the Rolls, with the Effects upon Tenacity, and Elastic Limit of Round Bars.

PER CENTS.

T.

T'.

E. L.

M W H to to CO CO 4^

•H-* w-iW» »4-*    *4-

►Lł-fc4“* 44» M

M tO CO CO CO cocococo MUiO'O'CMOCOOO

►Li-

15-70

12.03

8.83

6.13

4-36

6.68

4.90

3-41

3-14

3.60

CO sO

2.17

1.60

47,344

47,872

50,547

50,820

52.729

51,921

50,673

52,275

57,ooo

46,322

47,000

47,761

47,428

48,280

48,792

51,838

49,801

51,128

50,276

52,775

59*585

29,758

35,864

35,954

35,o87

39,608

39,066

33,933

38,445

Lost

23,430

24,961

26,400

29,941

31,892

38,992

36,467

40,534

38,596

35,933

39,126

Lost

TABLE VII.

Another Table showing Similar Results, with T' and E' L, for

Core, omitted.

D.

A.

PER CENTS.

T.

E. L.

ii

27

iii

15

IS

27

11.63

11.78

10.22

II

15

9.90

14

27

8.90

iS

15

8.18

iS

27

7.68

15

6.62

51,848

53,550

54,034

54,277

55,oi8

56,478

56,344

56,143

COCOCOCOCOCOCOCO to Cn CO 4^ LO CO 4^ ^0

'io CO to 'to    04^

Q\ CO Ox CO 10 M O O ^TOMLOtOOOH

The opinion of the committee on the effect of underkeating

or overheating is thus given in the abridgment of their report by Wm. Kent, M.E. : “ The indications are that if a bar is underheated it will have an unduly high tenacity and elastic limit, and that if overheated the reverse will be the case.”

In the words of the report : “ The evidence submitted is of sufficient value to justify us in asserting that variations in the amount of reduction by the rolls of different bars from the same material produce fully as much difference in their physical characteristics as is produced by differences in their chemical constitution.”

The committee also made some valuable experimental investigations with the object of ascertaining the influence of the

'/

Fig. 3

Fig. 4

relative dimensions of the test piece, already remarked upon in connection with Fig. 2. Eighteen specimens were prepared, of which Figs. 3 and 4 represent types.

Fig. 3 represents a specimen whose middle portion is turned down to a uniform diameter. Seventeen of the specimens were of this kind, with lengths of cylindrical portions varying from y₂ inch to 10 inches. Fig. 4 represents the eighteenth specimen with simply a groove in the centre, in which, at ab, the fracture took place. In this latter specimen, the reduction of area at the section of failure must necessarily be much less than in those like Fig. 3 ; hence, the ultimate resistance will be correspondingly greater.

Table VIII. is taken from the report already cited, and contains the results of the experiments on the eighteen specimens prepared in the manner indicated above.

L — original length in inches ;

I — per cent, of elongation ;

a = per cent, of contraction of fractured area ;

t — stress in pounds per square inch at first stretch ;

T = ultimate tensile resistance in pounds per square inch of original section.

TABLE VIII.

NO.

Z.

1.

a.

t.

T.

REMARKS.

IO

9ł

8ł

7ł

10

ii

12

«-tN    —

in co

23.1

24- 3

21.5

22.0

25.0

25.8

22.1 22.3

25- 4

21.2

38.2

36-5

31 I 31.2

39-9

38.6

40.0

34-7

39-3

25-7

32.2

37-4

26.7

36.6

13

14

15

16

Ił

27.0

27.0

26.0

38.3

36.2

34-0

17

!■

18

Groove

37-0

30.0

34-3

37-9

20.6

29,678

28,011

29.345

29.345 30,840

30,412

28,562

30,600

29,475

29,278

29.705

31,817

31,123

33,428

42,249

34,288

57,565

45,442

54,888

55,288

55,355

55,622

54,890

55,488

51,800

55,4i8

55,333

55,887

55,532

55,482

56,190

56,428

57,096

58,933

59,388

71,300

Slight seam.

Slight seam.

Bad seam.

Slight seam.

Seamy.

Seamy.

The diameters at the section of failure were nearly uniform and originally about 0.97 inch.

The values of /, a and T are as nearly uniform as could be expected until the length decreases to about 4 diameters (2 inches).

For the grooved specimen t and T are very large, and a very small.

Other experiments on a still softer iron were made with the same general results.

“ In conclusion,” states the committee, “ our results lead us to the decision that, in testing iron, no test piece should be less than one half inch in diameter, as inaccuracy is more probable with a small than with a large piece, and the errors are more increased by reduction to the square inch ; that the length should not be less than four times the diameter in any case ; and that, with soft, ductile metal, five or six diameters would be preferable.”

In Vol. II. of the “ Transactions of the American Society of Civil Engineers,” Mr. C. B. Richards has given a paper in which are recorded the results of some experiments exhibiting the influence of the relative dimensions of the specimens. The average of eight tests of Burden’s “ best ” iron, with “ long” specimens (similar to Fig. 3) varying from 5 to 5^ inches in length and 0.62 to 1.00 inch in diameter, gave :

T — 49,588 pounds ;

a = 46.7 per cent. ;

l — 30.4 per cent.

With “ short ” specimens (like Fig. 4) of the same iron, the average of six tests gave :

T — 62,089 pounds ;

a ~ 29.5 per cent.

The large value of T and small value of «, for the “ short ” specimens, are thus seen to be very marked in contrast with the same quantities for the “long” specimens.

Other experiments of Mr. Richards, showing the same results, will be given in connection with the resistance of boiler plates.

It has long been the impression that there exists a considerable difference between the ultimate tensile resistance of the

“ skin ” of a bar of iron and that of the portion of the bar underneath the skin. The U. S. committees, therefore, broke a number of bars first with the skin on, or “ in the rough,” and then with the skin turned off. In a large majority of the cases, the rough bars gave the highest ultimate resistance per square inch, by a small amount, while in a few cases the results were of the opposite character. On the whole, however, “ the accumulated evidence indicates that the strength of the skin of the bar is greater in proportion to its area than that of the rest of the bar.”

All the tests, of which the results have hitherto been given, were made on round bars, or on specimens turned from them. Results of tests on other iron will now be detailed, and it will be convenient to use the following and customary symbols for the various kinds of “ shape ” irons :

L , for angle irons :

X , for tee irons ;

E , for channel bars ;

I , for eye beams ;

Ö » for rectangular bars or “ flats ” ; © , for rounds ;

-+ , for star sections ;

in short, any shape iron, or steel, is represented by a skeleton of its section.

Table IX. contains the results of tests of a wide range of full sized eye bars as ordinarily manufactured for bridge building purposes. Some were made and tested in 1887.

TABLE IX.

SECTION.

LENGTH.

Inches.

Inches.

E. L.

T.

CONT.

STRAIN.

REMARKS.

'4- •'T

CO

tcf-

M W M

xxxxxxxxxxx

237

66

«0 -4—< —

to to H

230

173

251

251

370

304

304

300

300

27.500 25,318

32,590

30,360

33,000

28,010

26.500 26.750 26,000 24,200 24,030

51,000

48,070

51,870

49,040

50,350

47,440

51,200

49,000

4S,ooo

41,280

41,860

0.25 0.24 0.365 0.326 0.336 o. 17 o. 16 0 211

0.097

0.099

O.I43 o. 117 0.216 0.208 0.208 0-175

o. II o. 129 0.124 0.061 0.056

“ Strain

_    0*

lOtOlOtOtOHHHH    M

W CJ Ci C3 VÛ CO COLO C'4^ •‘"4

LO CO

E. L. — elastic limit in pounds per sq. in.

T. = ultimate resistance in pounds per sq. in. of original section.

Cont. — ultimate contraction or reduction of original area.

Strain = ultimate stretch in length given under “ Remarks.”

All bars except e and f were rolled and manufactured by the Phoenix Iron Co. The bars a were of single rolled iron, while all the others were of double rolled material. The bars e and /"were made by the Central Bridge Co. of Buffalo, N.Y., and were tested at the U. S. Arsenal, Watertown, Mass , as sample bars selected from those used in the Niagara Cantilever bridge ; and the results are taken from the paper on that structure by Mr. C. C. Schneider, Ch. Engr., presented to the Am. Soc. of C. E. March 4th, 1885.

Bars a were tested at the works of the Phoenix Iron Co.; bars b at the works of the Keystone Bridge Co., and bars c and d at the works of the Union Bridge Co., at Athens, Pa.

The bar d broke through the eye, for the reason that about ■§ inch in thickness of material was planed off the face of the head in order to get it into the testing machine.

Considered as a comparison between single and double rolled bars, the table possesses both interest and importance. The first three of the bars b show unusually high elastic limits, which may have been the result of the approximate methods used in the observation. On the whole, however, the elastic limits of the single rolled bars are not essentially different from those determined for the double rolled material.

With the exception of the last two ultimate resistances, which are very low, even for large bars, that column shows much more nearly uniform results for the two grades of bars than found among the elastic limits, and the values for the single rolled metal are fully equal to the best of the double rolled. As was to be expected, the smaller bars gave results appreciably in excess of those belonging to the larger ones.

The percentages of contraction for the single rolled bars are seen to be on the whole somewhat smaller than those belonging to the others, although the advantage is not maintained by the latter throughout the entire Table^ The last preceding observation holds, but less markedly, in the column of ultimate stretch or “ strains.” While the decision of such questions should be made only on a far greater number of tests than given in the table, it is proper to say that the latter shows precisely what is found in extended experience, i.e., that double rolled iron, as produced by the most reputable iron companies, possesses only a possible small and unimportant advantage in ductility and uniformity, but with less welding properties. Its cost is from twenty to twenty-five per cent, over that of single rolled iron ; which is out of all proportion to the very small advantage gained.

Two kinds of tests are usually required by engineers in determining the suitability for a given purpose, of the finished member and the material of which it is fabricated. Those tests which determine the character of the finished tension member {i.e., eye bar) have been exemplified by Table IX.; such tests fix the character of the finished member by showing the effect of the mode of manufacture. The other class of tests is made on specimens cut from the bar as it comes from the rolling mill and before it is manufactured into the bridge member; such tests simply fix the quality of the material.

Table X. shows the results of specimen tests from the full size bars given in the first column on the left. The material of these specimens was double rolled iron produced by The

TABLE X.

Double Rolled Iron.

ORIGINAL

TEST SPECIMEN.

LBS. PER SQ. IN. ORIG. SECTION. | PER CENTS OF FINAL

BAR.

INCHES.

Section.

Inches.

Ä Elastic Limit.

Ultimate

Resistance.

Contrac- Stretch in tion.    8 Ins.

5 x 5 x if 5 x if

5 x i-fg-5 x i|

5 x if 5 x i|

5 x 1,% 5 x i|

5 x i| 3! x iyV 3ł x iïV

I x 1.7 I x 1.63 IX 1.63 I X 1.5 I X 1-53

I X I .

I X I.

I X I .

I X I.14 IX I . 14 I X 1.2 I X 1.2

29,150 25,050 25,720 27,220 26,920 26,

990

27,030 26,

, 100 26,850 26,860 28,140 27,180

50,090

49.350

48,760

50,550

50.620 50,470 51,200 50,420 50,000 50,220 50,630

49.620

34-3

36.1

33-8

39-1

35-3

37-3

37-8

36.5

36.2

34-8

37-7

26.8

tO CO CO to CO CO CO to CO CO CO CO

CO o M o OOO co-p* M to to CO O CO CO CO o <^co CO coca CO

Phœnix Iron Company, and the tests were made at the works of that company in the latter part of 1886. The sizes of the specimens and the lengths for which the stretches were measured are seen at a glance at the Table. In preparing such test pieces care is always taken to have opposite sides of the test piece in exactly the same condition. It will be observed, for instance, that one dimension of each of the pieces is just equal to the original thickness of the bar from which the specimen was cut. Hence two opposite sides of each specimen had the rough surfaces of the original bar, while the other two sides were the machine-finished surfaces along which the cutting was done. The two machine-finished surfaces, moreover, should be smoothly cut. If all these precautions are not taken, the specimens may resist unequally on opposite sides, and fail in detail by pulling apart gradually from one surface ; or by rough cutting, the material in the vicinity of the machine surfaces may be so injured as to lose a large portion of its resistance.

The variety in the sizes of the bars shown in Table X. is not nearly so great as that given in Table IX., but for the same sizes the elastic limits as a whole run a little lower for the specimens than for the full size bars. While the ultimate resistances in the two tables are not very different, the general

TABLE XI.

Double Rolled Iron.

DIAMETER

TEST SPECIMEN.

LBS. PEE SQ. IN. ORIG. SECTION.

PER CENT. OF FINAL

ORIGINAL

BAR.

INCHES.

Diameter.

Inches.

Length. N Inches.

Elastic Limit.

Ultimate

Resistance.

Contraction.

Stretch in 8 Inches.

I«

il

if

ił

1.25

1.25

1.65

il

1-5

¹ • 51 i.41 i.41 14 i 1.26 1.26 1.13 x.13

if

if

ił

il

ił

26,820

26.740 30,300 28,858 30,119

29.740 26,800 28,760 26,390 31,880 27,200

50,480

5⁰.5So

51.340

5I,°40

52,429

50,546

48,527

50,130

5L340

5L400

49,900

45-7

46.1

31-5

45-5

35-3

32-3

33-3

21.0

39-5

43-8

32.8

32-7

44-3

43-4

48.7

30.2

31-7

25-5

26.7

30.0

28.7

312

result is a little higher for the specimens than for the bars, and more nearly uniform. The contractions and stretches for the specimens in Table X. are far larger as well as more nearly uniform in character than those given for the full size bars in Table IX.

No class of materials used by engineers possesses more widely varying characteristics of a physical nature than plates used in bridge construction. The very wide plates forming the webs of large plate girders give a high elastic limit, comparatively low ultimate resistance, final stretch and contraction ; these are always sheared plates. Narrow plates, either rolled in grooves or universal mill, approximate more nearly in character to bars in all respects of elastic and ultimate resistances and final stretch and elongation.

TABLE XII.

Bridge Plate Specimens.

ORIGINAL

TEST SPECIMEN.

LBS. PER SQ. IN. ORIG. SECTION.

PER CENT. OF FINAL

PLATE.

INCHES.

Section.

Inches.

Length.

Inches.

Elastic Limit.

Ultimate

Resistance.

Contrac- Stretch in tion. 8 in.

10 x

10 X 10 X 10 X 10 X IO X 10 X 12 X 12 X 14 X 14 X 20 X 20 X 24 X 24 X 27 X 30 X 30 X 69 X 72 X 72 X

(Xfw »(wajMML.-L,    a(Moc|MccfDi    *+"

i x 0.30 I X 0.3 I x 0.26 I x 0.26 I X o 32 I x 0.45 I x 0.45 I x 0.50 I X 0.57 I x 0.40 I x 0.63 I X 0 39 I X 0.39 I X 0.45 I x 0.45 I x 0.31 I x 0.31 I X 0.39 I x o.32 I x 0.31 I x 0.40 I x 0.3g I X 0.35

31,900

36,000

35,110

34,600

37,ooo

32,810

32,960

27,160

27,400

30,230

34,700 33,850 34,870 36,860 36,580 30,290 28,

100

39,470

33,330

34,290

35,8oo

44,38o

44,030

47.700 52,800

52.670

51,540

53.700

51.670 52,250 49,500 49,220 50,380 51,200

50,390

54,10°

53,930

53,720

46,900

46,920

52,370

49,210

49,680

49,870

53,060

55,960

22.4

23-4

20.6

ig.8

17-5

16.3

21.9

27.2 30.8

28.3

16.8

25.96

29 67 20.91 22.5 28.9

44.1

18.9

25.6

18.6

18.1

28.2

19.i 26.3

21.8 21.0 23.0 15-0 18 2

19.8 23.0

133

20.8

II.o

15.0

18.0

15-2 15 •¹

25.6

14-5

23.0

Table XII. gives results for bridge plates throughout a great range of width. All the specimens were of the same thickness as the original plates ; hence two sides were as they came from the rolls and two were machine finished. Although these re-

TABLE XIII.

Angle Iron Specimens.

SIZE OF

TEST SPECIMEN.

LBS. PEE SQ. IN. ORIG. SEC. PER CENT. OF FINAL

ORIGINAL ANGLE.

Ultimate Contrac- Stretch in Resistance. tion. 8 Inches.

INCHES.

Section.

Inches.

Length.

Inches.

Elastic

Limit.

Pounds.

6x4 —7i 6 x 4 —46 6x4 —46 6 x 3ł—36 6 x 3^—36 5 x —25 5 x —25 4 x —34 4 x —34 4 x —34 4 x —34 3ł x —23 3ł x —23 3 x —17 3 x —21 3 x —21 3 x —28 3 x —18 3 x —18 3 x 2J—17 3 x 2j—17 3 x 2\—18 3 X 2j—18

I x 0.7 i x 0.46 I X 0.46 I X 0.38 I X 0.38 I X 0.34 I x 0.34 I x 0.49 I X 0.47 I X 0.49 I x 0.49 I X 0.38 I x 0.38 I x 0.30 I x 0.38 I x 0.37 I X 0.46 I X 0.34 I x 0.33 I x 0.33 i X 0.33 I X 0.35 I X 0.35

CO CO CO CO CO CO CO CO CO co oo co CO CD CO CO CO OO CO CO CO CO co

26,800

28,570

28,850

29,890

30,520

29,200

29,370 28,250 29,780 34,440 31,680 26,960 28,600 30,660 30,260 29,030 31,080 32,220 33,230 33,030 32,320 29,760 30,430

49.420 49,460 49,340 47,090

48.420

46.900 47,770 48,860

49.360

52.900 50,620

45,790

49,080

51,000

51,050

48,920

51.300

50.300 52,430 51,350

51.360 50,280

50, x4°

25.6

24.6

27-3

x9-5

18.2

24-5

20.0

21.3

ii .8

22.7

20.7

26.8

13.0

14.4

15-6

22.8

25.1

18.5

31-9

26.8

28.1

34-3

30.8

29.0

35-4

30.1

36.3

27.6

34-7

32.1

29-5

17.8

18.8

21.3 13-75 23.00

22.3 23.1 26.0 27-3

20.8

22.8 23.0 23-3

21.8

x9-3

suits are somewhat irregular, the lowest ultimate resistances and relatively highest elastic limits are found with the greatest widths, with the exception of the 72 inch plates. It is altogether probable that both the high elastic limit and ultimate resistance for those were produced by the addition of steel to the piles from which the plates were rolled. This practice has of late obtained some footing in order to meet the extreme requirements of some very exacting specifications based upon insufficient knowledge regarding the actual capacities of plate iron in great widths.

Table XIII. gives the results of tests of specimens cut from all sizes of angles used in ordinary bridge work. These angles were all produced by the Phoenix Iron Company, and the tests were made at the works of that company in 1887. The results are most excellent, as well as being typical for the shapes tested. Although the results at one or two points are a little irregular, on the whole the elastic and ultimate resistances, as well as the final contraction and elongation, increase with considerable uniformity from the heaviest sections to the lightest, showing clearly the improved qualities in the smaller angle bars. The Table demonstrates in a very marked manner the varying characteristics which always accompany varying dimensions of bars of the same kind, even when produced of absolutely uniform material in the original piles.

Table XIV. exhibits some very interesting results obtained by testing full size bars to destruction, then allowing the portions to rest during the periods given, and finally retesting those portions. Some of the latter were heated previous to testing and allowed to cool in the air.

Those bars whose numbers are preceded by D were of double rolled iron and the others of single rolled material.

The tests were made in the government machine at Water-town, Mass., and are reported at page 205 of “ Ex. Doc. No. 1, 47th Congress, 2d Session.”

This subject of the effect of repeated stress separated by intervals of rest will again receive attention in a later section of this article ; it is sufficient here to observe the influence on the elastic limit and ultimate resistance of these full size bars. The intervals of rest after the first test varied from four to ten months, and in all these instances the elastic limit was raised to about nine-tenths the ultimate resistance found subsequently by the same test, and in every instance it had a value higher than the ultimate resistance found in the original test. The original ultimate resistance is, also, but about five-sixths of that found after a period of rest. It is a singular fact that the general effect of the different periods of rest is to reduce the final contraction about one-third of its original value, but to increase the final stretch from about fifteen to about forty per

TABLE XIV.

NO.

CHARACTER OF TEST.

SIZE OF BAR GAUGED

WHEN TESTED. l’gTH.

INCHES.

INCHES.

POUNDS PER SQ. IN. ORIGINAL SECTION.

PER CENT. OF FINAL

Elastic Ultimate Limit. Resist.

Cont.

Stretch

in

L’gth.

S 3 i

s 4 -i

D 6 -j

Original........

Rested 4 months Original Rested Original .

Rested 5

Original    ........r

Rested 8 months........

Heated cherry red and

cooled in air..........

Original................

Rested 10 months........

Heated dull red and cool-

ed in air...............

Original ................

Rested io months........

Heated to 370⁰ Fahr, and

cooled in air...... ....

Rested 15 days after 3d

test....................

Original ................

Rested 9 months.........

Rested 9 months.........

months

months

3.05 X 1.0

3.05 x 1,0

3.05 X 1*0 5.05 X 1.28

4.75 X 1.17 3.03 X 1.01 2.84 x 0.94

2.80 x 0.92 3 x I.01

7 x 0.95

2.83 x 0.92

3-0

2.8

3.03 x i.01 2.84 x 0.94 2.84 x 0.92

Cn 00 to cn go g go tg go go go g>

29.500    51,15° 60,000 51,110 61,960

50.390

60.390

27.500    5°,5°o

56.500    58,980

J    36,220    1

I    4¹,640    f

53,630

63,^o

28,500

53,440

28,000

21,670

29,500

53,920

22,800

29,000

53,920

41,240

53,56o

63,070

57,520    57,520

62,420 32,500    53,5oo

55,880    63,360

66,010

to HOO Hoo N O

H VO fO in w’ Ń CO fOCOfOO « H

H M to co h w oj to co

'O'O'O    -£*•    ^OS4*. M On

ó ö Cn b\ OsO O vb O

21.00 24.00 21.00 25-00 14-00 20.00 13 oo 18.00

33°°

ij5.o

23 o

32.0

15-0

19.0

19.0

SS&

b b b

cent, of the original. The qualitative effects are the same for both single and double rolled iron, but the amounts of change in the final contractions and elongations are markedly greater, as a rule, for the double rolled material. It must not be supposed that these increased resistances indicate improved material, for other investigations show that it has been fatigued, and has less capacity to resist a repetition of large stresses than before.

In all the preceding tables the length for which the “ Strain " is given should be carefully borne in mind. A considerable “ local ” strain takes place at the section of fracture, which causes the per cent, of elongation, or strain, to be much greater for a very short length than for a longer one.

Wrought-Iron Boiler Plate.

A committee of the Franklin Institute made a very extensive series of tests of boiler plate, and reported the results of their investigations to that body in 1837. The Report of that committee can be found complete in the “ Franklin Institute Journal ” for that year. The sectional area of the test specimens varied from 0.10 to 0.20 square inch. This fact, coupled with the hardness of much of the iron which they tested, rendered many of their ultimate resistances very high and extremely irregular.

These considerations deprive the results which they give in the many “ Tables ” of their Report of the greater part of their value for present practical purposes. They are therefore not given.

This committee made numerous experiments to determine the resistance of boiler plate in different directions in reference to the fibre of the iron. The results were by no means of a uniform character. In one set of forty strips cut in each direction (along the fibre and across it), the length strips showed an excess of resistance varying from one per cent, to twenty. This comparison was made principally on the minimum resistance of each bar, but the committee state that the result would not have been much different if the mean had been taken.

On reviewing all their experiments, the committee concluded that lengthwise of the fibre, the boiler iron which they tested was about six per cent, stronger than across the fibre.

BRAND.

cocficficococficficrj

M «WWW PQ « pq

c/5 H H

pq

ë ë ë ë ë

plifxîpiJfilpii^WWMWcficdcd

STRAIN.

0.19 O. 12

CONT.

N -t lO O lO M lO O' M C^HMmC^MMOVO

ddddddddd

T.

tNTt-cnMco moh cn r^o i^hooo m O'co OO od coh h co n-O O ^ M o ci o a rf co oo O O CI W 0^0 en m

Oco on o m Is en o co tN ej o O' en t O h o too h en o tŃtsH cf cd o' N 4o o* en cf cf 4 4 cô cô h en en cf cf

-t^lOlO'tTf'trrrN'eJ’^rt'tlOtOtOlOlOUttOU^lO XOO

TABLE XlVa.

Ti.

*t u^ O O coOmOOwomOOiO, rJ-O O'ino W co h in ocn o co o O w o co en    o    co m    •+ n    o    in

r^*co r> »o o od o 0s»-« Tft-i trt n* co    en    I    ho    i-cnenioM

o' o' o' h' rf cd o' of cf o' en rf en CO    en    I    t-T ccf    cf -f    en    cf cf

't’tiovoTtcn,t^tl>Tt't^tT}-toio    in Tt    uno    m    to to

to co co h co cooo OminOH O co    O O *ot rt-co h tJ-

T'.

co h O'O h-ci h »tcO'tinO'COM h    m 't co O' O' O'

r^wO'McnO^oociMCicnoon-ci od tj-h O'C'Nco d m cô w cf co in h i-N 4 4 ,td> to ! -fed' en rd en en d rtTttoioiO't^-rjTN^-^-iotoioto ui to to to io tn to

O'O'O'O'O O O O O N r^o o cc od co co co co oo co co co cn cjCJaacncncncncnc*ciMiHCjc*cjc<ciac*<NCłc*cn

Inches.

dddddddddddddddddddddddo

XXXXXXXXXXXXXXXXXXXXXXXX

voir>ir)\r>voir>iovnmr^r^.t^.i^^r^r^r^r^JT^i^r^r-«i>>xr)

M 01 CM Cl w <N wcococooooocooococococococooo a

HdwdHHHHModdodddddddddoH

HUjo^WUJjUi-lUJUJUJUhlUhlUiJ

nlHlc/}c/}h-]Hi-]i-4^^^(/]c/)c/}c/}a)c/}c/2c/}c/](/2(/}c/3c/}

O cno en Cl    Tt « W CO COCOM M a O »t ^ M w Cl H

HM    H

They also determined that the weakest direction of all was diagonally across the fibres, but their experiments did not enable them to determine quantitative results.

Table XIV«. is taken from the “ Transactions of the American Society of Civil Engineers,” Vol. II. It contains the results of some experiments on several different kinds of plate iron by C. B. Richards, M.E., and among other things it reveals the difference between “ long ” and “ short ” specimens.

Column “No.’ shows the number of tests, of which T is the average ultimate tensile resistance in pounds per square inch, T' the highest and Tt the lowest, all being referred to the original section.

Column “ Cont." shows per cent, (of original section) of contraction at section of failure.

Column “ Strain " shows per cent, (of original length) of elongation.

Column “ Spec." shows kind of specimen, i.e., “ long ” or “ short,” also direction of stress in reference to fibre ; “ LL" signifies “long and along fibre;” “LC" “long and across fibre;” while “.SX” and “SC" signify “ short and along ” or “ across fibre,” respectively.

In column “Brand'' “B. S." signifies “Bay State;”

B. S. H." “ Bay State Homogeneous Metal ; ” T." “Thorneycroft,” English; “Penn." “Pennsylvania ; ” “ S. F." “ Sligo Fire Box.’

Different brands of the same make, though given by Mr. Richards, have been neglected.

The lengths for which the “ Strains " existed are not given, although they should be. The long specimens were three or four inches between the shoulders.

In his “ Treatise on the Resistance of Materials,” Prof. De

Volson Wood gives the following results of some boiler-plate tests at the shops of the Camden and Amboy R. R. by Mr. F. B. Stevens.

“ ....57,012.00 “ ....51,813.00”

“ Av. breaking weight in pounds per square inch... . 54,123.00 Highest “

Lowest “

The experiments of Sir Wm. Fairbairn on English boiler plate (“ Useful Information for Engineers, First Series,” p. 259) along and across the fibres, gave irregular results, but other English experiments of Easton and Anderson would seem to make the resistance across the fibres from 5 to, 15 per cent, less than that along the fibres.

Effect of Annealing.

The Franklin Institute Committee determined the effect of annealing, at different temperatures, on about 56 specimens of boiler plate and wire iron. Table XV. is condensed from that giving their results on boiler plate.

The mean value of T for five specimens of iron wire 0.19 inch in diameter, before annealing, was :

T - 73,880.

After annealing by heating to redness and cooling in dry ashes, the mean of five specimens was :

T' = 58,101.

After annealing at red heat and quenching in water, the mean of another five specimens was :

“ 53>578.

TABLE XV.

DECREASE

NO.

T.

ANNEALING TEMP. FAHR.

T'.

BY ANNEALING.

57,133

CO

53,774

So cn to vO O'

53,185

52,040

48,407

1.037

i,iii°

i,i59°

i,237°

Bright welding heat.

56,678

52,186

46,212

44,i65

39,333

.025

.02g

•131

•151

.187

48,407

76,986

38,676

50,074

.201

•349

T is the ultimate tensile resistance in pounds per square inch at ordinary temperatures, before annealing.

T' is the same after annealing and cooling.

The means of sets of five, three and four specimens of wire 0.156 inch in diameter, exactly similarly treated, were, respectively :

T — 89,162.

T = 48,144-

27 = 50,889.

The process of annealing is thus seen to decrease the ultimate tensile resistance a very considerable amount. In many cases, however, this may make the iron very much more valuable, since annealing renders it much more ductile. If a structure or machine is subject to shocks or sudden applications of loading, a very stiff, hard iron, originally utterly unfit for the purpose, after being annealed might be used in its construction with safety.

Effect of Hardening on the Tensile Resistance of Iron and

Steel.

It has been seen that annealing reduces the ultimate resistance of wrought iron. Experiments have shown that hardening, on the other hand, increases the resistance of both iron and steel, provided the hardening is done in a proper manner. If the hardening is accomplished by heating and sudden cooling in water, without subsequent tempering, the resistance of hard steel is very much diminished. This is probably due to the internal stresses induced by the sudden cooling.

Knut Styffe (“ Iron and Steel ”) concluded from his experiments that “ by heating and sudden cooling (hardening), the limit of elasticity is raised while the extensibility is diminished, not only in steel but also in iron.” This results of the experiments by David Kirkaldy will be given hereafter.

Variation of Tensile Resistance with Increase of Temperature.

Table XVI. has again been condensed from a similar one given in the Report of the Franklin Institute Committee.

The third column gives the temperatures at which the ultimate tensile resistances in the fourth column were observed.

The committee observed that the resistance of many irons increased with the temperature, to nearly the boiling point of mercury in some cases, while others remained unchanged until a temperature of 572⁰ was reached. Above this point, however, as a rule, they found the decrease of resistance, below the greatest, to vary about as the 2.6 power of {Temp. — 8o°).

TABLE XVI.

TEMP. FAHR.

ULTIMATE TENSILE RESISTANCE

AT ORDINARY TEMPERATURE.

ULTIMATE TENSILE RESISTANCE

AT OBSERVED TEMPERATURE.

O en M

Cntnav^4--LWbJ tO tO tO m m O C'CO O vn hmWO bi h M U

oicnuicnoicnoi^Lnvj CX-rx ua O' Ln o oco too to o

M 4- M to to Ui ui «g tO O M L3 cn Vj Ln to Cbw h L W m w vj to cc-4^ co-L Co Co O' COO O to C M 4^ h ^ cn to to O' O'

212

394

440

552

562

572

574

576

578

630

732

819

1,022

1,142

1,187

i,3i7

67.939 67,765 59,085

55.939 59,623 66,620 65,387 66,065 53,465 6o,oio 53,378 55,892 37.410 18,672 21,910

18,913

In the London “ Engineering of 30th July, 1880, is given a synopsis of some German experiments by Herr Kollmann, which is reproduced in Table XVII. The resistance of the materials at o° Cent., or 32° Fahr., is taken as 100, and that at other temperatures as the proper proportional part of that number.

It will be noticed that these German experiments show a much earlier decrease of resistance than those of the Franklin Institute.

The results of some tests of a grade of charcoal boiler plate at three different temperatures are given in “ Effect of low and high temperatures on steel."    „

Some French experiments by M. Baudrimont are given in the “ Journal of the Franklin Institute ” for 1850, by which he found that at the temperatures 32⁰, 212° and 392⁰ Fahr., iron

TABLE XVII.

TEMPERATURE.

FIBROUS IRON.

FINE GRAINED IRON.

BESSEMER STEEL.

Cent.

Fahr.

o°

32°

IOO

IOO

IOO

IOO

212

IOO

IOO

IOO

200

392

95

IOO

IOO

300

572

90

97

94

500

932

38

44

34

700

1,292

16

23

18

900

1,000

1,652

1,832

12

wire gave the following tensile resistances, in pounds per square inch, respectively :

291,510.00;    271,602.00;    298,620.00;

These resistances are most extraordinarily high, but, so far as the influence of variation of temperature is concerned, show nothing discordant with the preceding results.

The same experimenter found the tensile resistances of gold, platinum, copper, silver and palladium to decrease, in every instance, as the temperature increased from 32⁰ to 3920 Fahr.

In his “ Useful Information for Engineers,” Second Series, Sir Wm. Fairbairn gives the results of numerous experiments made on “ short ” specimens of plate and rivet iron at different temperatures.

TABLE XVIII.

BREAKING WEIGHT IN POUNDS PER

STRESS IN REFERENCE TO FIBRE.

TEMP., FAHR.

SQUARE INCH.

o°

60

60

60

no

112

120

212

212

212

270

340

340

395

Scarcely red. Dull red.

49,009

40.357

43,406 50.219 44,160 42,088 40,625 39-935 45,68o 49.500 44,020 49,968 42,088 46,086 38,032 30,513

With.

Across.

Across.

With.

Across.

With.

With.

With.

Across.

With.

With.

With.

Across.

With.

Across.

Across.

In Table XVIII. will be found the results of his experiments on plate iron. On the whole, the table would seem to show a point of greatest resistance at about 270° to 300°, though so many irregularities exist that little or no law can be observed. In other words little or no decrease takes place at 395⁰ or below. Much diminution, however, is seen at “ scarcely red ” and more at “ dull red.”

Table XIX. shows the results of Fairbairn’s experiments on rivet iron at different temperatures. The irregularities are less than those seen in Table XVIII., and a maximum would seem to exist at about 325⁰.

The areas of the normal sections of the plate specimens varied from 0.6 to 0.8 square inch, while the sectional areas of the rivet-iron specimens were about 0.2 or 0.25 square inch.

Other results for wrought iron will be found in Table IX. of Art. 35.

TABLE XIX

TEMPERATURE,

BREAKING WEIGHT IN POUNDS

TEMPERATURE, BREAKING WEIGHT IN POUNDS

FAHR.

PER SQUARE INCH.

FAHR.

PER SQUARE INCH.

- 30° -j- 60

66

1x4

212

63,239

61,971

63,661

70,845

82,676

25O

27O

310

325

415

82,174

83,098

80,570

87,522

81,830

212

74.153

435

212

80,985

Red heat.

86,056

36,076

All the preceding results, while irregular to some extent, show conclusively that no essential decrease in the tensile resistance of wrought iron takes place below about 500° Fahr., while a possible increase at that temperature may exist over that at any below, but that at about i,ooo° it may lose more than a half of its resistance. These conclusions are of the greatest importance in the construction of boilers.

Effect of Low Temperatures on Wrought Iron.

It is a matter of common observation that many articles, large and small, are much more easily broken in very cold weather than at higher temperatures. These breakages are undoubtedly frequently due to the circumstances in which the piece broken is found at the time of failure, either partly or wholly.

The frozen, and consequently less yielding, condition of the ground in the winter is unquestionably a very potent factor in failures ot tires and axles of railway rolling stock, but it is at least an open question whether it is the sole cause.

A number of investigators have made numerous experiments with the object of determining the effect of low temperatures on the resistance of wrought iron in different forms.

From the results of these experiments, however, they have drawn the most discordant conclusions. In some cases this arises from the fact that the tests have not been made under the same circumstances, or have not been of the same kind.

Knut Styffe (“ Iron and Steel ”) made the following “ Résumé of Results of Experiments on Tension at different Temperatures : ”

I. “ That the absolute strength of iron and steel is not diminished by cold, but that even at the lowest temperature which ever occurs in Sweden it is at least as great as at the ordinary temperature (about 6o° Fahr.). . . .

3. “ That neither in steel nor in iron is the extensibility less in severe cold than at the ordinary temperature ; .    .    .

.4. “ That the limit of elasticity in both steel and iron lies higher in severe cold ;    .    .

He concluded from his experiments that the common impression of increased weakness and brittleness with a low degree of temperature is entirely erroneous. His tests, however, were wholly with tension gradually applied, and could support no conclusion in regard to other conditions.

The translator of Styffe’s work, Christer P. Sandberg, made some experiments in order to determine the effect of shocks at different temperatures, i.e., ordinary and low. These were also made in Sweden, and by dropping heavy weights, from different heights, on rails supported at each extremity. The records of these tests may be found in the translator’s Appendix to Styffe’s work.

The following are Sandberg’s conclusions, and they will be observed to be directly opposed to those of Styffe :

1.    “ That for such iron as is usually employed for rails in the

three principal rail-making countries (Wales, France and Belgium), the breaking strain, as tested by sudden blows or shocks, is considerably influenced by cold ; such iron exhibiting at io° Fa]ir. only one-third to one-fourth of the strength which it possess at 84° Fahr.

2.    “ That the ductility and flexibility of such iron is also much

affected by cold ; rails broken at io° Fahr, showing on an average a permanent deflection of less than one inch, whilst the other halves of the same rails, broken at 84° Fahr., showed a set of more than four inches before fracture.

3.    “ That at summer heat the strength of the Aberdare rails

was 20 per cent, greater than that of the Creusot rails ; but that in winter the latter were 30 per cent, stronger than the former.”

All these experiments were made previous to 1869, and with iron rails.

Prof. Thurston, from his own experiments and those of others, concludes (Trans. Am. Soc. of Civ. Engrs., Vol. III., p. 30), “ That with good materials, cold does not produce injury, but actually improves their power of resisting stress and increases their resilience.

“ That the influence of impurities, of various methods of manufacture, of changes of density with temperature, and of the causes which produce a concentration of the action of rapidly produced distortion and of quick blows, are subjects which still require careful investigation.”

He considers it probable that the cold-shortening effect of phosphorus is intensified at low temperatures.

After observing the failures on the railroads coming under their observation, the Railroad Commissioners of Massachusetts reported in 1874 that, in their opinion, neither iron nor steel attained any greater degree of brittleness, or became any more “ unreliable for mechanical purposes ” at low temperatures than at ordinary. They did not observe as a “ rule that the most breakages ” occurred “ on the coldest days.”

They further stated that “ the introduction of steel in place of iron rails, has caused an almost complete cessation of the breakage of rails.”

Thus it is seen that the subject is most thoroughly involved in confusion. It seems, however, to be established that the resistance of iron, at a low temperature, to a steady strain, is not diminished, while it may, perhaps, be increased.

Its resistance to shocks, at low temperatures, is probably very much affected by its quality, mode of manufacture or chemical composition, and these should always be taken into consideration when experiments are made.

The Report of the Mass. Railroad Commissioners would indicate that steel rails resist shocks at low temperatures better than iron ones.

Iron Wire.

Mr. John A. Roebling found by his tests that the English wire used in the Niagara Falls Suspension Bridge gave an ultimate tensile resistance of about 98,500.00 pounds per square inch (“ Papers and Practice Illustrative of Public Works.” John Weale, London, 1856). This wire was about 0.145 inch in diameter.

The Committee of the Franklin Institute made thirteen tests of some iron wire one-third of an inch in diameter, of which the highest, lowest and mean ultimate resistances in pounds per square inch of original section were as follows :

88,354.00 pounds. 84,186.00 pounds. 72,325.00 pounds.

Highest Mean.. Lowest

The results of other tests by the same committee have already been given under “ Effect of annealing.”

TABLE XX.

ULTIMATE TENSILE RESISTANCE IN POUNDS PER

ORIGINAL DIAMETER

SQUARE INCH OF

CONTRACTION OF

IN INCHES.

ORIGINAL AREA

OF SECTION.

Original Area.

Fractured Area.

O. 122 O.I23 o. 124 O.I25 0.122 o. 124 O.124

O. 122

o. 124 O.I24 O.125 o. 124

94,871

87,395

89,256

88,6i8

92,308

9¹» 735 90,082 92,308

735

179,032

162,500

145,946

137,974

168,750

156,338

170,313

168,750

173,437

164,063

156,522

152,174

O.47

O.462 O.388 o. 358

0.453

0.413

0.471

0.453

0.471

0.471

0.439

0.43

Table XX. is a condensed form of one given in the “ Transactions of the Am. Soc. Civ. Engrs.,” Vol. III., p. 212, and contains an account of the tests made by Prof. R. H. Thurston on some wires that had been in use 32 years in the cables of the Fairmount Suspension Bridge at Philadelphia. It is both interesting and important to observe that the long service cannot have appreciably injured either the ducility or ultimate resistance of the wire.

Table XXI. contains the records of tests on other wire, at the same time (1875), by Prof. Thurston. The small reduction of diameter at fracture shows the iron to have been not very ductile. It will also be noticed that the smaller diameters give much the highest resistances.

TABLE XXI.

ORIGINAL DIAMETER.

DIAMETER AFTER FRACTURE.

ULTIMATE RESISTANCE IN POUNDS PER

SQUARE INCH OF ORIGINAL AREA.

O.I34

O.I33

0.1205

0.1185

92,890

84,442

0.08

0.0795

0.071

0.068

94,299

90,384

0.0535

0.0532

105,871

0.029

0.029

113,546

According to Weisbach (“ Mechanics of Engineering, etc.,” Vol. I., 4th Edit.), Lagerhjelm and Brix found the mean value of the ultimate resistance, for a large number of tests of wrought-iron wire with diameters varying from 0.0833 to 0.125 inch, to be 98,000.00 pounds per square inch for unannealed wire, and 64,500.00 pounds for annealed.

Morin, in his “ Mécanique Practique,” gives the following for unannealed iron wire, after changing his results to pounds per square inch :

Mean for diameters of 0.039 to 0.118 inch....    85,000.00 (nearly).

Highest for diameters of 0.02 to 0.039 inch... 114,000.00 (nearly).

Lowest for large diameter......

For a special grade (“l’Aigle”)

71,000.00 (nearly). 128,000.00 (nearly).

SirWm. Fairbairn (“Useful information for Engineers, 3d Series,” p. 282) gives the following as the results of experi-

ments on various kinds of English iron wire. These experiments resulted from investigations relating to the fabrication of a submarine Atlantic cable.

KIND OF WIRE.

DIAMETER.

ULT. RESIST.

STRETCH.

Haematite...........

Homogeneous.......

Special Homogeneous

Charcoal............

Galvanized..........

Homogeneous.......

Homogeneous.......

Charcoal............

Homogeneous.......

Charcoal............

Haematite, S. 3......

Haematite, S. 4......

Homogeneous.......

Homogeneous.......

Homogeneous.......

Homogeneous.......

Special Charcoal.....

Inch. O.O87 O.O95 O.O97 O.O93 O.O98 O.O89 O.O9I O.O9I O.O88 O.O93 O.O89 O.O95 o. 180 o. 148 0.095 0.095 0.095

Pounds.

109,300

134,000

115,000

110,400

86.200 104,500 192,200

92.200 106,900

80,960

88,400

105,800

45.200 61,050

134,000

77,600

105,800

Inch. 0.280 0.366 0.267 O.173 O.I98 o. 190 0.712 o. 198 0.218 0.320 o. 171 0.366 0.480 0.550 0.346 o. 116 0.170

The ultimate resistance is in pounds per square inch, and the stretch is the total amount for 50 inches of length. Reviewing the values given, it appears :

1.    That wire is the strongest form in which iron can be used to

resist tensile stress ;

2.    That, as a rule, the ultimate tensile resistance increases as the

diameter of the wire decreases.

Tensile Resistance of Shape Iron.

The phenomena exhibited in the fracture of shape iron depend, to a great extent, on the character of the piles from which it is rolled. The webs of £s and Is are sometimes rolled from old rails in connection with double refined iron in the flanges. In such cases, specimens cut from the web will frequently, if not usually, show a high intensity of ultimate resistance, but very little ductility, while those cut from the flanges wili give good records of both kinds.

In general, shapes will offer less tensile resistance than either bars or rods, yet small specimens cut from good shape iron will give values ranging from 52,000 to 58,000 pounds per square inch, with ductility little less than that of Qs and Os*

English Wrought Iron.

A great number of experiments on English wrought iron have been made by Sir Wm. Fairbairn, David Kirkaldy, and others. A record of Fairbairn’s experiments may be found in his “ Useful Information for Engineers,” while an account of those of the latter is given in “ Experiments on Wrought Iron and Steel,” by David Kirkaldy, Glasgow, 1863.

B. B. Stoney, in his “ Theory of Strains in Girders and Similar Structures,” summarizes Kirkaldy’s results, in pounds per square inch, as follows:

57.555-00

54,729.00

50,737.00

46,171.00

Mean of 18S rolled bars...........

Mean of 72 angle irons and straps Mean of 167 plates, lengthwise .... Mean of 160 plates, crosswise.....

It should be stated that these means include some Russian and Swedish irons, also that the bars were small ones.

These results do not differ much from quantities for corresponding grades of American iron.

Fracture of Wrought Iron.

The characteristic fracture of wrought iron broken in tension, either directly or transversely, is rather coarsely fibrous.

not unfrequently exhibiting a few bright granular spots which, in rare cases, may possibly be crystalline. This characteristic (fibrous) fracture is always produced by the steady application of an external force, under the influence of which the piece is drawn out in jagged points at the place of failure.

The best of fibrous wrought iron, however, will exhibit a granular fracture if broken suddenly. In making tests, therefore, it is of the greatest importance to observe and direct the mode of application of the external forces producing fracture.

When some grades of iron in bars are broken transversely by shocks (such as are produced by falling weights), a phenomenon known as “ barking ” is produced. A skin of metal from a sixteenth to an eighth of an inch in thickness, on the tension side of the bent piece, tears apart and separates from the core of the bar. At the place of fracture and on each side of it, this skin or “ bark ” remains essentially straight. This kind of fracture shows remarkably well the fibrous character of wrought iron ; it is simply the separation of the fibres near the outside of the bar from those within.

Crystallization of Wrought Iron.

The subject of crystallization of wrought iron is one about which there is much dispute. In “ Strength of Wrought Iron and Chain Cables,” by Beardslee, as abridged by Kent, p. 36, the following is given as the opinion or view of the United States Testing Commission : “ The question as to whether crystallization can be produced in iron by stress, or by repetition of stress with alternations of rest, or by vibration, has been much discussed, and very opposite views are entertained by experts.

“We have met with but one unmistakable instance of crystallization which was probably produced by alternations of severe stress, sudden strains, recoils and rest.”

The committee then state the case of a connecting-rod, carefully made of the best quality of wrought-iron scrap, which had been used in a testing machine for forty years, in the Navy Yard at Washington. It was five inches in diameter, but one day, while in use it suddenly broke under a stress (total) of less than 200,000 pounds. “ The surface of the fractured ends showed well-defined crystallization, the facets being large and bright as mica.”

The data at hand, at present, are not sufficient for a decision of the question, but it may be confidently stated that in many cases granulation has been mistaken for crystallization.

Elevation of Ultimate Resistance and Elastic Limit.

It was first observed by Prof. R. H. Thurston and Commander L. A. Beardslee, U. S. N., independently, in this country, that if wrought iron be subjected to a stress beyond its elastic limit, but not beyond its ultimate resistance, and then allowed to “ rest ” for a definite interval of time, a considerable increase of clastic limit and ultimate resistance may be experienced. In other words, t'he application of stress and subsequent “ rest ” increases the resistance of wrought iron.

This “ rest ” may be an entire release from stress or a simple holding the test piece at a given intensity.

Prof. Thurston’s investigations were on torsion, while those of the United States Commission were on tension, and will be given here.

The Commission prepared twelve specimens and subjected them to an intensity of stress equal to the ultimate resistance of the material, without breaking the specimens. These were then allowed to rest, entirely free from stress, from twenty-four to thirty hours, after which period they were again stressed until broken.

The gain in ultimate resistance by the rest was found to vary from 4.4 to 17 per cent.

These tests, remark the committee, seem to indicate that the tough fibrous irons gained the most, while those which broke with a steel-like fracture gained the least.

Before the rest, the stress which produced the first permanent elongation was about 65 per cent, of the ultimate resistance, but after the rest the two were nearly identical.

The committee then took forty-two other specimens and subjected them to precisely the same operations, except that the rest periods varied from one minute to six months.

The gains were as follows :

In less than 1 hour..........

In less than 8 and over 1 hour

In 3 days...................

In 8 days...................

Between 8 and 43 days.......

In 6 months................

. 1.1 per cent., mean of 5 tests.

3.8 per cent., mean of 8 tests. ,16.2 per cent., mean of 10 tests.

17.8    per cent., mean of 2 tests. 15.3 per cent., mean of 5 tests.

17.9    per cent., mean of 12 tests.

After seven other experiments involving a rest of 24 hours, with an average gain of 15.4 per cent., the committee concluded “ that at the end of one day the result is, with very ductile irons, practically accomplished.”

The manifestation of this phenomenon in different grades of iron was then investigated.

“ Thirteen pieces were prepared, five of which were of soft charcoal bloom boiler iron, five of coarse contract chain iron, and three of a fine-grained bar of .    .    . very pure iron with

high tenacity.”

After testing these specimens subsequent to an eighteen hours’ rest, the committee state (Kent’s abridgment) :

“ These experiments confirmed the opinion already formed, and indicate that a bridge, cable, or other structure, composed of iron of either of the latter two varieties, will receive comparatively slight benefit from the operation of this law ; while ductile fibrous metal .    .    . gains .    .    . to a great extent

by the effect of strains already withstood.” The gain in these

specimens varied from about 3 per cent, (for the coarse iron) to about 18 per cent, (for the soft iron).

Again, two sets of specimens were prepared : one from the two portions of fractured bars after having been pulled asunder, the other from the bars in their normal condition. After a rest of several days the first set showed a gain over the second in ultimate resistance, varying from about 8 to 39 per cent., the higher values belonging to the more ductile irons.

Bauschingers Experiments on the Change of Elastic Limit and Coefficient of Elasticity.

In “ Der Civilingenieur,” Heft 5, for 1881, are contained the results of the experiments of Prof. Bauschinger, of Miinich. The observations in these experiments were made by the aid of a piece of apparatus which gave the elongations (all experiments were tensile) in ten-millionths of a metre, or approximately in ssöVüt °f an inch. An extraordinarily high degree of accuracy was therefore attained.

Prof. Bauschinger’s elastic limit was strictly a proportionality limit between stresses and strains. He also observed what may be called the “stretch-limit ” (Ger., Streckgrenze), at which point the stretching or elongation suddenly increases and continues to increase for more than a minute after the application of the stress. In ordinary experimenting this point has probably frequently been considered the elastic limit.

The test pieces were subjected to loads which gradually increased from zero by an increment a little less than 3,000 pounds per square inch, each load having been allowed to act one minute before adding the succeeding increment. At intervals of the loading separated by about 11,500 or 12,000 pounds per square inch, each piece was entirely unloaded and allowed to remain so for 15 or 20 minutes. After the “ stretch-limit” was found the piece was subjected to a final load somewhat greater than the “ stretch-limit,” and then entirely unloaded.

In some cases the piece was immediately put through the same process of testing either once or a number of times, and the results of such tests will be found in the columns of the

following tables, indicated by the contraction “ Ini y

In the remaining cases intervals of time, shown at the tops of the columns, were allowed to elapse between any one test and the succeeding one.

The tables, Nos. 1 to 7 inclusive, give the results of the experiments on seven specimens of a grade of iron called “ Schweisseisen ” (weld iron). These specimens were a very little less than 1 inch (25 millimetres) in diameter. Nos. 1 and 2 were about 32 inches long, and the others about 16 inches long.

Tables No. 8 to 13, inclusive, give the results obtained with Krupps “ Flusseisen.” These specimens were about one inch in diameter and sixteen inches long.

The tables have been condensed from those given by Bau-schinger and reduced to English measures.

The following is the notation :

E.    L. — elastic limit in pounds per square inch.

S.-L. = stretch limit in pounds per square inch.

F.    L. = final load in pounds per square inch.

E.

= coefficient of elasticity in pounds per sq. in.

Weld Iron.

IN ORIGINAL CONDI-

NO. I.

TION.

im’y.

im’y.

im’y.

E. L.

20,110

14,370

14,900

15,500

S.-L.

27,300

31,600

41,700

49,500

F. L.

31,600

40,200

47,700

IN ORIGINAL CONDI-

NO. 2.

TION.

E. L.

20,110

S.-L.

F. L.

28,700

31,600

E.

29,037,000

IN ORIGINAL CONDI-

NO. 3.

TION.

E. L.

23,164

S.-L.

29,000

F. L.

31,900

E.

29,208,000

IN ORIGINAL CONDI-

NO. 4.

TION.

E. L.

22,890

S.-L.

30,050

F. L.

3L470

Weld Iron.

AFTER 19 HRS.

AFTER 27 HRS.

AFTER 24 HRS.

2S,970

34,750

40,540

28,923,000

Weld Iron.

AFTER 51 HRS.

28,440

34,750

40,540

28,397,000

Weld Iron.

AFTER 80 HRS.

31,900

34,750

40,540

35,500

44,3oo

47,300

28,198,000

AFTER 41 HRS.

39,080

45,090

48,100

28,483,000

AFTER 68 HRS.

35,340

44,170

47,110

39,100

45,100

28,241,000

AFTĘR 45 HRS.

45,58o

28,170,000

AFTER 64 HRS.

43,800

IN ORIGINAL CONDI“

NO. 5.

TION.

E. L

21,070

S.-L.

F. L.

30,670

34,6xi

E.

29,293,000

IN ORIGINAL CONDI-

NO. 6.

TION.

E. L.

27,730

S.-L.

32,120

F. L.

35,040

E.

29,720,000

IN ORIGINAL CONDI-

NO. 7.

TION.

E. L.

20,110

S.-L.

30,160

F. L.

34,470

Weld Iron.

im’y.

14,720

35,320

42,700

28,312,000

Weld Iron.

AFTER 48.5 HRS.

26,720

37,1X0

43,040

28,639,000

Weld Iron.

AFTER 47 HRS.

26,720

38,590

43,040

AFTER 63 HRS.

42,090

48,110

51,110

28,056,000

AFTER 44.5 HRS.

33,350

45,48o

51,550

28,483,000

AFTER 50.5 HRS.

27,170

45,290

5i,320

im’y.

I5.260

51,860

26,705,000

AFTER 49 HRS.

24,940

28,881,000

AFTER 42.5 HRS.

18,540

NO. 8.

E. L.

S.-L.

F. L.

E.

NO. 9.

E. L.

S--L.

F. L.

E.

NO. IO.

E. L.

S.-L.

F. L.

Melted Wrought Iron.

IN ORIGINAL CONDI-

TION.

im’y.

im’y.

im’y.

35,340

36,750

46,910

45,230

52,770

31,256,000

8,990

53,890

59,8So

31,483,000

9-230

58,490

30,488,000

Melted Wrought Iron.

IN ORIGINAL CONDI-

TION.

im’y.

im’y.

im’y.

37,870

42,080

44,880

32,379,000

5,770

14,720

15,160

46,140

51,920

31,796,000

53,000

58,880

29,947,000

Melted Wrought Iron.

IN ORIGINAL CONDI-

TION.

33,790

36,600

42,230

60,630

28,213,000

AFTER 3 HRS.

AFTER 15 HRS.

AFTER 7 HRS.

11,490

17,730

15,260

43,090

53,210

61,020

5E704

59,130

NO. II.

E. L.

S.-L.

F. L.

E.

NO. 12.

E. L.

S.-L.

F. L.

E.

Melted Wrought Iron.

IN ORIGINAL CONDI-

TION.

33,930

39,570

42,404

32,536,000

AFTER 2.5 HRS.

AFTER 15.S HRS.

AFTER 5.3 HRS.

n,590

11,774

12,070

43,442

53,ooo

60,380

52,130

58,880

32,237,000

32,222,000

31,085,000

Melted Wrought Iron.

IN ORIGINAL CONDI-

TION.

36,900

39,731

42,630

32,479,000

AFTER 51 HRS.

AFTER 47 HRS.

AFTER 46 HRS.

43,800

49,640

52,560

31,754,000

39.820

40,950

55,940

58,880

31,696,000

60,630

31,568,000

Melted Wrought Iron.

NO. 13.

z o 2 2 h o o 5

2    >

r

AFTER

AFTER

AFTER

AFTER

AFTER

43.5 HRS.

54 HRS.

44.5 HRS.

45-5 HRS.

IO DAYS.

E. L.

35,340

5'. L.

36,745

38,930

47,600

41,740

42,720

61,560

56,640

F. L.

42,400

5i,9²⁰

59,630

During the progress of the various tests, the bars Nos. 6, 7, 9, u and 12 were subjected to shocks in addition to the static tests. These shocks were produced by striking the test piece on its end by a hammer. It does not appear that these blows of the hammer perceptibly influenced the results.

The ultimate resistance of the weld iron was found to vary from 55,300 to 58,870 pounds per square inch. That of the melted wrought iron was about 65,000 pounds per square inch.

Although there are some irregularities, the following general conclusions may be drawn from the tables:

By “immediate” testing the elastic limit'of weld iron is very much decreased.

With a rest (entirely free from load) between the tests, the elastic limit of weld iron is very much increased.

The greatest proportional gain, except in the case of previous immediate testing, seems to be acquired after a rest no greater than twenty hours.

Bar No. 6 is seen to give anomalous results.

In all cases of the weld iron the stretch-limit is considerably raised by repeated testing.

In no case is the coefficient of elasticity, after once testing, equal to its original value ; as a rule, a steady decrease is seen to take place by repeated testing, but there are some exceptions.

The elastic limit of “ Flusseisen,” after repeated testing, is found to be much less than its original value until the length of rest becomes about fifty hours.

The stretch-limit of the same metal is invariably raised by repeated testing, either with or without “ rests.”

In nearly all the cases of Nos. 8 to 13, the coefficient of elasticity is found to be slightly decreased by repeated testing.

For a very clear and detailed account of these experiments reference must be made to the “ Civilingenieur.’

Resistance of Bar Iron to Suddenly Applied Stress.

If tensile stress is suddenly applied to a bar of wrought iron, both its ultimate resistance and elongation will be very materially decreased.

As a mean of a number of tests, Mr. David Kirkaldy (“ Experiments on Wrought Iron and Steel ”) found with suddenly applied stress an ultimate resistance of 46,500 pounds per square inch, while with stress gradually applied it rose to 57,200 pounds.

In the former case the elongation was about 20 per cent., and as high as 24.6 per cent, in the latter.

It is thus seen that the mode of application of external force not only affects the character of the fracture of the iron, but also its ultimate resistance and elongation.

It will hereafter be seen that similar observations apply to other metals than wrought iron.

Reduction of Resistance Between the Ultimate and Breaking

Point.

It has already been observed that the ultimate tensile resistance of wrought iron is the greatest tensile resistance which it offers to being pulled asunder, and that a test specimen finally parts at much less than the ultimate resistance. This is due to the ductility of the iron, which allows it to “ pull out ” or stretch, thus decreasing the cross section as well as the actual resisting capacity of the metal.

The ultimate resistance, therefore, is not exerted on the final section of fracture, but on a section somewhat greater ; referring it (the ultimate resistance) to the section of fracture, then, may mean little or nothing.

The United States Commission made six tests, for the purpose of determining this reduction, on some specimens which had previously been stressed with a subsequent rest. The highest, lowest, and mean losses were as follows :

Highest Mean .. Lowest.

14.5 per cent.

13.8    per cent.

12.9    per cent.

It was observed from a number of specimens, by the same commission, that the reduction of area at the instant of ultimate resistance (or greatest resistance) was. about one-half, and the elongation or strain a little over three-quarters, of the corresponding quantities at the instant of fracture, supposing failure to be produced by a steady strain.

Some further observations seemed to show that if failure were produced by shock, the final contraction would be nearly the same as at the instant of greatest resistance in the case of a steady failure.

Effects of Chemical Constitution.

While it is well known that the resistance of wrought iron to tension varies greatly with the chemical composition, it is yet uncertain just what influence most of the foreign elements, found in iron exert, either individually or collectively. This will be apparent on examining Table XXII., taken from the report of the three committees of the United States Commission, to which allusion has here been so frequently made before.

The first part of the table represents the relative values of sixteen different irons in reference to their physical characteristics, one being the highest. The second part shows the amount of the various elements named in the left-hand lower column, found in the corresponding irons, i. e., each vertical column belongs to one iron.

An inspection of the table will make very evident the diffi-

en en

O.O72

O.O72

0.043

vo rj- O

vo vo

0.201

Ogi-°

0.026

'0.002:0.046

vO m I

M    m    M    I    M

0.202

O.27I

0.036

0.009

en Cl rf d

O.I78

O.I39,

;0.021

,0.172

0.02 3'0.04810.006 0.006 0.01810.032

0.031

1.071

0.028

'0.650

0.010

gSi-i

Cl    en    •3"    m O'

M    M    M

O.23I

O.I56

0.015

0.038

i£o-o/ioo

890-0

0.047,

0.078

0.037

1.210;

ii

0.193

,0.163

,0.032

10

ii

P.154,

iogi-o

P.033,

900-0

0.014

O.O39

ETC.

0.095

0.028

990-0

800-0

O.OO9 0.084

0.023)0.042

! o. 042

0.028

,0.546

ii

jo. 187

£91-0

,0.032,

io.oio

,0.020

0.023

I.2I4

i£ot>

O.O26

0.013

O O en

MM    W

VO

o ci o O

vn O    00 Cl M

M    MM

-t m ci co

M M

en o o* o vo

PERCENTAGES OF PHOSPHORUS,

|o. 191

0.250

0.169 0.225

0.203

I0.169

0.055

0.032

0.038

I0.182

p.033

0.081

0.033

0.03710.029

0.023

1.760

0.057 o. 848

hgi-o

0.044,

0*353

,0.020!

0.070

0.132,

1.044,

w ci en en ci

MMMM

M vo O I rt

M M    M

0.072 0.152

0.095 o. 149I0.147 0.154 0.350(0. o68jO. 029 0.042 0.009 0.058 0.011 0.046 0.014 0.022 0.030 0.021 0.008 o.oxg'o.026|o.o29 o. 014^0.040(0.027(0.031

0.722

0.455

0.331

NUMBER—

In tenacity

In reduction of area

In elongation. ..........

In welding value.........

In power of resisting shock.

Phosphorus

Silicon

Carbon.

Copper.

Manganese.

Cobalt.

Nickel.

Slag.

culty of drawing definite conclusions in regard to any one element.

For a detailed discussion of these results reference must be made to the report.

Kirkaldy s Conclusions.

The following conclusions were deduced by Mr. Kirkaldy from the results of his experiments. As will be seen, they belong to both wrought iron and steel in tension, and are taken from his “ Experiments on Wrought Iron and Steel,” 1861 :

X. The breaking strain does not indicate the quality, as hitherto assumed.

2.    A high breaking strain may be due to the iron being of superior quality, dense, fine, and moderately soft, or simply to its being very hard and unyielding.

3.    A low breaking strain may be due to looseness and coarseness in the texture, or to extreme softness, although very close and fine in quality.

4.    The contraction of area at fracture, previously overlooked, forms an essential element in estimating the quality of specimens.

5.    The respective merits of various specimens can be correctly ascertained by comparing the breaking strain jointly with the contraction of area.

6.    Inferior qualities show a much greater variation in the breaking strain than superior.

7.    Greater differences exist between small and large bars in coarse than in fine varieties.

8.    The prevailing opinion of a rough bar being stronger than a turned one is erroneous.

9.    Rolled bars are slightly hardened by being forged down.

10.    The breaking strain and contraction of area of iron plates are greater in the direction in which they are rolled than in a transverse direction.

11.    A very slight difference exists between specimens from the centre and specimens from the outside of crank shafts.

12.    The breaking strain and contraction of area are greater in those specimens cut lengthways out of crank shafts than in those cut crossways.

13.    The breaking strain of steel, when taken alone, gives no clue to the real qualities of various kinds of that metal.

14.    The contraction of area at fracture of specimens of steel must be ascertained as well as in those of iron.

15.    The breaking strain, jointly with the contraction of area, affords the means of comparing the peculiarities in various lots of specimens.

16.    Some descriptions of steel are found to be very hard, and, consequently, suitable for some purposes ; whilst others are extremely soft, and equally suitable for other uses.

17.    The breaking strain and contraction of area of puddled steel plates, as in iron plates, are greater in the direction in which they are rolled ; whereas in cast steel they are less.

18.    Iron, when fractured suddenly, presents invariably a crystalline appearance ; when fractured slowly, its appearance is invariably fibrous.

19.    The appearance may be changed from fibrous to crystalline by merely altering the shape of specimen, so as to render it more liable to snap.

20.    The appearance may be changed by varying the treatment, so as to render the iron harder and more liable to snap.

21.    The appearance may be changed by applying the strain so suddenly as to render the specimen more liable to snap, from having less time to stretch.

22.    Iron is less liable to snap the more it is worked and rolled.

23.    The “ skin ” or outer part of the iron is somewhat harder than the inner part, as shown by appearance of fracture in rough and turned bars.

24.    The mixed character of the scrap iron used in large forgings is proved by the singularly varied appearance of the fractures of specimens cut out of crank shafts.

25.    The texture of various kinds of wrought iron is beautifully developed by immersion in dilute hydrochloric acid, which, acting on the surrounding impurities, exposes the metallic portion alone for examination.

26.    In the fibrous fractures the threads are drawn out, and are viewed externally, whilst in the crystalline fractures the threads are snapped across in clusters, and are viewed internally or sectionally. In the latter cases the fracture of the specimen is always at right angles to the length ; in the former it is more or less irregular.

27.    Steel invariably presents, when fractured slowly, a silky fibrous appearance ; when fractured suddenly, the appearance is invariably granular, in which case also the fracture is always at right angles to the length ; when the fracture is fibrous, the angle diverges always more or less from 90°.

28.    The granular appearance presented by steel suddenly fractured is nearly free of lustre, and unlike the brilliant crystalline appearance of iron suddenly fractured ; the two combined in the same specimen are shown in iron bolts partly converted into steel.

29.    Steel which previously broke with a silky fibrous appearance, is changed into granular by being hardened.

30.    The little additional time required in testing those specimens, whose rate of elongation was noted, had no injurious effect in lessening the amount of breaking strain, as imagined by some.

31.    The rate of elongation varies not only extremely in different qualities, but also to a considerable extent in specimens of the same brand.

32.    The specimens were generally found to stretch equally throughout their length until close upon rupture, when they more or less suddenly drew out, usually at one part only, sometimes at two, and, in a few exceptional cases, at three different places.

33.    The ratio of ultimate elongation may be greater in short than in long bars in some descriptions of iron, whilst in others the ratio is not affected by difference in the length.

34.    The lateral dimensions of specimens forms an important element in comparing either the rate of, or the ultimate, elongation—a circumstance which has been hitherto overlooked.

35.    Steel is reduced in strength by being hardened in water, while the strength is vastly increased by being hardened in oil.

36.    The higher steel is heated (without, of course, running the risk of being burned) the greater is the increase of strength by being plunged into oil.

37.    In a highly converted or hard steel the increase in strength and in hardness is greater than in a less converted or soft steel.

38.    Heated steel, by being plunged into oil instead of water is not only considerably hardened, but toughened by the treatment.

39.    Steel plates hardened in oil, and joined together with rivets, are fully equal in strength to an unjointed soft plate, or the loss of strength by riveting is more than counterbalanced by the increase in strength by hardening in oil.

40.    Steel rivets, fully larger in diameter than those used in riveting iron plates of the same thickness, being found to be greatly too small for riveting steel plates, the probability is suggested that the proper proportion for iron rivets is not, as generally assumed, a diameter equal to the thickness of the two plates to be joined.

41.    The shearing strain of steel rivets is found to be about a fourth less than the tensile strain.

42.    Iron bolts, case-hardened, bore a less breaking strain than when wholly iron, owing to the superior tenacity of the small proportion of steel being more than counterbalanced by the greater ductility of the remaining portion of iron.

43.    Iron highly heated and suddenly cooled in water is hardened, and the breaking strain, when gradually applied, increased, but at the same time it is rendered more liable to snap.

44.    Iron, like steel, is softened, and the breaking strain reduced, by being heated and allowed to cool slowly.

45.    Iron subject to the cold-rolling process has its breaking strain greatly increased by being made extremely hard, and not by being “ consolidated,” as previously supposed.

46.    Specimens cut out of crank-shafts are improved by additional hammering.

47.    The galvanizing or tinning of iron plates produces no sensible effects on plates of the thickness experimented on. The result, however, may be different, should the plates be extremely thin.

48.    The breaking strain is materially affected by the shape of the specimen. Thus the amount borne was much less when the diameter was uniform for some inches of the length than when confined to a small portion—a peculiarity previously unascertained, and not even suspected.

49.    It is necessary to know correctly the exact conditions under which any tests are made before we can equitably compare results obtained from different quarters.

50.    The startling discrepancy between experiments made at the Royal Arsenal, and by the writer, is due to the difference in the shape of the respective specimens, and not to the difference in the two testing machines.

51.    In screwed bolts the breaking strain is found to be greater when old dies are used in their formation than when the dies are new, owing to the iron becoming harder by the greater pressure required in forming the screw thread when the dies are old and blunt than when new and sharp.

52.    The strength of screw-bolts is found to be in proportion to their relative areas, there being only a slight difference in favor of the smaller compared with the larger sizes, instead of the very material difference previously imagined.

53.    Screwed bolts are not necessarily injured, although strained nearly to their breaking point.

54.    A great variation exists in the strength of iron bars which have been cut and welded ; whilst some bear almost as much as the uncut bar, the strength of others is reduced fully a third.

55.    The welding of steel bars, owing to their being so easily burned by slightly overheating, is a difficult and uncertain operation.

56.    Iron is injured by being brought to a white or welding heat, if not at the same time hammered or rolled.

57.    The breaking strain is considerably less when the strain is applied suddenly instead of gradually, though some have imagined that the reverse is the case.

58.    The contraction of area is also less when the strain is suddenly applied.

59.    The breaking strain is reduced when the iron is frozen ; with the strain gradually applied, the difference between a frozen and unfrozen bolt is lessened, as the iron is warmed by the drawing out of the specimen.

60.    The amount of heat developed is considerable when the specimen is suddenly stretched, as shown in the formation of vapor from the melting of the layer of ice on one of the specimens, and also by the surface of others assuming tints of various shades of blue and orange, not only in steel, but also, although in a less marked degree, in iron.

61.    The specific gravity is found generally to indicate pretty correctly the quality of specimens.

62.    The density of iron is decreased by the process of wire-drawing, and by the similar process of cold rolling, instead of increased, as previously imagined.

63.    The density in some descriptions of iron is also decreased by additional hot-rolling in the ordinary way ; in others the density is very slightly increased.

64.    The density of iron is decreased by being drawn out under a tensile strain, instead of increased, as believed by some.

65.    The most highly converted steel does not, as some may suppose, possess the greatest density.

66.    In cast steel the density is much greater than in puddled steel, which is even less than in some of the superior descriptions of wrought iron.

Art. 33.—Cast Iron.

Coefficient of Elasticity and Elastic Limit.

Cast iron is a material of much less value to the engineer than wrought iron, and consequently has been the subject of much less experimental investigation.

The following table (Table I.) contains values of the coefficient of tensile elasticity for three (Nos. 1, 2 and 3) different irons used in the fabrication of cast-iron cannon. They are computed by the aid of Eq. (1), Art. 2, from data contained in “ Reports of Experiments on the Properties of Metals for Cannon,” etc., by the late Captain T. J. Rodman,

TABLE I.

NO. I.

NO. 2.

NO. 3.

NO. 4.

IV.

E.

E.

E.

E.

1,000

2,003

3,000

4,oco

5,000

IO,OCO

15,030

20,000

24,000

28,011,000

28,011,000

25,000,000

22,g62,000

23,031,000

20,960,000

16,773,000

13,384,000

10,150,000

50,000,000

28,571,000

23,810,000

22,727,000

20,833,000

17,000,000

13,204,000

7,370,000

3,454,000

33,333,ooo

28,571,000

27,273,000

25,000,000

23,810,000

20,000,000

17,241,000

14,085,000

11,060,000

25.OOO.OOO

16,667,000

15,000,000

15.385,000

13,889,000

12,195,000

10,000,000

8,000,000

IV and E are expressed in pounds per square inch.

U. S. A.

The iron was an excellent charcoal gun iron, and the specimens were from 30 to 35 inches long turned to a diameter of 1.382 inches. The data were selected at random (pages 158,

212 and 228 of the work cited) from the large amount accumulated by Captain Rodman.

Column No. 4 contains values of E given by Wm. Kent,

M. E. (Van Nostrand’s Magazine, Vol. 20); they belong to a piece of cast iron i | inches in diameter and 5 inches long.

The left-hand column, headed “ W,” gives the stress per square inch, while the three columns “ E” give the corresponding ratios between stress and strain for the three different irons. Such ratios are the “ coefficients of elasticity,” properly speaking, below the elastic limit only. It will be observed, however, that none of these specimens can really be considered to possess an elastic limit, unless possibly No. 1, whose elastic limit may be taken at, or a very little above 2,000 pounds per square inch.

In No.    i    first permanent    set    was    observed at 4,000 pounds per square    inch.

In No.    2    first permanent    set    was    observed at 4,000 pounds per square    inch.

In No.    3    first permanent    set    was    observed at 8,000 pounds per square    inch.

20000.

16000.

10000,

5000.

BO

1 inch

2 inches

3 inches

F-ig.l

Fig. I represents graphically the results of the experiments

on specimen No. 2. The constantly varying value of the ratio between stress and strain is shown in a very evident manner by the continually varying inclination of the curve. The strains (stretches) are laid down as if belonging to a bar 1,000 inches long.

The following results are deduced by B. B. Stoney (Theory of Strains in Girders and similar Structures, p. 369) from experiments by Eaton Hodgkinson on a bar of English cast iron 10 feet long.

These results show a limit of elasticity at about 6,000 pounds per square inch ; they also show much smaller values of E than those given in Table I. This last disagreement is undoubtedly due, to a great extent, to the fact that the values of E in Table I. probably «//belong to fine charcoal iron fabricated for a special purpose, while the others do not.

If A. = extension, or stretch in inches of a cast-iron bar when acted upon by a force W (in pounds), and if / represents the length of the bar in inches, Mr. Hodgkinson deduced the following formulae from his experiments :

A = /{.00239628 — ^.00000574215 — .000000000343946W} . (1)

For bars 10 feet long:

(2)

Permanent set, in inches = .0193A + .64À²

Although the preceding results are only a few of a great

many similar results that may be computed in the same manner, yet they give a fair representation of the general character of the elastic properties of cast iron. The metal is seen to be very irregular and unreliable in its elastic behavior. A large portion of the material can scarcely be said to have an elastic limit, although no apparent permanent set takes place under a considerable intensity of stress ; in other words, although perhaps all tested specimens resume their original shape and dimensions for small intensities of stress, yet the ratio between stress and strain is seldom constant for essentially any range of stress.

Ultimate Resistance.

On page 5 of Captain Rodman’s “ Reports ” are given the following densities and ultimate tensile resistances, expressed in pounds per square inch, of 16 specimens of warm-blast, charcoal Greenwood and Salisbury iron, taken from preliminary castings of second and third fusion pigs :

DENSITY.

ULT. RESIST.

DENSITY.

ULT. RESIST.

7.184.

7.198.

7-307-

7.099.

7-304-

7-273-

7.272.

7.219.

33,079

31,384

35,486

23,776

31,317

42,884

38,993

25,372

7.210.

7.172.

7-159-

7-137-

7.106.

7. TOO.

7.109.

7-191-

22,547

28,518

36,373

33,268

22,290

22,179

22,888

23,873

Again, Table II. is taken from page 261 of the same “ Reports.” The results are for specimens from trial castings of second-fusion pigs. The ultimate resistance is in pounds per square inch, while the strains are for an inch of length.

“ Ult. Ext." is the ultimate extension, or stretch, just before fracture, for one lineal inch. The specimens were 30 inches long and 1.382 inches in diameter.

TABLE II.

SPECIMEN.

DENSITY.

ULT. EXT.

ULT. RESIST.

7 • 207 7-274 7.178 7-202 7-255 7.280 7.221 7.230

30.117

31,681

23,617

24,260

28,220

27,147

25,627

24,767

•00303

•00334

.00291

.00161

.00287

.00382

.00424

.00223

On page 42 of “ Reports of Experiments on Metals for Cannon,” Major Wade gives the following for 15 proof bars cast with 8- and 10-inch guns and 6-pounder trial guns, at South Boston, 1844 :

31,027 pounds. 27,232    “

22,402    “

Greatest resistance per square inch. Mean Least

He states that these specimens show the general quality of the iron used at that time.

Again, on page 179 of the same “ Reports,” Major Wade gives for 25 specimens from 32-pounder cannon made at West Point foundry in 1850:

36,728 pounds. 32,023 28,990    “

Greatest resistance per square inch. Mean Least

He states that the character of this iron was “ that of good foundry iron, of the different grades of Numbers 1, 2, and 3 ; ” it was composed of first, or first and second fusion pigs.

The preceding results give correct representations of the character of the best quality of American cast iron, produced for use in cases requiring such a metal.

00

t-H

Three specimens, turned down to a diameter of about 0.625 inch, taken from the iron used in the Boston water mains, and broken at the Warren Foundry, Phillipsbu-rg, N. J., gave the following ultimate resistances in pounds per square inch :

13,070

15,470

18,300

As with all material, the character of cast iron affects, to a great extent, its resistance ; i. e., whether it is fine or coarse grained, gray or white, etc. It (the resistance) also depends upon the character of the ore from which .it is produced.

Major Wade (“ Reports,” pages 378 and 388) shows that the cold-blast iron which he tested gave much higher resistance than the hot-blast metal.

It is to be remembered that all the specimens from which the preceding results were deduced were what may be called “ small specimens.” Specimens with several square inches in area of normal section would probably give somewhat different results.

It is interesting to observe that, in experimenting upon cast-iron cannon, Major Wade (“ Reports,” pages 77 and 78) found that water was forced through the “ pores ” of the metal of one cannon at a pressure of 7,000 pounds per square inch, and through those of another with thicker metal (thickness equal to radius of bore) at a pressure of 9,000 pounds per square inch.

Capt. Rodman (“ Reports,” page 262) forced water through the pores of the metal of cylinders 5 inches long, 1 inch thick, and i inch bore, at pressures ranging from 15,276 to 25,464 pounds per square inch.

The experiments of Eaton Hodgkinson (“ Experimental Researches on the Strength and Other Properties of Cast Iron ”), on English metal gave the following resistances in pounds per square inch :

Cannon iron No. 2, hot blast........

I3.505 pounds. 16,683    “

17.755

14,200    “

21,907

13,434

17,466

16,676

18,855    “

14,535 16,542    “

Cannon iron No. 2, cold blast.......

Cannon iron No. 3, hot blast........

Cannon iron No. 3, cold blast........

Devon (Scotland) iron No. 3, hot blast

Buffery iron No 1, hot blast........

Buffery iron No. 1, cold blast.......

Coed-Talon iron No. 2, hot blast.....

Coed-Talon iron No. 2, cold blast....

Low Moor iron No. 3..............

Mixture...........................

Several of these results are the means of those of a number of tests. The areas of the normal sections of the test specimens varied from 1.54 inches to 4.27 inches, being considerably larger than those of the specimens tested by Major Wade and Captain Rodman.

The characteristic fracture of cast iron is granular and crystalline, with very little (scarcely perceptible by the unaided eye) reduction of area or elongation. Fracture takes place suddenly and without warning, and its ultimate resistance is influenced by many causes whose action may not be observed by any ordinary means ; for these reasons, it is a treacherous and unreliable material in tension, as indeed any brittle material must be.

Effect of Remelting.

Crude pigs are said to be “first-fusion ” metal. Once remelted pigs produce “second-fusion ” iron. Twice remelted pigs produce “ third-fusion ” iron.

etc.,

etc.,

etc.

On page 237 of Major Wade’s “ Reports,” the following values are given for Greenwood first-fusion iron (iron in original pigs) :

ULT. RESIST. IN POUNDS PER SQ. IN.

No. I iron No. 2 iron No. 3 iron

...... 15,129 (mean of 3 tests).

...... 27,153 (mean of 2 tests).

..... 34,923 (mean of 4 tests).

“ No. I is the softest gray iron,

“ No. 2 is intermediate,

“No. 3 is the hardest gray iron.”

Again on page 240 :

Greenwood, No. 1, 1st fusion Greenwood, No. 1, 2d fusion. Greenwood, No. 1, 3d fusion. Guns cast from 3d fusion ...,

ULT. RESIST.

. 20,900 pounds per sq. in. . 30,229 pounds per sq. in.

•    35,736 pounds per sq. in.

•    33,8i5 pounds per sq. in.

The last result is a mean of four tests. Finally on page 242 :

Nos. i and 2 mixed.

Nos. X, 2, and 3 mixed.

2d fusion 3d fusion. 2d fusion 3d fusion.

27,588 pounds per sq. in. 40,987 pounds per sq. in. 37,789 pounds per sq. in. 32,485 pounds per sq. in.

It is seen that “ the softest kinds of iron will endure a greater number of meltings with advantage, than the higher grades.” The greatest ultimate resistance, in pounds per square inch, is obtained with :

4th fusion, 3d fusion,

No. i iron at the Nos. I and 2 mixed at the Nos. I, 2 and 3 mixed at the 2d fusion.

These results probably indicate about the limits to which the remelting of this iron could be advantageously carried.

On page 279 of the same “ Reports,” is given the result of the test of a specimen of third-fusion iron, of a mixture of Nos. I, 2 and 3, taken from a gun. The ultimate resistance found was 45,970 pounds per square inch ; a most remarkable specimen of cast iron.

Effect of Continued Fusion.

Major Wade (“ Reports,” pp. 38-41) tested the effects of continued fusion on different grades of iron, both in relation to transverse and tensile resistance.

The general result was an increase of tensile resistance up to 3 hours in fusion, which was the longest period tried.

The following results are taken from pp. 40 and 41 of the “ Reports.”

IRON.

TIME IN FUSION.

ULT. RESIST.

0.5 hour,

Stockbridge

1.0

1-5

2.0

Proof bars..

' No. 5.. .0.5 No. 6 No. 7 No. 8

1-5

3-0

3-75

0.00

10-inch Howitzer, 2d fusion from pigs.

i .00 2.00 . 3-00

17,843    pounds per square    inch.

These tests show well the effect of continued fusion for a period not exceeding 3.75 hours.

Effect of Repetition of Stress.

Capt. Rodman (“ Reports,” p. 262) experimented on the effect of repeated stresses with the following results :

SPECIMEN.

Ao broke at 2301st repetition of 22,000 pounds per square inch.

“ 26,000    “    “    “    “

“ 20,000 “ 20,000 “ 22,500 “ 23,500 “ 21,600

Ao

Bo

Bo

Co

Co

Do

282d

252d

150th

651st

457th

I72d

The repetition of the letters representing the specimen indicates that duplicates were tested.

A reference to Table II. will show what single loads per square inch broke the same irons, and a comparison of the two will exhibit the “ fatigue ” of the metal.

On pages 166 and 167 he also gives some very interesting results of intermittent repetitions of stresses. He subjected a cylinder of cast iron, 1.382 inches in diameter and 35 inches long to intermittent repetitions of 15,000 pounds per square inch (about three-quarters of its ultimate resistance) as follows: 250 repetitions, then a rest of 40 hours ; next, 375 additional repetitions, then a rest of 30 days; next, 155 additional repetitions, then a rest of 29 days ; next, 1,020 additional repetitions, then a rest of 26 days; finally, 156 additional repetitions followed by breakage at the 1,956th repetition. In every case “ rest ” signifies entire freedom from load. Capt. Rodman’s table gives a detailed account of these experiments. He remarks upon them as follows : “ The most interesting point ... is the fact that at every interval of rest, of any considerable time, the permanent set, and the extension due to the last previous application of the force, diminished. And in some instances it required some fifty repetitions to bring up the extension and set to the same points where they had been at the beginning of the period of rest ; thus indicating clearly that the specimen was partially restored, by the interval of rest, from the injury which it had received ; and that it endured a greater number of repetitions, owing to the intervals of rest, than it would have done had the repetitions succeeded each other continuously, and at short intervals of time.”

These experiments show the “ fatigue ” of cast iron and the increase of the ratio of stress over strain produced by “ rest ”—so far as tensile stress is concerned.

An examination of the tables also shows that in any series of repetitions, between any two consecutive rests, both the extension and set were constantly increasing, consequently, that the ratio of stress over strain was constantly decreasing.

Effect of High Temperatures.

A few experimental results bearing on this point will be found in Table IX. of Article 35.

Art. 34.—Steel.

Coefficient of Elasticity.

The great number of the varieties and grades of “ steel ” renders possible the existence of a correspondingly great number of the mechanical quantities and coefficients used in its •consideration in connection with the “ Resistance of Materials.” In every case, therefore, the kind and character of the steel on which experiments are made, should be stated. In some cases, however, this is impossible.

In Table I. are contained the coefficients of elasticity of the hardened and tempered steel wire (see Table XXII.), supplied by the different makers named, in response to the call for bids for the steel cable wire for the New York and Brooklyn suspension bridge. (Washington A. Roebling’s “ Report,” ist Jan., 1877).

In the same “ Report,” page 72, the specifications state : “ The elastic limit must be no less than of the breaking strength. .    .    . Within this limit of elasticity, it must

stretch at a uniform rate corresponding to a modulus of elasticity of not less than 27,000,000 nor exceeding 29,000,000.”

TABLE I.

COEFFICIENTS OF ELASTICITY.

NUMBER OF

PRODUCER.

Greatest E.

Least E.

TESTS.

J. Lloyd Haigh.......

Cleveland Rolling Mills

Washburne & Moen,

Sulzbacher, Hymen, Wolff & Co

Jno. A. Roebling’s Sons Co.....

Carey & Moen................

29,817,067    28,815,797

30,142,026    28,917,715

28,887,006 30,389,946    29,103,238

30,231,929    28,788,6x9

31,261,041    29,418,025

29,757.300

CJ VO VO

Table I. gives the greatest and least results of these tests in pounds per square inch, in the columns headed “ E” together with the number of tests of the product of each maker. All the wire was No. 8 Birmingham gauge; i. e., 0.165 inch in diameter.

It is not evident from the “ Report ” whether these values were obtained for some particular intensity of stress, or whether they are mean values for the entire range below the elastic limit.

Table II. gives a very condensed statement of the results of a very elaborate investigation in the “ constants ” of steel by Prof. P. C. Ricketts at the mechanical laboratory of the Rensselaer Polytechnic Institute during the year 1886. The paper containing the detailed account of this series of very important tests was given to the Am. Soc. of C. E. in Feb. 1887. Although this table contains other values than those immediately desired, the opportunity of directly comparing different physical constants from the same quality of steel is a sufficient reason for inserting the entire table at this place. All the test

Rivet steel*

Bessemer +

5 5 s 3 M S S ? » Sf S> S? Ss> V 5 3 _3 ffPf&PS* Z ï 5 j -

os    h

w    £    a

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to

to

to

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TABLE II.

TENSION.

COMPRESSION.

Diam. Percen.

Reduc. Inches, of Area.

0.756 0.758 0-757 0-757 0.758 0.758 0.760 0.760 0.760 o. 760 0-759 0.758 0.756 0-755 0-754 0-757 0.758 0.758

0.757

0-755

0.757

0.750

0.749

0-751

0.7S2

0-754

0-749

0.752

0-757

0-753

0.748

0-754

0.750

0-751

0.750

0.750

0.747

0.750

0.751

0.763

0.760

0.760

0.763

0.760

0.756

0.747

0-745

0.745

0.746

0-744

61.7

61.7

60.8

65.3

Ó5.1

62.3

61.6

60.6

61.8

57 9

62.4 6i .0

65-7

64.7

64-3

63-4

64.o

64-3

51-7

49-4

51.2

62.1

Ć0.5

61.3

64.3

63.0

62.3

55-1

53-7

53-2

60.3

58.3

57-0

59-7

59-2

57-4

57-3

57-1

58.1

59-5

56.4

59-¹

56.6

58.3

54-8

55-7

55-0

56.3

POUNDS PER SQUARE INCH.

POUNDS PER SQUARE INCH.

Elastic

Limit.

Ultimate

Resist.

Coefficient of Elas.

Elastic

Limit.

Coefficient of Elasticity.

o.

TABLE II.—Continued.

SHEAR.

POUNDS PER SQUARE INCH.

SINGLE SHEAR.

DOUBLE SHEAR.

Elastic

Limit.

Ultimate

Resist.

Elastic

Limit.

Ultimate

Resist.

DOUBLE SHEAR ULTIMATE OVER SINGLE SHEAR ULTIMATE.

749

O.7ÖI

0.755

0-759

0.763

0.762

o 765

57-z

55-8

40 7

38.5

39-5

39-0

36.8

36.7

?5 855 844444 8,54 &44    8,5    8,5<85$455 8,8,8*8 *455555 8,55

O M    HOt    OvO O OHO h b CO^J H O O Cn 4* 10 tO 4*- O O Ó tO 4^ 4>. QwO m W m to 4»- ’O m Q    tO tO (> h >0 4- CWO In Cn

■39,600

38.800

37.800

37.800

38.600

39.400

37.400

36,900

39.100

38.100

37.¹⁰⁰

36.600

35.600

36.800

36.900 36,700 37,70°

37.¹⁰⁰

37.800

38.500

37.800

36.200

36.800

37.800

36.400

36.400 36,700

37.200

36.700

39.300

41.500

41.400

43.400

41.500

41.100

41.400 42,000

41.900

41.300

48.100

47.400

47.100

42.200

42.300 42,300 42,000

41.700 41,000

42.100

42.700

41.500

60.900

60.400

69.700

69.500

69.600

69.100

63.600

63.300 63,000 62,000

63.200

62.800

60.600

61.300 61,900

62.500

62.300

61.400 61,700

61.600

62.100

61.200

61.900

61.800

62.900

63.600

61.300

61.200

62.400 62,000

62.400

61.700

62.200

61.600

60.100 61,000

66.600

65.200 67,000

65.300

65.100

65.700

66.100

65.400

65.400

69.400

69.300

70.100

65.300

65.600

66.400

68.300

68.500

68.600

70.400

70.500

79.600

97.500

99.600

99.100

95.800

96.200

95.200

30,939,000 30,010,000 31,160,000 31,063,000 30,471,000 29,965,000 30,456, 30,885,000 ²7,335,ooo 30,618,000 30,172,000 30,424,000 29,696, 30,075,000 30,371,000 30,918, 30,801,000 31,091,000 30,032,000 31,646, 30,031,000 30,166, 30,415,000 30,232,000 30,030,000 30,556,000 30,011,000 30,210,000 32,965,000 30.097,000 28,950,000 29,391,000 29,890,000 29,186,000 29,252,000 29,464,000 29,907,000 29,899,000 29,270,000 29,706,000 29,500,000 29,238, ²9*439 29,678 29,390,000 30,083,000 30,266,000 29,442,000 29,375,ooo 30,158,000

30,784,

29,045,000

30,236,000

29,989,000

30,025,000

30,944,000

29,291,000

000

OOO

,000

,000

39,00°

39,5oo

39,ooo

41,100

41,100

41,000

40,200

40,200

40,400

41,600

41,600

41.600

38.600 38,600 38,600 3⁸,300 38,300

38,300

41,700

41.700 4¹,7°° 39,900 40,000 40,000 39,S°o 39,7°°

39.900 40,000 40,000

39.700

41.800

41.700 41,700 41,100

41.400

41.200

42.600

42.400

41.900

44,4°°

44.800 45,000 44,ICO 44,300

44.200 41,100

41.400 41,000

41.400

41.600

41.800

55.200

54.400

54.400

59.500

59.200

59.500

29,897,000

27,113,000

28,444,000

29,110,000

29,025,000

29,045,000

30,045,000

28,853,000

29,411,000

30,192,000

29,302,000

29,216,000

29,013,000

29,963,000

29,478,000

29,090,000

29,807,000

28,961,000

29,630,000

28,941,000

29,696,000

²9,437,co°

30,009,000

28,730,000

29,005,000

29,740,000

29,963,000

31,433,000

29,782,000

29,391,000

28,567,000

29,144,000

28,747,000

28,503,000

29,531,000

28,730,000

29,162,000

29,210,000

28,635,000

28,070,000

28,729,000

29,025,000

29,281,000

29,830,000

29,324,000

28,812,000

29,342,000

28,666,000

28,860,000

29,241,000

29,802,000

29,162,000

29,454,000

29,281,000

28,602,000

28,981,000

29,281,000

39,600

45,440

43,600

46,460

1.022

34,600

45,260

38,200

47,450

1.048

3i,5oo

46,020

33,800

47,590

1-034

3J,7°o

46,910

33,5°°

48,390

1.032

31,100

35,900

33,8oo

33,7oo

35,800

30,500

34,400

37,000

36,600

36,70°

4l,5°°

52,500

51,900

44,780

4/ ,60c

46,440

45A90

46.100

49,210

5L470

49,740

51,000

51,280

53,260

70,190

67,760

34,ooo

46,590

1.040

3⁸,50o

47,350

1.062

39,400

35,7oo

40,700

38,600

39,5oo

40,300

40,800

43,800

46,000

48,890

47,210

47,210

51,000

5i,47o

50,940

5U5IO

5²,550

53,39°

i.°53

1.045

1.024

1.036

i.000

1.024

i. 010

1.025

1.002

* Open hearth from Steelton, Pa.

t From Troy, N. Y.

pieces were uniformly about three-quarters of an inch in diameter, and the stretch was in all cases measured on eight inches. The elongations given are per cents of the original length of eight inches.

The reductions of area are the per cents of original sections of the test pieces which indicate the differences between the original and fractured areas.

As indicated, the first half of the Table belongs to specimens of open hearth rivet steel from Steelton, Pa., while the second half contains results drawn from tests on a comparatively wide range of metal from the Bessemer process of the Troy Steel and Iron Co. of Troy, N. Y. The open hearth steel is seen to vary only from 0.15 to 0.17 per cent, of carbon, while the Bessemer metal had carbon varying from 0.11 per cent, to 0.39 per cent, with a wide gap between 0.17 and 0.36 per cent.

The specimens i„ i₂ and i₃ were cut from the two ends and centre of bar 1, and those subjected to tension were located adjacent to specimens of the same name subjected to compression. Similar observations apply to other sets of specimens affected by the same figure or same letter. Hence there is shown in this Table, the relation of different physical quantities belonging to as nearly identically the same material as the possibilities of the case admit.

The coefficients of tensile elasticity exhibit unusual uniformity. Those for the open hearth steel show no variation with the small variation in carbon. Although the tensile coefficients for the Bessemer steel are slightly lower for the lowest per cents of carbon than for the highest, yet some of the lowest coefficients are found for the highest carbons, and it is difficult to determine any essential variation with varying proportions of that element.

While the average of the tensile coefficients is a very little more for the open hearth than for the Bessemer steel, there is really no sensible difference between them. The average ten-sile coefficient may be taken at 30,000,000 pounds per square inch.

Too much importance should not be attached to the percentage of carbon alone in these specimens, as the presence of other elements not given, such as manganese, phosphorus, etc., exert marked influences on the physical characteristics of steel.

The “ Report of the Naval Advisory Board ” prepared by Asst. Naval Constructor R. Gatewood, U. S. N., contains on pages 71 to 75, a large number of tensile tests.

The least coefficient of tensile elasticity given by Lieut. Gatewood is 24,360,000 pounds per square inch, while the greatest value is 30,890,000 pounds per square inch, and the mean 27,720,000 pounds. These values belong to 42 tests of accepted material, and were distributed about continuously over the range covered by the limiting values. The carbon varied from o. 11 per cent, to 0.24 per cent., and these extreme values belonged to about average values of the coefficient of elasticity.

Table III. has been computed from data obtained by David Kirkaldy during his experiments on Fagersta steel plates (“ Experimental Inquiry into the Properties of Fagersta Steel,” series D, Part 1). The test specimens were, in the clear, 2~y inches wide and 100 inches long. The thickness is given in the horizontal row, as shown. The values of the coefficient of elasticity (E) are the greatest and least, in pounds per square inch, for the various intensities    for five unan

nealed yx, y, y₂ and î^-inch (nominally) plates and five similar annealed ones.

These show very irregular elastic behavior. The y inch annealed specimen is the only one which can properly be considered as possessing a true “ coefficient of elasticity ” (about 29,000,000 pounds per square inch) above the stress intensity of 10,000 pounds, the ratios of stress to strain are so very variable. Prof. Bauschinger’s “ stretch-limit ” is clearly shown, for the different specimens, at that point of stress where the

i9

TABLE III.

ANNEALED.

UNANNEALED.

/■

Greatest E.

Least E.

Greatest E.

Least E.

10,000

14,000

18,000

22,000

26,000

30,000

34,000

38,000

42,000

46,000

50,000

45,455,ooo

38,889,000

36,000,000

34,375,ooo 33,333,ooo 31,915,000 20,631, 29,008, 13,084,000 4,670,000 2,294,000

000

33,333,000

30,435,000

29,032,000

28,205,000

25,000,000

1,714,000

1,107,500

821,000

37,037,000 34A46, 32,727,000 31,429,000 30,952,000 29,412,000 4,151,000 1,214,000

000

29,412,000 29,167,000 29,032,000 28,947, 20,968,000 1,765,000 1,066,000 805,000

000

Thickness.

0.125 inch.

0.380 inch.

0.255 inch.

0.625 inch.

values of E are almost annihilated. In these four specimens the first permanent sets were noted at 40,000, 20,000, 30,000 and 20,000 pounds per square inch respectively.

Table III«. contains a most valuable set of results obtained from the tests of full size steel bars for the Blair Crossing bridge by Geo. S. Morison, chief engineer, and it is taken from his report on that structure.

“ E. L.” is the elastic limit.

“ Ult.” is the ultimate tensile resistance.

“ E.” is the coefficient of elasticity in pounds per square

inch.

“ C.” is carbon and “ Mn.” manganese.

The bars were broken in the government testing machine at Watertown, Mass.^but the chemical analyses were made at the Pittsburgh testing laboratory.

The coefficients of elasticity are seen to vary from 28,089,-000 to 30,627,000 and thus sensibly coincide with the coefficients for test specimens found by Prof. Ricketts and Lieut.

Gatewood, with much lower percentages of carbon, phosphorus was determined for these bars, although it is not given in Table Ilia. ; it varied from about 0.06 to 0.2 of one per cent, with an average of about 0.09. It is a significant fact that two of the bars which broke in the head contained the the highest phosphorus.

The

Table VII. contains other coefficients of tensile elasticity for full size bars.

TABLE III«.

Full Size Steel Bars.

BAR.

INCHES.

GAUGED

l’ngth.

INCHES.

LBS. PER SQ. IN. PER CENT. FINAL

PER CENT. OF

E.

E. L. Ult. Stretch. Cont’n.

C.

Mn.

6.5 x 1.02 6.5X 1.38 6.5x1.54 6.5x1.14 6.0 X 1.02 7.ox 1.49 6.5 x 1.03 6.5 X 1.26 6.5 X1.26 6.5X1.40 6.5X 1.39 6.5x1.52 *6.5 X 1.02 6.5 XO.98 *6.5 x 0.98 6.5XO.98 6.5 X 0.98 6.5 x 0.97 6.5x0.97

34,»9°    69,25°

33,67°    66,700

34,47°    69,920

32,57°    64,74°

24,55°    52,39°

24,280    45,27°

36,270 broke

37>²5°

34,680 35,23°

37,44°

32,690 37,250    67,630

37,480    67,800

36,650    72,05°

37,600    68,720

35,810    65,850

33,230    64,410

37,640    68,290

12.2

22.0

16.6

20.0

44.2

45-°

42.1

46.6

18

•7

7-9

17-5

in

broke in

64,75°

64,020 70,470

16.5

20.7

13-1

15.°

15.8

13-8

12.3 12.0

16.4

139

head

head

45-9

43-9

37-9

43 ¹

33-7

38-4

34-1

39-2

49-5

30,456,000 28,846,

000

29,702,000 29,498,000 28,089,000 28,776, 29,940,000 30,120,000 30,627,000 30,030,000 29,585,000 30,030,000 28,901,000 .880.

0.27

0.25

0.295

0.24

broke

0.3°

0.25

0.295

0.23

bar

29,

30,270

29,630

29,

42.4    29,9

7°,

60,

,000

,000

000

,000

000

000

0.30

0.28

0.28

0.30

0.2g

0.28

0.63

0.885

0.68

0.78

in head.

0-745

0.82

0.75

0.76

not broken.

0.665

°.655

0.66

0.65

0.645

0.68

Prof. Alex. B. W. Kennedy (London “ Engineering,” Vol. XXXI., 1881) determined the coefficients of tensile elasticity of specimens of mild steel plates containing about 0.18 per cent, of carbon, and of some specimens of still milder rivet steel.

Twelve specimens of plates (i^4 X % ; 4X ; 2 x y ; 3^2 x ; and 2^ x l/>, all in inches) gave :

GREATEST.

33,670,000

MEAN.

LEAST.

29,882,000.

25,440,000

all in pounds per square inch.

* Broke in head and retested.

Eight other specimens of the same plates gave :

GREATEST.

MEAN.

31,940,000

2g,00I,000

LEAST.

20,110,000

all in pounds per square inch.

As a rule, the thinner plates gave the higher values of E. There were, however, some marked exceptions.

Eleven specimens of \\ inch round rivet steel, turned to about I inch diameter ; two each of } } and 1 A inch round, turned to \ and I inch diameter, respectively, gave :

GREATEST.

31,750,000.

MEAN.

30,670,000

LEAST.

29,790,000

all in pounds per square inch.

Hay Steel.

Some experiments on three different bars of the Hay steel used in the bridge at Glasgow, Missouri, by Gen. Wm. Sooy Smith, gave the following results (“ Annales des Ponts et Chaussées,” Feb., 1881) :

Experiment No. 1.

A bar of rectangular section 2.09 x 1.1 inches, reduced by hammering from a bar 2.6 inches square, was subjected to different intensities of stress varying from about 20,^00 to 54,000 pounds per square inch, at which the following values of the coefficients of elasticity (in pounds per square inch) were found :

GREATEST.

32,900,000

MEAN.

28,824,000

. LEAST. 26,094,000

At 54,000 pounds per square inch there was a “ trace ” only

of permanent elongation or set. The length of this bar, between the observation marks, was about 38.5 inches.

Experiment No. 2.

A round bar 1.04 inches in diameter was subjected to a stress of about 51,200 pounds per square inch, with a stretch of 1.66 millimetres per metre, at which a “trace ” only of permanent set was observed. The resulting coefficient of elasticity was :

E — 30,857,000 pounds per square inch.

The distance between observation marks was about 18.7 inches.

Experiment No. 3.

A bar 5.2 x 1.34 inches was subjected to a stress of about 49,200 pounds per square inch, with a trace only of permanent set and a strain of 0.00171 metre per metre. Consequently the resulting coefficient of elasticity was :

E — 28,772,000 pounds per square inch.

These experiments show that the coefficient of elasticity of Hay steel is not essentially different from that of other material of the same class.

In his “Report on the Renewal of Niagara Suspension Bridge,” Mr. Leffert L. Buck, C. E., gives the following values for the Hay steel used in that work :

GREATEST.

30,830,000

MEAN.

28,000,000

LEAST.

26,400,000

all in pounds per square inch. These results are for eighteen experiments on small specimens.

Ultimate Resistance and Elastic Limit.

In this section, it is to be observed, the “ elastic limit ” is seldom that point at which the coefficient of elasticity (stress over strain) ceases to be essentially constant, but more nearly Prof. Bauschinger’s “ stretch-limit,” at which the increment of strain, due to a constant increment of stress, very suddenly increases, involving a correspondingly great permanent set.

TABLE im.

MAKERS.

MEAN OF.

ULTIMATE RESISTANCE, POUNDS

PER SQUARE INCH.

REMARKS.

Greatest. Mean.

Least.

Coleman, Rahm & Co., Pitts

burg-.......................

Am. Tool Steel Co., Brooklyn Butcher & Co., Philadelphia.

Park Bros., Pittsburg.......

Steel Works, New York.....

Jessup, Sheffield, Eng.......

Anderson & Woods, Pittsburg Coleman, Rahm & Co., Pitts

burg........................

Miller, Barr & Parkin, Pitts

burg.......................

Miller, Barr & Parkin, Pitts

118,400

144,300 118.

000

86,000

100,000

91,200 106,500 112,100 118,

000 85,400

78,50°

100,000

68,000

74,000

93>5°°

118,000

74,000

100,000

“ Very poor steel.” For lathe tools.

2 Annealed.

2 Unannealed.

Annealed.

* îïtff

ill *

: 5'

«5

41 a.

si

16

90,000

103,200    102,200    101,200    Unannealed.

125,000 45,000 112,000 99,200 103,500

147,000 Chrome steel ingot. 69,800 Carbon rivet steel. 65,300 • Carbon steel.

100,000 Carbon rivet steel. 136,300

400    119,200    111,700

260    146,400    115,800

151,864    119,500'

217,400 165,500 141,900

128,000 81,300 i¹3,900 110,100 107,500 151,000 129,000 128,300 142,000 143,600 I35i

126,500

75,45°

112,600

io3,5°°

106,000

148,700

99,900

98,300

120,100

139,3°°

Steel plate.

Cast “machinery steel.” “ Gun metal.”

Chrome steel stave. Chrome steel stave.

190,2

227.500 176,

150.500

IOO

Chrome steel stave. Chrome steel stave. Chrome steel bar.

109.500    Steel normal untemp. 201,341 Temp, in oil at 82» F.

152.500    Temp, in water at 79⁰ F. 132,700 Temp, in water at 79⁰ F.

Table IIIb is condensed from Prof. Woodbury’s history of the St. Louis arch. The last four results are from the experiments of Chief Engineer Shock, U.S.N., while the “ N. Y. Chrome Steel Co.” result is from Kirkaldy’s tests.

The diameters of the (circular) specimens varied from 0.357 inch to 1.00 inch, and their lengths from 3 to 12 inches. The elastic limit varied from 45 to 55 (nearly) per cent, of the ultimate resistance.

TABLE IV.

Rail Steel.

ANNEALED.

UNANNEALED.

NO.

T.

STR.

T.

E. L.

STR.

CARBON,

PERCENTS.

00 CO Tl- *4- "^OO 00 ■'j- Tj- Tj- Tf

co co    to to

to to co CO 4- -P*

co*<r co^i co^j^vj    co ^4

O en O    CO tO o O tO CO en O tO w O

w hi to COHin O O O on ODW 01 M M tO O CO to cn vj 04^ O O O    O Ui cn cn to

coon Ocn^itoOOOOcncnOOOOOcn

37.625

36.625

32.250

38.250 36,500 37,ooo 36,625 38,000

34.250

40.250 36,750 40,250 35,833 37,958

36.625

38.625 36,031 38,125

19.6

15-6

22.5 ii - 5 12.8

M-5

17-5

14- 5 17.o

12.7

10.7

15- 0 ' 1S.9

14. i ii- 7

14.7

17.1

14.2

78,250

77,375

70.750 74,000

73.750

80.750 70,500 78,

69,500

76,000

74,250

76,000

72,958

76,792

74,000

78,375

73,219

77,188

2,S,S,88SKKg 8 8, &£•£-£££ S

r^M C4 ir>0 O co co 10 10 Cl    M o 00 C4 vr>

in N d'd' rf o~ fóo" O co com cóo co co c*

20.5

14-7

22.0

9-5

15.2

15.2

19.2 14.4 17-5

13.2 13-7

14-5

19.8

13-5

14.4

14.8

18.5

13-8

0.324 0-379 0.308 0.438 o. 405 0.384 0.282 0.381 0.367 0-394 0.378 0.388 0.314 0.302 0.391 0.386 0-334 0.390

Table IV. contains the results of one hundred and ninety-two tests of specimens from steel rails which had been in use on the Penn. R. R. during periods of time of different lengths, These results were given by Charles B. Dudley, Ph.D., Chemist to Penn. R. R. Co., in his “ Report to the Supt. of Motive

Power,” and published in the Journal of the Franklin Institute, March, 1881.

The specimens were circular and turned to a diameter of 0.75 inch between shoulders five inches apart.

The following is the notation :

“NoE is the number of specimens for which the other quantities are the average.

“ TE is the ultimate tensile resistance in pounds per square inch.

E. LE is the elastic limit in pounds per square inch.

StrE is the per cent, of original length of ultimate elongation or stretch (2. e., at instant of rupture).

TABLE V.

Bars Unannealed.

CARBON.

STRESS IN POUNDS PER SQUARE INCH.

PER CENTS.

Per cent.

Ultimate.

Elastic Limit.

Str.

Cont.

0.30

0.30

0.30

0.30

0.30

94,760

95,380

95,830 96,020 94,970 ,

Av. = 95,39°,

vnvO iń in in m in in in m

Av. = 55,635

I5-I

COLn LO O

14-3

Av.

30. 1

26.

31-

27-

20. J

28.6

Av.

OOOOO in in m m m

0 6 d 6 o'

112,340 1 112,470 I 111,980 113,320 113,040

Av. = 112,630

65,79° 66,040 66,160 65,550 65,980 ,

Av. = 65,904

M M M

O O O CO O O 01 O Co

Av. = 10. i

ig.

i6.

22.

21.

20.

Av. = 19.6

Str. is the elongation of original length. ContE is the contraction of original section.

TABLE VI.

Annealed Eye-Bars.

DISTANCE FROM PIN-

STRETCH PER-

TENSILE STRESS IN POUNDS PER

5 H

y x

s o

* 5 < 3

w « J w

s S

I foot 3 inches.

4 feet 34 inches. I foot 84 inches.

•ff = -Ay

Tj- O M Tf    ^

CENTAGE OF

ORIGINAL

LENGTH.

i’8

8.2 ]

SQUARE INCH AT

Ultimate.

't

87,400

•Ay

9.2

7.0

94,500

89,300 J >

Elastic Limit.

•fc8o fS = ’Ay

54,026 1

54,H3

f £iiTS

Head into 3 pieces. 5 feet 24 inches.

2 feet i| inches.

o feet 5^ inches, o feet 5^ inches, o feet 5I inches.

xn

Ü9

0.0

S‘6

29- y u I ►

40. j <*

•aV

£'z

•Ay

92,672 ^ o I 8.2 ]

9-3 j*

94,780 J > I ii.8 J

91,570

2.0

2.6

ZZ

69,140 ] eg 63,000 69,400 J >

¹¹J

O' 1

•igi'sS = 'Ay -Fi9‘9S = -Ay

51.762

54,065

52,518

58,473 " 56,059 ^

55,310 J

•aanxova

-nnvw ao aaow

•}3sdfp

"P³II°H

■pappAV

•3I°H

•jgpuiBip in ssipui §£

•PT33H

•3^3 SSOJ313 S3ipUI ‘łJDIip S3ipUI fl

■-

•tasis

•}33J Ol X tpUI £ X S3ipul £

CARBON

PERCENTAGE.

0.30

0.30

0.30

0.30

0.30

0.30

0.30

0.30

0.30

It will be observed that the process of annealing decreased both the ultimate resistance and elastic limit. The results are irregular, however, so far as the strains or elongations are concerned.

Tables V. and VI. are taken from “ Steel in Construction,” a paper read before the Engineers’ Society of Western Pennsylvania, by Albert F. Hill, C. E., 20th April, 1880. Table V. contains the results of tests of specimen bars 3" X fâ" X 30". These specimens were cut from rolled bars which were subsequently manufactured into eye-bars.

Nine eye-bars containing 0.30 per cent, of carbon were made, besides nine others containing 0.50 per cent, of carbon. Each group of nine was divided into three classes, with welded, rolled and upset heads, respectively. These eighteen eye-bars were then put into the testing machine, and the results belonging to the nine containing 0.30 per cent, carbon are given in Table VI.

The results of the tests of the bars containing 0.50 per cent, carbon, were not given, but it was stated by Mr. Hill that they verified the conclusions drawn from Table VI.

Mr. Hill considered that these experiments “ clearly establish the fact that welding of high grade steel for purposes of construction is out of the question in general practice.”

The metal in these bars was “ open hearth ” steel made by Messrs. Anderson & Co., of Pittsburg, Penn.

Table III«, contains the results of Mr. Morison’s tests, to which reference has already been made on pages 290 and 291. These bars are seen to be of high steel for tension members, and as they were annealed it is important to observe that their elastic limits average about half their ultimates, i. e., relatively much lower than the unannealed test specimens. Lower steel would be affected less.

Table VII. gives the results of tests of some full size eye bars of Pernot open hearth steel. The latter was made by the Cambria Iron Co., but the bars were formed by the Keystone

Bridge Co. and tested in the gov’t machine at Watertown, Mass. The tests are reported on pages 194 to 207 of Ex. Doc.

TABLE VII.

6^ Inch x I Inch Full Size Bars.

PER CENT. OF

POUNDS PER SQUARE INCH.

LENGTH

OF

BAR.

INCHES.

Elastic

Limit.

Ultimate

Resistance.

Coefficient

of

Elasticity.

Bearing

on

Pin Hole.

Final

Stretch

Final

Contraction.

222

222

37.480

36,650

67,800

64,000

29,880,000

30,270,000

86,100

79,430

15-9

33-7

Re-

test.

71,560

36.5

Re

test.

72,050

38.4

263

262

37,600

35,8io

68,720

65,850

262

33.230

262

37,640

64,410

68,290

29,630,000

29,960,000

29,670,000

29,960,000

83,650

80,060

79,060

83,960

12.6

34-1

¹¹ • 5

I5-I

13-5

39-2

49-5

42.4

No. 5, 48th Congress, ist Session. The carbon in these bars varied from 0.27 to 0.3 per cent., while the manganese ran from 0.58 to 0.71 per cent., and the phosphorus from 0.074 to 0.098 per cent.

No bar broke in the head. Tests 3 and 4 were the retests of the two parts of the second bar. The parts of the bar rested five days before being retested. The ultimate resistance is markedly increased in these two experiments.

Some of these bars showed considerable dishing of the head as failure took place, although no head failed. This shows that a bar whose width is 6^ times its thickness has reached about the limit of thinness, i. e., that a thinner bar of the same width would be liable to tear in detail through the eye with developing the full resistance of the bar.

Mr. Leffert L. Buck, C.E., in his “ Report on the Renewal of the Niagara Suspension Bridge,” 1880, gives the results of tests of plate specimens of Hay steel used in that work, seven to ten inches long between clamps. These specimens were subjected to various treatments, such as punching, annealing, blows while under stress, nicking on edges, etc., etc., and gave the following results under tension :

Hay Steel.

Greatest. Least . .. Greatest Least... Greatest, Least... Final contraction of j Greatest, ruptured section.. ( Least ...

Elastic limit

Ultimate resistance.

59.431

43.300

97,600

59.370

Ih

in

.S

’S:

Final stretch.

19.4

7.0

42.0

13.0

per cent.

The “stretch” and “contraction ’ are per cents of ten inches and original section, respectively.

Table VIII. contains the results of the experiments of Sir Wm. Fairbairn on the different varieties of English steel given in the left-hand column. The specimens were one inch square, and had previously been subjected to a transverse load. The per cents of strain or elongation are for a length of eight inches, which, it is presumed, included the section of fracture.

Table IX. contains the results of tensile experiments on Bessemer and crucible steel specimen bars by Mr. Kirkaldy for the “ Steel Committee ” (English).

The first part of the table gives the results of experiments on bars turned accurately to a diameter of 1.382 inches with a length “ in the clear ” of 50 inches. It is presumed that the per cents of elongation apply to that length.

The second part of the table gives the results of experiments on bars “in their natural skins,” with a diameter of 1.5 inches and length of 120 inches; to which length the per cents of elongation apply.

PRODUCERS.

TABLE VIII.

Bar Specimens.—1867.

ULTIMATE RESISTANCE

FINAL STRAIN OR

PER SQ. INCH.

ELONGATION.

Messrs. Brown dr³ Co.

Best cast steel, for turning tools . :......

Best cast steel, milder.................

Cast steel from Swedish iron for tools ...

Cast steel, milder, for chisels............

Cast steel, mild, for welding............

Bessemer steel.......................

Double shear steel from Swedish iron....

Foreign bar, tilted direct...............

English tilted steel....................

Pounds.

Per cent.

68,404

9L250

106,714

116,183

110,055

91.972

92,555

76,474

59,538

0.25

I.

I.

XT) O

■3.62

3-31

19.62

5-43

I3-5Ó

21.06

C. Cammel dr³ Co.

Cast steel, termed “ Diamond Steel

Cast steel, termed “ Tool Steel”........

Cast steel, termed “ Chisel Steel”.......

Cast steel, termed “ Double Shear Steel ”.

Hard Bessemer steel...................

Soft Bessemer steel....................

110,055

109,072

120,398

96,665

89,121

81,483

1.53

1.50

2.50 2-37

20.87

20.43

Messrs. Naylor, Vickers Co.

Cast steel, called “ Axle Steel ”.....

Cast steel, called “ Tire Steel ”.....

“Vickers Cast Steel, Special”......

“ Naylor & Vickers’ Cast Steel ” . ...

88,665 91,520 134, J45 118,066

vn O Q u"> 0* O O

VO O' M M

S. Osborne.

Best tool cast steel................

Best chisel cast steel..............

Sates-cup, shear blades, etc........

Best cast steel for taps and dies.... Toughened cast steel for shafts, etc

Best double shear steel............

Extra best tool cast steel...........

Boiler plate, cast steel.............

98,942

123,686

115,849

98,790

103,116

87,931

85,724

,676

0-93

3-18

2.12

1.68

5-25

2-43

in

0.43

13-50

TABLE VIII.—Continued.

PRODUCERS.

ULTIMATE RESISTANCE

FINAL STRAIN OR

PER SQ. INCH.

ELONGATION.

H. Bessemer.

Pounds.

Per cent.

Hard Bessemer steel. Milder Bessemer steel Soft Bessemer steel..

103,085

88,175

78,606

1.87

20.00

19.12

Sanderson Brothers.

83,484

107,940

107,182

75,199

103,960

2.25

3-31

2.81

1-25

3-43

Cast steel from Russia iron for welding..

Double shear steel................

Single shear steel......................

Fagot steel, welded....................

Drawn bar, not welded................

Messrs. Turton âr Sons.

Steel    for    cups........

“    “    drills.......

“    “    cutters.....

“    “    turning tools

“    “    machinery.. .

“    “    punches.....

“    “    mint dies....

“    “    dies........

“    “    taps........

Double shear steel....

100,155

87,552

95,372

80,273

102,915

102,567

106,237

87,471

97,994

73,266

2-75

1.06

1-37 o. 12

1-43

1.62

2.87

087

1.87

0.81

The “ Area of Fracture Section ” (Table IX.) is the per cent, of original sectional area, which, multiplied by that original area, will give the area of the fractured section. The per cent, of contraction will then be given by taking the difference between 100 and the number expressing the “ Area of Fracture Section."

The various grades of steel in the bar specimens of Table IX. exhibit, in the results, the great variations arising with different qualities of that metal.

TABLE IX.

Bar Specimens.—1868 and 1870.

LIM. OF ELAS.

PER SQ. INCH.

ULT. RESIST.

PER SQ. INCH,

PER CENT. FINAL

AREA OF FRAC

ELONGATION.

TURE SECTION.

5, Hammered, tires.....

5, Hammered, axles.....

4, Hammered, rails,.....

4,    Rol’d tires, axles, rails..

5,    Hammered, tires

4, Hammered, axles.....

1,    Hammered, rails.....

Rolled, axles........

3, Chisel...............'

3, Samples.............

Tires................

Rods...............

2,    Samples.............

3,    Gun-barrels..........

Hammered..........

Hammered..........

2, Rods................

2,    Rolled..............

3,    Faggoted............"

3, Samples.............

2,    Samples.............

3,    Tires and axles......

I.

Crucible. Bessemer.

Crucible.

Bessemer.

Pounds.

52.200 49,000 48,000

43.200

46.200 57.300 44,000 42,000 58,240 57,120 58,240 60,500

45.900 37,600 56,000 44,800

59.900

45.900 43,7oo

44,800

39.200 37,ooo

Pounds.

78.600

75,000

74.500

71.500

79.500

91.700

85.400

68.600 118,200 114,300

97.400

93.700 90,800 86,300 83,000

75,400

67,100

79.300 76,600

75.300 75,400

II.I

55-5

WWW

to to

cn co M

51-4

52-3

65.0

co co-p. H4isun o M COO

tnvO M H

O' O' to

62.5

72.1

96.9

89.9

O O M HMW W

o.9

2.0

II . I

ii.g

¹¹ • 5 13.6

94-3 80.0 94.8 100 o

95-9

95-7

98-5

98.7

97.1

55-8

54-4

80.8

58.8

Table II. contains a synopsis of the valuable series of tests of specimens by Prof. P. C. Ricketts. This Table has already been explained on page 287. The tension tests show remarkably uniform results in elastic limit and ultimate resistance, and characterize a most excellent material. With the exception of the two Bessemer specimens containing 0.36 and 0.39 per cent, carbon, all specimens were of very mild steel.

Table X. contains the results of twelve experiments by Mr. Kirkaldy (“Experimental Enquiry, etc., of Fagersta Steel,” 1873) on 2-inch square hammered bars of Fagersta steel, turned

TABLE X.

Fagersta Steel Bars.

PER CENT. OF FINAL

POUNDS OF STRESS PER SQ. INCH, AT

MARK.

FRACTURE.

Elastic Limit. Ultimate Resist. Contraction.

Elongation.

1.2

1.2

1.2

62,400

60,200

63,500

62,033

Av.

8i,952

81,424

92,224

85,200

Av.

coco co w <tw

co w co

2.65

Av.

to M H to

x.8

Av.

Granular.

63,600

62,400

63,200

Av. = 63,066

112,976 109,952 96,9¹²

106,613

Av.

-b. CO.£.

OWO

vO

6. xi

Av.

co co

OvO

i-S

Av.

Granular.

0.6    62,500

0.6    53,2oo

0.6    58,600

58,100

Av.

101,232

97,968

108,696

Av. — 102,632

O co m TT O

w d w

WWW

14-43

Av.

in m co

vJvJW

6.6

Av.

Granular.

co co co

6 6 6

44,200

41,500

40,600

43,100

6i,288

63,120

59.528

61,312

WWW

in TfO

w d w 000

61.52

Av.

Av.

Av.

M W 0*

W O W WWW

16.5

Av.

Silky.

to 1.128 inches diameter, with a length between shoulders of nine diameters or 10.15 inches.

“Mark" indicates the relative hardness, 1.2 being the hardest and 0.3 the softest.

The per cents are of the original sectional area {i.e., of one square inch) and of the original length, which, of course, includes the section of failure.

Sir Joseph Whitworth manufactures his compressed steel by subjecting the molten metal to an intensity of pressure of 13,000 to 14,000 pounds per square inch, immediately after it is taken from the furnace.

Table XI. contains the result of some tensile experiments on some specimens of this steel. Each specimen is turned to a diameter of 0.798 inch (0.5 square inch in normal section) for a length of two inches, for which the per cent, of final elongation is expressed (see “ Proc. Inst, of Mech. Engrs.,” 1875). The specimens are thus seen to be so formed as to give very high results, both for ultimate resistance and elongation.

TABLE XI.

Whitworth's Compressed Steel.

DISTINGUISHING COLORS FOR

ULT. RESIST. LBS.

PER CENT. FINAL

GROUPS.

PER SQUARE INCH.

ELONGATION.

REMARKS.

Red, Nos. X, 2, 3,

89,600

32.0

Blue, Nos. I, 2, 3

107,500

24.0

Brown, Nos. x, 2, 3.......

129,900

17.0

Yellow, Nos. I, 2, 3.....

Special alloy with Tungsten .................

152,300

10.o

Axles, boilers, cranks, propellor shafts, rivets, etc.

j Shafting, drill spin-\ dies, hammers, etc.

Large planing tools, Large shears, drills, etc.

Boring tools, finishing tools for planing,

etc.

161,300

14.0

20

TABLE XII.

Plates.— Unannealed.

0.

DIMENSIONS.

TENSILE STRESS IN

PER SQUARE

AT

fa

⁰ y

CHARACTER OF

FRACTURE.

Elastic Limit.

Ultimate.

0.30

0.30

0.30

0.30

0.30

0.40

0.40

0.40

0.40

0.40

0.50 o. 50 o. 50 0.50 0.50

g inch by 12 inches by 6 feet long, as they came from rolls.—50 inches between jaws of machine.

43,260

44,820

45,110

43,990

44,720

51,620

50.980

51.260

51,100

50,890

58,950

59,200

58,540

58,880

59,330

Av. = 44,380

51,170

Av.

58,960

Av.

79.120

77.840

78,390

77,970

78,280

81,990

81,720

83,730

81,830

83,130

85,790

86*

220

85,560

86,000

86,330

Av. = 78,320

Av. = 82,480

Av. = 85,980

19-3

13-9

10.5

Fine and silky.

Very fine.

Good ; slightly granular on edges.

Boiler Plate.

Table XII. was also taken from Mr. Hill’s paper, and contains results obtained from tests of large pieces of boiler plate of mild steel of the same character as that used in the bars the results of the tests of which are given in Table VII. The stress was in the direction of rolling.

TABLE XIII.

Plates.— Unannealed.

a*

I

o 2

s s à ° 3 5 1 I »

STRESS IN POUNDS PER SQ. INCH.

PER CENT. OF

STRETCH OF

Elastic Limit.

Ultimate.

orig’l LENGTH.

H tO W

CCfc© 44» tifr-

xxxxxxxxxxxxxxxxxx

ojco ccps ccjw co|a^-ccł»cct»#4J ^-afcocefco 44« b^-» oaejafco #4-» 44-,

bO W

bObOWHHMMM

ccf*

47,680

53,400

39,670

38,940

38,390

53,790

44,050

40.160 41,690 49,485 50,745 42,985

43.160 45,280 41,810 41,368 38,390

39,430

66,440

68,900

62,360

62,920

64,860

69,560

65,830

65,690

65,200

77,270

76,925

70,375

70,530

64,970

64,640

67,130

66,990

65,700

O OchNcO O cn O *-o<0

o 00 rj* 00 c^h n t om on c© HMOiciNHweieiciNWNCici

O H in CO CO

For 4 inches. For 10 inches.

Cl

CO

o

Greatest ult. resist, in all these experiments = 78,550

= 60,150

Least

Table XIII. contains the mean results of Prof. Kennedys experiments on mild boiler plate containing about 0.18 per cent, of carbon (London “Engineering,” Vol. XXXI., 1881). The left column shows the number of tests in each group, for which the other columns contain the mean results. The dimensions of specimens are not exact, but closely approximate. The length for which the per cent, of elongation is given, in all cases, contained the fracture. Consequently those '■'■Per cents.” include the “local” extension which exists at the section of fracture. This accounts for the larger values, as a rule, which are found for the 4-inch lengths ; 2^-inch lengths, containing the section of fracture, gave much higher values.

The first five groups were pulled from pins, and the next four from wedge grips. The manner of holding the test pieces, however, was not observed to have any influence on the results.

Within the limits of these experiments, also, the ratio of width to thickness of the specimen seemed to have no influ-It will be observed that Prof. Kennedy’s specimens

ence.

were all (what may be called) “ long ” specimens.

His experiments on some annealed specimens of this steel showed that the process of annealing reduced the ultimate resistance only 3 or 4 per cent.

TABLE XIV.

Fagersta Plate.

THICKNESS,

ELASTIC LIMIT IN LBS. PER SQUARE INCH.

ULT. RESIST. IN LBS. PER SQUARE INCH.

INCH.

Large.

Small.

Long. Mean. Large.

Small.

Long.

Mean.

Unannealed.

CC(31    ccji—

53,300    50,500    38,900    47,567    74,915

37,900    35,400    35,600    36,300    60,480

29,500    29,300    25,400    28,067    51,456

3T,ioo    30,800    27,500    29,800    55,803

28,000    28,300    26,100    27,467    52,924

71,940 ;    55,135    67,330

56,740    54,140    57,120

50,345    48,925    50,243

54,425 j    50,160    53,463

52,475 I    49,280    51,560

Mean.

35,960    34,860    30,700    33,840    59,¹¹⁶    57,185    51,528    55,943

Annealed.

cclco^h-ccl*-

35.500    33,200    26,700    31,800    57,485    55,459    45,4óo    52,801

33.800    30,500    29,800    31,367    54,543    53,715    ' 49,605    52,288

28,900    28,100    25,900    27,633    51,076    50,350    46,740    49,389

27.800    27,900    27,300    27,667    51,338    50,842

25.500    25,700    25,200    25,467    50,432

49,490    50,557

50,025    47,455    49.304

Mean.

30,300    29,080    26,980    28,787    52,975    51,878    47,750    50,868

Fractures all “ silky.”

TABLE XV.

Fagersta Plate.

THICKNESS,

PER CENT. CONTRACTION OF AREA.

PER CENT. FINAL STRAIN OR STRETCH.

INCH.

Large.

Small.

Long.

Mean.

Large.

Small.

Long.

Mean.

Unannealed.

ccl^i    oci»—•

43.I 47.I 48-5 54-2

37-9    42 7

59-3

62.5

50.0

55-1

58.6

61.7

59-7 71.0 61.2 60.7

Mean.

51-2

Annealed.

Hf*

h O'* "ń}“ o o

O nH ci ir>vO vO O vO

10.8

28.2

13-5

5.21

35.5 10.17 64.3 36.1 41.5 20.64 56 6 36.4 40.0 16.30 59.2 37.2 44.7 17.95

54-1

9-45

24.41

32- 57 30.78

33- 01

56.8    58.1    55-4    29.7    35.0    14.05    26.04

10.98 16.88 18.19 19-15

34-4 43-5 17.45

28.4 40. i 42.0

42.5

20.37

29.99

31.76

33-08

31-51

Mean.

60.9    63.5    65.8    63.4

33.I    39.3    X6.50    29.52

In his paper Prof. Kennedy explains in detail his “ elastic limit.” It is the point at which the material “ breaks down,” and considerably above the elastic limit as analytically defined in this work.

Tables XIV. and XV. exhibit the results of Mr. Kirkaldy’s experiments (in the direction of rolling) on some Fagersta

\A

Plate Specimen

Fig.l

steel plate specimens. The plate from which these specimens were taken was marked 0.15, and the material was a mild steel. The “Large” and “Small” specimens were shaped as shown in

Fig. I. The width BC or AD of the reduced portion was ten inches for the “ Large ” pieces, and one and one half inches for the “Small” ones. For the “Large” specimens, the length of the reduced portion (AB or CD) was ten inches ( = width), and four and one half inches ( = 3 widths) for the “Small.” The “Long” specimens were 100 inches by 2)fi inches “ in the clear.”

The results embodied in these two tables are of greater interest and value in consequence of the variety in the relative dimensions of the specimens. They show the important part played by “ lateral strains ” both in the ultimate resistance and final strains, or elongations, of test specimens.

With very few exceptions the following general principle may be deduced from Table XIV. :

Both the elastic limit and ultimate resistance increase with the ratio of the width over the thickness of the plates.

Nearly all the exceptions are in the results which belong to the unannealed, and the “ Long” annealed, specimens. It may be observed in connection with Table III., that the character of the former specimen (possessing a low and irregular value of E) is decidedly abnormal, to which, undoubtedly, its exceptions are due. Annealing the long specimens seems to cause the disappearance of essentially all influence of the relative dimensions of the cross section, where the ratio of width over thickness is, comparatively speaking, small.

One origin of the results above stated is plainly to be found in the lack of lateral contraction in the plane of the plate, in accordance with the principles shown in Article 32, “ Ultimate Resistance and Elastic LimitC

An examination of Table XV. shows the following general result, which, however, has more exceptions than the preceding :

The final contraction and elongation increase as the ratio of width over thickness decreases.

With the long specimens, this dees not seem to hold for less values of the ratio than A- = 6. Whether these principles may hold true, as general ones, or whether they may hold within certain limits (a possibility indicated in the “ Long" specimens), the number and character of these experiments does not permit to be decided. They show, however, that the partial prevention of lateral strains in one direction, whatever may be the cause, will affect, to a considerable extent, experimental results ; also, that in testing plates the shape and relative dimensions of the test piece should be carefully noted.

TABLE XVI.

Open-Hearth Steel Plates—1880.

CROSSWISE.

LENGTHWISE.

PER CENT, i

SPECIMEN,

OF

INCHES.

Stress in pounds per square inch.

CARBON.

Elas. Lim.

Ult. Resist.

Per cent, final stretch.

Stress in pounds per square inch.

Elas. Lim.

Ult. Resist.

Per cent, final stretch.

} X 1} X 18 I x ii x 15 rfc x i x 12

0.30

0.40

49,353

63,227

0.50    65,070

93,339

86,410

83,190

49.510

63,723

65,300

95.453

87,780

84995

H    M    M

cn    O'    00

In Table XVI. are found the results of tests by Mr. Hill (“ Engineers’ Society of Western Pennsylvania,” April 20th,

1880), on specimens of open-hearth steel plate. Each result is a mean of three, and each specimen was cut from unannealed plate in a planer. It is to be particularly observed that each thickness of plate gave essentially the same elastic limit and ultimate resistance, whether the direction of the testing stress was along or across the direction of rolling.

Although the elastic limit increases with the amount of carbon (consistently with the results in Table XII.), yet, it is very remarkable to observe that the ultimate resistance decreases as the carbon increases, which is not consistent with the results contained in Table XII.

TABLE XVII.

Siemens Steel Plate—1875.

THICKNESS POUNDS OF STRESS PER SQ. IN. AT PER CENT. PER CENT.

IN

FINAL

FINAL CON-

INCHES.

Elas. Limit.

Ult. Resist.

STRETCH.

TRACTION.

Unan

nealed.

0.37

0.71

34,600

30,400

72.900

66.900

22.3

24-5

37-5

44-7

Annealed.

OOOOOO

I OCri -P* -P* W O 10 O O O

31.500 31,200 29,800 29,400 26,300

24.500

67,500

66,400

66,

100

65.800

61.800 60,100

24.8 21 i

43-1

44-7

24.8

38.5

26.4

25-5

25.0

44-5

43-3

45-5

Unan-

nealcd.

0-37

0.71

34,300

3° 4°°

72,700

67,3°°

22.4

24.7

37-5

43-6

CROSSWISE.

Annealed.

0.37

0.40

0.42

0.52

0.62

0.70

31,200

31,000

30,000

29,800

26,300

24,500

66,900

66,900

65,800

66.600

60.600 60,200

26.4

26.3

20.4

20.2

22.7

26.0

46.6

49.6

39-°

46.7

35-3

50.7

The ratio of width over thickness of specimen increases from 2 (for the ^-inch, or, 0.30 per cent, carbon) to 5 (for the T\-inch, or, 0.50 per cent, carbon), and Mr. Hill considers this an explanation of this disagreement in the two sets of results. The results of a large number of tests on Fagersta steel specimens of considerable variety in the ratio of width over thickness (Table XIV.) showed a regular increase, in both elastic limit and ultimate resistance, with an increased ratio of width over thickness. Agreeably to these results, therefore, the increase of carbon, in Mr. Hill’s experiments, should have been accompanied by an increase in both elastic limit and ultimate resistance, since an increased ratio of width over thickness accompanied the increase of carbon. The disagreement seems inexplicable, but was probably due to the influence of some unnoticed peculiarity in the treatment of the material in the original plate, or of the specimens themselves.

Table XVII. contains the results of some specimen tests of Siemens steel plate, made by Mr. David Kirkaldy in 1875. The per cents of final stretch are for a length of eight inches, which contained the section of fracture.

Tables XIII., XIV., and XVII. show that, as a general rule, both the elastic limit and ultimate resistance, in mild steel plates, mcrease as the thickness of the plate decreases.

It is also seen that the process of annealing decreases both those quantities.    •

Although Table XVII. shows no very marked result in regard to final stretch and contraction, yet when it is taken in connection with Table XV., it is clear that the process of annealing considerably increases both the final stretch and contraction ; in other words, increases the ductility of the material.

Again, Table XVII. shows that the ultimate resistance of steel plates is essentially the same, both in the direction of rolling and across it. This result is in agreement with that of Mr. Hill’s experiments, as well as that of French experiments on Bessemer and Martin steel plates (Barba, on the “Use of Steel,” translated by A. L. Holley, pages 26 and 29).

Effects of Hardening and Tempering Steel Plates.

In connection with the results given in Table XVII., Mr. Kirkaldy found the following quantities by testing the same sized specimens of the same plates :

Annealed.

THICKNESS.

FINAL

CONTRACTION.

ULTIMATE

RESIST.

FINAL

STRETCH.

0.64 inch.... 57,100 pounds.... 24.1 per cent. ... 52.5 per cent. 0.62 inch. ... 60,500 pounds.... 20.2 per cent... . 48.7 per cent.

Hardened.

49.3 per cent. 45.5 per cent.

0.64 inch.... 64,700 pounds.... 22.4 per cent... 0.62 inch.... 65,050 pounds.... 18.0 per cent

The hardening was done by heating to a cherry-red and cooling in water at a temperature of 82° Fahr.

Table XVIII. exhibits the effects of simply annealing, or of first oil tempering and subsequent annealing, on specimens of gun steel manufactured by the Midvale Steel Company of Phila., for the Ordnance Dep’t U. S. Army, 1884. The results are taken from the Report of Chief of Ordnance for 1884. Oil tempering, or hardening in oil, may be said to almost universally increase both the elastic limit and ultimate resistance.

Annealing in all cases reduces the ultimate resistance and increases the final stretch and final elongation, i. e., increases the ductility. Oil tempering with subsequent annealing, however, is seen in Table XVIII. to produce a very irregular effect, although on the whole it slightly reduces the final elongation, except in the case of high steel, for which the opposite effect is produced. In all cases the combined operations are seen to produce a very material increase in the final contraction. Tempering or hardening increases both the elastic limit and ultimate resistance, but decreases the ductility.

TABLE XVIII.

Specimen Tests.

RESULTS BEFORE TREATMENT.

RESULTS AFTER TREATMENT.

TREATMENT.

Per Ct. of Final

Im „ . . Per Ct. of Final T71i .

Ult. Resist. _ Ult. Resist.

Pounds per    Pounds per

square inch. Elong. Con. Square Inch. Elong. Con.

Annealed..................

Oil tempered and annealed

U    ti    u

100,500

80.400 95,068 88,900 94,100

89.500

97.500 89,000 97,000 86,132 84,880

88.400

10.5 9.0

12.5

20.5 19-5

21.5 21.0 18.s 22.0

11.5 24.0

20.0

103,813

95,820

85,780

5-5

13.0

22.0

OJ H W^HMMMWWWH W 4*- OJ Cn ^ h 10 00>vj CO H tO OJ OJ OOCa

o *0 to b b bo b b b b b b on

1I P„ 11 § § § 11

tC t-T cTvcT cT o' t-C vo* tC ^ h*

OOf^OOOOOOO On On CO On On O

HHHHHHH    H

O to m in to m O O O O O O m O O VO in -t Ń O f>.00 On o 10 o' ’<*■ t>. Os On

H«NHHNHHHNHC4«HHH

20.0

39-7

22.0

35-0

47-0

35-0

4°. o

39-0

41.0

31.0

41.0

48.0

29.0

40.0

34.0

Rivet Steel.

In Table XIX. will be found the results of the experiments of Prof. Alex. B. W. Kennedy (“ Engineering,” 6th May,

1881). The steel was a very mild grade, for which coefficients of elasticity have already been given.

11    15

T¥> T¥

The specimens were turned down, as shown, from and iT¹^ inch “rounds.”

As in all other cases, the elastic limit and ultimate resistance are given per square inch of original section.

Effect of Reduction of Sectional Area, in connection with Hammering and Rolling.

Tables XX. and XXI. give the results of some of the experiments of Mr. Kirkaldy on Fagersta steel bars. The bars were originally three inches square, in normal cross section,

mu

AJ/EStKi

TABLE XIX.

Rivet Steel.

ORIGINAL DIA. OF

DIAMETER OF

POUNDS OF STRESS PER SQ. IN. AT

BAR.

SPECIMEN.

Elas. Limit.

Ult. Resist.

°    *2    Û

•    «    X

H    «    U

9    S    E

s    î    :

B.    E    «

Inch.

Inch.

Hr

0.512

0.505

0.507

43,400

45,200

45,780

44,790

Av.

64.770

65,500

65.770

65,350

e* coo

CM H

Coot, cn

6-os

Av.

Av.

-K:

łt

0.616

0.622

0.616

46,370

46,200

46,220

46,260

Av.

67,960

69,310

6g,2io

68,830

Av.

M tO M

O hO

H CO tO

19.9

Av.

O.804

O.804

O.786

o o

cn in

CO NÆl' ■V- ^3- "Cfr

47,690

60,280

60,750

63,500

61,510

23-3

Av.

Av.

Av.

and were hammered or rolled down to the dimensions shown in the second column in each table. Specimens were then turned down for testing to the diameters given in the third column, for a length of ten inches. The tables give results for duplicate specimens, one set having been unannealed and the other annealed. The fractures belonging to the 3x3 bars were all granular, and those belonging to the 0.5 x 0.5 bars

TABLE XX.

Fagersta Steel.— Unannealed.

DIA. OF

BARS IN INCHES. SPECIMENS,

POUNDS OF STRESS PER

SQUARE INCH AT

w

Oh

INCHES.

Elas. Limit. Ult. Resist.

Hammered.. Rolled......

O.5 X O. £

0.5 x 0.5

0-357

0.357

78,300

46,800

95,960

90,730

6.9

16.0

47.0

43.0

Hammered.. Rolled......

0.619

0.619

49,800    83,720

43-,100    87,760

16.0

16.2

44-7

29-3

Hammered. Rolled......

1.5 x 1-5 1-5 x 1.5

1.009

1.009

46,700 40,500

77,720

79,280

12.6

10.2

38.8

15.8

Hammered.. Rolled......

2x2

2x2

1.382

1.382

44,800 38,300

80,920

84,073

19.2

15-9

35-5

20.8

Hammered.. Rolled......

2.5 x 2.5 2.5 x 2.5

1.694

1.694

34,700

36,600

78,840

72,585

21.4

8.2

26.2

10.3

Hammered.. Rolled......

3x3

3x3

1.994

1.994

38,800    70,080

30,400    62,393

2-3

2-5

4.4

4.4

all silky ; the intermediate ones were partially silky and partially granular.

As a part of the hammering and rolling was done at such a temperature as to essentially amount to cold hammering or cold rolling, the annealed specimens show more truly the effects of the two kinds of treatment than the others.

The following results can be at once observed :

The elastic limit, ultimate resistance and final contraction at

TABLE XXI.

Fagersta Steel.—Annealed.

BARS IN INCHES.

DIA. OF

SPECIMENS,

INCHES.

POUNDS OF STRESS PER

SQUARE INCH AT

PER CENT.

PER CENT.

FINAL

FINAL

Elas. Limit.

Ult. Resist.

STRETCH.

CONT.

Hammered.. Rolled......

0.5 x 0.5 0.5 x 0.5

0.357

0-357

47,800

41,200

82,120

80,210

7-7

9.8

55-0 51 o

Hammered. . Rolled......

0.619

0.619

40,800

40,100

78,650

83,720

15.2 ¹¹ -3

54.0

39-7

Hammered. . Rolled......

1.5 x 1-5 1.5 x x. 5

1.009

1.009

42,300

37,800

77,810

82,780

13-7

15-2

47-7

38.7

Hammered.. Rolled......

2x2

2x2

1.382

1.382

41,300

36,100

78,893

80,330

17.7

16.8

41.2

38-9

Hammered.. Rolled......

2.5 x 2.5 2.5 x 2.5

1.694

1.694

31,300

32,700

66,140

71,630

14.7

13-8

45-4

35-5

Hammered.. Rolled......

3x3

3x3

1.994

1.994

29,800

27,600

69,640

60,193

7-7

8-4

3-8 I 5-4

sectioti of fracture increase very much with the decrease of sectional area for either the hammered or rolled bars.

Other and similar experiments verified these conclusions for both higher and milder Fagersta steels.

The per cents, of final stretch are the greatest for the intermediate sectional areas, whether annealed or unannealed, while the relative effects of rolling and hammering are irregular.

The hammered specimens invariably give the greatest final contraction, whether unannealed or annealed.

If unannealed, the hammered specimens give the highest elastic limit and ultimate resistance ; if annealed, while this holds true (essentially) for the elastic limit, the rolled specimens give the highest ultimate resistance in four out of the six tests.

Annealing decreases both the elastic limit and ultimate resistance ; this was also found to be the case for both higher and milder Fagersta steel specimens, which were similarly tested.

In a set of 24 experiments (precisely the duplicates of those whose results are given in Tables XX. and XXI.) with a higher grade of steel, the greatest final stretch was found to belong to the smaller cross sections ; while in a similar set with a milder grade of metal, the greatest final stretch was found with the larger bars, whether the specimens wTere unannealed or annealed.

Other relative effects of hammering and rolling were somewhat irregular, and seemed to depend on the grade of steel.

Effects of Annealing Steel.

It has not been convenient to separately classify the experimental results showing the effects of annealing, but it has been seen that the process, in general, decreases both the elastic limit and ultimate resistance, and increases the ductility ; the lower grades of steel being the least influenced.

Steel Wire.

Table XXII. contains the results of testing, to ultimate resistance, the wire for which the coefficients of elasticity were given in Table I., together with some belonging to the Chrome Steel Co.’s wire, also tested by the engineers of the New York and Brooklyn bridge. The diameter of this wire was about 0.165 inch (No. 8 Birmingham gauge). As will presently be shown, some of the material was cast steel and other Bessemer steel, all having been hardened and tempered.

TABLE XXII.

Steel Wire.

ULTIMATE RESISTANCE IN

PRODUCER.

rOUNDS PER SQ. IN.

Greatest. Mean. Least.

PRR CENT. FI-

NAL STRETCH.

DIA. FRAC-

TURE, INCH.

J. Lloyd Haigh.........

Cleveland Rolling Mills

,(l)    12

Washburn & Moen

John A. Roebling’s Sons Co

Johnson & Nephew.........

Carey & Moen..............

Chrome Steel Co............

182,450 182,'76 184,019 ³79i⁸33

179,019

206,170

194,227

170,150

175,340    166,169

178,400    172,984

¹761457    169,706

175,291    167,807

162,244    125,321

177,706    163,027

167,880    126,814

160,544    150,657

4-9

1-7

4.2

2.1

4-5

0.8

4-4

1.8

4.8

0.4

6.9

3-1

4.2

0.5

3-4

1.0

— Mr-iMMWIHMMHMMHH

o' o’ o’ d d o' d o’ o' o* o’ o* o' o'

The column “ Per cent, final stretch ” gives the highest values for the 5-feet lengths tested, and the lowest for the 100-feet lengths ; these were the greatest and least found.

The column “ Dia. fracture ” gives the greatest and least values of the diameter of the fractured section in decimals of an inch. There seemed to be no definite relation between the ultimate resistance and contraction of section of rupture.

Col. W. A. Roebling states that the character of the above steel was believed to be as follows :

(1)    English crucible cast steel.

(2)    Open-hearth steel.

(3)    English crucible cast steel.

(4)    Krupp’s Bessemer and cast steel.

(5)    Crucible cast steel and American Bessemer steel.

(6)    English crucible cast steel.

(7)    English crucible cast steel.

(8)    Crucible cast steel.

It is therefore seen that steel drawn into wire possesses an excess of resistance over that in larger masses, as bars ; it thus exhibits the same general phenomenon as wrought iron under similar circumstances.

Shape Steel.

The results given in Table XXIIa. belong to steel angles and deck beams rolled by the Phoenix Iron Co. for cruisers

TABLE XXIL*.

Steel Angles and Deck Beams.

ORIGINAL ANGLE OR DECK BEAM. INCHES.

TEST SPECIMEN.

Section.

Inches.

Length.

Inches.

POUNDS PER SQARE INCH OF ORIGINAL SECTION.

PER CENT. OF FINAL

Elastic

Limit.

Ultimate

Resistance.

Contrac- Stretch in 8    -

tion.

Inches.

2j X 2J

3 X2i 3 *2ł 3 *²ł 3

3 *3 3 *3 3 *3

3    *3

4    *3 4 *3 4 * 3 4 *3 4 *3 4 *3 4 *3

6 X3.25

6 X3.25 6 X3.25 6 X3.25.

2    x 2 2X2 2X2 2X2 2f X2j 2łx2£

2^X 2J

2ł*2£ 3łx3 3łx3 3łx3 3łx3 8 x5 8 x 5 8 x 5 8 x 5

a.

Yb.

b.

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOOCO

39,500

39,209

38,791

39,414

37,820

38,920

37,685

37.303

37,000

41,321

41,206

39,178

38,777

42,342

4!,3iS

43,600

42,012

40,596

40,900

41,616

42,460

40,850

40,380

39,230

39,57°

42,709

39,534

40,272

37,007

64,154

02,022

64,595

62,980

62,733

57,99°

61,991

62,943

66,538

67,356

66,307

66,307

67,878

62,390

64,842

63,256

6j,485

61,407

62,809

62,675

63,078

63,803

66,777

63,67!

64,632

62,513

65,237

63,984

66,873

60,310

63,720

62,770

62,933

63,300

66,730

65,190

62,440

63,130

65^77

63,375

66,677

61,924

52.7

51.5

2Ö.5

25.6

49-4

53-6

54.1

ći i

51.8

51.i

51

53-

51.7

52.

50.

52.2

49-0

51.8

47-7

53.1

5°-9

5!-4

47.0

47.5

48.2

5°-3

51-3

5¹ -6

47.0

48.5

45-0

43-7

48.6

44.6

48.1

47.6

5°-4

57-9

64.6

56.2

44.8

43-9

44-3

50.2

25.4

27.

25.

26.9

25-3*

27.9

26.3

26.6

25-5

25-7

23.1

26.4

24.4

24.6

28.6

25.6

22.8

25-7

25.5

23-3

27.0

28.2

29.0

30.3

27-5

27.8

26.0

25.0

27.8

27.8

26.5

23.8

25.0

25.8

26.5

28.3

26.5

31-3

The angles marked “a’ are of Bessemer steel ; all other members represented in the table are of open hearth steel. The members marked “ b ” are deck beams.

built by the U. S. Gov’t in 1887. The tests were made at the works of the Phcenix Iron Co., and show excellent metal ; the carbon varied from about 0.15 to 0.20 per cent.

Steel Gun Wire.

In 1875, W. E. Woodbridge, M.D., made a large number of tests on the mechanical properties of steel gun wires. The “ wires ” were about 0.3 inch square, having been drawn down from bars 0.375 inch square. The full, detailed account of these experiments is given in “ Report on the Mechanical Properties of Steel, etc., by W. E. Woodbridge, M.D.”

The results given in this section are abstracted from the “ Report ” mentioned.

TABLE XXIII.

Gun Wires—Annealed.

POUNDS OF STRESS PER

SQ. IN. AT

KIND AND MANUFACTURER.

w

B.

Elas. Lim. Ult. Resist.

Crucible steel ; Hussey, Welles & Co.

a    it    tt    11    <t

(io)

<<

it

Martin steel ; N. J. Steel & Iron Co

,(io)

“ Gun-screw wire ” iron ; Trenton Iron Co... Chrome steel ; Chrome Steel Co

(io)

(io)

Norway iron ; Messrs. Naylor & Co German steel ; Messrs. Park Bros. & Co Cemented cast steel ; Messrs. Park Bros. & Co.

41.100 26,800 34,000 39-700 42,300

37,500

39,200

49,000

24.700 43,400

39.200

43.200

39.100 26,000

22.100

35.100

26.700

92,300

50.700

61.700

71.600

72.100

71.800 71,500

94.600

52.600 89,000

77.700

71.100

89.100

47.800 5B700

61.700

57.700 74,900

5-6

45-0

22.0 16.0 15-0 17-3 21-5 18.0 14. i 21. i

67.0

59-0

40.0

46.0

46.0

46.0

9.1

6-3

14.2

8-7

28.5

14.2

15-7

17.0

19-5

37-0 57-0 61.0 6j .0

44.0

41.0

70.0

42.0

52.0

50.0

33-0

Table XXIII. gives results for wires which were annealed at bright red heat, without oxidation.

The per cents, of final stretch are for five inches of original length, except in the case of specimens marked “(10),” which indicates that the per cents, are for ten inches of original length.

Other tests of wires about 0.3 inch square and unannealed., gave the following ultimate resistances in pounds per square inch of original section. The wires were of different varieties of steel, including cast and Martin steel.

84,400.

58,700.

59,200.

130,800.

106,900.

108,200.

135,000.

The elastic limit varied from 35 to 92 per cent, of the ultimate resistance ; and the per cent, of final contraction varied from 11 to 43. The effect of annealing, both on resistance and ductility, is made very evident by comparing the two sets of results.

Effect of Low and High Temperatures on Steel.

The results of some German experiments and the experience of the Massachusetts Railroad Commissioners with steel rails for one year, have already been given in connection with wrought iron.

Table XXIV. contains the results of the experiments by Mr. Chas. Huston, as given in the “Journal of the Franklin Institute” for Feb., 1878.

“ U. R.” is the ultimate resistance in pounds per square inch, while “ C.” is the per cent, of contraction at the section of fracture.

Each result is a mean of three experiments.

„KIND

OF MATERIAL.

“ COLD.”

TABLE XXIV.

572q FAHR.

932⁰ FAHR.

U. R. C. U. R. C. U. R. C.

Charcoal boiler-plate, piled,

Siemens-Martin (exceptionally soft)...    54,600    47    66,100    38

Crucible steel (ordinarily soft).......

Crucible steel (not quite hard enough to temper).......................

55,400    26    63,100    23    65,300

64,400

64,000    36    69,300    30    63,600

M    M

<N CO W

78,400    27

82,800    16    77,300

The method of producing rupture at the desired place was such as to make the specimens partake, to some extent at least, of the nature of “ short ” ones, which, however, would not affect the comparative results.

It will be observed that the charcoal boiler-plate iron gave the highest resistance at the highest temperature, but that all the steels gave the highest “ U. R.” at the intermediate temperature 572⁰ Fahr.

It is somewhat remarkable that in every case but the last (the hardest steel) the contraction of fractured section decreased with the rise in temperature.

Other results for steel will be found in Table IX. of Article 35, and it will be seen that they tend to confirm the conclusions just drawn.

In the “ Annales des Ponts et Chaussées ” for Feb., 1881, page 226, are given the number of breakages of steel rails which occurred in Russia in 1879. The following is the table showing the number of failures for each month of the year.

These results conflict somewhat with those given by the Massachusetts Railroad Commissioners, in Art. 32.

The greatest number is found in the coldest half of the year, but the greatest number for any one month belongs to March, which is not the coldest month. It is probable that this is due to the effect of long wear on the frozen ground of the entire winter in connection with the possible alternate freezing and thawing of the ground in the month of March.

Effect of Manipulations common to Constructive Processes ; also Punched, Drilled, and Reamed Holes.

Table XXV. gives the results of the experiments of Mr. Hill (paper already cited in connection with Tables V. and VI.) on specimens of exactly the same size, and from the same steel plates, as those given in Table XVI.

yy

January .. February . March....

April.....

May.....

Tune.....

July......

August ... September October... November December.

59⁸

854

235

235

160

247

156

214

328

34=

692

The different methods of preparing and treating the specimens are shown in the column headed “ Treatment of specimen.

With the exception of those of the lowest two 0.30 per cent, specimens, the results are averages of a number of experiments.

From these results Mr. Hill concludes: ist. “That both shearing and punching are injurious, per se, to all grades of steel, and cold punching far more so than shearing.

2d. “ That both these operations affect the elastic limit ,    .    . far more than they do the ultimate resistance.

TABLE XXV.

Open-Hearth Steel.

POUNDS OF STRESS PER

s % i

S Ö

Pi

PER CENT.

SQUARE INCH AT

OF FINAL

Elas. Lim. Ult. Resist.

STRETCH.

00000

CO CO CO CO CD

6 6 6 6 6

Cut in planer.....................

Sheared .........................

Punched.........................

Punched and hammered cold.......

Punched, hammered and annealed ..

49-431

32,370

55,780

94,396

74,980

63,410

87,540

100,410

17.00

10.00

0.45

0-55

7-50

0.40 Cut in planer.........

0.40 Sheared..............

0.40 Punched .............

0.40 I Sheared and annealed . 0.40 I Punched and tempered

63,475

46,900

59-350

52,780

87,095

.75,330

68,890

86,160

103,560

w    M

^4 Ocn cn

b b b o b

00000

0.50

0.50

0.50

0.50

0.50

Cut in planer........

Sheared.............

Punched ...........

Sheared and tempered Punched and annealed

65,185

51,666

60,375

57,96o

84,092

79.900 78,400 87,293

84.900

12.50

5-00

4.00

17.00

12.00

3d. “ That apparently the lower grades of steel are proportionately more injuriously affected than the higher grades. .    .    .

4th. “That the injurious effects of shearing and punching can be almost entirely counteracted by subsequent annealing, or tempering in oil from a low heat.

5th. “ That annealing restores the elastic limit to a greater extent than the ultimate, while tempering as above, on the contrary, largely increases the ultimate resistance and ductility, but does not so fully restore the clastic limit.”

Table XXVI. shows the results of other experiments by

Mr. Hill, from the same paper, on the relative effects of drilling, punching and reaming, and punching (with and without annealing) rivet holes in steel plates. These plates were precisely the same as those from which the results given in Tables XVI. and XXV. were obtained. The dimensions of the different specimens are given in the second column from the left.

TABLE XXVI.

Open-Hearth Steel.

PER CENT. CARBON.

ULT. RESIST., LBS. PER SQ. IN. OF EFFECTIVE SECTION.

PER CENT. ELONGATION

OF HOLE.

⁰⁻³⁰    Y\, in. rolled plate, cut

0.30 in planer on all edges.

Strips 2l/₂ ins. wide, 18 0,30 inches long.

0.30

Drilled, 1 in. diameter.

EsaSifiyfri*—«

Punched and annealed, 0.935 in. diameter .. Punched, 0.935 in. diameter................

98.966

22.0

0.40    Drilled, 0.6 in. diameter

% in. rolled plate, cut Punched, 0.5 in.

°'4° asabove. Strips 1% ins. Reamed to 0.62 in.

15 inches long. Punched and annealed, 0.62 in. diameter ...

Punched, c.62 in. diameter.................

diameter

0.40

0.40

wide,

P P ? p

bitn hi <_n

OOOO

,    Drilled, 0.4 in. diameter ....

lb rolled plate, cut a9 Punched, 0.4 in. !.. t

above. Strips 1 inch Reamed to 0.5 in. falameter................

J wide 12 inches long. Punched and annealed, 0.45 in. diameter....

Punched, o 43 in. diameter.................

86,963

89,043

84,9s¹

82,330

w Oj    »0    10

Cn H    Ol    O

b b    b    b

The relative influences of the different operations of drilling, punching, etc., will be emphasized by comparing the results in Table XXVI. with those given in Table XVI.

The operation of punching is seen to considerably injure the material in the vicinity of the punched hole. In every case, the punched specimen gives very much less resistance than any other. It is further to be observed that the injurious effect of the punch is only partially removed by annealing.

Mr. Hill draws the following conclusions :

This

ist. “ That the ‘reamed’ hole is the strongest, and following in the order of strength come the ‘ drilled,’ the ‘ punched and annealed,’ and, lastly, the ‘ cold punched ’ hole, graduation is well defined in all three groups. That the reamed hole should be stronger than the drilled hole, I am unable to account for.

2d. “ That the injurious effect of punching is local, and can be entirely removed by enlarging the hole sufficiently with either drill or reamer. The amount of drilling or reaming required after punching varies with the thickness of the plate and grade of steel.

y yy

3d. “ That although annealing is in a measure beneficial in partially restoring strength and ductility to the punched plate, it will hardly be found available for bridge work ; for, if you attempt to anneal before riveting, the holes will not fit ; if after riveting, you create internal strains of which no account can be taken, and which may subsequently produce failure. Moreover, with proper machinery, punching and reaming will be found much cheaper than ‘ punching and annealing.

In regard to the excess of resistance with the “punched and reamed ” hole as compared with the drilled, it may be remarked that in every case the hole, as reamed, was greater in diameter than the drilled hole. There was, consequently, less material to be ruptured in the former case than the latter. This diminution of cross section makes the reamed specimen a “ smaller ” one than the other and intensifies the “ shortening ” effect of the rivet hole. Both these influences would increase the ultimate resistance, per square inch, of the reamed specimen beyond the drilled one ; but whether they supply the explanation for the whole difference is an open question.

It will be observed that the ultimate resistances for the drilled and reamed specimens, in Table XXVI., run considerably higher than the corresponding quantities in Table XVI., or, indeed, those in Table XXV. ; for which the explanation is simple and obvious. The specimens for Table XXVI. carried one rivet hole each ; and at this rivet hole failure took place. The effect of the hole in any specimen was the restriction of the contraction to its immediate vicinity, and the partial prevention of the latter, which reduced the specimen, to a great extent, to a “short” one. An increase of ultimate resistance was, consequently, to be expected.

The decrease of ultimate resistance with the increase of carbon has already been remarked upon in connection with Table XVI.

r\

The experiments of Prof. Alex. B. W. Kennedy, on the effect of punching and drilling holes, in mild-steel boiler plate, are well illustrated by Table XXVII., which is condensed from one given in London “ Engineering,” 6th May, 1881.

Fig. 2

None of these plates were annealed, but all were drilled or punched as received.

Within the limits of these experiments, Prof. Kennedy observes, neither the width of the test piece nor the different diameters of die, had any essential influence on the results.

The injurious effect of punching is shown by the fact that the punched specimens gave only 92 to 98 per cent, of the resistance of the drilled ones.

It will be noticed that both the drilled and punched specimens gave higher resistances than the natural plate. This is due to the “ shortening ” and other influence (/. c., the disturbance of the lateral strains) of the rivet holes, as before observed, and explained in Art. 32, “ Ultimate Resistance and Elastic Limit. ”

TABLE XXVII.

Punched and Drilled Holes.

TENACITY IN

TENACITY COMPARED WITH

HOLE, INCHES.

DIAMETER OF HOLE,

THAT OF

INS.

TONS, SQ. IN.

NET SECTION.

Nat. Plate. Dril’d Plate.

a(wx|osccwxfco 4m- 4^-44-*

Drilled........

Drilled........

2 punch, 34 die 1 punch, die

Drilled........

Drilled........

I punch, I die. g punch, -{(J- die

0.940

0.940

0.912 — 0.876 0.892 — 0.871 0.926

0.934

0.998 — 0.890 o 945 - 0.875

38.12

38.22

35-04

34- 44

35- 39 34-90

33- 91

34- 3»

1.105 1.108 1.000 1.025 i. 126 I.ill

1-073

1.096

0.918

0.902

0.965

0.978

Lach result is a mean of four, from plate specimens 2, 4, 6 and 8 inches wide. The pitch of rivet holes across the middle of specimens was 2 inches, and the width of each specimen was so chosen that each side passed through the centre of a rivet hole, as shown in Fig. 2. A “ ton ” is 2,240 pounds. The two diameters of punched holes are for the two sides of the plate.

A duplicate set of experiments on 32 specimens of a somewhat softer steel boiler plate, gave essentially the same results (see “ Engineering,” 6th May, 1881).

By experimenting on mild Fagersta steel plates with the thicknesses y, M an<^ % inch, Mr. Kirkaldy found the ratio of the resistance of drilled specimens over that of punched ones to vary from about i.i (for % and ^-inch specimens) to 1.5 (for specimen) when unannealed, and to be about i.i for all the thicknesses when annealed. All the specimens were

12.5    inches wide, with three rows of 0.77 inch holes, pitched

2.5    inches apart, running across the specimens. The average resistance for square inch of net section was greater than that

of the original plate for the drilled holes, but considerably less for the punched ones.

Mr. Kirkaldy states, “ the loss from punching is not constant, but varies with the thickness, and also with the hardness of the material.” He also concluded that punching hardens the material in the vicinity of the punch, and that the effect of punching is counteracted “ to a considerable extent ” by annealing.

The results of Mr. Hill’s experiments, as given in Table XXVI., show, for the thickness of plates there used, that by enlarging the diameter of the punched hole from o. 1 inch to 0.165 inch, by reaming, the injurious effect of the punch is entirely removed.

Experiments on French steel plates, produced by the Bessemer and Martin processes (métal fondu), confirm this result and form a basis for other conclusions, as follows (“ The Use of Steel,” by J. Barba, A. L. Holley, translator, p. 40) :

“ ist. That the effects of punching and shearing are essentially local and spread only over a very restricted region, less than 0.039 inch on the edges of the sheared or punched parts ;

“ 2d. That no cracks exist in this altered region ;

“ 3d. That tempering destroys the effects of shearing and punching by bringing the metal back to the state it would be in if drilling or planing had been substituted for punching or shearing ;

“ 4th. That annealing alone or after tempering destroys, as tempering alone does, the alterations caused by shearing and punching.”

These conclusions relate to plates from 0.27 inch to 0.46 inch thick.

In first-class practice, holes in steel plates and shapes are frequently first punched and then reamed to a diameter 0.125 inch greater.

Experiments on some narrow specimens of steel plate seem to indicate that conical punching (the die 0.16 to 0.20 inch greater in diameter than the punch) injures the material less than cylindrical punching (with a clearance of perhaps inch).

In the working of steel plates and shapes, during ordinary constructive processes, all local pressure of great intensity, and hammering while cold or at a low temperature, tend to produce internal strains of great intensity or other changes in molecular condition which cause the finished plate or shape to be liable to great brittleness and unlooked-for failure of a local character.

For these reasons M. Barba gives the following directions in regard to the working of steel :

“ ist. Avoid any local pressure of whatever nature it may be; 2d. If local pressures have been produced by blows of a hammer, the action of the punch, etc. (which may, as we have seen, cause ruptures), heat the piece to a cherry-red in a very regular manner and as much as possible in its entirety—the whole of it at once—and let it cool in the open air on a homogeneous surface, which has all over equal conducting power. This simple reheating, which may be considered as annealing for plates and bars, on account of their slight thickness, restores to the worked metal its original qualities, even if it was in a very unstable state of equilibrium.”

If a large amount of working (such as bending or curving) of a single kind is to be done to a single piece, it is best, if possible, to heat to a cherry-red and do the work by stages, rather than all at once ; and then anneal after the working is completed. If the working is local and the heating irregular, it may be necessary to anneal once or more during the progress of the work.

Local heating in the production of the ordinary steel eye-bar head, for example, frequently gives much trouble, unless resort is had to subsequent annealing.

These difficulties in the working of steel are found more pronounced in the higher grades, and much experience is still needed before they can be entirely overcome.

On account of the homogeneous character of the metal, upsetting processes, as in riveting, etc., seem to injure the molecular condition of steel much less than that of iron.

Bauschinger s Experiments on the Change of Elastic Limit and Coefficient of Elasticity.

The details of these experiments are given in “ Der Civil-ingenieur,” Part 5, 1881. The manner of application of the tests, and remarks on the quantities, elastic limit, stretch limit, and final load, will be found by referring to page 262. The following is the notation :

E.    L. = elastic limit in pounds per square inch.

S.-L. = stretch limit in pounds per square inch.

F.    L. = final load in pounds per square inch.

E. = coefficient of elasticity in pounds per square inch.

Bessemer Steel.

IN ORIGINAL CONDI-

TION.

AFTER 69 HRS.

AFTER 0.5 HR.

AFTER 68 HRS.

E. L.

S.-L.

25,970

40,380

43,272

8,760

14,970

51,920

55,470

71,850

F. L.

46,140

57,690

70,080

E.

29,848,000

29,549,000

29,009,000

30,146,000

A small specimen of this Bessemer steel, about an inch in diameter, gave an ultimate resistance of 75,800 pounds per square inch.

The elastic limit rises twice after two long periods of rest, and falls in a very marked manner after the short rest of 0.5 hour.

The stretch limit rises steadily while the coefficient of elasticity falls twice and then rises above its original value.

Prof. Bauschinger was the first to determine, in regard to Bessemer steel, that by stretching the metal beyond its elastic limit its elasticity is elevated, not only during the time of action of the load, but also during a longer period of rest, without load, of one or more days ; and that, in this manner, the elastic limit may exceed the load which caused the stretching. (Dingler’s Journal, Band 224.)

Fracture of Steel.

The character of steel fractures has been incidentally noticed, in some cases, in the different tables.

If the steel is low, or if some of the higher grades are thoroughly annealed, the fracture is fine and silky, provided the breakage is produced gradually. In other cases the fracture is partly granular and partly silky, or wholly granular.

In any case a sudden breakage may produce a granular fracture.

Ejfcçt of Chemical Composition.

The ten sets of results given in Table XXVIII. are taken from a great number of similar ones established by the United States Test Board, “ Ex. Doc. 23, House of Rep., 46th Congress, 2d Session.” The physical phenomena developed in connection with a given chemical constitution may be observed at a glance.

The amount of final contraction of fractured section may be accurately estimated by comparing the ultimate resistances of the original and final sections.

•ni "ös wad

-sa-i ‘iiwn oixsvaa

23,000

38,600

5i,i74

50,553

66,493

71,709

5°.522

75.294

28,500

28,35°

kc

-

25,945,000

29,090,000

24,581,000

26,076,000

27,056,000

25,398,000

33,054,000

27,542,000

28,085,000

26,282,000

►tj

M

»

32.67

23-17

15.92

8.17

8.00

5-83

10.75

7-33

34-33

29.00

ULTIMATE RESIST-

ANCE IN POUNDS PER

SQ. IN. OF

Final

Section.

Original

Section.

ooo‘i8

37,800

121,400

132,000

133,100

128,700

129,500

154,300

144,800

146,100

101,300

68,000

95,666

1x3,106

TABLE XXVIII.

Nickel.

Cobalt.

Copper.

0.010

800-0

0.002

£io-o

0.013

0.007

0.020

trace

0.003

0.023

trace

0.003

0.016 I 120,602

120,602

93,7H

135,269

52,000

48,245

0.009

0.003

810-0

0.013

£00 o

none

none

none

trace

trace

o 002

0.044

0.028

0.028

PERCENTAGES OF

Manga

nese.

Comb.

Carbon.

Graph-

ite.

Silicon.

Phos

phorus.

Sul-

phur.

0.020

090-0

0.038

0.046

0.045

0.273

0.262

o. 192

1S00

0.246

0.386

0.677

0.830

1.285

1.244

911 -o

0.042

O.OII

0.006

810-0

0.040

0.030

0.033

CSO’O

0.014

none

0.014 0.145

0.076

0.017

0.135

£io‘o

0.015 0.134

o. 190

901 -o

0.246

o. on

891-0

£100

Sio-o

£l0'0

0.045

o 109

0.004

0.003

0.003

trace

trace

trace

none

0.029

1100

The specimens were circular in section and either 0.625 inch or about 0.8 inch in diameter, while all possessed a length of 6 inches.

Art. 35.—Copper, Tin and Zinc, and their Alloys.—Phosphor Bronze.

Coefficients of Elasticity.

Table I. gives the coefficients of elasticity (E) of the various metals and their alloys, according to the various authorities. These coefficients were determined by experiments in tension, and E is given in pounds per square inch.

TABLE I.

METAL.

AUTHORITY.

E.

REMARKS.

Brass.....

Tin......

Zinc......

Gun Metal

T3 -<v

8,930,000

4,608,000

13,680,000

9,873,000

Cast metal.

Zinc............

Zinc............

Copper .........

Copper .........

Brass...........

“ Berlin Brass.”..

Wertheim.

12,828,000

12,420,000

17,702,000

14,958,000

12,148,000

13,192,000

Gun Bronze.....

Alloy...........

Alloy...........

Tobin’s Alloy ... Copper.........

Thurston.

11,468,000

13,514,000

14,286,000

4,545,000

9,091,000

Copper, 8 ; Tin, 1.

Ingot.

“Étiré,

Annealed.

ZnCu₂.

Zn^Cui 7.

Copper, 0.90 ; Tin, o. io (nearly). Copper, 0.80; Zinc, 0.20. Copper, 0.625 ; Zinc, 0.375. Composition, below table.

Cast metal.

Tobin’s alloy is a composition of copper, tin, and zinc, in the proportions (very nearly) of 58.2, 2.3, and 39.5, respectively. The value of E for this metal, and those for the two preceding and one following it, are calculated for small stresses and strains given by Prof. Thurston in the ‘‘Trans. Am. Soc. Civ. Engrs.,” for Sept., 1881.

TABLE II.

Cast Tin.

E.

I-

E.

1,950

1,147,000

3,200

96,400

2,360

472,000

4,000

41,540

2,580

172,000

Broke at 4,200 lbs.

TABLE III.

Cast Copper.

E.

I-

E.

800

10,000,000

12,000

2,000

9,091,000

13,600

18,750,000

8,193,000

4,000

9,091,000

16,000

2,235,000

8,000

14,815,000

22,000

137,000

Broke at 29,200 lbs.

The values of E (stress over strain) for different intensities of stress (pounds per square inch) for cast tin, cast copper, and Tobin’s alloy, are given in Tables II., III. and IV.

up ” is the intensity of stress in pounds per square inch, at which the ratio E exists.

Each of these metals is seen to give a very irregular elastic behavior.

Tables II., III. and IV. are computed from data given by Prof. Thurston in the United States Report (page 425) and “ Trans. Am. Soc. Civ. Engrs.,” already cited.

E.

2,000

4,000

6,000

8,000

10,000

14,000

TABLE IV.

Tobin's Alloy.

4,545,000

4,545,000

4,688,000

4,938.000

5,263,000

5,110,000

E.

18,000

24,000

30,000

40,000

50,000

60,000

5,455,000

5,941,000

6,250,000

6,390,000

4,744,000

3,436,000

Broke at 67,600 lbs.

Ultimate Resistance and Elastic Limit.

Table V. is abstracted from the results of the experiments of Prof. Thurston as given in the “ Report of the U. S. Board Appointed to Test Iron, Steel and other Metals,” and “Trans. Am. Soc. of Civ. Engrs.,” Sept. 1881. The composition of the various alloys was as given in the table, which also contains results for pure copper, tin and zinc. All the specimens were of cast metal.

The mechanical properties of the copper-tin-zinc alloys have been very thoroughly investigated by Prof. Thurston (“Trans. Am. Soc. of Civ. Engrs.,” Jan. and Sept., 1881). As results of his work he has found that the ultimate tensile resistance, in pounds per square inch, of “ ordinary bronze, composed of copper and tin ... ., as cast in the ordinary course of a brass founder’s business,” may be well represented by :

Tc — 30,000 + 1,000/ ;

“ where t isthe percentage of tin and not above 15 per cent.”

PERCENTAGE OF

Copper.

Tin.

IOO

OO

IOO

OO

IOO

OO

90

IO

80

20

70

30

62

38

52

48

39

61

29

21

71

10

79

00

90

100

00

00

Gun

IOO

Banca.

100

Bronze.

10

80

00

62.5

00

58.2

2-3

100

90.56

81.91

71.20

60.94 58-49 49.66 41.30

32.94

20.81

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

10.30

0.0

0.0

0.0

70.0

57-

45-

66.

58.

10.

60.

65-

8-75

21.25

23-75

23-75

2.30

50.00

10.00

20.00

TABLE V.

POUNDS STRESS PER SQ. INCH AT

Zinc. Elastic Limit.

Ult. Resist.

PER CENT. FINAL

Stretch.

Contract n.

uiinO

OOOOOOOOOOOOO    O    O    ooi^cjo

OOOOOOOOOOOOO    O    O    OMcoco

9.42

17.93 28.54 38.65 41.10 50.14 58.12 66.23 77-63 88.88

100.00

20.25

21.25 31-25

10 00

39    48

40    00 30.00 15.00

11,620

11,000

14,400

15,740

5,585

19,872

12,760

27,800

26,860

32,980

5,585

0.05

0.005

0.065

O.037

0.004

10.0

8.0

688

688

2,555

2,820

3,5oo

1,670

2,000

10,000

10,000 9,000 16,470 27,240 16,890 3,727 1,774 9,000 14,450 4,050 18,000 (?) 1,300 2,196 3,294 30,000 (?)

5,000 (?) 21,780 (?)

2,555

2,820

1,648

4,337

6,450

3,5oo

2,760

3,500

31,000

33,140

48,760

67,600

29,200

32,670

30,510

41,065

50.450

30,990

3,727

i,774

9,000

14.450 5,400

31,600

1.300 2,196 3,294

66,500

9.300 21,780

3,765

O.07

0.36

O.36

4.6

32-4

31.0

4.0

7-5

31-4 2g. 2

20.7 10.1

5-0

v0 O OO

M coo co

0' 6 0 d

co xr> m r>. m

co 6 6

15-0

13-5

00.0

00.0

00.0

00.0

00.0

00.0

00.0

150

75-0

47.0

86.0

40.0

29-5

8.0

16.0

43-0

38.0

28.0

17 o

II-5

0000

d d d d

000

Ńo o

TABLE V. — Continued.

PERCENTAGE OF

POUNDS STRESS PER SQ. INCH AT

PER CENT. FINAL

Copper.

Tin.

Zinc. Elastic Limit. Ult. Resist.

Stretch.

Contract’n.

70.00

75.00

80.00

55-00

60.00

72.50

77-50

85.00

OOOOOO O O O immn in

OinOOdi^NC* M W    MM

M W 4^ H M 10

MOMM^OOO

tn O Otntn O O O OOOOOOOO

24,000 (?) 12,000 (?) 12,000 (?) 22,000 22,000 11,000 20,000 12,000 (?)

33,i4o 34,96o 32,830 6S,c)oo 57,400 32,700 36,

4-0

25-0

i,, o

ii .o 0.0

34,5oo

3-0

The values of the elastic limit in the lower part of the table were not at all well defined.

“For brass (copper and zinc) the tenacity may be taken

as :

Tz = 30,000 + 500-3-.

where z is the percentage of zinc and not above 50 per cent.”

He found that a large portion of the copper-tin-zinc alloys is worthless to the engineer, while the other, or valuable portion, may be considered to possess a tenacity, in pounds per square inch, well represented by combining the above formulas as follows :

Txt — 30,000 + 1,000/ + 500,3-.

These formulae are not intended to be exact, but to give safe results for ordinary use within the limits of the circumstances on which they are based.

Prof. Thurston found the “ strongest of the bronzes ” to be composed of :

Copper Tin ... Zinc ..

55-0

0.5

44-5

100.0

This alloy possessed an ultimate tensile resistance of 68,900 pounds per square inch of original section, an elongation of 47 to 51 per cent, and a final contraction of fractured section of 47 to 52 per cent.

The first and sixth alloys of copper, tin and zinc, in Table V., are called by Prof. Thurston “Tobin’s alloy.” “This alloy, like the maximum metal, was capable of being forged or rolled at a low red heat or worked cold. Rolled hot, its tenacity rose to 79,000 pounds, and when moderately and carefully rolled, to 104,000 pounds. It could be bent double either hot or cold, and was found to make excellent bolts and nuts.”

As just indicated for the particular case of the Tobin alloy, the manner of treating and working these alloys exerts great influence on the tenacity and ductility.

Baudrimont found for a copper wire 0.0177 inch in diameter, an ultimate resistance of about 45,000 pounds per square inch* the wire being unannealed, while for a diameter of 0.064 inch, Kirkaldy found about 63,000 pounds per’square inch.

Prof. Thurston states : “ brass, containing copper 62 to 70, zinc 38 to 30, attains a strength in the wire mill of 90,000 pounds per square inch, and sometimes of 100,000 pounds.”

All of Prof. Thurston’s specimens were what may be called “long ” ones, i. e., they were turned down to a diameter of 0.798 inch for a length of five inches, giving an area of cross section of 0.5 square inch.

Gun Metal.

Major Wade (“ Reports of Experiments on Metals for Cannon,” 1856) made many experiments on a gun metal composed of copper 89 and tin 11 (very nearly), called gun bronze.

He found that different methods of manipulation of the molten metal and of treatment, as in cooling, affected to a great extent its resistance.

TABLE VI.

Gun Bronze.

ULTIMATE TENSILE RESISTANCE, POUNDS PER SQUARE INCH.

MINUTES IN

LADLE.

Gun-heads.

Small bars.

10    rH

O    en    O

Highest....    17,698

17,825

17,761    50,973

31,132

Mean

29,2x6 Lowest. ...    23,381

28,775

28,995    52,330

28,153

24,064

23,722

56,786

28,082

Density varied from 7.978 to 8.823.

Table VI. gives the average results of a large number of experiments made by Major Wade. It shows the great range in the tenacity of the different specimens.

General Results.

Table VII. gives general results' of various European experimenters. T represents the ultimate tensile resistance in pounds per square inch.

Some of these results are from the experiments of early investigators, who attached little importance to the size and form of the test specimen. In all the cases the results would be more valuable if the circumstances of testing were given. Those belonging to the more unusual alloys, however, possess considerable general interest in spite of the uncertainty surrounding their experimental origin. The presence of a little phosphorus in copper is seen to increase its resistance in a marked manner.

METAL.

TABLE VII.

EXPERIMENTER.

T.

Copper, wrought...........................

Copper, cast...............................

Copper, bolts, with phosphorus o.oi..........

Copper, bolts, with phosphorus 0.015.........

Copper, bolts, with phosphorus 0.02..........

Copper, bolts, with phosphorus 0.03..........

Copper, bolts, with phosphorus 0.04..........

/—Proportions.—,

Gun metal, copper 12, tin 1.................

Gun metal, copper II, tin I.................

Gun metal, copper 10, tin 1.................

Gun metal, copper 9, tin 1.................

-Weights in 100.——,

Alloy, copper 84.29, tin 15.71...............

Alloy, copper 82.81, tin 17.19...............

Alloy, copper 81.10, tin 18.90...............

Alloy, copper 78.97, tin 21.03, brasses.......

Alloy, copper 34.92, tin 65.08, small bells.... Alloy, copper 15.17, tin 84.83, speculum metal Tin.......................................

Anderson.

Mallet.

33,600

19,000 to 26,100

16.900

38.400

45.400

47.900 50,000

29,000 30,700 33,000 38,100

36,100 34-050

39>⁶5o 30,500 3A40 6,950 5,600

,-Proportion.-,

Aluminium bronze, copper 90, Al. 1..........

Aluminium bronze, greatest.................

Tin, cast..................................

Zinc, cast.................................

Brass, yellow..............................

Brass, yellow, copper 67, zinc 33..........,..

Brass, tube, copper 62, zinc 38...............

Brass, tube, copper 70, zinc 30...............

Brass, wire ...............................

Muntz metal, copper 60, zinc 40..............

Sterro-metal, copper 10, iron 10, zinc 80......

Sterro-metal, copper 60, iron 3, zinc 39, tin 1.5 Sterro-metal, copper 60, iron 4, zinc 44, tin 2.0

—cast in sand........................

—cast in iron, annealed.................

—cast in iron, forged red hot............

Copper 60, iron 2, zinc 37, tin 1.............

Copper 60, iron 2, zinc 35, tin 2.............

Copper 55, iron 1.77, zinc 42.36, tin 0.83—

cast..................................

—forged red hot........................

—drawn cold..........................

Anderson.

Rennie.

Stoney.

Rennie.

Anderson.

Everitt.

Dufour.

Anderson.

73,000

96.300 4,740 3,ooo

18,000

28,900

103,000

80,600

91.300 49-300

7,100

53,8oo

43.100

54,300

69,400

76,200

85.100

60,500

76,

85,100

Phosphor Bronze, and Brass and Copper Wire.

Table VIII. contains the results of the experiments of Mr. Kirkaldy on phosphor bronze, with two results each for brass and copper wire.

TABLE VIII.

Phosphor Bronze.

ULT. RESIST., POUNDS PER

METAL.

E. L.

SQUARE INCH.

FINAL STRETCH.

Annealed.

Unannealed.

Phosphor bronze

wire

55,800

55,200

40.500

26,300

21,700

Copper wire Brass wire..

75,000 74,000 63,700

54.100

50.100 102,750 121,000 121,000

139.100 159 500

151.100

63.100 81,200

49.400

47,800

53.400 54,200 58,900 64,600 37,coo 51,500

37-5

34- x

42.4

44.9

46 6

42.8

34-1

36. 5

The diameter of the phosphor bronze wire varied from about 0.06 inch to 0.11 inch ; that of the copper wire was 0.064 inch, and that of the brass wire 0.0605 inch.

The final stretch is the per cent, of the original length, and belongs to the annealed wire.

The contraction of fractured section for the phosphor bronze specimens varied from about four to thirty-two per cent, of original area.

The first five results belong to metal of the same composition but subjected to different treatment.

■* Some specimens tested by Mr. Kirkaldy gave as low as about 21,700 pounds per square inch.

Experiments on Rolled Copper by the “ Franklin Institute Committee."

The results of these experiments are contained in the “Journal of the Franklin Institute,” for 1837.

That committee found, as a mean of 66 experiments, the ultimate resistance of rolled copper to be 32,826 pounds per square inch. The temperature of the copper varied from 62° to 82° Fahr. “ The irregularities of strength in the different specimens varied from 1.9 to 4.8 per cent, of the mean tenacity.”

The resistance was found to be the greatest at ordinary temperatures, and to decrease with acceleration as the temperature increased.

Variation of Ultimate Resistance and Stretch at High Temperatures.

The results contained in Table IX. were obtained at Portsmouth (England) Dockyard, and were published in the Engineer, 5th Oct., 1877.    “ R" is the ultimate tensile resistance in

pounds per square inch, and “ St." is the per cent, of stretch for a length of 10 inches in all except the last (steel) specimen.

At 250° to 350° the gun-metal specimens lose about half their ultimate resistance and nearly all their ductility. Phosphor bronze loses about one-third of its resistance and two-thirds of its ductility at 300° to 400°. Muntz metal and copper are not much affected, nor is cast iron. Wrought iron and steel gain in ultimate resistance but lose in ductility. These

CO

vO

94-5

5-0

2-5

St.

20.0

R.

35,840

28,160

23,040

16,320

83.00

2.00

15.00

St.

26.O

26.0

25.5

26.3

26.0

23.°

25.0

25.O

1.2

0.0

'S

31,040

30,720

28,800

29,440

28,160

27,840

27,840

27,840

9,728

9,728

TABLE IX.

GUN METAL RODS I INCH IN DIAMETER.

85.00

5.00

10.00

91.00

7.00

2.00

87-75

9-75

2.5

St.

St.

R.

7S-

R.

21.0

o-gi

19-5

19.0

16.0

18.3

17.0

2.0

2.0

2.0

33,6oo

0*91

33,6oo

8.8

36,800

35.200

33,920

33,47°

32,960

096‘ee

31,680

16,640

16,000

14,720

15-5

14.0

0*6

o-e

0.0

0*0

0*0

33,6oo

33,600

29,440

16,320

16,960

16,640

17,600

co o    q

oo 10 o o

34.240

24,640

18,880

18,880

»7-75

9-75

2.5

st.

0*01

TI.O

0*01

io.o

O r-ł co O oo O

Copper Tin.... Zinc. ..

R.

34,240

32,320

33,6°°

31,040

32,320

32,000

28,800

15,680

16,960

000*91

•HHva ‘aanxvHajwHX

Atmos.

Ocoi

i5°°

200°

₀°Ss

₃oo°

ooSC

oOofc

45°°

500⁰

LANDORE STEEL.

0.74 in. x 0.49in.

St.

26.0

22.5

IT-3

£•01

0*01

R.

76,920

76,920

8I,770

83,200

WROUGHT IRON.

Yorkshire. Dia. = 0.70 inch.

■ts

R.

o⁻£z

24-3

£ ii

7-5

o-£i

13.8

60,030

60,030

60,680

62,640

60,680

o£g‘\L

Remanufac

tured

Dia. = o 74 inch.

St.

R.

22.0

18.8

o-£i

15.0

15.0

£•£1

12.5

12.0

o-£i

O’03

58,500

62,010

62,010

62,010

63.180

76,050

77,220

63,180

79,560

79,560

TABLE IX.—Continued.

CAST IRON.

COPPER.

MUNTZ

PHOSPHOR BRONZE.

oS*

Dia. = i

Welsh.......25

Scotch,

Scrap..

inch.

• 25

St.

R.

0.72

St.

Dia.

62

38

St.

0.0

0.0

0.0

0*0

0*0

0*0

0*0

0*0

0*0

0*0

30,720

27,070

28,220

28,160

30,080

30,080

25,660

oo9'£z

25,600

2.5

2-5

4.0

5-0

7.0

°‘9

60

o    o

vO    VO*

56.810

57,43°

54,960

54'34°

53,¹¹⁰

oii‘££

on‘££

oZg‘i£

51,260

48,220

2.5

2.5

3-9

3-9

2-5

2.3

2-3

S-o

Dia. = o. Copper. Zinc....

R.

81,900

79,56°

84,240

80.145

78,39°

76,050

76,050

ooz‘oZ

71,955

72,54°

7.0

Dia. = i inch. Copper ... 92.5 Tin

Phosphor.. 0.5

St.

V7-5

17.0

18.0

O'gl

c *5i

12.0

7.0

5°

5.0

'X

38,980

39,300

39,°4°

ozZ‘g£

37,I2°

36,800

30,080

27,140

24,320

26,880

•HHVd ‘aunxVHadlAIHX

Atmos.

IOO°

,,o£i

,.ooz

fp£z

300"

„o££

400°

45°’’

500°

results would probably be somewhat varied by different processes of, and treatment in, manufacture and construction.

The Muntz metal and copper specimens were rolled.

Bauschingers Experiments with Copper and Red Brass.

Prof. Bauschinger extended his experiments on the repeated application of stress so as to cover not only wrought iron and steel, the results of which have already been given, but also copper and red brass.

The notation is that already used :

E.    L. — elastic limit in pounds per square inch.

S.-L. — stretch limit in pounds per square inch.

F.    L. = final load in pounds per square inch.

E. = coefficient of elasticity in pounds per sq. inch. Ini y = “ immediately.”

The copper specimens were of rolled material about 16 inches long with a cross section about 2.4 inches by 0.64 inch. These specimens gave an ultimate tensile resistance, per square inch, of 28,900 to 32,000 pounds and a final contraction of 27 to 46 per cent.

The red brass specimens were turned to about one inch in diameter and 16 inches long. They gave ultimate tensile resistances, in pounds per square inch, varying from 19,600 to 23,460.

With one exception, in the second case of red brass, the elastic limit and stretch limit were elevated by repeated application of stress, whether immediately or at the end of following periods of rest.

The effects on the coefficient of elasticity are seen to be somewhat irregular.

IN ORIGINAL CONDI-

TION.

E. L

5,475

S.-L.

F. L.

E.

16,651,000

IN ORIGINAL CONDI-

TION.

E. L.

2,560

S.-L.

F. L.

E.

10,011,000

IN ORIGINAL CONDI-

TION.

E. L.

5,840

S.-L.

F. L.

E.

16,097,000

Copper.

im’y.

im’y.

im’y.

8,030

11,670

14,600

17,249,000

Copper.

AFTER 18 HRS.

7,320

14,650

16,295,000

Copper.

AFTER 43 HRS.

8,030

11,670

14,600

16,780,000

S,79°

14,650

11,450

22,880

21,970

16,154,000

T5,770,000

AFTER 23 HRS. AFTER 24 HRS.

8,080

14,680

22,010

15,197,000

AFTER 44.5 HRS.

10,340

14,760

22,160

16,069,000

11,520

23,040

15,756,000

AFTER 51.5 HRS.

15,390

23,080

15,472,000

E. L.

S.-L.

F. L.

E.

E. L.

S.-L.

F. L.

E.

E. L.

S.-L.

F. L.

E.

Red Brass.

IN ORIGINAL CONDI-

TION.

im’y.

im’y.

7,680

9,090

9,260

13,960

16,770

16,070

19.550

19,240

12,030,000

12,485,000

12,727,000

Red Brass.

IN ORIGINAL CONDITION.

AFTER 17.5 HRS.

AFTER 21 HRS.

5,600

9>¹¹5

8,550

14,020

16,820

12,322,000

16,130

19,240

19,640

12,314,000

12,485,000

Red Brass.

IN ORIGINAL CONDITION.

AFTER 53 HRS.

3,48o

9,090

i3,9!0

16,690

13,239,000

16,070

12,940,000

The explanation of the method of applying these repeated stresses will be found in connection with the results for wrought iron on page 262.

Art. 36.—Various Metals and Glass.

Coefficients of Elasticity.

The following values of the coefficients of elasticity, in pounds per square inch, contained in Table I. are taken from Wertheim’s “ Physique Mécanique,” pages 57 and 58. The cotable I.

COEFFICIENT OF ELASTICITY.

METAL.

EXPERIMENTER.

Drawn.

Annealed.

Lead

Wertheim.

2,564,000

2,457,000

Cadmium

7,713,000

7,555,000

Gold

11,564,000

7,942,000

Silver

10,463,000

10,155,000

Palladium

it

16,721,000

13,920,000

Platinum

24,237,000

22,067,000

efficients are the means of a large number of tensile experiments, with the exception of that for cadmium, which was derived from experiments on transverse vibrations. This last method gave results which differed, in most cases, from the direct tensile ones not more than the latter did from each other.

Wertheim also gives for the tensile coefficients of elasticity of some different glasses :

Ultimate Resistance and Elastic Limit.

Wertheim determined the elastic limit of many of the more rare metals, such as those named in Table L, and they are here given in pounds per square inch :

DRAWN.

Mirror glass.....

Goblet (common) Goblet (fine) Goblet (violet)... “ Crystal ”......

.. E — 8,792,000 po • • E — 9,559,000 . . E — 8,589,000 .. E = 7,110,000 .. E — 5,830,000

p-

in

<u

P-»

c¹

IT.

-    3*

-    3-

ANNEALED.

Lead Cadmium Gold Silver Palladium . Platinum .

284

142

4,266

4,266

7,110

20,600

to

to

to

to

to

to

355

171

19,200

16,350

25,600

37,000

His “ limit of elasticity ” is that force which will permanently elongate the metal 0.000,05 of its original length, and all his experiments were made on wires of very small diameters.

The following ultimate resistances were found for wires about ₇Vth inch in diameter by Baudrimont (“ Annales de Chimie,” 1850) :

Gold.....

Silver.... Platinum. Palladium

on LO 4-0. M M JO O ^4 LO CO

'S 8 8 8

ÏÔ5J

W u ł. IO (0 N O O'

fiii

.. .1 * - ~ 3 8*

J . , *0

-t

I

: : ; g

5'

The ultimate resistances of some other metals are :

METAL.

EXPERIMENTER.

ULT. RESIST.

Cast lead. Sheet lead

Rennie. . Navier , Jardine

Soft solder (| tin, lead). Rankine

Pipe lead

1,824 pounds per square inch. 1,926    “

2,240

7,500

Sir Wm. Fairbairn (“ Useful Information for Engineers,” second series, pages 226 and 267) found the following ultimate resistances in pounds per square inch by direct pull on straight tensile specimens :

Flint glass .. Green glass Crown glass,

2,413 pounds.

2,896

2,546

The specimens were of circular section and about 0.53 inch in diameter.

By subjecting spherical glass shells to internal pressure he found the following ultimate resistances in pounds per square inch :

Flint glass.., Green glass , Crown glass

4,200 pounds. 4,800    “

6,000    “

The thickness of these shells varied from about 0.02 (crown and green glass) to 0.08 (flint glass) inch.

Art. 37.—Cement, Cement Mortars, etc.—Brick.

The ultimate tensile resistance of these materials depends upon many circumstances, and only a few out of a great number of experimental results will be given. These results will be so chosen as to be representative, but a full and detailed knowledge of the action of cements and cement mortars, under different circumstances of testing and variety of composition, must be acquired by an examination of the original memoirs.

Mr. Bremermann, during the construction of the St. Louis bridge, found in 18 experiments with pure “ Fall City” (Louisville) cement : Elastic limit, 16 to 104 pounds per square inch, with a mean of 72 ; ultimate resistance, 35 to 147 pounds per square inch, with a mean of no; coefficient of elasticity, 800,000 to 6,930,000 pounds per square inch, with a mean of 2,239,000.

TABLE I.

KIND OF CEMENT.

Toepffer, Grawitz & Co., Stettin, Germany

Hollick & Co., London. ... ..............

Wouldham Cement Co., London.........................

Saylor’s Portland Cement, Coplay, Pennsylvania Wampun Cement & Lime Co., Newcastle, Pennsylvania.

Pavin de Lafarge, Teil, France..........................

A. H. Lavers, London .    ................................

Francis & Co .........................................

Wm. McKay, Ottawa, Canada................... .....

Borst & Roggenkamp, F Louquéty & Co., Boulon

Riga Cement Co., Riga, Russia.......

Scanian Cement Co., Lomma, Sweden........

Bruno Höfmark, Port-Kund, Esthland, Russia..........

Coplay Hydraulic Cement, Coplay Cement Co., Coplay, Pa ..

Charles Tremain, Manlius, N. Y.............................

Allen Cement Co., Siegfried’s Bridge, Pennsylvania .

P. Gouvreau, Quebec, Canada........................

Riga Cement Co., Riga, Russia......................

Anchor Cement Co., Coplay, Pa ....................

Cumberland Hydraulic Cement Co., Cumberland., Md

Société Anonyme, France.......................

Howe’s Cave Cement, No. 1, Howe’s Cave, N. Y “    “    “ No. 2,

“    “    “ No. 3,    “

Società Anonima, Emilia, Italy, ist quality ...

ly, Limehouse, Ontario, Canada London .......................

Thomas Gourd A. H. Lavers,

Scott’s Selenitic Cement (Howe’s Cave lime and plaster) Parian Cement, Francis & Co., London, ist quality. ... “    “    “    “    “2d quality.....

A. H. Lavers,

NO.

TESTS.

12

10

12

12

12

14

10

12

12

14

12

12

12

12

10

12

12

20

12

c.

NO.

T.

TESTS.

1.439

I.330

1,140

1,078

968

2IĆ

216

199

ia4

168

931

i5⁸

i,-0

907

882

192 ! 163 I

826

141

764

⁶93

580

292

276

276

234

230

208

196

184

183

170

170

181

*54

126

122

298

i,i75

696

205

132

ic8

134

112

154.

38

47

43

47

44

41

41

29

28

43

31

27

22

23

2iJ

52

181

169

51

Portland cements.

Roman and other cements.

Table I. contains the results of the tests made by General

Q. A. Gillmore at the Centennial Exposition, Philadelphia, in 1876 (Van Nostrand’s Magazine, March, 1877). The specimens were prepared “ by mixing them dry in each case with an equal measure of clean sand, tempering the mixture with water to the consistency of stiff mason’s mortar, and then moulding it into briquettes of suitable form for obtaining the tensile strength in a sectional area 1 ]/₂ inches square, equal to 2square inches. The briquettes were left in the air one day to set, then immersed in water for six days, and tested when seven days old. After thus obtaining the tensile strength

in each case, the ends of the broken specimens were ground down to i y₂ inch cubes, which were used the same day for obtaining the compressive strength by crushing.” The columns “ No. tests ” give the number of experiments from which were obtained the mean values contained in the columns C and T.

C — Ult. Compressive resistance in lbs. per sq. in.

T — “ Tensile

The former is given here in order to avoid the repetition of the makers’ names hereafter.

TABLE II.

AGE.

N. R.

N. N. J.

N. N. J.

■h

T.

T.

T.

16 hours.....

24    hours.....

36 hours .....

48 hours.....

60 hours.....

72 hours .....

4 days ......

7 days ......

10 days ......

14    days ......

15    days ......

19 days ......

21 days ......

25    days ......

1    month

64 days ......

69 days ......

2    months....

3    months.... 3i months. ..

4    months.... 6 months....

1    year ......

year .....

2    years.....

47

55

63

58

44

6o

68

97

120

140

257

64

50

49

134

71

73

77

105

97

92

108

117

157

202

221

175

177

250

33i

241

385

266

All specimens of neat cement mixed with fresh water at about 60 Fahr.

Table II. contains the results of a large number of tests of Rosendale cement (Trans. Am. Soc. of Civ. Engrs., Vol. VII., Feb. 1878—“ Improvement of the South Boston Flats,” by Edward S. Philbrick), also that from Howe’s Cave.

“ N. R.” signifies “ Newark & Rosendale Cement Co.”

“ N. N. J.” signifies “ Newark, N. J., Lime & Cement Co.” “ N. Y. R.” signifies “ New York & Rosendale Lime & Cement Co.”

“L. H.” signifies “ Lawrence Cement Co.—Hoffman Rosendale Cement.”

“ H. C.” signifies “ Howe’s Cave Cement.”

T — ultimate tensile resistance in pounds per square inch.

The values of T are averages of from 1 to 2,217 tests.

Table III. gives some results of the same neat cement specimens mixed wÿth salt water at about 6o° Fahr.

TABLE III.

N. N. J.

T.

AGE.

N. N. J.

L. H.

T.

T.

24 hours....

39

2 days

H

LTV vO

2 months..

134

3ï months. .

275

7 days

56

6 months. .

237

io days.....

20 days.....

I month ...

105

•<1 O' cn

VO CO o

X year .... 1} year

367

205

2 years....

311

234

The notation is the same as that of Table II.

Experiments and Conclusions of Wm. W. Maclay, C. E.

The following tables and conclusions are abstracted from “ Notes and Experiments on the Use and Testing of Portland Cement,” by Wm. W. Maclay, C. E. (Trans. Am. Soc. of Civ. Engrs., Vol. VI., Dec., 1877).

He made many valuable experiments in order to determine the effect of different temperatures at different periods in the life of the cement.

TABLE IV.

Portland Cement.

TEMP. (FAHR.) OF WATER IN WHICH BRIQUETTES WERE IMMERSED.

AVERAGE TENSILE STRENGTH PER SQ. IN.

7 days old.

21 days old.

Temperature of cement paste when bri- Temperature of cement paste when briquettes were moulded.    quettes were moulded.

32°    40°    50°    60°    70°    32⁰    40° i 50°    6o°    70⁰

4°°

50°

6o°

70°

Lbs. Lbs. 265    307    236

292

313

403

WW M M r co o o\4^ a* OvO 0 4^“

CO to to to t"⁴

CO COL/l M Ö* O' M H to “

40°    40⁰    45⁰    45°    45⁰    46°    46°    43⁰    43    43

Temperature of air during last 24 hours.

Table IV. contains some of the results of Mr. Maclay’s experiments which were made to determine the effect of the temperature of the water in which the specimens of neat cement were mixed. After the briquettes were moulded at temperatures shown in the upper horizontal column, and immersed in water either 7 or 21 days at the temperature shown in the left vertical column, they were taken out and dried in air at the temperature shown in the lower horizontal row. The resistances in pounds are averages of five or more results.

From these and many other similar results, Mr. Maclay concluded that the ultimate resistance follows “ very closely the temperature of the water in which the sample briquettes were kept immersed, the warmest water giving the greatest tensile strength ; that a change from 40° to 70° in the water, increases the tensile strength of the briquettes of neat cement, seven days old, from 63 to 168 pounds per square inch, or from 33 to 127 per cent. ; of the briquettes of mortar, gauged 1 to 1, the same change in temperature increases them from 32 to 59 pounds per square inch, or from 87 to 133 per cent.; of the briquettes gauged 1 cement to 2 of sand, from 19 to 37 pounds per square inch, or from 95 to 176 per cent. After an interval of three weeks the changes in tensile strength, .    .    . become

less marked, and, in some cases, an increase in the temperature of the water diminishes the tensile strength.”

Other experiments seemed to “ show quite conclusively that the tensile strength increases directly as the temperature of the air when the cement is being gauged, and inversely as the temperature of the air to which it is exposed for the last 24 hours before breaking.”

“ Exposing the briquettes after six days’ immersion in water to a high drying temperature weakens them so invariably, that some interference with the setting seems clearly demonstrated.....”

Some further experiments led him to conclude “that Portland cement gauged with either fresh or salt water, hardens more rapidly when immersed in salt or sewer water than in fresh water for the first seven days, and that this increase in the tensile strength probably continues for at least a year, more rapidly in the one than in the other.’’

Table V. contains the results of Mr. Maclay’s tests on the relative influence of fine and coarse sand.

TABLE V.

Portland Cement.

Showing difference in mortars with fine and coarse sand.

g § r

w n > M

§ I

H

I VOL. CEMENT.

2 VOLS. SAND.

I VOL. CEMENT.

3 VOLS. SAND.

Fine sand. Coarse sand. Fine sand. Coarse sand. Fine sand. Coarse sand.

I week

T month...

tn

CO VO

95

202

en m

00

VO O' 4^ GO

4^ M vO O

w rf

TABLE VI.

Portlatid Cement.

W.

BRAND.

T.

P. C.

101.5

108.0

112.0

113.0

114.0

115-0 116.0 117.0 118.0 119.0 120.0 121.0 122.0 I23.O I24.O I26.O I28.O I32.O

Alsen & Son, Itzehoe, Germany.

<<    <<    U    <4

Burham.

Gibbs.

Burham.

C<

Saylor’s American Portland.

a    n    11

326

340

289

317

285

2S0

316

301

276

305

232

269

281

272

260

265

369

322

93

91

87

87

>S

90

S3

85

85

85

84

90

90

90

90

90

87

W — weight in lbs. per bushel.

T = ultimate tensile resistance in lbs. per sq. in.

P. c. = percentage that passed through sieve of 2,500 meshes per sq. in.

“ The deduction from this table is, that by increasing the fineness of the sand of which the mortar is made the tensile strength is diminished, and that this reduction in tensile strength increases with the amount of sand used in the mortar.” An English experimenter, Lieut. W. Innés, R. E., was led to the same conclusion.

Table VI. shows the results of experiments by Mr. Maclay, made to determine the connection between the weight per bushel and tensile resistance of specimens seven days old. Commenting on the results he says, “ The close connection between the weight .    .    . and the tensile strength .    .

is now proved to be very uncertain, if not entirely fallacious.” As will hereafter be seen, these experiments invalidate a contrary conclusion reached by the English experimenter, John Grant, M. Inst. C. E., in 1864.

TABLE VII.

FRANCIS.

GILLING-

HAM.

PORTLAND.

}. B. WHITE BROS.

FRANCIS

SAYLOR'S

BROS.

AGE.

Port

land.

Port

land.

Port

land.

Port- Neat    i cement

land.    cement, i sand.

Roman.

Medina.

American

Portland.

1    week.

278

256

184

212

359

332

5³4

52 5

513

554

17S

182

v ■_> 374

416

423

432

306

301

378

383

428

8 »

i “

I yr

3⁶5

304

459

327

310

426

264

270

ił

443

611

414

326

497

367

355

339

224

I    I    J    J    I cn Cn I -P*.    1    I    M    M

I    J    I    J    I ov^o'vo' | -2    J    J    S wKh 8 o'00O

I a|&| |£! |i|s| I“

j| I si I rill sHsH

94

*35

132

136

200

183

203

212

123

122

128

136

162

168

"J- I    I    moo m -t-vo m    I    vo    I    O    I

VO    M OPtCO NN    CO    O    I

co J    Tf if iq ui m m    m    m

Vertical columns give tensile resistance in pounds per square inch.

Table VII. shows the variation of ultimate tensile resistance, per square inch, of various neat cements (with one exception) with age. The left vertical column shows the time during which the specimens were kept under water, and the other vertical columns the ultimate tensile resistance in pounds per square inch.

The results for the Burham, Francis, Tingey, Gillingham and Saylor Portland cements are from Mr. Maclay’s paper; the others are from“ Experiments on the Strength of Cements,” London, 1875, by John Grant, M. Inst. C. E. ; all are means of great numbers of experiments.

The mean results of Mr. John Grant’s experiments on Keene’s and Parian cements are given in Table VIII.

TABLE VIII.

keene’s cement.

PARIAN CEMENT.

AGE AND TIME IM-

In water.

Out of water.

In water.

Out of water.

MERSED IN WATER.

T.

T.

T.

T.

I week ..

242

243

264

216

260

267

in co

CO    O

c*    w

224

258

242

310

i month

218

260

242

332

202

288

222

322

226

320

232

380

..A

T = ultimate tensile resistance in pounds per square inch.

As the result of his experiments on Portland and Roman cements Mr. Grant was led to the following conclusions :

1.    Portland cement, if it be preserved from moisture, does not, like Roman cement, lose its strength by being kept in casks, or sacks, but rather improves by age ; a great advantage in the case of cement which has to be exported.

2.    The longer it is in setting, the more its strength increases.

3.    Cement mixed with an equal quantity of sand is at the end of a year approximately three-fourths of the strength of neat cement.

4.    Mixed with two parts of sand, it is half the strength of neat cement.

5.    With three parts of sand, the strength is a third of neat cement.

6.    With four parts of sand, the strength is a fourth of neat cement.

7.    With five parts of sand, the strength is about a sixth of neat cement.

8.    The cleaner and sharper the sand, the greater the strength.

9.    Very strong Portland cement is heavy, of a blue-gray color, and sets slowly. Quick setting cement has, generally, too large a proportion of clay in its composition, is brownish in color, and turns out weak, if not useless.

to. The stiffer the cement is gauged, that is, the less the amount of water used in working it up, the better.

u. It is of the greatest importance that the bricks, or stone, with which Portland cement is used, should be thoroughly soaked with water. If under water, in a quiescent state, the cement will be stronger than out of water.

12.    Blocks of brick work, or concrete, made with Portland cement, if kept Tinder water till required for use, would be much stronger than if kept dry.

13.    Salt water is as good for mixing Portland cement as fresh water.

14.    Bricks made with neat Portland cement are as strong at from six to nine ’months as the best quality of Staffordshire blue brick, or similar blocks of Bramley Fall stone, or Yorkshire landings.

15.    Bricks made of four parts or five parts of sand to one part of Portland .cement will bear a pressure equal to the best picked stocks.

16.    Wherever concrete is used under water, care must be taken that the water is still. Otherwise, a current, whether natural or caused by pumping, will carry away the cement, and leave only the clean ballast.

17.    Roman cement, though about two-thirds the cost of Portland, is only about one-third its strength, and is therefore double the cost, measured by strength.

18.    Roman cement is veiy ill adapted for being mixed with sand.

Mr. Don J. Whittemore has proposed the following formula for the ultimate tensile resistance of cements :

T - A y ]Sf ;

in which T is the ultimate tensile resistance in pounds per square inch ; A, an empirical coefficient, and iV the age of the

cement in days. For Portland cement (up to two years old) he gives ;r = 10, and A —

267 to 356, by the aid of Mr. Grant’s experiments.

(See Trans. Amer. Soc. of Civ. Engrs., Vol. VIL, Sept.

1878).

Fig.1

Fig. I shows the briquette used by Mr. Maclay;

Fig. 2, that used by Mr.

Grant, while that shown in Fig. 3 is the one generally used at the present time.

Each briquette is 1 y₂ inches thick, giving a breaking section of X 1% = 2.25 square inches. In such testing it is very necessary that the pull should be central.

J-—7>ź--—3

!£1

Fig-2

Fig-3

Artificial Stones.

The tensile resistances of many artificial stones and some natural British ones, can be found in “ A Practical Treatise on Natural and Artificial Concrete,” by Henry Reid, London, I879.

On page 198 he gives the following results of Professor Ansted’s experiments, T being the ultimate resistance per square inch :

Ransome stone (artificial)

Portland stone..........

Bath stone.............

Caen stone.............

old,

He also gives for “ Victoria ” (artificial) stone, three months

T = 740 pounds per square inch.

From 35 experiments on “ rock concrete” pipe two years old, for drainage and sewage purposes, Mr. Reid found :

HIGHEST.

MEAN.

LOWEST.

T — 700,

444

213 pounds per square inch.

Bricks.

Mr. Francis Collingwood, C. E. (Trans. Amer. Soc. of Civ. Engrs., Vol. VIL, Sept., 1878), found, as a result of twelve experiments on “ good Haverstraw stock brick,” the following values :

HIGHEST.

MEAN.

LOWEST.

169

90 pounds per square inch.

Adhesion between Bricks and Cement Mortar.

General Q. A. Gillmore (“ On Limes, Hydraulic Cements and Mortars ”) cemented Croton bricks together crosswise and then separated them by a pull. He used pure cement paste and mortars of various proportions, by volume, of cement to sand, but never more sand than 1 volume of cement to 2 volumes of sand. Nearly all the cement was Rosendale, although some specimens were prepared with Hancock (Maryland) or James River cement. Bricks so cemented in pairs were kept 320 days and then separated. Reviewing the results, Gen. Gillmore says, “ In tearing the bricks apart, at the expiration of the time specified, in a majority of cases the surface of contact of the brick and mortar remained intact, the adhesion to the brick overcoming the cohesive strength either of the bricks themselves, or of the mortar composing the joint between them. The results, therefore, although interesting for other reasons, furnish no entirely satisfactory measure of the power of adhesion.”

Also, “ At the age of 320 days (and perhaps considerably within that period) the cohesive strength of pure cement mortar exceeds that of Croton front bricks. The converse is true when the mortar contains fifty per cent., or more, of sand.”

TABLE IX.

RATIO OF ADHESION

MORTAR OR PASTE.

MATERIALS CEMENTED.

ADHESION PER

SQ. INCH IN LBS.

TO RESISTANCE OF

PURE CEMENT.

Pure cement ..........

I vol. cement I vol. sand 2 “ “

Croton bricks.

30.8

15-7

12.3

6.8

5-2

4-3

Pure cement...........

I vol. cement I vol. sand I “

I “

I “

Fine cut granite.

3-3

27-5

20.8

12.6

1.00 0.51 0.40 0.22 0.17 o. 14 O. II 1.00

0.76

0.46

9-2

7-9

0-33 o. 29

Table IX. contains the results of another series of experiments by General Gillmore, made for the purpose of determining the adhesion to Croton front bricks and fine cut granite, of mortars containing different proportions of sand. “ The bricks were used wet, and were well pressed together by hand. They were wetted with fresh water every alternate day for 29 days, the age of the mortar when tested. Each result is the average of five trials.”

Art. 38.—Timber.

Table I. contains the results of experiments made by Che-vandier and Wertheim (“ Mémoire sur les Propriétés Mécanique du Bois;” by E. Chevandier and G. Wertheim, 1846). The

TABLE I.

KIND OF WOOD.

COEFFICIENTS OF ELAST.

ELASTIC LIMIT.

ULTIMATE TENSILE RESIST.

Hornbeam Aspen.... Alder .... Sycamore. Maple ...

Oak

Pounds.

l,335,ooo

1,329,000

1,021,000

1,616,000

i,459,ooo

1,765,000

Pounds.

3,060

4,380

2,570

3,270

3,870

Pounds.

IO,

240

»O

>0

to

Birch

1,214,000

2,431,000

3,340

8,530

to

Beech

Ash

1,263,000

1,450,000

1,798,000

2,300

3,300

to

6,110

5,o8o

Elm

Poplar

1,364,000

1,436,000

1,027,000

2,890

2,620

to

2,100

9,640 (i) 9,940 (i) 2,800 (1)

Acacia

(    901,000 1

( 2,206,000 ) ■; to

Fir

Pine

2,018,000 j 2,218,000 ^

1,319,000 \ 1,088,000

4,540

3,060

2,320

11,280

5,940

3,530

results are means, and were obtained from small, well seasoned rods, with cross-sectional areas varying from 0.30 square inch (some nr specimens) to 1.50 square inches (one oak specimen), and are given in pounds per square inch. The results indicated thus “(1),” belong to one tree only, others, to several.

The limit of elasticity is that force per square inch which will produce a permanent elongation of 0.00005 of the original length.

These experimenters found that the elongations produced by different weights were composed of two parts, one permanent and one elastic ; the latter being essentially proportional

ULT. RESIST. IN

KIND OF TIMBER.

EXPERIMENTER. SP. GRAY. POUNDS PER SQ. IN POUNDS PER

INCH.

TABLE II.

SQ.INCH.

Oak, English.................

Oak, English.................

Oak, French..................

Oak, Dantzic...............

Oak, American White........

Oak, American, Baltimore.... Oak, African (or Teak).......

Teak, Moulmein .............

Iron Wood, Burmah .........

Chow, Borneo................

Greenheart, Guiana.........

Sabicu, Cuba...............

Mahogany, Spanish... Mahogany, Honduras Mahogany, Mexican. . Eucalyptus, Australia :

Tevvart .........

Mahogany ..........

Iron-Bark...........

Blue Gum............

Ash, English.............

Ash, Canadian............

Beech ..................

Elm. English ............

Rock Elm. Canada......

Hornbeam. England.....

Fir, Dantzic.............

Fir, Riga................

Fir. spruce, Canada.....

Larch, Russia...........

Cedar, Cuba ...........

Red pine, Canada......

Yellow pine, Canada ... Yellow pine. Canada .... Pitch pine, American.... Kauri pine, New Zealand Georgia pine, American. Locust, American ....

White oak, American.

Spruce, American.....

White pine, American Hemlock.. .

Laslett.

Hatfield.

0.858 0.893 0.976 0.838 0.9Ć9 0.742 0.971 0.777 i. 176 1.134 1.141 0.917 0.765 0.659 0-655

3,837

7,57i

8,102

4,217

7,021

3,832

7,°5²

3>3°i

9,656

7,I99

8,820

5,558

3,79i

2,998

3,4²7

1.169 0.996 1.150 i 049 0.750 0.588 0.705 0.642 0.748 0.819 0.603 0.553 O.484 0.6

49 0.469

0.553 O 551 0.552

0.659 O 544

10,284

2,940

8,377

6,048

3,780

5,495

4,853

5,460

9,182

6,405

3,231

4,051

3,934

4,203

2,870

2,705

2,759

2,259

4,666

4,040

¹6,ooo

24,800

I9,5°o

I9,5°°

I2,ooo

8,700

Values for this column will be found computed in Art. 63.

to the load and the former measurable even for small loads and variable not only with the load but also with the time during which the load acted : that the coefficient and limit of elasticity augmented with the seasoning, but that the greatest elongation diminished under the same circumstances ; that if the coefficient of elasticity and hltimate resistance along the fibre be taken as units, the coefficients of elasticity along the radius and tangent to the tree, will be 0.165 and 0.091 respectively, while the ultimate resistances in the same directions will be respectively o. 163 and 0.159, these results being considered averages.

The ultimate tensile resistances of many woods, domestic and foreign, are given in Table II., as well as the specific gravities.

The column “A” will be explained hereafter, in the chapter on transverse resistance or bending.

TABLE III.

WOOD.

White Pine

Yellow Pine ....

Locust

Black Walnut ..

White Ash

White Oak

Live Oak

ULT. RESIST.

ELASTIC LIMIT

IN POUNDS PER

IN POUNDS PER

SQ. IN.

SQ. IN.

6,880

3,900

20,700

28,930

9.790

15,490

13,210

10,310

13,200

19,200

5.700

9,700

8,100

6,300

COEFFICIENT OF

ELASTICITY IN

POUNDS PER

SQ. IN.

183,490

240,240

373,830

213,520

206,540

220,130

247,510

PER CENT. OF EXTEN-

SION AT

Elas. Limit.

Fracture.

0.40

0-73

0.63

1.10

0.53

0.78

0.77

0.58

1.65

1.85

0.85

1.48

1.30

I-I5

The table gives average results. Those determined from experiments of Mr. Laslett are of English origin (“ Timber and Timber Trees, Native and Foreign,” by Thomas Laslett, 1875) ; the others are from American experiments by the late

R. G. Hatfield (“ Transverse Strains,” 1877). Mr. Laslett’s specimens were 2 inches squafe in cross section, and geneatily were 30 inches long, while those of Mr. Hatfield were about 0.35 inch round.

It will be observed that Mr. Hatfield reached far higher results than Mr. Laslett. This disagreement may be due to the larger cross-sectional area of the latter’s specimens, which certainly brings his (Mr. Laslett’s) results more nearly in accordance with what might be expected from such pieces as are ordinarily used by engineers. Mr. Hatfield’s specimens were far too small for technical purposes.

Table III. is taken from a paper “On the Strength of American Timber,” by Prof. R. H. Thurston (Jour. Frank. Inst., Oct., 1879). The specimens were turned down to about 0.5 inch diameter for a length of 4.00 inches.

The small values of the coefficient of elasticity, as compared with those given in Table I., are probably due to the fact that they were found at the elastic limit. Smaller intensities of stress would probably give much larger values.

Prof. Thurston also states that timber in tension takes a permanent set however small the intensity of stress.

The values given in Table IV. were found by Col. Laidley, U. S. Army, in the Government machine at Watertown, Mass. (Ex. Doc. No. 12; 47th Congress, 2d Session). Two of the specimens were about 0.63 inch in diameter, and one 1.25 inches. All the rest possessed diameters of about one inch each.

Such small specimens as those of Hatfield, Thurston, and Laidley, which were probably selected, give much larger results than would be found for large pieces of ordinary lumber ; these considerations are highly prejudicial to the technical value of the results.

Far more importance attaches to the matter of size and character of timber specimens than to those of metallic ones. In the latter there is at least an approach to homogeneity of material, which the presence of knots, conditions of growth, seasoning, and other influences effectually prevent in timber

specimens. Hence it is the more necessary to test timber in circumstances of condition and size as nearly identical as possible with those which attend its actual use.

TABLE IV.

Diameter of Test Specimens, 1 inch.

ULTIMATE ESISTANCE PER SQUARE INCH IN

NO.

KIND OF WOOD.

POUNDS.

Greatest.

Mean.

Least.

10

ii

12

13

14

15

Yellow Pine... Oregon Pine . . Oregon Spruce White Pine ...

Spruce........

White Wood .. i Gum Wood ... White Maple .. Black Walnut . Red Birch White Ash .... Brown Ash... . White Oak

Red Oak......

Yellow Oak . .. Hickory......

17,922

11,299

17,044

7,466

19,400

15,71-1

14,650

22,838

27,532

11,733

22,703

12,133

20,520

19,610

15.478

13,810

16,160

8,916

14,283

6,787

14,313

11,164

11,492

18,682

24,120

11,632

17,410

10,124

20,390

15,995

I2,o66

O O r^O m O vn m r^O O OQ O O O co w cO cO vO vO i^OO O ChO - to to O CO C'vO OO N t

ir> o" rfr^C'C^co o w r'» O d

M    MMMM d M

All specimens were of well seasoned wood.

CHAPTER VI.

Compression.

Art. 39.—Preliminary.

With the exception of material in the shape of long columns, but few experiments, comparatively speaking, have been made upon the compressive resistance of constructive materials.

Pieces of material subjected to compression are divided into two general classes—“ short blocks ” and “ long columns ; ” the first of these, only, afford phenomena of pure compression.

A “ short block ” is such a piece of material, that if it be subjected to compressive load it will fail by pure compression.

On the other hand, a long column (as has been indicated in Art. 25) fails by combined compression and bending.

Short blocks, only, will be considered in the articles immediately succeeding, while long columns will be separately considered further on.

The length of a short block is usually about three times its least lateral dimension.

It has already been shown in Art. 4 that the greatest shear in a short block subjected to compression, will be found in planes making an angle of 45⁰ with the surfaces of the block on which the compressive force acts, i. e., with its ends. If the material is not ductile, this shear will frequently cause wedge-shaped portions to separate from the block. But the friction at these end surfaces, and in the surfaces of failure will prevent those wedge portions shearing off at that angle. In fact the friction will cause the angle of separation to be considerably larger than 45⁰ ; let it be called a. Then, in order that there may be perfect freedom in failure, the length of the block must not be less than its least width or breadth multiplied by 2 tan a. In some cases, a has been found to be about 55⁰, for which value

2 tan a = 2 X 1.43 = 2.86.

It was shown in the first section of Art. 32, that the “ ultimate resistance ” to tension is in reality a mean, and not the greatest intensity which the material exerts. The same course of reasoning will show that it is, also, in general, impossible to subject a short block to a uniform intensity of compression throughout its mass, and that the “ ultimate resistance to compression ” is a mean, usually considerably less than the greatest intensity which exists at the centre of a normal section. As the inner portion will be supported laterally by that outside of it, large blocks of brittle material may give greater intensities of ultimate resistance than small ones.

Art. 40.—Wrought Iron.

It is difficult to fix the point of failure of a short block of wrought iron or other ductile material. An excessive compressive force causes the material to increase very considerably in lateral dimensions, or to “ bulge ” out, so that every increase of compressive force simply produces an increased area of resistance, while the material never truly fails by crumbling or shearing off in wedges.

A short block of wrought iron is usually considered to fail when its length is shortened by five to ten per cent.

If px is any intensity of stress while lx is the compressive strain, or shortening per unit of length caused by px, then according to Eq. (2) of Art. 2, the coefficient of elasticity for compression at the intensity px, will be

•>

This ratio is not constant for all degrees of stress and strain, though for wrought iron, within the elastic limit, the divergences from a mean value are not great. Table I. contains coefficients of elasticity calculated by Prof. De Volson Wood, in the manner shown by Eq. (1), from the data determined by Mr. Eaton Hodgkinson and given in his work before cited. (See Prof. Wood’s “ Treatise on the Resistance of Materials ”).

TABLE I.

AĄ.

LI,.

LI,.

Pounds.

5,098

9,578

14,058

16,298

18,538

20,778

23,018

²5,258

27,49s

29,738

31,978

Inch. O.028 O.052 O.073 0.085 0.096 o. 107 o. 119 0.130 0.142 0.152 0.174

Pounds.

20,796,500

21,049,000

21,979,000

21,343,000

22,156,000

22,160,000

23,587,000

22,095,000

22,111,000

21,938,000

20,979,000

0000000

.    . o Q cDco n en O

X. OJ TfO I CO û w M Tj-vO O'

y OOO OWMMWWW

Pounds.

21,864,000

23,595,000

24,273,000

24,108,000

24,038,000

23,587,000

23,679,000

22,259,000

21,139,000

19,478,000

The results belong to two square bars, and E, is in pounds per square inch. A is the area of cross section ; it was 1.0506 square inches for the first bar and 1.0363 square inches for the other. Hence the bars were about one inch square. They were also ten feet long (L — 10.00 feet) and required lateral support to be kept in alignment so as to act like short blocks.

The table shows that the values of E, increase with /„ when the latter is small ; an opposite result was found for tension. What may be called the elastic limit is found for p, =

30,000.00 pounds per square inch (nearly). Hence, it is seen that the greatest value of E₁ is found for px equal to one-half to two-thirds the elastic limit.

The same general remarks in regard to the elastic limit, which were made in connection with tension, may be also applied to the compressive elastic limit.

The “ Steel Committee ” of British civil engineers, in 1870, made some experiments on twelve bars of Lowmoor wrought

TABLE II.

POUNDS PER SQUARE INCH FOR

Elastic Limit.

Coefficient of Elasticity.

Pounds.

29,800

Pounds.

29,091,000

25,800

29,091,000

29,100

26,200

28,718,000

28,000,000

iron, 1.5 inches in diameter and 120 inches long. These twelve experiments were divided into four sets of three each, and the table gives the means of each of these sets or groups. The coefficients are computed at the elastic limit. Judging from the results in Table I., smaller values of px would have given larger values of Ev

As a mean value, the coefficient of elasticity for wrought iron in compression may be taken at 28,000,000 pounds per square inch. For every ton (2,000.00 pounds) of compression per square inch, therefore, a piece of wrought iron will be shortened by an amount equal to

₀    --—-— of its length.

2,000

28,000,000    14,000

Table III. contains the results of some experiments made by Mr. Kirkaldy on some specimens of Swedish iron, in 1866. The last column gives the per cent, of compression of original length which the piece suffered at the point called the “ ultimate compressive resistance.” The results show well the great increase of resistance which a short block of ductile material offers with the increase of compression.

TABLE III.

SECTION OF

POUNDS PER SQUARE INCH FOR

PF.R CENT.

SPECIMEN.

LENGTH.

Elas. Lim.

Ult. Resist.

COMPRESSION.

In.

1-5 O

Lbs.

24,050

Lbs.

148,800

1-5 O

u\

21,200

28,100

1-5 O i .0 □

23,300

84,900

184,100

LO Tl* CO CO Tf    CO *0

Table IV. gives the results of experiments on some very-short lengths of Phoenix and Keystone columns. The first six results are for Phoenix sections from experiments by the Phoenix Iron Co., in 1873 ; the two following are for the same section from experiments made at Watertown, Mass., in 1879; while the last result belongs to a Keystone section experimented upon by Mr. G. Bouscaren, in 1875. Unfortunately the amount of compression or shortening, in each instance, was not recorded.

Reviewing the results given in Tables II., III. and IV., it is seen that the “ elastic limit ” of wrought iron in compression, in

TABLE IV.

LENGTH, INCHES.

RATIO OF LENGTH TO

AREA OF SECTION,

ULT. RESIST. IN POUNDS

DIAMETER OF SECTION.

SQ. IN.

PER SQ. IN.

8.00

8.00

4.00

4.00

4-00

4-00

8.00

8.00

9.00

1.46

1.46

0.92 0.92 i.or

i. 01 i .00 i .00 1.12

6.97

6.97

5.62

5.62

2.92

2.92

11.90

11.90

14-25

60,570.00 60,390.00 65,870.00 65,870.00 56,89 55,560.00 57,130.00 57,300.00 51,500.00

0.00

short blocks, may be taken from 0.4 to 0.5 its ultimate compressive resistance, while the latter may be taken at about 60,000.00 pounds per square inch.

Art. 41.—Cast Iron.

The irregular elastic behavior of cast iron, as seen in tension, will also be discovered in compression. Table I. contains results computed from the data obtained by Captain Rodman by testing solid cylinders 10 inches long and 1.382 inches in diameter. The second column belongs to a specimen cylinder taken from a 10-inch columbiad, and the third or last to a trial cylinder of remelted Greenwood and Salisbury iron. Neither specimen can be considered to possess a true elastic limit, but what is ordinarily so termed may be taken at about 20,000 pounds per square inch.

In the first specimen the first permanent set took place at 3,000.00, and in the second at 5,000.00 pounds per square inch.

TABLE I.

INTENSITY OF STRESS.

COEFFICIENT OF ELASTICITY IN POUNDS PER SQUARE INCH.

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

10,000

11,000

12,000

14,000

16,000

18,000

20,000

22,000

26,000

30,000

35,ooo

40,000

6,896,600

8,888,900

9,836,100

10,666,700

10,752,700

11,320,800

ii,3S2,ioo

11,510,800

11.920.500

12.121.200

12.290.500 12,182,700 12,444,400

12.260.500

11.920.500 11,695,900

11.253.200

10.236.200 8,596,

000

33.333.300 18,750,000

13.793.100

13,888,900

12,766,000

13.725.500

13.559.300

13.432.800

13.333.300

13,095,200

13.186.800

13.207.500

12.903.200

12.857.100 12,578,600

12.290.500

11.607.200

10,101,000

7,658,600

5,333,300

For a bar ten feet long and one inch square, Mr. Eaton Hodgkinson found the following values:

GREATEST.

13,216,000.00

MEAN.

12,134,100.00,

LEAST.

10,837,100.00 ;

all in pounds per square inch. The greatest value was found at 2,240, and the least at 38,080 pounds per square inch.

Since the coefficient of elasticity measures the stiffness of a body, and since the coefficient of elasticity for wrought iron in compression has been seen to be at least twice as great as that of cast iron in the same condition, wrought iron is at least twice as stiff, compressively, as cast metal. A bar of the latter material will be compressed by 2,000 pounds per square inch, about

—-’-OOQ— =    — of its length.

12,000,000    6,000

If l is the length of a bar in inches, W the compressive stress in pounds per square inch, then Hodgkinson found the total decrement in inches for 10-feet Low Moor cast-iron bars to be

V — l(0.012363359 — Vb.000152853 — 0.00000000191212 W). (1)

and the permanent set, in inches :

(²)

0.543 A.'³ + 0.0013

Major Wade tested a number of specimens of cast iron of different numbers of fusions, in order to determine the ultimate compressive resistance. His specimens were from 0.5 to 0.6 inch in diameter, and from 1.25 to 1.5 inches (nearly). The results were as follows :

FUSION.

NO. OF EXPS

GREATEST.

MEAN.

LEAST.

2d

3d

2d and 3d 2d......

3d

2d

3d

• 114.504-. 140,415.

. 169,427.

■ 140,415 . *68,251. . 163,528. . 174,120.

99,770.

139,540.

168,589.

136,868.

168,251.

154,576.

167,030,

84,529

138,666

167.752

133,321

168,251

144,141

156,863

All results are in pounds per square inch.

As the specimens gave way, portions sheared off along planes making angles with the normal sections of specimens varying from 46° to 62.5°. This is the characteristic compressive fracture of cast iron.

The 3d fusion iron gave the highest resistance.

Mr. Hodgkinson (“ Report of the Commissioners appointed to Inquire into the Application of Iron to Railway Purposes,” 1849) took specimens of 16 different kinds of British irons, 0.75

inch in diameter and 0.75 and 1.5 inches long, with the following results :

GREATEST.

MEAN.

117,605

86,284

LEAST.

56,445 pounds per square inch.

As a rule, the short specimens gave from 5 to 10 per cent, greater resistance than the longer ones. From another set of experiments with 22 different kinds of iron (specimens 0.75 inch in diameter and 1.5 inches long) he found :

GREATEST.

MEAN.

H5,995

84,200

LEAST.

54,761 pounds per square inch.

Mr. Hodgkinson found that the hardness and ultimate crushing resistance of thin castings were greatest near the surface, but that in thick castings the surface and heart gave essentially the same results. He also found that thin castings gave considerably greater ultimate resistance to crushing than thick ones.

Sir Wm. Fairbairn tested the effect of remelting on “ Eg-linton ” No. 3, hot-blast iron with the following results :

REMELTINGS.

ULT. RESIST.

REMELTINGS.

ULT. RESIST,

98,560

97,66o

92,060

91,170

92,060

92,060

91,620

92,060

123,420

10

ii

12

13

14.

15

16

17

18

129,250

156,350

163,740

147,840 defec.

214,820

171,810

157,920

197,190

All results are in pounds per square inch. It is observed that the 14th remelting gives the highest resistance.

From what precedes, it is seen that the ultimate compressive resistance of cast iron, in good ordinary castings, may safely be taken from 85,000 to 100,000 pounds per square inch.

Art. 42.—Steel.

Table II. of Art. 34 contains the results found by Prof. Ricketts in testing cylindrical specimens of mild steel in compression. These specimens were six inches long between carefully faced ends, and, as the table shows, their diameters were about 0.75 inch. The coefficients of elasticity for compression were found by measurements very carefully made with a micrometer on a length of four inches. The elastic limits, however, were determined by operating with a cylinder two inches long, and were taken at those points where the material of the specimens ceased to hold up the scale beam, and may have been somewhat above that point where the ratio between stress and strain ceases to be essentially constant.

The coefficients of elasticity are seen to be quite uniform» irrespective of the per cents of carbon, within the limits of the Table, and they are seen to be a very little less than the coefficients for tension. Yet the difference is so small that no essential error will arise if, for all engineering purposes, they are assumed the same.

A comparison of the elastic limits for tension and compression presents some irregularities ; yet with the exception of the high percentages of carbon in the last two grades of Bessemer metal, the two sets of elastic limits as wholes are not very different from each other. In the Bessemer steel with the two high per cents of carbon, the tensile elastic limits are materially higher than those for compression. The following very important conclusion results from this comparison of the elastic limits for the mild structural steels : since these elastic limits are essentially equal, it is not only permissible but wholly rational to increase the working resistances of mild steel bridge columns over those for iron in at least the same proportion that the tensile working stress of the same steel is increased over that of iron in tension. Experiments on a sufficient number of full size steel columns are yet lacking to verify this conclusion.

Chief engineer Wm. H. Shock, U. S. N., 1868, gives the following results for Parker Bros. “ Black Diamond ” steel :

Normal untempered steel: Ult. Resist, from 100,100 to 112,400 pounds per square inch.

Heated to light cherry-red and plunged in oil at 82° Fahr.'. Ult. Resist, from 173,200 to 199,200 pounds per square inch.

Heated as before, and plunged in water at 79⁰ Fahr., with final temper (plum-blue) drawn on heated plate : Ult. Resist. from 325,400 to 340,800 pounds per square inch.

Heated as before and plunged in water at 79⁰ Fahr., and tested at maximum hardness : Ult. Resist, from 275,640 to 400,000 pounds per square inch. In each of these cases there were three tests.

The following values (each is a mean of 8 tests) were found by the United States Test Board, “ Ex. Doc. 23, House of Rep., 46th Congress, second Session,” for small annealed specimens of tool steel, of about one inch in length and 0.715 inch in diameter :

Ult. comp, per sq. in. ) of original section. )

175,99- ; 174,586; 183,938 ; 193,413; 193,197; 174,586;

I93,5I7 ! i74,895 pounds per square inch.

127,579; I49,88x; 139,196; I45,75i; 128,834;

Final comp, per sq. ) 134,717 ;

in. of final section. )    125,126 ; 140,489 pounds per square inch.

The final lengths varied from 56 to 89 per cent, of the originals.

Kirkaldy’s “ Experimental Inquiry into the Mechanical Properties of Fagersta Steel,” 1873, furnish data from which may be computed a series of values of the ratio (A,) of stress over strain, or coefficient of elasticity, for different intensities of stress.

All the specimens were cut from plates of mild steel of the thickness shown in the table, and were iOO inches long and 2.25 inches wide. They were laterally supported in a trough arrangement designed by Mr. Kirkaldy.

TABLE I.

COEFFICIENTS OF ELASTICITY IN POUNDS PER SQUARE INCH.

INTENSITY

OF STRESS.

Unannealed.

Annealed.

% inch plate.

%-inch plate.

%-inch plate.

%-inch plate.

10,000

14,000

18,000

22,000

26,000

30,000

34,000

38,000

42,000

46,000

50,000

,000

58,824,000

56,000,000

54,545,000

48,889,000

45,614,000

42,254,000

40,000,000

38,384,000

35,000,000

30,872,000

25,000,000

38,462.000

33,333,ooo

31,034,000

29,730,000

28,261,000

24,000,000

3,795,000

50,000,000 48:276 45,000,000 38,596,000 23,214,000 7,792,000 5,207,000 4,265,000 3,717,000

34,4⁸3.ooo

33,333,000

31,034,000

26,190,000

19,118,000

2,542,000

Below 18,000 pounds per square inch annealing does not much change the coefficients for the ^-inch specimens, but affects the thin ones more decidedly. In all the specimens the elastic behavior is very irregular, and none of them can be said to possess a true elastic limit. In the unannealed thin and thick plates, the first permanent sets took place at 40,000 and 20,000 pounds per square inch, respectively ; in the corresponding annealed ones, at 30,000 and 20,000, respectively.

By referring to Table III. of Art. 34, it will be seen that the coefficients for compression are considerably larger than those for tension.

Table II. contains coefficients of elasticity computed from the results of experiments made under the supervision of the “ Steel Committee of Civil Engineers ” (English). All are computed for the “ limit of elasticity.” The upper portion of the table belongs to round specimens 1.382 inches in diameter and 50 inches (36 diameters) long, tested in 1868 ; the lower

TABLE II.

QUALITY.

GREATEST £,.

MEAN jEj.

LEAST Ex.

Hammered

and

rolled.

Bessemer.

Crucible.

26,130,000

34,461,000

35,000,000

33>939,ooo

34,461,000

31,464,000

Chisel, tire, rod, rolled, etc.

Bessemer.

Crucible.

30,270,000

31,550,000

29,091,000

29,474,000

28,718,000

28,000,000

E\ = coefficient of compressive elasticity in pounds per square inch.

portion to 1.5 inch round specimens and 120 inches long, which were tested in 1870. Table I. shows that considerably different results might be expected with lower intensities of stress.

TABLE III.

LENGTH.

LIMITS OF ELASTICITY IN POUNDS PER SQUARE INCH.

Greatest.

Mean.

Least.

Inches.

Bessemer ; tires, hammered, rolled, fagot-ted, etc.

Diam. = 1.382 ins. _

53,520

50,250

42,500

1.38

52,240

49,800

48,540

46,700

40,990

40,470

2.76

5-53

Crucible ; hammered, ' rolled, chisel, tires, rods, etc.

Diam. = 1.382 ins.

60,260

59,ooo

53,740

55,760

53,780

49,860

48,990

43,99°

42,000

1.38

2.76

5-53

The upper Bessemer results are for a set of 18, and the

lower for a set of 11 tests ; the upper crucible results are for a set of ii, and the lower for a set of 20 (?) tests.

The “ limits of elasticity ” of specimens of the same steels to which the upper portion of Table II. belongs (and for the same number of experiments) are shown in Table III.

The following “ limits of elasticity,” in pounds per square inch, correspond to the lower portion of Table II. :

GREATEST.

MEAN.

LEAST-

Bessemer. Crucible..

35,840 pounds. 36,290 pounds.

47,490,

60,480.

39,870

52,190.

TABLE IV.

ELASTIC LIMIT IN POUNDS PER SQUARE INCH.

LENGTH.

1.2.

0.9.

0.6.

0.3.

M e* TtOO

64,000

63.330

62.330

61,670

62.670

58.670

58,670 58,000

6o,ooo

57.330

53.330

52,670

39,000

42,000

41,000

40,670

Means.

62,833

59,500

55,830

40,670

ULTIMATE RESISTANCE IN POUNDS PER SQUARE INCH.

I diam.

169,910

133,330

102,170

I73,29P

117,560

95,2io

156,000

105,330

84,830

121,330

81,760

47,5io

Means.

135,140

128,690

115,387

83,540

The results of the experiments of Mr. Kirkaldy on specimens of different grades of Fagersta steel and of various lengths in terms of diameter, are given in Table IV. All the specimens were turned to 1.128 inches (1 square inch area) in diameter, and were of the lengths shown.

The numbers 1.2, 0.9, 0.6 and 0.3 were used to indicate the different grades of steel, the larger numbers belonging to the higher steels.

The specimens of one diameter in length shortened, under a load of 200,000 pounds per square inch, 21, 22, 26 and 48 per cent., respectively, for the marks 1.2, 0.9, 0.6 and 0.3. Three of the “ 2 diam.” specimens failed by detrusion, or by portions shearing off obliquely ; all the others either bulged or took a skew form, though one of the “ 8 diams.” finally broke.

Table V. contains the results of Major Wade’s experiments

TABLE V.

DESCRIPTION.

LENGTH OVER DIAMETER.

ULT. RESIST. IN POUNDS PER

SQUARE INCH.

Not hardened

2 55

198,944

Hardened, low temper.....

Hardened, mean temper ... Hardened, high temper

2.47

2.52

2.48

354,544

39L985

372,598

All specimens about x inch long and 0.4 inch in diameter.

on specimens of cast steel, in 1851. The results are seen to be very high.

A piece of the Hay steel used by Gen. Smith in the Glasgow, Mo., bridge, about 1]^ inches square and 3*43 inches long, gave an ultimate compressive resistance of 139,350 pounds per square inch (“ Annales des Ponts et Chaussées,” Feb., 1881).

Art. 43.—Copper, Tin, Zinc, Lead and Alloys.

Table I. shows some coefficients of elasticity (i.e., ratios between stress and strain), computed from data determined by Prof. Thurston, and given by him in the “ Trans. Amer. Soc. of Civ. Engrs.,” Sept., 1881. The gun bronze contained copper, 89.97 ; tin, 10.00 ; flux, 0.03. The cast copper was cast very hot.

TABLE I.

COEFFICIENTS OF ELASTICITY IN POUNDS PER SQUARE INCH.

STRESS IN POUNDS PER SQUARE

INCH.

Gun Bronze.

Cast Copper.

1,620

3.260

6,520

9,780

13,040

16,300

19,560

22,820

26,080

29,340

32,600

48,900

3,622,000

4,075,000

0,113,000

6,520,000

5,433,000

5,148,000

3,935,000

2,308,000

1,073,000

463,600

1,254,000

1,415,000

1,651,000

1,795,000

1,824,000

1,842,000

1,845,000

1,735,000

1,503,000

1,144,000

815,000

332,500

The ratios of stress over strain are far from being constant. Strictly speaking, therefore, there is no elastic limit in either case. In that of the gun bronze, however, it may be approximately taken at 20,000 pounds per square inch (Prof. Thurston takes it 22,820), and in that of the copper at 25,000 pounds. The test specimens were two inches long and turned to 0.625 inch in diameter.

At 38,000 pounds per square inch the gun bronze specimen was shortened about 41 per cent, of its original length, while its diameter had become 0.77 inch.

TABLE II.

COMPOSITION.

POUNDS

CAUSING

fa

X o

Copper. Tin.

5%

97-83

95.96

92.07

9°-43

87-15

80.99

76.60

69.90

6S-3¹

61.83 47-72 44.62

38.83 38.37 34.22 25.12 20.21

1.92

3.80

7.76

9.50

I5»I2

11.48

8.57

3.72

12.77 18.92 23 23 29.85 34-47 37-74 51-99 55.1S

60.79 61.32

65.80

74.51

79.62

84.58

88.50

9I-39

96.31

99.02

29,34°

39,200

31,500

32,000

39,000

65,000

101,040

0.74 0.32    99

Cast copper.

.46

Cast tin.

19,560 17,930 16,300 6,520 10,100 6,500 6,520 6,520 6,520 26,000 33,ooo 34,000 30,000 30,000 35,000 6,030

rîilîlifff ilffl 11111111 iiiffif &

GREATEST LOAD IN LBS. PER SQ. IN.

* g

ill

. d ë

h W

Hi

fa

w

fa

•J

ULT. CRUSHING RESISTANCE IN LBS. PER SO. INCH.

MANNER OF

FAILURE.

46,260

52,15°

84,100

61.930 89,640

103,490

114,080

146,680

84,750

39,”°

84,750

35,850

39,”°

29,340

19,560

17.930

16,300

9,450

14,020

O O O O'

0-37

0.30

0-45

0.34

°*39

0.20

0.09

0.04

0.03

0.02

0.02

0.01

0.02

0.01

0.06

0.28

0.29

0.51

0.50

0.06

0.34

0.36

0.38

74,970 78,230 71,710 104,300 91,270 97,79°

7,5oo

0.45

0-43

0.32

0.52

0.48

0.41

0.44

34,000 42,050 42,000 38,000 53,000 78,000 114,080 146,680 84,750 39,110 84,750 35,850 39,110 29,340 19,560 17,930 16,300 6,520 10,100 9,780

9.780

9.780 9,78o

39,000

45,5oo

42,000

36,000

OOO

37,

48,000 6,400

Flattened,

Crushed.

Flattened.

The copper specimen failed at 71,700 pounds per square inch, having been shortened about one-third of its length.

The results of a series of tests by Prof. Thurston, in connection with the United States testing commission, are given in Table II. ; they were abstracted from “ Mechanical and Physical Properties of the Copper-Tin Alloys,” United States Report, edited by Prof. R. H. Thurston, 1879. All the specimens were 0.625 inch in diameter and 2 inches long. Scarcely one of them can be said to possess an elastic limit.

The series of alloys presents some interesting results. About the middle third of the series are seen to be brittle compounds giving (as a rule) high ultimate compressive resistances, while the other two-thirds are ductile, and give at the copper end high results, and low ones at the tin end.

It will be observed that Prof. Thurston took the load per square inch which gave a shortening of io per cent, of the original length as the ultimate resistance to crushing of the ductile alloys and metals, since such materials cannot be said to completely fail under any pressure, but spread laterally and offer increased resistance.

TABLE ill.

PER CENT. OF

POUNDS PER SQUARE INCH FOR

PER CENT. OF

MANNER

Copper.

Zinc.

Eu

Ult. Resist.

SHORTENING.

OF FAILURE.

96.07

90-56

89.80

76.65 60 94 55-15

49.66 47-56 25-77

20.81 14.19 10.30

4-35

0.00

3-79

9.42

10.06

23.08

38.65

44.44

50.14

52.28 73-45 77-63 85.10 88.88

94-59

100.00

305,500

342,100

656,500

1,772,500

1,345,500

1,500,000

4,232,800

2,485,000

897,000

318,500

29,000

30,000

29,500

42,000

75,ooo

78,000

117,400

121,000

110,822

52,152

48,892

49,000

48,000

22,000

10.0

10.0

10.0

10.0

10.0

10.0

10.0

10.0

5-85

2-75

10.8

10.0

10.0

10.0

Flattened.

Crushed.

Flattened.

Table III. contains the results of Prof. Thurston’s tests of the copper-zinc alloys made while he was a member of the United States Board. The data are taken from “ Ex. Doc. 23, House of Representatives, 46th Congress, 2d Session.” The specimens were two inches long and 0.625 inch in diameter of circular cross section.

The values of El (ratios of stress over strain) are computed for about one-quarter the ultimate resistance. This ratio is so very variable for different intensities of stress that these alloys can scarcely be said to have a proper “ elastic limit.”

In the “ Philosophical Transactions” for 1818, Rennie gives the following as the results of his experiments on 0.25 inch cubes :

Fine yellow brass (io per cent,    shortening)...    12,852    pounds    per square    inch.

Fine yellow brass (50 per cent,    shortening)...    41,216    pounds    per square    inch.

Cast lead

(50 per cent,    shortening)...    1,932    pounds    per square    inch.

Art. 44.—Glass.

The following results are taken from Sir Wm. Fairbairn’s “ Useful Information for Engineers,” second series. The cylinders were about 0.75 inch in diameter and annealed.

TABLE I.

KIND OF GLASS.

SPECIMEN.

HEIGHT OF SPECIMEN.

CRUSHING RESISTANCE,

LBS. PER SO. INCH.

Fiint

Cylinder.

66

Green

Crown

Flint

Cube.

Green

Crown

OOO^-HMMMCnOOcnOOC^OOQ. OOOOO C>cn OOOOO^nOOO:T

23,43o

34,850

20,780

3²,8oo

22,580

35,030

38,’

020

23,180

38,830

14,240

13,200

13,260

11,820

20,470

19,950

21,870

It will be observed that the cubes give considerably less resistance than the cylinders.

All the glass was annealed, but Fairbairn remarks that the cubes may have been only imperfectly so, since they were cut out of the interior of larger masses, while the cylinders were cut from rods as they were drawn. The latter, also, thus retained their natural skins, which may have increased their resistances.

At the instant of failure the specimens were shattered into a great number of small pieces.

Art. 45.—Cement—Cement Mortar—Concrete—Artificial Stones.

Table I. of Article 37 contains the ultimate compressive resistances of a great number of pure cements, as tested by General Gillmore under the circumstances related in connection with the table. The results are given in pounds per square inch.

TABLE I.

THESE RESULTS ARE LBS.

CEMENT,

SAND,

LENGTH,

PER SQ. IN.

AGE, IN

VOL.

VOL.

INS.

DAYS.

Greatest.

Mean.

Least.

12

E. L.

R.

Ei.

E. L.

R.

Ei

1,502

1,889

1,500,000

587

783

800

1,320

800,000

36s

424

62O

500,000

191

261

E. L.

R.

Ei.

1,910,000

424

489

6,633,330

607,000

182

213

1,285,000

2,7,333

98

220,450

123 to M3-

117 to 141.

127 to I35-

Table I. contains the results of tests of “Fall City” (Louis-ville) cement and cement mortar. The tests were made by-Mr. Bremermann, by direction of Cant. Eads, during the con-

struction of the St. Louis bridge. “ E. L.” is the elastic limit; “ R.” the ultimate resistance to compression ; and “ E,” the coefficient of compressive elasticity, all in pounds per square inch.

The following results are from the same source :

Akron Cement.

24-inch cubes ; 1 vol. cement, o vol. sand ; ult. resist. = 2,140 lbs.

= 1,105 “

{<

it

“ Fall City,' Louisville, Cement.

2|-inch cubes ; I vol. cement, o vol. sand ; ult. resist. = 1,587 lbs.

=    640 ‘ ‘

tf

=    400

=    240

Louisville Cement from Beach âf Co.

3'

M 04 M C4 M

0)

--Ł

J

r.

--

i,6i5 lbs. 1,280

560

400 “ 280 “

Louisville Cement from Hulme àf Co.

24-inch cubes ; 1 vol. cement, o vol. sand ; ult. resist. = 2,320 lbs.

= 740 = 600

(t

<<

<«

(«

These ultimate resistances are in pounds per square inch. All specimens were “ moulded under hand pressure only, left 12 days in water, and exposed six months to the air.”

TABLE la.

DESCRIPTION OF PORTLAND CEMENT AND MORTAR.

ULT. RESIST. IN

POUNDS PER. SQ. IN.

Neat Portland cement

X Portland cement to I pit sand “    2    “

Made 3 months.

04 Of hT i-T

Neat Portland cement.....

i Portland cement to I sand

Made 6 months.

Neat Portland cement

UT CO cf of ł-T ł-T

959

i Portland cement to I pit sand.

44    2    “

Made g months.

678

The results of a large number of experiments on the compressive resistance of Portland cement and mortar, at different ages, by Mr. John Grant, C. E. (“ On the Strength of Cement,” 1875), are given in Table I a. The specimens were made into bricks 9 X 4.25 X 2.75 inches, and were compressed on their flat sides of 9 X 4.25 = 38.25 square inches area. The results are in pounds per square inch.

Table II. is taken from the same work, and shows the circumstances under which Mr. Grant made his experiments. Two sets of blocks were made in each case ; one set was kept in air for one year, and the other in water for the same length of time. The cubes were then crushed with the results shown. It is to be observed that the results are pounds per square foot. Two series of the blocks were formed by compressing the material in layers one inch thick ; the others were not compressed.

TABLE II.

o s i

% ä s 3 > w 3 < u

2 g s

fc Q

O Z j

« g

ULTIMATE COMPRESSIVE RESISTANCE IN POUNDS PER SQUARE FOOT.

Compressed.

Not Compressed.

(/>

w

Kept in air.

Kept in water.

Kept in air.

Kept in water.

Kept in air.

Kept in

water.

(Exceptional.)

239,680

333,760

10

253,120

230,720

199,360

l80,320

l68,000

137,760

120,960

108,640

381,920

35S,4°°

258,720

243,040

222,880

203,840

180,320

170,240

153,44°

107,520

340,480

385,280

268,800

268,800

219,520

182,784

147,840

120,960

107,520

94,080

W M H CO K) W W W

OvO    to    4-    *-<01    m    OO

4^.    Cß    O    w    Ul    CO    o    COO    JH

O' O co O H O »H    CO    CO    CO    to    O

O    C    O    O    O

vO O O O O ►h vO O O 00

cn O m

§.

268,8oo

344,9⁶° 215,040 250,880 215,040 163,072 125,440 112,000 89,600 71,680

336,000

322.560 250,880 241,920

210.560 152,320 112,000

98,560

80,640

62,720

12'' x 12" x 12" blocks.

6" x 6" x 6" blocks.

6" x 6” x 6" blocks.

Table II«. is from the same source as Table I.

The concrete blocks were pressed evenly on 36 square inches until failure took place.

The following results for artificial stones are given by Mr. Henry Reid (“ A Practical Treatise on Natural and Artificial Concrete,” 1879):

A 4-inch cube of Ransome’s “ Siliceous Stone ” gave 4,200 pounds per square inch.

By experimenting with 2-inch cubes of “ Rock Concrete ” pipes, Mr. Reid obtained the following results from two series :

GREATEST.

MEAN.

4,340.

5.650,

5.650.

■3,454

.4,428

4,763

LEAST.

.2,684 pounds per sq. in. ■3,401 3,107

<<

<< iC

TABLE II«.

6" x 6" x 6" Concrete Blocks.

All blocks were under water 12 days, and then exposed to the air for 6 months.

S3    3

o    o

ULT. RESIST., LBS.

PER SQ. IN.

tOtOtOtOtOtOMMMMMtObOtOMtOtOtOMMM

MMMMMM

invnmmmnimrnmnin

000000666660006

OOOOOOOOOOOmmmhhOOOOOO

uioiuuncnainvLninmui

889

1,124

1,170

722

889

I,36l

1,194

95O

64O

89O

1,170

I,36l

i,445

1,280

1,250

918

1,000

1,140

1,361

611

I,hi

There were six experiments in each series.

Three-inch cubes of Victoria stone (six experiments in the first series, and ten in the second) gave :

GREATEST.

MEAN.

5,179

4,708

LEAST.

3,294 pounds per sq. in. ,3,578 pounds per sq. in.

4,422

3,955

These cubes were made in February, 1879, and broken in May of the same year.

“ Two-inch cubes of silicated stone made with 3 parts Thames ballast and 1 of Portland cement, gauged with water, and put in the silicate bath for 11 days, about 12 months old fractured as follows :

4,237 pounds per square inch. 5,650 pounds per square inch.”

No. i No. 2

Four “granitic breccia” cubes, 3” x 3", about 25 years old, gave the following results :

GREATEST.

8,886,

MEAN.

8,028

LEAST.

7,533 pounds per square inch.

Seven blocks of Sorel stone, varying from \]/^ x 1]/^ x 1 inch to 2 x 2¹/q x inches gave:

AGE.

INERT MATERIAL.

I year.

2    years....

3    “ .....

9 months..

2 years.....

Not known

Coral sand...............

Pulverized quartz..........

Washed flour of emery.....

Fine marble..............

Mill sweepings...........

Marble and sand........

Marble with colored veneer

ULT. RESIST.

it

66

The weight of the oxide of magnesium varied from 12 to 15 per cent, of the whole.

The results of a series of tests, by Gen. Gillmore, in 1870 and 1871, on coignet béton blocks, 3.5 x 5.5 x 3 inches, are given in Table III. Two blocks of each kind were tested. All the blocks were two months old. The results are in pounds per square inch.

With four 2-inch cubes of Frear stone, Gen. Gillmore obtained the following results :

Four weeks old.. Four weeks old.. Three weeks old Six months old .

66

4,500 pounds per square inch. 4,626 2,250 2,000

These blocks were composed of one measure of hydraulic cement, two and a half of sand, moistened with an alkaline solution of gum shellac of sufficient strength to furnish one

TABLE III.

PROPORTIONS BY VOLUME, LOOSELY MEASURED.

COMP. RESIST. IN

POUNDS PER SO. IN.

Cement, 1 ; common lime powder,

Boulogne Portland.

66

66

66

66

66

66

66

“I

gravel and pebbles, 5 0.4 ; sand, 5.6 gravel and pebbles, 13 0.8 ; sand, 5.6 gravel and pebbles, 5 0.8 ; sand, 5.6 gravel and pebbles, 13

CO coo OWW

O'

519

551

571

649

68i

675

831

649

623

649

753

ounce of the shellac to 1 cubic foot of the finished stone. Portland cement was used in the first three blocks and Louisville cement in the last.

Specimens of artificial stone made under the Van Derburgh system and used in the walls of the Howard University and Hospital buildings at Washington, D. C., in 1868 and 1869, varying in age from 3 to 16 months, gave resistances of 173 (4 months old) to 564 (10 months old) pounds per square inch. Another specimen of the same stone, ten years old, gave 1,455 pounds per square inch.

Addendum to Article 45.

The following Table exhibits results taken from “ Ex. Doc. No. 35, 49th Congress, ist Session.” The compression tests

Compression of Cubes of Cement, Cement Mortar and Concrete.

BRAND

OF

CEMENT.

NEAT CEMENT, MORTAR OR CONCRETE.

ULTIMATE COMP. RESIST. IN LBS. PER SQ. IN. FOR FOLLOWING CUBES.

1 Inch Cubes.

6 Inch. Cubes.

8 Inch Cubes.

12 Inch. Cubes.

16 Inch. Cubes.

WEIGHT PER CU. FT. LBS.

Dyckerhoff’s j Portland |

neat

cement.

cement.

4,860

4,272

National

Portland

3—sand.

cement. 3—sand. 6—broken stone.

3,45°

2,654

4,86s

2,478

5,436

2,434

2,520

Ï2S-I32

116-124

4,OI4

2,627

3,°27

2,690

2,979

i37-145

Newark Co’s Rosendale

Norton’s

Cement

Norton’s

Cement

1— cement. 3—sand.

-cement. 3—sand.

2— gravel. 4 —stone, i—cement, ij-sand.

i—cement, i J-sand.

6—broken stone.

_ -cement.

3— sand.

i—cement. 3—sand.

6—broken stone.

762*

800*

707*

685*

614*

113-118

1,032*

2,048

2,321

1,325

1,63s

j 1,025*

1 1,230

846*

1,194

831*

1,113

674*

1,039

127-136

1,340

1,746

1,346

1,247

117.2-122.6

963

75°

1,000

i,433

1,560

i,445

136-147.9

790

688

718

114-121

862

766

843

128-140

of the cubes were made in the Gov’t machine at Watertown, Mass. The compressed faces of the cubes were either made

* These results were with pine cushions or bearing surfaces. All other bearing surfaces were of steel. There is seen to be a loss of nearly 25 per cent, in passing from the latter to the former.

true in the original molds, or else were trued with plaster of paris. The ages of the cubes, and cushions between which they were crushed were as follows :

Dyckerhoff’s Portland,

National,    ‘ ‘

Newark Co.’s Rosendale, “ 22 Norton’s Cement,

age 22 months “ 46

cushion—steel.

“ 46

“ and pine.

Pyramidal or approximately pyramidal failure along planes of greatest shear took place very generally.

Each result for the Dyckerhoff Portland cement is a mean of six tests ; each of all the others is a mean of two tests.

The results are somewhat irregular, although the smallest cubes generally give the greatest. Those for the 8, 12 and 16 inch cubes run comparatively uniform for any given mixture.

The following table shows the mean results of a great number of tensile tests of Portland and Rosendale cements by Eliot

Ultimate Tensile Resistance in Pounds per Square Inch.

Age.

NEAT

CEMENT.

CEMENT, I. SAND, I.

CEMENT, I. SAND, 1.5.

CEMENT, I. SAND, 2.

CEMENT, I. SAND, 3.

CEMENT, I. SAND, 5.

R.

P. R. P. R. P. R. P. R. P. R. P.

i Day.. i Week i Month

12

H

NO ON W o O 4»- 00 tO CO tO

to

NO

CD CO to H 00 4^ 10 ON

'OSCn O

to M M Cn O h Cn ON O ON ON

4¹

95

155

230

NO CO O' CO «VO NW

H H N CO

24    95

6o

140

125    198

180    257

to COCO H H o Cn 4*.

1000 NO 1000 CO 10

H M

004»- h 0 ON ONCn

C. Clarke, as found in the “ Trans. Am. Soc. C. E.,” for 1885. “ P ” indicates Portland, and “ R ” Rosendale cement. The test briquette was quite similar to Fig. 3, page 363.

Art. 46.—Bricks and Brick Piers.

The first set of results given below is computed from data given by Gen. George S. Green, Jr., C.E., in Vol. II. of “ Trans. Am. Soc. of Civ. Eng’rs.

Nos. I, 2 and 4 cracked, but did not crush to pieces, as the others did.

NO.

I,

2.

3.

4-

5.

6.

SIZE OF BRICK.

Inches.

SURFACE.

Sq. ins. I5-5-. 15-6.. 15.8..

15-4

15.6

15-9

COMP. RESIST.

Lbs. per sq. in. ... 3,230

... 3,360 ... 2,750 ... 1,994 ... 2,050 ... 2,920

The pressure was applied on the two opposite largest faces of the bricks, giving blocks whose heights were only 0.7 their least widths.

In Vol. VII. of the “ Trans. Am. Soc. of Civ. Engrs.” Mr. Francis Collingwood, C. E., gives the following as the results of compressing ten whole bricks on end :

GREATEST.

MEAN.

LEAST.

3,060,

2,065

1,524 pounds per square inch.

For ten half bricks on small side :

6,400,

4,610.

2,900 pounds per square inch.

For ten half bricks on flat side :

4,150,

3,370.

2,670 pounds per square inch.

In regard to these tests Mr. Collingwood says, “ The bricks were selected to give a fair average of ‘ good Haver-straw stock brick,’ not the hardest burned. No packing was inserted in the machine between the bricks and the compressing surfaces ; so that the strength in compression represents the case of imperfect beds, etc., although it was found that it made but little difference.” He attributes the higher values for the “ ten half bricks on small sides,” over those belonging to the half bricks on flat side, to the imperfect bearing surfaces of the latter.

Table I. exhibits the results of testing piers of brick masonry-in the Gov’t testing machine at Watertown, Mass. ; it is taken from “ Ex. Doc. No. 35, 49th Congress, ist Session.” The

TABLE I.

Crushing Strength of Brick Piers.

HEIGHT SECTION NO. OF PIER

FT. IN. PIER.

OF

COMPOSITION

OF

MORTAR.

ULTIMATE

RESISTANCE.

WEIGHT PER CU.

FT. LBS. LBS. PER SQ. IN.

h N fori-mvo txoo 0»

O    H d ro-t lOVO f'.oo ON    OHM

H    HMMHHMHHH    WNW

2 o 2 o 10 o 10 o

2 O IO O

8 x 8 8 x 8 8 x 8 8 x 8 12 x 12 12 x 12 12 x 12 12 x 12 12 x 12 12 x 12

i lime, 3 sand, i    3    ⁴⁴

i Port, cement, 3 sand. I u u 3 ⁴⁴

i lime, 3 sand.

i 3 u

“ “    3 “

i Port, cement, 2 sand, i ⁴⁴    “2    ⁴⁴

I37-4

*33-5

*36-3

133-5

131-7

125.0

132.2

2,520 1

1,877

3,776

2.249

1,940

1,900

1.511

1,807

3,670 , 2,253 J

Face bricks.

2 o

9 9 10 o 10 o

10 o

12 x 12 12 x 12 12 x 12 12 x 12 12 x 12 16 x 16 16 x 16

i    lime,    3    sand.

135-6

r33-⁶

^■S

136.0

13¹

2,440 ^ 1,540

2,150

2,050

1,118

r,587

2,003

2,720

1,887

12 x 12 12 x 12 12 x 12

i lime, 3 sand.

H9.7

1,37°

i,i33

1,210

Bay Common bricks. State b’ks.

* +-

ro -<*• mvo t^oo 0\ o « N co Cl Ci N N M MfOtnn

CO to

7    10

10 o 10 o 10 o 6 o

12 x 12 12 x 12 12 x 12 12 x 12 8 x 12 12 x 16 12 x 12 12 x 12 12 x 12 12 x 12 12 x 12

i lime,    3    sand.

13

i    ⁴⁴    3    ⁴⁴    , 1 Rosendale    cement,

i Rosendale cement, 2 sand, i lime, 3 sand, 2 Port, cement, i Port, cement. 2 sand, clear Port, cement.

IIÖ.2

ii8.i

107

118

120

!,33i 1 1,211 I I,I74 I

924 I

940

1,646

1,972 1,411 1,792 2,375 J

dimensions of piers are shown in the table, also the kinds of mortar used and the grades of brick. The “common ” and “ face ” brick, both hard burnt, were from North Cambridge,

f Bricks laid on edge.

* Joints broken every 6 courses.

Mass. The other bricks were from the Bay State Brick Co., of Boston and Cambridge, Mass., and were medium burnt.

The crushing strength of three bricks of each kind, between steel compression platforms were first determined as follows :

BRICKS.

CRUSHING resistance; LBS. PER SQ. IN.

size; inches.

Face

11,056

13,984

16,734

Common

19.785

22,351

12,995

Bay State

11,120

12,709

10,390

13,925 A

V

18,337 s >

11,406 ^

7.75x3.7x2.0

8.0 X3.6X2.1

7.8 X3.6X2.1

Care was taken to make the bed faces of all three bricks bear evenly against the compression platforms, and in order to accomplish this result thin sheets of brass were used for packing.

The brick piers were built of bricks “ laid on bed, and joints broken every course, with the exception of two 12 by 12 piers, one of which had joints broken every sixth course, and one had bricks laid on edge.

“They were built in the month of May 1882,’’ and “their ages when tested ranged from 14 to 24 months.” They were all tested between cast-iron plates.

“ Loads were gradually applied in regular increments, . . . returning at regular intervals to the initial load, . . .” “Cracks made their appearance at the surfaces of the piers and were gradually enlarged before the maximum loads were reached. Final failure occurred by the partial crushing of some of the bricks, and by the enlargement of these cracks, which took a longitudinal direction and occurred in the bricks of one course opposite the end joints of the bricks in the adjacent courses. This manner of failure was common to all the piers.” •

It is very important to notice that the resistance of the piers varies with the strength of the mortar used in the joints.

The remarkably high results for the single bricks given above are probably due to the excellent quality of the material tested and to the great care exercised to have even bearings on the compressed beds.

Art. 47.—Natural Building Stones.

The ultimate resistances and coefficients of elasticity given in Table I. were determined in connection with the construc-

TABLE I.

FOUNDS PER SQ. IN. FOR

MATERIAL.

LENGTH IN DIAM. IN

INCHES.

INCHES.

Ultimate

Resistance.

Eu

Grafton Magnesian limestone

66

COCO mtnui O'

Portland granite 6 6 6 6

Richmond ‘ ‘ Portland

Missouri red granite << 66 66

(6

OO Li W W W C>cn Ui Ui W

OOOOOOOvOCoOQQOO CO-f^ OOOOOO^ICOCDOOOO    C>

1.14 1.06 i. 06

i 07

3x3

2.38

113

2.36

2.36

2.38

2.30

3x3

3x3

3x3

3x3

3x3

W to U3 to W    co O O O P w COVJ

'oil ’o ii ił R b tn o co h ^ o Q Ô» To

8888888888888888

10,500,000

8,400,000

8,500,000

6,000,000

12,000,000

5,000,000

5,500,000

6,400,000

5,000,000

13,500,000

tion of the St. Louis arch, and have been taken from Prof. Woodward’s history. The following results for Missouri stones are from the same soured :

u O

IŚ - .

ir ' |J: = = §

11=: I®

S

ft

>. >•

. rß w *

I t's I I

. vn u~> id co

?

; : ; 5*

Blue

TABLE II.

Two-inch Cubes.

POSITION.

COMP. RESIST. I.BS.

PER SQ. INCH.

POUNDS PER CUBIC

FOOT.

(/)

Staten Island, N.Y.... 1

Dix Island, Me........

Dark....... Quincy, Mass........

Light.....Quincy, M

Flagging . . North River........

Old Quarry.. Westerly, R. I........

Old Quarry.. "Westerly, R. I........

Up River.... Richmond, Va ... ...

Up River .. Richmond. Va.......

Niantic River New London, Conn ... Niantic River New London, Conn ... Porter’s Rock Mystic River, Conn ... t>—.—>_ t>—1. Cystic River, Conn ...

..... Westerly, R. I.........

.. .. Westerly, R. I.........

... Richmond, Va ........

..... Richmond, Va.......

..... New Haven, Conn.....

Gray........ New Haven, Conn.....

Gneiss....... Sachemshead Quarry,

Conn................

Sachemshead Quarry,

Conn..............

Duluth, Minn.........

........... Huron Island, Mich...

Bluish-gray.. Keene, N. H.........

Gray....... Pompton, N. J........

Glen’s Falls.. Glems Falls, N. Y.....

Glen’s Falls.. Glen’s Falls, N. Y.. ..

Lake....... Lake Champlain, N. Y.

mplain, N. Y. N. Y........

Gneiss

Dark . Dark

Lake

Lake Cha North River. Kingston,

North River. Kingston. N. Y........

White..... Joliet, Ill..............

White....... Joliet, Ill............

Drab........ Lime Island, Mich....

Drab........ Lime Island, Mich....

Drab ........ Marquette,    Mich.....

Drab........ Marquette,    Mich.....

Dark........ Bardstown,    Ky......

Dark........ Bardstown,    Ky.......

Drab........ Canton, Mo..........

Drab........ Canton, Mo..........

Caen ........ France...............

Caen ........ France...............

East Chester. Tuckahoe, N. Y.......

East Chester. Tuckahoe. N. Y.....

Vermont. ... Dorset. Vt..........

Vermont

Dorset. Vt

Granite.

Limestone.

Marble.

Bed.

22.250 15,000 ’7-75° ,4i75° 13.425 ¹7.750

17.250

21.250 20.000 12.500

14.175

18,125

Edge-Bed.

Edge. 22.250 Bed. 14,687 Edge.

Bed.

Bed.

Edge.

Bed.

Edge. 15.937

Bed.

Bed.

Bed.

Bed. 10,375 Bed. 24,040 Bed. 11,475 10,750 25,000 21,500 13.900 11,050

12,775

Bed. :6,ç°o Bed. 25,000 Bed. >5-425 7-825

7.600 16,250 Edge. 15,000 Bed.

Bed.

Bed.

Bed.

Bed.

Bed. 12. Bed.

Bed.

14,937

14,100

13,875

7,750

9,500

14,000

17.750

18,125

Edge.

Bed.

Edge.

Bed.

Edge.

Bed.

Bed.

Edge.

Bed.

9.250

5.650

3-65°

3-45°

12,050

050

612

8,670

OO^J sś '(»'000 OO-OWwi. Hui oo'ü ioûo COW 0    I O\b\*-t^»MOl0O

Cracked before bursting. Burst suddenly.

Cracked before bursting. Cracked before bursting. Broke suddenly.

Waxy-looking. Broke suddenly.

Syenitic.

Average of 3.

Burst without cracking.

Rather a clay stone.

KIND.

Drab........

Drab........

Drab........

Drab_____

Common Ital. Common Ital.

Brown.......

Brown.......

Gray........

Gray.......

Brown.......

Brown ......

Pink.........

Pink........

Drab .......

Drab........

Drab........

Drab.......

Drab........

Purple.......

Purple.......

Purple.......

Purple.......

TABLE II.—Continued.

LOCALITY.

POSITION.

COMP. RESIST. LBS.

PER SQ. INCH.

POUNDS PER CUBIC

FOOT.

--

n

Mill Creek Quari Mill Creek Quan North Bay Quarr North Bay Quarr

Italy...........

Italy...........

Little Falls, N. Y. .. Little Falls, N. Y.. . Belleville, N. J... . Belleville, N. J.. Middletown, Conn . Middletown, Conn .

Medina, N. Ÿ.....

Medina, N Y. .....

Berea, Ohio........

Berea, Ohio.......

Berea, Ohio .......

Vermillion, Ohio . Vermillion, Ohio Fond du Lac, Wis.. Fond du Lac. Wis Marquette, Mich... Marquette, Mich .. Red-brown .. Seneca Freestone, O Red-brown .. Seneca Freestone, O Olive green.. Cleveland, Ohio....

Olive green.. ] Cleveland, Ohio.. Brown Brown Pink. .

Pink. .

Albion, N Y .......

Albion, N. Y.......

Kasota, Minn .....

; Kasota, Minn......

Light buff . j Fontenac, Minn

Light buff, .J Fontenac, Minn.....

Freestone. .. Dorch’ter, New Brun

I wick..........

Freestone.... Dorch’ter, New Brun

wick...........

Yellow drab. Massillon, Ohio .. .. Yellow drab. Massillon. Ohio . Craigleith .. Edinburgh, Scotland Craigleith.... Edinburgh, Scotland

w u>

Marble.

Sandstone.

en Q* û^crq* Cl (7^ 0.73* CLcrç*    Q-crq* P* P* P* P* P* p*7q P-crq P*Cfq P* P* p*Grq O-crq p*

M 5 ONOOO\ N « p r ÿ ^ a o w o. ÿ U pv a M 3 LL o,_o H 0 Ç w » w Ö ą if)

VOC^OONNVOVOOOiO lOVO CO O' H lO CO CO 00 M O O COCOO OWN t-tCOO HMHHHMMI-IMHMMMMHMMMMMMł-iMMi-IMMMHlHHMH

Broke sud’ly. by years of

Calcareous.

Hardened

exposure.

ii

00 CO CO

Table II. and the other tables of this article contain the results of tests given in the “ Report on the Compressive Strength, Specific Gravity and Ratio of Absorption of the Building Stones in the United States,” by Gen. Q. A. Gillmore, 1876.

The specimens, whose tests are given in Table IL, were 2-inch cubes. “ Each cube was placed between two cushion blocks ot soft pine wood, 2 inches by 2 inches square, and slightly more than 0.25 inch in thickness ; one on the top and the other under the bottom ; the grain of the wood being parallel in each

to the other—though no difference was observed when this was changed, as regards amount of record.” . cubes .    .

.    “ The

.    . were brought to a true, smooth and regu

lar, but not a polished surface.” The third column shows whether the specimen was crushed “ on bed ” or “ on edge.”

TABLE III.

Berea Sandstone Cubes.

EDGE OF CUBE.

EDGE OF CUBE.

COMP. RESIST., LBS. PER

SQUARE INCH.

COMP. RESIST., LBS. PER

SQUARE INCH.

Inch.

O.25

Pounds.

4>99²

Inches.

2.00

Pounds.

8,955

0.50

6,080

2.25

9^30

0.75

6,347

2.50

1.00

6,990

2-75

8,856

9,838

1.25

7,342

3-00

10,125

1.50

8,226

4.00

11,720

1-75

9>3io

General Gillmore showed that the size of the cube tested, affected very greatly the ultimate compressive resistance per unit of area of face of cube. Table III. shows the results of gradually increasing the size of cubes of Berea sandstone, crushed “ on bed ” between wooden cushion blocks increasing (with size of cube) from about 0.0625 inch to about 0.4 inch in thickness. The general result is very marked in spite of two or three irregularities.

These results are natural consequences of the character of stone and the cubical form of the specimens. A few of General Gillmore’s experiments showed that such results would probably not appear if the length of the specimens had been two or three times the width or breadth.

The effect of different bearing surfaces on the ultimate compressive resistance of stone cubes is well shown by the results given in Table IV. All the results are in pounds per square inch, and belong to two-inch cubes, with the exception of the “Sandstone, drab” specimens, which were 1.5 inch cubes. Each result is a mean of two to five tests.

TABLE IV.

ULT. COMP. RESIST., POUNDS PER SQUARE INCH.

KIND OF STONE.

Steel

Wood.

Lead.

Leather.

Granite, Millstone Point, Conn....

Granite, Keene, N. H. ..........

Marble, East Chester, N. Y..... .

Sandstone, Berea, Ohio...........

Vermont marble, Vt..............

Limestone, Sebastopol............

Sandstone, drab..................

Sandstone, Massillon, Ohio........

Sandstone, Massillon, Ohio (softer).

23,190

24,000

19,125

11,260

13,280

1,075

4,000

8,500

5,660

22,880

19,830

0,540

10,290

10,850

1,075

4,000

8,750

6,730

15.730

14.480

11.560

7.380

9,200

1,075

4.000

7,250

5,500

15,730

6,730 8,190

1,075

3,640

The steel cushion gave the highest results by a little. A soft cushion seems to be driven into the small cavities and interstices of the specimen, and thus to produce a splitting action at the bearing surfaces. “ The beds of the granite and marble cubes were rubbed to the border of polish ; those of sandstone were rubbed smooth.”

Again, polished and unpolished cubes give different resistances per square inch, as shown in Table V. The results there given are for two-inch cubes pressed upon by wooden cushions.

It is at once evident that the polished cubes gave considerably the highest resistances. This is probably due to the fact that the splitting action of the wooden cushions was reduced to a minimum on the polished surfaces.

TABLE V.

ULT. COMP. RESIST., PER SQUARE INCH.

KIND OF STONE.

Polished.

Unpolished.

Granite, Quincy, Mass.......

Granite, Staten Island, N. Y. .

Granite, Garrison’s, N. Y.....

Granite, Tarry town. N. Y.... Granite, Millstone Point, Conn

Granite, Keene, N. H........

Granite, Westerly, R. I .....

Marble, East Chester, N. Y... Marble, Vermont, Vt.........

Pounds.

24.750 25,000 21,630

23.750 22,880 19,830 23,500 I7.540 10,850

Pounds.

17.750

22.250 13,330

18.250

18.750

12.750

17.750 12,950

8,750

General Gillmore’s experiments show, in a very conclusive manner, that variety in circumstances of testing will produce a variety of results for the same section of stone specimen. Attending circumstances and dimensions of specimens, therefore, should always be given.

Art. 48.—Timber.

Table I. is based upon results of experiments made at the Stevens Institute, which were given by Prof. Thurston in the Journal of the Franklin Institute for Oct., 1879. The specimens were well seasoned and turned to about 1.125 inches in

WOOD.

White pine.....

Yellow pine. ...

Locust

Black walnut. ...

White ash

White oak

Live oak

WOOD.

Georgia pine White pine. Locust.....

White oak

Spruce.

TABLE I.

POUNDS OF STRESS PER SQUARE INCH AT

PER CENT. OF

Ult. Resist.

21as. Lim.

Coefficient of Elas.

FINAL SHORTENING.

9-590

5,600

354-400

3-5

11,950

14.820

7,000

8,150

7,140

10,410

NO. OF EX

TERIMENTS.

7,000

9,800

5,700

5,180

5,600

6,300

TABLE II.

469,800

604,950

1,079,500

713,300

361,300

594,350

2.9

3-3

X.25

2-3

3-3

3-4

ULT. RESIST. IN FOUNDS FER SQUARE INCH-

Greatest.

11,500

7,500

12,580

9,780

8,410

Mean.

Least.

9,520

6,640

11,720

8,000

7,860

8,170

5,880

11,010

6,530

7A70

diameter with a length of 2.25 inches ; they were compressed in the direction of the fibre. The coefficients of elasticity were computed at the “ elastic limit,” i. e., at the point at which permanent set began.

Table II. contains the results of experiments made by R. G. Hatfield (“Transverse Strains,” 1877). The specimens were from one to two diameters high, and were compressed in the direction of the fibres.

The mean results of numerous English experiments by Thomas Laslett (“ Timber and Timber Trees, Native and Foreign,” 1875) are given in Table III. He found very little difference in the results for i-inch, 2-inch, 3-inch and 4-inch cubes ; those for the smaller cubes, as a rule, gave a slight excess over the others. The cubes were crushed in the direction of the fibre.

TABLE III.

TIMBER,

i, 2, 3 and 4-inch Cubes.

ULT. RESIST. IN 1 LBS. TER SQ. IN.

ULT. RESIST. IN

TIMBER,

i, 2, 3 and 4-inch Cubes. lbs. per sq. in.

Oak, English (unseasoned). Oak, English (seasoned)

Oak, French...............

Oak, Tuscan...............

Oak, Sardinian.............

Oak, Dantzic..........

Oak, American, white......

Oak, American, Baltimore..

Teak, Moulmein...........

Iron wood ....

Chow.........

Greenheart ...

Sabicu.......

Mahogany, Sp;

Mahogany, Ho

Red pine.....

Yellow pine...

anish.. nduras

4^ Cn 0\ CT\pO"F tO M Ul La JTs'S UiUiv)

M 'OSO) 4*. -L Cn bv-'J "bob -L 004*- vb 4* ó MO CO OS NvOS WvOS OOOJ SU 000

000 OOOOOOOOO t/iQOOO

Mahogany, Mexican Eucalyptus, Tewart. Eucalyptus, mahogany Eucalyptus, iron-bark . Eucalyptus, blue-gum. Ash, English.

Ash, Canadian Elm, English.

Elm, rock .... Hornbeam....

Fir, Dantzic..

Fir. Riga.....

Fir, sp Larch Cedar.

Pitch pine Kauri_____

ruce

5,6°o

9.350

7A7°

10,300

6,900

6,970

5.490

5.78°

8,580

8,310

6,940

5.240

4.850

5,820

4,480

6,470

6,43°

The results of the compressive tests of short blocks of timber, during the construction of the St. Louis bridge, are given in Table IV. These are especially valuable, both in consequence of the large size of the blocks and the fact that the pressure was applied with and across the fibre.

The blocks are seen to be from two to eight times as strong with the fibre as across it.

TABLE IV.

KIND OF TIMBER.

White oak White oak Black oak

Gum.....

Cypress ....

Ash........

White pine. White pine. Yellow pine Yellow pine Cypress .... Cypress .... White pine. Yellow pine White oak..

WITH OR TER-

TENDICULAR

DIMENSIONS

TO FIBRE,

Perp.

With.

Perp.

Perp.

Perp.

Perp.

Perp.

With.

Perp.

With.

Perp.

With.

With.

With.

With.

IN INCHES.

4x4x4

4x4x4

6x6x6 6x6x6 6x6x6 6x6x6 6x6x6 6x6x6 6x6x6 6x6x6

ULTIMATE RESISTANCE IN

ROUNDS TER SQ. IN.

REMARKS,

Greatest.

Mean.

Least.

2,200 3:5°° 2,000 2,700 440 3.100

722

3>3r>!

1,222

4,9'7

3.166

3,694

4,722

3,778

1,730

3,375

1,800

2,250

38.5

2,550

610

3,241

1,092

4,796

426

3,m

3,²9r

4,611

3,764

1,300

3,200

1,600

1,800

330

2,000

555

3,083

1,000

4,722

417

3,000

2,889

4,500

3,750

I Not well ( seasoned.

Well seasoned.

Table V. contains the results of tests by Colonel Laidley, U.S.A., “ Ex. Doc. No. 12, 47th Congress, 2d Session.” A few other tests of short blocks from the same source will be found in the article on “Timber Columns.” Unless otherwise stated, all the specimens were thoroughly seasoned.

In this table, the “ length ” of all those pieces which were compressed in a direction perpendicular to the grain might, with greater propriety, be called the thickness, since it is measured across the grain.

In the tests (24-60), the compressing force was distributed over only a portion of the face, of the block on which it was applied ; thus the compressed area was supported, on the face of application, by material about it carrying no pressure. In some cases, this rectangular compressed area extended across

the block in one direction but not in the other. In all such instances the ultimate resistance was a little less than in those in which the area of compression was supported on all its sides.

TABLE V.

NO.

LENGTH, INS.

ULT. RESIST.,

LBS. PER SQ.

PERP. TO OR

INCH.

WITH GRAIN.

REMARKS.

16.5

10

ii

12

*3

»5

16

17

18

20

21

22

23

24

25

2'..

27

28

29

30

31

32

33

34

35

36

37

38

39

40

4*

42

43

Oregon pine .

Oregon pine .

Oregon pine .

Oregon maple

Oregon spruce..... 24.02

California laurel...    8.0

Ava Mexicana. ...

Oregon ash........

Mexican white mahogany. ........

Mexican cedar.....

Mexican mahogany

White maple.......

White maple.......

Red birch .........

Red birch.........

Whitewood.......

Whitewood........

White pine.........

White pine.......

White oak..........

White oak..........

Ash....... ........

Ash................

Oregon pine.......

Oregon maple......

Oregon spruce.....

Oregon spruce. ... California laurel...

Ava Mexicana.....

Oregon ash........

Mexican white mahogany ........

Mexican cedar Mexican mahogany

White pine........

White pine.......

Whitewood........

Whitewood........

Black walnut......

Black walnut......

Black walnut......

White oak..........

Spruce.............

Yello

19.9

. 19-9

8.0

8.0

8.0

8.0 8.0 8.0 12.0 12.0 13.0 13.0 12.0 12.0 12.0 12.0 12.0

12.0

12.0

12.0

1.95

3-⁶3

3-92

3-9²

3-5⁸

3-⁶9

3-ć4

3-77

3-75

3-75

3.06

2.90

3-!5

3 15

0.875

0.875

0.875

2.46 X 2.0 X.2X X 1.2r I.21 X I.21

3.63    x 3.63

3-92 X 5.7s

3.58 X 3.58 3.69 X 3.69

3.64    X 3.64

3-77 x 3-77 3-75 x 3-73 3*75 x 3.7s 4.00 x 4.00 4.00 x 4.00 4.26 X 4.26 4.26 X 4.26 4.00 X 4.00 4.00 x 4.00 4.00 x 4.00 4.00 X 4.00 4.00 x 4.00 4.00 x 4.00 4.00 x 4.00 4.00 x 4.00 3.4s x 3.00

3.63    x 3.00 5-75 x 4.75 4.75 x 4.00 3.58 x 3.00 3.69 x 3.00

3.64    x 3.00

44

45

w pine.....

Black walnut

46

2.40 3-7° 3 90 0.75 1.00 1.25

3.77 x 3.00 3-75 x 3.00

3.73    x 3.00 6.20 x 4.75

4.73    x 4.00 4.75 X 6.20 4.75 x 4.00 4.75 x 4.00 4.00 x 3.94 4 00 x 2.50 4.75 x 4.00 4.75 x 4.00 4.00 x 4.00 4 05 x 4.00 4.03 x 4.00 4.05 x 4.00

?3-?! RH2 Kjfg?2-S giJfgj&g.Kg 11^1! § § § 8>|8R2 gj 8,8

00    lovo vo~ »o yo cT tC t-CccT    w cT cT cT    *-T    *-T sf cf of ro weTcTcT

With.

Perp.

Unseasoned.

Worm-eaten.

Unseasoned.

Unseasoned.

Mean of two. Mean of two. Mean of four. Mean of two. Mean of two. Mean of two.

Mean of four.

TABLE V.—Continued.

NO.

LENGTH, INS.

COMPRESSED ULT. RESIST.,

SECTION IN

LBS. PER SQ.

PERP. TO OR

INCHES.

INCH.

WITH GRAIN.

REMARKS.

47    Black Walnut

48

49    “    “

50    White pine ..

1.50

I*75

2.06

51

52

53

54

55

0-75

1.00

1.25

1.50

¹-75

2.00

56    Yellow birch

57    Yellow birch

58    White maple

59    White maple

60    White oak ..

4-25

4*25

4.00

4.00

3-95

8 8 8 8 8 8 8 8 8 8 8 8 8 8 xxxxxxxxxxxxxx

2,500

2,400

2,360

1,320

1,100

I.160

1,070

I,o6o

1,000

2,000

1,650

1,700

1,900

2,500

Perp.

Mean of two.

The “ ultimate resistance ’ was taken to be that pressure which caused an indentation of 0.05 inch.

Nos. (44-55) show the effect of varying thickness of blocks. Within the limits of the experiments, the ultimate resistance is seen to decrease, somewhat, as the thickness increases.

The results of the experiments given in this article Show conclusively that the ultimate compressive resistance of short blocks of timber depend upon a number of conditions, such as method of compression, quality of material, size of block, etc., etc. These reasons account for the different results obtained by different experimenters for the same kind of timber.

CHAPTER VII.

Compression.—Long Columns.

Art. 49.—Preliminary Matter.

THERE is a class of members in structures which is subjected to compressive stress, and yet those members do not fail entirely by compression. The axes of these pieces coincide, as nearly as possible, with the line of action of the resultant of the external forces, yet their lengths are so great, compared with their lateral dimensions, that they deflect laterally, and failure finally takes place by combined compression and bending. Such pieces are called “ long columns,” and the application to them, of the common theory of flexure, has been made in Art. 25.

Two different formulae have been established for use in estimating the resistance of long columns ; they are known as ‘‘Gordon’s Formula” and “ Hodgkinson’s Formula.” Neither Gordon nor Hodgkinson, however, gave the original demonstration of either formula.

The form known as Gordon’s formula was originally demonstrated and established by Thomas Tredgold (“ Strength of Cast Iron and other Metals,” etc.), for rectangular and round columns, while that known as Hodgkinson’s formula (demonstrated in Art. 25) was first given by Euler.

In 1840, however, Eaton Hodgkinson, F.R.S., published the results of some most valuable experiments made by himself, in cast and wrought iron columns (Experimental Researches on the Strength of Pillars of Cast Iron and other Materials; Phil.

Trans, of the Royal Society, Part II., 1840), and from these experiments he determined empirical coefficients applicable to Euler’s formula, on which account it has since been called Hodgkinson’s formula.

Mr. Lewis Gordon deduced from the same experiments some empirical coefficients for Tredgold’s formula, since which time, Gordon’s formula has been known.

The latter is now in almost, if not quite, universal use among engineers, and will be completely given in the next Article. Hodgkinson’s coefficients and formula will be given farther on.

Before taking up either, however, it will be useful and convenient to determine the moments of inertia and squares of the radii of gyration of the various forms of cross sections of the columns now in common use.

o\c

It will also be both convenient and important to determine the conditions which exist with an isotropic character of section in respect to the moment of inertia.

f-G

Fig. 1. a

In Fig. la let BC be any figure whose area is A, and and whose centre of gravity is at O. In the plane of that figure let any arbitrary system of rectangular co-ordinates X', Y' be chosen and let XY be any other system having the same origin ; also, let x, ÿ and x, y be the co-ordinates of the element D of the surface A, in the two systems. There will then result :

x — x cos a -f- y sin a. y — y cos a — x sin oe.

The moments of inertia of the surface about the axes y and x will then be:

x²dA — cos² ct I x²dA -f- 2 sin a cos a

jy²dA

— cos² a

jy'²dA

— 2 sin a cos a

jx'y'dA -j- sin² a jy'²dA. x’y'dA -j- sin² a jx²dA.

If x and y are to be so chosen that they are principal axes, then must jx_y dA

xy dA — sin a cos a jy'²dA + (

= o, or :

cos² a — sin

² a) jx';y'dA

— sin a cos a x’²dA

(I a)

tan 2 a —

2 ^xy’dA

jx'²dA — y²dA

Hence, since tan 2a — tan (180 + 2a), there will always be two principal axes 90° apart.

Now, if

tan 2a = o. This makes a — o or 90° ; i.e., X’ Y' are the principal axes.

If, however, x'y'dA = o, while a is neither o nor 90°, Eq. (1«) becomes :

^x'y'dA = o, while no other condition is imposed,

jy'²dA —

x’²dA — o;

or :

—, i.e., indeterminate,

o

tan 2a =

This shows that any axis is a principal axis ; also, that :

= y'*dA.

J.x²dA z=. fldA = ^x'²dA

Hence the surface is completely isotropic in reference to its moment of inertia ; or, its moment of inertia is the same about every axis lying in it and passing through its centre of gravity.

It has been seen that this condition exists where there are two different rectangular systems, for which

jxy dA = Ix'y'dA = o ;

but the first of these holds true if either x or y is an axis of symmetry, and the latter, if either x or y is an axis of symmetry.

Hence, if the surface has two axes of symmetry not at right angles to each other, its moment of inertia is the same about all axes passing through its centre of gravity and lying in it.

Eqs. (la) and the two preceding it also show that the same condition obtains, if the moments of inertia about four axes at right angles to each other, in pairs, are equal.

In the case of such a surface, therefore, it will only be necessary to compute the moment of inertia about such an axis as will make the simplest operation.

Since a column fails partly by flexure, it is manifest that the moment of inertia of its cross section should be the largest possible about an axis passing through its centre of gravity, and normal to the plane of flexure.

(s 4- t) d³

bD

.    .    (I)

Box Column of Plates and Angles.

Fig. i shows the cross section of a box column composed of 4 plates and 4 or 8 equal legged |_s. FB and CD intersect at the centre of gravity of the cross section.

If the 4 Ls shown in dotted lines are omitted, the moment of inertia about FB will be :

k---5---1---

--

F-ł-

J..

I..

x___L

*-w-\--->

hi r-

r*

■L

ol

_.P 1

Fig, 1.

(s — a) (d — 2d)³ 4- a {d — 2s)³

If the dotted Ls are not omitted :

bt' 3

(d 4- t'Y (2 s 4- t) d³

4-

• • (2)

— a) (d — 2a)³ + a (d — 2s)³

If the 4 Ls shown in dotted linps are omitted, the moment of inertia about CD will be :

t'b³

f~~6 ⁺

a {w 4- 2t 4- 2s)³

(s — a) (w 4- 2t 4- 2d)³

(d — 2^) (w 4- 2 f)³

dw³

12

12

414

MOMENTS OF INERTIA.

[Art. 49.

If the dotted angles are not omitted :

tP

/=T ⁺

a [(w + 2t + 2sf-\w - 2s)³]

• (41

(s—d)[fw + 21 + 2af—(w—2af\    (d—2s) [fw + 2tf—w'i~]

12

If latticing is used instead of the two plates bt', t' becomes equal to zero, and the first term in the second member of each of the above equations disappears.

If A represents the area of the cross section, and r the radius of gyration:

(5)

r³ = —7

-7--c=

l---b---j------~»j

--=—

Box Column of Plates and Channels.

s— w-X~ —

F---1-----

6 Ä v

Fig;. 2.

Fig. 2 shows a normal cross section of this column. FB and CD -B intersect in the centre of gravity of the cross section. As in the pre-2 ceding Fig., these lines are lines of symmetry. The moment of inertia about FB is :

bt'³

—7- + bt

6

(d + ty

(s + t) d* — s (d — 2#)³

. (6)

The moment of inertia about CD is :

t’b³

/=-6 ⁺

2a {w + 2t Ą- 2s)³ + (d — 2a) (w + 21)³ — dzv³

12

•(7)

If latticing takes the place of the two plates bt’, all terms in the second members of Eqs. (6) and (7) involving t' will disappear. The moment of inertia about FB then becomes :

(s + t) d³ — s {d — 2d)³ '

. . (8)

and that about CD :

_ 2a {w + 2t + 2s)³ + (d — 2a) (w + 21)³ — dw³

12

• (9)

(Radius of gyration)² = r² = — ; in which A

A

is the area of whole section.

Eqs. (7) and (9) may also take the forms given in Eqs. (15) and (16).

Built Column of Plates and Angles.

Fig. 3 shows a normal cross section of this column with the two axes of symmetry, FB and CD, intersecting at its centre of gravity. The moment of inertia about FB is :

nil

•t“

t- —

ill

dl

CO

<-S

Fig. 3.

(s + }4t) d³

/=v(t ⁺ ^ ⁺ ^) ⁺

— a) (d — 2of 4- a (d — 2s)³

.    . (10)

The moment of inertia about CD takes the value :

t'b³

a (2s + t)³ (s — a) (2 a + t)³ (d — 2s) t3 + ----------^--— +

12

.(ii)

If the two plates bt¹ are omitted, the terms involving t' in Eqs. (10) and (11) reduce to zero.

(Radius of gyratiorif — r² =    ; in which A is area of sec-

tion.

-o-

False Channel Section.

Let FB and CD intersect in the centre of gravity, G, of the section. The distance xx of G from the back of the channel, is :

k-

j*---]--£--->j

f'-i

Fig. 4.

of the channel.

\b²d- (b²- t²){d- 2t')]

• (I2)

In which A is area of the cross section of the channel. This is usually found by taking one-tenth of the weight, in pounds, per yard Analytically :

A = 2bt' + t (d — 2t )

• • (I3)

The moment of inertia about CD then becomes :

21’ (b — xf + dx³ — (d — 2t') (xt — iy

• (h)

About FB, it has the value :

bd³ — [b — /) {d — 2t')³

12

. . (14«)

(Radius of gyration)² = r² =    .

The line CD can be very quickly and accurately located by balancing the section, cut out of manilla paper, on a knife edge.

Eqs. (7) and (9) may now take the forms :

i=‘-^ + 2r+2A (f+ ¹,)“' • • • (15)

(16)

i=2\i’ + a(=■ + ¹,)

In Eqs. (15) and (16) A represents the area of one channel section.

Angle Iron Section.

Fig. 5 represents this section

TT1

-o

with the two axes taken parallel l    J

to the legs, passing through the    ' jp B

centre of gravity G. The area F !    | "pB

of cross section is usually found__

from the weight per yard. Analytically :

jL

jf----•)---

D Fig. 5.

A - It + (/' - t)t — (/ + /' - t)t . . . (17)

Again :

. .    (18)

„    - (/¹ - r) (/ - /)]

t (/ - xj + l'x'* - (/' - t) (x' - t)3

.    . (21)

About FB :

If the angle iron is equal legged, / becomes equal to /'.

(Radius of gyratioiif — r* = ~ .

A

As in the case of the C, x₁ and x' may easily and accurately be found by balancing a model of the L section on a knife edge.

[T 'll

L JÜ

Latticed Column of Four Angles.

The four jb,s are held in the relative I positions shown in Fig. 6 by latticing, _j_B the latter being riveted to the legs of the l_s, but not shown in the Fig. The |,s are equal legged.

From either Eq. (20) or (21), the moment of inertia of the section of any one L> about an axis passing through

^-----0------j|

Fig. 6.

its centre of gravity and parallel to b, is :

t (l — xf + lx,³ — (/—/) (xx — tf

Hence the moment of inertia of the column section of Fig. 6, about FB, is :

.    .    (22)

A is the area of the column section, or four times the area of one L section.

If b is different from b', the moment of inertia of the column section about an axis passing through its centre and parallel to b' will be found by simply ćhanging b' to b in Eq. (22).

(Radius of gyration)² = r² = — .

—J-

A'

F—]------f-

Latticed Columns of Plates and Angles.

Fig. 7 represents a normal section of one of these columns. By the aid of Eq. (22), the moment of inertia of the section about FB may be written :

-B

2 \₃

i=r + b±(l-_ + (v + A-,.(2 ₃)

I*---*⁻D---**i

Fig. 7.

and that about CD, remembering that in /', b' is to be changed to b :

tb³

/=/ +-6'

(24)

If the plates are on the sides parallel to b', then b is to be changed to b' and b’ to b in Eqs. (23) and (24).

Cv<

Fig. 8 represents the normal section of the other of these columns, in which there is no latticing, the column being perfectly closed.

TX

Again, using Eq. (22), the moment of inertia about the axis FB is :

f-4-A.L

ir

t'(b'+ 2t;³

• • (²5)

_Jr_.

The moment of inertia about CD is :

K---b—I-----Ą

Fig? 8.

• (26)

= /' + 'Æ±lû (C+\b + f^ .

In the /' in Eq. 26, b' is to be changed to b. Ordinarily, b — b' and t — t'.

(Radius of gyration)² = r² — ~ , A being area of cross-

A

section.

Tee Section.

The axis FB is taken parallel to the head of the tee section,

and CD perpendicular to it,

____b___^______H while G is its centre of gravity.

Analytically, the area of the section is :

~T"

F“

-B

A = bt + dt' . . .    (27)

.Ł

Fig.9

The area may also be taken from the weight in the usual manner.

^ [{b — t )t² + {d -f- fft ]

. . (28)

The moment of inertia about FB is :

. . (29) •

The moment of inertia about CD is :

/ - tlP + dt

12

(30)

(Radius of gyration)² — r² — ~ .

A

As in the other cases, FB may be located by balancing on a knife edge.

False Eye Section.

If the area is not taken from the weight per yard, it may be written :

|c

A — bd — (b — t') (d - 2t) . . (31) C

---Ô-i----■*{

The moment of inertia about CD is :

• • (32) I

2tb² -f {d — 2t)t³

12

F—j----j-------B

> \t\

■l

Fig.10

About FB it has the value :

j _ bd³ — (b — f) (d — 2if

12

•    •    (33)

(Radius of gyration)³ — r³ = —

Star Section.

Fig. ii shows this section with the different dimensions.

The area of cross section is :

• • (34)

A = bt + b't' — tt' .

K-

4-

■b-

The moment of inertia about

F--

+----

■tr

-B

FB is :

• (35)

J—

Fig.1l

About CD the moment of inertia has the value :

tbß 4- [R - t)D

• (36)

12

Ordinarily, t = f.

(Radius of gyration)² — r² —

Solid Rectangular Section.

F-

f-

<p

-1---b-

In Fig. 12 A — bh.

The moment of inertia about

FB is:

•    •    (37)

Fig. 12.

and about CD :

(Radius of gyration)² = r² —

h²

— —-or — .

12

12

If the rectangular section is square, b — h.

Hollow Rectangular Sections.

of the section shown in Fig. 13 is: A — bh — b'h'.

The

The moment of inertia

area

bte - b'JC

b'

12

• •    (39) F

Q -t---.    i

---■(---1--------\~ B

and that about CD is :

Fig. 13.

hfo — kb"⁶

12

(40)

(Radius of gyration)a = r² =    .

All the equations of this case (except Eq. (40)), just as they stand, apply directly to the rectangular cellular section of Fig. 14, considered in reference to the axis

If there were 11 cells instead —

FB.

of 3, the space between any adjacent two would have the width

< -f >

b'

Fig.14

Solid and Idollow Circidar Sections.

First consider a solid cylindrical column whose cross section has the radius r₂, as shown in Fig. 15. The moment of inertia about any diameter is :

.    .    (41)

■f—ri—

(Radius of gyration)² =

nr₂⁴

4nrd

r₂²

- = r\

Fig. 15.

Next consider a hollow circular column whose interior and exterior radii are r₁ and r₂ respectively. The moment of inertia about any diameter is:

(A = area) . .    (42)

77-(V₂⁴ — rf) _ Afrf + rf) _

(.Radius of gyration)² —

n{r² ~ r²)

= r\

As tables of circular areas are very accessible, it may be convenient to write :

r² =

7rr₂

12.566

; or r²

_ n(r₂³ + r²) 12.566

Phœnix Section.

Fig. i6 shows the section of a 4 segment Phoenix column.

Let CD represent any axis taken through the centre of the column. The moments of inertia of the rectangles bl about axes through their centres of gravity and parallel.to CD will be very small indeed compared with the moment of inertia of the whole section. The moment of inertia of any one of these rectangles, therefore, about CD, will be taken as equal to the product of its area by the square of the normal distance from its centre of gravity to the axis CD. The moment of inertia of the section about CD will then be :

/ rĄ- ⁷

_//7

f    :/

-*----”4.

/c

$ 0 k

Fig.16

_ tt(V₂⁴ - r*)

/D

-f- 2ÖI

(r₂ + j) sin³ a+ (r₂ + j j

cos² a

+ 2 bl (r₂ + 0

(43)

The moment of inertia is thus seen to be the same about all axes, a result of the general principle established in the first part of this Article.    x

The area of the cross section is :

A = 7T(r₂² — rf) + 4 bl.

(43a)

Ms

(Radius of gyrationy — r³ =

The moments of inertia of six and eight segment columns may be found in precisely the same manner. The moments of inertia of the rectangular sections of the flanges about axes passing through their centres of gravity, being veiy small indeed when compared with the moment of inertia of the whole section, may be neglected without sensible error.

True Eye Section.

Let r — -₇- 2S.......; r is then the

b — t₁

batter, or slope, of the under side of each flange to the top or bottom of the beam ; it ranges from about one-third to essentially nothing.

7T

"TfT

v.

~TC

U-

If the area of the cross section is not deduced from the weight :

Area of section

= A — 2 bt + Jitj -f- s{b — /j) . (44)

S/

-b-

"7T

The moment of inertia about

Figi17

CD is:

j _ 2tb* + hyj r{b⁴ — tf)

12

48

• ■ (45)

If t^ is very small as compared with b, remembering that j r is then essentially equal to s, there will result :

Area of section

= A = 2 bt + hti + s(b —    . . . . (49)

Fig.18

(21 + y₂sfb* + hyi

12

. .    (46)

This formula is sufficiently accurate for all wrought-iron and steel beams.

The moment of inertia about FB is:

bd*--— (h* — hf)

12

• •    (4 7)

In any of these three cases :

(Radius of gyration)² = —

(48)

True Channel Section.

!c

< X, >j

In Fig. 18 let r =    ^

it is the batter or slope of the under side of the flange.

If the area of the section is not de-

¥---1

h, I

—-j——j-ér³----j---—b duced from the weight :

; as before,

The centre of gravity, G, can be found by balancing a manilla, or other, pattern on a knife edge ; or, analytically :

_ b²t +■ fi ht² + s{b — tï) (b + 2/j)

ihe moment of inertia about CD is :

— Ax?

• (51)

If tz is very small compared with b, and remembering that br is then essentially equal to s ; this last equation will become :

- Ax? ....    (52)

The moment of inertia about FB is :

bd'³ — ~ (A⁴ — A?) 8 r

12

• « (53)

In any of these three cases :

(Radius of gyrationy —

Deck Section.

The head of this section will be considered circular in outline, as shown in Fig. 19. Let a be the area of the circle C.

If the area of the section is not deduced from the weight :

Area of section

— A — a + (d — k)t₁ + ip ~ tz) (t + s) . . (55)

If the centre of gravity, G, is not found by balancing a pattern on a knife edge, there will result, analytically :

a (2d - h) + tt(d - h)² + {b- t? (f + st + H s*)

• (56)

ad-

-*T~>

-HG-

-tr

-d-

Fig.19

The moment of inertia about FB is :

(* + *)⁴ -

6r

--Ax*; . .

• • (57)

2S

in which equation r = ~^ .

The moment of inertia about CD is :

^/ą. al? -J- /j³ (d — h — t — s) -j- tfà -(- — (JA —

O

12

•(53)

br

If t. is small as compared with b, so that essentially —

j _ 3^² + 4^³ (d — h — t —s) + (41 + s)b*

• (59)

In all cases :

('Radius of gyration)² — —T

(60)

Angle Section about Oblique Axis.

The angle iron is here supposed to be equal legged, and the axis about which the moment of inertia is taken, passes through the centre of gravity (before found in this Art.) and cuts the sides / at an angle of 45⁰. In Fig. 20, G is the centre of gravity and HK the axis.

'7

K_

Fig.20

The moment of inertia about HK is :

/ - ²1*,⁴- (*1 - t)*\ + /{/ - (2x₁ - y₂1)\*

. (61)

If A is the area of cross section :

(Radius of gyration)² =

VO

If a long column has the same degree of fixedness or freedom in all directions, the least value of the square of the radius of gyration must be taken for insertion in Gordon’s formula, because in the plane of that radius the column will offer the least resistance to flexure.

Art. 50.—Gordon’s Formula for Long Coluynns.

Since flexure takes place, if a long column is subjected to a thrust in the direction of its. length, the greatest intensity of stress in a normal section of the column may be considered as composed of two parts. In fact, the condition of stress in any normal section of a long column is that of a uniformly varying system composed of a uniform stress and a stress couple. In order to determine these two parts let A represent the area of the normal cross section ; /, its moment of inertia about an. axis normal to the plane in which flexure takes place ; r, its radius of gyration in reference to the same axis ; P, the magnitude of the imposed thrust ; / the greatest intensity of stress allowable in the column, and a , the deflection corresponding to /. Let p' be that part of /caused by the direct effect of P, and p” that part due to flexure alone. Then, if h is the greatest normal distance of any element of the column from the axis about which the moment of inertia is taken, by the “ common theory of flexure

cPa

cP a h

If the column ends are round, ć — 1 ; but if the ends are fixed, the value of ć will depend upon the degree of fixedness. Also,

11

11

c'S A 1i

. . (2)

Hence,

fS

Eq. (3) may be considered one form of Gordon’s formula. Before deducing the more common and useful form, it will

l²

be necessary to show that A = a ; in which expression a is considered constant.

Let p be the greatest intensity of bending stress in any section, whose greatest value in the column is p". By the “ common theory ” (taking the origin of co-ordinates at the centre of gravity of the cross section at one end of the column, and the axis of x along the centre line before flexure) :

77 T d²y_ pi

dx² ~ h *

Also,

Mh

' = -r>

and p" =

MJp.

(4)

in which equations E is the coefficient of elasticity M the bending moment for any section, and M₀ the value of M corresponding to p".

Hence,

M₀

and

d²y __ p" M ~~ ~ Wi Mq ‘

dx²

Consequently,

The section located by /₀ is that at which the deflection is

is considered con-

greatest, and for which = o, while

dx    hn

stant. The ratio

M₀

is numerical, though variable, being one be-

tween quantities of the same degree. M₀ is exactly the same as M, except that x, in the latter, is displaced by l₀ ; there are the same number of terms in each, and those terms are multi-

A₀ rx

plied by the same coefficients. Now Mdx² may be so ar-

o J h

ranged as to have the same number of terms as Mw but the coefficients of those terms will be different, and the exponents of l0 in the former will be greater by 2 than the exponents of l₀ in M₀. Hence Ip = dl² (c being some constant) will be a factor in all the terms of the definite double integral. From these considerations it follows that,

A f*

o J ^0

M dx²

M₀

= all² ;

(6)

in which a! is some constant. Consequently,

a’p"

l²

Eh /a ~ A

(7)

It is seen therefore that the quantity a, depends upon both p" and E, and it is ordinarily considered constant.

Since / = Sr², Eqs. (1) and (7) give :

P" — ai

ĆPI² P P

— a “C ~1 ' * •

o r*

II

Ns

Ni.

►H

00

Eq. (8) sliows that aff — a. Hence,

fS

The integration by which Eq. (y) is obtained, being taken between limits, causes everything to disappear which depends upon the condition of the ends of the column. Consequently Eq. (9) applies to all columns, whether the ends are rounded or fixed. Let the latter condition be assumed, and let it be represented in the adjoining figure. Since the column must be bent symmetrically, there must be at least two points of contraflexure. Two such points, only, may be supposed, since such a supposition makes the distance between any two adjacent points the greatest possible and induces the most unfavorable condition of bending for the column.

Fig.1

If B and C are the points of contraflexure supposed, then BC will be equal to a half of AD, for each half of BC must be in the same condition, so far as flexure is concerned, as either AB or CD. Also, the bending moment at the section midway between B and C fnust be equal to that at A or D. Consequently, the free or round end column BC must possess the same resistance a» the fixed or flat end column AD. In Eq. (9), therefore, let / = 2BC — 2:

/S

¹ + 4a

Eq. (10) is, consequently, the formula for free or round end columns with length /,.

The flat, or fixed end column AD, is also of the same resistance as the column AC, with one end flat and one end free or round. Hence in Eq. (9) let there be put / = f AC = f and there will result, nearly,

fS

I 4- 1.8 a —-A

Eq. (11) is, then, the formula for a column with one end flat and the other round. A slight element of approximation will ordinarily enter Eq. (11) on account of the fact that C is not found in the tangent at A just as Eqs. (9) and (10) are based on the supposition that A and D lie exactly in the line of action of the imposed load.

If the column is swelled, as shown in Fig. 2, the the moment of inertia /, and distance, h, become vari. able. Hence :

Mh

, and p" —

MX

Consequently,

* = JL x* ■

P f.MJ,. I ’

and,

Fig-2

⁷° f* P" 4 M o )i₀Eh₀M₀ I

If, in the reasoning applied to Eq. (5), there be written —=■

M₀

for M, and for M₀, it will at once be seen that Eq. (12) will

-*o

give precisely the same general form of result as Eq. (5), but the coefficient a will have a different value. Farther, since 4 Ar I can never be less than unity, but is in general greater, it follows that, for swelled columns, a is greater than for columns that are not swelled. Although these considerations show that the value of a will be different in the two classes of columns, yet they also show that the general form for the breaking weight P, whatever may be the condition of the ends, will be precisely the same whether the columns are swelled or straight.

Since the swelling of a column will give it a greater resistance to bending,/" will take a correspondingly less value, while

A and .S' remain the same. Eq. (8), then, shows that if f and 5 are unchanged, P must be increased. In other words, a swelled column will sustain a greater load than one not swelled but possessing the same kind and area of cross section. This is indeed true of solid columns, but may not be, and usually is not, for reasons to be assigned hereafter, true for built columns of shape iron. These reasons are not introduced in the hypothesis on which the formulae are based.

Although the quantities f and a, in Eqs. (9), (10) and (11), are usually considered constant, they are strictly variable. Eq. (7) shows that a is a function of p" -V E. It is by no means certain that p" is the same for different forms of cross section, or even for different sections of the same form, and the very variable character of the coefficient of elasticity is well known. It (the latter) is known not only to vary with the products of different iron mills, but even with the different products of the same mill.

Again, the greatest intensity of stress, f, which can exist in the column varies not only with different grades of material, but there is some reason to believe that it must also be considered as varying with the length of the column. The law governing this last kind of variation, for many sections, still needs empirical determination. It is clear, therefore, that both f and a must be considered empirical variables.

The expense necessarily attending experimental researches on the ultimate resistance of long columns built of American material, has prevented the attainment of many desirable results. Yet much very valuable work of this kind has been done.

umns.

In the “ Report of Progress of Work,” etc., made by Thomas

D. Lovett, consulting and principal engineer to the trustees of the Cincinnati Southern Railway, Nov. 1, 1875, are found the records of some valuable experiments on wrought-iron long col-The results of these experiments will be used in fixing values of a and f

If the number of experiments were sufficiently great, the results should be combined by the “ Method of Least Squares.” In the present instance, however, the use of the method is altogether impracticable in consequence of the small number of experiments of any given class. It will be seen, however, that the combination of the experimental results is not altogether of a random nature.

Since/and a are to be considered variable quantities, let y

¹

• (19)

The moment of inertia about CD is :

. . (20)

27

²

'b'³ + {b - t')t³

³

12

p

take the place of / and x that of a ; also, let / = — represent the mean intensity of stress. Eq. (9) then takes the form :

<43)

I -f-ex'

in which c = l² ~ r². For round or free end columns x will take the place of 4a, and of 1.8a for columns with one end round and one end flat.

In Eq. (13) there are two unknown quantities,^ and ^ consequently two equations are required for their determination. If two columns of different ultimate resistances per unit of section, and with different values of c, are broken in a testing machine, and the two sets of data thus established separately inserted in Eq. (13), two equations will result which will be sufficient to completely give y and x. Those two equations may be written as follows :

(H)

(15)*

/=/(!+ ć x) .    .

/=/'(! +c"x)

The simple elimination of y gives:

,r - P" -/

cp' — c'p'

Either Eq. (14) or (15) will then give y.

In selecting experimental results for insertion in Eq. (16), care should be taken to make the differences/" — p' and c — c" as large numerically as possible, in order that the errors of experiment may form the smallest possible proportion of the first.

Before applying Eq. (16) it would be well to recognize the condition of the end of a column resting on a pin, as in pin connection trusses. The end of a column resting on a pin might, at first sight, be considered round or free in a plane normal to the axis of the pin. The compressive strains existing in the vicinity of the surface of contact between the pin and soffit of the pin hole, produce a considerable surface on which the frictional resistance to any relative movement is very great. This resistance to movement is not sufficient to produce a “ flat ” or “ fixed ” condition of the column end, but causes a degree of constraint intermediate between the flat and round condition; so that a column with two “pin ends ” gives an ultimate resistance approximating to that of a column with one round end and one fixed end. The following two cases will then hereafter be recognized :

Two Pin Ends,

One Pin End and one Flat End.

All the necessary data for the treatment of the experiments given in the report of Mr. Lovett, are found in the following table. The column “ Area ” gives the areas of normal cross sections in square inches. The column r² gives the squares of the radii of gyration, in inches, about axes normal to the plane of bending. It is inferred from the table and the report under consideration that the radii of gyration for swelled columns belong to sections at middle of columns. The c column contains the squares of / divided by r, both being taken in the same unit ; it is a matter of indifference what that unit may be.

The quantities x, y and p are found by the formułce (16), (15) or

(14) and (13). The column headed Exp. contains the ultimate resistances in pounds per square inch, determined by experiment.

Keystone

VA

Uh?

Closed

Open

Square

Khoentx

Jltn.Kr. Co.

An “ open ” column is one in which the flanges of the segments that compose it are separated by an open space; a closed column is one in which no such spaces are found. All the columns treated in the table are closed except Nos. 2, 3, 4, 8, 25,31, 24, 26, 30, and 5.

The columns 13 and 19 failed about axes giving thq greatest • moments of inertia or radii of gyration. This was probably due to some cause equivalent to a less degree of constraint at the ends than was intended. For this reason those two results are not used in determining x and y. They will be noticed again.

An examination of the table shows that the flat end swelled

and open straight Keystone columns give about the same ultimate resistance, by experiment, per square inch, so long as c remains the same, though the straight columns give the largest results by a little. Hence p' was taken as an arithmetical mean of the experimental results of Nos. 4, 25, 31, 24, 26, and 30, and c' at 9,208. In the same mannerp" was taken a mean of the experimental results for Nos. 3 and 8, and c" at 3,060. The arithmetical means mentioned arep' = 25,517 pounds, and p" — 32,850 pounds. Substitutions in Eqs. (16) and (15) or (14), then give :

x = 0.00005455 and y — 38,300.00 pounds.

auoisXajj

■uui

O JC

nl°3

nbs

'xiuaoijci

•03 -Jg -ray

y Flat ends, swelled.

J- Straight

Flat ends, open ...

Flat ends, closed..

Pin ends, swelled

Flat ends,

Pin ends

Flat ends

Round ends

Flat ends

Special round ends,

Pin ends

Special pin ends.

Exp.

33,600

28.800

24.100 36,900

21.100 25,40° 25,000 27,5°° 30,000 30,000 32,000

27.800

22,000

30,000

33.200

30.200

25,500

37,500

»0

31,000

34*8

36,600

21,700

23,700

31,5°°

24,000

26,700

26,500

22,000

27,800

Wrought Iron Columns.

I-

35,35°

30.900 23,600

30.700 24,000 24,000

26.700

26.100 26,300

33.100 32,200

24.900

21,950

30,000

32,400

30,600

25,800

39,°oo

34,000

33,500

33,500

21,650

23,700

31,500

24,600

*4,¹⁰⁰

27,400

600

,4°°

23,

26.

y-

X.

AREA.

36,000

36.000

36,000

36,000

36,0

o<.

36,000

39,500

39,500

39,500

39.500

39.500 39,5°° 36,000

39,00°

39,000

39,000

39,o°°

42,000

42,000

42.OOO

42,000

42,000

36,000

36,000

36,000

36,000

36,000

36,000

36,000

.00005455 or

326

11.044

14-25

OOO

9gsoooo* jo—-oooSi

tl

000Z1

0000S

46000

OO

215

.0000588 or

.00002 or

goooo*

.0000217 or

II5OO .000047 or

2.99¹

9,646

3,*3° 9,189 9,157 8,718 9,39i 9,x57 3,5i9 4^36 i°,7I4

i6S‘6

10,414

7,*33

9,623

8,680

3,796

n,749

12,635

12,635

n,749

24,053

6,657

5,344

EiS'oi

6,596

11,147

7)77°

10.834 10.883 10-353 11.424 n .464 12.041 11.178

464

9.206

7-833

9.798

10.945

ii

9-347

ii.628 10.909

000*11

8-536

8-935

8-935

8-935

8-935

5-388

8.653

18.215

5-479

8-733

8-733

I3-5I

14.84

12.96

14.80

18.83

15-13

19.20

14.49

15-13

14.62

23.67

18.83

13.12

13.60

I3-7°

26.05

13.60

14.09

13-7°

1358

13- 58

13.89

14- 97

20.10

25-05

12.50

19.90

20.72

20.10

^8 WöW8V8'8 8'8 8

t ’O

00'

9"

//

(..,

00"

//

00'

,oo

00'

OO

OO

OO"

OO

ïV!?»V»iî!?îr S? 'S JT> S' !? 8f S> 8 'S 8 S'« *

a ro ^co

10 H T*-vo 0

N rot) « ro

On

lO N    CO N    H

(N    N    N CO W

vo O CO O H LO 00 CO    vo

H N N M H    HH    H    H H H

y is taken at the nearest hundred. These values of x and y, placed in Eq. (13), give the values in column

Since, however, the resulting values of were a little too large for the swelled columns, and a little too small for the straight open ones, x was allowed to remain as determined, and y was made 36,000.00 for swelled, and 39,500.00 for straight open columns. The resulting values of u p " are given in the table.

The differences between the results in the columns “p” and “ Exp!' are not greater than experimental differences.

Since x depends on the condition of the ends of the columns, as well as on the character of the iron, it is reasonable to give it the same value for all flat end Keystone columns. Then taking^ at 39,500.00 pounds, it will be seen that Eq. (13) gives results agreeing, as nearly as could be expected, with those of experiment for straight closed Keystone columns with flat ends.

No. 5 is the only experiment with a pin end Keystone column. As it was also swelled, y has been taken at 36,000.00 and x at j-§hn>* so that p would be a little less than the result of experiment. As these values depend on one pin end experiment only, they should not be considered very satisfactory. At the same time corresponding values for other columns show that they cannot be very erroneous.

Precisely the same general principles and considerations governed the selection of x and y for the several remaining classes of columns shown in the table. The agreement between the columnsp and Exp. is as close as could be expected.

The extraordinary character of Nos. 13 and 19 has already been noticed. No. 13 was intended to be a pin end column, but the plane of flexure contained the axis of the pin. Now if it be considered a round end column in the plane of failure, x will have the value 4 X s^ooir = tt}wo> and the resulting value of p will be 24,600.00 pounds. The result of experiment was 24,000.00 pounds. Again, No. 19 was intended to be a flat end column, but it failed in the direction of its greatest radius of gyration. Using the values of and y for pin ends, there will result p — 26,400.00 pounds. The result of experiment was 27,800.00. The effect of defective fitting, etc., would therefore seem to be the lessening of the end constraint by what may be termed one degree.

Expressing all the results in concise formulae, they may be written :

Keystone Columns.

Flat Ends—Swelled

■P =

36000

l² ’    *

• • (U)

18300 r²

Flat Ends— Straight..

OU

Ui

CD

CL •

39500

■ • (18)

18300 r²

Pin Ends—Swelled

36000

.    . (19)

15000 r²

Square Columns.

Flat Ends

39000

l² ’ *

. . (20)

35000 r²

Pin Ends

39000

• • (21)

17000 r²

Phœnix Columns.

Flat Ends

■P =

42000

.    . (22)

50000 r²

Round Ends

42000

l²

• • (23)

12500 r²

Pin Endą (hypothetical) ..../> =

42000

• • (24)

22700 r²

American Bridge Co. Columns.

Flat Ends

■P =

36000

• • (23)

46000 r²

Round Ends

•/ =

36000

. .    (26)

11500 r²

Pin Ends

■P =

36000

. . (27)

21500 r²

The pin end formula for the Phoenix column is based on the hypothesis that the relation between the values of x for flat and pin ends is the same as that existing in the American Bridge Co. columns, which last is shown by experiment. This is a very unsatisfactory method, and should not be implicitly relied upon.

All values of x for round end columns are found by multiplying the corresponding flat end quantities by 4, according to Eq. (10).

Eqs. (17) to (28), inclusive, give the ultimate resistances of the various classes of columns. With great variations of stress a safety factor as high as six or eight may be used, or it may be as low as three or four if the condition of stress is uniform or essentially so.

For a complete account of the details of the foregoing experiments, the original “ Report ” must be consulted. The consideration of the shades of influence exerted by the different devices to produce a given end condition have here been neglected on the ground that such degrees of influence are too small to be involved in a practical formula.

Some important deductions bearing on built columns of all forms of cross section may be drawn from the results of these experiments. It has already been noticed that the swelled columns Nos. 2, 3, 4, 8, 25, 31, do not give as great ultimate resistances as similar straight ones; a result perhaps not to be expected, though the explanation is simple. Both internal tensile and compressive stresses are induced in the originally straight segments when they are sprung to their proper curvature in the swelled column. Consequently this internal compressive stress causes a portion of the material to reach its ultimate resistance much sooner than would be the case if the columns were straight. Again, a slight increase of direct compressive stress is caused by the inclination of the segments of the column to its axis. If the segments could be prepared for the column without initial internal stress, the ultimate resistance would probably be considerably increased.

A consideration of these experiments would also seem to indicate that a closed column is somewhat stronger than an open one. This is undoubtedly due to the fact that the edges of the segments are mutually supporting if they are brought in contact and held so by complete closure, but not otherwise.

Thus the crippling or buckling of the individual parts of the column is delayed, and the ultimate resistance increased.

The general principles which govern the resistance of built columns may, then, be summed up as follows:

The material should be disposed as far as possible from the neutral axis of the cross section, thereby increasing r ;

There should be no initial internal stress ;

The individual portions of the column should be mutually supporting ;

The individual portions of the column should be so firmly secured to each other that no relative motion can take place, in order that the column may fail as a whole, thus maintaining the original value of r.

These considerations, it is to be borne in mind, affect the resistance of the column only; it may be advisable to sacrifice some elements of resistance in order to attain accessibility to the interior of the compression member, for the purpose of painting. This point may be a very important one, and should never be neglected in designing compression members. It may be observed, however, that the sole object is to prevent oxidation in the interior of the column, and if the column is perfectly closed this object is attained. Phoenix columns which have been in the most exposed situations (in one case submerged in water at one time for several hours) during periods varying from twelve to twenty years, without the slightest oxidation in the interior of the columns, have come within the observation of the writer. Different results, however, in other cases have been found.

In the experiments detailed in Mr. Lovett’s report it is to be noticed that all deduced values of y are less than the ultimate resistance of wrought iron in short blocks, and some, though not nearly all, would seem to indicate that this difference increased slightly with the length of the column. Further experiments, therefore, may show that the quantity /"has some such value as the following :

C being a constant quantity, and fa function of the reciprocal of the length.

In connection with the experiments already detailed, Mr.

G. Bouscaren, C.E., has given an account, in the Trans, of the Am. Soc. of Civ. Engs, for Dec., 1880, of other experiments, the results of which are given in the table below.

Column No. 33 was composed of four angle irons,

arranged as shown in the figure. It was swelled from 8%" X 8]/f' at the ends to io" X io" at the centre. There was only one experiment

lently the val-not be deter-

'•I

10 I

iL-~4 — ¹⁰"—*|

ues of x and y in Eq. (13)

mined. The angle irons, however, were of the same manufacture as the iron of which the Am. Br. Co.’s columns were built. As a mere matter of trial, therefore, y is

taken at 36,000.00 pounds, and is then found to be    •

This result seems to indicate considerable advantage in such a form of column, but one experiment alone furnishes insuf-cient basis for such a deduction.

The columns 35 and 36 illustrate the effect of repeated stress. The columns 37 to 43, inclusive, were intended to furnish information in regard to the distance between the rivets in the zigzag bracing and the thickness of the metal, in order that the column may fail as a whole and not by “ local buckling.”

Columns 39 and 40 were each composed of a single short piece of channel bar ; the others were composed of two channel bars held together by zigzag bracing.

NO. LENGTH. AREA.

r’¹.

C.

X.

y-

Exp.

4^4^-NOJOJOJ-ł^U>UJOJ »0 H O <0 CO '■J Ui Q\Ul Ow

"A

rj¹

13

00 CC CO O CO 00 VO -t F) o '^-v

°y § S

cb

c|

36,000    31,700

31,720    Pin    Ends.

The following forms of cross section, and observations, are taken from Mr. Bouscaren’s account :

rT7 f rr-"

At?. 35.—Gave way by pin crushing and splitting web of channel. Column not injured otherwise.

No. 36. — Column No. 35 tested again after crushed ends had been cut off and thickening plates riveted on with pin holes 34 feet from centre to centre. Column failed by deflection.

No. 43.—Failed by bending sideways at right j

8"

----

K---10«"---,

TU

mT~

gles to pins, without buckling of metal. Bracing u^j i x y±" ; rivets 20" apart in same flange and 10'' j i in opposite flanges.

No. 37.—Webs buckled in both directions,

-r-^j⁵ in middle and one end of column ; column did

uf ¹² Ï⁷ not bend. Bracing 2" x TV' ; rivets 24" apart

#£222-in same flange and 12" in op- —^

posite flanges.

No. 38.—Failed in same manner as No.

37 and by deflection, simultaneously. Bracing -lit ^ and rivets same as in No. 37.

an

H”

lb

JJT

___à

ib

-12-

=/ić'

8" I

jj

fjLk. flanges.

No. 39.—Failed by buckling of web and

No. 40.—Same as No. 39. Failed by buckling of web and flanges.

No. 41.—Column same as No. 37 with rivets spaced 20", in same flange, instead of 24".

Failed by buckling of web and bending in j 1

__ijiF

lf>

both directions, simultaneously.

No. 42.— Failed by buckling in plane of lat-ticing, without buckling of metal.

! From these experiments Mr. Bouscaren J concluded that, for the ratio of length to diam-r eter used, “ the thickness of metal should not

be less than — of the distance between supports transversely,

TU

J *-----10" 4—4

0.'353

f!

8"

Jif

SO

.    .    .    . and that the distance between rivets longitudinally

should be such that the length of channel spanning it, considered as a column, .... shall give the same resistance per square inch of area as the column itself, treated in the same manner with the same constant "f",” {y).

These conclusions are agreeable to that reached by Mr. B. B. Stoney : “ When the length of a rectangular wrought-iron tubular column does not exceed 30 times its least breadth, it fails by the bulging or buckling of a short portion of the plates, not by the flexure of the pillar as a whole.” (Theory of Strains, 2d Edit., Art. 334.)

It should be stated that the experiments whose results have been given were made in hydraulic machines in which the forces were not weighed, consequently the results involve the “packing” friction, which was probably not great, however.

In applying Eqs. (9), (10), and (11) to solid cast-iron columns, there may be taken, approximately:

f — 80000.00 pounds, and a — Tfï¹TiT7.

For solid wrought-iron columns, approximately:

f — 36000.00 pounds, and a —    •

Experiments on steel columns are still lacking. Mr. B. Baker, in his “ Beams, Columns, and Arches,” gives for

Mild Steel, / = 67000.00 pounds, and a = Strong Steel, / = 114000.00 pounds, and a —

These, however, must be considered only loose approximations for the ultimate resistance.

In the “ Trans, of Am. Soc. Civ. Engrs.,” for Oct. 1880, are given the following formulae for ultimate resistance of wrought-iron columns, designed several years since by C. Shaler Smith, C.E. :

Square Column.

FLAT ENDS.

ONE PIN END.

TWO PIN ENDS.

385OO

385OO

l²'

375OO

l² ’

5820 d²

3000 d²

1900 d²

Phœnix Column.

42500

l²‘

40000

l² '

36600

l²'

4500 d²

2250 d²

1500 d²

American Br. Co. Column.

36500

l²'

36500

36500

l² '

I*, l²’

3750 d²

2250 d²

1750 d²

Common Column.

cdi

lattice

3650O

l²'

365OO

l²'

365OO

/**

2700 d²

1500 d²

. 1 L

I —J-* -- ——

1200 d²

The formula for “ square columns ” may be used, without much error, for the common chord section composed of two channel bars and plates, with the axis of the pin passing ' through the centre of gravity of the cross section.

Compression members composed of two channels connected by zigzag bracing, may be treated by the same formula after putting 36,000.00 for 39,000.00 in Eqs. (21) and (22).

Art. 51.—Experiments on Phoenix Columns,* Latticed Channel Columns

and Channels.

In May and July, 1873, some experiments were made at Phoenixville, Penn., on full sized Phoenix columns, by the Phoenix Iron Co. The results of these experiments are given in column headed “ Experiment,” while the column headed “p” contains the results of the application of the formula established in the preceding Article : ¹

42000

l²

or =

42000

/²’ *

.    .    (I)

50000 r²

50000 r²

* The preceding Article was written as a lecture and read to the Class in Civil Engineering at the Rensselaer Polytechnic Institute nearly a year before this Article was written ; it has, therefore, been allowed to stand without change.

according as the ends are “ flat ” or “ round.” “4 segment ” ones.

All columns are

TABLE I.

DATE.

ENDS.

AREA. LENGTH.

r².

Experiment.

May 3, 1873... Flat ... May 3, 1873... Round.. May 3, 1873... Flat ... May 3, 1873... Flat ... ’ July 19, 1873. . Flat ... July 19, 1873.. J Flat ...

Sq. Ins. ¹ Feet. 5.84 i 23.81

4.10

19.950

Lbs.

30,274.00

Lbs.

30,000.00

5-95

10.21

8.50

13-31

24.00

23-38

22.71

23.20

12.85    23.20

4.10

8.68

8.00

8.47

8.47

20,230

9>⁰65

9,282

9,i5i

9,i5i

16,387.00*

36,419.00

38,235.00

32,742.00

35,408.00

16,040.00

35,600.00

35,430.00

35,500.00

35,500.00

In applying the formula the length was reduced to inches, in order to bring it to the same unit as that in which the radius of gyration, r, is expressed.

The columns “ Experiment" and ¹¹ p ” are each, of course, per square inch.

It is seen that the experimental results, and those by Gordon’s formula, give a very close and satisfactory agreement. It is also seen that the analytical relation between flat and round ends is a true one.

The square of the radius of gyration, 4.10, was taken the same for the first and second columns because their normal sectional areas are so nearly the same. The value 4.10 belongs to a 4 segment column, whose area is 5.88 sq. ins.

The same observation applies to the last two columns. The value 8.47 belongs to a column whose area of cross section is 13.08 square inches.

A most valuable and instructive set of experiments on Phoenix columns was also made in the large testing machine at the

U. S. arsenal at Watertown, Mass., under the direction of Messrs. Clark, Reeves & Co., the results of which were presented to the American Society of Civil Engineers at the 13th annual convention, June 15, 1881. The value of these experiments is enhanced by the fact that they were made on full sized columns, such in reality as are used in ordinary bridge construction.

In the following table are given the results of these experiments, as well as those of several formulae presently to be explained.

The following is a portion of the notation :

l — length in inches ; r — radius of gyration in inches ;

E. L. = elastic limit in pounds per square inch ;

Exp. = ultimate resistance in pounds per square inch.

TABLE II.

NO.

LENGTH.

AREA.

r³.

I a- r.    A -t- r².

E. L.

Exp.

A-

/>'■

Feet.

28

28

25

25

22

22

19

19

1?

l6

l6

II

12

13

*3

*4

13

10

10

*5

l6

»7

18

*9

20

21

8 ins.

8 ins. 25' 2.65" 8' 9"

Sq. in. 12.062 12.181

I2-²33

12.100

12.371

12.311

12.023

12.087

12.000

12.000

12.185

12.063

12.248

12-339

12.265

11.962

12.081

12.119

n-9°3

11.903

18.300

18.300

Ins.

8.94

8.94

8.94

8.94

8.94

8.94

8.94

8.94

8.94

8.04

8.94

8.94

8.04

8.94

8.94

8.94

8-94

8.94

8.94

8.04

*9-37 '9-37

Lbs.

112

112

IOO

100

88

£3

76

76

64

64

52

52

40

40

28

28

l6

16

2.7

2.7

68.8

24

12,544

12,544

10,000

10,000

27,960

7,744

7,744

5,776

5,776

4,096

4,096

2,704

2,704

1,600

1,600

Lbs.

35,i:o 34,'50 35,270 35,040 35,57° 34,36o

365

29,290

28,890

784

784

256

256

7.29

7.29

4,7

57

33

76

26,940

28,360

29,350

29,590

28,050

29,5’°

35,

36,90 > 36,580 36,580 36,8:7 37,200 36,480

36,397 38,' 57

43.300 49,500

51,240

57,130

57.300 36,010 42,180

Lbs.

32,550

32,550

34,000

34,000

35,420

35,4:o

36,800

36.80O

38,130

38,13°

39,4oo 39,-00 40,700 40,700 42,200 42,200 44 >77° .44.77° 69,600

69.600

37.600 42,840

Lbs.

34,488

34,488

35i°4°

35,040

35,592

35,592

36,144

36,144

36,696

36,696

37,²48

37,248

37,800

800

38,352

36,666

In determining r² for Nos. 1 to 20, inclusive, a column whose

area of cross section was 12.23 square inches was taken. The areas of the actual cross sections varied so little from this quantity, that the resulting value of r² was assumed to belong to all of the first 20 columns. All the columns were tested *vith flat ends.

An application of Eq. (1) to these columns reveals considerable discrepancies between the results of that formula and the quantities given in the column “ Exp." of the table, when the values of / A- r become comparatively small, as was anticipated in the preceding article. Instead of the constant 42,000 jn the numerator of Gordon’s formula, these experiments show that a variable quantity must be used, which shall increase as / -f- r decreases, or as r -r l increases.

After several trials it was found that the following modified form of Gordon’s formula would give tolerable results throughout the entire range of the experiments :

(2)

40000 (l + y)

/²

50000 r²

The results of Eq. (2) are given in the column of the table headed pv The agreement between the two columns is not as close as could be desired, yet the discrepancies are not sufficiently great to vitiate the safe use of the formula.

In the following figure, the Watertown experiments, as well as those of Mr. Bouscaren and the Phoenix Iron Co. (given in this and the preceding Article), are shown by diagram. The different classes of experiments are indicated as shown. The experimental curve is drawn with particular reference to the Watertown experiments, for it is then found to be properly located in reference to the others. The other curve expresses Gordon’s formula according to Eq. (2). It would not be difficult to find an equation which would fit the experimental curve very closely throughout the range of the experiments, but it would not be as simple as Eq. (2), or as two others to be shortly given.

on or

It is interesting and important to observe that each experimental value in the diagram (which is a mean of two, belonging to columns of the same length, in the table), lies exceedingly close to the curve, with the exceptions of those shown at a and b. a corresponds to a mean of Nos. 17 and 18, and is abnormally high ; b shows the mean of Nos. 13 and 14, and is abnormally low.

72000

r_Li

® Watertown Experiments

I .....-1 —I — !■— I - I —I

F Bouscaren’s

a Phoenix

T44

iLdJi

V.

48000

33

_“lS

litown Exf Aiurpj__

!T

CL

36000

NT

24000

Flat end Phoenix Columns'

12000

140

120

100

80

7 4-

60

40

20

It may be observed that the experimental curve is nearly a straight line from a point just above b to the extreme left of the diagram. For that portion of the curve, therefore, the following formula applies very closely :

(3)

P' - 39>64° - 46 -

The results of this formula are given in the column headed • The table, in connection with the diagram, shows that

this formula may be used with accuracy for values of l a- ? lying between 30 and 140, and further experiments may possibly show that it is applicable above the latter limit.

For values of l a- r less than 30, the following formula will be found to give results approximating very closely to the experimental curve :

■c

On

•    -    (4)

The results of the application of this formula are given in the column headed “p"

The extreme simplicity of Eqs. (3) and’(4) makes it a matter of great interest and importance to determine, by other experiments covering extended ranges of / -=- r, whether those forms, with different constants, may not apply to shapes other than that of the Phoenix column.

The inapplicability of the true long column formulae, when

— is found below certain limits, which is shown in Art. 25, furnishes a proper foundation for thoroughly empirical formulae, such as those expressed in Eqs. (3) and (4).

By Eq. (4), the ultimate resistance of Phoenix wrought iron to pure compression would be about 60,000 pounds per square inch.

The results of the application of Eqs. (3) and (4) to Bou-scaren’s and the Phoenix experiments are not given, but the diagram shows clearly that they would be satisfactory. Data sufficient for the application are given in this and the preceding article.

The following is the record of the Phoenix tests of the very short columns shown at c, d and e in the diagram. It is a question whether the degree of distortion which accompanied the extremely high result of 65,867 pounds per square inch, was not considerably greater than that which would characterize the condition of “ failure ” in an actual structure. This important point cannot receive too much attention in connection with short column tests, where the relative distortion, in the condition of “ failure,” is far greater than that in long columns.

Latticed Columns and Channels.

----ji'-----4

NO.

1.

AREA.

rK

I r.

EXP.

*v

P".

Ins.

Sq. in. 6.98 i

3-95

15-6

60,573

51,500

55,500

6. c_)3

4. ii

3-95

15.6

60,387

5i,5oo

55,500

5-63

2-37

6.76

65,867

55,Soo

57,300

5-63

2-37

2.6

6.76

65,867

55>Soo 57,300

2.93

2.25

2.67

7-13

56,889

55,5oo

57,200

2-93

2.25

2.67

7-13

55,555

55,5oo

57,200

During 1880 and 1881 Col. T. T. S. Laidley, U.S.A., tested a large number of long columns composed of two channel bars latticed in the ordinary manner (Ex. Doc. No. 12, 47th Cong, ist Session). These columns were furnished with 3^2-inch pin ends, and were tested at Watertown, Mass., in the large government machine. The adjoining figure shows the relative positions of ] the channels and pin. 6”, 8", 10” and 12” ^s were employed, and all the columns, the results of whose tests are given in Table III., preserved the uniform distance of 8 inches between the channels.

The radius of gyration, r, of the cross section, given in that table, is in reference to the axis of the pin.

All the posts were single latticed, and the pitch of the latticing (the distance apart of rivets in the same flange of a E) was 18 inches for the 6 and 8-inch channels, and 22 inches for the 10 and 12-inch. 2" x Y" latticing was used for the 6-inch Es; 2" x yis" for the 8 and 10-inch, and 2lY" x Y" f°r the 12-inch.

The area of cross section for the Es of the same depth in different columns varied slightly, consequently about an average area was taken.

TABLE III.

Pin Ends.—3%" Pin.

AREA OF

LENGTH OVER

NO.

E. SECTION IN SQ.

INCHES (2 CS).

LENGTH IN

RADIUS OF

INCHES.

GYRATION, INS.

RADIUS ; OR

/ r.

Inches.

Pounds.

io

io

ii

12

13

14

15

i6

i7

iS

19

20

vO CO OOCOvOoOvOOvO

O O O O' O O O'^4 O' 0**4 O O 0*^1 O O 0*-4 O in in oi oi ui oi 01 O ltv ui O O cm ui Q ui Ln en Q

i6o

200

144

150

200

250

1S0

240

360

300

210

280

350

240

320

270

360

300

330

360

3.00

3-65

2-35

2-35

3.00 3 65

2-35

3.00

4.44

3-65

2- 35 ■ 3-00

3- 65

2-35

3.00

2- 35

3- 00

2-35

2-35

2-35

53-3

54-8

61.3

63 -9

66.7

6S.5

76.7

80.0

81.0

82.2

89-5

93-3 95-9 102.2

106.7 115 -o 120.0

127.8 140.6 153-4

35,025

33,9²°

34,450

34,130

33,790

33,770

34,i8o

32,375

31,475

33,oi5

31,935

31,800

30,780

30,085

29,600

30,820

25,885

24,355

21,33°

15,320

⁴p is the ultimate resistance per square inch, in pounds.

All these columns failed as wholes, and each result is

a mean of two. Other columns of the same set, “ and tested at the same ~ time, failed by buckling of the channels ; they cannot, consequently, be classed among long columns which are so constructed as to fail as wholes.

lË-V+i

rg

VO

The values of p in Table III. are shown graphically in Plate I. The ratio l~r is laid off along the horizontal line and the ultimate intensity p on the vertical line, as shown. The full curved line is then the experimental curve and possesses great value of a practical nature. Within the limits of the diagram, when the ratio

fcd

_j

EE

⁼⁼5[⁼ff

iièttü

s^+-v,

m-

ft-t

l+r

IE

=\\

is known, the ultimate resistance of the column per square inch (/) can be at once accurately read from the plate without calculation or scale.

Hi

FF

U40-

tion :

The following equa

..o

ID

39OOO

/²

(5)

30000 r²

probably gives as accurate results as any form of Gordon’s formula. The dotted curve is constructed from it. Its results are seen to be only tolerably approximate between the

limits — = 50 and 135. It possesses little value when compared with the plate.

Table IV. contains results for columns of the same set which failed by buckling of the individual channels of which they were composed.

TABLE IV.

NO.

E.

Inches.

10

12

12

12

12

LENGTH,

RADIUS OF

INCHES,

120

120

126

120

180

240

300

GYRATION IN

I , ULT. P

IN LBS. FER

INCHES, r.

SQ.INCH.

CONDITION OF ENDS.

2.35

51 -1

36,025

Flat.

2-35

5¹ •¹

33,740

One flat ; one pin.

3-65

34-5

35,450

Pin.

4.44

4-44

4-44

4.44

27.0

40.5

54-0

67-5

34,245

34,660

33,985

33,590

Pin.

Pin.

Pin.

Pin.

If r is the radius of gyration in reference to an axis through the centre of gravity of a single channel section, and parallel to the web, the following values will hold for the present cases :

6" E ; r' = 0.58 inches. 8" E ! r' — 0-48 inches. 10” E ! r' — 0.69 inches. 12" E i r' — 0.87 inches.

Although the lattice rivets were alternate in the same channel, each flange was unsupported for a distance equal to the pitch, z>., 18” for the 6" and 8” Es> and 22" for the 10" and I2" Es* Hence :

Tl ^ ^ ^ 0000

■—t -n •—t —t

U o co 0\

nrmn

M K) H w W U CO CO

31.0

374

31-9

25-3

Table IV. shows that the column of io" Es commenced to fail by buckling of the g^s when

r - 34-5,

and when

22 4- r = 31.9 ;

that the column of 12" Es commenced to fail similarly when the length became so small that

l ~ r — 27.0,

while

22 -4- r = 25.3.

These results would seem to show that pin end columns with single but alternate latticing will begin to fail by buckling of the channels when l-r r, for the column as a whole, becomes so small that it is about equal to the same ratio for a single channel between two adjacent rivets in the same flange.

Nos. I and 2 of Table IV. show that if the ends possess a greater degree of fixedness, the value of l — r is much greater when buckling begins to take place, but the number of experiments is not sufficient to indicate the exact amount.

As would be anticipated under the circumstances, / maintains about the same value for all the columns in Table IV.

Hence when l a- r becomes so small that buckling takes place, the ultimate resistance of the column is independent of the length.

The graphical representations of the results given in this Article show that the curve of ultimate resistances has a very sharp declivity for small values of / -f- r, but that it becomes nearly straight and horizontal for larger values, and that it again increases in declivity with a still father increase in that ratio. These phenomena seem to be much more pronounced in the tubular variety of columns. They find a simple and obvious explanation in the fact that in columns of moderate length the deflection at the centre of the column about keeps pace (in the same direction) with the movement of the centre of pressure at the ends.

Plate I. shows (what was to be anticipated) that this effect is also much less pronounced with pin ends than with flat ones, it being borne in mind that the phenomena here considered do not produce the horizontal straight line which would be seen if Plate I. included less values of / a- r than 50. The latter represents the buckling of the individual parts of the column, and not the failure of the column as a whole.

A few experiments by Col. Laidley with columns of the same Es as the above, but with pins only three inches in diameter/gave uniformly less ultimate resistance than those with three and a half inch pins. Although this result was to be expected, the number of experiments was not sufficient to justify any quantitative conclusions ; it can only be stated that the smaller the pin the less will be the ultimate resistance.

ULT. RESIST., IN

AREA OF

TABLE V.

Flat End

NO.

E.

SECTION IN SQ.

l.

r'.

7-

I.BS. PER SQ. INCH

INCHES.

-/■

10

ii

12

13

14

15

16

17

18

J9

Inches.

10

10

10

10

10

12

12

12

12

12

O'* O'* O' 0 4^    to to to to

O O O O O cococoodcococococo colo www O O O O O w uv ui w ui O O O O O w w w w

Inches.

6.00

17.60

23 QO

4S.OO

8.00

17.90

23.90

29.90 48.00 10.00

17.90

23.90

29.90 48.00 12.00 17.80

23.90 29 90 4S .00

Inches.

O.58

0.58

0.58

0.58

0.48

0.48

0.48

0.48

0.48

0.69

0.69

0.69

0.69

0.69

0.87

0.S7

0.87

0.87

0.87

IO.3

30-3

41.1 82.6

16.6

37-2

49-7

62.2

99.8

14-5

26.0

34-7

43-4

69.6

13.8

20. 5

27-5

34-4

55-2

42,293

36,835

33,910

28,140

43,295

35,280

35,975

33,400

30,620

35,o8o

33,820

34,355

34,050

34,o8o

37,240

36,590

36,605

35,150

36,040

Table V. contains the results of Col. Laidley’s tests of portions of the Es used in the columns which have just been treated. These portions had flat ends.

The moment of inertia of the section, from which the radius of gyration r was computed, was taken about an axis parallel to the web of the channel and passing through its centre of gravity.

Many of the results are means of two tests each.

The results given in Table V. are shown graphically in Plate II. The values of the ratio l a- r are laid off on the horizontal base line, to the left from 0 ; while the values of p in

Plate H,

=:§

_lu

—rf+H-Rl

r¹

rr

“T

ri

r+r-T-ff

OT

H+4

m

Of!l

HA

OT

|i

te

mi

T''

ot

F-

S:=H

pounds per square inch are laid off vertically from O, as shown. The full curve then represents with great accuracy the experimental* results.

The dotted curve represents the following form of Gordon’s formula for the ultimate resistance in pounds per square inch :

J±

UJ—

36000

L²

(6)

63000 r'²

v:

t±

jyF

in

₁i_UZ

~i i

This formula is sufficiently accurate for all ordinary purposes, between the limits

ttd

OT

I + r' = 15

FfF

and

ii

l r' = 90,

fa

but does not compare in value with the experimental

(fun

curve-

Addendum to Art. 51.

Sinjpe the issue of the first edition of this book, the series of tests of full sized columns, of which Table III. gives the results of the first 20, has been continued at Watertown, Mass., and the test records are given in “ Ex. Doc. No. 5, Senate, 48th Congress, ist Session,” and “ Ex. Doc. No. 35, Senate, 49th Congress, ist Session.” Table VI. shows the digested records put in shape to be of some value to engineers. These columns had 3^ inch pin ends, and the results belong to failures in the plane normal to the pin axes.

Columns 21, 22, 23, 24, 25, 26 and 37 to 48 inclusive were

The word “ built ” in the Table indicates that the

closed box columns composed of two channels and two plates ; the remaining columns, except those of the Wilson section, were composed of two channels latticed together in the usual manner.

LH J

_!JIL

ro 1

channels were built of plates and angles ; otherwise they were rolled. The Wilson column is that used so frequently by Jas. M. Wilson, C.E., formerly Eng’r Bridges and Buildings Penna. R. R. It has the section shown by the sketch in the margin. In these columns the pin was always placed parallel to the plate between the channels, i.e., normal to the webs of the channels and as shown by the broken lines.

In columns 25, 26, 39, 40, 41, 42, 47 and 48, the pins were placed through (z. e., normal to) the webs of the channels, as shown in the Fig. on page 455 ; in all the other channel columns the pins were placed parallel to the webs of the channels.

The results given in Tables III., and VI. are shown graphically on Plate A. All results are brought together on one

plate in order to obtain the most probable curve for ordinary wrought iron columns with 3^ inch pin ends.

TABLE VI.

3-2 Inch Pin End Columns.

NO.

INCHES.

SECTION SQ. INS.

LENGTH IN INCHES.

RADIUS

OF

GYRATION.

INCHES.

LENGTH OVER RADIUS. I A- r.

ULTIMATE IN LBS. PER SQ.

IN.

REMARKS.

7.6

160

8.1

i6o

4-5

36

7.6

160

4-5

36

10

10

10

10

7.6 ii.g 12 3 12.4 12.7

160

200

200

5-23

5-23

::°6

3¹

3¹

44

10

7-5

7-5

7.6

12

10

¹4

10

*5

IO

l6

10

7.6

12.1

12.2 ii.9 ii .9

17

18

7-7

J9

7-7

20

7-7

21

22

23

24

25

27

28

29

30

31

32

33

34

55

36

37

5-6

38

5-6

39

6.0

40

6.0

41

8.0

42

8.0

43

8.0

44

8.0

45

5-6

46

5-6

47

48

6.0

6.0

4Q

8.0

50

8.0

7-7

16.2

16.3 21.0 20.6 17.9

19.4 9.8

10.2 10.o

10.    o

16.1

16.3 9-7 9-8

16.2

16.1 9.21 9.44

11.42

11.42

17.8

17.2 12.65 12.76

9.24

9-36

11.42

11.    31 15-34 15-40

200

200

24O

24O

24O

24O

3OO

3OO

3OO

3OO

32O

32O

32O

32O

32O

32O

32O

3²°

320

320

5-98

5-98

5-23

5'23

4-5

4*5

4.6

5-98

5-98

4-5

4-5

5-²3

5-²3

3.78

3-⁸

2.7

2.7

120

120

180

180

240

240

240

240

320

420

180

180

180

180

240

240

240

240

240

240

240

240

160

160

1.87

1.87

1.87

1.87

2.44

2.44

1.87

I 87

2-44

2-44

2.9

2.9

2-73

2- 73

3- 6

3-6

3-6

3-6

2.9

2.9

2-73

2-73

4.2

2.5

44

34

34

46

53

53

65

65

SO

5°

71

71

61

61

85

85

s4

84

n₉

n₉

64

64

'/•

98

98

128

128

131

131

62

62

66

67

67

76

7r‘

83

83

88

38

64

33.910

3⁶.s8o

34.34°

33.53°

33.74°

34.67°

31.13°

3r.99°

39° 34.39° 34.12° 33>4IQ 33.63° 32,44° 32,83° 32,74° 31,610 29,870 . 30,840 3°,77°

28,020

27,91°

25,77°

25,95°

26,480

25,290

-30,220

31,380

25,160

21,050

26.430

22,540

19,38° 16, 19,700 '7,57°

31,050

30,720

33,205

32,329

32,077

32253

37,668

3°,596

28,950

29,879

29.947

29,186

30,965

3i,494

33,

220

Swelled.

Swelled.

Swelled.

Swelled.

Built

Wilson column

Built.

Built.

Wilson column.

TABLE Via.

Flat End Channel Columns.

NO.

E SECTION. INS. SQ. INS.

LENGTH

IN

INCHES.

RADIUS LENGTH OVER

GYRATION. RADIUS.

/ *+• r.

OF

ULTIMATE IN LBS. PER SQ. IN.

REMARKS.

INCHES.

TO

IT

12

47

47

51-5

Si-5

75

75

76

76

10

10

17

76.

17.4

12.6    •

12.7

76.

78.

78.

tntn O'O'

13

4.8

121

14

4-7

121

2-3

2-3

53

53

34,950 35,595 33,682'

33,o6i 33,003 34,505 33,019 33,943,

34,279'

33,333 32,666

33,862 j

36,720

35,33° f latticed both sides.

Built channels.

These columns latticed on one side. All above are complete box columns. Rolled channels

The results given in this table are a digest of the records of flat end column tests given in “Senate Ex. Doc. No. 35, 49th Congress, ist Session.” Within the limits of these tests, a comparison with the 3^ inch pin end results shows that the difference in the end conditions exerts no influence on the ultimate compressive resistance per square inch up to at least / r = 80. The number of tests in this Table is quite insufficient to establish any law between pin and flat end columns of this character. Hence no diagram is drawn or formula

given.

The various kinds of columns covered by these experiments are seen to possess about the same resistance, except the Wilson column, which falls from io to 20 per cent, below the others. This is due to the fact that in this section the greater portion of the metal is but slightly supported.

The full line on the Plate is drawn as a mean of the channel columns only, and is of great practical value. The upper broken line expresses Eq. (5) of page 458, which is probably as good a pin end formula for channel columns as can be devised.

Results of experiments will be given below which show that the resistance per square inch of a pin end column increases with the diameter of the pin, and inasmuch as pins ordinarily

F late A .

mm

i*

aü

rrn

»!

used in columns of the dimensions of those tested, usually considerably exceed 3^ inches, the mean value of these tests may probably be a little too low for ordinary bridge practice.

HM

- Tl

iTt r8Q

•mm

mm

• •i '

Tables VII. and VIII. contain the mean of a large number of most valuable tests of full size iron and steel angle, tee, channel and beam struts with the various end conditions indicated, by James Christie Esq., Supt. of the Pencoyd Iron Co. The detailed account of

ma

+r

j+rf

BÖS

i i I ; 1 i i fi TT 11 lii: i I

-T

no

this complete series of tests should be carefully consulted ; it may be found in the “ Trans, of The Am. Soc. of C. E.,” Vol. XIII., 1884. All sizes of angles and tees up to 4 inches by 4 inches by f inch and over 15 feet in length were used in these The “ hinged ends ”

The

mm

tests.

were either one inch or two inch pins in semi-cylindrical bearings or one inch or two inch balls in sockets, “round ends" were the above described balls resting on flat

The

mm

or plane surfaces.

The “ flat ends ” were secured by simply resting the carefully faced ends of the struts

a ma

FR

Li.

TTTT

£?

ttt

Es^hr:5c0---220-|--240-U-260-

Hr

-|-r

ËÉ

•äS

L_L 1

-160-

Jfc

-H-

-OH—

xaxc;

rj-H-iao-

sto

ffl

sffi

i±

rPm

ÏÈŒ

LT

sea

t±

WÊÊÊt

ś. -T

F

be

- —160-

m±

ifc

i—t

=p

_L

□i

44-4-

±ht

TTT

ffl

p±t

Ü1

«#s

i i i ;

500001

......

00. FCl

Plate D.

—T

IIIIIII⁻riirrr

jjjjJJj I LLL

ip;

f I

:

FF

_L

rr

î

I

TT

T

I

ie

,

f ■

%

—r

ffl

—~h'”——r

EEffiïT

//

v

:n::|:î:0

Li

:< //

;7

£

y-

-1—5r—/'

li

—

ti

n

=-

±

/

±

i

-î-

h

i

H-?

§

-—s

on the plane bearing surfaces of the testing machine, while the “ fixed ” strut “ ends ” were obtained by clamping the ends of the struts rigidly to those bearing surfaces.

Every imaginable means was taken by Mr. Christie to secure the utmost accuracy in all details of these tests.

In Tables VII. and VIII., I is the length of strut and r the

TABLE VII.

Mean Results of Wrought Iron Strut Tests.

20

40

60

80

100

120

140

160

180

200

220

240

260

2S0

300

320

340

360

380

400

420

440

460

480

FLAT END ANGLES.

49,000

40,000

35,ooo

32,000

29,000

26,000

23.500 21,000 19,000

16.500 14,000 12,000

10.500 9,000

7.500

6,000

4.800

3.800

3.200

2.900

2.500

2.200 2,000

1.900

HINGED END ANGLES.

45,000

40,000

36,000

32,000

29,000

26,000

22,000

17,000

13,000

11,000

9,000

8,000

7,000

6,000

5,ooo

4.500

4,000

3.500 3,ooo

2.500 2,300 2,100 1,900 1,700

FIXED END ANGLES.

45,000

38,000

34,000

32,000

30,000

28,000

25.500 23,000 20,000

17.500 15,000 13,000 11,000 10,000

9,000

8,000

7,000

6,500

5.800 5,200

4.800

4,3oo

3.800

FLAT END

FLAT END HINGED END ROUND END CHANNELS TEES.

TEES.

TEES.

AND

BEAMS.

OOQOOOOOOOOOQO

8888&8888888§a_

c5 M co m w Ń60    w OocTr^iô'rf^fcôcôcr

888880

ir> ir> O m Ö m

M w w to D L3 W -L JOJOCOLO^-J^cncn Ocoo to cn coto^J m

ooo

ooo

ooo

ooo

500

500

500

500

500

44,000

36.500

30.500 25,000

20.500

16.500 12,800

9.500

7.500 6,000 5,000

4.300

3.800 3,200

2.800

2.500 2,100 1,900 1,700

1.500

1.300

38,000

35,000

34,000

31.500 29,000 26,000 24,000 21,000 18,000 15,000

12.500 ir.ooo 10,000

9,000

7,500

6,000

5,000

4,000

least radius of gyration of its normal cross section. In order to get the least radius of the angle sections, the moment of inertia was taken about an axis through the centre of gravity of the cross section and parallel to a line through the extremities of the legs.

All results in Table VII. belong to wrought iron struts, while Table VIII. belongs to struts of Bessemer steel. The “mild” steel contained from o. 11 to 0.15 per cent, carbon, but 0.36 per cent, carbon was found in the “ high steel.” The ultimate tensile resistance of the former varied from 60,000 to 66,000 pounds per square inch with 26 to 24 per cent, stretch

TABLE VIII.

Flat End Steel Angle Struts.

ULTIMATE RESISTANCE, POUNDS PER SQUARE INCH.

ULTIMATE RESISTANCE. POUNDS PER SQUARE INCH.

20

30

40

50

60

70

80

90

100

no

120

I30

I40

ISO

I60

Mild Steel.

High Steel.

Mild Steel.

High Steel.

72,000

51,000

46,000

43,000

41,000

39,000

38,000

36.500

35,000

33.500

31.500 29,000 27,000 25,000 23,000

100,000

74,000

65,000

61,000

58,000

56,000

54,000

51,000

47,000

43.500

40,000

36.500

33.500

30,800 28,300

I70

ISO

I90

200

210

220

23O

24O

25O

26O

r^co O' N W N

3OO

MWMWWMMI-iMLO

^4 000 0> O M to    ^ COO M

o    hOOOBwOw’o

88888888888888

20,000

•23,8oo

21.800 20,000

18.400 16,900

15.400 14,000

12.800

11.800 11,000 10,200

9,500

9,000

in 8 inches, while the high steel possessed an ultimate tensile resistance of about 100,000 pounds per square inch and a stretch of about 16 per cent, in 8 inches.

It is to be observed that up to 80 radii of gyration the resistance of the fixed end angles falls below that of both the hinged and flat end struts, but beyond that limit it exceeds

them both until it reaches over double their values at and about 400 radii of gyration.

The flat and hinged end conditions approach each other in their resistances until they become nearly equal at the highest values oi l a- r.

f TH-$

^!#S

i=:f

T:2

11

'-H

iS

if

:::iE

Plates B and D represent graphically the results given in Tables VIL,

■ -Kir

-ri

—I—0Î

CO

Ts-

VIII. and IX.; Plate D being devoted wholly to Table VIII.

The curve

for flat end channels and beams has beeif moved to the right in order to separate it from the others. This curve includes not only Mr. Christie’s data, but that of Table V. and results of later tests found in “ Senate Ex. Doc. No. I, 47th Congress, 2d Session,” and given in Table IX.

Plates B and D and the precedingTables show that at and above 200 radii of gyration the iron and mild steel angle struts possess the same ultimate resistance per square inch. The iron and high steel

(f)

to

TFtâ

±5=

On

St

So

hH

ft

TT

ft

a pH

ii

si

Wg

edge;

ear

cbffli;

ui

If

to

—3

f-

\~’X

:i+l

lift

it

iff

..

-o-

■k

continually approach each other, and undoubtedly become equal in unit resistance at a length a little above 300 radii of gyration. This is due to the fact that the coefficients of elasticity for the three metals are essentially identical, since it has been shown in Art. 25 that long column resistance varies directly with the coefficient of elasticity.

TABLE IX.

Flat End Eye Beam Struts.

NO.

BEAM.

AREA OF SECTION.

SQUARE IN.

LENGTH IN INCHES.

LEAST RADIUS GYRATION. INCHES.

ULT. RESIST. IN LBS. PER SQ. INCH =

O h o m

MMMM

OOOOOOO 00 oo^r O' nH taM »M tcf- ^    ^    ^    ^    ^    ^    ^

u-> vn w w

4-18

6.05

6.65

6-59

14.4

6.85 7-15 10.26

9-3

10.19

10.46

14.8

14-74

120

180

192

193

57

192

192

155

216

264

264

264

264

0.6

200

0-75

0.8

0.8

1.24 0.72 0.72 # 0.92 0.92 0-93 0-93 i .00 1.00

240

241

242

46

267

267

169

234

284

284

264

264

24,210

13,99°

12,540

14,000

37,^-jO

12,460

11,920

20,170

16,020 (?)

11,100

10,300

12,400

12,690

Table VII. also shows that, from 40 to 120 radii of gyration, the resistance per square inch of hinged and flat-end angle struts are identical.

Although no formulae can be found that will exactly fit the curves of plates B and D, those of the form of Eq. (3), on page 453, most nearly accomplish that result. Inasmuch as it is generally impossible, in engineering design, to separate the conditions of flat and fixed ends, one formula only is given for these two conditions, the influence of the former predominating. Round end members are seldom or never found in engineering

structures, hence a formula is given for pin or hinged end angles and tees. If round end members should be used, the table and plate will show how much the pin end resistance must be reduced for a given value of l -f- r, in order to get the round end resistance.

The straight broken lines on plates B and D represent the following formulae : *

Flat and fixed end iron angles and tees.

p — 44000 — 140 -

(7)

Hinged end iron angles and tees.

p = 46000 - 175 -

(8)

Eqs. (7) and (8) are to be used only between the limits of l -7- r = 40 and l -7- r — 200.

* Although the above formulae possess great advantages, both in accuracy and simplicity, over the old Gordon or Tredgold forms, it is not amiss to state that the curved broken lines on plate B represent the following formulae :

Flat and fixed end iron angles and tees.

40000

(a)

30000

Hinged end iron angles and tees.

P-

40000

(à)

20000

These formulae can be used with fairly good results between the limits of / -s- r = 40 and / -4- r — 180. They are given simply in deference to an old usage, with the decided opinion that they should be abandoned.

Flat end iron channels and eye beams.

p — 40000 — 110 -

(9)

Eq. (9) is to be used only between the limits of / r = 20 and / — r — 240.

Flat end mild steel angles.

p — 52000 — 180 -

(10)

Flat end high steel angles, p — 76000 — 290

(ii)

Eqs. (10) and (11) are to be used only between the limits of l a- r = 40 and / -4- r — 200.

TABLE X.

Solid 2-Inch Square Columns — i-.V Inch Pin Ends.

ULT. COMP. RESIST. LBS. PER SO. IN.

LENGTH.

INCHES.

ULT. COMP. RESIST. LBS. PER SO. IN.

30

10

42

35

14

49

54

18

63

60

66

20

22

70

77

72

24

84

78

26

84

28

91

98

90

95-5

101.7

107.6

113.6 119 6

125.6

131-7

30

32

34

36

38

40

42

44

105

in

11S

125

132

139

146

153

30,125

28,160

26,515

26,475

25 430 27,245

26,800

24,630

24.705

25,050

23,365

21,415

20,395

20,430

19,085

20,150

137-6

143.8

149.8 155-7 161.S

167.8 173-6 179-5

89.6

II9-4

89.7

119 5

46

48

50

52

54

56

58

160

167

174

181

188

195

202

210

17.780

17,600

17.180

17,670

16,725

16,900

14-525

14,355

Hat ends.

30

40

104 j 26,180 22,730

139

One flat and one pin end. 104

40    139

30

25 155

22,160

Plate E shows the results of tests of solid 3 inch square wrought iron columns with ends bearing on pins 1.5 inches in diameter, as given in Table X., which has been digested from the records of tests found in “ Senate Ex. Doc. No. 5, 48th Congress, ist Session.” According to the usual notation, / in the Table is the length in inches; r, the radius of gyration (in inches) of anormal section, and d the length of aside (3 inches). As all bars are here 3 inches square, there is a constant ratio between d and r.

The formula shown by the broken line on Plate E is as follows : for pin end solid wrought iron columns :

p = 32000 — 80 -

(12)

p = 320OO - 277 -=

Eq. (12) is to be used only between the limits of / -5- r — 20 and l -i- r — 220, or l ■— d = 6 and / — d = 65.

TABLE XI.

Three-inch Square Solid Columns.

Is

O X Z U

w 2

hJ ~

83

ULT. REST. LBS. PER SO. IN.

120

4O

I39

10,285

ił

120

40

I39

18,335

Ił

120

40

I39

20,430

Is

120

40

I39

21,440

at

120

40

139

22.250

The “ flat end ” and “ one pin and one flat end ” results in Table X. are both interesting and important—as showing that

the resistance of the latter end condition is essentially a mean between those for pin and flat ends.

Table XI., taken from the same source as Table X., also possesses no little importance as showing the influence of pin diameter. An increase of f inch in pin diameter below inches increases the column resistance over 2,000 pounds per sq. in. Above that limit the increment of resistance for the same increase in pin diameter is continually less, although very material. As a general principle, it may be said thait an increase in pin diameter will produce a corresponding increase in column resistance.

Formulez for Engineering Practice.

If the greatest allowed working stresses in columns be taken at one one-fifth the ultimate resistance, as is usual for railway structures, the following formulae will result :

Flat end latticed channel columns.

8000

(13)

40000 r

Pin end latticed channel columns.

(14)

7800

Or,

30000 r% p = 8500 — 28 -

(15)

Eqs. (13), (14) and (15) should be used only between the limits of / -4- r — 40 and / -r r = 140 ; and Eq. (13) is given only as a formula which is quite generally used among engineers, but which, as yet, has no foundation on a series of tests of

full sized columns ; it gives results which are probably too high.

Flat and fixed end iron angles and tees.

p = 8800 — 28 -    .....

r

(16)

Hinged end iron angles and tees.

P = 9200-35-

(17)

Eqs. (16) and (17) are to be used only between the limits of l a- r = 40 and l — r — 200.

Flat end iron channels and eye beams.

(18)

p — 80OO — 22 -r

Eq. (18) is to be used only between the limits of / -=- r = 20 and / 4- r — 240.

Pin end solid wrought iron square columns.

p = 6400 — 16 -

P = 6400 - 55 ^

Eq. (19) is to be used only between the limits oi l A- r = 20 and l -T- r — 220, or / d — 6 and l a- d = 6$.

Flat end mild steel angles.

. I

p = 10400 - 36 -

(19)

(20)

Flat end high steel angles.

P = 152OO —

58-

r

(21)

Eqs. (20) and (21) are to be used only between the limits of l a- r — 40 and / -f- r = 200.

For columns with one flat and one pin end, in all cases use a mean of two pin ends and two flat ends.

For hinged end angles of steel, in the absence of experimental data, the proper reduction from flat end angles of the same material may be assumed to be the- same percentage, or ratio, as that between flat and hinged end iron angle columns

with an equal value of

It is important to observe that the new form of column formula (Eqs. (15) to (21) inclusive) is better adapted to forms of section in which the metal is near the neutral axis, than to those in which the metal is placed at the greatest possible distance from that axis. It is yet a question whether the old Tredgold form (Eqs. (13) and (14) ) is not the best for channel columns and those of similar section. The new form, on the other hand, is much the best for angles, tees, eye beams, solid sections, etc. No formula, however, which can be devised, is to be compared in value with the experimental diagram, like Plates A to E.

Steel Latticed Channel Columns.

Although some interesting tests of full sized pin end channel columns of Bessemer steel have been published by Mr. James Dagron in the Trans, of the Am. Soc. of C. E. for 1887, yet the

number was only 8, and the range of — too limited for the deduction of any law or formula, had the design of the columns been satisfactory. Again, it is not stated whether the rivet holes were drilled, or punched, or punched and reamed, while the resistance of the columns would probably be materially affected by those processes. Tests of full sized steel latticed columns are therefore still needed in order to positively fix their resistance.

Such tests as have been made, however, indicate that properly designed and fabricated steel columns, of metal ranging in tensile strength, in specimens, from 65,000 to 73,000 pounds per sq. in., will give a resistance from 25 to 33 per cent, in excess of that of wrought iron columns with the same value

of , provided that ratio does not exceed 135 to 140. The

working stresses for such columns, therefore, can be found by increasing those given for wrought iron 25 percent, for ordinary railway practice and usual lengths of span, or 33 per cent, for spans of, say, 300 feet and over. These limits represent about the general engineering practice of the present time (1887).

Details of Columns.

In addition to the data already given in another portion of this article, the tests cited in this Addendum show that the unsupported width of no plate in a compression member should exceed 30 to 35 times its thickness. These tests have usually been made with plates or metal k to & inch in thickness, and it is altogether probable that the above ratio of width over thickness would be increased with greater thicknesses.

In built columns, however, the transverse distance between centre Imes of rivets securing plates to angles or channels, etc., should not exceed 35 times the plate thickness. If this width is exceeded, longitudinal buckling of the plate takes place, and the column ceases to fail as a whole, but yields in detail.

The same tests show that the thickness of the leg of an angle to which latticmg is riveted should not be less than 1 of the length of that leg or side, if the column is purely and wholly a compression member. The above limit may be passed, somewhat, in stiff ties and compression members designed to carry transverse loads.

The panel points of latticing should not be separated by a greater distance than 60 times the thickness of the angle leg to which the latticing is riveted, if the column is wholly a compression member.

The rivet pitch should never exceed 16 times the thickness of the thinnest metal pierced by the rivet, and if the plates are very thick it should never nearly equal that value.

Art. 52.—Euler’s and Tredgold’s Forms of Long Column Formulae.

The form of the general formula given in the preceding Article, as will presently be shown, does not seem to be as well adapted to the expression of accurate results as that of Euler, given in Art. 25.

It has already been observed that the coefficient a (Eq. (9) as a factor, in which p" is the great-

of Art. 50), contains

est intensity of bending stress, i.e., a part of the quantity “p ” wdiich is sought. The possible use of the formula is based on the fact that E is very large in respect to p".

The existence of p" in a is due to the redundant form of Eq. (8) of the Article cited.

_ cććp"

E (see £q-

p

S’

Since, in that Article,/⁷ = — and a — af

(7)), Eq. (8) gives :

_ PI² _ a’c'p" PI2 P - a Sr* ~ E Sr* ’

ii

S'! hi

<>•

.    . I)

This is Euler’s formula as given in Eq. (6) of Art. 25. In this equation b has the analytical values \n²E, n²E and 2.25 n²E for ends fixed, rounded and one fixed one rounded, respectively, as shown in Art. 25.

It would seem, therefore, that, since Eq. (1) involves nothing variable in the second member but r -r /, it ought to give more accurate results than Tredgold’s form of Art. 50.

It was shown, however, in Art. 25 that the common theory of flexure is analytically applicable only to fixed end columns of wrought iron, in which the ratio of length over radius of gyration is somewhat greater than 140; and to round end columns in which that ratio is somewhat greater than 70. Since the implicit assumption of an indefinitely small cross section underlies the analytical treatment of long columns, it is possible that the analytical coefficients and exponent may not obtain far above the limits indicated in Art. 25. Now, since other conditions of ends will lie between these limits, it is seen that both long column formulae are strictly inapplicable to a large portion of the columns designed by engineers.

Fortunately, a sufficient number of experiments have been made with full sized columns to show that either form of formula, when holding empirical quantities properly determined, will give excellent results. This has already been shown for Tredgold’s form, and it will now be seen that Euler’s form may be expected to give still better results.

If, as is usual, r is the radius of gyration and / the length (both in the same unit), and if both the coefficient and exponent

T .

of y , in Euler’s general formula, be considered variable, the following equation (see Art. 25), may be written :

For other values f and /') of r and /, the mean intensity becomes :

r’\ x

(3)

Dividing Eq. (3) by Eq. (2), then taking logarithms and solving for x :

(4)

los (i%)

Subtracting Eq. (3) from Eq. (2) and solving for y:

(5)

P -P'

These formulae will first be applied to results of the experiments made on Phoenix columns at Watertown, Mass. These results are contained in Table II. of the preceding Article, and

-and “ Exp." are reproduced in Table I. of this

the columns

Article. In the latter, however, the column “ Exp." contains the means of the various pairs of experiments whose results are given in the former.

TABLE I.

Phœnix Columns.

112

100

88

76

64

52

Exp.

Ext.

P-

34.650

34,550

40.0

36,440

39,000

35,150.

35,000

28.0

40,700

40,630

35.000

35,530

16.0

50,400

43,400

36.130

36,150

2.7

57,200

53,400

36,580

36,900

68.8

36,000

36,570

37,ooo

37,800

24.0

42,200

41,400

Now, let there be taken :

10

00

P — 40,700»

P = 34,650.

Inserting these values in Eqs. (4) and (5), there will result :

x = 0.117 and y = 59,723.

Then let there be written :

II

VO

0.II7

(6)

The various values of

^ in Table I., inserted in Eq. (6),

give the results shown in columns ” of that Table. They are seen to be much more satisfactory, as a whole, than those given by any form of Tredgold’s formula in the preceding Articles; although Eq. (2) of Art. 51 is a little closer to the

I

experimental results for values of — less than 24.

So much of the curve represented by Eq. (6) as does not coincide with the experimental curve, is shown by the dotted line in the Fig. of the preceding Article.

That curve, together with the results given in Table I., shows the close agreement of Eq. (6) with experiment for all

T    I

values of — from 1 to — .

/    112

= I,

V

It is interesting and important to observe that when ~

Eq. (6) gives :

p — 60,000 ;

or about the ultimate compressive resistance of wrought iron in cubes.

An application of Eqs. (4) and (5), in the manner already shown, to the results of Bouscaren’s experiments on Keystone columns, given in the large table of Art. 50, gave the following results for swelled Keystone columns :

.*■ = 0.25 and y — 78,000; or:

(7)

fY \ ^

78,000 (7)

TABLE II.

Keystone Columns.

SWELLED.

STRAIGHT.

c.

Exp.

C.

Exp.

/•

326

33,600

37,800

8,718

25,000

28,000

2,991

9.391

28,800

28,700

27,500

27,700

9,646

9-I57

24,100

24,800

30,000

27,800

3A30

3,5i9

36,900

28,500

30,000

3r,35o

9,i89

4A36

21,100

24,900

32,000

30,700

9,!57

25,400

24,900

10,714

27,800

27,300

Also, for straight Keystone columns :

x — 0.25 and y = 87,000; or:

. (8)

p = 87,000

The results of the application of these formulae, and the ex-

perimental results, are given in Table II. The lengths and other data can be found in the table just cited.

By the same operations with the square column results (Bouscaren’s) of the same table, there were found :

x = 0-5, and y — 303,000; or:

P = 303,000

r\¹ A

(9)

The following columns, “ Exp. and “p ” contain the experimental square column results and those computed from Eq. (9).

c.

10,414

7,133

9,⁶²3

Exp.

30,000

33.200

30.200

P-

30,000

33,000

30,600

Square

Columns.

Only “ flat end ” experiments have been treated, for the others are utterly insufficient in number for the determination of the empirical quantities.

In fact, with the exception of the Watertown experiments on the Phoenix columns, the number of those with “flat ends” is not sufficiently great, nor the range of / -f- r sufficiently extended, to establish reliable formulae.

In all cases, however, it is to be observed that the formulae of this Article give results more nearly agreeing with the experimental ones than those computed from any form of Tred-gold’s or Gordon’s formula. It would seem that this form of formula has not heretofore received the attention to which its importance and value entitle it.

Each of the three Eqs. (7), (8) and (9), become inapplicable compressive resistance per square inch of wrought iron in short blocks.

These empirical results tend to give experimental confirma

tion to Euler’s formula, for the exponent and coefficient of

are seen to increase very much as the lowest value of c, in the different sets of experiments, increases.

Art. 53.—Hodgkinson’s Formulae.

The detailed account of the experiments on which Eaton Hodgkinson based his various formulae is given in the Phil. Trans, of the Royal Society of London, for 1840. His cast-iron columns were small ones, the greatest length of which was 60.5 inches. The greatest value of the length divided by the radius of gyration was :

6a 5

0.25

= 484 ;

while the least value of the same ratio was :

-£*■

O Oo

• : • Ol 'VJ

00

CO

to

|-t.

The greatest diameter was about two inches.

Let d = diameter of column in inches. Let l = length of column in feet.

Then for, the breaking weight (P) of solid cylindrical cast-iron columns, when expressed in pounds, Hodgkinson’s formulae take the shape :

ć/3-76

P = 33,379 JGÏ ; (for rounded ends) .    .    .    (1)

P = 98,922

d**⁵

Z¹⁷

; (for fixed ends) .    .

. . (2)

For hollow cylindrical columns of cast iron :

; (for rounded ends) .    .    (3)

P — 29,120

y>76    ć/3.76

Z¹-⁷

P = 99,320

Z>55 _ ć/3-55

Z¹-⁷

; (for fixed ends) ...    (4)

In Eqs. (3) and (4), D is the greater, or exterior, diameter of the column, while d is the interior diameter. It is to be observed that P is the total breaking weight in pounds.

The longest wrought-iron solid cylindrical column tested by Hodgkinson had a length of 90.75 inches and a diameter of about 1.02 inches. Hence the greatest ratio of length over radius of gyration was about 90.75 X 4 = 363.

His formulae for the total breaking weight of solid cylindrical wrought-iron columns, in pounds, are :

P = 95,848

d±*⁶

Z²

; (for rounded ends) ...    (5)

d 3-55

P — 299,617 —j^~ ; (for fixed ends) ....    (6)

In his experiments on square pillars of Dantzic oak, the greatest dimensions were : length = 60.5 inches, and side of square section = 1.75 inches.

His longest red deal pillar was 58 inches in length, and the cross sections were 1 x 1, 1 X 2 and 1x3; all in inches.

Hodgkinson used Lamandé’s experiments on French oak in establishing a formula for that material. In those experiments, the longest pillar had a length of 76.5 inches and a normal section of 2.13 inches by 2.13 inches.

Retaining the same notation, the following are the total breaking weights, in pounds, of solid square timber pillars with flat ends :

4->

II

'i

Ui

• • (7)

Red deal (dry) ;

• • (9)

French oak (dry) ; P— 15,455

In Eqs. (7), (8) and (9), “d” is the side of the square section of the column in inches, while l is the length in feet.

All the preceding formulae are to be used only in those cases in which the length exceeds 30 times the diameter or side of square, if the ends are fixed ; or 15 times the length, if the ends are rounded. Between these limits and a short block, in which the length is 4 or 5 times the diameter or less, the following formula is to be used : Let C be the ultimate compressive resistance of the material, per unit of area, in short blocks, and let A be the area of the normal section of the column ; then Hodgkinson’s formula for these columns of intermediate lengths is :

PCA

P' P+HCA ■ '

.    . (10)

The small size of the columns experimented upon by Hodgkinson militates very strongly against the practical value of his formulae, unless it should be shown experimentally that the same formulae maybe equally applicable to large and small columns.

With the greatest ratio of l over r, the ratio of the resistance of a fixed end pillar over that of one of the same length and with rounded ends was about 3.34. With the lowest value of l over r, the same ratio was about 1.63. According to Euler’s formula, that ratio should have been 4. It is seen, therefore, that with these columns the common theory of flexure failed far above the limit given in Art. 25.

From his experiments Hodgkinson drew the following conclusions :

The strength of a pillar with one end round and the other flat, is the arithmetical mean between that of a pillar of the same dimensions with both ends rounded, and with both ends flat.

A long uniform pillar, with its ends firmly fixed, whether by disks or otherwise, has the same power to resist breaking as a pillar of the same diameter and half the length, with the ends rounded or turned so that the force would pass through the axis.    ^

Long uniform cast-iron pillars with both ends round break in one place only—the middle ; those with both ends flat, near each end and at the middle ; those with one end round and one end flat, about one-third the length from the round end.

The resistance of solid pillars with round ends was increased about one-seventh by increasing the diameter at the middle. Flat-end pillars (solid) had their resistances increased very slightly by the same means, but hollow pillars seemed to derive no benefit at all by enlargement at the middle.

The resistance of flat-end pillars was increased slightly by the application of disks to their ends.

Irregular and imperfect fixedness of the ends may cause a loss of two-thirds, or more, of the resistance with ends perfectly fixed.

Solid square cast-iron pillars failed in diagonal planes.

The relative resistances of columns of the same length and area of cross section were about as follows :

Long, solid, round pillar.... “    square pillar ....

triangular pillar

IOO

93

no

Art. 54.—Graphical Representation of Results of Long Column Experiments.

If the values of / over r (length over radius of gyration), for

TABLE I.

Tubes.—Flat Ends.

NO.

LENGTH.

EXT. DIA.

THICKNESS.

AREA.

ULT. RESIST. PER

SQUARE INCH.

Inches.

120

Inches.

120

1-5

120

120

120

60

60

120

IO

60

ii

60

12

60

13

89

90

15

30

16

60

17

18

30

2.00

2- 35

2.50 3.00

1.50

3- 04

2.00

4- 05 2-35 2-34

2.50 4.00 4-05

1.50 4.00 2.00

19

30

20

30

2-35

21

29

2-34

22

29

2-37

30

23

28

24

28

2-34

2.50

3.00

4.00

Inch. O. IO O. IO

0.23

o. II

0.15

o. IO

0.17

o. IO o. 16 0.22 0.21 O II

O.24

O. 12 O. IO

O.24

O. IO

O.24 0.21 O.23 0.20 O. II

O.I5

O.25

Sq. ins. O.44 O.61

I.50 O.80 1-35 0.44 1.41 0.61

240

179

160

141

120

120

90

89

1.9

1.47

1-37

0.80

2.87 i .61

0.44

2.85

0.61

1.60

1.44

1-55

1.36

0.80

1.41 2.85

87

80

80

71

67

65

60

45

45

40

40

39

39

35

28

21

Pounds.

14.670 23,206 21,900

29.800

27.670

3I,l8o

29,790

33,300

26,960

29.330 30,000 35,ioo

26.800

33.330

34,220

32.200 36,980 35,66o 36,000 36,910 39,570 36,490

37,39°

48.200

a series of columns which have been tested to breaking, be accurately laid off on a horizontal scale, and if the breaking weights per square inch be laid off with equal accuracy on a vertical scale, the resulting curve will represent the resistances of all columns for which l over r lies within the limits of the experiments, with far more accuracy than any simple and practicable formula that can be devised. Such a curve for the Watertown experiments on Phoenix columns has already been incidentally constructed in Art. 51.

TABLE II.

Solid Rectangular Pillars.—Flat Ends.

NO,

LENGTH.

SECTION.

AREA.

l r.

ULT. RESIST. PER

SQ. IN.

vO OO^J O'    to H

Inches.

120

90

120

120

60

90

90

120

60

O h o    vnvo t^co O O

WMWWWWMMMWC4

60

60

30

90

60

30

30

30

15

3-8

Inches.

2.98 X 0.5 2.98 X 0.5 3.01 x 0.77 3.00 x 1.00

2.98    x 0.5 5.86 x 0.99 ) 3.00 x 1.0 f 1.02 x 1.03 3.00 x 1.51 3.01 x 0.77 3.01 x 0.99 5.84 X 1.00

2.99    x 0.50 3.00 x 1.53 I.03 X I 02 3-or x 0.76 3.00 x i.00 1.02 x 1.02 1.02 x 1.02 1.02 x 1.02 1.02 x i 02

Sq. Ins.

1-5

1-5

2.31

3.00

1.50

3.00

1.05

4-53

2.31

2.99

5-84

1.50 4-59 1-05 2.30 3 00 i .04 1.04 1.04 i .04

822

643

540

4M

400

311

3°o

272

270

207

207

206

204

200

135

104

100

50

25

13

Pounds.

8,160

2,410

3,380

4,280

5,630

9,600

9,750

10,170

12,970

18,070

17,700

16.850

19,990

17,270

27,770

29,660

25,330

34,550

48,680

50,400*

Bore this without failure.

Tables L, II. and III. contain the results of some English experiments on small flat-end wrought-iron columns of different

w?

tfjzf

-rtJrr

II

30':Ci(

feitH

g-fl

K Fh

m

:S

3-H-t5CH4-1-0CM4

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:^+ff==ff{=

ÜË

|5.0i

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müzui

i⁰' 'l‘ ⁵I°: ;

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'S3Cl.n_L r i?!J

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IeIt

il

J—L

Plate III.

FLAT END COLUMNS.

trrin

TT

4-

..

-

L Q......

800    750    7-00    6-50    600    5-50    500    450    400    3-50    300

I (-j~j    M|~~ T

All these experiments were on small cross sections. Were little more than models.

In reality the columns

forms of cross section. These results are taken from the “ Proceedings of the Institution of Civil Engineers,” of London, Vol. XXX.

The experiments on tubular and angle-iron columns

(Tables I. and III.) were made by Mr. Davies, while those on solid rectangular columns (Table II.) were made by Mr. Hodg-kinson. The graphical representation of these results is shown by a very accurate construction in Plate III. Fig. 1 belongs to Table I. ; Fig. 2 to Table II. ; and Fig. 3 to Table III. The result shown at a (No. 1 of Table II.), Fig. 2, is most anomalously high, as is very evident, and has been neglected

The horizontal scale shows the ratio of l over r, while the vertical scale shows pounds per square inch, to a scale of 30,000.00 pounds to the inch.

TABLE III.

3" X 3" X tV LS.—Flcit Ends.

NO.

LENGTH.

AREA.

/ -t- r.

ULT. RESIST. PER

SQ. IN.

Inches.

60

Sq. Ins.

1.7s

71

Pounds. 23,600 .

48

I.78

56

36

1.78

42

29,480

35,380

18

1.78

21

39,400

There are only four results with angle irons, but so far as they extend, they are less, for the same value of l over r, than those for either tubes or solid rectangular sections. This was to be expected, since the legs of angles are comparatively thin and give very little lateral support to each other. A single unsupported angle iron, therefore, does not make a good compression member.

These results, in connection with those of Art. 51, show very clearly that an empirical curve (or formula) may be constructed to cover, with sufficient accuracy for practical purposes, columns of different forms of cross section, provided they are so built that their component parts are mutually supporting.

As compression members of single angle irons with fixed ends are quite common in some riveted bridge and roof trusses, it would be desirable to frame a formula on an extended series of numerous experiments. In the present instance this is impossible, but the following formula may be used with safety for equal legged angle iron columns with flat or fixed ends, so long as l -f- r lies between 20 and 100:

(I)

p — 200,000

in whichp is the ultimate resistance per square inch. An application to the columns of Table III. gives the following results :

No.

/•

71

56

42

21

23,740 lbs. per square inch. 26,760 lbs. per square inch. 30,860 lbs. per square inch. 43,600 lbs. per square inch.

By comparison with the results in Table III., the deviations from the actual resistances given by experiment may be seen at a glance.

Art. 55.—Limit of Applicability of Euler’s Formula.

The great range of l -f- r in the experimental results of Tables I. and II. of the preceding Article, furnishes means of testing the applicability of Euler’s formula with high values of that ratio.

Mr. Hodgkinson determined the mean value of the compressive coefficient of elasticity for some wrought iron of presumably the same grade as that to which Table II. belongs, at about 23,250,000 pounds per square inch. That value gives :

ą.tc²E — 917,920,000.

Taking l a- r from No. 1, Table I. :

.    . (2)

• • (3)

= 47i²e(^J = 16,000 (nearly)

Experiment gave 14,670

Taking l A- r from No. 7, Table II. :

= 4n²E(^) = 10,200 (nearly)

Experiment gave 9,750

Taking l ~ r from No. 5, Table II. :

p = 47i²E

= 5»74° (nearly)

•    • (4)

Experiment gave 5,630

Taking l — r from No. 3, Table II. :

p — 47z3e(^ — 3,150 (nearly) Experiment gave 3,380

Taking / — r from No. 2, Table II. :

. . (6)

= 471²eÇ^ = 2,220 (nearly)

Experiment gave 2,410

In Eq. (2), / -f- r is 240, yet the result by formula- is only a little too large. With / -r r ranging from 300 to 643, the formula gives very satisfactory results. These tests would seem to show, therefore, that only when / -r- r becomes equal to about 250 for flat-end columns, does Euler’s formula become applicable to wrought-iron compression members, but that above that limit it gives very satisfactory results.

This is an interesting and striking confirmation of the correctness of the formula, which, as was stated in Art. 25, is based on the supposition that the lateral dimensions are very small compared with the length.

Art. 56.—Reduction of Columns at Ends.

When columns are built of angle irons, channel bars, or I beams, it is frequently the practice to cut off, for some distance back from the ends, the flanges of bars or beams, or one of the legs of angle irons, in order to give clearance for other members of the structure. In such cases the whole compression to which the column is subjected is carried, at the ends, by the webs of the bars or beams, or legs of the angles, which are thus solid rectangular columns of great comparative breadth and little thickness, even when reinforced by plates of the same thickness as the webs or legs. In such cases, the angle iron experiments of Mr. Davies (a part of which are given in Art. 54), and a most valuable set of full sized, latticed, channel-bar column tests, made at the works of the Keystone Bridge Co., Pittsburgh, Penn. (“ The American Engineer,” 4th Feb.,

¹

hen the value of-; is such that up” approaches the ultimate

1882), show that the full resistance of the column is not developed, but that they fail at the ends where the cutting away of the flanges and legs reduces the column to two thin, weak, rectangular columns. Columns, therefore, should never be cut away in the manner indicated unless the circumstances render it absolutely necessary, and then the ends should be reinforced by extraordinarily heavy thickening plates, so that the sum of the resistances of these rectangular columns, at each end, shall be equal to that of the column as a whole.

Art. 57.—Timber Columns.

Tests of this class of members, the results of which have been published, although of great value, have not been made with sufficiently large ratios of length to radius of gyration to produce true “ long column ” failures. This renders impossible the establishment of a long column formula or diagram for practical use in connection with the use of long timber columns.

Some very valuable experiments, however, have been made with full sized columns having lengths as great as fourteen feet. The first results to be given are those of a large number of tests by Prof. Lanza, of Boston, in which he used the United States testing machine at Watertown, Mass. These tests were made during 1881, on such members as are commonly used in the construction of cotton and woollen mills.

Table I. contains the results of Prof. Lanza’s tests. A large majority of the columns had cores bored out of the centre, which varied in diameter from 1.5 to 2.0 inches. The absence of material did not affect, in any way, so far as could be observed, the resistance per square inch.

Column 20 had the force applied 2p[ inches out of centre at one end, and column 35, 1.9 inches. These tests were made in order to observe the effect of eccentricity in the application of loads. They show a marked decrease in ultimate resistance.

NO.

FORM.

LENGTH,

TABLE I.

Timber Mill Columns.

Top. Mid. Base. Top. | Base.

ULT. RESIST., LBS. PER SQ. INCH.

ENDS.

i Round

5 Cylindrical..

12.01 12.02 12.01 12.02 12 00

Yellow pine ; partially seasoned. 9-3¹

8.30    9.71

54    8.88

10.65

p cs-4

40

7.80

10.55

IO.O7

8.99

7-79

ii

10

ii

12 Square......

34 Cylindrical..

92 11.98 2.01 2.00 2.01 2.00 12.44 ¹² 57 ¹¹-93

8.q(

7.70

10.46

9.98

8.9I

7- 79

8- 43

8.30

9-9²

10.45

8.96

7.70

65.89

5i-84

42.38

29.98

8375

61.19

44.72

85.92

85.22

77-3⁶

61.21

45-47

9-è² . 75-45

80

co in Oi d »•'•vo -tNinOwnThN

O'VO W o LOCO M 01 O OH 100 H

O vo r^vo vo O u-) Tf N cn 10 -*t-oo

■4-co'4--4’'^r'4-'4'4-'4'4-4-tororo

Flat

15    Round

16    Cylindrical..

13-99

2.00

7.70

7.70

Yellow pine ; air seasoned.

-    j 7 90 j 44.56 I 47.0X

-    I 7.Q0 I 44.56 I -

Yellow pine : dock seasoned.

17 Cylindrical.. 19 Square......

11.92 1.98 *¹ *93

8-00

8!o8

8-00

7.98 8 08

i j.

21

22

23

•94

12.85

2.00

2.01

8-75

lo.os

3-98

10.20

8.92

10.13

9.02

VC    O 00    TOO    -

O    O N    O    NO    N

06    06 rövö    O' «    ci

rr    vo    o 00    O

-N- O M O-L- Os Os

'S 'S 2    5 S'

One flat : one round. Flat.

One flat ; one round. One flat ; one round. Flat, pintle.

Flat.

24

25

Round

12.01

n.97

IV/iite 7uood ; partially seasoned.

9.65154.02170.95 7.72129.78144.62

8.46

6.38

9.61

7.40

White oak : partially seasoned.

26

27

28

Round

29

3°

31

Cylindrical..

3²

33

34

12.01 12.01 12.01 12.01 1200 6-33 2.00 2 00 j.98 n-93

M3

•37

7 55

6 6o

10.13

9.40

8-75

II.OI IO.23

10.00

10.00

9.98

8.18

7-79

00*0

I I I 8,5

63.28

52-83

42.58 32.02 76 70 76.70 78.23 52-55 4'’■93 50.92

39-01

80.00

62.14

48.83

3,°°3 3,786 3,758 3,435 2,478 2,738 3>!32 3,I4° 3'3°3 1.964

TABLE I.—Continued.

NO.

FORM.

LENGTH,

DIA. IN INCHES AT

SECT. AREA IN

SQ. INS. AT

FEET.

Top. Mid. Base. Top. Base.

ULT. RESIST., LBS. PER SQ. INCH.

ENDS.

White oak ; used in mill 6% years.

36

Round

37

38

39

40

41

42

12.08 12.08 12.u 12.05 11.72 12.01 12.07

O.Ö4

6.85

Oven

4,604

6,029

4,

029

680

2,945

3,45i

4,225

3,264

Flat.

White oak ;

43

44

Cylindrical

45

46

47

44-

49

5°

51

Round .... Cylindrical

52

53

Round

1387

14.00

13.89

13.80

13.92

13.89

13-67

1385

13-65

13.90 ii.51

10.56

10.54

10-54

10.50

10.20

10.80

9-25

9-55

9.40

9-3;

5-9*

74    I    V

10    cn

C    I    O

Tj-foo m r^oo vo mvo « m ’I-nhvo rovo moo O 4- ■4- 4- m öv ci 4-oó vo' mvo

00000000 CO VO VO VO VO N

68.04

38.66

4,6o2

4-951

4.266 3,881 4,⁶74 4-1838 3*434 4,618 3,9Sl

3.266

6,147

<<

Flat, pintle. Flat.

Pintle ends. Flat.

White oak ; thoroughly seasoned; 1 year old.

54

Cylindrical..

56

7-74

10.95

10.91

-    45.04

-    91.16

-    95-48

3,219

1.865

4,450

Flat.

One flat : one round. Flat.

Although the ends of Nos. 39 and 44-51 were flat, they were not parallel.

All the columns indicated by “ Round ” were tapered, and they almost invariably gave way by the crushing of the fibres at the small end. In all such columns the ultimate resistance is per square inch of the small end.

Some of the square columns had their corners slightly beveled.

In his report to the Boston Manufacturers’ Mutual Fire Ins. Co., Prof. Lanza says: “The immediate location of the fracture was generally determined by knots;” .    .    . but states

~V

that, whether knotty or straight grained, failure took place in the tapered columns at the small ends. Tapering a column, therefore, to the extent shown in these cases, is a source of weakness.

TABLE II.

Yellow Pme.

NO.

LENGTH,

FORM OF SECTION.

SECTION DIMENSIONS,

ULTIMATE RESIST-

INCHES.

INCHES.

ANCE PER SO. IN.

Lbs.

20.4

119-95

119.90

20.0

16.0

Circular.

Square.

10.2 Diam. x II x II

ii

ii

10.4 x

8.0

O CO Tf

•4

3-0

6.0

6.0

10

ii

3-0

12

3-0

13

3-0

14

15

16

17

18

14.0

17.2

19.1

180.0

180.0

180.0

Rectangular.

1.5 x 1.5

3x3 3x3 1.5 x 1.5 1.5 X 1.5

1.5    X T.5

4.6    X 46 4.6 X 46 5-3 x 53

16    x 13. 16.2 x 7.

17    x 8.

vn co O O

W    00 O' 00 00 00 00 U\ CO^J O' O' O'

Straight grained and seasoned 20 years.

Nos. 13, 14 and 15 were pme of very slow growth. Nos. 16, 17 and 18 were very green and wet.

Tables II. and III. contain the results of Col. Laidley’s tests, some of which belong to short blocks. These tests were made during 1881, and a detailed account of them is given in “Ex. Doc. No. 12, 47th Congress, ist Session.”

These experiments give some very important deductions.

In the first place, within the limits of the ratio of length to diameter, or shortest side of rectangular section, appearing in these tests, the ultimate resistance is essentially independent

TABLE III.

Spruce, thoroughly seasoned.

NO.

LENGTH,

INCHES.

FORM OF SECTION.

SECTION DIMENSIONS, ULTIMATE RESIST-

INCHES.

ANCE PER SO. IN.

Lbs.

24

Rectangular.

24

36

36

DO

60

120

120

60

io

XI

30

12

15

121.2

Circular.

12

un in invo o m \n xn m m §

xxxxxxxxxxxß

754

of the length. This is the result of the action of causes noticed in the consideration of wrought-iron columns composed of £ ’s. The ultimate resistance of any such column, therefore, is to be obtained by multiplying the area of its cross section by the ultimate resistance, per square inch, of short blocks.

In Prof. Lanza’s experiments, the greatest ratio of length to radius of gyration was about 86. Below this value the general conclusion just given may be expected to hold, but probably not much above it.

In Cob Laidley’s tests the greatest value of the same ratio was about 90 (No. 17 of Table II.), at which there seemed to be a little decrease in ultimate resistance.

Again, it is to be observed that Prof. Lanza’s results are much less than those of Col. Laidley for the same timber. The columns of the former were of ordinary merchant material, with the usual accompaniment of knots, weak spots, crooked grain, etc., while the latter experimented with fine, straightgrained timber.

The slow growth specimens (13, 14 and 15, of Table IL), gave much the highest results, while the wet and unseasoned ones (16, 17 and 18) gave the lowest of all.

Hence, the ultimate resistance of timber columns will depend upon quality and condition of material, mode of growth, degree of seasoning, etc., etc.

Table II. also shows, what has been observed elsewhere, that smaller specimens give higher results than larger ones.

Formula of C. Shaler Smith, C. E.

During the winter of 1861-62, Mr. C. Shaler Smith conducted a series of over 1,200 tests of full size yellow pine square and rectangular columns for the Ordnance Dept, of the Confederate Government. The results of these tests have never been published, but Mr. Smith has kindly furnished the writer with the following summary :

The tests were grouped as follows :

“ ist. Green, half-seasoned sticks answering to the specification, ‘ good merchantable lumber.’

“ 2d. Selected sticks reasonably straight, and air-seasoned under cover for two years and over.

“ 3d. Average sticks cut from lumber which had been in open air service for four years and over.”

and

If l — length of column in inches ; d — least side of column section in inches ; p = Ult. Comp, resistance in lbs. per sq. in. ;

then the formulae found for these three groups were :

For No. i :

5 >400

. I l3 ¹ ' 250 d²

8,200 ■ I / 300 d²

For No. 2 :

1² °

For No. 3 :

5,000

l² '

250 d²

But in order to provide against ordinary deterioration while in use, as well as the devices of unscrupulous builders, Mr. Smith recommends the formula for group No. 3 as the proper one for general application. He also recommends that the

until 25 diameters are reached, '

factor of safety shall be

and five thenceforward up to 60 diameters. This last limit he regards as the extreme for good practice.

Mr. Trautwine computed his tables from tests of group

No. 3.

Addendum to Article 57.

Tables IV. and V. have been formed by digesting the results of tests of timber columns made at Watertown, Mass., and found in “ Ex. Doc. No. 1, 47th Congress, 2d Session.”

Each result in both tables is usually a mean of from two to four tests, although a few belong to one test only. All timber, both of yellow and white pine, was ordinary merchantable material, with about the usual defects, knots, etc., and failure frequently took place at the latter ; it was all well seasoned, and all columns were tested with flat ends.

Flat end yellow pine columns were observed to begin to fail with deflection at a length of about 22 d, d being the width or least dimension of the normal cross section. All columns were of rectangular section, and l in the following table is the length.

TABLE IV.

Yellow Pine Columns with Flat Ends.

Length.

Size of Stick. Inches.

Ult. Comp. Resistance.

Pounds per So. In.

Length.

Size of

Stick.

Inches.

Ult. Comp. Resistance.

Pounds per Sq. In.

Ft. Ins.

15    O

IO o

16    8 15 o

12 6

15 o

20 o

15 O 15 O 15 O

in m r^vo in O O    m

cs

O m O' in m ci O' w in in

HH    M    M    H H

xxxxxxxxxx

in

oi in    in oo r-^o m

cd m    in in L. in in in

21.7

22

26.7

27.O

27-3

30.8

3I.2

31-9

32. i 32.8

loco rt* co o o o

rfr COCO Q'N -t O' O inO

•t co m o m T h o o' COTfrfcÔuTCOCOCOCOCO

Ft. Ins,

15 O 23 4 17 6 15 o

26    8

15 o

20 o 22 6 25 O

27    6

5.ox 12 7.7 x 9.7

5.5    X 5.5

4.5    x II.6 7.4 x 9.4 4.0 x II.4 5-4 x 5-4 5-5 x 5-5 5-5 x 5.5 5-3 x 5-3

35-6

36-4

38.2

41

43

44

44-3

50

55

62.3

M W M 05 tO    W LO U)

co O co O co 4-^ to Lo ui O' O' O'O O 4-* O O' vO C'en sjmoj 10 to •£> 4^

Table IV., therefore, includes no short column, i. <?., one which failed by compression alone with no deflection.

About sixteen of the latter were tested with the following results :

Short yellow pine columns ; / -4- d below 22.

maximum = 5,677 lbs. per sq. in. mean = 4,442 “ “ “ “ minimum = 3,430 “ “ “ “

Each of the preceding tests was made on a single rectangular stick. A number of tests, however, were made on compound columns formed by bolting together from two to three rectangular sticks, with bolts and packing or separating blocks at the two ends and at the centre. The bolts were parallel to the smaller sectional dimensions of the component sticks. As was to be expected, those compound columns possessed essentially the same ultimate resistance per square inch as each component stick considered as a column by itself, as the following results show. I is the length of the column and d the smallest dimension or width of one member of the composite column. All had flat ends.

Table V. gives the results for white pine columns, and corresponds with Table IV. in that it shows only the failures with deflection, which was observed to begin with those columns at a length of 32 d. I and d possess the same signification as in Table IV., the column l -f d, showing the ratios between the lengths and least widths.

Thirty columns with lengths less than 32 d were tested to destruction. These sticks failed generally at knots by direct compression and without deflection. The results of these thirty tests were as follows :

maximum

mean

minimum

Short white pine columns ; l -s- d below 32.

3.700 lbs. per sq. in. 2.414 “ “ “ “ 1,687 “ “ “ “

All the preceding white pine columns were single sticks, but a large number of built posts composed of two to four white pine sticks bolted together, with spacing blocks at the two ends and at the centre, were also tested with the results shown below. I — d is the ratio of length over least width of a single stick of the set forming the composite column.

I + d.

486$

TIMBER COLUMNS.

[Art. 5 7.

I +d.

Number of Tests.

32.I

I8 .

maximum mean

minimum = 2,756 “

4,559 lbs. per sq. 3,841 “    “    “

.3

3b-

I8

maximum 3,357 “ mean minimum

— 3,122 “ = 2,942    “

Number 0/ Tests.

32.I

15

maximum

mean

minimum

36

maximum

mean

minimum

Il I! II

40

maximum

mean

minimum

2,273 lbs. i, 980 “ 1,661 “

2,255 “ i,999 “ 1,804 “

2,021 “ 1,830 “ 1,419 “

.3= =

A comparison of these results with those given in Table V. shows that these composite or built columns were the same in

TABLE V.

White Pine Columns with Flat Ends.

Length.

Size of Stick. Inches.

Ult. Comp. Resistance.

Pounds per Sq. In.

Length.

Size of Stick. Inches.

Ult. Comp. Resistance.

Pounds pee So. In.

Ft Ins.

15 O 20    3

15 O 15 3 23 4 15 o

5-6 x 15.6

7.4    x 9.3

5.6    x ii.5

5-4 x 5-4

7.7    x 9.6

4.5    x II.6

1,874

32.4    I 2,448

32 7    2,432

2.744

36.4    2,072 1,672

32

33

4O

Ft. Ins.

17 6

26    8

20 O 22 6 25 O

27    6

5.4 x 5.4 7-5 x 9.3 5-3 x 5.3 5.2 x 5.2 5 3X5-3 5-4 x 5.4

40

42.7

45

52

57

62

1,841

2,113

1,455

1,501

952

1,081

strength per square inch with the single sticks of which they were composed, the latter being considered single columns.

All the white pine composite columns were tested with flat

Plate F.

7T

3bód-ius!rj-H

-i-l-j-u 1 }—h 20QQ-PH--

d:l

ft

1000--V-

-WHITE

E STICKS,

-H

-H

t:

j±

ip

ht

th

+H+

±ht±

-r

Ü4

soi 0

lg®#

N:E; STICKS] -FLAT -EN DS A + Ir

1T1~ T1—r~T~~i~ i~

4-

ftffl

Fii.Orf:

H*

+1.01+

1321

48 6d

TIMBER COLUMNS.

[Art. 57.

ends and were built up with the greatest widths of individual sticks adjacent to each other.

The results in Tables IV. and V. are shown graphically in Plate F. One ordinate gives the values of l d, and the other the ultimate resistance in pounds per sq. in.

The full curved lines running into horizontal tangents at the left represent about mean lines through the points indicating the actual column tests.

The broken lines represent the following empirical formulae ; in which p is either the ultimate resistance or working stress in pounds per sq. in.

For yellow pine .    .

“ white “    .    .

00 00

“ 70 (/

-4 7 V +

For wooden railway structures there may be used : For yellow pine .    .

“ white “    .    .

• • P = 750-9 {1+ d)

.    . p = 500 — 6 (/ d)

For temporary structures, such as bridge false works carrying no trafic :

For yellow pine .    .

“ white “    .    .

ha-

— 1500 — 18 (/ -f-

= IOOO — 12 (/ -r-

The preceding formułce are to be used only between the limits

of 20 rand 60 4 for yellow pine and 30 4-and 6o4- for white d    d    d    d

For short columns below 20 ~ and 30^ there are to be used for yellow and white pine respectively :

pine.

Temporary

Structures.

. .1100 lbs. per sq. in. .. 600 “    “ “ “

Railway Bridges.

.......SSO.....

.......3OO.....

Ultimate, p — 4400, p — 24OO

All the preceding values are applicable to good average lumber for the engineering purposes indicated.

Yellow pine White “

CHAPTER VUE

Shearing and Torsion.

Art. 58.—Coefficient of Elasticity.

It has already been shown in some of the Articles of the first portion of this book, on shearing and torsion, that the coefficients of elasticity for those two stresses are the same ; and, indeed, that those two stresses are identical in character. The coefficients of elasticity, given in this Article, are then derived chiefly from experiments in torsion.

In his “ Leçons de Mécanique Pratique,” 1853, Gen. Arthur Morin gives the following résumé of the results of experiments up to that time, in which G is the coefficient of elasticity, for shearing, in pounds per square inch.

G., lbs. 8,571.000 9,523,000

8,571,000

14,300,000

2,857,000

6,237,000

1,523,000

571,400

618,600

MATERIAL.

Soft wrought iron,

Iron bars........

German steel.....

Fine cast steel....

Cast iron........

Copper..........

Bronze ..........

Oak ............

Pine...........

The above value for cast iron must, however, be much too small, as will presently be seen.

In “ Der Civilingenieur,” Heft 2, 1881, the results of some very interesting and important experiments on cast-iron rods or prisms of various cross sections, by Prof. Bauschinger, are

given in full detail. The rods or prisms were about 40 inches long, and were subjected to torsion, while the twisting of two sections about 20 inches apart, in reference to each other, was carefully observed. The results for four different cross sections will be given—i. <?., circular, square, elliptical (the greater axis was twice the less) and rectangular (the greater side was twice the less). In each case the area of cross section was about 7.75 square inches. The angle a is the angle of torsion—i. e., the angle twisted or turned through by a longitudinal fibre, whose length is unity, and which is at unit’s distance from the axis of the bar.

SECTION.

a.

G.

Circular

Elliptical

Square..... Rectangular.

The formula by which G is computed, when the torsional moment and angle a are given, is the following :

(0

G = — . c-r

0/    A⁴

in which M is the twisting moment ; A the area of the cross section ; 1^ the polar moment of inertia of that cross section ; and c a coefficient which has the following values :

Ą71² — 39.48 for circle and ellipse ; 42.70 “ square ;

42.00 “ rectangle ;

as was shown in Art. 10.

B-auschinger’s experiments show that the coefficient of shearing elasticity for cast iron may be taken from 6,000,000 to 7,000,000 pounds per square inch ; also, that it varies for different ratios between stress and strain.

It has been shown in Art. 4, that if E is the coefficient of elasticity for direct stress, and r the ratio between direct and lateral strains, for tension and compression, that G may have the following value :

(2)

2 (1 + r)

Prof. Bauschinger, in the experiments just mentioned, measured the direct strain for a length of about 4.00 inches, and the accompanying lateral strain along the greater axis of the elliptical and rectangular cross sections, and thus determined the ratio r between the direct and lateral strains per unit, in each direction. The following were the results :

Compression.

SECTION.

Circular.

Elliptical........0.23

Square.....

Rectangular

r.

0.22

0.24

0.24

G.

6,541,000 lbs. per sq. in. 6,541,000 “    “

6,442,000 “

6,499,000 “

Tension.

Circular.. .. Elliptical...,

Square.....

Rectangular

0.23

0.21

0.26

0.22

i i il

The values of E are not reproduced, but they can be calculated indirectly from Eq. (2) if desired.

It is seen that the values of G, as determined by the different methods, agree in a very satisfactory manner, and thus furnish experimental confirmation of the fundamental equations of the mathematical theory of elasticity in solid bodies.

The fact that Cis essentially the same for all sections is also strongly confirmatory of the theory of torsion, in particular.

These experiments show that, for cast iron, the lateral strains are a little less than one quarter of the direct strains. If rwere one quarter, then G = -§ E ; or E = § G.

In the “Journal of the Franklin Institute,” for 1873, Prof. Thurston gives the following values of G, as determined from experiments with his torsion machine.

White Pine.......

Yellow Pine, sap.. Yellow Pine, heart

Spruce...........

Ash.............

Black Walnut.....

Red Cedar.......

G =

G =

G =

G =    2x1,000

G =    410,000

G —    582,000

G =    890,000

Spanish Mahogany...... G —    660,000

G =    570,000

G —    910,000

G —    1,225,000

G —    355,oco

220,000 pounds per sq. in.

495,000    ........

495,000    “    “ “ “

Oak

Hickory . Locust... Chestnut.

The specimens were small ones, and the timber was seasoned.

Art. 59.—Ultimate Resistance.

Before considering the ultimate shearing resistance of special materials it will be well to notice the two different methods in which a piece may be ruptured by shearing.

If the dimensions of the piece in which the shearing force or stress acts are very small, i.e., if the piece is very thin, the case is said to be that of “ simultaneous ” shearing. If the piece is thick, so that those portions near the jaws of the shear begin to be separated before those at some distance from it, the case is said to be that of “ shearing in detail.” In the latter case failure extends gradually, and in the former takes place simultaneously over the surface of separation. Other things being the same, the latter case (shearing in detail), will give the least ultimate shearing resistance per unit of the whole surface.

In reality no plate used by the engineer is so thin that the shearing is absolutely simultaneous, though in many cases it may be essentially so.

Wrought Iron.

The following averages (each result being an average of six tests), are from Chief Engineer Shock’s experiments, in 1868, on ordinary commercial “ rounds ” (“ Steam Boilers,” by William H. Shock, Chief Engineer, U. S. N.), in which S' is the ultimate shearing resistance in pounds per square inch :

AM. OF ROUND.

o. 5 inch. 0.625 inch. 0.75 inch. 0.875 inch. I.00 inch.

SINGLE SHEAR.

44,150 lbs. 39,250 lbs 39.550 lbs 41,500 lbs 40,700 lbs

DOUBLE SHEAR.

41,090 lbs. 38,670 lbs. 39,770 lbs. 37,890 lbs. 37,650 lbs.

Although these figures show some irregularities, the general result is unmistakable, and shows a decrease of 5 with an increase of diameter.

The results of experiments at Bristol, England, by Mr. Jones (“Proc. Inst. Mech. Engrs.,” 1858), on punching plate iron, are as follows :

THICKNESS OF PLATE.

0.437 inch.... 0.625 “ • • • 0.625 “ ... 0.875 “ ... I.000 “ ...

DIAM. OF HOLE.

0.250.

0.500

0.750,

0.875

X .000,

s.

(t

Mr. C. Little found the following for English “ hammered scrap bars and rolled iron,” with parallel cutters or shears :

AREA CUT.

o.50 x 3.00 ins. 0.50 x 3.00 ins 1.00 x 3.00 ins. 1.00 x 3.00 ins 1.00 x 3.02 ins 1.00 x 3.02 ins. 1.80 x 5.00 ins,

DIRECTION.

Flat

Edge

Flat

Edge

Flat

Edge

Edge

.S'.

49,950 lbs. per sq. 51,750 “

51,750 ‘

50,850 ‘

44,350 ‘

46,150 ‘

46,150 ‘

in.

In these experiments the edges of the shears were always parallel to each other, thus tending to produce simultaneous shearing. In ordinary workshop practice, however, the jaws of the shears make a constant angle with each other, thus shearing successive portions of the material as the jaws approach, whatever may be the dimensions of the piece, and consequently always producing shearing in detail. In the experiments (by the same authority, i. e., Mr. C. Little, “ Proc. Inst. Mech. Engrs.,” 1858) from which the following results were deduced, the angle between the jaws of the shears was an inclination of i in 8 :

BARS.

x i.5 ins 4-5 x 1-375 ins. 3.0 x i.00 ins 5.25 x 1.75 ins 6.00 x i.50 ins

FLATWAYS.

S — 40,800 S = 32,000 S = 35,200 5 = 37,400 .S' = 33,600

EDGEWAYS.

These

As was to be expected, the “ Edgeways ” results are much the largest, as with that position of the bar the shearing approached more nearly the simultaneous condition, results show that it is much more economical to shear a bar flatways than edgeways.

Mr. Edwin Clark (“On the Tubular Bridges”) found the resistance of ^-inch rivet iron, in single and double shear, to vary from 49,500 to 54,100 pounds per square inch. The tensile resistance of the same iron was about 53,800 pounds per square inch.

Reviewing all these results, the ultimate shearing resistance of wrought iron may safely be taken at 0.8 of its tensile resistance, as stated by Mr. D. K. Clark.

Cast Iron.

Very few experiments on the resistance of cast iron to shearing have been made, as this metal is seldom or never used to resist such a stress.

Mr. Bindon B. Stoney (“Theory of Strains in Girders and Similar Structures,” p. 357 of 2d Edit.) has found, by experiment, that the ultimate shearing resistance of the cast iron with which he experimented varied from about 17,900 to 20,200 pounds per square inch. He concluded that the shearing and tensile resistances might be taken the same.

Steel.

The results of Prof. Ricketts’ shearing tests on both open hearth and Bessemer steel rounds with different grades of carbon are given in Table II. of Art. 34. The elastic limit is the point at which the metal first fails to sustain the scale beam. The double shear resistance in one case exceeds the single by over six per cent. According to these tests, the ultimate shearing resistance of mild steel may be taken at three-quarters of its ultimate tensile resistance. Each shear result is a mean of three tests.

Mr. Kirkaldy investigated the ultimate shearing resistance of four grades of Fagersta steel, and the following results are taken from “ Experimental Enquiry into the Me-

chanical Properties of Fagersta Steel,” by David Kirkaldy, 1873. The test-piece, in each case, was turned from a 2-inch square bar, to a diameter of 1.128 inches, and each result is a mean of three experiments. S is the ultimate resistance to shearing, in pounds per square inch ; r is the ratio of ultimate shearing over ultimate tensile resistance of the same steel ; while “d” is the detrusion or relative movement of one part of the specimen in respect to the other at the instant of separation over the entire surface.

MARK.

r.

d.

1.2

0.9

0.6

0.3

, 61,400.00 lbs.. . . . 79,740.00 “ ... . 71,650.00 “

. 45,410.00 “ . . .

... 0.19 inch. ... 0.25 “

... 0.28 “

... 0.32 “

As is evident, the lower “Mark” numbers belong to the softer steels.

In each case two surfaces were sheared, as the “round” was a pin for three links, two of which pulled one way, and one the other.

All of Mr. Kirkaldy’s experiments seem to show that the ultimate shearing resistance of steel is about three-quarters the tensile.

Table I. contains the results of the experiments of Prof. A. B. W. Kennedy, as given in “ Engineering ” for May 6, 1881. The tensile resistance of the same steel was given in the chapter on “ Tension.”

The specimens were round and of mild rivet steel. The ratio of the ultimate resistance to shearing over that to tension varied from 0.80 to 0.89.    /

In the “ Journal of the Franklin Institute,” for March, 1881, Charles B. Dudley, Ph.D., gives the results of 192 tests of rail steel, the specimens, 0.625 inch round, having been taken from rails which had been subjected to service for considerable periods of time on the Penn. R. R. The tests were made by

TABLE I.

Rivet Steel.

DIAMETER IN INCHES.

MEAN.

ULTIMATE RESIST. IN

LBS. PER SQ. IN.

RATIO OF ULT. SHEAR

OVER ULT. TENSION.

i .00 i. 00 1.00 i .00 1.00 i .00 0.62 0.62 0.62 0.62 0.62 062 0.62 0.62 0.62 0.62

54,110' 54,930 55,240 52,830 56,660 53,530, 60,260' 59,400 59,600 > 59,220 59,740, 53,670' 51,290 ■ 52,670 53,000 51,620,

54,550

59,640

52,450

0.89

0.87

0.80

Mr. J. W. Cloud, engineer of tests for the Penn. R. R. Co. The following is a summary of the results :

63,560.00 pounds per sq. in. {greatest'). 59,880.00 pounds per sq. in. {mean).

53,380.00 pounds per sq. in. {least).

The percentages of carbon and ultimate tensile resistances are given in Table IV. of Art. 34. By reference to that table it will be observed that 5 is not far from three-fourths the tensile resistance.

Copper.

From some English experiments, Mr. Bindon B. Stoney concluded that the ultimate shearing resistance of copper was about two-thirds of that of wrought iron.

Timber.

In treating the shearing resistance of timber, it is very necessary to consider whether the shearing takes place along the fibres, or in a direction normal to them.

TABLE I.

Along Fibres.

s.

KIND OF WOOD.

GREATEST.

MEAN.

LEAST.

Georgia Pine,

934

843

713

White Pine

530

482

433

Locust

1,389

1,165

970

White Oak.

L474

1,250

1,076

Spruce

647

542

4^3

Hemlock

410

369

322

TABLE II.

Across Fibres.

KIND OF WOOD.

5-.

KIND OF WOOD.

Ash..................

Beech ................

Birch.................

Cedar, white..........

Cedar, Central Amer.. . .

Cherry...............

Chestnut..............

Dogwood .............

Ebony*................

Gum.................

Hemlock..............

Hickory..............

6,280

5,223

5,595

1,372 to 1,519 3,4io 2,945 i,535 6,510

7.750

5,890

2.750

6,045 to 7,285

Locust................

Maple...............

Oak, white............

Oak, live.............

Pine, white............

Pine, yellow, northern.. Pine, yellow, southern.. Pine, yel., very resinous.

Poplar................

Spruce...............

Walnut, black.........

Walnut, common......

to Co 4- on en -p>* JO JO-J*

'b    4- 4- CO w

tn W GO CO to cn Co en o O O on cn On

M N) 4-to en h O coon 00

CO

Table I. contains the results of experiments on the shearing of small specimens along the fibres, by the late Mr. R. G. Hatfield (“Transverse Strains,” 1877). S' is the ultimate shearing resistance in pounds per square inch. There were about nine experiments for each kind of timber.

Table II. contains the results of experiments by Mr. John C. Trautwine on round specimens 0.625 inch in diameter, and across the fibres (“Journal of the Franklin Institute,” Feb.

TABLE III.

Along Fibres.

-

::

1    Oregon pine..................

2    Oregon pine.................

3    Orégon maple ..........

4    Oregon maple ..............

5    California laurel.............

6    Ava Mexicana..............

Ava Mexicana ...........

Oregon ash .................

9 Oregon ash .............

xo Mexican white mahogany.... u Mexican cedar ............

12    Mexican cedar............

13    Mexican mahogany .......

14    Mexican mahogany.........

15    Oregon spruce ..............

16    Oregon spruce..............

17    White pine..................

18    White pine ..................

19    Whitewood.................

20    Whitewood ...............

21    Whitewood ..... ..........

22    Yellow pine. ......... ......

23    Yellow pine.................

24    Ash.......................

25    Ash........................

26    Red oak......................

27    Red oak..    ..................

28    White oak..................

29    White oak ..................

30    Yellow birch...............

31    Yellow birch............

32    White maple...... ..........

33    White maple.................

34    Spruce......................

35    Spruce......................

SHEARING AREA ULT. SHEAR IN

IN SQUARE

INCHES.

5. o and 10.7 14.4

it .0 a

0.9

nd

14.8

11.0 14.6 8.2 ii.o and 15.0 9.8

15.0

ii . i 22.9 and 5 5 3.0 and 16.0 “

7 3“

10.8    “

21.6    “ 13.0 “ 17.0 16.3 “ 16.2 “ 16.0 “ 16.0 “ 16.2 “

15.8    “ 17,0 “

25.6    “ 16.0

15.9    and

is-8 “

16.0 “

POUNDS PER SQ.

DIRECTIONS TO RINGS

INCH.

OF GROWTH.

14-0

14-2

I5-¹

£££ Ä.5£.SJä<2SJ2«5ü:ißBaj2S> t

o 00o ono o b bo bii Mwi boo ót

442 and 1,096 820

436

1,028

549 and 1,204

346

700

443

1,126

438 and 1,000

M    M Tt- fvCO O tv VC <N 00 rovo MN nN    Cv 10 VO

vo oocNoicNfNi-cooLOTfcio m«o h N    fOfT)H fOfOMM mtN tv CO tv lOVO    ro 0* CO

423

814

566

M33

and 356 ami⁵ 423

352

37°

“    481

647

385

399

4°9

700

745

91 9

846

815

672

537

347

374

Perpendicular (2 exps.) Oblique. Perpendicular. Oblique.

Oblique (2 exps.)

Perpendicular.

Parallel.

Parallel.

Perpendicular.

Oblique (2 exps.)

Perpendicular.

Parallel.

Parallel.

Perpendicular.

Parallel (2 exps.) Perpendicular. Perpendicular (2 exps.) Parallel (2 exps 1 Oblique (2 exps.) Parallel (2 exps.) Perpendicular (2 exps.) Oblique (2 exps.) Perpendicular (2 exps.) Parallel (2 exps.) Perpendicular (2 exps.) Perpendicular (2 exps.) Parallel (2 exps.) Parallel (2 exps ) Perpendicular (2 exps.) Oblique (2 exps.) Perpendicular (2 exps.) Oblique.

Perpendicular (2 exps.) Parallel (2 exps.) Perpendicular (2 exps.)

1880). As before, A is the ultimate shearing resistance in pounds per square inch.

Table III. has been condensed from the results of Col. Laidley’s tests at the Watertown Arsenal (Ex. Doc. No. 12, 47th Congress, ist Session). Usually, two such results have been selected as will give a correct idea of the resistance. In all cases except Nos. 19, 20, 23 and 33, the smaller resistance belongs to the larger shearing surface. In No. 33 the smaller resistance belongs to an unsatisfactory experiment.

Art. 60.—Torsion.

Coefficients of Elasticity.

The coefficients of elasticity for torsion or shearing have been given in Art. 58, and need not be repeated here.

Ultimate Resistance and Elastic Limit.

Wrought Iron.

In 1866 Mr. Kirkaldy tested four hammered Swedish iron bars turned to a diameter of 1.5 inches for a length of seven diameters. The average ultimate moment of torsion was produced by a weight of 2,677 pounds with a leverage of 12 inches; hence, in Eq. (83) of Art. 10 ; M — 2,677 X 12 = 32,124. Putting 2rQ — d — 1.5 inches in that equation, there will result :

Tm — 5.1 -y- = 48,540 pounds per square inch. d³

This is the greatest intensity of torsional shear in the sec

tion.

If Tm be taken at 48,000 the diameter of a wrought-iron shaft required to resist an ultimate moment M, will be :

• •    (I)

If the working moment be taken at one-eighth the ultimate, then the diameter required will be :

d = 0.047 V⁷⁸    — 0.094    ...    (2)

in which is the working moment.

If H is the number of horse powers per minute to be transmitted by the shafting, and n the number of revolutions which it is to make :

Mx =

12 x 33,000

2it

(3)

Putting this value in Eq. (2) :

d = 3742

JH

(4)

This value of d will be much too small in the case of long shafting required in the distribution of power, in consequence of the bending caused by the belting.

The mean torsional moment at the elastic limit, in Mr. Kirkaldy’s four experiments, was about 0.4 the ultimate.

In 1846 Major Wade (“ Experiments on Metals for Cannon ”) tested three wrought-iron circular cylinders about 1.9 inches in diameter and 15 inches long, with the following results :

28,325 lbs. per sq. in.

27,525 “

27,800 “ 83,650

Mean = 27,900 (nearly). If the mean be taken at 28,000 :

•    • (5)

d = 0.056 $~M    .    .

It is seen that Major Wade found Tm much less than Kirk-aldy’s value for Swedish iron, and d in Eq. (5) is correspond-ingly greater. If H and n carry the same signification as before, and if 8 is the safety factor :

d = 4.49

*1^1

. . (6)

In all these results, the moments are supposed to be given in i?ich-pounds, and the resulting values of d are consequently in inches.

Cast Iron.

Major Wade also made tests on circular cylinders of cast iron about 1.9 inches in diameter and 15 inches long.

If d is the diameter = 2r₀ in Eq. (83) of Art. 10, he found the following results with the grades of iron shown :

2d fusion.......

3d fusion.......

2d and 3d fusion

2d fusion.......

3d fusion.......

2d fusion.......

3d fusion.......

= 45,000 (nearly).

Hence the diameter in inches, for the ultimate moment M in inch-pounds is:

5-1

45,000

M — 0.048 a³

'M

• • (7)

These values of Tm are very high, because the iron with which Major Wade experimented was evidently of a special character and extraordinarily strong.

The same experimenter tested some square sections, for which, by Eq. (73) of Art. 10 :

. . (8)

5 — ; (b = side of square)    .    .

The following are from Major Wade’s results :

b — I 00    inches ;    M —    8,750 inch-pounds    ;    Tm    — 43,750 pounds.

b — 1.42    inches ;    M —    23,000 inch-pounds    ;    Tm    — 40,210 pounds.

b = 1.75    inches ;    M —    54,000 inch-pounds    ;    Tm    = 50,370 pounds.

The mean of these results is : T — 44,800 (nearly). Hence for the ultimate moment in inch-pounds :

• • (9)

b = ißSio = °-⁰⁴⁸¹ i/M

It is to be observed that, according to these experiments, Tm is the same for circular and square sections ; a result very different from that of Prof. Bauschinger’s experiments, as will presently be seen.

Four of Major Wade’s experiments on hollow circular cylinders are next to be given.

Since Txr = o, in Eq. (78) of Art. 10, the resisting moment of such a cylinder, if d is the external and dt the internal diameter, will be :

Tmd* - Tmd? __ Tm

5-1

(ii)

5.1 dM

d* — d*

For the first case :

d = 3.25 ins. ; dt = 2.61 ins.; M = 95,000 in.-lbs. Substituting in Eq. (11) :

Tm = 24,170 lbs. per sq. in. (nearly).

For the second case :

‘d — 2.21 ins. ; dt = 1.54 ins. ; M — 49,500 in.-lbs.

Substituting in Eq. (11) :

Tm = 30,610 lbs. per sq. in. (nearly).

For the third case :

d = 1.81 ins. ; dx = 0.91 in. ; AT — 37,250 in.-lbs.

Substituting in Eq. (11):

Tm = 34,220 lbs. per sq. in. (nearly).

For the fourth case :

d — 1.30 ins. ; dt = 0.65 in. ; M — 13,000 in.-lbs.

Substituting in Eq. (11) :

Tm = 32,180 lbs. per sq. in. (nearly).

These results indicate that Tm decreases as the thickness of the wall of the hollow cylinder decreases and as the exterior diameter increases.

Professor Bauschinger (Der Civilingenieur, 1881, heft 2) tested cylinders about 40 inches long, and with the following cross sections and approximate dimensions :

Circle.... Ellipse...

Square... Rectangle Rectangle

Diameter =    3-²5    inches.

2.30 inches. 4.40 inches. _    3.00    inches.

— ( 3-00 inches.

_ (2.04 inches.

~ (4.10 inches, i .02 inches. 4.10 inches.

Diameters =

Sides

Sides

Sides

The ultimate twisting moments substituted in Eqs. (£3), (41), (73), (75), and (77) of Art. 10, give :

Tm

For Circle.

For Ellipse For Square For Rectangles (sides 2 to i). . Tm For Rectangles (sides 4 to r).. T,

Tm

Tni

27,730 pounds per square inch. 36,120 pounds per square inch. 37,160 pounds per square inch. 36,370 pounds per square inch. 37,090 pounds per square inch.

These experiments give Tm considerably less value for the circular cross section than for the others.

The U. S. Board, however, found the following values for four cast-iron cylinders one inch long and 0.565 inch in diameter:

Tm - 35*980 ; 34,no; 34,280, and 33,770 lbs. per sq. in.

Elas. Lim. — 60; 55 ; 64, and 62 per cent, of Tm, respect

ively.

Steel.

In connection with the torsional resistance of steel, tests of circular cylinders only are to be considered. Those to first receive attention were made by Mr. Kirkaldy on English steel, in 1866-1870, and the results have been deduced from his data.

As the sections are all circular, Eq. (83) of Art. 10 is the only one needed:

(12)

_ 5.1 M

In this equation Tm is the greatest intensity of torsional shear, in any section, in pounds per square inch; ild” the diameter of the shaft or cylinder in inches ; and M the twisting moment in inch-pounds.

In all the following experiments the lever arm of the twisting couple was 12 inches; hence, if P is the twisting force, M = 12P, and Eq. (12) becomes

(13)

61.2P

The mean of four experiments with Bessemer steel gave for the ultimate resistance

P = 2,307 lbs., with d — 1.25 inches ;

/. Tm ~ 72,298 lbs. per sq. in.....(14)

The length was 10 inches.

The mean of some results with Krupp’s cast steel in specimens 1.25 inches in diameter, and 2.5 inches for torsion length, gave:

The following set of results were obtained from 2-inch square bars turned down to 1.382 inches in diameter for a length of 11 inches, and gives the means of the number of tests indicated.

SPECIMENS.

ELASTIC

STRAIN.

5 Hammered tires, 5    “ axles,

rails,

4 Rolled tires, axles and rails.

Bessemer

steel.

... 3,450 lbs ••• 3,293 “

... 3,248 “

5 Hammered tires, axles, rail,

i Rolled rail.

Crucible

steel.

... 3.226 ... 3.562 ... 3,786 ••• 4,054

.. • 3,002

Tm (ultimate).

... 80,006 lbs..

... 76,365 “ ..

... 75,321 “ ..

... 74,811 “ ..

... 82,603 “ • •

... 87,797 “ .... 0.013 ... 94,012 “ .... 0.016 .. 69,616 “ .... 0.012

. 0.014 . 0.011 . 0.012

. 0.008

. 0.014

.    .    (I6)

The elastic strain is the fraction of a complete turn made by the specimen at the elastic limit.

The mean of the Bessemer steels in (14) and (16) give:

Mean Tm = 75,760 lbs. per sq. in.

Hence, if M is the breaking moment of the twisting couple in inch-pounds, the following will be the diameter of the shaft in inches :

<¥

3/ 5-liïf ₌

/ 75,760

Or, if 11 is the safety factor, and Ml the greatest working moment :

(18)

d - 0.040

The mean of the crucible steel results in (16), with the exception of the last, is :

Mean T„t = 88,140 lbs.

Hence the diameter (in inches) of the shaft which will just sustain the breaking moment M, in inch-pounds, is :

d = <]

5-iM

88,140

= 0.03....    (19)

Or, if 11 is the safety factor, and Mt the greatest working moment :

(20)

d — 0.03 Sy^nJVp

In all the preceding experiments the elastic limit varied from 40 to 47 per cent, of Tm (ultimate) as given in (14), (15) and (16).

In 1873 Mr. Kirkaldy made some experiments on specimens of Fagersta steel which possessed a length of about 9 inches and a diameter of 1.128 inches, the length of the twisting lever being still 12 inches. Eq. (13) then gives the following results, each being a mean of three tests:

MARK.

1.2

O.9

0.6

0.3

P (ultimate).

. 2,120 lbs. . 2,336 “ . . 2,261 “ . . 1,520 “ .

Tm (ultimate).

STRAIN.

90,397 lbs 99,607 “ 96,409 “ 64,813 “

O.29

O.79

1.02

3-22

The “ strain ” is the number of complete turns made by the specimen at the place and instant of rupture.

The specimens with the higher “mark” numbers were the higher steels.

The elastic limit varied from 46 to 58 per cent, of the ultimate Tm.

The diameter of a shaft for any of these grades may readily be computed by the use of these values of Tm in equations similar to Eqs. (17) to (20).

The following values were determined by the Committee on Chemical Research of the U. S. Board, “ Ex. Doc. 23, House of Rep., 46th Congress, 2d Session/’ with specimens 1 inch long turned to diameters of 0.625 and 0.565 inch, and tested in a Thurston machine:

Tm-

100,990 lbs.    per    sq. in,

ELAS. LIMIT IN PER CENT.

0F Tm.

ULT. ANGLE OF TORSION,

34

34

33

42

34

45

39

35

* 40

44

39-

36

29

32

32

149 -o

I42°.3 68°.4 56°. i 278°.2 220°.8

99°-5 165°.o 84° -9 180'.7 53⁰ to 113° ' 48° to 84° 6i° to 1430 42⁰ to 123° 73⁰ to 141⁰ ,

Tool steels :

annealed.

Each of the last five results is a mean of eight tests.

The first portion of these results would possess more value if the test specimens had been larger.

With these values of 7W, the diameter of a shaft, with the torsion moment M in inch-pounds, becomes:

Tm

1.721 A³

'M

Copper, Tin, Zinc, and their Alloys.

The following values of Tm have been computed by the aid of Eq. (12) from data determined by Prof. R. H. Thurston, and given by him in the works already cited in connection with tension and compression. The test specimens were 0.625 inch in diameter, with a torsion length of 1.00 inch, and were tested in his torsion machine. The ultimate shearing resistances of these alloys in torsion are thus seen to vary as widely as their tensile resistances.

table I.

COMPOSITION.

ULTIMATE TORSIVE

Cu.

Sn.

SHEAR, Tm.

ELASTIC LIMIT ; PER CENT. OF Tm.

ULTIMATE TORSION

ANGLE.

Pounds.

Degrees.

IOO

OO

100

00

00

IOO

00

IOO

90

10

80

20

70

30

62

38

35,910 28,430

3,196

3,297

43,943

47,671

4,407

35

40

45

33,

41

62

TOO

153-0 52 to 154

557-0 691. o II4-5

16.3

1,770

IOO

1-5

1.0

52

48

686

39

61

29

71

10

90

90

10

90

10

5,881

5,257

5,761

25,027

31,851

IOO

1.0

IOO

IOO

1-7

63

49

57

2-34

131.8

57-2

72.6

Tm is in pounds per square inch.

Table I. relates to alloys of copper and tin, and Table II. to alloys of copper and zinc.

None but specimens with circular sections were tested. With the preceding values of Tm, the following expression for the diameter in inches may be written, if M is given in inch-pounds :

JM

= 1.721

TABLE II.

PERCENTAGE OF

ULTIMATE TORSIVE

Copper.

Zinc.

SHEAR, Tm.

ELASTIC LIMIT ; PER CENT. OF 7^.

ULT. TORSION

ANGLE.

Pounds.

90.56 81.90 71.20 60.94 55-15 49 66

41.30 32-94 20.81

10.30 0.00

9.42

17.99

28.54

38.65

44.44

50.14

58.12

66.23

77-63

8S.88

100.00

35.100

41.575

41,000

48,520

52,320

43A54

4,588

7,241

16,374

22,500

9,186

17.2

27-5

24.0

29.4

32-7

36.0

100.0

100.0

100.0

85.6

38.1

Degrees.

458.0

345-0

269.0

202.0

109.0

38.0

1.8

1.2

0.8

7-1

I4I-5

Timber.

In the July, 1873, number of Van Nostrand’s Magazine, Prof. Thurston gave the results of some experiments on timber test specimens of circular section, 0.875 inch in diameter. Eq. (12) may be written as follows :

(21)

T

M — d* — Cd*

5-1

Prof. Thurston determined the values of C, and the values of Tm = $.iC have been computed from them :

Tm (per sq. in.) 1,530 pounds. 2,142 2,448    “

White pine......

Yellow pine, sap.. Yellow pine, heart

Spruce ..........

Ash.............

Black walnut.....

Red cedar.........

Spanish mahogany .

Oak...............

Hickory..........

Locust............

Chestnut...........

It is presumed that the axis of torsion was parallel to the fibres, which would cause the shear to take place across the latter.

It is interesting to observe that Tm is generally considerably less than the ultimate resistance to simple shear as given in Table II. of Art. 59.

If d is in inches and M in inch-pounds, there may again be written :

7 —- = 1.721

If M is given in foot-pounds, 12M is to be written for M. If M₁ is the greatest working moment, and n the safety factor, nM₁ is to be written for M.

Relation between Ultimate Resistances to Tension and Torsion.

In the “ Trans. Am. Soc. of Civ. Engrs.,” Vol. VII., 1878, Prof. Thurston gave the results of some of his experiments which were made with a view to the determination of this relation. If M is the ultimate torsional moment in foot-pounds of specimens one inch long and 0.625 inch in diameter ; 6 the angle of torsion corresponding to this greatest moment M ; and T the ultimate tensile resistance in pounds per square inch ; he deduced from a large number of steel specimens of wide range in grades the following formula :

Art. 60.]

TENSION AND TORSION.

511

'9OO - ff

No experiments were made in which 6 was greater than

T is thus seen to increase as M increases and as 6 decreases.

CHAPTER IX.

Bending or Flexure.

Art. 61.—Coefficient of Elasticity.

The coefficient of elasticity, as determined by experiments in flexure, can scarcely be considered other than a conventional quantity. If the coefficients of elasticity for pure tension and compression were exactly equal to each other, and if all the hypotheses involved in the common theory of flexure were true, then, indeed, the coefficient of elasticity for flexure .would possess actual existence, and be the same as that for either tension or compression.

These conditions, however, never exist, and the quantities found in this chapter under the name “ coefficient of elasticity” possess value chiefly as empirical factors which enable the deflections in the different cases to be estimated with sufficient accuracy for all ordinary purposes.

The formulae to be used in the determination of the coefficients of elasticity for flexure have already been established, and their use will be shown in succeeding Articles.

Art. 62.—Formulae for Rupture.

As with the formulae for the coefficient of elasticity, so with the formulae for rupture in bending ; they are all deductions from the common theory of flexure, and, strictly speaking, are subject to all the limitations involved in it.

If K and K' are the greatest intensities of stress in the section of rupture and at the instant of rupture ; y the variable normal distance of any fibre from the neutral surface ; yz and y the greatest values of y ; b the variable width of the section (normal tojy) ; and M the resisting moment at the instant of rupture ; then the general formula for rupture by bending, as given by Eq. (1) of Art. 27, is:

■yi    JC

ÿ*b dy H--H y²b dy .    .    .    .    (i)

«    y J-y

This equation is based on the supposition that the coefficients of elasticity for tension and compression are not equal. Although this supposition is strictly true, yet equality is almost invariably assumed ; particularly in the treatment of solid beams. Fortunately, this assumption is not far wrong in those materials which are most valuable to the engineer.

Eq. (i), however, will hereafter be applied to some cast-iron flanged beams.

If the tensile and compressive coefficients of elasticity are

K K'

equal, — = —- . Or, if K is the greatest intensity of stress in

y* y

K K ,    ,

= -j,and Eq.(i)

the section which exists in the fibre at the greatest normal distance, diy from the neutral surface, then becomes :

KI

(2)

This is Eq. (14) of Art. 18, and is the one almost invariably used in engineering practice.

In Eq. (2) I is the moment of inertia of the cross section of the beam about its neutral axis. By introducing the value of I for each particular shape of section, simple working forms of Eq. (2) may easily be obtained. This will be done for two sections in the following Article.

Art. 63.—Solid Rectangular and Circular Beams.

While the rectangular form of cross section almost invariably characterizes timber beams, similar ones of iron, steel and other metals are only occasionally seen. Beams of iron and steel with circular cross sections, however, are quite common as pins in pin connection bridges.

If 2Px represents the moment of the external forces about the neutral axis of any section, Eq. (2) of the preceding Article becomes :

KT

dx

(O

The following are the values of / and d, for rectangular and circular sections, h being the side of the rectangle normal, and b that parallel to the neutral axis, while r is the radius of the circular section, and A the area in each case :

blfi    Ah²

12

12

Rectangular : -

nr*

Ar³

Circular :

dx — r.

If the beams are supported at each end and loaded by a weight W at the centre of the span (or distance between supports), which may be represented by /, then the moment at the centre of the beam becomes:

2Px = M = -

Wl

(2)

There will then result from Eq. (1) : For rectangular sections :

M= —

Wl Kbit² KAh

(3)

For circular sections :

M= -

I! 7

nKr³

(4)

The quantity K is called the modulus of rupture for bending, and if experiments have been made, so that W is known, Eq.

(3) gives :

(5)

3 Wl _ 3 Wl 2 Ah 2 bl?

And Eq. (4) :

K=

Wl Wl

Ar

nr³

. (6)

If the rectangular section is square, bh² = b³ = h³.

Wrought Iron.

If the beam is simply supported at each end and carries a load W at the centre, while E is the coefficient of elasticity and w the deflection at the centre, Eq. (28) of Art. 24 gives :

Wl*

zu =--

48 El

If, in any given experiment, w is measured, E may then be found by the following form of Eq. (7) :

Wl*

E — —— 48 wl

(8)

If the section is rectangular :

(9)

Wl*

4 wbh*

Mr. Edwin Clark tested a one-inch square wrought-iron bar with the following results at the “ elastic limit : ”

l — 12 inches. w = 0.09 inch.

W — 2,636.00 lbs. b = h — I inch.

Eq. (9) then gives :

E = 12,652,809.00 pounds per square inch.

The mean for 2 one and a half inches square bars was as follows :

I — 36 inches. w = 0.305 inch.

W = 2,766.00 lbs. b — h = 1.5 inches.

E — 20,894,600.00 pounds per square inch.

A mean of 4 two inches square bars of Swedish iron, tested by Mr. Kirkaldy, in 1866, gave the following results at the “ elastic limit : ”

W = 6,625.00 lbs. b = h = 2 inches.

/ = 25 inches. w = 0.082 inch.

E = 19,725,000.00 pounds per square inch.

By “ weighting ” these results in proportion to the number of tests of which each is a mean, the mean of all becomes :

E — 19,049,000.00 pounds per square inch.

It is very probable that if w had been measured for smaller loads, E would have been materially increased.

Mr. Kirkaldy tested the same four square Swedish iron bars to rupture. By the aid of Eq. (5), and the data given above, the greatest, mean, and least results were as follows :

w.

K.

FINAL DEFLECTION.

Greatest Mean.. -Least..

15,885 lbs 14,516 lbs 13,338 lbs

74475 lbs. per sq. in 68,044 lbs. per sq. in 62,522 lbs. per sq. in

5.85 ins. 5-35 ins. 4.98 ins.

The ultimate tensile resistance of the same iron was about 45,000 pounds per square inch. These experiments would seem to show that K, for square bars under similar circumstances of span and depth, may be taken about 1.5 times the ultimate resistance to tension.

The results in the following table were computed by the aid of Eq. (6), for some circular beams of “ Burden’s Best ” iron, which were tested at the Rensselaer Polytechnic Institute in November, 1882. As beams cannot be actually broken under such circumstances, the “ ultimate ” value of K was taken with a final deflection of one to one and quarter the diameter.

The “ elastic limit ” is taken at that point beyond which the metal “ flows,” and is indicated by the incapability of the specimen to hold up the scale beam beyond it, under a small increase of stress ; in other words, it is that point at which the specimen “ breaks down.”

These experiments show conclusively that “ultimate" K decreases as the ratio of span over diameter increases, but they

Circular Beams of “ Burden's Best ” Wrought Iron.

K.

KIND.

DIAMETER.

SPAN.

Elastic.

Ultimate.

Elastic.

Ultimate.

Turned .. Turned .. Turned .. Turned .. Rough... Rough... Turned .. Turned .. Rough... Rough... Turned . . Turned . . Turned . . Turned ..

Ins. I.25 1-25 I.25 I.25 I .OO I. OO I .OO I .OO I .OO I. OO

0-75

0-75

0.75

0.75

Ins.

12

12

12

12

8„

12

12

12

Lbs.

3,000

4,400

Lbs.

6,000

10,500

Lbs.

46,950

45,900

54,76o

52,150

55,000

57,000

55,ooo

3,000

4,8oo

1,100

1,900

1,100

1,900

5L950

57,ooo

47,100

53,88o

47,100

58,370

Lbs. 93,900 109,500 93,870 114,700 g¹,700 101,900 91,600 107,000 91,680 97,800 74,050 85,310 74,050 85,310

are not sufficiently extended to establish the limits of application of the observation.

Cast Iron.

All the following results for cast-iron beams are found from Major Wade’s experiments (“ Strength and other Properties of Metals for Cannon,” 1856). His test bars were nearly two inches square in section or two inches in diameter, and were twenty-four inches long. They were loaded at the centre, and the distance between supports was twenty inches. The following table gives results for square bars. K is given in pounds per square inch, and is found by the aid of Eqs. (5) and (6). “ DefC is the final deflection.

Although Major Wade made many other experiments of the same kind, these may be considered representative ones.

Bars with square section, Eq. (5).

HOURS

KIND OF IRON.

IN

W.

K.

DEF.

FUSION.

Lbs.

Lbs.

In.

Richmond iron ; 2d fusion

Stockbridge iron ; 2d fusion.

Franklin iron ; 3d fusion

WMMMmOOMh^

11,587

12,487

15,019

15,525

11,812

14,512

16,48

19,462

12,987

13,365

15,363

14,616

13,788

14,850

16,056

42,130 45,110 52,870 55,930 42,760 52,670 60,500 69,680 49,070 50,120 57,330 54,550 48,730 50,720 56,050 60,410

0.156 0.152 o. 152 0.147 o. 162 0.195 0.202 0.230 0.250 o 217 0.220 O.I95 O.152 o. 170 0.175 0.170

Bars with circular section, Eq. (6).

Franklin iron ; 3d fusion.

Franklin iron ; 2d fusion,

MndOOOHO cqo h o 7) NCO ci T^-vO H VÛ O C4 rf C->

p 00" t-T 0“ O' c? F'

M MM

70,600

57,720

70.740

71.740 52,36o 63,670 64,820 52,36o

0.237 o. 166 0.254 0.240

0.240

0.262

It is both interesting and important to observe that K and the final deflection are materially larger for circular beams than for square ones.

By comparing these values of K with the ultimate tensile resistances found by Major Wade, and which have been given under the head of “ Tension,” it will be seen that no great error will be involved if K is taken at twice the ultimate tensile

resistance for square bars, and two and a quarter times the same quantity for bars with circular section.

Whether these ratios will hold for iron of inferior quality to that used by Major Wade, can only be determined by farther experimenting.

Steel.

Some circular Bessemer steel beams with 12 and 8-inch spans were tested at the Rensselaer Polytechnic Institute in Nov., 1882, with the results which are given in the next table. The “ elastic limit ” is that point at which the specimen “ breaks down.’’ The “ ultimate ” value was that for which the deflection was equal to one or one and a quarter the diameter.

Circular Bessemer steel beams, Eq. (6).

K.

IV.

KIND.

SPAN.

Elastic.

Ultimate.

Elastic.

Ultimate.

Turned

OOOOmmwm ^

MMMM Q Q OO ? tnuuntn Ö O O O

Ins.

Lbs.

Lbs.

Lbs.

12

12

12

12

2,500

3,750

1,150

1,800

1,150

1,800

4>5°°

7.500

1,800

3.300

1,700

3.300

86,000

85,300

76,400

76.400

77.400 80,800 77,400 80,800

Lbs.

146,750

i52,Soo

137,520

152,800

122.200

148.200 114,400 148,200

The “ ultimate ” K is seen to decrease as the ratio of length over diameter increases.

The following table contains results computed from the experiments of the “ Steel Committee ” of the British Institution of Civil Engineers; the experiments were made in 1868.

The bars were 1.9 inches square in section, and the distance between supports was twenty inches.

Bessemer Steel, Eq. (5).

KIND AND NUMBER OF TESTS.

K.

ELASTIC OVER

ULTIMATE.

Lbs. per sq. in. 120,030 129,325 125,900 115,120

0.573

0.615

0.612

0.563

5, tires, hammered........

5, axles, “    ........

4, rails, “    ........

4, tires, axles, rails ; rolled

Crucible Steel, Eq. (5).

5, tires, hammered 4, axles, “

I, rail,

I, axle, rolled....

143.530

152,055

175,470

118,160

0.574

0-539 o 436 0.538

M m m *t co coo 00 cô cô cô CO

Each result is an average of the number of tests shown in the left column.

The ratio “ elastic over ultimate ” is the value of K at the “ elastic limit ” divided by its ultimate value as given in the table.

Table IX. of Art. 34 gives the ultimate tensile resistances of these same steels. That table, taken in connection with the results just given, shows that K is about 1.66 times the ultimate tensile resistance for square Bessemer steel bars, and about 1.85 times the same quantity for square crucible steel bars.

Mr. J. W. Cloud, of the Penn. R. R. Co., made bending tests of the Bessemer rail steel whose ultimate tensile resistances are given in Table IV. of Art. 34. His test pieces were 12 inches long, 1.5 inches wide, and 0.5 inch thick. The load was applied in the direction of the thickness, and midway between supports 10 inches apart. The greatest, mean and least results of the 18 means of the groups shown in Table IV., Art. 34, are the following:

K.

iv.

Greatest Mean . .. Least ...

3,349 lbs. 3,026 lbs. 2,765 lbs

133,960 lbs. per sq. in . 121,040 lbs. per sq. in. 110,600 lbs. per sq. in.

With these rectangular specimens of Bessemer rail steel, supported flatwise, therefore, K may be taken about 1.6 the ultimate tensile resistance.

The following table contains the results of Mr. Kirkaldy’s experiments on square bars of Fagersta steel. These bars were 1.9 inches square in section, and rested on supports 20 inches apart. W is the breaking weight at centre, and K is

Fagersta Steel Square Bars.

MARK.

IV.

ELASTIC OVER ULTI-

FINAL DE

POUNDS.

LBS. PER SQ. IN.

MATE.

FLECTION.

1.2

1.2

30,496

32,896

35.376

Mean

32,589-

133,380 143,88o 154,7!o

Mean

144,000.

0.669

0.695

0.616

Mean :

o. 66.

Ins.

o 75 0.72 0.87

0.9

0.9

0.9

43,820

44,552

43.I2S

Mean

43,833-

191.640 194,850

188.640

Mean

191,700.

10 rj-

060

OO N O w

VO

Mean

0.496.

1.46 i. 62

1.38

oop

O' O' O'

U) CO 4^ CO O' p H to to to c Cb

000

Mean

38,i45-

176,100

158,310

166,740

Mean

167,040.

0.467

0-491

0.4S2

Mean : 0.48.

3-15

3-56

3.22

000

WWW

24,420

23,280

28150

Mean

25,283.

106,800

101,820

123,120

Mean — 110,580.

0.561

0-653

0.654

Mean

0.623.

5.22

5-05

5 05

computed by the aid of Eq. (5). The column “Elastic over ultimate" contains the ratios of the values of K at the “elastic limit ” divided by the ultimate values given in the table.

The “Mark" shows the character of the steel; 1.2 is the hardest, and 0.3 the softest.

K is about 1.6 times the ultimate tensile resistance for the grades 1.2 and 0.6, and 1.8 times the same quantity for the grades 0.9 and 0.3.

Combined Steel and Iron.

In Sept., 1881, some interesting and valuable experiments on the transverse resistance of pins (solid circular beams) were made at Phœnixville, Pa., by the Phoenix Iron Co.

The pins were of combined iron and steel, the core of the pin being of steel, and the outside of iron. In such a pin the iron seems to change gradually to the steel, but the shell of iron may perhaps be considered one quarter to one half an inch thick.

These pins are supported at each end and loaded in the centre. The results of the experiments are given in the following table :

D = diameter of pin.

/    = length in inches between supports.

W = weight (pounds) at centre.

K' — intensity of stress per sq. in. on extreme fibre, in general.

K — intensity of stress per sq. in. on extreme fibre, at rupture.

K is the greatest value of K' for any one pin. Either K or K', by Eq. (6), has the value :

Wl

K or K' ~ .

Ar

The mean of the two values of K is :

99,260 -f 144,360

PIN.

D.

1.

IV.

K' OR K.

REMARKS.

Ins.

4iV

4iV

4.V

24

24

24

4i

24

24

24

20.5

20.5

20.5

36,000 60,000 100,000 60,000 84,000 148, 68,000 84,000 252,000

32,815.

54,692.

9IiI54 40,241. 56,337-99,260. = K 38,955 • 48,121.

144,360. = K

Elastic limit.

Not broken. Deflection = f ins.

Elastic limit. Broken.

Slight permanent self. Broken.

— 121,810.00 pounds.

Copper, Tin, Zinc, and their Alloys.

In the following table are given the data and the results of the experiments of Prof. R. H. Thurston, as contained in his various papers, to which reference has already been made. The distance between the points of support was twenty-two inches, while the bars were about one inch square in section, and of cast metal.

The modulus of rupture, K, is found by Eq. (5), in which, however, in many of these cases, W is the weight applied at the centre, added to half the weight of the bar. When K is large and the specimens small, this correction for the weight of the bar is unnecessary ; otherwise, it is advisable to introduce it.

The coefficient of elasticity, E, is found by Eq. (9), in which W is the centre load added to five-eighths of the weight of the bar.

The manner in which both these corrections arise, is completely shown in Case 2 of Art. 24.

E, for any particular bar, has a varying value for different degrees of stress and strain. Those given in the table may be considered average values within the elastic limit.

As usual, “ elastic over ultimate ” is the ratio of K at the elastic limit over its ultimate value.

An examination of the ultimate tensile and compressive resistances of these same alloys, as given in preceding pages, shows that the ratio of K over either of those resistances is very variable. It is usually found between them, but occasionally it exceeds both.

Square Bars.

PERCENTAGE OF

ELASTIC OVER FINAL DEFLEC-

Cu.

Sn.

Zn.

LBS. PER SQ. IN. ULTIMATE.

TION.

LBS. PER SQ. IN.

IOO

0.00

0.00

IOO

0.00

0.00

IOO

IOO

90

90

?o

70

61

48.

39 •

28.

9-

o.

MM MOM

0.00

0.00

10.00

10.00

20.00

30.00

38.3

52.0

60.8

71.6

90.3

IOO

80.

IOO

62.

58.

55-

92.

82.

7i-

63-

58

54-

0.00

0.00

2.30

0.50

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

20.00

37-5°

39-48

44-5°

7.68

16 98 28.54 36.36

41.10

44-78

29,850

25,920

21,251

29,848

49,400

56,375

56,715

12,076

2,761

3,600

8,400

8,067

5,305

3,740

4,559

21,193

43,216

95,620

72,308

2i,7S4

23,197

24,468

43,216

63,304

47,955

0.14 to 0.41 0.346 o. 140 0.400 0.41 0.657 1.00 1.00 1.00 1.00

0-583

0.25

0.273

0.267

Ins. 8.00 1.38 to 8.00 2.31 Bent. Bent. 3-36 0.492 0.062 0.032 0.019 0.060 o. 121 Bent. Bent. Bent.

9,000,000 j- 10,830,600

13,986,600

10,203,200

14,012,135

0.30

0.41

o 51

0-53

0.48

0.39

3-27

3 • T3

1.99

Bent.

Bent.

Bent.

Bent.

Bent.

Bent.

13,304,200

15,321,740

9,663,99°

17,039,130

12,302,350

9,982,832

7,665,988

6,734,840

5,635,59°

11,000,000

14,000,000

11,000,000

13,842,720

14,425,150

14,035,33°

14,101,300

11,850,000

10,816,050

Square Bars.— Continued.

PERCENTAGE OF

ELASTIC OVER FINAL DEFLEC

Cu.

Sn.

Zn.

LBS. PER SQ. IN.

ULTIMATE.

TION.

LBS. PER SQ. IN.

43-36 36.62 29.20 20.81 10.30 0.00

70.22 56.88

45-00

66.25

10.00

58.22 60.00 65.00 70.00 75.00 80.00 55-00 60.00 72.52 77-50 85.00

0.00

0.00

0.00

0.00

0.00

0.00

8.90

21.35

23-75 23-75 50.00 2.30 10.00 20.00 10.00 I 5.00 10.00 5-00 2.50 7-50 12.50 12.50

56.22

62.78

70.17

77-63

88.88

100.00

20.68

21.39

31-25

10.00 40.00 39 48 30 00 15-00 20.00 20.00 10.00 44-50 37-50 20.00 10.00

17,691

4,8g3

16,579

22,972

4i,347

7,539

50,541

2,752

6,5¹²

8,344

21,525

95,623

24,700

n,932

36,520

55,355

67,117

1.00 ^ 1.00 1.00 i .00

0-73

0.57

on mco

2-5

61,705

62,405

Ins.

O.O982

O.0245

o.0449

O.1254

O.5456

o.1244

0.4019

0.0146

0.0150

0.0162

Bent.

2.000

o. 1267

0.0514

0.1837

Bent.

Bent.

Bent.

1.500

Bent.

0.705

Bent.

12,918,210

14,121,780

14,748,170

14,469,650

12,809,470

6,984,644

14,400,000

14,800,000

7,000,000*

12,000,000*

9,000,000

10,600,000

14,500,000

17,000,000

15,000,000

13,000,000

13,500,000

11,000,000

13,000,000

12,000,000

13,500,000

12,500,000

* These bars were about half the length of the others.

Timber Beams.

As timber beams are always rectangular in section, Eq. (3) only will be needed. Retaining the notation of that equation, if the beam carries a single weight W at the centre of the span / :

(10)

2    KAh

3    I *

If the total load W is uniformly distributed over the span :

- 4 KAh

(II)

As K is supposed to be expressed in pounds per square inch, all dimensions in Eqs. (10) and (11) must be expressed in inches.

In the use of timber beams it is usually convenient to take the span / in feet, and the breadth (b) and depth (h) in inches. Placing 12/ for /, therefore, in Eqs. (10) and (n);

and’ ⁼

KAh

² 18/    ‘ '

• (12)

in which formulæ / must be taken in feet and A and h in inches.

If B be put for — , Eq. 12 becomes :

Ah

Ah

W = /; ~~ ; and, W = 2B ~

• • (13)

Hence, when W and W have been determined by experi

ment :

For single load W at centre :

Wl

B — -A h* =

Wl

Ah'

Bb

(14)

For total load IV uniformly distributed :

W'l

2Ah *

\h* =

W'l

2 Bb

OS)

If the beam has a section one inch square and is one foot

W

long, B — W — — — . By therefore, may be considered the

unit of transverse rupture ; it is sometimes called the coefficient for centre breaking loads.

Table I. is a condensed statement of the result of experiments by the late R. G. Hatfield, a complete account of which may be found in his “ Transverse Strains,” 1877. All the test

TABLE I.

MATERIAL.

B.

K= rtB.

MATERIAL.

B.

K = 18 B.

Georgia Pine. Locust......

Lbs.

850

1,200

Lbs.

15.300

21,600

Ash.

Lbs.

900

White Oak.

650

11,700

Spruce .... White Pine

550

500

Hemlock

450

Whitewood

600

Chestnut

480

9,900

9,000

8,100

10,800

8,640

Maple.......

Hickory.....

Cherry......

Black Walnut Canadian Oak

New England Fir...

1,100

1,050

650

750

59°

370

Lbs.

16,200

19,800

18,900

11,700

13,500

10,600

6,6lO

specimens were of American woods with cross dimensions varying from one to two inches and span of 1.6 feet.

Table II. contains the results of experiments on specimens of American timber, given by Prof. R. H. Thurston in the “Journal of the Franklin Institute,” Oct., 1879. The test specimens were 3 inches square and 4.5 feet between supports. The coefficient of elasticity is in pounds per square inch, and is found by Eq. (9).

Later experiments by Prof. Thurston (“ Jour. Frank. Inst.,” Sept., 1880), on a great variety of yellow pine specimens, both in respect to dimensions and degree of seasoning, induced him to draw the following conclusions in regard to that timber :

The elasticity of yellow pine timber as used in construction is very variable, the coefficient varying from one to three millions, the average being about two millions for small sections, and a little above one and a half millions of large timber.

TABLE II.

Specimens 3 ins. x 3 ins, x 4.5 ft.

ELASTIC

DEFLECTION IN INCHES.

COEFFICIENT

MATERIAL.

LIMIT.

K.

B.

OF

Elastic.

Ultimate.

ELASTICITY.

White Pine..

Lbs.

4,3-0

Lbs.

Lbs.

Lbs.

5,280

293

0.86

1.28

883,636

Yellow Pine.

12,720

Locust

8,400

j 16,740 13,680

930

760

084

0.82

1.96

2 70

3,534,727

2,046,315

Black Walnut

5.640

7,440

413

0.50

0.72

1,944,000

White Ash ..

6,360

9,72c

540

1.50

2.50

1,080,000

White Oak..

7,200

9,840

547

0.90

1.76

1,620,000

Live Oak

9,040

11,280

627

0.94

1.3S

1,851,428

The highest values are as often given by green as by seasoned timber.....

The density of the wood does not determine the coefficient ;    .    .    .    .

A high coefficient usually accompanies high tenacity and great transverse strength, but it is not invariably the fact that maximum ultimate strength is accompanied by initial stiffness.....

K varies from 10,000 to 17,000 pounds per square inch (or B, from 556 to 944) with a mean value of about 13,000 (or about 722 for B).

In “Van Nostrand’s Magazine” for Feb., 1880, Mr. F. E. Kidder, B.C.E., gives the following results of experiments with 5 yellow pine specimens about 1.25 inches square in section and 8 white pine specimens about 1.5 inches square; all on supports 40 inches apart :

34

Yellow Pine.

GREATEST.

MEAN.

LEAST.

Coefficient of elasticity. .. 1,926,160 lbs.... 1,821,630 lbs.... 1,707,282 lbs.

12,280 lbs. 682 lbs.

K.

B,

14,654 lbs.... 813 lbs....

13,048 lbs.... 725 lbs....

White Pine.

Coefficient of elasticity ... 1,461,728 lbs ... 1,388,497 lbs....

8,297 lbs.... 461 lbs....

K.

9,440 lbs.... 524 lbs....

1,251,252 lbs. 7,578 lbs. 421 lbs.

Table III. contains values of B which have been computed from data determined by MM. Chevandier and Wertheim (“ Mémoire sur les Propriétés Mécaniques du Bois,” 1848). The timber was from the Vosges. The great variations in the length of span and dimensions of beam render these especially valuable.

TABLE ill.

Vosges Timber.

4»

BREADTH.

DEPTH.

SPAN.

W AT CENTRE.

B.

Ins. 11.4 10. o

Fir.

8.8

6.7

365

9-7

9-5

9.2

8.6

7.6

6-3

Ins. 12.8 II.2

9.6

7-7

4-85

2.16

i. ii

10.9

9-3

7-4

Ft.

42.64

36.08

29.52

29.52

29.52

9.91

9.91

Lbs.

Lbs.

Oak.

5-4

6-3

18.04

18.04

18.04

18.04

18.04

M M M    MM

4^. P H 'J\ Vj to M 4-» M 4-

120

867

581

3.26

3-2

339

356

287

355

415

445

500

293

392

376

643

3-07

3.16

9.84

8.20

421

354

11.50 564 l 9-5

2.15

1.66

18.04

9.84

825

433

343

532

i. ii

9.84

715

614

The weights of the beams were allowed for in the manner already shown in that section of this Art. which is headed “ Copper, Tin, Zinc, and their Alloys.”

TABLE IV.

Laslett ’s Tests.

Sections 2x2 inches with span of 6 feet.

KIND OF TIMBER.

W, IN LBS.

3, IN LBS.

FINAL DEFLEC-

COEFFICIENT

TION.

OF ELAS., OR E.

Oak, English................

Oak, English................

Oak, English................

Oak, French.................

Oak, French ................

Oak, Tuscan.................

Oak, Sardinian..............

Oak, Dantzic................

Oak, Spanish................

Oak, American, white.......

Oak, American, Baltimore...

Oak, African (or teak).......

Teak, Moulmein.............

Teak, Moulmein ............

Iron wood, Burmah..........

Chow, Borneo...............

Greenheart, Guiana..........

Sabicu, Cuba................

Mahogany, Spanish..........

Mahogany, Honduras.......

Mahogany, Mexican.........

Eucalyptus, Australia :

Tewart................

mahogany.............

iron-bark..............

blue-gum.............

Ash, English................

Ash, Canadian...............

Elm, English................

Rock elm, Canada...........

Fir, Dantzic.................

Fir, Riga....................

Fir, spruce, Canada..........

Larch, Russia...............

Cedar, Cuba............. ...

Red pine, Canada............

Yellow pine, Canada ........

Yellow pine, Canada.........

Yellow pine, Canada.........

Pitch pine, American........

Pitch pine, American........

Pitch pine, American ......

Kauri pine, New Zealand ....

562

422

4° 7

8i3

877

831

758

758

474

562

804

723

1,108

913

843

Ii²73

975

1-333

I)²93

856

802

783

1,029

r,.'

1,407

712

862

638

393

)20

77

600

670

Ó26

560

653

627

483

3°4

1,049

93°

744

719

3°5

610

658

623

5⁶9

569

356

422

Ć03

54-

831

685

632

955

731

1,000

970

642

587

772

515

1,055

534

647

479

-’"5

690

658

450

503

470

420

490

470

362

228

787

6₉3

558

539

Inches. 5.10 3-95

Pounds.

7.71

6.00

7 58

7.66

6.50

6.46

6.62

8.83

7-I3

5.14

3.38

6.49

4 25

2.83

4.Ć2

3-75

3-45

4.06

3-92

4-75

4.71

3.81

4.21

8.63

7-37

5 29

8-79

5-14

3-63

5-19

4-33

4-37

4.63

1.66

3 39

3-45

4-79

4.67

4.42

4.00

902,600

1,536,800

1,440,000

605,000

871,400

1,184,600

1,547,200

1,010,880

1,378,500

1,172,400

2,369,30°

2.472.300 1,057,900

2.369.300 1,882,800 1,187,100 2,021,800

1,791,000

2,420,000

1,805,100

1,404,000

1,299,700 i>395-4°°

1.763.200

1.849.200

2,030,800

1,834,30°

1,602,000

1,636,30°

E has been computed only for those cases in which W exceeds 700.

In the cases of the fir specimens, B increases very considerably as the depth of the beam decreases, and with little irregularity. The same general result seems to hold with the oak specimens, although there are very marked irregularities-On the whole, therefore, these experiments would seem to show unmistakably that B or K has much larger values for small depths of beam than largë.

The modulus of rupture, K, may of course be found by taking 18B, but its values are not given in the table.

Tables IV., V., VI. and VII. contain values of B and E which have been computed from data determined by the English experiments of Messrs. Laslett, Maclure, Fincham, Edwin Clark and G. Graham Smith. These experiments are among the latest and most valuable ever made.

In all these tables W is the total load applied, including the weight of the beam, wherever that correction is made.

In Table IV. the coefficient of elasticity is computed, in all cases, for a centre load of 390 pounds. In Table V. the centre load for the same computation is 1,680 pounds ; and in Table VII. the elastic load had different values for different beams.

In all cases, except the four noted in Table VII., the applied loads were placed at the centre of the span.

Although these experiments do not embrace a great variety of cross section for all kinds of timber, yet Tables IV., VI. and VII. give much larger values of B for small depths of pine and fir beams than for large ones. This is a very important consideration in connection with the ultimate resistance of beams, and probably obtains for all kinds of timber. In fact, Table III., as has been observed, indicates the same results for Vosges fir and oak.

These experiments also showed that the coefficient of elasticity, E, varied materially in the same specimen for different deflections, and that values among the greatest may be found with large deflections ; also that the “ elastic limit ” for flexure in timber beams is more conventional than real, since with

TABLE V.

Fincham's Tests.

3x3 inches, section ; 4 feet span ; very dry timber.

KIND OF TIMBER.

W.

B.

COEFFICIENT OF ELAS-

TICITY.

Riga fir

Pounds.

Pounds.

4,530

670

Pounds.

2,293,760

Red pine... Yellow pine Norway fir.. Scotch pine Kauri pine.

3,78o

2,756

3,292

2,520

4,110

TABLE VI.

559

408

487

373

608

Maclure's Tests.

Specimens of Memel Fir.—1849.

1,593,000

1,550,000

1,850,000

925,000

1,977,400

BREADTH.

DEPTH.

SPAN.

IV.

B.

FINAL DEFLECTION,

INCHES.

Inches.

Inches.

Feet.

Pounds.

Ił

Ił

483

Pounds.

644

2f

450

GOO

2|

12

12

W W Ht

to to to

1,910

1,311

1,104

1,482

34,720

38,080

61,600

637

437

368

494

482

353

O.75

O.75

I. OO

1.125

3-5

4-5

2.0

2-5

TABLE VII.

Tests by Edwin Clark and G. Graham Smith.

TIMBER.

DEPTH. SPAN.

W.

B.

FINAL DE- COEFFICIENT

FLECTION.

OF ELAS.

American red pine American red pine American red pine ....    6

Memel fir..............

Memel fir..............

Baltic fir...............

Baltic fir...............

Pitch pine.............

Pitch pine.............

Pitch pine.............

Pitch pine.............

Red pine..............

Red pine..............

Inches. Inches. Feet. Pounds.

Lbs.

12

12

12

12

iS

15

13.5    135 IO-5

13-5    13-5

14

14

33)

29.9

497

qo8

291

260

12

12

12

12

15

15

12

12

15

15

15

15

Inches. Pounds.

4.00    1,443,830

1,155,100 1,015,900

3.10

1.68

2,150,500

E. Clark.

10-5

12.25

12.25

12.25

12.25

10.5

10.5

12.25

12.25

10-5

10.5

IQ-5

10.5

293

271

448

442

229

286

i.ii

1-93

1-31

1-31

1.14

1.94

1.561.300

1,573,4°°

1.442.300 3,125,000

1.431.300 i,935,4co 1,693,400 1,247,000 1,247,ooo

1,329,750

1,329,75°

1,270,000

G. Graham Smith.

loads about half the breaking weight, not only the deflection but the “ set ” varied with the time.

The quantity ordinarily termed the load at the “ elastic limit ” may be taken from 0.5 to 0.6 the breaking weight. In Table VII. it varied from 0.50 to 0.78.

The latest experiments on timber beams are those of Col. Laidley and Prof. Lanza; both experimented during 1881. Col. Laidley’s results are given in Table VII«.

As was to be expected, in accordance with conclusions already drawn, the sticks of Oregon pine with the smallest depths gave values of K and B considerably larger than the others. These results emphasize the fact that for large beams K or B must be taken from tests on beams equally large if accurate computations are to be made. With these consider-

NO.

IO

NO.

10

12

13

TABLE Vila.

Seasoned Sticks, Loaded at Centre.

KIND OF WOOD.

WIDTH.

DEPTH.

K = 18 B.

LBS. PER SQ.

INCH.

B.

REMARKS.

Oregon pine...............

Oregon pine...............

Oregon pine...............

Oregon maple............

California laurel...........

Ava Mexicana.............

Oregon ash................

Mexican white mahogany.

Mexican cedar.........

Mexican mahogany........

SPAN.

Feet. 15.°° 6.6 o

15*⁰⁰

6.67

4.00

10.00

15.00

20.00

10.00

16.00

7.00

7.0°

.67

6.67

16.00

Ins.

44

22

22

44

44

44

44

44

44

44

Ins.

3-48

Ins.

3-48

11,900

661

1.22

1-23

1.21

3-63

358

369

364

3-77

3-75

3-75

1.20

3.63

3-58

369

3-64

3-77

3-75

3.75

TABLE VIM.

13,210

16,570

10,560

8,920

9,93°

8,460

9,610

7,935

15,830

Seasoned Spruce Beams.

734

Cross grained.

921

587

496

552

470

534

441

879

Worm eaten.

Cross grained.

K = 18 B.

WIDTH.

DEPTH.

MANNER OF LOADING.

LBS. PER SQUARE    B.

INCH.

Inches.

2.00

2.00

2.00

2-75

3.00

3.00

3.00

3-9°

2.50

3-75

7.00

1-75

3.00

3.00

! 8    8    8    8    8 8 8    8    ° 8    8 K    8    8    8

y Ci    O'    Ci    Ch    O O' Ot    N    CON    N VO    O'0\    N

Cm    h    m    m m    m

At centre.

it    it

4.5 feet from one end. At centre.

At 4 points, 16 ins. apart. 4.5 feet from one end.

5,S2ö

5,389

5,237

4,082

3,285

4,508

5,651

4,253

3,787

3,²7i

8,748

7,562

4,931

4,961

5>2i8

307

299

291

226

183

250

314

237

210

182

486

420

III

ations in view, Prof. Lanza’s results for large spruce beams, which are given in Table VII^., possess great value.

With the exception of Nos. 11 and 12 the material was common merchantable lumber.

Timber Beams of Natural and Prepared Wood.

on

Table VIL. contains the results of some experiments by A. M. Wellington, C.E. (“ R. R. Gazette,” Dec. 17, 1880) small specimens i¹/^ inches square and 15 inches between supports. “ All the woods, except as specified, had been cut six to eight months and were partially seasoned.”

TABLE VIE.

Specimens 1.25 inches square, 15 inches long.

NATURAL.

PREPARED.

KIND OF TIMBER.

LOSS, PER

IV. in Lbs.

By in Lbs.

IV. m Lbs.

B. in Lbs.

CENT.

White oak, well seasoned..

White ash..............

Beech ...................

Elm....................

Pin oak..................

White oak, green..........

Soft maple...............

Black ash................

Sycamore................

989

926

864

763

941

747

742

685

628

633

593

8125

553

8oi

489

602

479

476

439

401

763

755

4- L cn cn CO CO H 10 too W vj

11.2

7-2

0.0

20.0

643

640

411

13-7

550

4°9

332

6.9

17.2

The “ prepared specimens had been treated by the Thil-meny (sulphate of baryta) process ; and all specimens of the same kind of wood were cut from the same stick.

The column “ Loss " is the per cent, of loss caused by the preservative process employed.

Cement, Mortar and Concrete.

Table VIII. and Table IX. contain values of K computed from data given by Gen’l Gillmore in his “ Limes, Hydraulic Cements and Mortars,” 1872. All the prisms were 2 inches square in cross section and 8 inches long, and were broken by the weight W, wrhich was applied at the centre of a 4-inch span. K is computed by Eq. (5), all dimensions being in inches. The composition is shown in the tables. The pure mortars of Table VIII. were kept 24 hours in a damp place, and then immersed in salt water until broken. Nos. 1, 2, 3 and 4 were 59 days old ; the others, 320. As a rule, those which set under pressure were considerably stronger than the others.

In Table IX., all the prisms set under a pressure of 32 pounds per square inch, and were kept in sea water, after the first 24 hours, until broken.

Many reliable experiments, such as those which follow, show that when masonry is built in a strictly first-class manner, its transverse resistance is very considerable.

Table X. is taken from a paper entitled “Notes and Experiments on the Use and Testing of Portland Cement,” by Wm. W. Maclay, C.E., in the “ Trans. Am. Soc. of Civ. Engrs.,”

1877.

The concrete prisms were six inches square in cross section and two feet long, and rested on supports one foot apart. W was applied at the centre of the span. If W₁ is the weight of the prism whose length is equal to the span, Eq. (5) becomes :

bh²

K ₌-3 (w+y₂wy

in which b, h and l are to be taken in inches.

N.

X.

Pounds.

211

373

217

188

308

* 294

485

oo£

519

477

459

3*4

654

TABLE VIII.

Section of Prism 2 inches square. Supports 4 inches apart.

PRESSURE PER

AT CENTRE.

SQUARE INCH

WHILE SETTING.

COMPOSITION OF MORTAR.

KIND OF CEMENT

Pounds.

281.5

497-5

288.8

250.8

409.8

392.4

646.0

400.0

692.5

635-5

613.0

418.0

871-5

00*0

32.00

OO'O

OOO

32.00

32.00

0.00

0.00

32.0°

32.00

OO'O

00*0

32.00

4 vols, dry cement, 2.6 vols, water,

4 vols, dry cement, 1.4 vols, water

4 vols, dry cement, 1.4 vols, water

4 vols, dry cement, 2 vols, water

Pure cement and water, thin

Pure cement and water, stiff

Pure cement and water, thin

Pure cement and water, very thin

Pure cement and water, thin

Pure cement and water, stiff

Pure cement and water, thin

Pure cement and water, very thin

Pure cement and water, stiff

James River

James River

James River

James River

James River,

James River.

Rosendale, Hoffman Brand

Rosendale, Hoffman Brand

Rosendale, Hoffman Brand

Rosendale, Hoffman Brand

Rosendale, Delafield & Baxter,

Rosendale, Delafield & Baxter,

Rosendale, Delafield & Baxter,

“ Ä' ” is a mean of the number of experiments shown in column N.

ro

io vo

TABLE IX.

Section of Prisms 1 inches square. Supports 4 inches apart.

K, rOUNDS PER SQUARE INCH.

KIND OF CEMENT.

Pure

cement.

i vol. cement, i vol. cement, i vol. sand.

2 vols. sand.

English Portland (artificial).................

Cumberland, Md............................

Newark and Rosendale.....................

Delafield and Baxter (Rosendale)............

“ Hoffman,” Rosendale .. .................

“ Lawrence,” Rosendale....................

Round Top, Md............................

Utica, Ill...................................

Sheperdstown, Va..........................

Akron, N. V................................

Kingston and Rosendale....................

Sandusky, Ohio.............................

James River, Va...........................

Lawrenceville Manf. Co. (Rosendale).......

Sandusky, Ohio..........................

Kensington, Conn...........................

Lawrence Cement Co., “ Hoffman ” Brand.. Round Top, Md.............................

320 days old.

i year old.

On    4*- Cn.Cn Cn Cn C/i C\ Os    w

Cn w O    H4.VJ 0\4* COCO K) CO m cn

G\ 0\    1 I ChOOJ OO I W S) N H 0\ M

945

( O' i

420

5r9

456

45°

404

489

417

348

468

683

Ć84

630

i    CO    4>-    OJ4-W4*.    WOJ4.V3

I    00    !    N    Hiuicj to    vO'vjmh

I    O    i    1*0    I C/l Co 00 N    I I I    vO Ü1VO OJ

In Mr. Maclay’s experiments, since the span was twelve inches and the ends overhung six inches, K was computed by the formula :

(17)

K — - — = —

2 bh²    12

Table XI. contains the results of some French experiments cited by Gen. Gillmore in his “ Limes, Hydraulic Cements and Mortars.” The concrete prisms were of Boulogne Portland cement, about 5.91 inches square in section, and broken by a weight (W) at the centre of a span of about 31.5 inches. K was computed by Eq. (16).

Table XII. gives the results of trials of concrete prisms by Gen. Totten, in June, July and August, 1837, the prisms hav-

TABLE X.

Concrete Prisms 6" x 6" x 2'. Supports 1 foot apart.

T.

T.

DISPOSITION OF PRISMS AFTER

BEING MADE

W.

K.

Fahr.

18⁰

Fahr.

Pounds.

Pounds.

18⁰

40°

18⁰

40⁰

18°

40°

18⁰

98°

18⁰

98°

98°

24⁰

40⁰

24°

97^

Placed in North River. 'I o'

Exposed outside......    £

Kept indoors.........

Placed in North River.

Exposed outside......

Kept indoors.........

Exposed outside......

525

775

VO VO VO

ci ci

750

i,8oo

8oo

44

6o

94

¹5

27

63

150

67

32°

40°

32°

98e

u

bfl

1,475

700

123

58

All prisms were of Portland cement concrete ; 1 vol. cement, 2 vols, sand, 5 vols, small broken stone.

T — temperature of air when concrete was mixed. T' — temperature of concrete when mixed.

ing been made in Dec., 1836. The cement was from Ulster Co., N. Y.

The lime (slightly hydraulic) was from Fort Adams, R. I., where the tests were made. W (the centre breaking weight) and K are in pounds.

These experimental results on the flexure of solid beams in cement, cement mortar and concrete, in connection with those of Gen. Gillmore on the adhesion of bricks and cement or ce

It should be borne in mind, however, that such a conclusion is implicitly based on the assumption of perfect manipulation of the cement and mortar and the most conscientious

ment mortar, show that masonry beams may have considerable transverse resistance ; and such resistance may be an important element of strength in some arches or similar masonry structure.

care in laying the masonry. These ends were attained in the test specimens, but it is probably safe to say that such is not the case even in what is termed first-class masonry.

*3

u

VO

rô

CEMENT, MORTAR AND CONCRETE.

541

After

60 days.

co O OJ (M Om O' ONco m m rf m in r^co co O N O u~) d 0'tr) h O w rf en h O O co Ooo O

CnCOCOMC*C*C4MMMMMMM

K, in pounds.

After

20 days.

CM O'NO in m in N n O f^O O O vO m vO ^ O cn too m COO m m O' r^vo co vO m in in m

C^COWMC^CMMMM

TABLE XI.

Portland Cement of Boulogne-Sur-Mer.

RESISTANCE OF CONCRETE.

Wx .

After 10 days.

Pounds.

After 60 days.

After 20 days.

After 10 days.

m o co co O O co co co co co vnco M    m e* w

vO O O cm o O cm O co r^vo ^t't'Tcocom^-in

C*M01MMMMM

CM Om IN O CM CJ CO 01 CO Om im co co co O O O O'O O'O'O'O' OcO O'CO O O' O'CO CO CO O O O'

'R3'₍g£3;vS'g&£g4£3'S^'g5£R

cn m rf* oi O O    O m Tf

H H h-T M~ M~

CO CO CO CO co co

1,093

1,322

1,065

1,087

oog

646

889

954

66S

448

463

359

346

289

240

304

218

202

192

961

ISI

856

492

778

773

492

404

sie

271

Lzz

I49

163

141

114

114

161

136

I7Ó

CONCRETES.

•paonpojd ojara -U03 jo auinjo^

vO c<") m O

Comp, in vols.

*S3iq

-qod 13 ^

MMMM

MMMMMMMM

•jbuoj\[

•poonpoad juj -joui io ammo a

i. 69

1.24

1.12

So-i

I.

96-0

MORTARS.

Composition in vols.

•jsibjYV

•pires eas

'IU3UI33

0.62

0.43

0.38

0.35

0.34

0.32

Sz’O ‘DUIiq oo*z co-1 ‘;uauio3

‘puns

rf O' W I O' O' CO m O'    ’t W O O

vo O' O W CO CO

co

rF

oo⁻o ‘axni'i ‘pućg oo-i ‘juouisq

co I r^co O O rt o O' W rt

O vo CO M

SZ'O ‘DUIiq oS-i ‘puvs oo-i ‘411311133

2,056

130

3,537

O -to W co co vo rf

M M C/D W

ci CO

TABLE XII.

Prisms 6 ins. x 6 ins. x 12 ins.—Supports 9 ins. apart.

COMPOSITION OF MORTAR.

oo-o ‘arniq oS-i ‘pućg

OO-I ‘1U3UI33

Sz-o ‘311117 OO-I OO-I ‘5U3UI33

‘puns

‘OOII-J

‘puns

00-0 OO-I 001 ■ ÎUOUI03

Sz-o ‘amj3 oS-o ‘pui’S 00-1 ‘}U3UI33

oo-o ‘stu]-! oS-o ‘ pui’S oo-i ‘JU3UI33

‘3011-7

‘pućs

00*0 00*0 00*1 ‘;U9UI33

COMPOSITION OF CONCRETE.

2,045

129

1,547

CO co vo O' CO M VO W

^ - <1 w

w CO

h m o O rf vnco -to r^. M 0 0 0 O' oco Tf O w i^O nm O    r-< o o h cj O' 00    i-'

nh cocon m h o co w w o m w r-> cT vn co i-T w* co "i- W m~ w~

O' m co CO O' W vnvo O' vn W co voco HH w m no mt> O^co O N O CO h N f 1 N M CO    rf rf    Th —

o co coo m

*t O' m N ci

M r-i M 1^« M

CO

cf

co -t -t- t>. 1^. r^O m m o co cT inj

4,142

260

4,983

312

2,117

O i * O co vo r^co Tf O con O' O Cl N rflN voO CO O'CO TtH Niflco Cß Cl 01 OmOmtJ-cOOmcOm

ci

££1

5,047

316

1,049

66

CO VO CJ

ci

w w m cn cn co m co m

VO VO M

cT

co m co vow —r vo o r^O'î^r^.r^M ioncoo rt co i^mO 10 -f O O    O'O rj-rj-corfw r^l^»0 COO

OcoOWWWcomO    com    i- co O    cow    WO    m

rf rf CO w" M*    W~    vo o' CO M

Granite fragments..

1    measure of mortar Granite fragments..

2    measures of mortar Brick fragments. .

1    measure of mortar Brick fragments.. .

2    measures of mortar

Stone gravel.......

1    measure of mortar

Stone gravel........)

2    measures of mortar. \

Brick gravel......

1    measure of mortar

Brick gravel.......

2    measures of mortar Stone fragments

grouted.........

Stone fragments grouted.........

;4

Stone Beams.

But few experiments have been made on the transverse resistance of the different kinds of stone. The following values of K have been computed from the experiments of R. G. Hatfield (“Transverse Strains”) and Gen. Gillmore (“Building Stones”).

B.

K = 18 B.

200 lbs

Blue Stone Flagging

Sandstone........

Brick, common.....

Brick, pressed.....

Marble, Eastchester Granite, Millstone Point (doubtful).... 133 lbs Marble, Eastchester ..

Granite, Keene, N. H

3,600 lbs. '

59 lbs...... 1,062 lbs.

33 lbs 37 lbs 147 lbs

128 lbs 103 lbs

594 lbs. y Hatfield. 666 lbs.

2,646 lbs. __

2,390 lbs. '

2,300 lbs.

1,860 lbs.

r Gillmore.

All beams were broken by centre weights. The last three tests were with prisms 2 ins. x 2 ins. X 6 ins., over a span which was taken at 3 inches.

Practical Formułce for Solid Beams.

The quantities B, K and E, which have been established, form a practical basis on which the deflection and ultimate resistance of solid beams are to be computed.

Breaking weight (in pounds') at centre of circular beam, Eq.

(6):

KAr

W=-=7

nPK

(18)

If W is a uniform load :

JV =

2 KAr 27cAK

(19)

In Eqs. (18) and (19), A (the area), r (the radius) and / (the span) are to be taken in inches.

Breaking weight (in pounds) at centre of rectangular beams,

Eq. (₅):

W= -

2    KA/i __ 2 bldK

3    /    ~ 3 /

(20)

If IV is a uniform load :

W--= 1

4 KAh __ 4 bldK

I ~ 3 I

(21)

In Eqs. (20) and (21), A (the area), b (the breadth), d (the depth) and l (the span), are to be taken in inches.

If / is expressed in feet, and all other dimensions in inches, Eq. (20) becomes :

bid

B -A-

(22)

and Eq. (21) :

Ah

W = 2B -If - 2B

bid

(²3)

Deflection (in inches)    centre of circular beams :

.    . (24)

_ ( w + y₈piy³ ₌ (jv+ÆZ³

\2Eirr⁴    \2EAr²

Deflection (in inches) at the centre of rectangular beams :

• • (²5)

(w+ s/₈piys _ (w + Yspiy³

\EbId    \EAh

In Eqs. (24) and (25), W is the centre load, and pi the total uniform load, expressed in pounds ; while A (area), /³ (cube of span), r (radius), b (breadth), and d (depth), are to be taken in inches. If there is no uniform load,// is zero ; and if there is no centre load, W is zero.

Comparison of Modulus of Rupture for Bending with Ultimate

Resistances.

The experiments on solid beams which have been cited, show the somewhat remarkable result that, in general, K has neither the value of the ultimate resistance to pension nor of that to compression ; nor, indeed, in some cases, is there anything like a well defined relation between those quantities. If those ultimate resistances have widely different values, K may be found between them ; in other cases it may considerably exceed either. In no case, however, it may safely be asserted, will it be found less than both. These investigations show that K varies with the kind of cross section, and it is altogether probable that it also varies with varying proportions of the same kind of cross section. Experimental data for the determination of this point, however, are still lacking.

In the absence of experiments conducted in a manner proper to the solution of this problem, it is difficult to assign confidently the reason for the facts as they appear.

The explanation will probably be found in the effects of the following causes, while it is borne in mind that with the small ratios of span to depth usually found in connection with solid beams, the common theory of flexure is only loosely approximate, and hence, that the greatest intensity shown by the common formulae is probably considerably different from the actual.

The varying intensity in adjacent fibres prevents perfect freedom in lateral strains, and causes a corresponding increase in resistance. In the experiments which have been made, the place of greatest intensity of stress is exceedingly small, thus placing the part first ruptured somewhat in the condition of a very short specimen. Again, after the elastic limit is passed, in consequence of the flow of the material, it is highly probable that the law of the variation of stress intensity changes and

becomes such that, with the same greatest intensity at the surface of the solid beam-, the resisting moment is considerably increased.

Finally, it has been shown that the experimentally determined ultimate resistances to tension and compression are, in reality, mean intensities, and not the greatest which the material is capable of exerting at any one point, or along any one line, as in the extreme fibres of a bent beam. On this ground alone, K ought to be considerably greater than either T or C, as determined from the usual cross sections.

Art. 64.—Flanged Beams with Unequal Flanges.

In the beams which are to follow, the material is distributed in a much more advantageous manner, in respect of its resisting moment, than in the solid beams which have been heretofore treated. In these beams, it will be found, in almost all cases, that the ultimate intensity of bending stress, at the point which first ruptures, is equal either to the ultimate resistance to tension or compression. In this respect, at least, therefore, the ultimate load for flanged beams is more easily and exactly determined than for solid ones.

In Fig. i is shown a “ flanged beam.” The “ flanges ” are the two horizontal parts above and below; the “web” is the vertical part uniting the two flanges so as to form J the perfect beam.

In order that there may be economy b of material in the beam, neither flange I must begin to fail before the other ; in other words, the two exterior layers of fibres, above and below, must begin to fail at the same time.

The intensities, then, in these two

4----b----

"T"

b'

c_

t,

Fig.1

layers must, at the instant of rupture, equal the ultimate resistances to tension and compression in bending.

Equal Coefficients of Elasticity.

By the common theory of flexure, if the two coefficients of elasticity are equal, it has been shown that if C is the centre of gravity of the cross section, the neutral axis of the section will pass through that point. Let it now be supposed that the lower flange is in tension while the upper is in compression. Also let T represent the ultimate resistance to tension in bending, and let C represent the same quantity for compression in bending. Then, since intensities vary directly as distances from the neutral axis,

• • (0

The ratio between ultimate intensities is represented by ri. If d — h + hx is the total depth of the beam, and hence h — d — h,\

hl — n'(d — h,) =

dri

1 + ri

. - (2)

If, as an example, for mild steel there be taken :

ri = ~ = 0.75 ; hx

The relation between h and hx shown in Eq. (2) is entirely independent of the form of cross section. But according to the principles just given, in order that economy of material shall obtain, the cross section should he so designed that h and hi shall represent the distances of the centre of gravity from the exterior fibres.

The analytical expression for the distance of the centre of gravity from DF is :

• - (3)

y₂ga* + (b - by\d - y₂t') + y₂(bt- b%>

b'd f{b - b')t' + {b, - by,

The meaning of the letters used is fully shown in the figure. In order that the beam shall be equally strong in the two flanges, the various dimensions of the beam must be so designed that

(4)

It would probably be found far more convenient to cut sections out of stiff manilla paper and balance them upon a knife edge.

The moment of inertia about the axis AB, thus determined, is :

This value is to be substituted in Eq. (2) of Art. 62, now changed to

M= —

CI TI

For various beams whose lengths are l and total load W, the greatest value of M becomes :

Cantilever uniformly loaded :

in

Cantilever loaded at end :

M - Wl.

Beam supported at each end and uniformly loaded:

Beam supported- at each end and loaded at centre :

Wl

M= —

The last two cases combined :

Sometimes the resistance of the web is omitted from consideration. In such a case the intensity of stress in each flange is assumed to be uniform and equal to either T or C. At the same time the lever arms of the different fibres are taken to be uniform, and equal to h for one flange and /z, for the other, h and /Zj now representing the vertical distances from the neutral axis to the centres of gravity of the flanges, while d = h + hv On these assumptions, if f is the area of the upper flange, and fl that of the lower, there will result:

M = fC.h+ f'T.hj

(5)

But since the case is one of pure flexure :

fC — f'T

(6)

M — fC{h + ht) = fCd — f Td ... (7)

Also, from Eq. (6) :

Si's

00

Or, the areas of the flanges are inversely as the ultimate resistances.

Unequal Coefficients of Elasticity.

All these results presuppose equality between the coefficients of elasticity for tension and compression. In some cases this presumption is not permissible. To the formulae of Art. 27 resort must then be made.

The neutral surface must first be located. If d is the total depth of the beam, h^~ d — h\ h, then, must be found. Eq. (5) of Art. 27, when applied to Fig. 1; becomes :

E,Cb_lf _ (b - b') (h - t')

(K - b') {d-h- tfl

E representing the coefficient of elasticity for compression, and E that quantity for tension.

Performing the operations indicated and reducing, writing n for E -r- E :

(n — i)b'h² + 2 [nt'(b — b') +    — b') + b'dfli

• (9)

= nt'\b — b') + (2d — /,) (bl — b')tt + b'd².

h is to be measured on the compression side of the beam.

This is a quadratic equation of condition for the determination of h. It is best to leave it as it is until the numerical substitutions are made and then to solve it. hx immediately results from the equation = d — h.

Frequently there is no compression flange, the section being like that shown in Fig. 2. In such a case b is equal to b', or f is equal to zero ; hence the two terms nt'(b — b') and nf\b — b') in Eq. (9) disappear. No other change occurs.

Fig.2

Eq. (1) of Art. 27 then gives the following resisting moment of the section :

.    .    (10)

(b)p - (b - v){h - ty + bĄ _ (6, - b')(K - o*

C is the greatest intensity of stress in the section of the same kind as £'.

If the section is like Fig. 2, b again equals b' and the term (b — b') (h — t')³ in Eq. (10) disappears, but nothing else is changed.

If T is the greatest stress on the other side of the neutral surface from C :

[nbh³ — n{b — b') {h — t')³ + bji³

(ii)

- (bt - b') (kt - A)³]

ess.

In order that the beam may be equally strong in the two flanges, the ratio between h and /q, as determined by Eq. (9), should be the same as that determined by the following proc-If zz is the rate of strain at units’ distance from the neutral surface :

If there is no waste of material, the cross section must be so designed that the ratios given by Eqs. (9) and (12) will be the same.

If the thicknesses of the flanges t' and /, are small compared with the depth d of the beam, and if b' also is small, i. e., if the flanges are assumed to give the whole resistance to bending while the web takes up the shear, Eqs. (10) and (11) may be much simplified.

C T

Making, therefore, b' = o in Eq. (10), putting -— = — and

then expanding the quantities (h — t')³ and (hx — Z,)³ :

= Ćbt’ Çh - i + -0 + TbA (hx

Under the conditions taken, Cbt' — Tb₁t₁ ; also, — and -;-are very small and may be neglected. Hence,

• (13)

M — Cbt’ {d — t' — th = Tbltl (d — t' — Zt)

may

Euh —

CE

Euh, rr:

TE' *

. . (12)

But both of these approximations have made M too small. As an approximate compensation, therefore, — be written for — (? + /,). The moment then becomes :

M = Cbt' ( d —

t’ + c

• • (14)

The quantity within the parenthesis of the second member of this equation is evidently the distance between the centres

of gravity of the flanges, while the quantity Cbt' — Tbxt„ is simply the flange stress. Eq. (14) is, then, identical with Eq. (7), as was to be anticipated. The equality of flange stresses gives :

Oil ■H

a relation identical with Eq. (8).

If desirable, an approximate correction for the neglect of the web may be introduced in Eq. (14). It has been seen that that equation is precisely the same as if E' were equal to E, i.e.y as if the two coefficients of elasticity were equal. Now, it will be shown in the next Article that if E' = E, the resistance of the web to bending is equal to that of one-sixth of its area of normal section concentrated in each flange. Hence, if a is the area of the normal section of the web, since bt' and b₁t1 are areas of the normal sections of the upper and lower flanges, there may be approximately written :

• • (15)

Values of C and T may be determined by experiment.

In the case of solid beams, it has been seen that if r and r' are certain ratios, K — rT or r C. Hence, since the web of a flanged beam is really a solid beam subjected to flexure, Eq.

(15) may be written :

In which,

f 4- /

d--^—- = depth between flange centres ;

a! = bpï = area of bottom flange ; a!' — bt' = area of top flange.

Cast-Iron Flanged Beams.

In the preceding Article it has been seen that r is equal to about 2 for a solid bar with square cross section. This would make r — 6 = 1^. A few imperfect experimental indications, however, seem to indicate a decrease of r for a greater ratio of depth to breadth. Let, therefore, r ■— 6 = 0.25. Eq. (16) then becomes :

(17)

TD (*' + f )

If W — centre breaking load in pounds ;

Wj = total uniform breaking load in pounds ; l = span in feet ;

12 l — span in inches :

W • 121

= 3 Wl =

TD (a¹ + £) :

(iS)

TD («' + £)

In the same manner

En

Wx = 2

CO

• * (19)

Or, if pi is the weight of the beam, supposed uniformly distributed,

(y+4) -

. .    (20)

It has been shown under the head of “ Tension ” that T

varies from 15,000 pounds per square inch, for ordinary castings, to 30,000 for those of extra quality. In Eqs. (18), (19) and (20),

D must be taken in inches ; a and a in square inches ; and / in feet.

Those equations have been verified in a most satisfactory manner by the numerous English experiments of Hodgkinson and Cubitt (“ Experimental Researches,” etc., by Eaton Hodgkinson, F.R.S., 1846), and Berkley (“ Proc. Inst, of Civil Engineers,” Vol. XXX.), as is shown by the following table. This table gives the actual centre breaking weights W, of the different beams, together with the values of W computed by the formula of Mr. D. K. Clark (“ Rules, Tables and Data”), which is essentially identical with Eq. (18) ; Mr. Clark taking the total depth minus the depth of the lower flange instead of “ D,” and “ 0.28#,” or “ 0.29a," instead of “ 0.25#.”

As the results are given to confirm the accuracy of the formulae under consideration, they are stated in tons of 2,240 pounds. Nos. 17, 27 and 34 were of the form shown in Fig. 2 ; the others had sections like Fig. 1. The results for those three beams are not satisfactory, and Eq. (io) should therefore be used in all such cases where anything more than a very loose approximation is desired. In that Eq. n may be taken equal to unity, on account of the great irregularities in the ratio of the two coefficients of elasticity. Since, in this case (see Fig. 1), b = b' Eq. (10) becomes :

ÿ‘

M = T, Wfi + bj,? -.(b, - V) (K - >,)³] • • (21)

NO.

SPAN.

Feet.

Cast-Iron Flanged Beams.

CENTRE DEPTH.

PROPORTION, UPPER

COMPUTED

ACTUAL

FLANGE TO LOWER.

W (TONS).

W (tons).

H N tort I/Ko t^co O'

WWWWWWWWtOtOtOWlOtOtOlOtOtOHHHHMHNHMH

CMn 4- W to m OO CO*<| C'en 4- CO to m O O CO^ C'en 4^ CO to m O

CO-Ê*.

ox tn in oi ut uun

in xn

m xn O O O imn tmmmn

to to

UX b) vj

cnOOOOOCOOw-^^C

Inches.

5-125

5-125

5-125

5-125

5-125

5125

5-125

5-125

5-125

5-125

5125

5-125

5- 125

6- 93

4.10

10.25

5-125

5-125

5-125

5-125

5-125

5-125

5-125

17.0

9.0

30.5

36.1

7-15

xn xn xn

xn

7-17

10.75

10-75

12.75

12.8

14.0

17-25

715

10.75

I to I I to 2 I to 4 I to 4 I to 4.5 X to 4 i to 5.5 I to 3.2 1 to 4.3 I to 5.6 i to 6 I to 7 i to 6.7 I to 6 i to 6 i to 8.3 none I to 4 I to 4 I to 5.5 I to 5.5 I to 7 i to 7 i to 4.6 i to 1.33 I to 2.1 none I to 3.6 I to 3.6 I to 2.3 I to 2.3 I to 2.7

i to 2.25 none I to 2.2 1 to 3.4 i to 2.25

mmmmmm    coo to to M m m M    t-1 M MM

vp 0)4- M M msjsj p -p H 4. M O' m MOOCOCOÇnpO M pvp CO C'pl C'-£* CO CO CO tO

oo

02

85

93

24

02

71

95

1563

21.76

tOMMMMMMM    Cn^ItOtOMMMMMM    M    MM

CO en O' to en p    to H    M    M    eo    p    o    en    CO 4^    m m    p    O    CO    to    C"0    O    p P    CO pi    ÇCpJ    C'-Ê*    CO    CO    CO    to

00 b O CO O tO    ben    M    Q    O    C>    O    O    O tO    00^1    O    Q    b    CO    O    ô    4-    i b    C'en    m    i    4^ vj    C'b'tOO

VJ CO O coen Ut    c OCO    O    O    O    O    O    O en    en en    o    O    Co    C    en    Q    Q    O en    004^.    m    ^4    C'O    4-^    O    O    CO

If the weight of the beam is taken into consideration, as in

Eq. (20) :

A mean of three of Mr. Hodgkinson’s beams of 4.5 feet span, 5.125 inches depth, gave:

Pi

W -\-r— = 8,766 lbs., and C = 45,700 lbs.

One of Mr. Cubitt’s beams of 15 feet span and 14 inches depth, gave :

W + r-2

P±._

28,100 lb?., and C = 30,850 lbs.

The bottom flange of this beam was unsound :

C must necessarily depend upon the span, since that portion of the web which is subjected to compression is somewhat in the condition of a long column. This, indeed, is true of the compression flange of any flanged beam, but the effects resulting from such a condition are much more marked in the class of beams shown in Fig. 2.

If, then, W is the centre breaking weight and W₁ the total uniform breaking load (not including the weight of the beam), Eq. (21) becomes :

In this equation, l must be taken in feet and other dimensions in inches.

For 5 feet span C may be taken at 45,000 lbs, For 15 feet span C may be taken at 35,000 lbs.

In order that a beam with top and bottom flanges may give the best result, i.e., reach its ultimate resistance in each flange at the same time, Mr. Hodgkinson found that the area of the lower flange section should equal about six times that of the upper. That relation has been anticipated in Eq. (8).

Deflection of Cast-Iron Flanged Beams.

If W is the centre load in pounds, / and w the span and centre deflection, respectively, in inches, and I the moment of inertia of the cross section, Eq. (8) of Art. 24 gives :

(-5)

Wl³

4 8wl

Or, if / is in feet, which is more convenient :

E = 36 W/3

zol

.    .    (24)

A mean of two of Mr. Berkley’s beams gave :

l = 4.5 feet ; w = 0.284 inch ; W = 20,160 lbs. :

I — 18.74. Hence: E — 12,424,600 lbs.

A mean of two of Mr. Cubitt’s beams gave :

l — 15 feet ; w — 0.465 inch ; W = 11,200 lbs.;

/= 227.03. Hence: E = 12,886,720 lbs.

The four preceding beams had top and bottom flanges, as in Fig. I. Another of Mr. Cubitt’s beams, without top flange, as in Fig 2, gave :

I — 15 feet ; w = 0.41 inch ; W — 13,440 lbs. ;

/ = 373. Hence: E — 10,679,400 lbs.

This last beam had a defective bottom flange, hence there may be taken without essential error:

E = 12,000,000 lbs.

Taking l in feet, Eq. (24) now gives for the centre deflection :

• (25)

3W'fi

1,000,000/

in which W’ is either the centre load, or five-eighths (^ths) the total uniform load, as the case may be.

The formula by which / is to be computed is the one which immediately follows Eq. (4).

Wrought-Iron T Beams.

Fig.3

The wrought-iron T beam is the most important beam 01 that material with unequal flanges. In the case of wrought iron the two coefficients of elasticity are essentially equal to each other ; consequently the axis about which the moment W37 of inertia of the section is to be taken passes through the centre of gravity of the latter.

All the experiments cited in this section are those of Sir William Fairbairn, given in his “ Useful Information for Engineers,” first series.

Experiment I.

A section of the beam is shown in Fig. 3. It was composed of two 2)4-inch Ls riveted to a 5 x %-]] inch plate. AD was horizontal, and the flange, BE, downward ; hence F was in tension.

W — centre breaking weight = 3,409

lbs.

Fig.4

/, by Eq. (29) of Art. 49, = 1.738. xt = distance of centre of gravity from F — i.91 inches.

Span = l — 7 ft. = 84 inches.

K = T' = apparent intensity of tensile stress at F.

Hence :

k= r =

Wlxt

4/

- = 78,400 lbs.

Experiment II.

Beam and data the same as before, except:

W = 7,750 lbs.

/ = 27 inches.

Hence :

K= V =

WlxT

4/

= 57,344 lbs.

Experiment III.

Beam and data the same as before, except : BE was upward, causing compression at F.

W — 10,777 lbs.

/ = 27 inches.

K — C' — apparent intensity of compressive stress at F.

Hence:

K = C = 78,400 lbs.

Experiments II. and III. were made by testing portions of the same beam used in Experiment I.

Experiment IV.

A section of the beam is shown in Fig. 4., but it was tested with the rib or web upward, as shown in Fig. 2.

AD = 2.85 inches.

BF =2.5 inches.

Thickness of rib = 0.29 inch.

Thickness of flange = 0.375 inch.

W = 3,019 lbs.

I = 48 inches.

x₁ distance of centre of gravity from F = 1.86 inches.

I = 0.989.

Hence :

4/

K= C = Blip ₌ 68,100 lbs.

Experiment V.

Beam and data same as for IV., except: Rib was downward, as shown in Fig. 4:

W — 3,153 lbs.

K = T' = 71,000 lbs.

In all these experiments half the weight of the beam was included in W.

.These results show that the apparent ultimate intensities of resistance to compression and tension in bending of T beams may be taken equal to each other; also that there may be taken :

K — C — T' — 70,000 lbs. per sq. in.

The ultimate tensile resistance (T) of this iron probably ranged from 45,000 to 50,000 pounds per square inch. Hence, nearly :

From the equality of C and T', it follows that the beam is equally strong whether the web or rib is up or down.

Deflection of Wrought-Iron T Beams.

If w is the centre deflection of a beam loaded with the centre weight W, E the coefficient of transverse elasticity, and l the span, then, as has been seen :

(26)

(27)

Hence :

IV/³

W “ 48El

or,

Wl³ 48W

A mean of the experiments II. and III. gave :

Art. 64.]

WROUGHT IRON.

563

W = 4,040 lbs.,

0.17 + o 18

W —---- =

0.175 inches.

/= 1.738.

Hence :

Wl³

48 wl

= 5,446.5oo-

This is a small value for E, but is due to the fact that the beam was a built one.*

A mean of the experiments IV. and V. give:

0.15025 in., /= 0.989.

W — 1,400 lbs., w

_ 0*135 + 0.17 __

Hence:

E — 21,706,000.

This last value of E is about four times as large as the other. Hence the rolled beam would deflect only one-quarter as much as the built one. All values of W were within the elastic limit.

These values of E, inserted in Eq. (26), will give the deflection for a load W (including five-eighths the weight of the * beam) at the centre. If is the total uniform load, 5/& Wz is to be put for W in the equation. Eq. (26) requires /, w and I to be in inches.

If, however, / is in feet and other dimensions in inches :

36 Wl3 w — -.....„ , ■

El

(28)

The foregoing formulae, both for breaking weight and deflec-

* It is probable that the riveting was done by hand. The improved modern machine riveting would make a much stiffer beam.

tion, may be used for the bending of angle irons with sufficient accuracy for all ordinary purposes.

Art. 65.—Flanged Beams with Equal Flanges.

Nearly all the flanged beams used in engineering practice are composed of a web and two equal flanges. It has already been seen that the ultimate resistances, T and C, of wrought iron, to tension and compression are essentially equal to each other; the same may be said also of its coefficients of elasticity. While these observations may not be applied with precisely equal force or truth to the milder forms of steel now working their way, to a considerable extent, into engineering construction, they certainly hold without essential error.

-o

_T__ô4-----m

In Fig. i is represented the normal cross section of an equal-flanged beam. It also represents what may be taken as the section of any wrought iron or steel I beam. Although the thickness t' of the flanges of such beams is not !    uniform, such a mean value may be taken

I j I    as will cause the transformed section of

h J_____I—b--}----Fig. i to be of the same area as the original section.

Fig.1

Unless in very exceptional cases where local circumstances compel otherwise, the beam is always placed with the web vertical, since the resistance to bending is much greater in that position. The neutral axis //A will then be at half the depth of the beam. Taking the dimensions as shown in Fig. i, the moment of inertia of the cross section about the axis HB, is :

(I)

while the moment of inertia about CD has the value :

(2)

2/'$³ -f- hP

12

With these values of the moment of inertia, the general formula, M —    , becomes (remembering that dx — ^ or ;

bd³ — (b — t)h³

6d

(3)

Or;

M' = C

2/'$³ + hP

(4)

Ć7 is written for W, since W = T — C.

Eq. (3) is the only formula of much real value. It will be found very useful in making comparisons with the results of a simpler formula to be immediately developed.

d

Let dx — y₂ (d + h). Since t' is small compared with - ,

the intensity of stress may be considered constant in each flange without much error. In such a case the total stress in each flange will be : Cbt’ — Tbt', and each of those forces will act with the lever arm y₂dx. Hence the moment of resistance of both flanges will be :

Cbt' . dt.

tłfi

The moment of inertia of the web will be : — . Conse-

12

quently, its moment of resistance will have very nearly the value :

Ct/1²

The resisting moment of the whole beam will then be:

C{bt'dx +

(5)

A still further approximation is frequently made by writing dji for h* ; then if each flange area bt' — f Eq. (5) takes the form :

(C)

Eq. (6) shows that the resistance of the web is equivalent to that of one-sixth the same amount concentrated in each flange.

If the web is very thin, so that its resistance may be neg

lected :

M — Cfd, = Cbt’d,

(7)

Or:

(7a)

Cases in which these formulae are admissible will be given hereafter. It virtually involves the assumption that the web is used wholly in resisting the shear, while the flanges resist the whole bending and nothing else. In other words, the web is assumed to take the place of the neutral surface in the solid beam, while the direct resistance to tension and compression of the longitudinal fibres of the latter is entirely supplied by the flanges.

Again recapitulating the greatest moments in the more commonly occurring cases :

Cantilever uniformly loaded :

Cantilever loaded at the end :

M = Wl.

Beam supported at each end and uniformly loaded :

Beam supported at each end and loaded at centre :

Wl

Beam supported at each end and loaded both uniformly and at centre :

- («•+?)•

In all cases W is the total load or single load, while p, as usual, is the intensity of uniform load, and / the length of the beam.

In “ Useful Information for Architects, Engineers and Workers in Wrought Iron,” issued by the Phoenix Iron Co. of Phœnixville, Penn., are the record of some experiments by which the value of C or T maybe determined. These will now be used.

Example /.

A 7-inch I was subjected to successive loads at the centre of the span, the ends being simply supported. The beam weighed 60 pounds per yard ; consequently the area of the cross section was 6 square inches. The span was 21 feet, or 252 inches. The dimensions represented in Fig. 1 are the following:

t — 0.36 inches.

.*. IP — 210.63. = 343.

h - 5*95 d — 7.00 b = 3.67    “

(b- t)= 3.31

t' — 0.525    “

= i-9²7

d₁ = y₂ {d-\- k) — 6.475 inches. f — bt'

The following table gives all the recorded results.

CENTRE

DEFLEC-

PERMA-

LOAD.

TION.

NENT SET.

REMARKS.

w = 7 (IV+P/) ■

Lbs.

8 8 8 8 8 8 8

0000000

Ins.

0.468

O.743

1.020

1.298 1-578 I.887 2.3OO 3-540

5.298

Ins.

O.029

0.030

0.060

O.X83

.ËP- - -

<V - - -

pr

O

Cl-

Ins.

w = 0.537 w — 0.775

W = I.012

w = 1.250

The coefficient of elasticity, i?, is taken at 30,000,000

lbs.

With the load of 10,000 pounds at the centre the “ beam sunk slowly, top flange yielding.” The beam, therefore, may be considered as essentially failing with a load of 10,000 pounds at its middle point. As the top flange yielded, the ultimate resistance to compression, or C, will be given by the experiment.

In reality, the beam carried a uniform load of 20 pounds per foot (its own weight), besides the single load of 10,000 pounds at the centre. Hence, Eq. (22) of Art. 24 will give the value of M. It is as follows :

20 X 21 -I- 2 = 210; W — 10,000, and / = 252.

But — =

/ is taken in inches because the dimensions of the cross section are in the same unit. These values give :

M - 643,230.

Also the data given above, placed in Eq. (3), give :

M = C x 13.37.

Equating these values :

C — 643,230 -f- 13.37 — 48,110 pounds ...    (8)

Again, the proper data inserted in Eq. (6), the approximate formula, give:

M — C x 14.79.

Hence :

C = 643,230 -4- 14.79 = 43490 pounds .... (9)

The first permanent set was observed with a centre load of 5,000 pounds. This gives a bending moment at centre of

- 328,230.

= (5.000+^)

Hence :

C = 328,230 -4- 13.37 = 24,550 pounds.

As the permanent set with this load was very small, and as

there was none at all observed with a centre load of 4,000 pounds (nearly corresponding to C = 20,000 pounds), the limit of elasticity may be taken at about :

20,000 -f- 24,000

= 22,000.00 pounds.

In the right hand column of the table are calculated the deflections by Eq. (21) of Art. 24, the coefficient of elasticity being taken at 30,000,000 pounds. By Eq. (1), using the data already given :

/ = 46.795-

Hence :

/³ -r 48El — 0.0002375.

Also :

^pl — 262.5.

These values inserted in the formula give the results shown in the table. The experimental- quantities are seen to increase much more rapidly than the results given by the formula. The agreement, however, is sufficiently close for ordinary purposes.

Example II.

The second example, derived from the same source as the first, is that of a 9-inch I, 87 pounds per yard. The data to be used in connection with Fig. 1 are as follows :

.*.    / = bt' = 2.88.

.*. d³ = 729.000. h³ = 432.581.

1—21 feet =252 inches ; p 29 pounds per foot.

dl = 8.28 inches.

W — 17,500 pounds.

The bending moment at centre, as before, is :

-H*“ I

Art. 65.]

EXPERIMENTS.

571

1,121,683.5.

The above data inserted in Eq. (3) give :

M — C x 25.08.

Hence :

C — 1,121,683.5 -4- 25.08 = 44,724 pounds. .    .    (10)

Again the approximate formula Eq. (6) gives :

M — C x 27.92.

Hence :

C — 1,121,683.5 -r- 27.92 = 40,175 pounds. .    .    (11)

The results of this experiment are given in the following table, exactly as in Ex. I.

CENTRE

DEFLEC-

FERMA-

LOAD.

TION.

KENT SET.

REMARKS.

ZU

Lbs.

2,000

4,000

6,000

8,000

10,000

12,000

13,000

14,000

16,000

17,000

17,500

Ins. o. 228 0.474 o. 720 0.962 1.201 1.432 1.580 1.863 3-256 5-233 5-602

Ins.

0.048 0.050 0.117 o. 269

Weight removed.

Ins.

0.257

0-454

0.651

0.848

1.045

E is taken at 30,000,000

lbs.

The beam may be considered as having yielded, in failure, with a centre load of 17,50x3 pounds. That number was consequently taken above in the greatest value of M.

If it be assumed that permanent set was just at the point of beginning with the centre load of 9,000 pounds, which cannot be far wrong, the corresponding moment will be:

M = - (9,000 + ~^ = 586,152:

.-. C = 586,152 -4- 25.08 = 23,370 pounds (limit of elas.).

Taking a mean of the results of the two examples :

By exact formula [Eq. (3)] :

'C ~ 46,417 pounds.

By app. formula [Eq. (6)] :

Ci — 41,833 pounds.

For the limit of elasticity:

Ct = 22,700 pounds (nearly).

These results may be considered accurate for the Phoenix Iron Co.’s beams. These experiments were made in 1858.

It is interesting to notice that these beams failed in the compression flanges.

It is also important to observe that the ultimate resistance,

C, is fully equal to the ultimate tensile resistance of good wrought iron in large bars. This serves to confirm the opinion that the ultimate tensile and compressive resistances of „ wrought iron are not far, at most, from being equal to each other, and that these quantities may be used for C or K in the

formulae for flanged beams. If the approximate formula, Eq. (6), is used, however, according to these results C or K should be taken about 0.90 (nine-tenths) of the value used in the exact formula, Eq. (3).

The last column of the second table is calculated by the formula, as shown, taking E at 30,000,000 pounds. The same general observations apply to these results as in the preceding example.

Example III.

The data for this example are taken from the hand-book for 1881 published by the N. J. Steel and Iron Co., Trenton,

N. J., where the beams were broken. The breaking weight is the mean of two results for light 6-inch wrought iron Is.

d — 6.00 ins.

t — 0.25 in.

t' = 0.456 in.

I — \2 ft. = 144 ins.

/ = 23.815, by Eq. (1).

Since the beam weighed 40 pounds per yard :

W — 14,000 -f 80 — 14,080 lbs. (centre breaking load').

Hence :

C= -

Md

= 63,840 lbs. per square inch.

By approximate formulât :

on|

= 0.21.

f= 1.368 .'.

tli

-g- +/= 1-578.

— 5-544 ins.

M — 506,880.

Hence, by Eq. (6) :

Ci = 57>93° lbs. per square inch.

Example IV.

A 9-inch heavy Trenton beam, 85 pounds per yard. The data are taken from the same source as were those in Ex.

III.

d — 9.00 ins.

t = 0.38 in.

t’ — 0.68 in.

/ = 15 ft. = 180 ins.

I — 108.47, by Eq. (1).

W = 32,000 -f- 212 = 32,212 lbs. (at centre).

Hence :

C —    = 60,120 lbs. per square inch.

By approximate formula :

th

-g = 0.484.

th

f= 2.72    .-.    — +/= 3.204.

d₁ = 8.32 ins.

M= 1,449,540.

Hence by Eq. (6) :

Cx = 54,370 lbs. per square inch.

Taking the means of these two sets of results :

By exact formula [Eq. (3)] :

C — 61,980.

By app. formula [Eq. (6)] :

Cx = 56,150.

All the conclusions reached in connection with Exs. I. and

II. are confirmed by the results of Exs. III. and IV.

C and CT are much larger, however, for the Trenton than for the Phoenix beams, and both are very high for beams of such length of span with no lateral support for the compression flange.

In calculating the deflection of rolled wrought-iron beams E may be taken from 28,000,000 to 30,000,000.

The exact formulae of this Article are strictly applicable to rolled beams only, but the approximate formula finds extensive application in cases of built beams.

Experiments by U S. Test Board.

Table I. contains the results of a valuable series of tests by the U. S. Board, “ Ex. Doc. 23, House of Rep., 46th Congress, 2d Session.”

The values of K and E at elastic limit are computed from data contained in that document in the manner already shown in detail, and which it is not necessary to repeat. It is both interesting and important to observe the considerable, though irregular, increase of the intensity of stress in the exterior fibre, at the elastic limit, with the decrease of depth. E is seen to vary from 26,099,400 to 36,664,400, with a mean value of 31,128,260. As a general result, E is slightly larger for the smaller beams than for the larger.

*X333 NI NVdS

ONNNHNMHNNttNNN'l-tN't^HMHM

CSCSCSCSCSCSHMCSCSMWCSCSCSHMCSMMHMMM

*HDNI 3HVAÖS HHd SQNAOd NI ‘g-

on,

34,328,500

26,099,400

28,903,000

OOO

OOO

000

35,409,000

30,720,000

31,859,000

29.270.200 26,863,750 28,8*6,710

32.834.200

3° >836,

33.333'®°°

30,128,700

30,099,300

33.799.5°°

34,057,600

32,453,4°°

34,064,800

31,631,800

30,976,000

36,064,400

32,687,000

29,915,

3^167,

32,628.

•QVOT 3HJLN3D IVNId

i2 HNO H H00 N NO N NH M\û rONO M m 0'O'O'0Ns#-t^0N0' m rc tj- o co m rt- ro O no oo mm tj-vO vo vo 00 ^ sf es vo Nn n G ro'tN moo^ roNO_ rooo m moc^oo^ omoOnNN- rooo vo vo 2 es- es*" es*" m c> --*'oo es ocT tC ro ro mvo vo O O ^ *>■ tCv^Tvo m *4 4-

*Y"CSCSMmMmmCS    mm    mm

TABLE I.

A, IN POUNDS PER SQUARE INCH AT

Final Load and Deflection.

Elas. Lim.

•NOI1VHÄD

30 SAIQVH 30 3HVAÖS

'/ HO

‘VIXHaNI JO XN3IVOlï

•(aVAXDV) sa NA O J

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30 V3HV a3j.n«nvo3

Inches.

0.56

2.5

1.14

Buckled lat.

1-43

°-5

°*37

2.05

¹ *79 °-93 0.89 1.79 2.04 1*97

i.01 1.03 2.08 1.08 1.10 0.92 0.90 0.74 0.94 1.05

Pounds. 19,120 35,840 26,097 32,20 o

43.100

32,43°

31.100

31,360

38,280

37,3°°

37,7°°

37,10°

3°,i5°

32,33°

32,400

36,800

36,800

34,393

40,30°

39,7°°

39,600

4I,I7°

38,

120

41,640

41,640

Pounds.

18,210

886

29,

26,097

28,721

25,191

28,713

29,478

29,500

30,508 -j 33,957

30,000

27,413

32.575

34.393

33,207 36,504 34,74° 38,116 38,116

37,75°

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All beams were loaded at centre, and deflections

In this table, / is exact. Hence K is computed by the exact formula : Zf=-.

were taken at same point.

A chemical analysis of six specimens from these beams gave the following results.

These experiments were conducted by Gen’l Wm. Sooy Smith, who kindly gave to the writer the final centre loads and deflections.

PERCENTAGES OF

Sulphur.

Phosph’us.

Silicon.

Total Carb.

Manganese.

Cópper.

Cobalt.

Nickel.

0.010

0.436

0.189

0.031

0.031

0.012

0.029

0.029

0.008

0.447

0.190

0.038

0.028

0.008

0.021

0.023

0.010

°-453

0.203

0.037

0.028

0.010

0.015

0.018

0.012

0.423

0.182

0.039

0.022

0.022

0.010

0.015

0.005

0.271

0.177

0.027

0.028

0.052

0.031

0.026

0.011

0-375

0.197

0.039

0.028

0.010

0.018

0.016

Col. T. T. S. Laidley, U. S. A., has also completed the tests of a few beams to failure, the results of which are given in “Ex. Doc. 12, 47th Congress, ist Session.” Table II. contains values of K at failure, computed from Col. Laidley’s data. Beams No. 1 carried a load uniform from end to end by

TABLE II.

NO.

DEPTH,

SPAN IN

INS.

MEAN OF

FEET.

LOAD.

LATERAL SUP-

TOTAL LOAD

K IN LBS.

PORT.

IN POUNDS.

PER SQ. IN.

15-0

28.5

3 Exps.

Uniform.

Uniform.

118,000

54,260

15-0

28.0

10.5

17.0

10.5

17.0

2 Exps. 2 Exps. i Exp.

Centre.

Centre.

Centre.

None.

None.

None.

28,650

25,890

21,020

32,210

22,020

38,070

means of brick masonry arches, which thus also gave to them a uniform lateral support. This lateral support produced a very high value of K, i.e., 54,260 pounds, which fell to 25,890 with no lateral support. In the latter case nothing prevented the compression flange yielding laterally like a column. The 10.5-inch beams were much shorter, and the long column influence less marked ; consequently the values of K are correspondingly higher. The tests are not sufficiently numerous to fix the law of the decrease of K with the increase of span.

Beams Nos. 1 and 2 weighed 200 pounds per yard, with a moment of inertia (/) equal to 706.6. Beam No. 3 weighed 105 pounds per yard, and gave I = 174.75; while No. 4 weighed 92 pounds per yard, with I — 154.9.

Art. 66.—Built Flange Beams with Equal Flanges.—Cover Plates.

A “built beam” is a beam built up of plates and angles like that shown in Fig. 1. As shown in that figure the web is composed of a single plate, called the “ web plate,” supported by “stiffeners,” if necessary, as is usually the case. These stiffeners are vertical pieces of Ls or J^s riveted to the web plate, in accordance with principles to be shown hereafter. The flanges, as shown by the heavy lines, are composed of Ls and plates so arranged as to give the requisite area of cross section at any point.

The method of designing such a beam, and the calculation of the elements of its resistance, will be given in detail. The beam is supposed to be of wrought iron, and one of a system for a double track railway bridge ; the stringers under the two tracks, which rest on the beam, are placed at A and B, and D and H. The weight of the beam, taken uniformly distributed, is 5,600 pounds. The concentrated load at each of the points A, B, D and H, composed of the train weight added to that of the stringers, is 42,000 pounds.

The following are some of the dimensions of the beam: Span RR = 28.0 feet. Depth of web plate = 48 inches. RH = RA = 4.50 feet. DH = AB = 6.00 feet. BD — 7 feet.

The web plate will be taken T\ inch thick. The method of determining this thickness will be shown hereafter.

In this case resistance to flexure of the web will be neglected ; the web will be assumed to resist the shear only, as is assumed in Eqs. (7) and (7a) of Art. 65. The “depth,” du of

Ao⁰o⁰oüo°,.'o

o'.o

IÎ.

cu

1°

lo⁰o°oV,o

Fig. 1.

beam will then be the vertical distance between the centres of gravity of the sections of the flanges, and each flange is to be considered as composed of two Ł.S and the “ cover ” plate or plates only ; no part of the web is to be included. Strictly speaking, then, the depth is variable ; but this variation is so slight that no essential error will be committed if it be considered constant and equal to the depth of web plate, or 48 inches. This procedure, which saves much labor and time, is always permissible where cover plates are used, and the bending resistance of the web plate neglected. The next example will exhibit a case in which they are not used.

The direct stresses of tension and compression existing in the flanges must be carried through the rivets which unite the flanges to the web ; hence the necessary number of those rivets will first be determined.

The reaction at R, using the data already given, will be :

R = 2 x 42,000 +    = 86,800 pounds.

The weight per lineal foot of floor beam is :

200 pounds = w.

5,6oo _

28.0

The bending moments for the two sections A and B will next be found.

Moment at

A — (86,800 — 100 x 4.50) 4.50 = 388,600 nearly.

Moment at

B = 86,800 x 10.5 — 42,000 x 6 — 100 (10.5)² = 648,375.

Since the depth of the beam is 4 feet :

Flange stress at

A = 388,600 4- 4 = 97,100 pounds.

Flange stress at

B = 648,375 4- 4 = 162,100 pounds.

The allowable intensity of pressure between the rivet and its hole (see Art. 73) will be taken at 10,000 pounds. The diameter of rivets is a matter of judgment ; it will be taken at -J inch. Rivets for built beams usually range from ■§ to 1 inch in diameter.

The selection of the Ls for the flanges is also, to some extent, a matter of judgment. In the present instance, 5" x 4" Ls, 60 pounds per yard, will be taken. These will be found to answer the purpose.

The effective bearing surface between each rivet and the web plate will then be :

t^iOO

OJ

00

ro

cr

O'

.5

Hence each rivet may carry :

0.383 x 10,000 = 3,830 pounds.

Consequently the number of rivets between R and A should be :

97,100    3,830 = 26 (nearly).

The increase of flange stress between A and B is :

162,100 — 97,100 = 65,000 pounds.

Hence the number of rivets required between A and B is :

65,000 4- 3,830 = 17 (nearly).

Since 26 rivets are required between R and A, the corresponding pitch would be but a little more than two and one-tenth inches, which is somewhat too small. With a |-inch rivet, a three-inch pitch is about the least advisable. If the rivets be placed at a pitch of three inches between R and B, forty-two will thus be located, and this is sufficiently near the desired number. The four-inch leg of the angle is placed against the web plate, but if necessary the five-inch leg could be so placed and still more rivets staggered in. In such methods as these, nearly the full number of rivets required between R and A may be supplied, while the two or three lacking will be found, without danger to the beam, adjacent to A on the side towards B. Three or four in excess of the number required will be found between A and B.

No central bending moment at C has been computed, because the only difference between such a one and that at either B or D is due to the weight of the beam only. This difference is essentially nothing. The proper support of the Ls in compression, however, requires that the rivets be pitched at about six inches between B and D. In ordinary floor beams a proper bond between the flanges and web requires that the pitch should never be greater than about six or eight inches.

The shearing of the rivets is not considered, because they sustain double shear in the flanges, and their bearing capacity is by far the least of the two.

The rivets, of course, should be pitched alike in both top and bottom flanges.

The greatest allowable intensity of tensile stress in the bottom flange will be taken at 8,000 pounds per square inch, and an equal intensity will be taken for the compressive stress in the upper flange. The area required in the bottom flange at A is :

97TOQ

8,000

= 12.1 sq. ins. (nearly).

That required at B is :

= 20.3 sq. ins. (nearly).

162,100 8,000

The area of the two 5" x 4" Ls, 60 pounds per yard, is 12.00 square inches. The thickness of the angle iron where it is pierced by the rivets binding it to the web is about 0.6 inch. Hence the area of metal taken out by one rivet is :

0.875 x 0.6 x 2 = 1.05 sq. in.

Or, the effective area of the Ls at A is:

12.00 — 1.05 = 10.95 square inches.

Now, since the weight of the beam itself is small, compared with the weight of the train, the flange stress, or moment, varies almost uniformly from R to A. Hence, an increased section is first needed at

(10.95 A- 12.1) x 4.50 = 4.1 feet (nearly),

from R. Since, however, the cover plate to be added must take its stress through the rivets which bind it to the Ls, it should overlap the necessary distance by one and a half to twice its width. In the present case, then, instead of beginning the cover plate at just 4.1 feet from R, a 12" x cover plate will begin at 30 inches from R and extend along the beam to a point at the same distance from R'. The length of this cover plate will then be 28.0 — 5.0 = 23 feet. This cover plate will be bound to the angle irons by f" rivets, which should, so far as possible, be pitched half way between the -J" rivets in the other legs of the angle irons. The effective area of this cover plate, for tensile stress, will then be :

(12 — 2 x 1) x Ą = 5.6 sq. in. (nearly).

The available area of two Ls and one cover plate is, since two rivets now pierce each angle :

9.90 4- 5.6 — 15.50 sq. ins.

For the reason already given, the moment, or flange stress, varies nearly uniformly between A and B, but at a different rate than between R and A. Since AB is 6.00 feet, the point at which another increase of section must begin is at the distance

[(15.50 — 12.1) -T- (20.3 — 12.1)] x 6.00 = 2.5 feet (nearly) from A. Again, as in the previous instance, a second cover plate, 12" x , will be put on, and it will begin, not at 2.5

feet from A, but at one foot from that point. The available area of this plate will be :

(12 — 2.0) x £ = 5.00 sq. ins.

The total area at the centre of the beam available for tension will then be :

15-5° + 5.00 = 20.50 sq. ins.

It is to be observed that in deducting metal taken out by a rivet in a tension flange, a diameter greater by an eighth of an inch than that of the rivet has been assumed. This should always be done, for the punch is always larger than the rivet, and the punched hole is still larger on the die side of the plate, and for the further reason that the metal is injured for some distance around the hole. In the compression flange no deduction need be made for rivets, as the latter completely fill the holes. Otherwise the method of designing the compression flange is precisely that just followed, and the two flanges will be taken alike.

The number of rivets required in a cover plate is yet an important question. .Since all stress carried by the cover plates must be given to them by the rivets, the number of rivets between the end of any cover plate and that point at which a further increase of flange section is necessary, must be sufficient to carry all the stress in the cover plate itself.

Applying this principle to the first cover plate found necessary : The load which each f rivet in the 12" x flfl cover may carry is :

0.875 x A x 10,000 = 5.000 pounds.

The total tensile stress carried by the 12" x ^ cover is;

5.6 x 8,000 = 45,000 pounds. Hence the number of rivets required is :

45,000 -r- 5,000 = 9.

According to the design it is 4.5 feet from the end of this cover to a point 2.5 feet from A toward B, where the next increase in section is required ; and over this 4.5 feet these 9 rivets must be distributed. But in order that a proper bond between the component parts of the flange may be obtained, it is seldom advisable to make the pitch over 6 ', and at the end of the cover plate this pitch should be halved for about three rivets. Proceeding in this manner, that part of the bottom of the beam at the end nearest R in Fig. 1, which includes the 4.5 feet of cover under consideration, will present the appearance of the sketch in Fig. 2. RG is 2.5 feet and GF 4.5 feet. In

OOO

000 Q

Fig.2

this manner 20 rivets are introduced instead of 9, but it is advisable to put in the extra number.

In the compression flange other considerations appear besides the simple bearing capacity of the shaft of the rivet. Between any two consecutive rivets the cover plate forms a solid rectangular column with essentially fixed ends, whose length is the pitch of rivets. The pitch, therefore, must not be sufficiently great to allow the existence of any material amount of long column flexure. Unless plates, therefore, are very heavy, the greatest pitch should not exceed about six or eight inches.

The bearing capacity of a rivet in 12" x f cover is :

0.875 x 0.5 x 10,000= 4,375 pounds.

The full tensile capacity of the cover plate is :

5.00 x 8,000 = 40,000 pounds.

Hence, the number of rivets required is :

40,000    4,375 = 9 (nearly).

The end of the cover plate, as designed, is one foot from A towards B ; and the nine rivets are nearly all required between that end and B, a distance of 5 feet. Hence, if the rivets are pitched in this cover plate, near the ends, as shown in Fig. 2 for the other cover, and at six inches over the intervening space, more than the number just determined will be introduced. For the reasons already given, however, the number will really be not too great.

In each flange, then, there will be found the following pieces properly joined:

5" x 4" Ls, 60 pounds per yard.

2 —

i — 12" x    plate, 23 feet long,

i — 12" x    plate, 17 feet long.

At the ends of the beams R and R, Fig. 1, provision must be made for the reaction. In this example the reaction is 86,800 pounds. The transverse shearing resistance of the web should at least equal this at the ends. The area of a transverse section of the web is :

48 x = 21.00 sq. ins.

If the greatest allowable shearing intensity in the web be taken at 5,000 pounds, its shearing resistance will be :

21.00 x 5,000 = 105,000 pounds.

This result is about 20 per cent, greater than is required.

Hence safety, so far as shearing is concerned, is amply secured. Rut the end of the beam is also subject to an upward pressure of 86,800 pounds, which must also be provided for. Two 6" x 4" X y Ls will be riveted to the ends as shown in Fig. I, one on each side of the web, and the 6'' legs lying against it. By pitching •£" rivets at 3” (nearly), in a zigzag manner, 20 rivets can be introduced to hold these 4'' X 6' x i" Ls to the web. The carrying capacity of each f" rivet against the web plate has already been found to be 3,830 pounds. These 20 rivets therefore will carry 3,830 X 20 = 76,600 pounds. Since the area of the cross section of two 4" X 6" x Ls is about 10 sq. ins., the bearing of the rivets against the web plate is all that need be considered in this connection.

A proper bearing for the difference 86,800 — 76,600 = 10,-200 pounds remains to be found. As each rivet will carry 3,830 lbs., three only are required to take up the 10,200 pounds remaining. For some distance in the vicinity of R, Fig. I, in the lower flange of the girder, the rivets will be pitched at three inches. Since some portion of these ends must rest on shoes or brackets, three of the rivets near the ends may be utilized to carry the 10,200 pounds in question. It is to be remembered that in such an instance as this, the lower ends of the f X 6" Ls must bear fairly and truly against the angle irons composing the lower flange, in order that they may take up their proper amount of the reaction.

In some cases the ends of the beam are to be secured to vertical surfaces without any supporting shoe or bracket. The entire reaction of such a beam must be carried by the vertical angles at the ends. The number of f rivets required to hold these angle irons to the web would then be 86,800 -f- 3,830 = 23 (nearly). By simply making two rows, these could easily be worked into the longer legs of the 6'' x 4" X Ls- 12 rivets would then be put through each of the two 4” legs and the vertical surface to which the beam is secured.

No account has heretofore been taken of the shearing resistance of the rivets, because that has been much greater than their bearing capacity, but instances may occur in which such a condition of things does not exist. Hence the shearing and bearing capacities should always be estimated, and security taken in reference to that which is least. As an example : at 7,500 pounds per sq. in. the shearing resistance of a f rivet is (0.875)² X 0.7854 X 7,500 = 4,600 pounds (nearly) ; while the bearing capacity of the same rivet in the 6" x 4" X f L is only :

0.875 x 0.5 X 10,000 = 4,375 pounds.

Precisely the same operations are required in determining the number of rivets in the vertical |_s at A and B, Fig. 1, as in those at the ends of the beam ; consequently it is not necessary to repeat them.

Thus, there is completed the operation of designing the beam, with the exception of finding the thickness of the web, which will be given hereafter.

In general two or three things are to be observed. The number of rivets actually required by these calculations should always be, as they just have been, somewhat exceeded. In the best of riveted work the rivets will not exactly fill the holes, and the beam will not act perfectly as one continuous whole.

Again, stress is given to the flanges along the line of the rivet holes, which is some distance from the centre of gravity of the cross section of the flange. Consequently, some bending will be induced in both flanges, and this necessitates some extra material. This excess may be estimated if desirable, but ordinarily it is entirely unnecessary. The existence of this bending demonstrates the advisability of putting on as few cover plates as possible. It is far better to use heavier Ls with a little waste of material at the ends.

It is also better to use one heavy cover plate than two thin ones having an equal combined thickness, even though the use of the former entails a little waste ; for the heavy plate between two consecutive rivets will resist far more bending as a column than the two others each of half the thickness.

If the end of the beam were made as shown in Fig. 3, no web plate would be required between R and It: ... 1 A, for all shear would be carried by the inclined flange.

Fig.3

The upper flange, being in compression, would require riveting, but none would be needed in the lower, except in the immediate vicinity of R. The flange stresses between A and R would also be uniform, instead of uniformly varying as in Fig. 1.

Art. 67.—Built Flanged Beams with Equal Flanges.—No Cover Plates.

The flanged beam represented in Fig. r is supposed to carry a portion of the floor of a highway bridge. In this case, also, the bending resistance of the web plate will be neglected. The beam proper is the portion RR' R'R, supported at RR and RR' ; while the portions ARR and HRR' form cantilevers for the support of the sidewalks.

The following are the dimensions :

RR' = 28 feet.

AR = HR = 6 feet.

RR — R'R' — 31 inches.

AH — 40 feet.

RB = BM — MF — FR' = 7 feet.

The depth RR has been taken at 31 inches, so that the effective depth to be used in finding the flange stresses will be about 2.5 feet.

The weight of the beam proper, RR'R'R, added to the flooring which it supports, is taken at

14,650 pounds.

The greatest uniform load between R and R' will be taken at

37,440 pounds.

JL

HE

Fig.1

R'

Hence the total uniform load to which the beam is subjected is :

37,440 + 14,650 = 52,090 pounds.

The weight of one cantilever, with the flooring which it supports, will be taken at

3,100 pounds

The total moving load on AR, or HR\ will be taken at

8,640 pounds.

The total load, therefore, carried by one cantilever is :

3,100 + 8,640 = 11,740 pounds.

The beam proper, RR', may sustain its greatest load when

the sidewalks carry nothing but their fixed weight. This condition of things will cause the greatest compression in the upper flange and tension in the lower, and will be assumed in designing the beam.

The fixed weight of a cantilever will cause stresses in the flanges of opposite kinds to those produced in the beam, but of such small amount that they will be neglected ; the neglect originating a very small error on the side of safety.

The total load per linear foot of RR is :

52,090 -T- 28 = 1,860 pounds.

The flange stress in the beam at R will be nothing ; it will be found at the two points B and M. Strictly, the “ depth ” to be used should be the vertical distance between the centres of gravity of the flanges. It will not be far wrong to take this depth at 2.5 feet, since the web plate is 31 inches deep. The reaction at R is :

52,090 -f- 2 = 26,045 pounds.

The flange stress at B is :

(26,045 x 7 — 1,860 X (7)'² -r- 2) -r- 2.5 = 54,700 pounds.

The flange stress at the centre M is :

(52,090 X 28 -r- 8) -f- 2.5 = 72,926 pounds.

If, as in the preceding Article, the greatest allowable stress in the flanges is 8,000 pounds per square inch, a flange area of 9.115 square inches is required in the present case. If each flange is composed of 2—4'' X 6" X Ls> 5¹ pounds per yard, there will be a very little excess of flange area, as there should be ; these Ls will then be taken for the flange, the 4" legs being riveted to the web plate ; rivets will be used in riveting the flanges to the web. Where pierced by the rivets, the legs of the Ls are about thick. Hence a rivet hole will cut out 2 X I X i.00 = 1.00 sqare inch. There will then still remain 10.2 — 1.00 = 9.20 square inches of effective area, which is a little in excess of the 9.115 required.

A web plate f" thick will be assumed. Taking 10,000 pounds per square inch as the greatest allowable intensity of pressure between shaft of rivet and plate, the bearing capacity of each rivet will be :

0.875 x 0.375 X 10,000 = 3,280 pounds.

In this case all the moving load rests upon the top of the beam, and since the edge of the web plate is only 0.375" wide, that moving load must be taken as resting on the Ls of the upper flange, and hence indirectly on the rivets. Also, since nearly the whole of the fixed load -rests upon the upper flange, the entire load of the beam will be taken as resting on that flange. Consequently, between R and B the rivets will be subjected to the action of a vertical force equal to 1,860 X 7 = 13,020 pounds, and a horizontal one equal to 54,700 pounds. The resultant force will then be :

V(13,020f + (54,7oo)² = 56,230 pounds.

Between B and M the vertical force will then be the same, but the horizontal one will be

72,926 — 54,700 = 18,226 pounds.

The resultant, therefore, is :

^(13,020)² + (18,226/ = 22,400 pounds.

Hence the number of rivets required between R and B is:

56,230 -4- 3,280 = 18 (nearly).

The number between B and M is :

22,400 -4- 3,280 = 7 (nearly).

If, therefore, commencing at R or R', the rivets be pitched at 3 inches for a distance of 4.5 feet, then at 6 inches to the centre M, about 36 or 37 rivets will be found in each half of each flange. This number is in excess of that required, but for the reasons given in the preceding Article, it is probably not too many. Thus the flanges are designed without the use of cover plates.

In this case the beam will be suspended from hanger loops at R and R', which carry resting plates or shoes for the beam at their lowef extremities.

The total reactions at the lower R and R' will be half the total weight of the entire beam with the moving load, or:

Reaction = (52,090 + 23,480) -4- 2 = 37,785 pounds.

At R and R' 2—4" x 4" x |_s will be riveted to the beam as shown. The lower ends of these angles should abut firmly and squarely against the angles of the lower flange.

Since the greatest allowable pressure between a rivet and the web plate is 3,280 pounds, the number of rivets required at each end of the beam in each pair of vertical Ls is:

37»785 -f- 3,280 = 12 (nearly).

If, consequently, these rivets be pitched at 3 inches, a sufficient number will be obtained, if it be remembered that three or four of the lower flange rivets near R or R' may be available for bearing.

The pitch in the stiffners (3" x 3” x -fe" Ls) at C and D may be taken at 6”, with an extra rivet at each end.

The horizontal flange stress for the cantilevers at R and R'

is:

(11,740 x 3) v 2.5 = 14,088 pounds.

The secant of the angle which the inclined flange makes with the horizontal is about 1.05. Hence the inclined flange stress is :

14,088 x 1.05 = 14,800 pounds.

Hence, if in each flange at A there are

14,800 -7- 3,280 = 5 (nearly)

rivets, securing the flanges to the piece of plate shown, ample security will be obtained.

The cantilever flanges possess a large excess of material.

Calculations on the shearing of the rivets between the web and flange have not been made, because the resistance of a rivet to double shear is much in excess of its bearing capacity.

The excess of material in the Ls of the flanges is not as much as it really should be, because the line of horizontal stress along the rivet holes is somewhat below or above the centre of gravity of the flange, and some bending is consequently induced. This bending, however, is not as great as if cover plates had been used, and the neglect of the bending resistance of the web plate is somewhat of an offset. Besides, as has already been stated, in this particular case, the fixed weight of the cantilevers relieves a little of the flange stress of the beam as actually found.

Since the transverse section of the web plate has an area of

0.375 X 30 = 11.25 square inches, transverse shearing at the points of support is more than provided for.

If either a railway or highway floor beam has a variable depth, the operations are in no manner changed. The depth, however, to be used in finding the flange stress at any point must be the vertical depth at that point. The stress thus determined must be multiplied by the secant of the inclination to the horizontal at the same point for the inclined flange.

Art. 68.—Box Beams.

The class of beams known as box beams in engineering practice are represented in Figs. 1 and 2. In Fig. 1 the upper and lower flanges are each composed of a plate whose thickness is f and two Ls whose lengths of legs and thickness are s and a, respectively. If it be assumed that the web plates,

“T

t;

fî

Pî

L_a

< lü

I—

k--—b-

—TT!

sst'

>t «

Ç---20--i

rf

r¥T

i

<S>{

---B

A____

---B

__Jl

t' i

k—

Fig.1

-b

Fig.2

Fig.3

the thickness of each of which is t, offer no resistance to bending, then the effective depth of the box beam will be the vertical distance between the centres of gravity of the flanges. If f is the area of one of these flanges, and d₁ this effective depth, the resisting moment of the beam, as has already been shown, will be :

in which C — K — intensity of stress at the distance ]/₂d, from the neutral surface. If the flange area is desired :

(²)

In other words, the methods and all the operations regarding rivets, etc., as well as the values of C and T, or K, are precisely the same for the box beams as for the other built beams of the preceding Articles.

If each flange is composed of several plates and 4 Ls (as shown by one in dotted lines), then t' is to be taken as the combined thickness of all the plates, while f will be the combined area of the several plates and 4 Ls.

Fig. 2 shows a box beam composed of two channels and one or more plates in each flange. The general observations applied to Fig. 1 apply with equal force to Fig. 2. The bending resistance of the webs of the channels may be neglected if very thin, or when desired in any case, but the exact formula to be given in the next Article is well adapted to this beam.

Art. 69.—Exact Formulae for Built Beams.

The exact formulae for the built sections already given are simply the special forms of the general formula :

KI

d,

(0

The moment of inertia, /, is to be taken about a horizontal line through the centre of gravity of the normal section, i.e., about a line parallel to the side b in the three Figs, of the preceding Article.

The moment of inertia of the cross section shown in Fig. 3, about AB, can be either directly written from an examination of that Fig., or derived from Eq. (2) by simply writing

— for t. It has the value :

(s +

In Fig. i of that Article the moment of inertia of the cross section about AB is:

«►

(s — a) (d — 2a)³ + a{d ~ 2S)³

. . (2)

If there are four Ls in each flange, one only of which is shown in dotted lines:

bt'³

~6 + bt'

(s — a) (d — 2d)³ + a{d — 2s)³

• • (3)

The moment of inertia of the cross section shown in Fig. 2, about AB, is :

bt'³

6 +bt

t {d + t'y

(s + t)d³ — s(d — 2d)³

•    (4)

/=t(j + 0*+*')■) +

r(j “ a) {d — 2af + a{d — 2j‘)³1

- [_-    ■ g    j • •

If the plates are omitted from the flanges in Fig. 3, as in the Article on built beams without cover plates, t' = o, and

t _ (s + y*t)d3 6

(s — a) (d — 2a)³ -f- a(d — 2s)³

. (6)

In all these cases d₁ = J/₂d + or y₂d, according as there are or are not cover plates.

These several values of I and diy substituted in Eq. (1), will give the resisting moments for the various sections. It is an open question, however, what degree of accuracy may be expected to result in the application of these formulae. It is to be remembered that the very best of riveted work does not secure that degree of continuity presupposed by the Eq. (1). It may be stated, however, that Eq. (4) is better applicable to its cross section than the others, for there is perfect continuity between the w*b and a part of the flange.

Art. 70.—Examples of Built Beams Broken by Centre Weight.

Example I.— Wrought Iron Beam.

This beam was tested by Sir William Fairbairn (“ Useful Information for Engineers,” first series), and was composed of four 2-inch |_s riveted to a 7 by y~inch web plate. The distance between supports was 7 feet or 84 inches.

A section of the beam is shown by the section, only, of Fig. I in Art. 67 ; there were no cover plates.

The |_s in the bottom flange were a very little heavier than those in the upper, but the difference was so small that it has been neglected ; or, rather, the small excess has been assumed to supply the loss caused by the rivet holes.

Centre breaking weight = 24,380 + 80 = 24,460 lbs.

1=7 feet = 84 inches. .\ M =--= 513,660.

Wl

Referring to Eq. (6) of Art. 65 :

th

d = 6.5 ins. f — 2.083.    = 0.30.

15-5.

••• *(/+") =

15-5

33,140 lbs. per sq. in.

dx was taken as the depth (nearly) between the roots of the Ls.

The beam gave way in the top, or compression, flange by the twisting of the Ls at a comparatively low compressive intensity. This indicates that the discontinuous riveted connection between the web and flange, although the pitch of the rivets was only 4.5 inches, fails to give such perfect support to the top flange, as a column, as the perfect continuity of the connection in a rolled beam.

The condition of the top flange, as a column, in a built beam, therefore, exercises a very important influence on the ultimate resistance of the beam, and should not be neglected.

It is probable, however, that the high compressive resistance of American wrought iron of the present day would give a much higher value of C under the same circumstances.

When the centre load added to five-eighths the weight of the beam was 8,400 pounds, the centre deflection, or w, was 0.18 inch. Hence the coefficient of elasticity was:

E _ (jv±

48 wl

= 12,321,000 pounds.

/³ must be taken in inches. I was computed by Fairbairn at 46.77.

Example II.—Steel Beam.

The data for this beam were given by Albert F. Hill, C. E., in “Steel in Construction,” Engineers’ Soc. of West. Penn., April, 1880. Each flange was composed of two 2[ X 2[ X steel angles, and one 54- X cover plate. The web was a 12 x tV“⁰-5°C” rolled steel plate. The clear span was 5 feet; pitch of rivet, 4.5 inches; total effective area of section, 8.51 square inches. The rivet holes were drilled T⁹¥ inch in diameter.

Referring to Eq. (6) of Art. 65 :

dl = 12 ins.,

/= 3.13 sq. ins.,

= 0.375 sq. ins.

W — centre weight = 130,000 + 70 = 130,070 pounds.

M — 3 Wl (/ in feet) = 1,951,050.

Hence :

c = h9*2.oi~ ⁼ 46,387 pounds.

The centre load did not break the beam, but caused a deflection of 0.9375 inch, and permanent set of 0.50 inch, with beginning of side deflection.

Very closely approximate, / = 252. Hence, with / in feet and a centre load of 70,000 pounds with the corresponding deflection of 0.25 inch:

E = 3-——7.9,99° * l?A ₌ 5,000,000 pounds.

O.25 X 252

This low value of E is undoubtedly due to the fact that the beam was a built one.

The results of all the tests of built beams given in this chapter show that they are much less stiff than rolled ones of the same section. In fact, in computing deflections with the best designs and best quality of riveted work, E should probably never be taken at more than about half its value for similar rolled sections, or say at 12,000,000 to 15,000,000.

After E is determined, the deflection at once results from the usual formula :

w = 36(^+ %pl)E

El

(0

/³ is here in feet. W is the load at centre, and pi the uniform load (2. e. weight) of or on the beam.

Art. 71.—Loss of Metal at Rivet Holes.

As has been indicated in all examples, the metal punched or drilled from parts of beams in tension should always be deducted from the total tension area in order to obtain the effective area for computation of the ultimate resistance. In estimating this loss the actual diameter, as punched or drilled, should be taken, and not that of the cold rivet before driving, since the latter is always at least one sixteenth inch less than the former.

In the compression portions of the beam, if the work is done in a first-class manner, no deduction need be made.

Art. 72.—Thickness of Web Plate.

The following approximate method of determining the thickness of the web plate in a flanged beam is based upon the principles established in Art. 28.

It was shown in that Article that on two planes which make angles of 90° with each other and 45⁰ with the neutral surface, and whose intersection forms the neutral axis at the section considered, there exists on one a tension and on the other a compression, each of whose intensities is equal to that of the longitudinal and transverse shear at the same point. It was also shown in Art. 17 (see Eq. (38)) that the intensity of these shears is f the mean intensity of shear of the whole section.

No essential error is committed (especially in built beams) if it be assumed that the whole shear is taken up by the web. In the Article just cited it was shown that the intensity of shear at the top and bottom surfaces of the beam is zero, as well as f the mean at the neutral surface. Now, if this shear be assumed uniform in intensity throughout the transverse section of the web, the shear will be made much too large at the top and bottom surfaces, and only two-thirds its proper value at the centre or neutral axis.

In accordance with these assumptions on one hand, and the established principles on the other, the web may be considered as composed of small columns with ends fixed (at the flanges), and with sections rectangular, whose axes lie at 45⁰ with the neutral surface.

The assumption of the uniformity of shear in respect to these elementary columns causes two errors in opposite directions, with the resultant error, in most cases at least, on the side of safety.

In rolled beams, if t' is the mean thickness of a flange, and d the total depth, then the length of these elementary columns may be taken as :

or :

/ = (d — 21") sec 45⁰. I — 1.414^ — 2t ’ ).

(O

In built beams, if d' is the depth from centre to centre of rivet holes, there may be taken :

(2)

If .S' is the total shear at any transverse section, A the area of that section of the web, taking the depth at d — 2t' or d\ and s the mean shear, or : then these elementary columns will be subjected to an intensity of compression equal to s. Hence if t, the thickness of a wrought-iron web, is sufficiently great, there may be taken, by Gordon’s formula :

/²

(3)

3,0001²

Solving this equation for t :

(4)

3,000(/- s)

For the ultimate resistance of wrought-iron rectangular columns, f may be taken at 40,000. If a safety factor of 5 be taken, the value of t becomes :

t — 0.0183/

8,000 — ^

■ • (5)

Eq. (5) is for wrought iron only. The empirical constants for steel yet remain to be determined.

These formulae show that t decreases with the depth of the beam, and that it also varies in the same direction with s. If, therefore, the depth of the beam is constant, Eq. (5) need only be applied at the section where s is the greatest, i. e., at or near the points of support.

If, however, the depth is variable, it may be necessary to apply the formula at a number of sections in order to find the greatest value of t.

Eq. (5) frequently gives much larger values of t than are required. It could be made an accurate and valuable formula if the empirical quantities which enter it were determined by experiments on flanged beams.

The data of Art. 66 give :

d' — 48 ins.,

4 in.,

16

A — d’t — 21 sq. ins.

5 = 86,800 lbs.,

s = 2~⁼ 4>I3° (nearly).

I — d' x 1.414 — 67.9.

Hence :

t — 1.2 inches (nearly).

This value with a safety factor of five is evidently excessive, though it applies only to the portions RA and HR' of Fig. 1 in Art. 66. Yet the result may be accepted as indicating that the web needs support for those portions, and the necessity of the stiffening pieces shown.

The data of Art. 67 give :

d! = 27 inches, t — inch, A — d't = 10.1 sq. ins.

.S' — 26,000 pounds. /. s — — = 2,600 (nearly).

l — d' x 1.414 — 38.2.

Hence :

t — 0.49 inch (nearly).

The thickness taken, therefore, is probably ample, even without the aid of stiffening pieces.

The amount of assistance to be derived from stiffeners cannot be computed with any certainty. They are very essential however, and should be introduced in all large beams.

However small the built beam, or light its load, the web plate should never be less than 0.25 inch in thickness.

Before leaving this subject it may be well to observe that the excessive thickness given by Eq. (5) was, in some measure at least, to be anticipated. It has already been stated that the assumption of uniform compression throughout the length of the elementary column leads to an error on the side of safety. Again, the equal tension at right angles to the greatest compression in the material of the web, as well as the decreasing compression toward the centre of the beam, gives support to the elementary columns throughout their entire lengths. These causes give rise to an excess of safety, in the formula, whose amount can only be determined by experiment. Three-quarters of the thickness given by the formula would probably be ample.

The experiments of the late Baron von Weber showed that a very thin web will give a remarkably large supporting power.

CHAPTER X.

Connections.

Art. 73.—Riveted joints.

ALTHOUGH riveted joints possess certain characteristics under all circumstances, yet those adapted to boiler and similar work differ to some extent from those found in the best riveted trusses. The former must be steam and water tight, while such considerations do not influence the design of the latter, consequently far greater pitch may be found in riveted truss work than in boilers. Again, the peculiar requirements of bridge and roof work frequently demand a greater overlap at joints and different distribution of rivets than would be permissible in boilers.

Kinds of Joints.

Some of the principal kinds of joints are shown in Figs. 1 to 6. Fig. i is a “ lap joint,” single riveted ; Fig. 2 is a “ lap joint,” double riveted ; Fig. 3 is a “ butt joint ” with a single butt strap and single riveted ; while Figs. 4, 5 and 6 are “ butt joints with double butt straps, Fig. 4 being single riveted while the others are double riveted. Fig. 5 shows zigzag riveting and Fig. 6 chain riveting. All these joints are designed to resist tension and to convey stress from one single thickness of plate to another. Two or three other joints peculiar to bridge and roof work will hereafter be shown.

OO O c )

O O O

OO O ( )

ooo

ooo

Fig. 1

Fig. 2

Fig. 3

ooo

TU

TO

ooo

zu

o jo o ]o o io

OOIOO

O oj O O ooioo

W W TO ’C7'

Fig. 4

Fig. 5

Fig. 6

In the cases of bridges and roofs these “ butt straps ” are usually called “ cover plates.” '

Distribution of Stress in Riveted Joints.

A very little consideration of the question will show that only an approximate determination of the distribution of stress in a riveted joint can be reached.

In order that rivets, butt straps or cover plates, and different portions of the same main plate may take their proper portions of stress, an absolutely accurate adjustment of these different parts must be attained ; but all shop work must necessarily be more or less imperfect, and the requisite condition can never be maintained during and after construction. Hence the amount of stress carried by each rivet, or each cover plate, and hence each portion of the main plate at the joint, cannot be found.

In the cases of lap joints with three or more rows of rivets (frequently found in truss work), or in similar work when two rows of rivets join a small plate to a much larger one, the outside rows, or row, in consequence of the stretching of the material at the joint, must take far more than their portion of stress, if, indeed, they do not carry nearly all. The same condition of things will exist in butt joints if two or more rows are found, under similar circumstances, on the same side of the joint.

If a strip of plate in which the ratio of width over thickness is very considerable, be so gripped in a testing machine that the applied stress be approximately uniformly distributed over its ends, and if it be tested to breaking, it will be found, if the broken pieces be joined at the place of breaking, that the central portions of the fracture are widely separated, while the edges are in contact. This is due to the cause explained in Art. 32, “Coefficient of Elasticity.’’ Now if a hole or holes be made in or near the centre of the specimen, a portion of the material in the front and rear of these holes will be relieved from stress, and the total stress in the central section of the specimen will be more nearly uniformly distributed in the remaining material. And again, these holes will “ neck ” the specimen down to a short one. The influences noticed in Art. 32, “ Ultimate Resistance and Elastic Limit," will thus be called into action. For both these reasons the existence of the hole, or holes, in itself, will increase the intensity of the ultimate resistance of the plate.

On the other hand, the effect of the punch, if the hole is punched, as will presently be shown, is to decrease the resistance of the metal about the hole. If the hole is in a joint, also, the bearing pressure between the rivet and plate is very great, and as this pressure must be carried as tension to the material adjacent to the rivet hole, and through that in its immediate vicinity, the latter (i.e., the material at the extremities of diam-ters parallel to the joint) will receive much greater tension than that in the central portion between the holes.

These last two influences tend to reduce the mean intensity of ultimate resistance of the material of the joint, and sometimes more than counterbalance the increase caused by the existence of the holes simply as such. In other cases the resultant effect can only be determined by experiment.

In Figs. I and 2 it will be observed that the stresses in the plates of a lap joint act excentrically, and, let it first be assumed, with a lever arm equal to half the sum of the thickness of the two plates. If, however, a specimen joint is put in a testing machine, the resultant stress may be made to pass through the centre of the joint, thus making the lever arm for each plate about half its thickness.

If, therefore, t is the thickness of one plate and /' that of the other, while T and V are the mean intensities of tension in the plates, p the pitch of the rivets and d the diameter ; in the first case each plate will be subjected to the bending moment:

rt{p - d) (t±£) = Tf(p - d) (iiÛ) . (i)

And in the second :

M= TV

HT

10

to I

tO

If K is the greatest intensity of tensile bending stress, then :

tM

2/

or,

t'M

2/

.(3)

be :

The greatest intensity of tension in the plate will therefore

T + K, or, T + K .    .    .

• • (4)

The moment of inertia / will have the value :

12

» or,

(/ - d)t'*

12

If each plate has the same thickness, t — t' and T = T' ;

hence :

By Eq. (1) By Eq. (2)

K = 6T

K= 3 T

(5)

(6)

These values of K are very large and appear excessive. It is to be remembered, however, that the formula used Eq. (3) is strictly applicable only within the elastic limit.

There is no reason to doubt, therefore, that within that limit the greatest intensity of tension in the plates of the joint may reach from Ą.T io jT.

From these considerations it is to be expected that the true elastic limit of the joint, as a whole, would be very low.

The preceding investigations in the flexure of the joint are based upon the virtual assumption that the plates remain straight after the application of external stress. In reality such a condition of things does not obtain. Even below the elastic limit the plates begin to take positions which are shown in an exaggerated manner in Fig. 7. On account of the

p[

]>p

Fig. 7

bending, the material at the points AA stretches much more than that at the points BB (with low values of T that at the latter points may be in compression), so that the centre lines of the plates P and

P' are brought more nearly into coincidence, thus lessening the bending moment to which the joint is subjected. After the elastic limit of the material at AA is passed, a considerable increase of strain or stretch takes place at those points for the same increment of stress. Two important results follow this increase of strain between the elastic limit and failure : the joint becomes very markedly distorted, so that the plates P and P' become much more nearly in line, and the stress becomes much more nearly uniformly distributed in the sections AB,

AB. This is equivalent to saying that the joint is subject to a greatly decreased bending moment.

If the plates are thin, the excess of strain at AA over that at BB, requisite to bring the plates PP' essentially into line, may easily be within the stretching capacity of the material. If, however, the plates are thick, that condition will not hold, and the material at A A will begin to fail before PP' are nearly in line. Hence, the mean intensity of stress in a thick plate, other things being equal at the instant of rupture, will be considerably less than that in a thin one. It might thus happen that a lap joint with thin plates would be found stronger, even, than one with thicker plates.

Reference will hereafter be made to experiments which verify these conclusions.

It will now be well to turn back a moment to the consideration of Eqs. (5) and (6). Those equations show the effect of bending to be dependent on T only, and entirely hidependent of the thickness of the plates, which apparently contradicts the conclusion just drawn. But, as has already been intimated, those equations involve the virtual assumption that the plates remain continually straight, and do not contemplate the altered conditions of the joint which exist just at and before rupture. Again, they presuppose no passage of the elastic limit. There is thus no real contradiction.

Although a single riveted lap joint only has been treated, precisely the same considerations apply to a double riveted lap joint, a butt joint with single butt strap or cover plate, and all butt straps or cover plates of butt joints. The main plates of butt joints with double cover plates are not subjected to flexure.

The rivets of all riveted joints are subjected to heavy flexure, the greatest of which usually occurs in single lap and butt joints like Figs. 1 and 3. An approximate value of the bending moment, in any case, may be found as follows :

Let n be the number of rows of rivets in one plate. In Figs. I, 3, 4, n is i ; and 2 in Figs. 2, 5 and 6. Then if t and t' are the thickness of the two plates or of one plate and one cover, T and T' the mean intensities of tension in the same pieces, and if M be taken from Eq. (1), the approximate bending moment will be :

M KAd

; (From Art. 63) ;    .    .

•    - (7)

in which A is the area of the cross section of one rivet, K the greatest intensity of tension or compression due to bending, and d the rivet diameter, as before. From Eqs. (7) and (1):

K= 4Tt(p - d)

(¹+0

nAd

• • (8)

lit = t':

K= 8 77

AP~d)

11 Ad

•    - (9)

This equation is approximate because it is virtually assumed that the pressure on the rivet is uniformly distributed along its axis.¹ This is a considerable deviation from the truth, particularly as failure is approached. The true bending moment is much less than that given by Eq. (7) after the rivet has deflected a little.

When the joint takes the position shown in Fig. 7, it is clear that the rivet is also subject to some direct tension.

There is a very high intensity of pressure between the shaft of the rivet and the wall of the hole. This intensity is not uniform over the surface of contact, but has its greatest value at, or in the vicinity of, the extremities of that diameter lying in the direction of the stress exerted in the plate. At and near failure this intensity may be equal to the crushing resistance of the material over a considerable portion of the surface of contact.

The intricate character of the conditions involved renders it quite impossible to determine the law of the distribution of this pressure. The bending of the rivets under stress tends to a concentration of the pressure near the surface of contact of the joined plates, while the unavoidably varying “fit” of the rivet in its hole, even in the best of work, throws the pressure towards the front portion of the surface of the rivet shaft. The intensity thus varies both along the axis and around the circumference of the rivet.

If any arbitrary law is assumed, the greatest intensity of pressure is easily determined. Such laws, however, are mere hypotheses and possess no real value. All that can be done is to determine, by experiment, the mean safe working intensity on the diametral plane of the rivet which is equivalent to a fluid pressure of the same intensity against its shaft.

Thus, if f is this mean (empirically determined) intensity, d the diameter of the rivet, and t the thickness of the plate, the total pressure carried by one rivet pressing against one plate is:

(10)

R = fdt

There yet remains to be considered the condition of that portion of the plate on which the pressure R = fdt is applied, and which is situated immediately in front of the rivet.

This portion of the plate is really in the condition of a beam fixed at each end, with a span equal to the diameter of the rivet. The beam, however, is not a straight one. At each end of the diameter the direct bending stress will be tension ; and, on account of the position of the material, its direction will be approximately, at least, that of the proper tension of the plate. At those points, therefore, the proper and bending tension will act to some extent together, and the metal will usually be more highly stressed than anywhere else. This accounts for the usual manner of tensile fracture of a joint, in which the metal begins to tear on each side of the rivets, the metal between (generally in a diagonal direction in zigzag riveting) being the last to give way.

In the interior of the joint it is quite impossible to determine the value of this tensile bending stress on each side of the rivet. On the exterior of the joint, however, an approximate result may be reached ; and hence, the depth h, Fig. 2, from the centre of the outside row of rivets to the edge of the

plate. The depth of the beam will be taken as i^h —

, and

the pressure or load will be considered concentrated at the middle of the diameter or span. If t is thickness of the plate, p the pitch of the rivets and T the mean intensity of tension between the rivets, the load on the beam will be (p — d) 77, and the moment of inertia of the cross section will be :

CO

From what has been shown in the chapter on bending, the modulus of rupture in the present case may be safely

taken at — T.

In Art. 24, the moment at the centre and end of a span fixed at each end and loaded in the centre was shown to be equal to one-eighth the load into the span.

Hence, by the usual formulae:

d\*

2 Kf

- d)Tt =

h--

¹

In accordance with this assumption, strictly speaking, \t (thickness of main plate) should be taken instead of t in the sum (/ + /') in the above formulae foi bending, when applied to the double butt joints, Figs. 5 and 6.

k — 0.71 V(J> — d)d + 0.5^. . . . (11)

Reviewing the results of this section, it may be concluded that the bending of the plates about axes parallel to them, or normal to them in the interior of the joint, and the bending of the rivets, as well as the law of the distribution of pressure against them, cannot be expressed by formula with any useful degree of accuracy; but that such influences must be recognized in the empirical determination of the shearing and tearing resistances of the joint and the mean intensity of pressure against the diametral plane of the rivet.

Effect of Punching.

The effect of punching wrought-iron plates has been found to be injurious. The tensile resistance of the remaining material will be considerably less than that of the plate before punching. Yet the injurious effect of the punch does not extend far into the plate. If the punched hole is reamed, so that the diameter is increased an eighth of an inch, the remaining plate will usually give the normal resistance per unit of section, or essentially so.

It has been found by experiment that effect of the punch is less injurious as the die hole is increased in diameter, although there is probably a limit to the application of this principle.

The diameter of the die hole is usually from to } larger than that of the punch. This excess should depend upon the thickness (t) of the plate, and it is sometimes taken as 0.2t.

Numerous foreign experiments (chiefly English) by Barnaby, Stoney, Fletcher, etc., show that the loss of tensile resistance due to punching wrought-iron plates runs usually from 10 to 15, and may vary from 5 to 33 per cent, of the original resistance.

The loss of resistance due to punching and its remedy, in steel plates, have already been treated in Art. 34.

Wrought-iron Lap Joints, and Butt Joints with Single Butt

Strap.

A butt joint with single butt strap, similar to that shown in Fig. 3, is really composed of two lap joints in contact ; since each half of the butt strap or cover plate with its underlying main plate forms a lap joint. It is unnecessary therefore to give it separate treatment.

From these considerations it is clear that the thickness of the butt strap or cover plate should be the same as that of the main plate.

Let t — thickness of plates.

“ d = diameter of rivets.

“ p — pitch of rivets (i. e., distance between centres in the same row).

“ T = mean intensity of tension in plates between rivets.

“ T — mean intensity of tension in main plates.

“ f = mean intensity of pressure on diametral plane of rivet.

“ S = mean intensity of shear in rivets.

“ n — number of rivets in one main plate.

“ q = number of rows in one main plate.

“ h — amount of extreme lap as shown in Fig. 2.

If all the dimensions are in inches, then T, T',/"and S are in pounds per square inch.

The starting point in the design of a joint is the thickness t of the plate. The rivet diameter is then expressed in terms of t, and the pitch in terms of the diameter.

The thickness t of boiler plate depends upon the internal pressure, and is to be determined in accordance with the principles laid down in Art. 9, after having made allowance for the metal punched out at the holes and the deterioration or other effect caused by the punch.

In truss work the thickness depends upon the amount of stress to be carried, and the same allowances are to be made for punching and deterioration.

The relation existing between T and T' is shown by the following equations :

t{p-d)T= tpT :

L - / .

' T p-d’ or,

(12)

T _ p -_d

In order that the joint may be equally strong in reference to all methods of failure, the following series of equalities must hold :

~ tpT' — — t (p — d) T = nfdt = o.yS$Ąnd²S.

tpT' = t(p — d) T = qfdt — 0.7854qd²S. . (13)

It is probably impossible to cause these equalities to exist in any actual joint, but none of the intensities T', T, for S should exceed a safe working value.

In ordinary American boiler practice d varies from 1.5^ to 21\ the latter for thin plates and the former for thicker ones, the extreme limits being about -f inch and ij- inches.

The following are some rules given by the best foreign authorities for wrought iron :

Browne .

Fairbairn

Fairbairn

d = 2t (or 1.25/ with double covers). .. . (14) d — 2t for plates less than | in d — i.$t for plates greater than f in. .. . (16)

(15)

Lemaitre

d — 1.51 + 0.16

(17)

(,8)

Antoine

Pohlig

d — 2t for boiler riveting

09)

Pohlig

d = 32? for extra strength

(20)

(21)

Redtenbacher .. d — i.$t to 21

Unwin

Unwin

d = 0.75/ + t⁵¥ to I / + I d — 1.2 V t .............

(22)

(23)

As the results of some of his experiments on ^-inch steel plate joints, Prof. A. B. W. Kennedy gives in “ Engineering,” loth June, 1881, the following rules for rivet diameter :

.    .    (24)

Single riveted lap joint.... d = 2.25/ Double riveted lap joint... d — 2.21/

These rules are for mild steel plates and for greatest strength, but are not to be applied to plates over jj in. thick; as the diameters would then become excessive. He therefore

DIAMETER OF RIVETS.

OF

PLATE.

Lloyds’

Rules.

Liverpool

Rules.

English

Dockyard

Rules.

French Wilson’s Hovrez’s

Veritas.

Rules.

Rules.

Hall’s

Rules.

In.

î.

■<

i-

14

W W M M

oci—xJh-oc,—

W M ł—I

ol.-x;-

i I I I I I ff* P*

concluded that thicker plates than in. would give proportionally less resistance.

It has been found by experiment that there is a very decided interdependence existing between the values of T and f in cases of failure by tearing. This is probably due far more to the bending action of the rivet, which was considered in detail in one of the preceding sections, than to the direct influence of the pressure between the rivet and its hole.

Table I. contains values of T and f at the instant of failure, which were tabulated by Prof. Unwin in “ Engineering ” for Feb. 20th, 1880. All the plate was English material. The results show very clearly the increase of T with the decrease of f. They are, however, somewhat discordant. The punched single riveted lap joints of Mr. Stoney’s experiments show an apparently abnormally low value of the tenacity T for a given intensity of compression f ; but the drilled holes show less disagreement.

Fairbairn

Kirkaldy

Browne

Stoney

Lap, single riveted...................

Lap, single riveted ...................

Lap, double riveted...................

Lap, double riveted...................

Lap, double riveted...................

Lap, double riveted...................

Butt, double riveted, one cover.......

Butt, single riveted, two covers.......

Butt, single riveted, two covers.......

Lap, single riveted..

Lap, double riveted.

Butt, double riveted Butt, double riveted Butt, double riveted Butt, double riveted Lap, single riveted..

Lap, single riveted..

Lap, single riveted.... ......

Butt, single riveted ..........

Butt, single riveted...........

Butt, single riveted...........

Lap, single riveted, punched..

Lap, single riveted, punched..

Lap, single riveted, punched..

Lap, single riveted, punched..

Lap, single riveted, punched..

Lap, single riveted, punched..

Lap, single riveted, punched..

Lap, single riveted, drilled....

Lap, single riveted, drilled....

Lap, single riveted, drilled....

Lap, single riveted, drilled....

Lap, single riveted, drilled....

Lap, single riveted, drilled....

Lap, single riveted, drilled....

83,776

66,860

78,290

76,830

58.460 5i,3°° 58,020

94.210 65,180 58,580 36,740

74,700

71,750

63,170

62,6lO

93,640

86,950

84,980

101,150

94,240

92,840

66.210 55,660

49.460 47,260 43,680 42,110

38,770

64,400

59,020

54,650

48,370

47,520

46,140

45.92O

39,650

44,580

52,190

48.830

58,460

55-330

53,98o

53,540

60,700

47,260

57,340

43,900

45,570

45,020

39,200

29,120

27,100

26,300

29,120

28,880

31,9*0

32,930

37,630

35,840

45,920

44,350

40,770

46,820

34,940

41,440

36,740

47,490

48,380

48,270

EXPERIMENTER.

FORM OF JOINT.

TABLE I.

Wrought Iron.

f, IN LBS. PER

SQUARE INCH.

iT, IN LBS. PER

SQUARE INCH.

Reviewing all the results, it would seem that , the following values may safely be given single riveted lap joints with punched holes in first-class work :

f — 55,000 to 60,000 f - 55,000 to 50,000

T — 45,000 to 40,000. T — 45,000 to 50,000.

The following values of f> T and S, at the instant of failure,

are from the experiments (English) of Messrs. Greig and Eyth and the Master Mechanics’ Association.

/, IN LBS. PER SQ. INCH.

T, IN LBS. PER SQ. INCH.

IN LBS PER SQ. INCH.

Single riveted lap joints... -1

"64,400. . . . .

59,49°.....

59.96°.....

62,400.....

_66,280.....

46,820.

43,650.

43,970.

45,76o.

47,690.

40,990] 41,300 I

41,680 j-. .(A)

43,340

38,77oJ

All the holes in these joints were drilled, consequently, as will hereafter be shown, S is a little low. Further, all the joints broke by simultaneous shearing of the rivets and tearing of the plates : they may therefore be considered well designed.

Now, if f— T= 50,000, which is experimentally showm to be correct in single riveted lap joints, for which q — 1, the second and third members of Eq. (13) give :

p — 2d.

But this pitch would scarcely give sufficient room for heading the rivets. It has just been seen that the results in group (A) belong to well proportioned joints. An examination of those results will show that /varies from 1.337" to 1.4 7] nearly; which is not an essential disagreement with the results of Table I. Hence, putting these values in Eq. (13) :

(²5)

p — 2.33d to 2.4^

This agrees with good ordinary practice in boiler making, which makes :

p = 2.3d to 2.75/ nearly.

The preceding results are for single riveted lap joints in wrought iron.

TABLE II.

Wrought Iron Double Riveted Lap Joints.

POUNDS PER SQUARE INCH

EXPERIMENTER OR AUTHORITY.

MODE

HOLES

FOR

OF RIVETING.

MADE BY

/■

T.

Sir Wra. Fairbairn

Hand.

Punch.

David Kirkaldy...............

Easton and Anderson........

Machine.

Greig and Eyth, R. V. J. Knight

Drill.

Punch.

68,580 70,090 60,860 69,490 58,350 5’,030 36,710 56,380 53,4°° 59,97° 57,°3° 34,090 22,020

21,540

21,500

22,220

30,330

31,230

5M5°

SS-^o

45.670

52,060

58,350

54,680

57,270

36,470

34,^70

38,770

45,790

49,060

24,440

23,630

28,650

29,610

27,280

28,740

In the second preceding section considerations were adduced which show that for a given value of the mean intensity of compression between the rivet and its hole, in a double riveted lap joint, an increased value of T over that for a single riveted lap joint should be expected. So far as comparison can be made, Tables I. and II. verify this conclusion, although

the increase is not very great. This arises from Ap the fact that the increased length of a double joint requires less bending at A, A, Fig. 8, than a single one to bring the plates P and P' nearly into line.

The tables show that for thin plates f is equal to T, at the instant of rupture, for an intensity not far from 55,000 pounds

P[

c:

O'

-A

Fig. 3

per square inch. This will reduce somewhat the allowable ratio between /'and T.

A careful examination of the results given in the tables seems to make it perfectly safe to take f from 1.1 T to 1.257". These values in the second and third members of Eq. (13) give (remembering that q is here equal to 2) for double riveted lap joints :

Or, say :

p — 3.2 to 3.57 p — 3.25 to 4.0/

. .    (26)

The smaller values of p belong to thick plates and the larger values to thin ones, both because the increased thickness brings a greater proportional load on the rivet and because the lever arm of the bending moment is greater.

It should be stated that in some apparently good boiler practice p is sometimes taken as high even as 5d. The ease with which a double riveted lap joint is made steam tight may tempt a decrease in expense of riveting. It is probable that the rivets of joints in which the pitch exceeds about 4d carry an excessive compression and a corresponding liability to weakness.

In Table II. the experiments of Mr. Knight were made on plates one inch thick, which are excessively heavy, and the values of /"and T are remarkably small. It has already been demonstrated that great thickness of plates would produce results of such a character, although the sufficiency of such an explanation has been doubted. There seems little reason to doubt, however, that the cause just cited, together with the normal decrease of resistance with an increase of thickness, is a complete explanation.

It is to be observed that in the preceding deduced values of f and T, the bending of the plates about axes both parallel and normal to their surfaces, have been recognized and provided for.

If the accuracy of the experiments cited be assumed, and they are the most reliable and valuable that have ever been made, there may be taken :

For i-inch plates, T — 30,000 to 35,000 lbs. per sq. in.

For i^-inch plates, T = 50,000 to 55,000 lbs. per sq. in.

And for intermediate plates proportional values.

For single riveted lap joints, f — 1.33 to 1.4 T. For double riveted lap joints, f = 1.1 to 1.25 T.

As y and 7" have been found to be dependent on the pecu-iar circumstances attending the use of the material in the joint, so, in the same general manner, the determination of the ultimate shearing resistance of the rivets must involve a similar recognition of environment.

It has been found by experiment, as might have been anticipated, that rivets in drilled holes offer less resistance to shearing than those in punched holes. This arises from the fact that the edges of drilled holes are much sharper than those formed by a punch.

Table III. gives the mean results of a large number of experiments by the authorities named. It has been condensed, and the results converted to pounds per square inch, from a similar one given by Prof. Unwin, in “ Engineering ” for 26th March, 1880.

These results are for single riveted lap joints, and therefore for single shear. They are only a very little larger than the values determined by Chief Engineer Schock for single shear, as the apparatus of the latter was essentially equivalent to a drilled hole.

For plates 0.25 inch to 0.375 inch thick, there may be taken, as is usually done, 5 = 0.8 T. It has been seen (Table II.) that a plate an inch thick can be expected, in lap joints, to

TABLE III.

Shearing of Wrought Iron Rivets.

EXPERIMENTER OR AUTHORITY.

KIND OF HOLE.

S IN POUNDS

PER SQ. IN.

RESISTANCE (TENSILE) OF PLATE OVER S.

Fairbairn.................

Stoney.....................

Stoney ....................

Fairbairn..................

Fairbairn................. .

Master Mechanics’ Association Greig & Eyth...............

Punched.

Punched.

Drilled.

Punched.

Drilled.

Drilled.

Drilled.

50,180

42,200

40,920

45,820

43,610

46,590

41,280

0.783 0.910 1.061

1.071

Mean result Mean result

0.846

1.066

Punched.

Drilled.

46,030

43,100

give T not much over 35,000, and as the thickness does not seem to appreciably affect S, for this inch plate there may be taken 5 = -f T. The ratio of f over T has been seen to vary from 1.33 to 1.4T. Let a mean value of 1.36 for this last ratio be inserted in the third member of Eq. (13) ; then, by inserting the other values just found in the fourth member of the same equation, there will result for single riveted lap joints :

• • (²7)

For thin plates, d = 2.1/ For thick plates, d = 1.5/

For double riveted lap joints these results would be diminished only slightly. Hence Eq. (27) may be taken as applicable to both single and double riveted lap joints in wrought iron.

It will be observed that Eq. (27) is included within the limits of the Eqs. (i4)-(23).

A great number of results by the experimenters already cited in this chapter show that the total resistance of a single riveted lap joint, as a whole, for plates not over 0.5 inch thick, may vary from 44 to 58 per cent, of the solid plate in its normal condition, and that the mean value may be taken from 50 to 52 per cent.

In a double riveted lap joint this mean may be taken at 60 per cent, of the resistance of the original plate, for moderate thicknesses. In Mr. Knight’s experiments with inch plates (double riveted), the resistance of the joint, as a whole, ranged from 33 to 36 per cent, of that öf the plate.

It is clear, from the preceding investigations, that this “efficiency” of the joint must decrease as the thickness of the plate increases. In fact, Mr. Bertram found, in i860, that some joints in ^4-inch plates were stronger than those in either tV or /^-inch plates. Although such results do not involve impossibilities, they are certainly remarkable, and have not since been obtained.

As has before been observed, all the preceding results apply directly to butt joints, in wrought iron, with single butt strap'or cover plate.

The width of overlap (h) from the centre of the outside line of rivets to the edge of the plate (see Fig. 2) may now be determined in terms of d, by the aid of Eq. (u). Since the load on the rivet is represented by (p — d)Tt, p must be taken in terms of d for a single riveted joint, in which p — 2lf₎d to 2'Sjd. As a margin of safety, hnd as it will, at the same time, simplify the resulting expression, let p — 3d.

Eq. (11) then gives:

(28)

h = i.$d.

Experience has shown that this rule gives ample strength, and is about right for caulking, in boiler joints.

The distance between the rows of riveting is not susceptible of accurate expression by formulae, although the considerations involved in the establishment of Eq. (11) would lead to an ap-

proximate value. It is evident, however, that this distance should never be as small as h. Apparently, in more than double riveted joints, this distance should increase as the centre line of the joint is receded from, in consequence of the bending action of the rivet. There are other reasons, however, besides that of inconvenience, why such a practice is not advisable.

In chain riveting the distance between the centre lines of the rows of rivets may be taken equal to the pitch in a single riveted joint, or, as a mean, at 2.5 the diameter of a rivet.

In zigzag riveting (Fig. 5) this distance may be taken at three-quarters its value for chain riveting.

Steel Lap f oints and Butt joints with One Cover.

The general phenomena attending the tests of steel joints are precisely the same in kind with those observed in connection with riveted iron plates ; they do not, therefore, need particular consideration in this section.

Table IV. contains results communicated to the “Committee of the Institution of Mechanical Engineers ” by Messrs. Parker and Sharp (“ Engineering,” 16th April, 1880). The joints failed by tearing, and gave the values of T shown in the table. The intensity of pressure,/⁻, existed at rupture.

The following values of T and f under precisely the same circumstances, i.e., failure, were found by Prof. A. B. W. Kennedy (“ Engineering,” 20th May and 10th June, 1881,) for single riveted lap joints.

T.

THICKNESS OF PLATE.

|-inch

TABLE IV.

Steel Joints.

POUNDS PER SQ. IN. FOR

Joint.

hole.

X    b.

H    O

T.

/■

Treble riveted (chain)

Drilled.

Punched.

Drilled.

Quadruple riveted (zigzag).......

Double riveted butt (one cover)... Double riveted butt (one cover) ...

Drilled.

Punched.

.2.5.2.2.2.2* .2.2* .2.2.2.2* .2' .2

cîico scjx H~ ■-+>« '-h- H"*    H®* '“T f M o-

79,220

52,280

50,330

73,36o

70,040

80,890

78,400

66,940

67,520

80,380

75,78o

6S,250

65,950

56,760

87,920

97,730

6o,OIO

39'38o

37,740

57,700

55,o8o

54,470

52,790

52,220

53,330

73,830

49,500

47,58o

35,4C>o

42,360

76,200

83,840

The holes in these plates were all drilled, and each result is a mean of two tests.

These experiments do not present a sufficient range to show clearly the relation existing at failure between T and f It is clear, however, that no recorded intensity f has been large enough to decrease T to any appreciable amount. In some of Prof. Kennedy’s tests, in which failure took place by shearing, f was not far from 1.2 T (with T = 65,000 to 75,000), and it would appear from his experiments that such a ratio may properly be taken for thin plates in single riveted joints. At the same time, with the mild steel used by Prof. Kennedy, T may he taken at 70,000 poimds for plates f to Jq inch thick.

Putting 1.2 T for f in the third member of Eq. (13):

Or, say,

p — 2.2d p — 2.25 d

(²9)

for single riveted lap joints. It will probably be best to allow this pitch to stand for thick plates also, although experiments to verify such a conclusion are yet lacking. For very thick plates in single riveting, however, T should not be taken over 50,000 to 55,000 pounds at the highest.

Experiments on double riveted lap joints by Martell, Kir-kaldy and Easton and Anderson, show that it will be essentially correct, and certainly safe, to take f and T as in the single riveted joints. With q equal to 2, Eq. (13) will then give for double riveted steel lap joints :

Or, say,

P = 34d

P = 3-5^

• (30)

Although relating to treble and quadruple riveted joints, Table IV. shows in a marked manner the decrease of T with the increase of thickness, and verifies the conclusion drawn in the preceding section in regard to that phenomenon.

The results cited by Prof. Unwin, in the report so frequently referred to heretofore, indicate that for treble riveted joints f may be taken essentially equal to T for thin plates, and 0.97" for thick ones. Hence, using Eq. (13) as before :

TREBLE RIVETING.

• • (31)

Thin plates (0.25 and 0.375 in.), P = Ąd Thick plates (0.875 and 1.00 in.), / = 3.?d

Some experiments of Mr. Kirkaldy on joints with ^-inch Siemens steel plates quadruple riveted, seem to show that the pitch should be about the same as in treble riveted. This is undoubtedly due to the fact that with such a great number of rivets it becomes impossible to obtain even an approximately proper distribution of load among them.

In treble and quadruple riveting the tests cited show that T may be taken at 70,000 to 75,000 for thin plates, and 55,000 to 60,000 for thick ones.

In all the preceding investigations it is supposed that the holes are drilled, or that the plates are subsequently annealed if punched.

In nearly all the experiments cited by Prof. Unwin, the value of T, as found in the actual joint, exceeded the ultimate resistance of the original plate ; a result which finds its explanation in the drilling of the holes and the “ shortening ” effect produced by their presence, aided by their equalizing effect.

Table V. gives the ultimate shearing resistance of steel rivets as determined by Sharp, Martell, Kirkaldy and Greig and Eyth. A very considerable reduction is noticed with the increase in plate thickness, due probably to increased bending and size of rivet.

Prof. Kennedy found the following values in single riveted lap joints :

RIVET DXAM.

0.75 in.. 1.00 “

1.00 “ . . O.75 “ . . 0.75 “ . . 0.75 “ . . 0.75 “ . . 0.75 “ ..

s.

Each result is a mean of two or three tests.

In Mr. Kirkaldy’s four tests of J^-inch treble and quadruple riveted lap joints, with i^-inch rivets, the ultimate shearing resistance 5 varied from 41,110 to 46,260 lbs. per sq. in.

TABLE V.

Shearing of Steel Rivets.

THICKNESS

S IN POUNDS PER

JOINT.

MEAN OF.

OF PLATE. I

SQ. IN.

57.570

53>69⁰

53-310

50,650

60,930

56,220

57,120

53.540

53,9⁸o

43,56o

46,140

43,010

: : S'

Single riveted........

Double riveted (chain)

Treble

“ (zigzag)... Quadruple riveted (zigzag)

Four experiments by Mr. Kirkaldy on single riveted lap joints, during 1881, gave S varying from 52,106 to 54,042 lbs. per sq. in.

Prof. Kennedy’s results give nearly:

Tables IV. and V., plates not over y₂ in. thick :

5 = 0.87.

Mr. Kirkaldy’s for treble and quadruple riveting :

5 = 0.7T.

For ordinary plates therefore in single and double riveting, for which f = 1.2 T and S as a mean = 0.75 T, the third and fourth members of Eq. (13) give :

For thick plates in treble and quadruple riveting, for which f— 0.9 T, and 5 = 0.7 T\

•    •    (33)

d — i .6t (nearly).

The rivet pitch, therefore, for steel plates, may be said to vary from 21 for thin plates to i.6t for thick ones, with a maximum diameter of 1^ to ix³¥ inches.

Prof. Kennedy’s best designed single riveted lap joints gave from 55 to 64 per cent, the strength of the solid plates.

Well designed double riveted lap joints should give from 65 to 75 per cent, the resistance of the solid plate.

Equally well constructed treble and quadruple riveted joints should have an efficiency of 70 to 80 per cent, of the solid plate.

It is therefore seen that there is little economy in more than double riveting ordinary joints.

The distance between the centre lines of the rows of rivets, and the distance from the edge of the lap to the outside centre line of holes, may be taken the same as for wrought-iron joints, according to the rules given in the last part of the preceding section.

All rivets have heretofore been supposed to be steel. In the case of steel plates and iron rivets, there may be taken, at least approximately, 0.9.S for S, and f — T for thin plates, or 0.8 T for very thick ones. These values are to be inserted in the preceding formulae for all steel joints, and the results for p and d taken.

Wrought-iron Butt Joints with Double Covers.

Butt joints with double butt straps or covers differ in two respects, and advantageously, from lap joints and butt joints with a single cover ; i. e., in the former the rivets are in double shear and the main plates are subjected to no bending. The

cover plates, however, are subjected to greater flexure than the plates of a lap joint, for there is no opportunity to decrease the leverage by stretching. As the covers form only a small portion of the total material, these, with economy, may be made sufficiently thick to resist this tendency to failure.

Let /' — thickness of each cover plate.

And let the remaining notation be the same as in the preceding section. The intensity of compression between the walls of the holes in the cover plates and the rivets, and the tension in the former, will be ignored on account of the excess in thickness of the two cover plates combined over that of the main plate. This excess in thickness is required on account of the bending in the covers noticed above.

The thickness of each cover should be from to the thickness of the main plates, or t' = 0.75/ to 0.875/.

The combined thickness of the covers will thus be from 1.50 to 1.75 that of the main plates.

The four principal methods of rupture in the main plate will then lead to the following equations, corresponding to Eq. (13):

nfdt — \.^Qd,nd²S.

-tpT — — t(p — d) T =

q    q

.*. tpT' — t {p — d) T — qfdt = 1.5708qd³S . (34)

The experiments of Kirkaldy, Fairbairn, Greig and Eyth and Knight, show that in well proportioned joints f — 1.25 to 1.5 T (the higher values belonging to the thinner plates), with a mean value of about \.\T. As no bending exists in the main plates, this value holds in single or double riveting.

Hence for single riveting, the second and third members of

Eq. (34) give

In double riveting, for which q — 2 :

•    •    (36)

p = 3.8d ; or, say, p = 4.0^ .    .

On account of the essential impossibility of even an approximately proper distribution of the load among the rivets, and the consequent liability of failure of the joint in detail, in treble riveting the pitch should probably not exceed 4.5*7, nor 5d in quadruple riveting.

There may be taken, according to the experiments just

cited :

For punched inch plates :

T — 40,000 lbs. per square inch.

For drilled *^-inch plates :

T = 55,000 lbs. per square inch.

Other thicknesses and conditions give approximately proportional values, allowing about 10 per cent, for the deterioration of the punch ; i.e., T, for a ^ punched plate, may be taken at 45,000 pounds.

It has already been observed that the value of 5 may be taken at 0.8 7" for lap joints, but the few experiments that have been made on shearing in butt joints with double covers, show that the ratio must be taken somewhat less, in consequence probably of the double shearing which takes place.

Hence, let 5 be taken at 0.757".    '

Using the third and fourth members of Eq. (34), therefore, and making S = 0.75 T :

For thin plates in which f— 1.5 T :

For thick plates in which f — 1.25 T :

(38)

d — 1.1/

It is hardly worth while, however, to make any rivet less than ^4 inch in diameter. Hence there may be taken the limits :

For i^-inch plate ; d — 0.375 inch. For i-inch plate ; d = 1.125 inch.

These results are verified by good boiler practice.

The distance from the centre line of outside row of rivets to the edge of the cover plate, or from the edge of the main plate to the centre line of the first row of rivets in the same, may be taken at |\d as in lap joints, since the calculation is precisely the same. This rule frequently gives a considerable margin of safety over that of any other portion of the joint.

The distance between the centre lines of the rows of rivets may be taken at 2.5 to 3.0d for chain riveting, and ^ that distance for zigzag riveting.

Steel Butt Joints with Double Cover Plates.

For the same reasons stated in the preceding section, considerations touching the stress in the cover plates will be omitted. And also, for the reasons there given, these cover plates should each possess from ^ to the thickness of the main plate ; or :

t' = 0.75 to 0.875/.

Table VI. gives the results of a large number of tests in which the joint failed by the tearing of the plates. The intensities of tension and compression, T and f₎ existed at failure.

TABLE VI.

Double Riveted Butt Joints.

POUNDS PER SQUARE INCH

EXPERIMENTER OR AUTHORITY.

HOLES BY

FOR

T.

/•

Henry Sharp Martell.....

Drill.

Punch.

Drill.

Boyd

Kirkaldy, annealed plates

Punch.

Greig- and Eyth Parker.........

Drill.

Kirkaldy, \ inch Siemens steel plates. Mean of two “    A    “    Landore    “    “    ||    rivets......

44    U    U    44    4k    44    4k    4k

Bored.

Punch.

Bored.

96,160

87,600

55,i°°

5¹,74°

64,290

58,690

55.200 5T,²3° 64,32° 68,990 75,49° 82,45°

83,180 76,59° 78,220 74,03° 7°,54° 73,92°

72.560 72,39° 76,520 67,670 57,36° 49,37° 50,92° 62,140 61,800

66.200

63.260

63.560 69,590

67,54°

66,750

67.260

8.3,33°

75,i7°

76,205

71,680

88,890

89,13°

76.160 70,800 88,930

93.160 101,900

99,660

100,510

90,460

92,380

92,850

88,500

84.630 83,080

107,110

112,780

92,270

7i,44°

49,100

50,760

107,410

74.520

112,000

107,700

104,100

II7ł3°°

98.630 108,000 121,060

The first of the last set of results in the table, by Mr. Kirkaldy, was found with zigzag riveting in which the distance between the centre lines of the rows of rivets was too small.

These results are quite irregular, but it would seem to be as safe a deduction as possible to take f ~ 1.25 z¹, with T equal to 70,000 to 75,000 pounds per square inch for thin plates, and 55>ooo to 60,000 for thick ones.

With this value of f, and q — 2, the second and third members of Eq. (34) give for doidde riveted butt joints with two covers :

(39)

p = 3-5^

If the same value of f be preserved, there will result for single riveted butt joints with two covers :

• • (40)

p = 2.5 d. . .

Experiments on treble and quadruple riveting are yet lack

ing.

But few experiments on the shearing of rivets in butt joints with double covers have yet been made. Four tests by Messrs. Sharp and Kirkaldy give :

THICKNESS OF PLATE.

s.

Single riveted Double “

0.875 in

0.55 in 0.875 in,

.    42,000    lbs.    per sq. in.

All the holes were drilled.

These values of 5 range about 0.7 T. Putting this ratio, therefore, in Eq. (34), and taking f = 1.25 T} the third and fourth members of that equation give :

(41)

d — 1. 14/.

It is probable that this is a little too small for thin plates, and a little too large for thick ones. Hence there may be taken :

For thin plates, d — l]jt For thick plates, d — \l/%t

Double riveted butt joints designed in accordance with the foregoing deductions should give a resistance ranging from 65 to 75 per cent, of that of the solid plate.

Single riveted joints will give an efficiency somewhat less; perhaps from 60 to 65 per cent.

It is to be supposed, in applying the rules just established, that all steel plates are drilled, or subsequently annealed if punched.

As in the preceding cases, the distance between the centre lines of the rows of rivets may be taken at 2.5 to 3d for chain riveting, and three-quarters that distance for zigzag.

Efficiencies.

The values of the quantity which * has been termed the “ efficiency ” of the joint, i.e., the ratio of the resistance of a given width of joint over that of an equal width of solid plate, in the preceding investigations, are those actually determined by experiments with the joints themselves. They may, therefore, be relied upon. Some values which have for many years been considered as standard, but which, in reality, are of a

TABLE VII.

Buti Joints with Two Covers—1877.

NO. OF

PLATE

RIVET DI

PITCH OF

TESTS.

THICKNESS.

AMETER.

RIVETS.

HOLES.

RIVETING.

EFFICIENCY.

A in.

A ini' in. i in.

» in. §• in. I in. I in.

2^ in. 3 in. 24 in.

Punched.

Punched.

Drilled.

Chain.

Zigzag.

Chain.

0.672

0.669

0.662

3 in.

Drilled.

Zigzag.

0.633

somewhat arbitrary nature, and at best belonging to a limited class of joints, have been disregarded.

Table VII. gives the results of Mr. Kirkaldy’s experiments in reference to the comparative resistance of chain and zigzag riveting. The difference is not great, but what there is is in favor of the chain riveting.

TABLE VIII.

Kirkaldy's Tests—1872.

JOINT.

RIVETING.

HOLES.

RIVET DIAMETER

PITCH IN

IN TERMS OF t.

TERMS OF d.

EFFICIENCY.

Lap.........

Lap.........

Lap.........

Lap ........

Butt, I cover . Butt, i cover . Butt, i cover . Butt, i cover . Butt, 2 covers. Butt, 2 covers. Butt, 2 covers. Butt, 2 covers.

Single.

Single.

Double.

Double.

Single.

Single.

Double.

Double.

Single!

Single.

Double.

Double.

Punched.

Drilled.

Punched.

Drilled.

Punched.

Drilled.

Punched.

Drilled.

Punched.

Drilled.

Punched.

Drilled.

Il II II II II II II II II II II II

HHHHbOtOtOtOtOtOtOtO

P — 3« p = 2%d P — 4\d p—\d P — 3d p = 2 U P = 4a^ P=*4 P =

P = jd P = SW P — 44 d

0-55

O.Ó2

O.69

O.75

O.55

O.62

O.69

O.75

O.57

O.67

O.72

O.79

Table VIII. gives the results of the same experimenter on the relative value of punched and drilled work.

The drilled work is seen to give decidedly the greatest efficiency in every case.

The joints to which Tables VII. and VIII. belong were of wrought iron.

Experiments by Mr. Kirkaldy during 1881 show that well-designed double riveted steel butt joints with two covers may be expected to give efficiencies varying from 0.65 to 0.75.

Riveted Truss Joints.

The circumstances in which riveted joints are used in truss work, render permissible many special forms which can find no place in boiler riveting. If joints are found under the same circumstances, so far as the transference of stress is concerned, precisely the same forms would be used, except that caulking is, of course, only required in boiler work.

Fig. 9 shows a common form of chord construction in riv-

r\ r\ r\ r\ r\ B

r\

r\

nt¹

■■

000

000

000

000

SŁ

Fig. 9

CO □ Û LÜ

eted truss work, with the relative proportions A|~r^— exaggerated.

The lower portion of the figure shows a section of the chord, in which the cover plates are shaded. The joint is supposed to be in tension.

AB is a horizontal cover plate, under which the horizontal component plates form lap joints at C, D and E. As the distance MN must necessarily be much greater than the allowable pitch in boiler work, these lap joints, considered in themselves, should be at least treble riveted. On the other hand, the preceding investigations show that even with treble riveting there is great disparity in the loads carried by the different rivets and consequent tendency to detailed rupture ; there would seem, therefore, to be little or no benefit in more than treble riveting.

The distance between the centres of rivets along the line of the chord—i.e., along AB in the upper figure—may be taken at three diameters. The overlap CD = DE (upper Fig.) would then be taken at 9 diameters, and from A, C, D or E to the centre of the first hole, at 1 y₂ diameters. The cover AB should extend 9 diameters also on either side of C and E.

In this work the diameter of the rivet may usually be taken about the same as for boiler work. In estimating the resistance of the whole joint, however, it is to be borne in mind that the rivet holes take metal out of all the plates, and that they are usually punched.

It is impossible to follow the stresses in such a joint or to compute its efficiency. If tested to failure, the latter would probably be found pretty low.

The joint in the vertical plate should be formed as at EG— i.e., it should be a double cover butt joint. The principles already established in a preceding section, in regard to the thickness of covers and diameter of rivets, should be observed here.

The two rows of rivets on either side of the joint may as well be chain riveted with a pitch 3yz to 4 diameters. Other rivets should then be staggered in until the group of rivet centres on each side is brought to a point, as shown in the up-part of Fig. 9. In this manner the available section of a width of plate equal to that of the cover, becomes approximately equal to the total, less the material from one rivet hole. Hence the efficiency of the joint becomes correspondingly increased.

If the joint is in compression the preceding observations hold without change, except that all covers should have the same thickness as the plates covered.

Even if the joints C, D, E and H are of planed edges, little or no reliance should be placed upon their bearing on each other, since the operation of riveting will draw them apart more or less, however well the work may be done. Melted zinc, or

other similar metal, has been poured into compression joints with the intention of insuring good bearings, but the results are not satisfactory.

In the case of very wide chords, four longitudinal rows of rivets should be used in such joints as are exemplified in Fig. 9.

Unless great caution is observed and excellence of design secured, there will frequently be excessive bending in the riveted joints of trusswork, on account of the great variety of connections required.

Diagonal Joints.

It has been proposed to form riveted joints, the edges of whose plates are neither perpendicular nor parallel to the stress transferred. In this manner a greater number of rivets and a greater section of metal will resist the stress exerted in the body of the plate.

Mr. Kirkaldy made some tests on such lap joints, single ' riveted, with ^-inch plates, the joints of which lay at 45⁰ with the applied force, with the following results :

to 4-

Entire plate.. Square joint.. Diagonal joint

The diagonal joints are thus seen to give by far the best results. They are, however, much the most expensive also.

Friction of Riveted Joints.

There are not lacking experiments to show that the friction between the plates of a riveted joint is very great. This, however, cannot be relied upon to give additional resistance to the joint, since a sensible relative movement of the plates takes place in advance of its greatest resistance and essentially destroys the friction.

The experiments of Edwin Clarke, Harkort and Lavelley show that this friction may range from 8,330 to 22,400 lbs. per sq. in. of rivet section.

The specimens were prepared with one slotted plate, so that friction was the only resistance to the parting of the plates.

Hand and Machine Riveting.

Pneumatic, steam and hydraulic riveting machines have lately been brought to such a degree of perfection, that machine work is now very generally preferred to hand riveting.

The resistances of joints will vary to some extent with the method of riveting. Usually, however, the variation will not be greater than may be found for the same kind of riveting in different places and under different circumstances.

As a rule, machine riveting is much more reliable than hand, in that the hole is better filled and the rivet more quickly headed, in consequence of the great excess of pressure exerted. There is thus much less liability of loose rivets.

Many of the preceding experimental results were obtained from machine work.

Addendum to Art. 73.

The following series of valuable tests of riveted joints of both iron and steel were made in the government machine at Watertown, Mass. The results in Table IX. were taken from “Senate Ex. Doc. No. 1, 47th Cong., 2d Session,” while those in Table X. are found in “ Senate Ex. Doc. No. 5, 48th Cong., 1st Session.” The character of plates, rivets, and holes is shown in the tables, and the intensities of tension in net sec-

PITCH OF

RIVET.

TABLE IX.

Riveted Joints—Iron and Steel.

MAX. STRESSES ; POUNDS PER SQ. IN.

Tension on Compress’n Shearing net area of on diaçnet’l on

surface (/). Rivets (S)

plate (T).

EFFICIENCY OF JOINT P. CENT.

/•

On Or -MjJ tO m Q O OQv> On Ut

=1

..g

=1

2 Ins.

HHHHtOtOMH

OtfSlCojCA

n^-^^WWWW to tOOJOJ to to

O VQ 00 vj H O KO OO vQ 00^ ON VJ Qn

Is f I' !=

2 Ins.

to to

to to

lOO tv.00 CO CO m M

vo vo

1'

5s

“h “m

Single riveted lap joints ; i inch iron plates.

76,140

82,910 73,26°

79-360 65,420 68,890 87,670 83,940 78,230 84,660 66,778 60,886

Single riveted lap joints ; \ inch steel plates.

46,340 46,010 60,250 59,240 4°,950

42,370

63,190 61,310 66,860 70,000

62,496

58,338 60,184 57,439

Double riveted lap joints ; 4 inch iron plates.

38,535

41,750 50,592 49,950

43,230

45,520

38,580

41,790

52,160

54,930

49,420

47,260

45,890

49,720

41,005

37,500

34,900

38.640 34,870 38,660

33,4²°

35,200

39.640 40,610 45,3°° 48,420 44,204 42,038

57-7

61.4

52.8

57 •¹

60.6

64.0

4J" punched holes. “ drilled “

“ punched “

82,480

81,780

107,260

105.290

77,870

80,200

120,160

116,090

90.000

94.230

101,180

94,800

114,603

109,650

37,890

37,860

49,270

48.750

36,350

36,710

56,100

52,460

41.790

43.750

65,220

60,382

52,742

50,645

64,120

69,710

42,118

41,660

65-9

63.1

60.3

65.5

53-1

48.3

53-2

52.8

69.2

68.0

48.2

49.6

74-

₇T.B 68.8

72.0

69.0

64.8

70.6

67.6

drilled

II" punched holes.

“ drilled

ir ;

43,110    60.3    V    drilled    holes.

N n^io ri^co rr co ro moo oc

■3- ^    ■«-

N N 10 M 10

Double riveted lap joints ; | inch steel plates.

54,640 53,7i5

61,510

60,300

65,400

64,600

56,944

69,130

25,400

25-53°

70.4 punched holes.

64,600    30,430    74.9

tm    m    76:i    4“    drilled

98,360    61,130    79.5    “    “

IO NO M M NO VO

Double welt butt joints ; } inch u'on plates.

I 53,475 I    50,959

67,321 I 16,944 I 62.2 Ipunched holes. 64,138 I 16,719 I 59-3 I “

SIZE OF

PITCH OF

NO.

RIVET AND

KIND.

RIVET.

TABLE IX.—Continued.

MAX. STRESSES ; POUNDS PER SQ. IN.

Tension on Compress'n Shearing net area of on diamet’l on plate (7’). surface (/).

Rivets (S;

EFFICIENCY OF JOINT P. CENT.

62

63

64

65

to to to to

rB

C/3

67

720 1" 721, 1"

Single riveted lap joints ; f inch iron plates.

60,340 58,150

60,730 57)53°

54,13°

53,4°°

52,97°

50,220

37,460

36,130

38,19°

36,210

41,750

41,290

61,703

58,510

38,280

35,520

37,530

36,050

34,23°

34,15°

26,180

24,830

49.0

47.2

49-7

47.1

50.0

49-3

f" punched holes. “ drilled “

“ punched “

60.4

57-¹

SM M M

M HWWüiCnüiüiüUn VO OO vO OO OM-n 4*OJ 10 h

steel

iron

2 Ins.

1}

Ił

Single riveted lap joints ; $ inch steel plates.

39,740 38,190 66,430 55,460 50,650 50,900 53,560 53,600 36,614 36)59°

39,220

37)7°°

55)215

54)74°

63.650

63,976

65,460

65,210

73,394

73,970

63,210

60,760

89,580

88,660

80,930

81,600

89,490

88,990

79,510

80,200

-«4-vo H lONN O'VO O

to m -<ł- rovo o O h N Tf- rl-VO VO VO VO

punched holes.

“ drilled punched

Iiy    "

68

69

2 Ins.

2    “

58

70

71

81

82

3ł ‘ 3Ï

Double riveted lap j'oints ; f inch iron plates. 39,160 4I,°7°

40,640 41,480 41,27°

58,700 57,34°

43,450

50,730

50,220

46,255

46,110

30,92°

3°,i3°

24,760

26,150

25,33°

27,55°

27,010

39,130

38,410

63-5

66.4

punched holes.

65-7

60.5

60.4

5°-4

49.1

ii

drilled

57

59

2 Ins.

2    “

60

61

83

84

steel

3i “

3l

Double riveted lap joints ; ” inch steel plates.

62,800 64,720 63,210 54,930

44,660

43,650    83,000

50,760

52,45°

56,860    34,710

49,53°    3°,83°

84,460    52,750

51,845

32,31°

32,93°

73-2

75-2

73-2

63.8

64.4

63.0

J" punched holes.

‘ drilled

M    M N

VO vO 4><-    4*.

^ G\ <-n.    -t*-

f" iron

i" “ i" “ ł" “

Reinforced riveted lap joints ; J inch iron plates. (.See Fig. below.)

) 2 Ins. joint.

I 4    “ welt.

••

38,870

43,770

44,840

42,680

59,080

56,640

57,9¹⁰

40,360

34,460

33,89°

67.6

74.0

75-7

drilled hole, f" welt

\r

“ i" “

55,35°

31,810

71.9

SIZE OF

NO.

RIVET AND

PITCH OF

KIND.

rivet.

TABLE IX.—Continued.

MAX. STRESSES ; POUNDS PER SQ. IN.

Tension on Compress’n Shearing net area of on diamet’l on plate (7¹). surface {/). Rivets (S)

EFFICIENCY OF JOINT P. CENT.

i:

VO    fx    OO    on

T    >t-    O'    G\

W    W    N    N

I" steel

i" “

Ï" iron

ł" “

Reinforced riveted joints ; » inch steel plates. (See above Fig.) 1 2 Ins. joint.!

14 “ welt.

62,050

62,880

61,020

61,710

67,320

68,135

67,300

68,040

32,960

33,9°°

34,250

34,750

89.0

90.1

87.8

88.9

drilled holes.

U) C/O to to to to to (OOłO^J*.

OO^I OJ to H o

•H»

OB

tO tO tO tO tO tO

Single riveted lap joints ; £ inch iron plates.

39-8 39-7 42.9 44-3

31,100

31,395

3²,370

33>l8°

39,90°

40,500

4I,5°°

4^955

47.8so

48,890

58,880

59,900

34,280

34,96°

38,020

39,220

47.020

47,830

j§" punched holes. “ drilled “

52.2

54-²

ill!

” 4

c/i

32q| 3n iron I2 Ins.

635I }" “ I2 “

6191 îl" iron 620I ft" “

12 Ins. |2 “

730I 1" iron I2# Ins. 731I r" “    |₂f    “

732! 1" iron |2| Ins. 7331 1"    “ U* “

steel I2* Ins.

“ Wi “

steel I2J Ins. “ |2| “

734| I" 7351 i

7361 i"

737I 1"

Single riveted lap joints ; 1 inch steel plates. 50,940 47,890 56,980 47,060

Double riveted lap joints ; i inch iron plates.

25,280    57.0

24.560 I 55-2

Double riveted lap joints ; \ inch steel plates.

19.670 I 53-8 19.644 I 53-8

Single riveted lap joints ; f inch iron plates.

Single riveted lap joints ; f inch steel plates.

49,650 I 56,760 I 52,770

Double riveted lap joints ; 4 inch steel plates.

39,780    I    30,470    I    70.9

38,300    I    29,340    I    68.3

38,204

ss^s

60,210

49,590

44,320

42,920

64,602

64,519

34,680

34,230

43,580

45,850

69,680

67,100

41,ICO 38,636 36,770

3°,54o

38.2 punched holes.

35-9

5Ï.2    1"

42.2 U

59,640

57,950

J Jf'' punched holes.

29,354 29,37¹

22.060

56.3

60,150

43,490 I 50.5 46,080 I 53-6

|i" punched holes.

I ijY' punched holes. jiïV' punched holes, jjp" punched holes. Jijjg" punched holes.

TABLE X.

Riveted Joints—Iron and Steel.

THICKNESS

DIAM. AND

PITCH

MAX. STRESSES ; POUNDS PER SQ. IN.

NO.

OF PLATE

KIND OF

OF

AND KIND.

RIVET.

RIVET.

Tension on net area of plate (T).

Compress’n on diamet’l surface (/)

Shearing

on

Rivets (5)

EFFICIENCY OF JOINT P. CENT.

■r

>n.

>n.

IE

is.

Single riveted iron lap joints.

39,300 41,000 35,650 35,150

50,850

53,°5°

47,35°

46,690

33,7r°

35,17°

37,3°°

36,780

47.0 l" punched holes. 49.° H" “

45.6 tf" «    ;

44-9

2 Ins

t, u

10

12

2jt “ ²i?

Single riveted iron butt joints.

46,360 46,875 46,400 46,140 44, 260 42,350 43,310 41,920

72,39°

73,°5°

61,040

61,740

60,33°

58,080

57,000

56,54°

25,38°

25,45°

24,630

24.310 23,010

22.310 21.870 22,140

O ŁO rt- d d O' H

Ô 6 Ô Ô t>* 't- Ci W

LOVO lO 10 IT) IT) 10 10

}" punched holes.

13

Î⁴

16

f" steel.

ł" “

*" u

; s «»î

«of* hM

1} Ins.

2    “

Single riveted steel lap joints. 61,270

60,830

47,530 49,840

65,760

65,32°

44,59°

46,960

40.390 39,9C0

29.390

31,070

punched holes.

1"

17

18

19

H" iron.

łl" “

“ steel.

20

21

22

23

24

Single riveted steel butt joints.

2 Ins.    62,770

“ “    61,210

63,920 66,710 62,180 62,590 54,650 54,200

97,940

95,210

62,220

ftllo

71,930

55,6ï°

55,84°

3r,²4°

31,02°

20.370

19,890

27,750

27,940

23.I9O

22,8lO

71.7 punched holes.

it    U    ((

4" :: ::

69.

57-r

55-°

⁶3-4

63.8

54.°

53-4

iA"

tion of plates, compression or bearing on diametral surface and shearing on rivets are those which existed at the instant of failure. The bold faced figures show the kind of failure, and when such figures are found, for the same test, in two or three columns, they show that the same two or three kinds of failure took place simultaneously.

It is important to notice that in general the highest ultimate resistances of tension and compression or bearing are

found with the thin plates, and that those quantities diminish appreciably as the thickness of plate increases, both for iron and steel. This law is not so well defined in reference to the diameter of rivet, if indeed these tests show it at all, except for steel.

The length of these test joints varied from 9.75 to 13

inches.

Although the results of these Tables are somewhat irregular, they confirm the general accuracy of the relations established between the values of T, f, and 5 in the preceding portion of this Article, as well as other general rules and conclusions for boiler work.

Some efficiencies are much lower than given for similar joints on pages 638 and 639, but such instances can, by the aid of the Tables, be traced either to indifferent design or a phe-nominally low value of some one of the three resistances. In general, the results compare well with those given on the pages named.

The pitches of rivets are seen to be adapted to boiler work, and much less than are ordinarily used in bridge work ; yet the corresponding resistances show what may legitimately be done and expected when rare and extraordinary conditions demand a departure from usual rules.

Before deducing from the preceding results of this Addendum working intensities for bridge construction, it is to be first explained that those results are as given in the government reports, and that the net section used is the gross section of the plate, less the actual metal removed by the punch or drill, with no allowance for deterioration by the former in the immediate vicinity of the hole. Again, in the Tables IX. and X. the diametral bearing surface and the shearing area of the rivet are taken to be those of the drill, or a mean between the punch and die in case of punched holes. In bridge work, in determining the net section, metal is deducted for a diameter equal to that of the cold rivet before driving plus one-eighth of an inch ; and the shearing and bearing are computed for the section and diameter of the cold rivet before driving.

With these explanations in view, the preceding tests justify the following working stresses for the plate girder floor beams and stringers of railway bridges with machine driven rivets.

Rivet shearing j

RńetbearinĄ\^I Pf ^ ^

8,000 lbs. per sq. in. for iron. 10,000 “    “    “    “    “ steel.

7,500 lbs. per sq. in. for iron. 10,000 “

“    “ steel.

Tensio7i in net section of plate |

The apparently low bearing resistances, especially for steel, are taken for the reason that very thick plates are frequently used in bridge construction, and the ultimate bearing resistance for them is appreciably less than for the thin plates used in most of the preceding tests.

The preceding working stresses are based on steel for rivets giving from 56,000 to 64,000 pounds per square inch tensile resistance, while the steel for plates, in test specimens, should offer from 58,000 to 66,000 pounds per square inch ultimate tensile resistance.

In the government report from which Table IX. is abstracted, can be found a large number of tests made for the purpose of determining the proper minimum distance from the centres of rivet holes to the edge of plates. As a result of those tests and other experience on the same subject, it may be stated that the least distance from the centre of a rivet hole to the edge of a plate may be taken at one and one-half the diameter of the hole for steel and one and five-eighths the diameter of the hole for iron, in cases where it is important to secure the maximum resistance of the joint.

Art. 74.—Welded Joints.

At the present time the process of welding can, with proper care and material, be made to give excellent results.

Scarf welds give much better results than lap welds, on account of the bending to which the latter are subjected.

Mr. Kirtley (Institute of Mechanical engineers of Great Britain) made some experiments with small strips, 7.5 inches long and inch thick, cut across welded joints. These strips were taken out of boilers whose longitudinal joints had been welded. Twenty-three experiments with strips varying from one to one and a half inches wide, gave the following results per square inch of plate section :

WELDED.

SOLID PLATE.

Greatest Mean .. Least..

53,310 lbs. 46,140 “ 36,960 “

57,790 lbs. 52,860 “ 46,370 “

Art. 75.—Pin Connection.

A pin connection consists of two sets of eye bars or links, through the heads at one end of each of which a single pin passes. Fig. 1 shows a pin connection; A, A, B, B, are eye bars or links, and P is the pin.

_p

Fig. 1

The head of the eye bar (one is shown in elevation in Fig. 2) requires the greatest care in its formation. It is imperfect unless it be so proportioned that when the eye bar is tested to failure, fracture will be as likely to take place in the body of the bar as in the head—in other words, unless its efficiency is unity.

In Fig. 2 the head of the eye bar, or link, is supposed to be of the same thickness as that of the body of the bar whose width is w.

.T

Fig. 2

XT'

%-i-V

A.

----------B4-+0-

r.

'R

If t is the thickness of the bar, so that wt is the area of its normal section, then t is almost invariably included between the limits of y-w and l/%w. In fact these extreme values are each too extreme for the full resistance of the bar, although they are sometimes used. These ratios, as well as the diameter of the pin in terms of w, can only be determined by experiments on full-sized bars. A large number of such experiments have been made both in this country and in Great Britain, and while the resistance of the bar as a whole depends to a considerable extent on the mode of manufacture or formation of the head, it has been found that for the best proportioned head t should range from y.w to %w, and the diameter, d, of the pin from 0.75W to w.

It is extremely difficult to reach more than a general idea of the condition of stress in an eye bar head, although an approximate mathematical treatment of the question may be found in the “Trans. Am. Soc. of Civ. Engrs.,” Vol. VI., 1877,

K.J

DAR &ÄDY PC'

Amerykami

in which the results agree essentially with those of experiment.

between the diameter of the pin and that of the

TOO

Before taking a general view of the stresses which may arise in an eye bar head, it must be premised that a difference of gV to

pin-hole is exceptionally good practice. Before the eye bar is strained, therefore, there is a line of contact only between the pin and eye bar head, but on account of the elasticity of the material, this line changes to a surface when the bar is under stress, and increases with the degree of stress to which the bar is subjected. The line and surface of contact is, of course, in the vicinity of Q, Fig. 2, i.e., on that side of the pin toward the nearest end of the bar. The consequence of this is, that when the bar is strained, the portion about QB, Fig. 2, is subject to direct compression and extension ; that about RL, NE and GS to direct tension and bending, while in the vicinity of T there is a point of contra-flexure, and the stress in the direction of the circumference changes from compression to tension as E is approached from Q.

As a result of many of the experiments which have been made, the following mode of proportioning the head has heretofore been very extensively used : Let r represent the radius of the pin, while reference is made to Fig. 2. Then take EN — o.66w. The curve DRBK is a semicircle with a radius equal to r -j- o.66w, with a centre, A, so taken on the centre line of the bar that QB = 0.8 Jw. G F is a portion of the same curve, with A' as the centre (A'C = AC); GH is any curve with a long radius joining GF gradually with the body of the bar. HG should be very gradual in order that there may be a large amount of metal in the vicinity of CG, for there the metal is subjected to flexure as well as direct tension. FD is a straight line parallel to the centre line of the bar.

As the preceding rule gives a head whose outline causes a more expensive die than a simple circle, at the present day eye bar heads are usually formed as shown in Fig. 3.

5T~T

Fig.3

ABD is a semicircle with a radius cqud to r -f- o.8w to r + 0.9w, and whose centre C is the centre of the pin-hole. The portions FA and HD are formed as before.

There should be no weld across the bar in the vicinity of FH. Consequently, heads are usually formed by placing proper sized pieces upon the upset ends of the plain bars, and then, after insertion in a heating furnace, forcing the head to the desired shape in a die under hydraulic or steam pressure.

The intensity of this pressure will affect, to a considerable extent, the permissible dimensions of the head. The greater the pressure, the better will be the results.

The unfinished head is sometimes rolled on the bar, as by the Kloman process.

The thickness of the head is sometimes made greater than that of the body of the bar. If the head is circular, as in Fig. 3, the section of metal on each side of the pin (through AC or CD) should be not far from eight-tenths that in the body of the bar.

This thickening of the eye bar head is an excellent thing for the bar, but subjects the pin to a great increase of bending, and, hence, requires increased pin diameter.

In pin connections, the pin is subjected to very heavy bending.¹

If M is the bending moment to which the pin is subjected, K the greatest intensity of bending stress developed, and A the area of the normal section of the pin, Eq. (4) of Art. 63 gives :

A A

M =    = o. i Kd³ (nearly) .    .    .    .    (1)

Or :

d = 2.16

,/K

vz

(2)

Values of K, for circular sections, may be found in Art. 63.

Art. 76.—Iron, Steel and Hemp Cables or Ropes.—Wrought-Iron Chain

Cables.

The following tables of resistance and other properties of cables are those published by John A. Roebling’s Sons Co.

It will be observed that the figures for hemp ropes are given in comparison with either iron or steel in each of the tables.

In considering the resistance of iron and steel cables composed of wire twisted into strands, it is of the highest importance to keep clearly in view the circumstances or conditions produced by the manner of fabrication, as they are peculiar to all classes of -ropes, whether of hemp or wire.

In this class of material the fibres or strands no longer lie parallel to the direction of the stress which they carry, but the process of twisting causes each fibre or wire to take a helical form, the pitch of which is not constant for the different portions of the rope. The consequence is that if the process of fabrication were absolutely perfect, so that each wire or fibre could take its proper portion of load, the stress in that wire or fibre would be its portion of load multiplied by the secant of its inclination to the axis of the rope. As a matter of fact, however, each wire does not take its proper portion of load ; the imperfections unavoidably incident to the processes of manufacture render such a result impossible. Hence the increased necessity of experimental determination of the ultimate resistances of metallic and hemp ropes.

The same composite character of these productions renders anything like an approximately elastic character, even, an essential impossibility. It is true that any rope will yield to a considerable extent while under stress, and then return nearly to its original condition, but this behavior is only apparently elastic ; it is almost entirely due to the increase of helical pitch of the strands caused by the external loading. During this

Standard Hoisting Ropes with 19 Wires to the Strand.

IRON,,

Trade No.

Circumference in inches.

Diameter.

Weight per foot in lbs. of rope with hemp cen.

Breaking strain in tons of 2,000 pounds.

Proper working load in tons of 2,000 pounds.

Circumference of hemp rope of equal strength.

CCf*

OOO

W M

nhr    r.ht COIto i-+m r-fco W-* »cfco H<?»

vOO min't’trtCOCOW 04 04 M M

24

IÎ

ił

li

■i

8.00

6.30 5-25 4-10 3-65 3.00 2-50 2.00 1-58 1.20 0.88

0.70

0.44

0.35

74

65

54

44

39

33

27

20

i6

Ilł

8.64

5-13

4.27

3-48

-ICI T-łN IT) Tf W W

co 04 w o O'

MMMM

U1 OW 00

rl- CO

Standard Hoisting Ropes with 19 IVires to the Strand.

CAST STEEL.

Trade No.

Circumference in inches.

Diameter.

Weight per foot in lbs. of rope with hemp cen.

Breaking strain in tons of 2,000 pounds.

Proper working ioad in tons of 2,000 pounds.

Circumference of hemp rope of equal strength.

10

i°i

c.:^    —|-i    c.M*    -frj—.[»    f

2ï

I?

If

If

if

Ij

8.00 6.30 5-25 4- io 3-65 2.50 2.00

1.58 i. 20

0.88

10^

i of

o. 70 0.44

0-35

M to to CO CO en O'^J O Co OCO O-fH oocn-i- coo O

tH tH M tO tO to Co cn O' 00 H CO in O'

Ln O'

tH tH

rh-t* -+M t-fcs    —I«*    -4?»—

in^nH O ooo O uvt -t mmmmm

operation the strands endeavor to place themselves more nearly parallel to the direction of stress, and give rise to a corresponding decrease in diameter. Since these influences preclude the existence of either coefficient of elasticity or elastic limit, ultimate resistances only will be given in this section.

The preceding observations evidently do not apply to suspension bridge cables which are built up of parallel wires. The operations leading to the production of such a cable are of such a refined and exact character that the total resistance of the cable may be assumed without essential error to be the sum of the resistances of all the wires taken separately : the coefficient of elasticity and elastic limit may, and usually do exist with perfect definition.

Galvanized Steel Cables for Suspension Bridges.

DIAMETER IN INCHES.

ULTIMATE STRENGTH IN TONS OF

2,000 POUNDS.

WEIGHT PER FOOT.

220

2f

200

13

2f

ISO

¹¹ -3

IO

ISS

8.64

no

6-5

100

5-8

it

if

95

5-6

75

ił

65

4-35

3-7

Transmission and Standing Ropes with 7 Wires to the Strand.

s:

Trade No.

Circumference in inches.

Diameter.

Weight per foot in lbs. of rope with hemp cen.

Breaking strain in tons of 2,000 pounds.

Proper working load in tons of 2,000 pounds.

Circumference of hemp rope of equal strength.

II

12

IS

IS

16

17

iS

19

20

21

22

23

24

•n HricHfH» .0 -nr-'-»    «4» c-^x —-H* —-he

•t T CMO n O) C4 <N ^    -4 w W M

-+N ••'Six —y-ipn M t—< W t-H W

3-37 2.77 2.28 1.82 1.50 X . 12 0.88 0.70 0-57 0-41 0-31 0.23 o. ig o. ió 0.125

36

30

25

20

i6

12.3

8.8

7.6

5-8

4.1

2.83

2.13

1.65

1.38

1.03

vO

«N*    — H

000

M i—ł

00

to to to t0O04^4^cncn O'

Transmission and Standing Ropes with 7 Wires to the Strand.

Trade No.

II

12

13

14

15

16

I?

18

19

20

23

21

24

Circumference in inches.

4ł

34

2|

2}

Ig

1!

ii

Diameter.

ił

Ij

I4-

la

Weight per foot in lbs. of rope with hemp cen.

3-37 2-77 2 28 i. 82 1.50 i. 12 0.8S 0.70

0.57

0.41 0.31 o. 19 o. 16

Breaking strain in tons of 2,000 pounds.

Proper working load in tons of 2,000 pounds.

Circumference of hemp rope of equal strength.

67

55

45

36

30

22

I?

I3ł

10

*>|C4

MMMMM

W W 4^ ui tn 0^1 co O O to W ui

tif-*

Wrought-Iron Chain Cables.

It might at first sight be supposed that the pull which the link of a chain cable could resist would be twice that offered by a bar of round iron equal in cross section to that of one side of the link. But a weld exists at one end of the link and a bend at the other, each requiring at least one heat for the portion of the link in which it is located. These manipulations produce a considerable decrease in the resistance of the link.

The United States Committee on “ Tests of Chain Cables,” of which Commander L. A. Beardsley was chairman, made many experiments on the iron of which chain cables are made, as well as on the finished cables.

The following conclusions and table are taken from the report of that committee : “ .    .    . that beyond doubt, when

made of American bar iron, with cast-iron studs, the studded link is inferior in strength to the unstudded one.

Ultimate Resistance and Proof Tests of Chain Cables.

DIAM. OF BAR.

Inches.

AV. RESIST. =

PROOF

163$ OF BAR.

TEST.

DIAM. OF BAR.

AV. RESIST. =

PROOF

163# OF BAR.

TEST.

Pounds.

71,172

79,544

88,445

97,731

107,440

H7,577

128,129

139,103

150,485

Pounds.

33,840

37,820

42,053

46,468

51,084

55,903

60,920

66,138

71,550

Inches.

If

iH

iff

iff

Pounds.

162,283

174.475 187,075 200,074

213.475 227,271 241,463 256,040

Pounds.

77,159

82,956

88,947

95,128

101,499

108,058

114,806

121,737

“ That, when proper care is exercised in the selection of material, a variation of five to seventeen per cent, of the strongest may be expected in the resistance of cables. Without this care, the variation may rise to twenty-five per cent.

“ That with proper material and construction the ultimate resistance of the chain maybe expected to vary from 155 to 170 per cent, of that of the bar used in making the links, and show an average of about 163 per cent.

“ That the proof test of a chain cable should be about 50 per cent, of the ultimate resistance of the weakest link.”

The decrease of the resistance of the studded below the unstudded cable is probably due to the fact that in the former the sides of the link do not remain parallel to each other up to failure, as they do in the latter. The result is an increase of stress in the studded link over the unstudded in the proportion of unity to the secant of half the inclination of the sides of the former to each other.

From a great number of tests of bars and finished cables, the committee considered that the average ultimate resistance, and proof tests of chain cables made of the bars, whose diameters are given, should be such as are shown in the accompanying table.

Manila and Hemp Rope.

The results given below were obtained in the Govt, machine at Watertown, Mass., and are found in “ Ex. Doc. No. 35, 49th Congress, ist Session.” “The rope was secured in the testing machine either with a hitch made by taking a round turn over a 4| inch pin held from turning, and then making fast the end to a smaller pin, or by passing the ends over 3^ inch thimbles, and securing to the standing part by means of seizing.”

KIND.

CIRCUM

FER

ENCE.

INCHES.

FASTENING.

ELONGATION JUST BEFORE FAILURE.

PER CENT.

BREAKING LOAD IN POUNDS.

Greatest.

Mean.

Least.

Manila.

31

3ł

<<

Hemp.

3Î

3-t

3-t

3?

3?

3?

Hitch.

Seizing.

Hitch.

Seizing.

Hitch.

Seizing.

Hitch.

Seizing.

Hitch.

Seizing.

Hitch.

Seizing.

14.6 to 18.1

14. i to 17.2

12.2 to 17.0

8.3 to 10.9

8.1 to 13.0

8.1 to ii.4

9,600 10,820 11,400 12,150

12.200

11.900 7,950

12.900 10.100 12,480 10,500

14.200

8,620

10,015

10,650

10,810

11.350

11,020

6,990

n,350

8,315

10,700

8,535

12,900

O ^4 CO 0"P po 00 OO CO

Go go    co O i uun O O 10

tn O en O tn O Ocn O tA ui u\ OOOOOOOOO

000

The rope used in these tests had never been in service, but was ten years old, having been made at the Boston navy yard, and kept in store that length of time at the Washington navy yard. All the manila rope had 3 strands, but of the hemp about half had 3 strands and the other half 4.

CHAPTER XI.

Miscellaneous Problems.

Art. 77.—Resistance of Flues to Collapse.

If a circular tube or flue be subjected to external normal pressure, such as that of steam or water, the material of which it is made will be subjected to compression around the tube, in a plane normal to its axis. If the following notation be adopted :

l — length of tube ;

d — diameter of tube ;

t — thickness of wall of the tube ;

p — intensity of excess of external pressure over internal,

then will any longitudinal section It, of one side of the tube, be

subjected to the pressure    Put let a unit only of length

of tube be considered. This portion of the tube is approximately in the condition of a column whose length and cross section, respectively, are nd and t.

The ultimate resistance of such a column is, Art. 25 :

tt'EI n²d³

As this ideal column is of rectangular section : (I)

EP

and

i=L,

12

P-

EP

12 d*

But P — pd, hence :

is the greatest intensity of external pressure which the tube can carry. But the formulae of Art. 25 are not strictly applicable to this ideal column. The curvature on the one hand and the pressure on the other tend to keep it in position long after it would fail as a column without lateral support. Hence, p will vary inversely as some power of d much less than the third.

Again, it is clear that a very long tube will be much more apt to collapse at its middle portion than a short one, as the latter will derive more support from the end attachments ; and this result has been established by many experiments. Hence,/ must be considered as some inverse function of the length /.

Eq. (1), therefore, can only be taken as typical in form, and as showing in a general way, only, how the variable elements enter the value of/. If x, y and z, therefore, are variable exponents to be determined by experiment, there may be written :

tx

p ~ C Ed*

• (2)

in which c is an empirical coefficient.

Sir Wm. Fairbairn (“ Useful Information for Engineers, Second Series ”) made many experiments on wrought-iron tubes with lap and butt joints single riveted. He inferred from his tests that y — z — 1. Two different experiments would then give :

(3)

(4)

pld — ctx

p'l'd' = ct'x

Hence,

log (pld) = log c Ą- x log t ;

log (p'l'd') = log c + x log t’ ;

in which “ log" means “ logarithm.” Subtracting one of these last equations from the other, the value of .*• becomes :

x= log (pld) - log (fid') _ log t — log t'

los (?)

• - (5)

As p, l, d, t, p', V, d and /' are known numerical quantities in every pair of tests, x can at once be computed by Eq. (5) ; c then immediately results from either Eq. (3) or Eq. (4). By the application of these equations to his experimental data, Fairbairn found for wrought-iron tubes :

(6)

p = 9,675,600 ~~ppp

in which p is in pounds per square inch, while t, l and d are in inches. Eq. (6) is only to be applied to lengths between 18 and 120 inches.

He also found that the following formula gave results agreeing more nearly with those of experiment, though it is less simple :

p-l9

• - (7)

p — 9,675,600 -jd — 0.002 ^ .

Fairbairn found that by encircling the tubes with stiff rings he increased their resistance to collapse. In cases where such rings exist, it is only necessary to take for l the distance between two adjacent ones.

In 1875 Prof. Unwin, who was Fairbairn’s assistant in his experimental work, established formulae with other exponents and coefficients (“ Proc. Inst, of Civ. Engrs.,” Vol. XLVI.). He considered x, y and z variable, and found for tubes with a longitudinal lap joint :

P — 7,36 3,000

t²-¹

l°*dl-ib

(8)

From one tube with a longitudinal butt joint, he deduced :

p — 9,614,000

/2.2I

7°-V¹¹⁶ ' '

• • (9)

For five tubes with longitudinal and circumferential joints,

he found :

p - 15,547,000

/2-:5

l°-9dul6 '    *

.    .    (10)

By using these same experiments of Fairbairn, other writers have deduced other formulae, which, however, are of the same general form as those given above. It is probable that the following, which was deduced by J. W. Nystrom, will give more satisfactory results than any other :

At the same time, it has the great merit of more simple application.

From one experiment on an elliptical tube, by Fairbairn, it would appear that the formulae just given can be approximately applied to such tubes by substituting for d, twice the radius of curvature of the elliptical section at either extremity of the smaller axis. If the greater diameter or axis of the ellipse is a,

and the less b ; then, for d, there is to be substituted -j- .

0

Art. 78.—Approximate Treatment of Solid Metallic Rollers.

An approximate expression for the resistance of a roller may easily be written. The approximation may be considered a loose one, but it furnishes a basis for an accurate empirical formula.

The following investigation contains the improvements by Prof. J. B. Johnson and Prof. H. T. Eddy on the method originally given by the author.

The roller will be assumed to be composed of indefinitely thin vertical slices parallel to its axis. It will also . be assumed that the layers or slices act independently of each other.

Let E' be the coefficient of elasticity of the metal over the roller.

Let E be the coefficient of elasticity of the metal of the roller.

Let R be the radius of the roller and R' the thickness of the metal above it.

JI.-C

i "I

ii

I!

Ti

'I

Fig. i.

Let w — intensity of pressure at A.

ft

any other point.

P — total weight which the roller sustains per unit of length.

x be measured horizontally from A as the origin. d = AC.

e = DC.

From Fig. i :

wR

AB = A'B' =

_ pR

A*

BC =

wR'

£'

d = AC = AB + BC

= w{ ⁺§')! ■ ■ • (I)-

hi I

And

A'C' = A'B' + B'C' = p

( +£)• • •

Dividing Eq. (2) by Eq. (1):

But

If the curve DAH be assumed to be a parabola, as may be done without essential error, there will result :

/:

A'C'dx =

ed.

Hence :

But :

P we

V 2Rd - d* = V 2Rd nearly.

By inserting the value of d from Eq. (1) in the value of ey just determined, then placing the result in Eq. (7) :

= y 2zosr ç

N^3

. • (4).

If R = R! :

EE'

(5).

The preceding expressions are for one unit of length. If the length of the roller is /, its total resistance is

¹

For a detailed treatment of this subject, the author’s “ Bridge and Roof Trusses ” may be consulted.

ly    • • • <⁶)-

Ox\{R—R!\

11

>3

EE

• - (7).

In ordinary bridge practice Eq. (7) is sufficiently near for all cases.

A simple expression for conical rollers may be obtained by using Eqs. (4) or (5).

As shown in Fig. 2, let z be the distance, parallel to the axis, of any section from the apex of the cone; then consider

■I

(ŚT-- &

----i~ —3.-

Fig. 2.

a portion of the conical roller whose length is dz. Let Rx be the radius of the base. The radius of the section under consideration will then be

^3

and the weight it will sustain, if Rx — R ;

Rx

2WZ

E + £'

EE'

Hence :

'=/:

dF =

R-ct

2/

2'vd

E 4- E

EE'

. . (8).

Eqs. (6), (7), and (8) give ultimate resistances if w is the ultimate intensity ol resistance for the roller.

It is to be observed that the main assumptions on which the investigation is based lead to an error on the sid« cJ safety.

If for wrought iron, w = 12,000 pounds per square inch, and E = E' = 28,000,000 pounds, Eq. (5) gives :

OO I CO

Art. 79.—Resistance to Driving and Drawing Spikes.

Some very interesting experiments on driving and drawing rail spikes were made by Mr. A. M. Wellington, C. E., and reported by him in the “ R. R. Gazette,” Dec. 17, 1880. He experimented with wood both in the natural state and after it had been treated by the Thilmeny (sulphate of baryta) preserving process.

“ The test blocks were reduced to a uniform thickness of 4.5 inches ; this thickness being just sufficient to give a full bearing surface to the parallel sides of the spikes when driven to the usual depth, and to allow the point of the spike to project outwards. It was considered that the beveled point could add

Spikes were Standard : 5.5 inches x A inch.

NATURAL WOOD.

PREPARED WOOD.

KIND OF WOOD.

To driving To pulling To driving spike, pounds, spike, pounds, spike, pounds.

To pulling spike, pounds.

Beech

White oak, green Pin oak.........

White ash

White oak, well seasoned.....

Mean.

5,953

7,

Jx On CCH to o\

Oneo on^4 Oo^j

NO COO O fC OJ

Mean.

[5,978

>23

[ 6,553 4,560

[4,281

7,288 I 7,656 f

6,117

4,589

6.588

5,978

Mean.

7,472

5,353

6,283

8,873

8,267

Mean.

[ 8,420

6,135

6,267

[ 6,201

(Split.)

Black ash

J 3,996 I I 4,202 j

4,090

Elm

Chestnut, green Soft maple.....

Sycamore......

Hemlock.......

4,453

4,758

3,99°

3,386

4,606

3,691 5;^ J 3,843

2,910

4,408

4,868

3,536

3,843

2,73°

3,790 2,578 i 3,645 f 3,188 3,188

[ 4,638 [ 3,69°

[ 3,260

4,453 4,453 4,3C1 3,38o 4,453 J. 4.148 i

4,i47

4,300

3-340 3,028 3,3°° 3,493. 4,*48 4,202

3,290

4,i75

3,ITI

[3,188

1,996

3,843

3,448

3,691

3,976

[ 3,645

[3,833

2,725

3,°3°

[2,877

1,968

very little to the holding power of the spike, and it was desired to press the spike out again by direct pressure after turning the block over. .    .    .”

The forces exerted in pulling and driving the spikes were produced by a lever. A few tests with a hydraulic press showed that the friction of the plunger varied from about 6 to 18 per cent.

The accompanying table gives the results of the experiments.

Art. 80.—Shearing Resistance of Timber behind Bolt or Mortise

Holes.

Col. T. T. S. Laidley, U.S.A., made some tests during 1881 at the United States Arsenal, Watertown, Mass., on the resistance offered by timber to the shearing out of bolts or keys, when the force is exerted parallel to the fibres.

S'

Fig.1

T7

X.5"

Fig.2

The test specimens are shown in Figs. 1 and 2. Wrought-iron bolts and square wrought-iron keys were used. All the timber specimens were six inches wide and two inches thick. The diameter of the bolts used (Fig. 1) was one inch for all the specimens. The keys were 1" x 1.5" and 1.125" x 1.5" as shown in Fig. 2. In all the latter specimens, failure took place in front of the smaller key where the pressure was greatest.

In many cases the specimen sheared and split simultaneously in front of the hole. By putting bolts through the pieces in a direction normal to the force exerted, so as to prevent splitting, the resistance was found (in most cases) to be considerably, though irregularly increased.

CENTRE OF

KIND OF WOOD.

HOLE FROM

TOTAL AREA

ULTIMATE SHEARING RESISTANCE PER

END OF

OF SHEARING.

SQUARE INCH, IN POUNDS.

SPECIMEN.

Inches.

Spruce (bolts)

\\

Sq. inches.

16

399

24

359

27s

202

White pine (bolts)

16

ill

24

450

32

327

Yellow pine (bolts)......

16

607

24

720

456

32

337

Yellow pine (square keys)..

16

599

369

24

572

28

438

White pine (square keys)...

16

550

412

24

332

28

236

Spruce (square keys)

16

24

4T0 (not thoroughly seasoned.)

329 *'    “    kk

242 (wet timber).

28

279

Unless otherwise stated, the wood was thoroughly sea

soned.

The accompanying table gives the results of Col. Laidley’s

tests.

Art. 81.—Bulging of Plates.

A plate offers resistance to “ bulging ” when it is simply supported, or firmly fixed, around its entire edge, and carries a single, or uniformly distributed load acting normal to its surface. The very complicated nature of the stresses and strains existing in a plate thus acted upon, together with the fact that its conditions just before rupture are entirely different from those accompanying the initial loading, give to the problem a character of unusual intricacy, and, indeed, preclude a solution possessing a degree of approximation commonly obtained in questions relating to the elasticity and resistance of materials.

An elegant analysis of the problem, considered as one of pure elasticity, may be found in “ Die Theorie der Elasticität Fester Körper,” by Clebsch. It is, however, of little value in connection with questions of ultimate resistance.

The following roughly approximate, but simple, analysis may be used to suggest the form of an empiri'cal formula which can be completed by the aid of experiments.

Let the length, breadth and thickness of a rectangular plate simply supported around its edges, be represented by a, b and t, respectively, and let it first be loaded by a uniformly distributed pressure whose intensity (per unit of ab) is zu.

If the plate is supposed to consist of two sets of small strips or beams parallel to a and b, those crossing each other at the centre must have the same deflection at middle. If, further, the uniform load zu be supposed to be so divided into two parts, w, and zu', that they would cause two rectangular beams whose spans are a and b to have the same centre deflection, the following equation (see Eq. (26) of Art. 24) must obtain :

5 zu,a⁴ _ 5 zu'b⁴

3 84ÉI ~ 384El ’

Then, since zu' -f- zu, = zu, there must result :

zu —

aAzu

aA + bA ’

; and zi\

bAzu

<7⁴ -j- b⁴ *

The bending moments at the centres of such beams would be (Eo. (27), Art. 24, and Eq. (14), Art. 18) :

; and

u\a² _ 2 KJ

w'b² 2 K'l

Since the beams are rectangular in section, I — ■—. Hence :

Kx

3 wxa² %

and K'

3 w'b2 ~ “4^ *

According to these hypothetical conditions the greatest intensity of stress at the centre of the plate will have the value :

K² + K'²y ₌

3 a²b²w

Ąt²V2 ([a⁴ -j- b⁴)

• (0

Hence :

ab

JA

² \K\/ 2 (a⁴ + b⁴)

.    . (2)

For square plates, a = b. Hence

^a²zv _ 8t² *

; and t — 0.615«

• - (3)

If the edges are fixed, the greatest bending will occur along those lines ; and for K₁ and K' then are to be put and

Hence :

: and K' =

• (4)

a²b⁴w

² ’ 4 i² (a⁴ + b⁴) '

b²a⁴w

41² (a⁴ + b⁴)

Since the greatest bending occurs along the edges, these

are the expressions for the greatest intensities of stress. If ab2 is greater than a²b, then is Kx greater than K' ; and vice versa. In the first case the expression for t is :

t = 0.70 Jab²

(a* + b4) AT, *    *

-    . (5)

But if K' > Klt or, a²b > ab¹ :

t = o.Joja²b

(a* + b⁴) K' *    *

. . (6)

If the plate is square :

jr a²w A =- ;

4?

ci

Cu

II

•    • (7)

If a plate is loaded with a single weight P, it may be supposed to be divided in the same manner as w ; so that

P, + P' = P.

The equation of middle deflections for ends simply supported then becomes :

/V _ p'fr

48El ~ 48El *

Hence :

a³P

a³ + b³'

; and, Pt —

b³P

a³ -j- b³ *

Proceeding in precisely the same manner as before :

fa⁴ -T b⁴ •    •    . .    (8)

K = 1.06

abP

t² {a³ + b³)

and

t = 1.03

/ abP ,_

\K{a³ + b³)    + bfi •    *

• • (9)

If the plate is square :

aP

075 — ; and, t = 0.87

• (10)

If the edges are fixed in position, the hypothetical beams are fixed at each end and loaded at the centre, and the greatest bending moments (at centre and ends alike) are thereby reduced to one-half their preceding values, or, what is the same thing, 21² is to take the place of t² in Eqs. (8), (9) and (10). Hence :

.....(11)

K = 0.53

abP

t² (a³ + b³)

t = 073

abP

X ( a³ + b³)

fa⁴ -f- b⁴)' . .

. . (12)

If the plate is square :

aP

0-37577; and * = 0.613

aP

• (13)

These equations are of little value as they stand, except as indicating a form of formula to which empirical coefficients are to be fitted. The hypothetical division of the plate into small beams is very far indeed from being correct. In the empirical determinations which follow, therefore, K will not be the greatest intensity of stress in the plate, but a coefficient or quantity partly analytical and partly experimental.

Circular plates have not been considered, because square ones furnish the requisite type of formula.

Experiments have thus far been made on square and circular plates only ; hence, oblong rectangular plates will not again be noticed.

same.

Kirkaldy’s experiments on Fagersta steel plates and Fair-bairn’s on wrought-iron ones would seem to indicate that the thickness t varies about as (zPf* or (A)0,8 ; but the variation in diameter or side of square was not sufficient to establish any relation between t and a, while other elements remain the Regarding, therefore, K as an empirical quantity which may have different values for square and circular plates, Eqs. (3)> (7)> (IO) and (13), may be written as follows :

w'-⁶ :

SP

and t = 0.615 a

ZifA

V~K

• (H)

a²

K — —■ zv¹-⁶ ;

4t²

VK

• (IS)

K — — P¹-⁶ ;

and t — 0.87

Va P°*

VK

.    (16)

K =l°L P¹-⁶ ;

and t — 0.613

VaPoS

VK

• (17)

Kirkaldy made twenty experiments with mild Fagersta steel circular plates, 12 inches in diameter. He forced these through an aperture 10 inches in diameter by the pressure of a very blunt point. The edge of the aperture on which the plate rested was rounded ; hence the initial diameter of aperture was somewhat more than 10 inches. Eqs. (16) are the ones to be used in connection with these experiments.

From the first member of that equation, K was computed for a number of different experiments, by substituting the numerical values of P, t and a. In this manner the following values were found to give good results :

For unannealed /niild Fagcrsta steel circular plates :

K = 6,760,000,000.

Hence :

■ • (18)

. . (19)

t = 0.000,010,6 Va PoZ .    .

For annealed mild Fagcrsta steel circular plates :

K — 5,710,000,000.

Hence :

t = 0.000,011,52 */aP°A .    .

Eq. (16) gives :

, t V/c °-*7 VaJ

Table I. contains the results of computation by this formula and those obtained in the tests. On account of the rounded edge of the supporting ring, K was so taken that P, as computed, is a little larger than its experimental value. None of these plates were cracked, but they were bulged at the centre from 3.00 to 3.45 inches.

In “ Engineering” for Sept. 28, 1877, Robert Wilson describes four experiments on unstayed flat boiler heads subjected to hydraulic pressure. These flat circular plates were

TABLE I.

Circular Plates simply Supported.

ANNEALED.

UNANNEALED.

P, in pounds.

P, in pounds.

t, in inches.

t, in inches.

Expertmental.

By formula.

Experimental.

By formula.

ccy-*    ałw uf-* «|y»

2I5,6c)0

219,420

198,000

162,740

166,000

154,330

196,530

148,690

104,850

115,860

95,600

103,780

71,800

69,800

59,430

62,520

35,400

29,350

25,430

26,290

Each “ experimental” result is a mean of two.

riveted to angles encircling the body of the boiler. The edges of the plates were thus fixed, and Eqs, (15) are therefore to be used. Proceeding in precisely the same manner as before, the following values were established :

For wrought-iron flat boiler heads, with fixed edges :

K — 11,000,000.

Hence :

t — 0.000,015«ze/⁰-⁸

w was taken in pounds per square inch ; it has the value :

0.62s

4 t²I<

a²

The results of the experiments, and of this formula, are :

DIAMETER,

INCHES.

34-5

34-5

26.25.

28.25.

5 O-f Kl» Kl» Kl» 2

(—TO, IN POUNDS PER SQ. IN.—,

Experimental.    By formula.

280

200,

371

300

349

211

296

270

The agreement, in this case, is not satisfactory. It is probably due to the lack of a proper exponent of a. These plates were fractured along the lines of rivet holes in the edges.

Two means of four experiments by Fairbairn remain to be considered. His plates were square ones of wrought iron, firmly fixed to a square frame 12 inches by 12 inches in the aperture. The force was applied by a blunt point at the centre, consequently Eqs. (17) are to be used.

By precisely the same method already used, the following results were established :

For wrought-iron 12-inch square plates, with edges firmly

fixed :

. . (21)

K = 390,000,000. t — 0.000,031 Va Poi .

The expression for the indenting force is :

P= t

SK

3a

The experiments and computations are :

DIAMETER,

INCHES,

INCH.

Experimental.

P, IN POUNDS.

By formula.

12

12

MH

16,780

37,720.

16,350

38,890

The plates gave way at the centre, under the blunt point. 43

Some experiments by Kirkaldy, in 1875, on wrouglit-iron circular plates simply supported around the edge, show that for 12-inch plates forced through a 10-inch aperture with rounded edge, there may be safely taken :

■ ■ (22)

t — 0.000013 Va FoS . .

In all the preceding formulae, a and t are to be taken in inches ; w in pounds per square inch, and P in pounds.

The investigations can only be considered provisional. Although they give, as a whole, tolerably satisfactory results, the range of the experiments is far too small for the establishment of thoroughly reliable formulae. Experiments on which a proper exponent of a can be based, are yet wholly lacking ; and as the only resort, that found in the rough analysis has been retained.

Art. 82.—Special Cases of Flexure.

There are a few cases of flexure which, while not frequently found in engineering experience, are of some practical importance, and are occasionally required. The two or three which follow involve the integration of some linear differential equations that are treated in all the advanced works on the integral calculus ; consequently the operations of integration will not be given here, but the general integrals will be assumed.

Flexure by Oblique Forces

In Fig. i let OX represent a beam acted upon by the

\ cC

Fig.l

oblique forces P, which make angles a with the axis of X. The origin 0 is supposed to be taken x anywhere on the axis of the beam. If right-hand

moments are positive and left-hand negative, the component Psin a will have the negative moment — Psin ax about O. The lever arm of P cos a, if the deflection w is positive, is + w, and its moment P cos a. w is positive. Hence the resultant moment of any force, P, in reference to the origin O is :

• - (O

d*W

El —— = — P sin a . x 4- P cos a . w ax²

If a is greater than 90°, cos a is negative, so that if

P cos a

El

and V = —

P sin a . x

El

the two cases may be expressed by the equation :

(²)

d²w

dx²

If a — + V — Ay and b = — V — A, the general integral of Eq. (2) is :

tL^Ve-t-dx-, (3)

w = er + Cłx H---—r Ve~ ax dx--

a — b    ci

in which C and C are arbitrary constants to be determined by the special conditions of any given problem, and e — 2.71828. When a is less than 90° and

a ~ \J

cos a

El

'P cos a .

El

Eq. (3) becomes :

C and C yet remain to be determined by the particular circumstances of a given case.

The conditions on which the determination of these con-

dw

stants rest are expressed by giving known values to w and -j--

for values of x, also known.

If a is greater than 90° :

P cos a

El

- . V — I, and b — —

P cos a

El

and Eq. (3) ¹ becomes :

x + C sin

tv — C cos

P cos a \ J/i

El

f P cos a \

\ËT~)

(5)

x — tan a . x

As before, C and C are to be determined by the circumstances of each case to which the equation is applied ; and the value of cos a, it is to be remembered, is to be substituted with the positive sign.

Let a column with one fixed and one free end, and with the force P acting parallel to its original axis, be considered. Since a = 18o° :

w = C cos

El

• x ] + C sin

El

. . (6)

With the origin of co-ordinates at the free end w must equal zero for x — o; hence C — o.

The value of w then becomes :

(7)

w — C sin

dx

• • (8)

dw

But if l is the length of the column, = o for x — L Hence :

cos

or if n is any whole number from o to infinity :

If the value of

. / = y₂{2n 4- i)7T .    .

El

• - (9)

HL be taken from Eq. (9) and inserted

El

in Eq. (8), there will result :

dw

dx

• (10)

times

Eq. (10) shows that for values of x equal to 1, 3, 5, 7 . . .

'PP- = 0. The most dangerous supposition,

dw

2 ti —I dx

i. that which requires the greatest value of P} is n — 0.

This value of n in Eq. (9) gives :

7i²EI

₄/²

(u)

The ultimate resistance of the column is thus seen to be independent of the deflection, as was found for a different case in Art. 25. The end of the column, in this case, which carries the load is free to deflect laterally, but in Art. 25 both ends were supposed to be fixed in a lateral direction in reference to each other. In the latter case the resistance is seen to be nine times as great as in the present.

Since :

cos

Hence, if wt is the deflection of the free end from a vertical tangent to the fixed, Eq. (7) becomes, for x — l :

w₁ = C.

In general, therefore :

w = wt sin

.    . (12)

For the same value of x, therefore, w varies directly as wlf and the relative deflections may be computed by the equation :

w

— = sin

wt

2/

'O

VD

or in the ordinary case :

w

— = sin

nx

zo₁

(H)

Eq. (1) was written for one force only. If any number of forces act :

/Du)

El d    — P sin a . x Ą- P cos a . w)\

and in place of w there is to be put ~2w.

General Flexure by Continuous Normal Load.

The most general case of flexure by a continuous normal load, is that in which the intensity (load per unit of length of beam) is a variable quantity. Let x be an abscissa measured along the original axis of the beam, and let w represent the deflection. Then the intensity of the load may be represented by f{x, w). It was shown in Art. 20 that :

d\M

^ =    =/(*,■*) .... (-5)

dx³

The integration of the equation :

d*w __ f( x, w)

El

dx*

will depend upon the form of the function f(x, w).

Let it be supposed that fix, w) — ex, c being a constant. Then if A, At, A₂ and A₃ are constants of integration, there will result :

Again, if f(x, w) =    £, as before, being a constant :

(17)

ex'*

120 El

. Ax³ , Axx² ,    .    .    .    . ..

+ ——+—^— + ^^ + ^3*    *    (l6)

d*zv

~dx*~ “ “Ä7

For simplicity of notation, let :

a⁴ = -i

then the general integral of Eq. (17) becomes :

w = Ac** -f- Axe~a* + A₂ cos ax + A₃ sin ax.    . (18)

In Eq. (18) ^ = 2.71828 is the base of the Naperian logarithms ; while in both Eqs. (16) and (18) A, Ax, A₂ and A₃ are arbitrary constants to be determined by the circumstances of each individual case.

CHAPTER XIÏ.

Working Stresses and Safety Factors.

Art. 83 —Definitions.

In all metallic and timber constructions the greatest (supposed) possible loads are determined from the attendant circumstances of the different cases, and then the stresses induced by these greatest loads are computed. These stresses are called the “ zv or king stresses."

The ultimate resistance of any piece in a structure divided by the working stress gives a number called the “ safety factor.” Occasionally the reciprocal of this number is called the safety factor, though but seldom.

The intensity of the ultimate resistance of any piece in a structure divided by the intensity of the working stress, will also give the safety factor. This is the more usual and convenient form, since it does not involve the cross section of the piece.

The values of safety factors depend upon many circumstances, such as kind and character of material, kind of stress, circumstances in which material is used and the amount of variation of stress in the piece, or the fatigue of the material. The safety factor is intended, also, to cover both computed stresses and others which are recognized, but are not within the reach of exact analysis. The latest practice among American engineers will be illustrated in the following Articles by extracts from specifications drawn for some first-class constructions.

The following clauses are from specifications prepared by the Phoenix Bridge Co., and accompanied their design for the river spans of the Cincinnati and Covington bridge. Both specifications and design were adopted, and the unprecedentedly long and heavy spans were constructed in accordance therewith.

“All parts of the structure shall be so proportioned that the combined effects of temperature and all the specified loads, except the wind pressure, shall not cause the stress per square inch to exceed the following limits:

For Iron.—In tension, ten thousand (10,000) pounds. In compression, for lengths less than fifty (50) times the least radius gyration, eight thousand (8,000) pounds. In shearing across fibres, seven thoüsand five hundred (7,500) pounds. In bending on the extreme fibres of pins fifteen thousand (15,000) pounds. On bearing surfaces, twelve thousand (12,000) pounds. The bearing surfaces of pins and rivets shall be reckoned from the diameter and not from the semi-circle. The • stress per square inch in compression for members whose length exceeds fifty (50) times their least radii of gyration, shall be reduced according to the following . formulae :

8000

For square bearings, R —

For pin bearings, R =

36000 \ r/ 8000

—(-Y

18000 \ r /

For the flanges of rolled beams, R —

IOOOO

x)o \ b )

For top flanges of built beams, R =

8000

¹

The symbolical method by which Eq. (3) was established shows that if a — — i B + /y/A — 4 and b — — \ B — \/A — -4 B'\ the complete integral of

the equation + B-E + Aw = Vis given by Eq. (3).

L/zy

5000 \b )

where R is intensity of working stress, / length in inches of member between supports, r least radius of gyration of cross section, b breadth of top flange of girder in inches.

For Steel.—In tension on chord bars and end main diagonals, sixteen thousand (16,000) pounds. On main diagonals nearest the middle of the spans, thirteen thousand (13,000) pounds tension. For intermediate main diagonals the tensile intensities are to be directly interpolated. In shearing on rivets and pins, ten thousand (10,000) pounds. In bending on the extreme fibres of pins, twenty thousand (20,000)

pounds. On bearing surfaces, fifteen thousand (15,000) pounds for rivets and eighteen thousand (18,000) pounds for pins. In compression on top chords and inclined end posts, provided that the ratio of length to least radius of gyration does not exceed fifty (50), fourteen thousand (14,000) pounds. For all other steel struts the intensities are to be found by the following formula :

14000

'Vi I k

20000

where R is the intensity, / the length of column in inches, and r the least radius of gyration in inches. Steel struts subject to alternating stresses of compression and tension shall be proportioned by the following formula :

(1-1

14,000

min. stress

max. stress

l—U)

20000 V r/

where R, l and r have the same signification as in the last clause.

An addition of fifty (50) per cent, to all specified intensities of working stresses shall be allowed for all wind stresses and for all combinations of wind stresses and other stresses.

The thickness of metal in compression shall not be less than one-sixteenth (1-16) of the distance between supports in line of stress, or less than one-thirtieth (1-30) of the distance between supports at right angles to the line of stress, or less than one-eighth (1-8) of the distance from the edge of plate of flange to line of support, or less than one-quarter (1-4) inch when both faces are accessible for painting, or less than five-sixteenths (5-16) of an inch when only one face is accessible for painting.

The ratio of length of strut between supporting points to its least diameter shall not exceed forty-five (45).

The limits of stress specified for shearing and for the pressure on bearing surface of holes shall determine the number and the size of the rivets.

Bed plates and bearing plates shall be truly planed on all sliding and rolling surfaces, and shall be so proportioned that the maximum pressure per square foot on masonry will not exceed thirty-six thousand (36,000) pounds. They will be securely anchored against upward and sideways motion.

The rollers shall be of steel ; the pressure per lineal inch on same shall not exceed V'540,000 x d, where d is the diameter of the roller in inches. .    .    . All wrought iron must be

tough, ductile, fibrous, and of a uniform quality for each class, straight, smooth, free from cinder pockets or injurious flaws, buckles, blisters and cracks.

As the thickness of the bar approaches the maximum that the rolls will produce, the same perfection of finish will not be required as in the thinner ones. No specific process or pro. vision of manufacture will be demanded, provided the material fulfills the requirements of this specification.

3.    The tensile strength, limit of elasticity and ductility shall be determined from a standard test piece, not less than one-quarter inch in thickness, cut from the full sized bar, and planed and turned parallel ; if the cross section is reduced, the tangent between shoulders shall be at least twelve times its shortest dimensions, and the area of the minimum cross section in either case shall not be less than one quarter of an inch and not more than one square inch. Whenever practicable, two opposite sides of a piece are to be left as they come from the rolls, but the finish of opposite sides must be the same in this respect. A full sized bar, when not exceeding the above limitations, may be used as its own test piece. In determining the ductility, the elongation shall be measured, after breaking, on an original length the nearest multiple of a quarter inch to ten times the shortest dimension of the test piece, in which length must occur the curve of reduction from stretch on both sides of the point of fracture, but in no case on a shorter length than five inches.

4.    All iron to be used in the tensile members of open trusses, laterals, pins and bolts, except plate iron over eight inches wide and shaped iron, must show by the standard test pieces a tensile strength in pounds per square inch of :

52,000 -

7,000 x area of original bar circumference of original bar

(all in inches),

with an elastic limit not less than one-half the strength given by this formula, and the elongation of 20 per cent.

Plate iron 24 inches wide and under, and more than 8 inches wide, must show by the standard test pieces a tensile strength of 48,000 pounds per square inch, with an elastic limit not less than 26,000 pounds per square inch, and an elongation of not less than 12 per cent. All plates over 24 inches in width must have' a tensile strength not less than 46,000 pounds per square inch, with an elastic limit not less than 26,000 pounds per square inch.

Plates from 24 to 36 inches in width must have an elongation of not less than 10 per cent. ; those from 36 to 48 inches in width, 8 per cent. ; over 48 inches in width, 5 per cent.

All shaped iron and other iron not hereinbefore specified must show by the standard test pieces a tensile strength in pounds per square inch of :

50,000 —

7,000 area of original bar circumference of original bar’

with an elastic limit of not less than one-half the strength given by this formula, and an elongation of 15 per cent, for bars five-eighths of an inch and less in thickness, and of 12 per cent, for bars of greater thickness.

All plates, angles, etc., which are to be bent hot, in the manufacture must, in addition to the above requirements, be capable of bending sharply to a right angle at a working heat without a sign of fracture.

All rivet iron must be tough and soft, and pieces of the full diameter of the rivet must be. capable of bending cold until the sides are in close contact, without sign of fracture on the convex side of the curve.

All iron specified in clause 4 must bend cold 180 degrees, without sign of fracture, to a curve the inner radius of which equals the thickness of the piece tested.

Specimens of full thickness cut from plate iron or from the

flanges or webs of shaped iron, must stand bending cold through 90 degrees, to a curve, the inner radius of which is one and a half times its thickness, without sign of fracture.

A variation in cross section or weight of rolled material of more than 2]/₂ per cent, from that specified may be cause for rejection.

Steel.

No specific process or provision of manufacture will be demanded, provided the material fulfills the requirements of this specification. The ultimate tensile resistance of the steel to be used in tension shall be 62,500 pounds per square inch, and the ultimate tensile resistance of the steel to be used in compression shall be 68,000 pounds per square inch, the tests to be made in the following manner:

16.    From one among the ingots of each cast a round sample bar, not less than three-quarters of an inch in diameter, and having a length not less than twelve diameters between jaws of testing machine, shall be furnished and tested by the manufacturer without charge. These bars are to be truly round, and shall be finished at a uniform heat and arranged to cool uniformly, and from these test pieces alone the quality and material shall be determined as follows:

17.    All the above-described test bars must have a tensile strength within 4,000 pounds per square inch of that specified, an elastic limit not less than one-half of the tensile strength of the test bar, a percentage of elongation not less than 1,200,000 ■— the tensile strength in pounds per square inch, and a percentage of reduction of area not less than 2,400,000 -f- the tensile strength in pounds per square inch. In determining the ductility, the elongation shall be measured after breaking on an original length of ten times the shortest dimension of the test piece, in which length must occur the curve of reduction from stretch on both sides of the point of fracture.

Finished bars must be free from injurious flaws or cracks, and must have a workman-like finish, and round or square test pieces cut therefrom when pulled asunder shall have a reduction of area at the point of fracture as above specified.

Rivet steel shall have a specified tensile strength of 60,000 pounds per square inch, and test bars must have a tensile strength within 4,000 pounds per square inch of that specified, and an elastic limit, elongation and reduction of area at the point of fracture, as stated in clause 17, and be capable of bending double, flat, without sign of fracture on its convex surface of the bend.

A variation in cross section or weight of rolled material of more than 2*/j per cent, from that specified may be cause for rejection.

Cast Iron.

Except where chilled iron is specified, all castings shall be tough gray iron, free from injurious cold shuts or blow holes, true to pattern and of a workman-like finish. Sample pieces one inch square, cast from the same heat of metal in sand moulds, shall be capable of sustaining on a clear span of 4 feet 6 inches a central load of 500 pounds when tested in the rough bar.

To determine the strength of eyes, full size eye bars or rods with eyes may be tested to destruction, provided notice is given in advance of the number and size required for the purpose, so that the material can be rolled at the same time as that required for the structure, and any lot of iron bars from which full size samples are tested shall be accepted :

ist. If not more than one-third of the bars tested break in the eye ; or,

2d. If more than one-third do break in the eye and the average of the tests of those which so break shows a tensile

strength in pounds per square inch of original bar, given by the formula-—

— 500 x width of bar

52,000 —

7,000 x area of original bar circumference of original bar

(all in inches),and not more than one-half of those which break in the eye fail at more than 5 per cent, below the strength given by the formula. Any lot of steel bars from which full sized samples are tested shall be accepted if the average of the tests shows a strength per square inch of original bar, in those which do break in the eye, within 4,000 pounds of that specified in Clause 17; but if one-half the full sized samples break in the eye, it shall be cause for rejecting the lot from which the sample bars were taken.

In all cases where a steel piece in which the full strength is required has been partially heated, the whole piece must be subsequently annealed.

All bends in steel must be made cold, or if the degree of curvature is so great as to require heating, the whole piece must subsequently be annealed.

Art. 85.—Specifications for the Blair Crossing Bridge.

The following specifications for this bridge are taken from the report of Geo. S. Morison, Chief Engineer, 1886.

The steel shall be manufactured by the open hearth process ; Bessemer steel will not be accepted. A small ingot shall be cast from every charge, and from this ingot a sample bar 3-4 of an inch in diameter shall be rolled ; if this bar fails to meet the requirements of the laboratory tests, the whole charge will be rejected.

Steel used in the compression members, bolsters, bearing plates, pins, and rollers shall contain not less than 34-100 nor more than 42-100 of one per cent, of carbon, and not more than I-10 of one per cent, of phosphorus. A sample test bar 3-4 of an inch in diameter shall bend 180 degrees around its own diameter without sign of crack or flaw. The same bar tested in a lever machine shall show an elastic limit of not less than 50,000 pounds, and an ultimate strength of not less than 80,000 pounds per square inch ; it shall elongate at least 15 per cent, in a length of 8 inches before breaking, and shall have a reduced area of 35 per cent, at the point of fracture. It shall be incapable of tempering.

Steel for rivets and eye bars shall contain not more than 25-100 of one per cent, of carbon, and less than 1-10 of one per cent, of phosphorus. A sample bar 3-4 of an inch in diameter shall bend 130 degrees and be set back upon itself without showing crack or flaw ; when tested in a lever machine it shall have an elastic limit of not less than 40,000 pounds, and an ultimate strength of not less than 70,000 pounds per square inch ; it shall elongate at least 18 per cent, in a length of 8 inches, and shall show a reduction of at least 45 per cent, at the point of fracture. In full sized bars this steel shall have an elastic limit of at least 35,000 pounds, and an ultimate strength of at least 65,000 pounds per square inch ; it shall elongate 10 per cent, before breaking, and for strains less than 30,000 pounds per square inch shall show a modulus of elasticity between 28,000,-000 and 30,000,000 pounds.

The steel plates for the chords and end posts shall be rolled in universal mill.

Steel for pins shall not be hammered, but rolled between gothic rolls.

The iron used in tension members shall be double refined iron, rolled twice from the puddled bar. Small samples having a minimum length of 8 inches shall be furnished by the contractor for testing as directed by the engineer ; these samples shall show an elastic limit of at least 26,000 pounds, and an ultimate strength of at least 50,000 pounds per square inch ; shall elongate at least 15 per cent., and shall show a reduced area of at least 25 per cent, at the point of fracture. The fracture shall be of uniform fibrous character free from crystalline appearance. When tests are made of full sized bars, a reduction of from 5 to 10 per cent., according to size of bar, from these requirements will be allowed, provided the iron is of uniform and fibrous character.

Small samples having a minimum length of 8 inches shall be furnished by the contractor from the iron used in shapes, plates, and other miscellaneous forms, as directed by the engineer ; these samples shall show an elastic limit of at least 24,-000 pounds, and an ultimate strength of at least 47,000 pounds per square inch ; shall elongate at least 10 per cent, before breaking, and show a reduction of area of at least 15 per cent, at the point of fracture. In plates more than 30 inches wide, an elongation of 8 per cent, and a reduction of 12 per cent, at the point of fracture will be considered satisfactory.

Cast iron shall be of the best quality of tough, gray iron.

The heads of iron eye bars, and the enlarged ends of screws in laterals and counters shall be formed by upsetting, or by dieforging with a plate welded on the side ; welds in the body of the bar will not be allowed. Six extra iron eye bars, of such size as the engineer shall direct, shall be furnished by the contractor to be tested; these test bars shall meet the requirements above specified for strength of material, and at least four of them shall break in the body of the bar. Should these test bars fail to meet the requirements of the specifications, the whole lot of bars may be rejected.

The heads of steel eye bars shall be formed by upsetting and forging into shape, or by such other process as may be accepted by the engineer ; no welds will be allowed. After the working is completed, the bars shall be annealed by heating them to a uniform dark red heat throughout their entire length, and allow them to cool slowly. Four sample bars of sizes required in the work shall first be manufactured by the contractor, and tested under the direction of the engineer ; these bars shall meet the requirements above specified, and at least three of them shall break in the body of the bar. If the tests of these four bars are satisfactory, the contractor shall proceed with the manufacture of the full order of steel bars for the work, and from the bars so manufactured the inspector shall from time to time select six bars to be tested to breaking, which bars shall also conform to the requirements of the specifications. Should these test bars fail to meet the requirements of the specifications, the whole lot of bars may be rejected. All steel bars shall be tested to a strain of 20,000 pounds per square inch before shipment.

Art. 86.—General Specifications for Iron Railroad Bridges and Viaducts, by Theodore Cooper, C. E., 1887.

The excerpts given in this article are from the general specifications of Mr. Theodore Cooper, C. E., consulting engineer, which have secured a wide adoption in American railway practice.

Proportion of Parts.

30. All parts of the structure shall be so proportioned that the maximum loads shall in no case cause a greater tension than the following (except as per 36) :

Pounds per Square Inch.

. 15,000

On solid rolled beams, used as cross floor beams and stringers....    9,000

On bottom chords and main diagonals (forged eye bars) .

On bottom chords and main diagonals (plates or shapes), net section. 8,ooo

8,000 6,500 8,000

On lateral bracing

10,000

On counter rods and long verticals (forged eye bars).............

On counters and long verticals (plates or shapes), net section.....

On bottom flange of riveted cross girders, net section...........

On bottom flange of riveted longitudinal plate girders, over 20 ft. long, net section.......................................

8,000

30,000 R²

S,ooo

for compression members with pin bearings.

20,000 R*

P = the allowed compression per square inch of cross section.

L — the length of compression member, in inches.

R = the least radius of gyration of the section in inches.

No compression member, however, shall have a length exceeding 45 times its least width.

32.    The lateral struts shall be proportioned by the above formula to resist the resultant, due to an assumed initial strain of 10 000 pounds per square inch upon the rods attaching to them, produced by adjusting the bridge or towers.

33.    In beams and girders compression shall be limited, as follows :

8.000

In rolled beams, used as cross floor beams and stringers.

In riveted plate girders used as cross floor beams, gross section... . 7,000 In riveted longitudinal plate girders, over 20 ft. long, gross section 7,000 In riveted longitudinal plate girders, under 20 ft. long, gross section 6,000

In riveted lattice girders, gross section

Pounds per Square Inch.

On bottom flange of riveted longitudinal plate girders, under 20 ft.

long, net section......................................

On floor beam hangers, and other similar members liable to sudden

loading (bar iron with forged ends)........................

On floor beam hangers and other similar members liable to sudden

loading (plates or shapes), net section......................

Angles subject to direct tension must be connected by both legs, or the section of one leg only will b*. considered as effective.

31. Compression members shall be so proportioned that the maximum load shall in no case cause a greater strain than that determined by the following formula (except as per 36) :

7,000

6,000

5,000

8,000

for square end compression members.

40,000 ip

p-

8,000

L-

for compression members with one pin and one square end.

7,000

34. Riveted longitudinal girders shall have, preferably, a depth not less than 1-10 of the span.

Rolled beams used as longitudinal girders shall have, preferably, a depth not less than 1-12 of the span.

35.    Members subject to-alternate strains of tension and compression shall be proportioned to resist each kind of strain. Both of the strains shall, however, be considered as increased by an amount equal to 8-10 of the least of the two strains, for determining the sectional area by the above allowed strains

(30, 30-

36.    For spans exceeding 150 feet, the above allowed tension (30) on bottom chords and main diagonals, and the compression on top chord sections (31) may be increased for each member by the following amount :

— 50 per cent.

150 x its strain from dead load

Its strain from dead and live loads

The strains in the chords from the assumed wind forces

need not be considered, except as follows :

1st. When the wind strains on any member exceed one-quarter of the maximum strains due to the dead and live loads upon the same member. The section shall then be increased until the total strain per square inch will not exceed by more than one-quarter the maximum fixed for dead and live loads only.

2d. When the wind strain alone, or in combination with a possible temperature strain, can neutralize or reverse the tension in the end panels of the lower chord.

38. The rivets and bolts connecting the parts of any member must be so spaced that the shearing strain per square inch shall not exceed 7,500 pounds, or three-fourths of the allowed tension per square inch upon that member; nor the pressure upon the bearing surface per square inch of the projected semi-intrados (diameter x thickness of piece) of the rivet or bolt hole exceed 12,000 pounds, or one and a half times the allowed tension per square inch upon that member. In the case of field riveting the above limits of shearing strain and pressure shall be reduced one-third part. Rivets must not be used in direct tension.

39.    Pins shall be so proportioned that the shearing strain shall not exceed 7,500 pounds per square inch ; nor the crushing strain upon the projected area of the semi-intrados of any member (other than forged eye bars, see article 69) connected to the pin be greater per square inch than 12,000 pounds, or one and a half times the allowed tension per square inch ; nor the bending strain exceed 15,000 pounds per square inch when the centres of bearings of the strained members are taken as . the points of application of the strains.

40.    In case any member is subjected to a bending strain from its own weight or from local loadings, such as distributed floors on deck bridges, in addition to the strain produced by its position as a member of the structure, it must be proportioned to resist the combined strains.

41.    Plate girders shall be proportioned upon the supposition that the bending or chord strains are resisted entirely by the upper and lower flanges, and that the shearing or web strains are resisted entirely by the web plate ; no part of the web plate shall be estimated as flange area.

42.    The iron in the web plates shall not be subjected to a shearing strain greater than 4,000 pounds per square inch ; but no web plate shall be less than three-eighths of an inch in thickness.

43.    The webs of plate girders must be stiffened at intervals, about the depth of the girders, whenever the shearing strain per square inch exceeds the strain allowed by the following formula :

Allowed shearing strain —

12,000

H-

3,000

where H — ratio of depth of web to its thickness.

44.    No wrought iron shall be used less than 1-4 inch thick, except for lining or filling vacant spaces.

45.    The compression flanges of beams and girders shall be

stayed against transverse crippling when their length is more than thirty times their width.

46.    The unsupported width of any plate subjected to com-presssion shall never exceed thirty times its thickness.

47.    The flange plates of all girders must be limited in width so as not to extend beyond the outer lines of rivets connecting them with the angles, more than five inches or more than eight times the thickness of the first plate. Wh*re two or more plates are used on the flanges, they shall either be of equal thickness or shall decrease in thickness outward from the angles.

48.    In members subject to tensile strains full allowance shall be made for reduction of section by rivet holes, screw threads, etc.

Quality of Material.

101.    All wrought iron must be tough, fibrous and uniform in character. It shall have a limit of elasticity of not less than 26,000 pounds per square inch.

Finished bars must be thoroughly welded during the rolling, and be free from injurious seams, blisters, buckles, cinder spots, or imperfect edges.

102.    For all tension members double rolled bars must be used. They shall stand the following tests:

103.    Full sized pieces of flat, round or square iron, not over 4 1-2 inches in sectional area, shall have an ultimate strength of 50,000 pounds per square inch, and stretch 12 1-2 per cent, in their whole length.

Bars of a larger sectional area than 4 1-2 square inches, when tested in the usual way, will be allowed a reduction of 1,000 pounds per square inch for each additional square inch of section, down to a minimum of 46,000 pounds per square inch.

104.    When tested in specimens of uniform sectional area of at least 1-2 square inch for a distance of 10 inches taken from tension members which have been rolled to a section not more than 4 1-2 square inches, the iron shall show an ultimate strength of 52,000 pounds per square inch, and stretch 18 per cent, in a distance of eight inches.

Specimens taken from bars of a larger cross section than 4 1-2 inches will be allowed a reduction of 500 pounds for each additional square inch of section, down to a minimum of 50,000 pounds.

105.    The same sized specimens taken from angle and other shaped iron shall have an ultimate strength of 48,000 pounds per square inch, and elongate 15 per cent, in 8 inches.

106.    The same sized specimens taken from plates less than 24 inches in width shall have an ultimate strength of 48,000 pounds, and elongate 15 per cent, in 8 inches.

107.    The same sized specimens taken from plates exceeding 24 inches in width shall have an ultimate strength of 46,000 pounds, and elongate 10 per cent.

108.    All iron for tension members must bend cold, for about 90 degrees, to a curve whose diameter is not over twice the thickness of the piece, without cracking. At least one sample in three must bend 180 degrees to this curve without cracking. When nicked on one side, and bent by a blow from a sledge, the fracture must be nearly all fibrous, showing but few crystalline specks.

109.    Specimens from angle, plate (106) and shaped iron must stand bending cold through 90 degrees, and to a curve whose diameter is not over three times its thickness, without cracking.

When nicked and bent, its fracture must be mostly fibrous.

no. Rivets and pins shall be made from the best double-refined iron.

in. The cast iron must be of the quality of soft gray iron.

115. The timber shall be strictly first class white pine, southern yellow pine or white oak bridge timber ; sawed true, and out of wind, full size, free from wind shakes, large or loose knots, decayed or sap wood, worm holes, or other defects impairing its strength or durability. It will be subject to the inspection and acceptance of the Chief Engineer.

Art. 87.—Standard Specifications for Iron and Steel Railway Structures by the Phoenix Bridge Co.

The following clauses are from the standard specifications of the Phoenix Bridge Co. for iron and steel railway structures.

Working Stresses for Iron.

The greatest working stresses in all wrought iron tensile members of railway spans 150 feet in length and under, shall be as follows :

8,000 lbs. per sq. in. 8,000    “    “

In counter web members. ........................

In long verticals ................................

10,000

7,000

7,000

15,000

In main web and lower chord members (eye bars)....

In suspension loops.............................

In suspension plates (net section)..................

In tension members of lateral and transverse bracing.

In counter rods and long verticals of lattice girders (net sect.) 7,000 In lower chords and main tension members of lattice girders

8,000

8,000

8,000

12,000

(net sect.).........................

In bottom flange of plate girders (net sect.)

In bottom flange of rolled beams.........

In angle iron lateral ties (net sect.)........

The greatest working stresses in wrought iron compression members of spans 150 feet in length and under, shall be the following, in which “ P” is in pounds per square inch:

Flat Ends.

8,400

Phoenix column

P-

Pin Ends. 8,400

Latticed or common column

.P =

50,000 r* 8,000

30,000 / 7,800

ri P =

30,000 r2 9,000 - 34 ~

Angle iron struts

40,000;2 p = 9,000 — 30 —

“ /” is the length of column, and “ r ” the radius of gyration of section, in direction of failure ; both are to be taken in feet or both in inches.

Upper chords shall be proportioned by the flat end formula.

A mean between flat end and pin end results shall be used for one pin end and one flat end.

Lateral and transverse struts shall be designed by taking working stresses, equal to one and four-tenths those given by the preceding formulae.

In spans over 150 feet in length, the greatest working tensile stresses per square inch of wrought iron, lower chord and end main web eye bars shall be :

8,000 ^1 +0.9 x

min. total stressX max. total stress/’

whenever this quantity exceeds 10,000.

In such spans the main web eye bars nearest the centre shall be proportioned for 10,000 pounds per square inch ; and the web eye bars between the end and centre shall be found by direct interpolation between the above values.

The greatest working stresses on the upper chords and end posts of spans exceeding 150 feet in length, shall be determined by increasing the results of the above column formulae by the same proportion that the preceding process gives to the lower chord eye bars of the same span. The proportionate increase of working stresses for the intermediate posts, shall be the same as that of the web eye bars, meeting their upper ends in through spans, and lower ends in deck spans.

In the compression flange of plate girders and rolled beams, the working stress shall not exceed 8,000 pounds per square inch of gross section. The greatest shearing stress due to combined dead and moving loads, shall not exceed 7,500 pounds per square inch, in any rivet or pin ; or 10,000 pounds per square inch for wind stresses ; or 9,000 pounds per square inch for wind stresses in combination with those due to moving load. A deduction of 20 per cent, from these values shall be made for field-driven rivets.

The greatest bearing stress on any rivet or pin due to combined dead and moving loads shall be taken at 12,000 pounds per square inch of diametral surface; or 17,000 pounds per square inch of diametral surface for wind stresses; or 15,000 pounds for wind stresses in combination with those due to moving load. A deduction of 20 per cent, from these values shall be made for field-driven rivets.

The bending stress of tension or compression on the extreme fibres of pins shall not exceed 15,000 pounds per square inch for combined dead and moving loads ; or 20,000 pounds for wind stresses ; or 18,000 pounds for wind stresses combined with those due to moving load.

Working Stresses for Steel.

The greatest allowed working stresses in steel tension members, for spans of 200 feet in length and less, shall be as follows:

In counter web members.

In long verticals.......................

In all main web and lower chord eye bars In plate hangers (net section)..........

10,500 lbs. per sq. in. 10,000    “    “

13,200    “    “

9,000

In tension members of lateral and transverse bracing. 19,000

In steel angle lateral ties (net sect.)

15.000

For spans over 200 feet in length, the greatest allowed working stresses per square inch, in lower chord and end main web eye bars, shall be taken at    ,

10,000 ( i +

min. total stress'

max. total stress/’ whenever this quantity exceeds 13,200.

The greatest allowable stress in the main web eye bars nearest the centre of such spans, shall be taken at 13,200 pounds per square inch ; and those for the intermediate eye

bars shall be found by direct interpolation between the preceding values.

The greatest allowable working stresses in steel plate and lattice girders and rolled beams, shall be taken as follows :

Upper flange of plate girders (gross section)

Lower flange of plate girders (net section). .

In counters and long verticals of lattice girders (net sect.) 9,000    “

In lower chords and main diagonals of lattice girders (net

section).................

In bottom flanges of rolled beams In top flanges of rolled beams....

The greatest allowable working stresses in steel latticed or common columns, or steel Phœnix columns, for spans of 200 feet in length and less, shall be determined by taking four-thirds the values given by the formula for iron columns, on page 697.

The greatest allowable working stresses for the same kind of columns, in spans over 200 feet in length, shall be found by increasing the values established by the preceding paragraph by the same proportion, for the upper chord and end posts, as the lower chord eye bar stresses are increased for the same length of spans.

The greatest allowable working stresses in the columns nearest the centre shall remain unchanged ; and those for the intermediate columns shall be found by direct interpolation.

The greatest working stresses, in pounds per square inch, allowed in steel angle struts, shall be as follows :

Fląt end steel angles

₀ /

P= 12,500 - 44 —

Pin end steel angles

P = 12,500 — 50 —

“/’’is the length of the column, and “ r ” the radius of gyration of section, in direction of failure ; both are to be taken in feet, or both in inches.

Upper chords shall be proportioned by the “flat end” formulae, in all cases.

A mean between flat end and pin end results shall be used for one pin and one flat end.

Lateral and transverse struts shall be designed by taking working stresses equal to 1.4 those given by the preceding formulae for both 200 feet pin spans and angle struts.

The greatest shearing stress on any rivet or pin, due to combined dead and moving loads, shall not exceed 10,000 pounds per square inch ; or 13,000 pounds per square inch for wind stresses; or 12,000 pounds for wind stresses in combination with those due to moving load. A deduction of 20 per cent, from these values shall be made for field-driven rivets.

The greatest bearing stress on any rivet or pin, due to combined dead and moving loads, shall be taken at 16,000 pounds per square inch of diametral surface ; or 22,000 pounds per square inch of the same surface for wind stresses ; or 20,000 pounds for wind stresses in combination with those due to moving load. A deduction of 20 per cent, shall be made from these values for field-driven rivets.

The bending stress of tension or compression on the extreme fibres of pins shall not exceed 20,000 pounds per square inch for combined dead and moving loads; or 26,000 pounds for wind stresses ; or 24,000 pounds for wind stresses combined with those due to moving loads.

General Clauses.

In case wind stresses combine with those due to dead and moving loads, no increase of section will be required, unless the wind stresses exceed one-third the sum of those caused by the dead and moving loads, in which case provision will be made for the excess, at a unit stress equal to four-thirds that allowed in the same member for the combined dead and moving loads; but, in no case, shall that unit stress exceed 15,000 pounds for iron, or 18,500 pounds for steel for tension; or 10,500 pounds for iron, or 14,500 pounds for steel for compression ; or 10,000 pounds for iron, or 12,000 pounds for steel for shear; or 19,000 pounds for iron, or 25,000 pounds for steel for extreme fibre stress in pin bending.

Art. 88.—Niagara Suspension Bridge.

In his “ Report on the Renewal of the Niagara Suspension Bridge,” Mr. Leffert L. Buck, C. E., has given some data and calculations, from which he deduces that the safety factor for the cables is :

11,000 4- (1,400 x 1.78) = 4.41,

the total load between the towers being 1,400 tons, and the ultimate resistance of the four wrought-iron cables 11,000 tons, while 1.78 is the ratio between the cable tension at the top of the towers and the vertical load between the towers.

The new iron and steel stiffening truss is designed for a safety factor of 5.

Art. 89.—Specifications for Boiler and Fire Box Steel, Penn. RR.

Co., 1883.

1.    A careful examination will be made of every sheet, and none will be received that show mechanical defects.

2.    A test strip from each sheet, taken lengthwise of the sheet, and without annealing, should have a tensile strength of 55,000 lbs. per square inch, and an elongation of 30 per cent, in section, originally 2 inches long.

3.    Sheets-will not be accepted if the test shows a tensile strength less than 50,000 lbs., or greater than 65,000 lbs. per square inch, nor if the elongation falls below 25 per cent.

4.    Should any sheets develop defects in working, they will be rejected.

5.    Manufacturers must send one test strip for each sheet (this strip must accompany the sheet in every case) ; both sheet and strip being properly stamped with the marks designated by this company, and also lettered with white lead to facilitate matching.

Under the maximum strains produced by any condition of the loads and wind pressures jointly, the strain on the steel composing the structure was limited to the following amounts per square inch :

X4,ooo lb. tension. 13,000 lb. “

12,000 lb. “

10,000 lb. “

20,000 lb. “

12,000 lb. “

12,000 lb. compression. 10,000 lb.

14,000 lb.

12,000 lb.

Art. 90.—The St. John Cantilever, St. John, N. B., 1885.

The general dimensions of the river spans are as follows :

812 ft. 6" 477 ft. o" 143 ft. 6" 287 ft. o" 382 ft. o''

Total length of structure on centres of end pins............

Length of centre opening from centre to centre of the piers......

Length of centre span....................................

Length of east cantilever....................................

Length of west cantilever....................................

(The two arms of each cantilever being of equal length)

Length of panels from centre to centre of pins, about...........

Depth of cantilever trusses, 27 ft. at the ends, 80 ft. at the centre for the west cantilever, and 65 ft. for the east cantilever. Depth of centre span ......................................

24 ft. o"

27 ft. 0"

For the upper chords...............

Diagonal ties.......................

Centre and counter ties..............

Suspension ties....................

Wind ties ..................... ...

Floor beam or stringer flanges........

For the lower chords and central posts

Intermediate posts..................

Wind struts.......................

Floor beam or stringer flanges........

The above values for compression being used for the value of .Fin Gordon’s formula, the value of A in said formula being taken as 1-4500 for the lower chords and central posts, and as 1-1500 for the intermediate posts and wind struts. The tensile strains in riveted connections were to be taken ten per cent, less than the above amounts, and the shearing strains were limited to 10,000 lbs. ; the bearing pressure on rivets or pin-holes not exceeding the diameter of rivet or pin multiplied by the thickness of bearing multiplied by 16,000 lbs., and the strains on the extreme fibres in pins from bending not exceeding 20,-400 lbs. No steel of a less thickness than 5-16 inch was allowed to be used, and no rods of a less diameter than 1 inch.

Art. 91.—Penn. RR. Specifications, 1887.

The following clauses are from the standard specifications of the Penn. RR. Co.

21. The maximum and minimum stresses in tension and compression, as found for the before-mentioned loads, are to be used in determining the permissible working stress in each piece of the structure according to following formulae :

For pieces subject to one kind of stress only (all compression or all tension) :

For pieces subject to stresses acting in opposite direc

tions :

a — u (1 — rt).

In the above formulae :

a = Permissible stress per sq. in., either tension or compression.

7,500 lbs. per sq. inch, for double rolled iron in tension (links or rods).

“ rolled iron in tension (plates or shapes).

“ compression.

7,000 “

6,500 “

_ Minimum stress in piece Maximum stress in piece

6C

((

Maximum stress of lesser kind 2 x Maximum stress of greater kind

22. The permissible stress “a ” for members in compression is to be reduced in proportion to the ratio of the length to the least radius of gyration of the section, by the following formulae :

For both ends fixed.

36000/

For one end hinged.

24000/

For both ends hinged

18000/

Art. 92.—Specifications for Steel Cable Wire for the East River Suspension Bridge.

3.    The general character of the wire is as follows : it must be made of steel ; it must be hardened and tempered ; and, lastly, it must all be galvanized.

4.    The size of the wire shall be No. 8 full, Birmingham gauęe.

5. Each wire must have a breaking strength of no less than

3,400 pounds. This corresponds in wire weighing 14 feet to the pound, to a rate of 160,000 pounds per square inch of solid

section. The elastic limit must be no less than —— of the

100

breaking strength, or, 1,600 pounds. Within this limit of elasticity, it must stretch at a uniform rate corresponding to a modulus of elasticity of not less than 27,000,000 nor exceed 29,000,000.    .......

Mode of Testing.

There will be four kinds of tests.

Firstly.—One ring in every forty (40) will be tested as follows : a piece of wire sixty (60) feet long, will be cut off from either end of the ring, and it will then be placed in a vertical testing machine. An initial strain of 400 pounds is now applied, which should take out every crook and bend. A vernier

of one foot, is so at-

10,000

Where “a”

“ b” “ l”

“g"

Il II II II

Permissible stress already found.

Allowable working stress per sq. in.

Length of piece in inches, centre to centre of connections. Least radius of gyration of section in inches.

gauge, capable of being read to

of its length.

100,000

The same wire will then be subjected to a breaking strain, and the total amount of stretch noted. The minimum strength required is 3,400 pounds, equal to an ultimate strength of 160,000 pounds per square inch. The minimum stretch, when broken, shall have been 2 per cent, in 50 feet, and the diameter

15

tached as to indicate the stretch of 50 feet of the wire. Successive increments of 400 pounds strain are then applied, and the vernier read each time, until a strain of 1,600 pounds is reached.

The conditions now are as follows : that the amount of stretch for each of these increments shall be the same, and that the total stretch between the initial and terminal strains shall

of one foot, equal to

not be less than

97

194

100,000

of the 50

1,000

And furthermore, on reducing the strain to 1,200

feet.

pounds there shall be a permanent elongation not exceeding

of the wire at the point of fracture shall not exceed —— of one r    100

inch.

Fourthly.—Every ring will be subjected to a bending test by cutting off from each ring a piece of wire one foot long, and coiling it closely and continuously around a rod one half inch in diameter, whenv if it breaks it will be rejected.

Straight Wire.

9. All the wire .    .    . must be “ straight ” wire ; that is

to say, when a ring is unrolled upon the floor the wire behind must lie perfectly straight and neutral, without any tendency to spring back in the coiled form, as is usually the case. This straight condition must not be produced by the use of straightening machines of any kind, as they only injure the strength and elasticity of the wire. .....”

Art. 93.—Specifications for Steel Wire Ropes for the Over-Floor Stays and Storm Cables of the East River Suspension Bridge.

3.    The steel from which the wire for these ropes is made must be of a uniform and suitable quality, and after drawn must be thoroughly and evenly galvanized throughout.

4.    The galvanized wire must have an ultimate strength of 150,000 pounds per square inch of full section. When tested in lengths of five feet it must stretch *no less than three and one-half per cent, of its length, and in lengths of one foot it must stretch no less than four per cent.

5.    It must be capable of being bent continuously around a rod of three times the diameter of the wire, without fracture.

6.    The modulus of elasticity must not vary more than 2,000,000 pounds, nor exceed 30,000,000 pounds.

7.    It must have a limit of elasticity of not less than 70,000 pounds per square inch.

Art. 94.—Specifications for Steel Suspenders, Connecting Rods, Stirrups and Pins, for the East River Suspension Bridge.

All of the steel used must be of a uniform and suitable quality, known as “ Mild Steel.” It must have an ultimate tensile strength of 75,000 pounds per square inch of full section, and an ultimate stretch of no less than 15 per cent, in one foot of length, including the fractured section ; and a reduction of no less than 25 per cent, of area at the point of fracture. It must have an elastic limit of no less than 45,000 pounds per square inch, and a modulus of elasticity between 26,000,00a and 30,000,000 pounds per square inch. Specimens turned down from full-sized rods to an area of one square inch, or less, must show a greater strength per square inch, and a greater elongation than that called for in the full section/’

Art. 95.—Specifications for Certain Steel Work . . . East River

Bridge, 1881.

All of the steel used in this work must be of a mild, uniform, elastic and ductile quality, suitable for bridge members. Siemens-Martin or open-hearth steel, or Bessemer steel under the Hay process, will be preferred.

Specimens of the steel proposed to be used must be furnished by each bidder. Two specimens, direct from the rolls, each i inch square and 24 inches long, are required.

All of the steel must be capable of sustaining a tensile strain in every full-sized round or flat bar of not less than 70,000,000 pounds per square inch of cross section. It must have an elastic limit in all shapes of no less than 40,000 pounds per square inch. A modulus of elasticity of not less than 26,000,000 nor more than 30,000,000 pounds per square inch.

An ultimate elongation of 10 per cent, of the full length of uniform sections, and 15 per cent, in one foot of length, inclusive of fractured section, is also required. The area of the reduced section at the point of fracture must not exceed 80 per cent, of the original section.

Small specimens of one foot in length, of even section of one square inch, or less, should reach in tensile strength 75,000 pounds per square inch, with a modulus and limit of elasticity, and reduction of area before mentioned, and an ultimate stretch of 15 per cent.

All round or flat bars, or flat pieces cut from the web of any shaped bars, must be capable of being bent cold for 180⁰ to a

curve whose diameter is no greater than the thickness of the bar, and that without cracking.

The rivets must be made of very ductile steel particularly adapted for that use.

The rods from which the rivets are made must, when tested, have a tensile strength of not less than 70,000 pounds per square inch, and elongate at least 20 per cent, in a length of one foot, and shall reduce at the point of fracture 30 per cent.

If the minimum is reached in any one of these requirements the others must be exceeded by at least 10 per cent. The rod must be capable of being bent cold under a hammer 180⁰, and the inner surfaces brought in contact without producing any fracture.

Cold straightening must be avoided, and when resorted to, the piece so straightened must be annealed afterwards, and all pieces, of which any portion for any cause is reheated, the whole must be annealed and very slowly cooled ; and all pieces in which, from test or otherwise, a want of uniformity is suspected, must be annealed if required by the engineer.

All rivet holes must be drilled, unless some system of punching and reaming approved by the engineer be followed, whereby all of the compressed section around the punched hole be cut away.

The spacing must be accurately done, as no gauging or drifting will be allowed.’’

CHAPTER XIII.

The Fatigue of Metals.

Art. 96.—Woehler’s Law.

In all the preceding pages, that force or stress, which, by a single or gradual application, will cause the failure or rupture of a piece of material, has been called its “ultimate resistance.” It has long been known, however, that a stress less than the ultimate resistance may cause rupture if its application be repeated (without shock) a sufficient number of times. Preceding 1859 no experiments had been made for the purpose of establishing any law connecting the number of applications with the stress requisite for rupture, or, with the variation between the greatest and least values of the applied stress.

During the interval between J 859 and 1870, A. Wohler, under the auspices of the Prussian Government, undertook the execution of some experiments, at the completion of which he had established the following law :

Rupture may be caused not only by a force which exceeds the ultimate resistance, but by the repeated action of forces alternately rising and falling between certain limits, the greater of which is less than the ultimate resistance ; the number of repetitions requisite for rupture being an inverse function both of this variation of the applied force and its upper limit.

This phenomenon of the decrease in value of the breaking load with an increase of repetitions, is known as “ the fatigue of materials.”

Although the experimental work requisite to give Wohler’s

law complete quantitative expression in the various conditions of engineering constructions can scarcely be considered more than begun, yet enough has been done by Wohler and Spangenberg to establish the fact of metallic fatigue, and a few simple formulae, provisional though they may be. The importance of the subject in its relation to the durability of all iron and steel structures is of such a high character that a synopsis of some of the experimental results of Wohler and Spangenberg will be given in the next Article.

Art. 97.—Experimental Results.

The experiments of Wohler are given in “ Zeitschrift für Bauwesen,” Vols. X., XIII., XVI. and XX., and those of Spangenberg may be consulted in “ Fatigue of Metals,” translated from the German of Prof. Ludwig Spangenberg, 1876.

These results show in a very marked manner the effect of repeated vibrations on the intensity of stress required to produce rupture.

Spangenberg states that “the experiments show that vibrations may take place between the following limits with equal security against rupture by tearing or crushing :

Wrought iron

Axle cast steel,

Spring steel not hardened..

-f- 17,600 and — 17,600 lbs. per sq. in. + 33,000 and + 48,400 and -f- 26,400 -f- 30,800 and — 30,800 -|- 52,800 and -f- 88,000 and -j- 38,500 f +• 55,00° and ! -f- 77,000 and -)- 27,500 -f- 88,000 and + 44,000 -f- 99,000 and -j- 66,000

And for axle cast steel in shearing :

<< «« <«

T 24,200 and — 24,200 lbs. per sq. in. -f- 41,800 and

Phcenix Iron in Tension.

POUNDS STRESS PER

NUMBER

POUNDS STRESS PER

NUMBER

SQUARE INCH.

OF REPETITIONS.

SQUARE INCH.

OF REPETITIONS.

From o to 52,800 From o to 48,400 From o to 44,000 From o to 39,600

800 rupture.

From

106,910 rupture. 340,853 rupture. 409,481 rupture.

o to 39,600 o to 35,200 From 22,000 to 48,400 From 26,400 to 48,400

From

480,852 rupture. 10,141,645 rupture. 2,373,424 rupture. 4,000,000 not broken.

Westphalia Iron in Tension.

From o to 52,800 From o to 48,400 From o to 48,400 From o to 44,000 From o to 44,000

4,700 rupture. 83,199 rupture. 33,230 rupture. ,136,700 rupture. 159,639 rupture.

From o to 39,600 From o to 39,600 From o to 39,600 From o to 35,200 From o to 35,200

180,800 rupture. 596,089 rupture. 433,572 rupture. 280,121 rupture. 566,344 rupture.

Firth Sons' Steel in Tension.

103,540 rupture. 12,200,000 not broken. 229,230 rupture. 692,543 rupture. 12,200,000 not broken.

From o to 66,000 From o to 60,500 From o to 55,000 From o to 55,000 From o to 55,000 From o to 55,000

83,319 rupture. 168,396 rupture. I33i9I° rupture. 185,680 rupture. 360,235 rupture. 186,005 rupture.

From o to 55,000 From o to 53,900 From o to 53,900 From o to 52,800 From o to 52,800 From o to 50,600

Krupp's Axle Steel in Tension.

473,766 rupture. 13,600,000 not broken. 12,200,000 not broken.

From o to 88,000 From o to 77,000 From o to 66,000 From o to 60,500

18,741 rupture. 46,286 rupture. 170,000 rupture. 123,770 rupture.

From o to 55,000 From o to 52,800 From o to 50,600

Phosphor Bronze (unworkea) in Tension.

POUNDS STRESS PER

NUMBER

POUNDS STRES^ PER

NUMBER

SQUARE INCH.

OF REPETITIONS.

SQUARE INCH.

OF REPETITIONS.

From o to 27,500 From o to 22,000 From o to 16,500

147,850 rupture. 408,350 rupture. 2,731,161 rupture.

From o to 13,750 From o to 13,750

1,548,920 rupture. 2,340,000 rupture.

Phosphor Bronze (wrought) in Tension.

From o to 22,000

From o to 16,500

53,900 rupture. 2,600,000 not broken.

From o to 13,750

1,621,300 rupture.

Common Bronze in Tension.

From o to 22,000 From o to 16,500

4,200 rupture. 6,300 rupture.

From o to 11.000

5,447,600 rupture.

Phcenix Iron in Flexure (one direction only).

3,420,000 rupture. 4,820,000 not broken.

From o to 60,500 From o to 55.000 From o to 49,500 From o to 44,000

169,750 rupture. 420,000 rupture. 481,975 rupture. 1,320,000 rupture.

From o to 39,600 From o to 35,200 From o to 33,000

4,035,400 ruptnre.

Westphalia Iron in Flexure (one direction only).

From o to 52,250 From o to 49,500 From o to 46,750

612,065 rupture. 457,229 rupture. 799,543 rupture.

From o to 44,000 From o to 39,600

i,493,51¹ rupture. 3,587,509 rupture.

Homogeneous Iron in Flexure (one direction only).

POUNDS STRESS PER

. NUMBER

POUNDS STRESS PER

NUMBER

SQUARE INCH.

OF REPETITIONS.

SQUARE INCH.

OF REPETITIONS.

4,035,400 rupture. 3,420,000 not broken. 48,200,000 not broken.

From o to 60,500 From o to 55,000 From o to 49,500 From o to 44,000

169,750 rupture. 420,000 rupture. 481,975 rupture. 1,320,000 rupture.

From o to 39,600 From o to 35,200 From o to 33,000

Firth <&■» Sons’ Steel in Flexure (one direction only).

From o to 63,250 From o to 60,500 From o to 55,000

281,856 rupture. 266,556 rupture. 1,479,908 rupture.

From o to 52,250 From o to 49,500 From o to 49,500

578,323 rupture. 5,640,596¹ rupture. 13,700,000 not broken.

From o to 77,000 From o to 66,000 From o to 60,500 104,300 rupture. 317,275 rupture. 612,500 rupture.

From o to 55,000 From o to 55,000 From o to 49,500

729,400 rupture. 1,499,600 rupture. 43,000,000 not broken.

Krupp's Spring Steel in Flexure (one direction only).

From

o to 110,000 o to 88,000 o to 66,000 o to 55,000 o to 49,500 From 88,000 to 132,000 From 99,000 to 132.000

From

From

From

From

39,950 rupture. 117,000 rupture. 468,200 rupture. 40,600,000 not broken. 32,942,000 not broken. 35,600,000 not broken. 33,478,700 not broken.

From 72,600 to 110,000 From 66,000 to 99,000 From 44,000 to 88,000 From 44.000 to 88,000 From 61.600 to 88,000 From 27,500 to 77,000 From 33.000 to 77,000

19,673.30° not broken. 33,600,000 not broken. 35,800,000 not broken. 38,000,000 not broken. 36,000,000 not broken. 36.600,000 not broken. 31,152,000 not broken.

Phosphor Bronze in Flexure (one direction only).

POUNDS STRESS PER

FOUNDS STRESS PER

NUMBER

SQUARE INCH.

SQUARE INCH.

OF REPETITIONS.

5,075,169 rupture. 10,000,000 not broken.

From o to 22,000 From o to 19,800

862,980 rupture. 8,151,811 rupture,

From o to 16,500 From o to 13,200

Common Bronze in Flexure (one direction only).

837,76° rupture. 10,400,000 not broken.

From o to 22,000 From o to 19,800

102,659 rupture. 151,310 rupture.

From o to 16,500 From o to 13,200

Phoenix Iron in Torsion (both directions).

56,430 rupture. 99,000 rupture. 479,490 rupture. 909,810 rupture

-    35,200 to + 35,200

-    33,000 to + 33,000

-    28,600 to + 28,600

-    26,400 to + 26,400

-    24,200    to    +    24,200

-    22,000    to    +    22,000

■-    19,800    to    +    19,800

-    17,600    to    +    17,600

3,632,588 rupture. 4,917,992 rupture. 19,186,791 rupture. 132,250,000 not broken.

English Spindle Iron in Torsion (both directions).

204,400 rupture.

147.800    rupture. 911,100 rupture. 402,900 rupture.

1,004,700 rupture.

384.800    rupture. ',317,7°° rupture.

-    37.400 to + 37.400

-    37>4°° to + 37>4°°

-    35,200 to + 35,200

-    35,200 to + 35,200

- 33,000 to + 33,000

" 33,000 to + 33,000 - 30,800 to + 30,800

-    30,800 to + 30,800

-    28,600 to + 28,600

- 28,600 to + 28.600

-    26,400 to + 26,400

-    26,400 to + 26,400

- 22,000 to + 22,000

- 22.000 to + 22.000

979,100 rupture. 1,142.600 rupture.

595,910 rupture. 3,823,200 rupture. 6,100,000 not broken. 8,800,000 not broken. 4,000,000 not broken.

Krupp's Axle Steel in Torsion (both directions').

POUNDS STRESS PER

NUMBER.

POUNDS STRESS PER

NUMBER

SQUARE INCH.

OF REPETITIONS.

SQUARE INCH.

OF REPETITIONS.

55,100 rupture.

797,525 rupture. 1,665,580 rupture. 4>iÖ3i375 rupture. 45,050,640 rupture.

-    44,000 to + 44,0^0

-    39,600 to + 39,600

- 37,400 to + 37,400

- 35,200 to + 35,200

- 33,000 to + 33,000

367,400 rupture. 925,800 rupture. 4,900,000 not broken. 4,800,000 not broken. 5,000,000 not broken.

-    46,200 to + 46,200

-    37i4oc> t0 + 37'²°°

-    35,200 to + 35,200

-    33,⁰⁰⁰ to + 33,000

-    33,⁰⁰⁰ to + 33,000

In Art. 33 will be found some experiments by the late Capt. Rodman, U. S. A., on the fatigue of cast iron, but they are sufficient in number and character to show the general effect only, and give no quantitative results.

The specimens used in all the preceding experiments were

small.

During i860, ’61 and ’62, Sir Wm. Fairbairn constructed a built beam of plates and angles with a depth of 16 inches, clear span of 20 feet, and estimated centre breaking load of 26,880 pounds.

This beam was subjected to the action of a centre load of 6,643 pounds, alternately applied and relieved eight times per minute ; 596,790 continuous applications produced no visible alterations.

The load was then increased from one-fourth to two-sevenths the breaking weight, and 403,210 more applications were made without apparent injury.

The load was next increased to two-fifths the breaking weight, or to 10,486 pounds ; 5,175 changes then broke the beam in the tension flange near the centre.

The total number of applications was thus 1,005,175.

The beam was then repaired and loaded with 10,500 pounds at. centre 158 times ; then with 8,025 pounds 25,900 times, and finally with 6,643 pounds enough times to make a total of 3,150,000.

In these experiments the load was completely removed each time.

It is thus seen that vibrations (without shock) with one-fourth the calculated breaking centre load produced no apparent effect on the resistance of the beam, but that two-fifths of that load caused failure after a comparatively small number of repetitions.

It is probable that the breaking centre load was calculated too high, in which case the ratios £ and f should be somewhat increased.

Art. 98.—Formulae of Launhardt and Weyrauch.

Let R represent the intensity (stress per square unit of section) of ultimate resistance for any material in tension, compression, shearing, torsion or bending ; R will cause rupture at a single, gradual application. But the material may also be ruptured if it is subjected a sufficient number of times, and alternately, to the intensities Pand Q, Q being less than Pand both less than R, while all are of the same kind. When Q = o let P — W, and let D — P — Q. W is called the “ primitive safe resistance,” since the bar returns to its primitive unstressed condition at each application. In the general case Pis called the “ working ultimate resistance.”

By the notation adopted :

¹

Accidental.

Krupp's Axle Steel in Flexure (one direction only).

(I)

But by Wohler’s law, P is a function of D ; or,

(2)

P = /P)

Â sufficient number of experiments have not yet been made

• • (5)

in order to completely determine the form of the function

f{D).

It is known, however, that :

For Q = o ; P — D = W;

and for

P = Q = R.

Provisionally, Launhardt satisfies these two extreme conditions by taking :

R - IV

(P - Q) •    • • (3)

Even at these limits this is not thoroughly satisfactory, for = o,P = <^-By solving Eq. (3) :

when D

W), or, indeterminate.

•    •    (4)

But if the least value of the total stress to which any member of a structure is subjected is represented by min B, and its

min B max B

greatest value by max B, there will result Hence :

p= w(i +

R — W min B' W max B,

which is Launhardt’s formula. In the preceding Article some values of W are shown. In applying Eq. (5) it is only necessary to take the primitive safe resistance, W, for the total number of times which the structure will be subjected to loads.

Eq. (5), it is to be borne in mind, is to be applied when the piece is always subjected to stress of one kind, or in one direction only. It agrees well with some experiments by Wohler on Krupp’s untempered cast spring steel.

Since bridges are expected to possess an indefinite duration of life, in such structures that number should be indefinitely large.

If the stress in any piece varies from one kind to another, as from tension to compression, or vice versa ; or from one direction to another, as in torsion on each side of a state of no stress, Weyrauch has established the following formula by a course of reasoning similar to that used by Launhardt.

If the opposite stresses, which will cause rupture by a certain number of applications, are equal in intensity, and if that intensity is represented by S, then will S be called the “ vibration resistance”; this was established by Wohler for some cases, and some of its values are given in the preceding Article.

Let -f- P and — P' represent two intensities of opposite kinds or in opposite directions, of which P is numerically the greater. Then if D — P -f- Pr :

P — D — P\

The two following limiting conditions will hold :

For P’ — o\ P — D — W: For P' = S; P = 5 = %D,

But by Wohler’s law, P — f (D\ and the two limiting conditions just given will be found to be satisfied by the provisional formula :

w- s

{PfP>) .    (6)

By the solution of Eq. (6) :

P= W i -

W - S P" W ' P,

• • (7)

If, without regard to kind or direction, max B is numerically the greatest total stress which the piece has to carry, while max B’ is the greatest total stress of the other kind or direc-max B’ max B

lowing, which is the formula of Weyrauch :

P’

tion, then will — =

Hence, there will result the fol-

. . (8)

P= W i —

W — S max Br W max B,

Eqs. (5) and (8) give values of the intensity P which are to be used in determining the cross section of pieces designed to carry given amounts of stress. If n is the safety factor and F the total stress to be carried, the area of section desired will be :

nF

in which — is the greatest working stress permitted.

n

If for wrought iron in tension W = 30,000 and R = 50,000,

Eq- (5) gives :

2    min B \

3    max BJ

- 30,000 ^1 +

Hence, if the total stress due to fixed and moving loads in the web member of a truss is max B — 80,000 pounds, while that due to the fixed load alone is min B ~ 40,000, there will result :

= 30,000 (\ + j ■

40,000

80,000,

— 40,000.

In such a case the greatest permissible working stress with a safety factor of 3 would be about 13,300 pounds.

For steel in tension, if W = 50,000 and R = 75,000:

50,000 ^1 +

1    min B \

2    max By

For wrought iron in torsion, if .S' = 18,000 and W — 24,000, Eq. (8) will give :

„    / i max B'\

P = 24,000 I---.

\    4 max B J

Other methods based on Wohler’s experiments have been deduced by Müller, Gerber and Schaffer, of which synopses may be found in Du Bois’ translation of Weyrauch’s “ Structures of Iron and Steel.”

Art. 99.—Influence of Time on Strains.

In the section “ elevation of ultimate resistance and elastic limitf in Art. 32, the effect of prolonged tensile stress and subsequent rest between the elastic limit and ultimate resistance, was shown to be the elevation of both those quantities. It is a matter of common observation, however, that if a piece of wrought iron be subjected to a tensile stress nearly equal to its ultimate resistance, and held in that condition, that the stretch will increase as the time elapses.

Experiments are still lacking which may show that a piece of metal can be ruptured by a tensile stress much below its ultimate resistance. It may be indirectly inferred, however, from experiments on flexure, that such failure may be produced, as the following by Prof. Thurston will show.

A bar 10 parts tin and 90 parts copper, 1 X 1 x 22 inches and supported at each end, sustained about 65 per cent, of its breaking load at the centre for five minutes. During that time its deflection increased 0.021 inch. The same bar sustained 1,485 pounds at centre for 13 minutes and then failed.

A second bar of the same size, but 90 parts tin and 10 parts copper, was loaded at the centre with 160 pounds, causing a deflection of 1.294 inches. After 10 minutes the deflection had increased 0.025 inch; after one day, 1.00 inch; after two days, 2.00 inches ; and after three days, 3.00 inches, when the bar failed under the load of 160 pounds.

Another bar of the same size showed remarkable results ; it was composed of 90 parts zinc and 10 parts copper. It gave the same general increase of deflection with time, but eventually broke under a centre load which ran down from 1,233 t° 911 pounds, after holding the latter about three minutes.

A bar of the same size and 96 parts copper with 4 parts tin, after it had carried 700 pounds at centre for sixty minutes was loaded with 1,000 pounds, with the following results :

DEFLECTION.

3.118 inches. 3-540 3.660    “

4.102 7-634

AFTER

o minute...............

5 minutes..............

15 minutes..............

45 minutes..............

75 minutes..............

Broke under 1,000 pounds.

A wrought-iron bar of the same size gave, under a centre load of 1,600 pounds :

DEFLECTION.

0.489 inch. 0.632    “

0.650 “ 0.660 “ o. 660 ‘ ‘

AFTER

o minute. 3 minutes 6 minutes 16 minutes 344 minutes

It subsequently carried 2,589 pounds with a deflection of 4.67 inches.

During 1875 and 1876 Prof. Thurston made a number of other similar experiments with the same general results.

Metals like tin and many of its alloys showed an increasing rate of deflection and final failure, far below the so-called “ ultimate resistance.” The wrought-iron bars, however, showed a decreasing increment of deflection, which finally became zero, leaving the deflection constant.

Whether there may be a point for every metal, beyond which, with a given load, the increment of deflection may retain its value or go on increasing until failure, and below which this increment decreases as the time elapses, and finally becomes zero, is yet undetermined, but seems probable.

It does not follow, therefore, that the principle enunciated in the section named at the beginning of this Article, is to be taken without qualification. If “ rest ” under stress, too near the ultimate resistance, be sufficiently prolonged, it has been seen that it is possible that failure may take place.

In verifying some experimental results by Herman Haupt, determined over forty years ago, Prof. Thurston tested three seasoned pine beams about inches square and 40 inches length of span, and found that 60 per cent, of the ordinary “breaking load” caused failure at the end of 8, 12 and 15 months. In these cases the deflection slowly and steadily increased during the periods named.

Two other sets of three pine beams each, broke under 80 and 95 per cent, of the usual “ breaking load,” after much shorter intervals of time.

In all these instances it is evident that the molecules under the greatest stress “ flow ” over each other to a greater or less extent. In the cases of decreasing increments of strain, the new positions afford capacity of increased resistance ; in the others, those movements are so great that the distances between some of the molecules exceed the reach of molecular action, and failure follows.

In many cases strained portions of material recover partially or wholly from permanent set. In such cases a portion of the material has been subjected to intensities of stress high enough to produce true “ flow ” of the molecules, while the remaining portion has not. The internal elastic stresses in the latter portion, after the removal of the external forces, produce in time a reverse flow in consequence of the elastic endeavor to resume the original shape.

It is altogether probable that the phenomena of fatigue and flow of metals are very intimately associated. Some of the prominent characteristics of the latter will be given in the next chapter.

CHAPTER XIV.

The Flow of Solids.

Art. ioo.—General Statements.

ALTHOUGH there is no reason to suppose that true solids may not retain a definite shape for an indefinite length of time if subjected to no external force other than gravity,'⁵⁵' many phenomena resulting both from direct experiment for the purpose, and incidentally from other experiments involving the application of external stress of considerable intensity, show that a proper intensity of internal stress (in many cases comparatively low) will cause the molecules of a solid to flow, at ordinary temperatures, like those of a liquid. And this flow, moreover, is entirely different from, and independent of, the elastic properties of the material ; for it arises from a permanent and considerable relative displacement of the molecules. Nor is it to be confounded with that internal “ friction ” which, if an elastic body is subjected to oscillations, causes the amplitudes to gradually decrease and finally disappear, even in vacuo. This latter motion is typically elastic and the retarding cause may be considered a kind of elastic friction.

It is evident that if a mass of material be enclosed on all its faces, or outer surfaces, but one or a portion of one, and if external pressure be brought to bear on those faces, the material will be forced to move to and through the free surface ; in

This, perhaps, may be considered a definition of a true solid.

other words, the flow of the material will take place in the direction of least resistance.

The theory of the flow of solids to be given is that developed by Mons. H. Tresca in his “ Mémoire sur l’Écoulement des Corps Solides,” 1865. He made a large number of experiments on hard and soft metals, ceramic pastes, sand and shot.

These different materials *all manifested the same characteristics of flow, which are well shown in Fig. 2. AB CD, Fig. 1, is supposed to be a cylindrical mass of lead with circular horizontal section, confined in a circular cylinder, MN, closed at one end with the exception of the orifice 0.

This cylinder is supported on the base PN, while the face AB of the lead receives external pressure from a close-fitting piston. When the pressure is sufficiently increased, the face AB in Fig. 1 sinks to AB in Fig. 2, while the column hkHK, in the latter figure, is forced to flow through the orifice 0.

In Tresca’s experiments with D lead, the diameter AB was about 3.9 inches ; the diameter HK of the orifice, from 0.75 in. to 1.5 ins., whiie the length of the column or jet JiK varied from 0.4 in. to about 24 ins. The total pressure on the face AB varied from 119,000 to 198,000 pounds. The initial thickness AD varied from 0.24 inch to 2.4 inches.

Fig-1

1 r

-Tj iy

A Til iii *

Fig.2

-B-

J3

H'

M-

Fig.3

--K

JEl

Some experiments exhibiting in a remarkably clear manner the flow of metals in cold punching were made by David Townsend in 1878, and the results were given by him in the “Journal of the Franklin Institute” for March of that year. If the dotted rectangle ABFG, Fig. 3, shows the original out-* line of the middle section of a nut before punching, he found that the final outline of the same section would be represented by the full lines. The top and bottom faces were depressed by the punching, as shown ; the upper width AB remained about the same, but the lower, GF, was increased to CD. Although the depth of the nut, AC, was 1.75 inches, the length of the core punched out was only 1.063 inches. The density of this core was then examined and found to be the same as that of the original nut. Hence a portion of the core equal in length to 1.75 — 1.063 = 0.687 inch was forced, or flowed, back into the body of the nut. Subsequent experiments showed that this flow did not take place at the immediate upper surface AB, nor very much in the lower half of the nut, but that it was chiefly confined to a zone equal in depth to about half that of the nut, the upper surface of which lies a very short distance below the upper face of the nut. The location of this zone is shown by the lines HK and' MN in Fig. 3.

Tresca’s experiments on punching showed essentially the same result.

Art. ioi.—Tresca’s Hypotheses.

The central cylinder FGKH, Fig. 1 of Art. 100 was called by Tresca the “ primitive central cylinder.” As the metal flows, this cylinder will be drawn out into the volume of revolution, whose axis is that of the orifice and whose meridian section is FGkKHh, Fig. 2, the diameter FG being gradually decreased.

It was found by experiment that if the original mass AC,

Fig. I, was composed of horizontal layers of uniform thickness, the reduced mass in Fig. 2 was also composed of the same number of layers of uniform thickness, except in the immediate vicinity of the central cylinder.

Tresca then assumed these three hypotheses : i°.— The density of the material remains the same whether in the cylinder or in the jet ; in other words, the volume of the material in the jet and in the cylinder remains constant.

Let R — radius of the cylinder.

Let R₁ = radius of the orifice.

Let y = variable length of the jet (i. e., hH).

Let D — original depth of material (BC = AD, Fig. 1) in the cylinder.

Let d — variable depth of material {BC — AD, Fig. 2) in the cylinder.

Then by the hypothesis just stated :

(I)

R²d — R²D — R*y

2°. The rate of compression along any and all lines parallel to the axis of the primitive central cylinder, and taken outside of that limit, is constant.

If, then, the material lying outside of the central cylinder be divided into horizontal layers of equal thickness, a very small decrease in the variable depth equal to d{a) will cause the same amount of material to move or flow from each of these layers into the space originally occupied by the central cylinder, thus causing a portion of the material previously resting over the orifice to flow through the latter. If d(d) is the indefinitely small change of depth, and dR₁ the indefinitely small change in the radius of the cylindrical portion resting over the orifice, then the equality of volumes expressing this hypothesis is the following :

7i(R² — R²) . did) = 27tR₁d . dRx ;

or :

(2)

did') _ 2Rx dRx ~ R² - R*

3⁰.— The rate of decrease of the radius of the primitive central cylinder is constant throughout its length at any given instant during flow.

Let r be any radius less than Rlf then if the latter is decreased by the very small amount dRt, the former will be shortened by the amount dr \ and by the last hypothesis there must result :

dRt

dr

(3)

This is a perfectly general equation, in which r may or may not be the variable value of the radius of that portion of the primitive^central cylinder remaining above the orifice at any instant during flow.

These are the three hypotheses on which Tresca based his theory of the flow of solids. It is thus seen to be put upon a purely geometrical basis, entirely independent of the elastic or other properties of the material.

Art. 102.—The Variable Meridian Section of the Primitive Central

Cylinder.

The meridian curve haH, or hbK, Fig. 2 of Art. 100, may now easily be determined.

Eq. (1) of Art. 101 may take the first of the following forms, while its differential, considering d and y variable, may take the second :

d= D-

R?

R²

y-

did) = -

R²

dy.

Dividing the second by the first :

d{d) _ d ~

dy

R²

2 Rt dR1 ~ R² - Rj '

The last member of this equation is simply Eq. (2) of Art. 101 ; and if the value of dR„ in Eq. (3) of the same Article, be inserted in the third member of this equation, there will result :

2 R²

dr

dy

R² - R² ‘ r

R²

y - -z,-, D

R²

Integrating between the limits of r and Rv and remembering that r will be restricted to the representation of the radius of that portion of the primitive central cylinder which remains, at any instant, over the orifice, by takings = o for r = Rt :

R²

2 R²

■fc, __ R 2 l°g~g — loS

Ri

R-¹

-S- D

R?

“log” indicates a Napierian logarithm.

Passing from logarithms to the quantities themselves, and reducing :

This is the desired equation of the line, in which r is measured normal to the axis of the cylinder or jet, while y is measured along that axis from the extremity of the jet. When the material is wholly expelled :

R²

R²D

a*

(I)

y = D, and r — o.

Eq. (2) is applicable to the jet only. For the line hF or Gk, resort will had to the equation :

d(d) _    2R? dr

~cT ~ R² - R² r

Again integrating between the limits d and D, or r and Rl} and reducing :

1)

2 Ä]²

(2)

This value of r is the radius of that portion of the primitive central cylinder which remains over the orifice when D is reduced to d.

Art. 103.—Positions in the Jet of Horizontal Sections of the Primitive

Central Cylinder.

That portion of the primitive central cylinder below ab in Fig. i of Art. 100, will be changed to abKH in Fig. 2 of the same Article.

If, in the latter Fig., y' is the distance from HK to ab,

measured along the axis, then the volume of HKab will have the value

r/

nr² dy.

J o

If d' is the distance aF—bG, in Fig. 1, the equality of volumes will give :

■/

r² dy ■=■ R²{D — d).

Eq. (1) of Art. 102 gives :

r² = R²

(■ - S)

R'²

I r² dy

= R²D - R²D{ i -

A² A

R²(D -d').

R²

r² L¹ 'd’\ R*

J)

If A" is the number of horizontal layers required to compose the total thickness D, and n the number in the depth d' :

Hence :

71

,A

D.....(2)

Tresca computed values of y for some of his experiments, and compared the results with actual measurements. The agreement, though not exact, was very satisfactory. Within limits not extreme, the longer the jet the more satisfactory was the agreement.

Art. 104.—Final Radius of a Horizontal Section of the Primitive Central

Cylinder.

Let it be required to determine what radius the section situated at₄the distance d' from the upper surface of the primitive central cylinder will possess in the jet.

It will only be necessary to put for_y in Eq. (1) of Art. 102, the value of ÿ taken from Eq. (1) of Art. 103. This operation gives :

,, *.a

d \ &

R¹ - R?

Hence :

R^ - Rf

r’ = Rx

2R2

(i)

If 7?x is small, as compared with R, there will result approximately :

til

• - (2)

Art. 105.—Path of any Molecule.

The hypotheses on which the theory of flow is based enable the hypothetical path of any molecule to be easily established. In consequence of the nature of the motion there will be

three portions of the path, each of which will be represented by its characteristic equation, as follows :

First : let the molecule lie outside of the primitive central cylinder.

Let R and H be the original co-ordinates of the molecule considered, measured normal to and along the axis of the cylinder, respectively, from the centre of the orifice HK (Fig. 1 Art. 100) as an origin, while r and h are the variable coordinates.

The first hypothesis, by which the density remains constant, then gives the following equation :

7t(R - R²)H = n(R - r²)h ;

or :

.    .    (I)

JiR - hr² = {R - R²)H . .

This is the equation to the path of the molecule, in which r must always exceed RT.

As this equation is of the third degree, the curve cannot be one of the conic sections.

Second : let the molecule move in the space originally occupied by the central cylinder.

While h and r now vary, the volume nr*(p — h) must remain constant. When r = RI let h — 4* Hence :

(2)

r\D - h) = R²{D - 4) But if h — 4 and r — Rt in Eq. (1) :

H.

R² - R'²

R - R²

Placing this value in Eq. (2) :

r*(D - h) = R²(d -H

R² - R'² \ R² - R² ) ‘

. - (3)

Third : let the molecule move in the jet.

After the molecule passes the orifice, its path will evidently be a straight line parallel to the axis of the jet. Its distance r₁ from that axis will be found by putting h — o in Eq. (3). Hence :

(4)

HR² - R'² \* DR² - R² )

ADDENDA.

Addendum to Art. 34.

Both Tables in this Addendum show tests of steel used in the St. John Cantilever covered by the specifications of Art. 90. It will be noticed that this steel is of a very mild character, but very ductile, uniform and reliable. Table I. shows tests of specimens from a great variety of bars and shapes, while Table II. gives the results of tests of full size eye bars. With one or two exceptions, the tests of the latter were suspended before failure took place, which accounts for the incomplete record. Both tables are from London “ Engineering,” of Oct., 1886.

TABLE I.

Test Specimens of Steel.

ORIGINAL FORM OF MATERIAL.

SECTION OF TEST-PIECE.

STRENGTH PER SQUARE INCH. PER CENT.

OF ELON-

- GATION IN

8 INS.

ELASTIC.

ULTIMATE.

PER CENT.

OF REDUCTION AT FRACTURE.

In.

i /₆ square steel bar. IiV “

2i*₆    “

I round

Ii~q    x    square

II rivet

OOOOwmmOOwOmMm ^

m in L cnO W i    OoO w r

4- ui 0 4^0 ui -L COCoO^W w O O 4- coca h owui coco 0>C0 co 4^ O r⁵

Pounds.

33.092

32,373

30,000

34,800

34,99²

36,896

35.135

33,010

32,225

36,482

29,944

37,284

42,652

40,441

Pounds.

55.9^

58,212 63,448 56,494 56,981 55,637

27-5

32.4

ï7.i

29.6

21.0

28.1

65-3

60.3

62.0

67.0

64.4

61.9

55-8

ORIGINAL FORM OF MATERIAL.

TABLE I.—Continued.

SECTION OF TEST-PIECE.

STRENGTH PER SQUARE INCH.

ELASTIC.

ULTIMATE.

PERCENT. OF ELONGATION IN 8 INS.

PER CENT.

OF REDUCTION AT FRACTURE.

42

42

ni

ni

21

21

20

5i

5è

5i

5i

si

si

III

III

ni

I5ł

I5Î

I5Î

I5Î

oo|cac»lM3otM»Mx|MacjM*j«oo)MoclMcc,cocc|M c4-|«r«0Cpi tcp

Steel angle.

steel channels.

0.568 0.603 0.788 0.980 i.215 1.227 0.908 1.232 1.050 0.833 0.990

0.583

0-754 0.866 1.072 0-994 0-973 i. 140

0.572

0.800

0.629

34,826 39,344 35,220 33,49s 39,120 46,806 36,556 43,801 46,458

39,623

39,¹²5 40,0:85 33,862

36,720

36,382

38,158

36,699

37,625

31,804

65,446

62,189

60,285

60,526

59,745

60.456 65,462 61,841 63,473 69,900

56.303

60,703

61,857

64.457

54011

58    330 57,605 65,939

59,4i8

59    625 60,111

18.7

25.0

88.4

48.6

29 • 5

28.8

33-0

23.9

27.1 14.0

16.9

28.1

307

28.6

29.8

22.4

33-0

22.6

18.3

35-5

²3-9

300

23 7

47.6

21.4

5²-3

33-5

48.7

50.0

46.6

51-3

52.7

55-1

55-3

44.0

60.2

34-6

25.2

51.8

44-7

46.0

32.0

Totals .. Average

1,215,138

36,810

2,135,882

61,025

909.0    1.779.4

26.7

50.8

TABLE II.

Steel Eye Bar Tests.

DIMENSIONS OF BAR.

SIZE OF BAR HEADS.

STRAINS PER SQ. IN.

Size.

Length.

Diam- Thick-eter. ness.

Pin-

Hole.

Elastic.

Maxi

mum.

GREATEST

REDUCTION.

00 COCOCOvJvJtA ł 0*^00»hmmM^

how OG3 4- 04^ ‘

xxxxxxxx

mOOOOOOOk^

OOOOvOvOvOO ? CO coo O' co^i. CO

Ft.

16.

16

16

16

16

16

16

Total...........

Average........

37.350 34.090 35,010 32,120 33,070 32,870 34,390 32,780

271,680

33,9⁶o

o 05    om Vj Ul O

O tn 44 (41 Co vO M O OOOOOOOO

16.2    per cent,

22.2    “

45-4

34-2

32-5

32.5

<<

Addendum to Art. 65.

In the autumn of 1883 an extensive series of tests of wrought iron eye beams, subjected to bending by centre loads, was made by G. H. Elmore, C. E., and the writer, at the mechanical laboratory of the Rens. Pol. Inst. The object of these tests was to discover, if possible, the law connecting the value of K for this class of beams with the length of span when the beam is entirely without lateral support. The means by which the latter end was accomplished, and a full detailed account of the tests will be found in volume I., No. 1, “Selected Papers of the Rensselaer Society of Engineers.” The main results of the tests are given in Table III. All the tests were made on 6 inch eye beams with same area of normal cross section of 4.35

TABLE III.

w ? CL.

nil

Elas. Lim. Ultimate. Pounds. Pounds.

PERMANENT | PERMANENT VERTICAL DEFLECTION.

INCHES.

LATERAL DREFLECTION. INCHES.

POUNDS PER SQ. IN.

OcovO ■'TC* O CO O

C* W w w M w

cn

tli

5 8    &£££<§ 8 !?!■«««

o C4    M M M n C4 O rt OCOn CO O ^2

4 T? 4 rf 4 »OVO «tîtCfÜÔH HOO CO N CÔ C><»

27,720

29,623

28,264

28,264

26,564

29,596

31,191

31,164

30,221

31,314

33,082

33,082

29,736

31,936

35,497

36,617

34,136

34,136

32,6x9

32,619

31,094

32,885

30,791

32,020

29,579

32,632

33,049

33,023

32,907

33.817

35,358

37,064

35,010

36,527

4i,737

41,396

43,434

43,8i3

45,532

44,744

o. 14 0.30

0.2

0.18

0.28

0.48

0.30

0.30

0.35

0.33

0-39

0.50

0.30

0.29

0.605

0.67

0.67

0.67

0.96

0.60

0.5

0.4

i.00 1.25 1.20 1.10 1.08 1.09 1.08

1.50

0.90

1.05

1-53

1.88

1-75

1-75

1.70

1.86

24,170,000

26,374,000

24,520,000

24,313,000

25,771,000

25,003,000

26,082,000

23,373,ooo

25,287,000

24,022,000

25,1x5,000

24,218,000

21,611,000

21,987,000

23,040,000

20,935,000

22,023,000

25,272,000

24,315,000

21,275,000

square inches. Actual measurement showed the depth d of the beams to be 6.16 inches. The moment of inertia of the beam section about a line through its centre and normal to the web was I = 24.336. The radius of gyration of the same section in reference to a line through its centre and parallel to the web was r — 0.6 inches. / was the length of span in inches.

If M is the bending moment in inch-pounds, W the total centre load (including weight of beam), and K the stress per square inch in extreme fibre, the following formulae result :

Md

Wl

K — —5- and M —-

(12)

(13)

Wld

8/

The experimental values of IV, /, d and / inserted in the above formula give the values of K shown in the table. The coefficient of elasticity, E, was found by the usual formula :

• (H)

Wl³

A$wi

in which w is the deflection caused by W.

The full line is the graphical representation of the values of K given in Table III. Since K must clearly decrease with the

Plate I.

■SOOCiaibs:

4000a

length of span, and increase with the radius of gyration of the section about an axis through its centre and parallel to the web (the latter, of course, being vertical), K has been plotted in reference to / -4- r as shown. No simple formula will closely represent this curve, but the broken line covers all lengths of span used in ordinary engineering practice, and is represented by the formula:

(15)

K = 51,000 - 75 y

For railway structures the greatest allowable stress per sq. in. in the extreme fibres of rolled beams may be taken at :

(16)

k -- 10,000 —

i5T

Values of k taken from a large scale plate, like Plate I., are, however, far preferable to those given by any formula.

Rolled Steel Beams.

The researches of Mr. James Christie, Supt. of the Pencoyd Iron Co., on the transverse or bending resistance of rolled deck and eye beams, are given in great detail in the “ Trans. Am. Soc. C. E.” for 1884, from which the results in Table IV. are abstracted. All beams were rolled at the Pencoyd Iron Works. The “mild steel” contained from o. 11 to 0.15 per cent, of carbon, and the “ high steel ” about 0.36 per cent, of carbon. These steels are the same as those referred to in “Addendum ” to Art! 51.

No. 14 is the only test of a “ high” steel beam ; all the remaining tests being with mild steel shapes. Tests 3 to 9 inclusive were of deck beams, and the skeleton sections show whether the bulbs were above or below in the experiment. The values of K were computed by Eq. (13) in the manner already explained for the preceding iron beams, while E was computed by Eq. (14).

TABLE IV.

Transverse Tests of Steel Beams.

NO.

KIND OF BEAM.

SPAN

IN

MOMENT

OF

INCHES. r INERTIA.

FINAL CENTRE LOAD IN POUNDS.

K IN POUNDS PER SQUARE INCH AT

Elastic.

Ultimate.

COEFFICIENT OF ELASTICITY E, IN POUNDS PER SQ. IN,

VO CO^J C'en    U N

8,-3 g 2 sw 3 3 3

Vnio«î VV.V0,« 00 co 00"o"b"o'n '«'«    "«"n'« 'n '«

■“VH h°°H zi-00-1 H¹

o\ 0co 00    ^05^0, 0 0 0 0 ^vo^ co'invo O'    'S o* O' O\oo vo v£n

2.76

2.76

12

12

37-6

37-6

84.8

82.9

70.2

7°-3

70.2

70.2

2.74

I5°-5

150.5 150-5 150-5

264.7

267.6

273.8

263.7

256.7

257.8

262.6 262.4 264.0

261.7

5.500

8,300

8,800

8,400

14,860

34,000

34,000

14.500

13.500 13,000 12,930 19,480

31,300

11.500

22.500 21,000 19,5°° 18,000

24.500 24,200 22,000 29,000 27,000 34,000 34,ooo 36,700 38,000 43,000

41.100

40.800 50,000 46.900

51.200

47.100 47,IO° 46,000

39.800

37.600

37.500

32.800

40.300

54.300

35,000

35.200

35,000

34.400

33.400

32.500

27.500

35.600

32.100 38,000 37,3°°

37,7oo

36.300

38.400

45,200

45.100 55,000 52,500

54.300

59.3°o

59.300 5T.3°o

48.800

44.400

44.100 39,900

42.800

30,890,000 25,011,000 27,718, 25.489,000 23,692,000 18.765,000 23,040,000 29,923,000 30,209,000 28,889,000 29,055,000 3i,3i3!⁰⁰° 23,689,000 27,5*5,000

28,414,000 27,182,000 29,160,000 29,727,000 30,749,000 29,568,000 29.164,000 30,219.000 30,030,000 29,709,000 28,234,000 27,717,000 28,784,000 27,818,000

Beams 3 and 4 were rolled from the same ingot ; as were also 6 and 7 ; as were also 10, 12 and 13, and, as were also 16, 17, 18 and 19. All beams were broken by a centre load, and they were unsupported laterally in either flange. The moments of inertia were computed from the actual beam sections. The length of span is represented by l while r is the radius of gyration of each beam section about an axis through its centre of gravity and parallel to its web. The values of r were as follows :

5 inch JL

vO -a

•s.

.s *

SS¹

0 o' o d

3 inch I ... .r ... .r

<< «

-4

0.59 inch. 0.88    “

o 95

i.01

The values of K, both for the elastic limit and ultimate, are so erratic in relation to those of l -4- r that no law such as is revealed by Table III. and Plate I. for iron beams, can be discovered. The most marked feature of the Table IV. is the very considerable excess of ultimate and elastic K for deck beams over the same quantities for eye beams. These tests, however, show that for mild structural steel, containing 0.18 or 0.20 per cent, carbon, the working stress in the extreme fibres of eye beams may be taken at 10,000 pounds per sq. in. in railway structures when the length of span does not exceed 150?*, and when the resistance is computed by the exact formula with the moment of inertia.

Addendum to Art. 70.

In “The American Engineer” for March 14th and 21st, 1884, is given a detailed account of some valuable tests of wrought iron and steel built beams. These tests were made under the supervision of Mr. C. L. Strobel, C. E., who gave the data cited. All these girders had the same span of 12 feet, and they were tested in a vertical or natural position. All web plates were nominally 14 inches deep and £ inch thick, and the flange angles were all 3 inches by 3 inches by nominally inch thick. All these beams were broken by a load applied at the centre, at which point vertical stiffening angles were riveted to the web. Five beams were thus tested to destruction with the results given in the following Table.

Tests of Plate Girders by C. L. Strobel, C. E.

12 ft. span, 14 inches depth, 14" x 4" web. 3" x 3" x -fa" flange angles.

BEAM.

CENTRE LOAD.

FLANGE STRESS PER SQ. IN. AT FAILURE.

MODE OF FAILURE.

Upper.

Lower.

Wrought

Iron.

Hard Steel 0.34 C.

:: :: j

Soft Steel 0.11 C.

<«    K    I

a    a f

63,440 lbs. Punched rivet holes.

110,500 lbs. Punched rivet holes.

115.700    lbs. Punched and reamed.

76.700    lbs.

Punched and reamed. 81,900 lbs. Punched rivet holes.

48,400 lbs.

53,000 lbs.

Tearing lower flange.

79,500 “ 80,900 “

87,100 “

88,700 “

Buckling upper “

56,600 “

62,000 “

59)3°° “

65,000 “

Tearing lower “

The flange stress per sq. in. at failure given above was computed in the extreme fibres by the exact formula from the moment of inertia of the entire section less the rivet holes taken out in the tension flange. The chemical analyses of the two steels were as follows :

C.

Mn.

Si.

s.

P.

Soft steel

.11

•39⁶

.019

.046

.088

Hard “

•34

•954

.063

•113

•175

Test specimens from the hard steel gave ultimate resistances from 94,300 to 107,700 lbs. per sq. in. ; elastic limit from 59,-300 to 63,300 lbs. per sq. in. ; stretch from 14 to 26 per cent. ; contraction, 17 to 38 per cent. The corresponding quantities for the soft steel were—ultimate from 60,400 to 62,700 lbs. per sq. in.; elastic limit, from 40,200 to 49,700 lbs. per sq. in.; stretch, from 21 to 28 per cent. ; contraction, from 55 to 60 per cent.

These tests show that, for ordinary railway practice, with such mild structural steel as is generally used for plates and angles, the working stresses in plate girder stringers and floor beams may be taken at 10,000 lbs. per sq. in. gross section of compression flange or net section of tension flange. For the same members in wrought iron, the corresponding value would be 8,000 pounds per sq. in.

Addendum to Art. 73.

A butt joint with a set of single or double cover plates or butt straps may be formed in such a manner that the rivets and cover plates will take very nearly or exactly their proper proportional loads. Each set of cover plates is composed of a series uniformly decreasing in length, the longest of the series lying adjacent to the main plates or members joined. One row of rivets parallel to the joint is then put through each end of each cover plate, and, of course, also through those lying underneath. In this manner the number of rivets from the end of the longest or lowest cover plate to any section parallel to the joint is proportional to the sectional area of the covers against which they pull ; the joint is consequently of nearly uniform resistance.

The number of butt straps or cover plates in a set depends upon the size of the members joined.

In most cases the rivets cannot take exactly their proportional loads, for the reason that those portions of the members joined which lie within the limits of the joint are not of uniform resistance, as the system of covers is.

wmm

fv.-'izmmi

K ft ą ft ó VV

INDEX.

A.

Actual energy of elasticity.................

Adhesion between bricks and mortar........

Alloys of copper, tin and zinc in compression

Alloys of copper, tin and zinc in tension.....

Aluminium bronze in tension...............

American Bridge Co. column...............

Angle iron column........................

Angle irons as columns...................

Annealing, effect of, on wrought iron.......

Annealing, effect of, on steel...............

Artificial stones in compression............

Artificial stones in tension.................

PAGE

................... 95, 96

................. 364, 365

................. 387-389

336, 339-341, 343, 346, 347

343

438, 439, 442, 448

445

475-477

245

298, 313, 319, 331 • • • • 393, 395, 396

........ 363, 364

B.

Bauschinger’s experiments on steel............

Bauschinger’s experiments on wrought iron

Beam, continuous...........................

Beam, one end fixed and other simply supported

Beam, fixed at both ends.....................

Beam, non-continuous with uniform load.......

Beam, non-continuous with single weight......

Beams, solid................................

Beams, solid, cast iron.......................

Beams, solid, cement, mortar and concrete......

Beams, solid, combined iron and steel..........

Beams, solid, copper, tin, zinc and alloys......

Beams, solid, practical formulae for............

Beams, solid, steel...........................

333

262-268

177

182-184

188-190

139, 174 138, 174 514, 515

518-520 537-542 523, 524 524-526

543, 544 520-523

Beams, solid, stone..........................

Beams, solid, timber.........................

Beams, solid, wrought iron...................

Bending by continuous normal load............

Bending by oblique forces....................

Bending in riveted joints......................

Bending moments............................

Bending moments, greatest...................

Bending moments, greatest or least............

Bending moments, greatest, with four weights ..

Bessemer steel, coefficient of elasticity..........

Bessemer steel, ultimate resistance.............

Box beams..................................

Bouscaren’s experiments (columns).............

Brass, fine yellow, in compression..............

Brass, in tension ............................

Brass, red...................................

Brick in compression.........................

Brick in tension.............................

Buckling of latticed columns..................

building stones, natural, in compression........

Built beams, steel...........................

Built beams, wrought iron....................

Bulging of plates............................

Butt joints, double covers....................

Butt joints of uniform resistance...............

Butt joints, single butt strap. See “ Lap Joint.”

PAGE

543

........ 526-536

........ 515-518

........ 679-680

........ 674-679

609-612, 614, 615 ... 129, 132, 133

137

134

•    • • • 735. 736 288, 290, 291 301-303, 321

•    • • • 595, 596 446, 447, 453

389

336, 340-344

350

397

364

... 458, 459 ... 398-402 ... 600, 601 ... 578-601 ... 665-674 606, 633-639

737

c.

Cables or ropes..................

Cadmium.......................

Cantilever.....................

Cantilever with single load.......

Cantilever with uniform load......

Cast-iron columns, hollow round .. Cast-iron columns, solid round....

Cast-iron flanged beams..........

Cast iron in compression..........

Coefficients of elasticity Ultimate resistance .... Effect of remelting.

648-654 351, 352

132

172

.... 142, 172 .... 470, 472 447, 470, 472 •••• 554-559

376

377

378, 379

Cast iron in shearing, Cast iron in tension .

Coefficient of elasticity and elastic limit

Ultimate resistance..................

Effect of remelting..................

Effect of continued fusion............

Repetition of stress.......

Effect of high temperatures

Cast steel, ultimate tensile resistance ....

Cement in compression................

Cement mortar in compression..........

Cement mortar in tension..............

Portland....................

Various brands.............

Cement, pure, in tension...............

Variation of strength with age

Maclay’s experiments........

Grant’s experiments.........

Keene’s cement.............

Parian cement.............

Portland cement............

Chain cables.........................................

Channels as columns.................................

Chemical constitution of wrought iron...................

Chrome steel, coefficient of elasticity....................

Circular cylinder, torsion of............................

Coefficient of elasticity..............................

Coefficient of elasticity, wrought iron in flexure...........

Coignet béton in compression..........................

Collapse of flues......................................

Columns, ends round, flat or fixed......................

Columns, limit of application of flexure formulae.........

Columns, long, flexure of..............................

Columns, reduction of end sections....................

Columns, with flat ends..............................

Columns, with one round and one flat end...............

Columns, with round ends.............................

Common column.....................................

Common theory of flexure....................'.........

Common theory of flexure, unequal coefficients of elasticity

Compression, greatest in bent beam.....................

Concrete in compression...............................

PAGE

487-489. 493

276

276

279

282

284

284

Cast iron in torsion

.... 286, 347 487, 500-503 ... 301, 302 .... 390-392 354. 39°-392 354. 359, 360

360

354

353

355, 356, 360, 361 ........ 357, 359

M W

VO VO

361

• • 357, 359, 360

....... 652-654

». 455, 461, 462

270

286

........ 75, 76

3, 208, 209, 512 ...... 516, 517

396

655-659

191

195, 196

190

479

193

194, 195

194

449

122

199

204

393, 394

Connections.................................

Continuous beam............................

Contra-flexure...............................

Co-ordinates, equations in polar...............

Co-ordinates, equations in rectangular..........

Co-ordinates, equations in semi-polar...........

Copper in tension...........................

Copper, tin, zinc and alloys in flexure..........

Copper, tin, zinc, lead and alloys in compression Copper, tin, zinc, lead and alloys in shearing....

Copper, tin, zinc, lead and alloys in torsion......

Crucible steel, coefficient of elasticity...........

Crucible steel, ultimate tensile resistance........

Crystallization of wrought iron.................

Curved beams, flexure of......................

Cylinders, thick, hollow......................

Cylinders, thin, hollow.......................

PAGE

606

132

131

27

14

20

336, 337, 339. 340, 343-445, 349

.....................525, 526

..................... 386-389

.................... 487, 495

................. 487, 507-509

291

294, 303, 324

259

124-129, 143, 144

•    ....... 36, 38

.......... 36, 37

D.

Deflection by common theory of flexure Deflection of cast-iron flanged beams.. Deflection of wrought-iron "J" beams .. .

Diagonal riveted*joints..............

Diameter of rivets...................

Distribution of stress in riveted joints ..

Drilling holes in steel................

Driving and drawing spikes...........

130

•    558

562, 563

642

618, 619, 625, 631, 632, 634, 635, 637, 639

.............................. 607-615

.............................. 325-332

.............................. 663, 664

E.

Efficiency of riveted joint..........

Elastic limit, elevation of..........

Elastic limit, Phoenix iron specimens

Elasticity........................

Elasticity, coefficient of...........

Elasticity, energy of

Elasticity, limit of.................

Elliptical cylinder, torsion of......

English wrought iron..............

638, 639

2ÓO

JO

to o o -j- 00 O' O in in N N

Eye-bar heads, formation of............

Eye bars, steel, ultimate tensile resistance

Eye beams, wrought-iron...............

Euler’s formula........................

Euler’s formula adapted to columns......

Euler’s formula, limit of applicability ....

PAGE

645-647, 738 ... 297, 738 ... 567-578

193

463-468

477-479

F.

Fatigue of cast iron........................

Fatigue of metals..........................

Fixed-end column.........................

Flanged beams of cast iron................

Flanged beams with equal flanges............

Flanged beams with unequal flanges.........

Flat-end column. See “ Fixed-end Column.”

Flexure................................

Flexure by continuous normal load..........

Flexure by oblique forces... ................

Flexure, coefficients of elasticity for.........

Flexure, coefficients of elasticity unequal......

Flexure, common theory of.................

Flexure, formulae for rupture................

Flexure, graphical method ..................

Flexure, normal stress in ..................

Flexure, theory of..........................

Flow of solids.............................

Flues, collapse of..........................

Flusseisen................................

Fracture of wrought iron...................

..................... 284, 285

...................... 708-722

I9¹» Ï93, 433- 437, 439, 44*, 442

..................... 554-559

..................... 564-601

..................... 546-564

106

676, 680 674-679

512

199

122

512

196

112

106

••• 723-733 • • • 655-659 263, 266-268

258

G.

Glass in compression.......

Glass in tension...........

Gold.....................

Gordon’s formula..........

Grant’s conclusions........

Graphical method for flexure Gun bronze in compression. . Gun bronze in tension......

389

••• 352, 353 ••• 351, 352 409, 430-442

362

196

386

Gun metal in tension Gun wire, steel.....

PAGE

34G 343, 346, 347

322

H.

Hammering, effect on steel ........

Hardening, effect on steel...........

Hardening, effect on wrought iron ... Hay steel, coefficient of elasticity... . Hay steel, ultimate and elastic limits.

Hemp ropes......................

Hodgkinson’s formula..............

Hooke’s law......................

247, 315-319

314

247

....... 292, 293

....... 299, 300

........ 648-652

193, 409, 469-472

I.

Influence of time on strains Intensity of continuous load

719, 722

136

K.

Keene’s cement in tension

Keystone column.......

Kirkaldy’s conclusions....

361

438, 439, 44r

272

L.

Laidley’s tests of timber columns Lanza’s tests of timber columns .

Lap joints.....................

Lateral strains.............

Latticed columns..............

Launhardt’s formulae...........

Lead in compression ..........

Lead in tension...............

Length of test piece, influence of

Limit of elasticity..............

Load, intensity of continuous.... Long column..................

....... 483, 484

........ 481,    482

606, 616-632, 639

455-460 715, 716

389

351, 352 230, 231

209

136

371, 409, 430

Long column, flexure of Longitudinal oscillations

PAGE

... I9O IOO-IO5

M.

Martin steel shapes in tension................

Mill columns, timber.......................

Modulus of resilience........................

Modulus of rupture for flexure or bending......

Moment, greatest bending...................

Moment, greatest or least bending............

Moments, bending.........................

Moment of inertia..........................

Moment of inertia, angle section..............

Moment of inertia, angle section, oblique axis..

Moment of inertia, box column..............

Moment of inertia, channel section, false......

Moment of inertia, channel section, true.......

Moment of inertia, circular section............

Moment of inertia, column of plates and angles

Moment of inertia, deck section..............

Moment of inertia, eye section, false..........

Moment of inertia, eye section, true...........

Moment of inertia, latticed columns..........

Moment of inertia, Phoenix section............

Moment of inertia, rectangular sections.......

Moment of inertia, star section...............

Moment of inertia, tee section . ..............

Mortise holes, shearing behind ..............

Muntz metal in tension......................

32I

48I, 482

97

129, 515, 545

137

134

129, 132, 133 ... 410, 411

417

429

4i3> 414

416

426

423

415

427

421

425

418, 419

424

422, 423

421

42O

664, 665 343, 347

N.

Neutral axis......................

Neutral curve, cantilever.........

Neutral curve, continuous beam ... Neutral curve, non-continuour, beam

Neutral curve, special cases.......

Neutral surface..................

Non-continuous beam.............

177

174

T7ï

114

O.

Open-hearth steel in tension. Oscillations, longitudinal ...

Oscillations, torsional......

Overlap in riveted joint.....

PAGE

297, 306, 311, 326, 327 ............ 100-105

78

615, 626, 627, 635

P.

Palladium.............

Parian cement in tension

Permanent set .........

Phoenix columns.......

Phosphor bronze.......

Pin connection.........

Pin end columns........

Pitch of rivets.........

Plates, bulging of......

Platinum..............

Portland cement and mortar in tension

Potential energy of elasticity.........

Practical formulae for solid beams.....

Pressure on rivets...................

Punching, effect of in riveted joints ... Punching steel, effect of.............

351. 352 354, 361

211

..................... 438, 439, 442, 448-455

..................................    344, 347

..................................    644-648

.................................    437, 439

616, 617, 621, 623,    626,    629,    632-634,    637, 639

.......................    665, 674

........................    351, 352

................ 354, 357, 359, 360

...........................94, 9⁶

.........................    543, 544

613,    619-624, 627-629,    632,    634-637

.................... 615, 624, 634

......................... 325-332

R.

Rail steel .............................................

Reactions under continuous beams.........................

Reaming holes in steel...................................

Rectangular cylinder, torsion of..........................

Red brass in tension..................................

Reduction of column ends...............................

Reduction of resistance between ultimate and breaking points

Resilience..............................................

Resilience, modulus of....................... ..........

Rivet steel............................................

Riveted joints...........................................

Riveted joints, friction of................................

295

157, 171

325-332

59

350

479

269

9^> 97

97

315

606-643

Riveted truss joints........

Rollers, resistance of......

Rolling, effect of, on steel .. Ropes, iron, steel and hemp Round end columns.......

PAGE

........ 64O-642, 737

.... 659-662, 693, 694

............ 315-319

............ 648-652

igr, 194, 433, 439, 442

s.

Safety factor....................................

Sandberg’s conclusions regarding low temperatures ..

Shape iron in tension.............................

Shape steel in tension...........................

Shapes, tensile resistance of......................

Shear, counter...................................

Shear, greatest bending intensity in rectangular beam

Shear, greatest total in non-continuous beam........

Shear, in bent beam..............................

Shear, main.....................................

Shearing behind mortise holes.....................

Shearing, coefficient of elasticity...................

Shearing, greatest in bent beam...................

Shearing of rivets................................

Shearing, ultimate resistance......................

Short blocks.....................................

Siemens-Martin steel.............................

Siemen’s steel...................................

Size of test piece, effect of........................

Skin of bar, resistance of.........................

Solid beams, rectangular and circular..............

Spangenberg’s experiments........................

Specifications, Franklin Square bridge.............

Specifications, Greenbush bridge......•............

Specifications, Menomonee bridge.................

Specifications, Niagara suspension bridge...........

Specifications, Plattsmouth bridge.................

Specifications, railway bridges.....................

Specifications, Sabula bridge.....................

Specifications, steel cable wire. East River bridge ... Specifications, steel wire rope, East River bridge ....

Specifications, steel work for East River bridge.....

Spheres, thick, hollow............................

Spikes, driving and drawing......................

681

253

257

321

238

137

121

137

133

137

.... 664, 665 4, 6, 487-490

206

621, 624, 625, 630-632, 637

................ 490-498

371

324

312

212, 224, 225

233

514, 544

709

694-698

686-688

689-694

688

701-703 699, 700 682-686

703-705

705,    706

706,    707

84

663, 664

750

INDEX.

Square columns Steel columns .,

PAGE

438, 439» 441, 448

448

Steel in compression

380

Coefficient of elasticity............

Elastic limit and ultimate resistance

380, 382, 383 .... 380-385

Steel in shearing Steel in tension.

Effect of tempering.......................................

Effect of annealing.........................................

(For various grades and varieties see text under preceding heads.)

381

382

487, 493-495

286

Coefficient of elasticity......................

Ultimate resistance and elastic limit...........

Boiler plate...............................

Hardening and tempering steel plate..........

Rivet steel...............................

Reduction of section by hammering and rolling,

Annealing steel.............................

Steel wire..................................

Shape steel.................................

Gun wire................................

286

294

306

314

315

315

319

319

321

322

Effect of low and high temperature on steel

323

Constructive manipulations, such as punching, drilling and reaming. 325 Bauschinger’s experiments Fracture of steel........

333

Effect of chemical composition

334

334

Steel in torsion...........................

Steel plate, coefficient of elasticity...........

Steel plate, ultimate tensile resistance........

Sterro-metal in tension.....................

Stones, natural building in compression......

Strain...................................

Strains, influence of time on...............

Strains, lateral.............................

Stress ...................................

Stresses, expressions for tangential and direct

Stresses, greatest at any point in a beam.....

Stresses, plane of greatest normal in a beam.. Stresses, plane of greatest shearing in a beam. Styffe’s conclusions regarding low temperature

Suddenly applied forces or loads.............

Suddenly applied stress, resistance of iron to . Swelled columns..........................

487, 504-507

........ 290, 291, 292

294, 306, 314, 326, 327

343

398-402

719-722

202

204

206

252

98, 100

269

434» 439» 441

T.

Temperature, effect of increase on wrought iron ..

Temperature, effect of low on wrought iron.......

Temperatures, effect of high and low on steel.....

• Tempering, effect on steel......................

Tension, greatest in bent beam.................

Theorem of three moments......................

Theorem of three moments, common form........

Theory of flexure, general formulae..............

Thick, hollow cylinders .... ..................

Thin, hollow cylinders................ ........

Thick, hollow spheres ........................

Thickness of web plate in flanged beams..........

Thurston’s conclusions regarding low temperatures Thurston’s relation between tension and torsion...

Timber beams................................

Timber beams of natural and prepared woods.....

Timber columns, C. Shaler Smith’s formulae......

Timber columns or pillars.....................

Timber in compression........................

Timber in shearing............................

Timber in tension.............................

Timber in torsion .............................

Tin in compression...........................

Tin in tension.................................

Tobin’s alloy in tension.........................

Tobin’s alloy in flexure.........................

Torsion, coefficient of elasticity.................

Torsion, general observations...................

Torsion, greatest shear in circular sections.......

Torsion, greatest shear in elliptical sections......

Torsion, greatest shear in rectangular sections ....

Torsion, greatest shear in triangular sections.....

Torsion in equilibrium.........................

Torsion, moment of circular sections.............

Torsion, moment of elliptical sections............

Torsion, moment of rectangular sections.........

Torsion, moment of triangular sections...........

Torsion of circular section.....................

Torsion of elliptical section....................

Torsion of rectangular section..................

Torsion of triangular section...................

PAGE

247-251, 324, 347

....... 251-254

........ 323, 324

314

204

146

154, 158

143

36, 33 36, 37

84

601-603

253

510

526-536

536

........ 485, 486

.... 471, 480-486

........ 403-408

487, 488, 496-498

........ 365-370

487, 4S8, 509, 510

............ 387

336, 337, 339, 343 336, 338, 339, 341

........ 525, 526

.. 8, 487-490, 498

77

76

56

71, 74, 75

59

43

76

••• 55, 56 70, 74, 75 •• 58, 59

75

54

59

Torsion pendulum...................

Torsional oscillations.................

Townsend’s experiments, flow of solids . Tresca’s experiments, flow of solids

Tresca’s hypotheses, flow of solids......

Triangular cylinders, torsion of........

Tubes as columns...................

Tubes, collapse of...................

PAGE

82, 83

78

725

724

725-727

57 *

473, 475 655-659

u.

Ultimate resistance

Ultimate resistance, elevation of......................

Ultimate resistance of wrought iron along or across fibres

Ultimate resistance, Phoenix iron specimens.........

Unequal coefficients of elasticity, flexure with..........

210

260

242, 243

240

199

w.

Web plate of flanged beam, thickness of......

Welded joints.............................

Weyrauch’s formulæ.......................

Whitworth’s compressed steel................

Wire, brass...............................

Wire, copper... ..........................

Wire, Fairbairn’s tests......................

Wire, phosphor bronze ....................

Wire, Roebling’s tests on wrought iron.......

Wire ropes................................

Wire, steel...............................

Wire, steel gun............................

Wire, Thurston’s tests on old wrought iron . ..

Woehler’s experiments .....................

Woehler’s law............................

Working stress............................

Wrought-iron chain cables. .................

Wrought-iron columns, solid rectangular......

Wrought-iron columns, solid round..........

Wrought iron in compression...............

Coefficient of elasticity..........

Elastic limit and ultimate resistance

601, 603 643, 644 717, 718

305

341, 344 341, 344

257

344

254

648-652

319, 320

322

255

710-714

708

681

.... 652-654

447, 474, 475

.... 447, 470

372

373, 374 374-376

PAGE

Wrought iron in shearing .................................

Wrought iron in tension.................................

Coefficient of elasticity..........................

Effect of size..................................

Rounds and flats...............................

Plates by Franklin Institute committee...........

St. Louis bridge specimens.....................

Values of coefficient to ultimate resistance.........

Graphical representation........................

Ultimate resistance and elastic limit..............

Influence of size and dimensions.................

Values for large bars...........................

Reduction of piles.............................

Influence of length............................

Influence of skin of bar.........................

Shapes.......................................

Large bars and rounds.........................

Specimens from bars, plates and angles..........

Boiler plate...................................

Effect of annealing............................

Effect of hardening..........................< .

Variation of resistance with increase of temperature

Effect of low temperature.......................

Iron wire.....................................

Resistance of shape iron........................

English wrought iron..........................

Fracture of wrought iron........................

Crystallization of wrought iron ... ..............

Elevation of ultimate resistance and elastic limit. . .

Bauschinger’s experiments on the change of elastic limit and coefficient of elasticity......................

Resistance of bar iron to suddenly applied stress Reduction of resistance between the ultimate and breaking point... 269

270

Effect of chemical constitution

Kirkaldy’s conclusions

Wrought iron in torsion.........

Wrought iron in tee beams......

z.

487, 49i-493

........ 212

... 212-223

212

213

215

216

217

220

223

224-226

227

228, 229 230, 231

233

233

234. 235 237-239

241

245

247

247

251

254

I''* CO CO to m in

259

260

262

269

272

487, 498-500

559-563

Zinc in compression Zinc in tension.....

388

336, 339» 343

48

S, 61

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V

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v

■

ł

ł

7

t

f

■

f»

■

-S

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