WYDZIAŁY POLITECHNICZNE KRAKÓW

BIBLIOTEKA GŁOWNA

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STATE OF RHODE ISLAND'S VICTORY BOOL CAMPAIGN FOR THE ARMED FORCES

RHODE ISLAND STATE LIBRARY EXTENSION SERVICE

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ARTHUR H. BLANCHARD.

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IM.

ARTHUR ¥ '«OH.-.nu.

SYMMETRICAL

MASONRY ARCHES

INCLUDING

NATURAL STONE, PLAIN-CONCRETE, AND REINFORCED-CONCRETE ARCHES

FOR THE USE OF TECHNICAL SCHOOLS, ENGINEERS, AND COMPUTERS IN DESIGNING ARCHES ACCORDING TO THE ELASTIC THEORY

BY

MALVERD A. HOWE, M.' Am. Soc. C. E.

Professor of Civil Engineerings Rose Polytechnic Institute

FIRST EDITION

FIRST THOUSAND

NEW YORK

JOHN WILEY & SONS London: CHAPMAN & HALL, Limited 1906

Copyright, 1906

BY

MALVERD A. HOWE •

BIBLIOTEKA POLITECIIHIGZHA

KRAKÓW

ÄOBERT DRUMMOND, PRINTER, NEW YORK

PREFACE.

The object of this book is to present in a simple form the method to be employed in the designing of masonry arches according to the elastic theory.

The entire subject of arches has been fully treated in the author’s Treatise on Arches, in which formulas for special cases and conditions are given. Considering the fact that masonry arches are constructed of materials and under conditions which are more or less uncertain in character, the use of comprehensive or rigid formulas is not necessary or warranted. Consequently the formulas and methods here presented are somewhat approximate, but quite accurate enough for the purpose for which they are intended.

The greater portion of the book is taken up with the solution of examples, giving each step in detail so as to be easily followed by the undergraduate or the engineer who has not the time to review the theory of arches in a comprehensive manner.

The first and second examples have been solved by a somewhat longer method than necessary. This method was used in order to show clearly the several processes and checks.

In the third example will be found the simplest solution of the formulas for the horizontal thrusts and bending moments at the supports presented up to this time.

The numerical and graphical work has been given with such discrepancies as may be expected unless extraordinary care is exercised and many decimal places used. The discrepancies are of no practical importance, as the results are much nearer being exact than any masonry structure -can be built, so as to fulfil the conditions upon which the ■calculations are based.

For the benefit of those who desire to follow precedents and as an aid in making preliminary calculations and estimates, the general data for over five hundred arch bridges have been given in tabular form with references to periodicals, etc., where more extended descriptions can be found Without any doubt many errors exist in this table, which is quite incomplete in some particulars. The data have been derived from many sources and in some cases supplied from drawings by scaling and in others by calculations.

M. A. H.

Terre Haute, July 1906.

CONTENTS

CHAPTER I.

Fundamental Formulas for the Elastic Arch.

i . Angular Distortion Produced by Bending.....................

2.    Changes in the Coordinates # and y Produced by Bending only..

3.    Changes in x and y Produced by a Direct or Axial Stress......

4.    Changes in s, x, and y Produced by a Rise of Temperature. . . .

5.    The Combination of Bending, Axial Thrust, and Temperature

Effects.............:................................

6.    Neglecting the Axial Stress and Assuming the Modulus of Elas

ticity as Constant.....................................

PAGE

CHAPTER II.

Symmetrical Arches Fixed at the Ends.

7.    Conditions which must be Satisfied.........................

8.    Determination of the Horizontal Thrust H1 Produced by Ver

tical and Horizontal Loads and Changes of Temperature. .

9.    The Horizontal Thrust Produced by a Single Vertical Load

Placed at any Point upon the Arch.....................

10.    The Horizontal Thrust Produced by a Single Horizontal Load

Placed at any Point upon the Arch.....................

11.    The Horizontal Thrust Produced by a Change of Temperature.

12.    Determination of the Bending Moment at the Left Support

Produced by any Single Load and Changes of Temperature.................................................

13.    Formulas which Apply for Vertical Loads only...............

14.    A Graphical Determination of mx in (2a) for Vertical Loads. ..

15.    A Graphical Determination of Some of the Factors in the Equa

tion for H1 for Vertical Loads........................

16.    A Graphical Representation of the Second Term in the Expres

sion for Mi for Vertical Loads.........................

17.    A Graphical Representation of Mx for Vertical Loads only.. . .

18.    The Loads Producing a Maximum Mx and the Ordinates Locat

ing the True Equilibrium Polygon for a Single Vertical Load................................................

10

12

!3

16

!7

18

20

19.

20.

21.

22.

The Effect of the Axial Stress for Vertical Loads only........

Loads which Produce Maximum Values of Tx or Radial Shear.

Formulas which Apply for Horizontal Loads only............

A Graphical Representation of Mx for a Single Horizontal

PAGE

23

25

26.

23-

24.

25-

26.

A Graphical Representation of Mx for Two Equal and Symmetrical Horizontal Loads.............................

Arch Ribs for which A is Constant..........................

Formulas for Hi and Mi for Vertical Loads when A is Constant. Formulas for hi and Mi for Horizontal Loads when A is Con-

CO On

27.

28.

29.

3°-

3¹-

32.

33-

34-

35-

36.

3 7-

38.

39-

40.

41.

42.

43-

44-

45-

Formulas for Hi and Mi for Changes of Temperature when A is

Constant.............................................

Effect of the Axial Stress when A is Constant................

Determination of Nx, the Normal or Axial Stress, and Tx, the

Radial Shear at any Point.............................

A Graphical Determination of Nx and Tx for Vertical Loads.. .    33

Fiber Stresses for any Section Reliability of the Elastic Theory when Applied to Steel Ribs. .    37

Reliability of the Elastic Theory when Applied to Ribs Composed of Natural Stone Voussoirs.......................

Reliability of the Elastic Theory when Applied to Plain Concrete Ribs...........................................

Reliability of the Elastic Theory when Applied to Reinforced

Concrete Ribs........................................

Reliability of the Elastic Theory : Summary.................

Depth of the Arch Rib....................................

Empirical Formulas for the Thickness of the Ring at the

Crown in Stone Arches................................

Thickness of Arch Ring of Stone at the Support.............

Thickness of Abutment....................................

Thickness of Piers........................................

Remarks concerning Empirical Formulas.....................

Albula Railroad Practice..................................

The Dead Load..........................................

Dead Load E d librium Polygon following the Axis of the Arch Ring................................................

3i

3²

33

37

41

42-

43

43

46

47

4&

48.

49

49

51

CHAPTER III.

Examples showing the Application of the Formulas, etc.

46.    Preliminary................

47.    First Example: Data.......

48.    Subdivision of the Arch Axis,

49.    Computation of x and y.....

53

53

54

PAGE

56

50.    Computation of Hi for Unit Loads..................

60

51.    Computation of Mi, Vu yu y₂, and y₀ for Unit Loads,

65

cj 2 r Depth of Ring and the Dead Load.................

68

53.    Live Loads and Loads Producing Maximum Moments,

69

54.    Mi, Fj, and Hi for Live Loads.....................

55.    Maximum Moments at Point o Produced by the Live Load... .    69,

56.    Maximum Moments at the Crown Produced by the Live Load. 70

57.    Moment at the Crown Produced by Live Loads 1-10 Inclusive. 72

72

58.    Moment at the Crown Produced by Loads 8-8' Inclusive......

59.    Maximum Moment at Point 6' Produced by Live Loads i-8'

78

Inclusive...........................................

60.    Maximum Moment at Point 6' Produced by Live Loads 7'-i'

73

Inclusive............................................

73

74

75

76

61.    Moment at Point 6' Produced by Live Loads i-xo Inclusive.. .

62.    Moments at all Points Produced by Live Loads 1-8' Inclusive

Determined Graphically...............................

63.    Maximum Moment at Point 6' Produced by Loads j'-i' Inclu-

sive

64.    Fiber Stresses Produced by Dead and Live Loads

65.    Effect of Temperature Changes.................

66.    Effect of the Axial or Direct Stress.............

Graphical Determinations........................

• 79

81

67.    A Check upon the Effect of the Axial Stress for Dead Loads...    84

68.    Effect of Making Spandrel Filling of Uniform Material Weigh

85

ing 100 Pounds per Cubic Foot.........................

87

69.    The Radial Shear.........................................

88

70.    Second Example: Data...................................

88

71.    Subdivision of the Arch Axis...............................

89.

72.    Computation of Hi for Unit Loads..........................

89.

73.    Computation of Mi.......................'...............

96

74.    Values of V1, yu y₂, y₀, etc., for Unit Loads..................

96

75.    Values of Hi and Mi for the Dead Load.....................

96

76.    Location of the Equilibrium Polygon for the Dead Load......

77.    Maximum Fiber Stresses Produced by the Dead Load at Point 1 101

78.    Maximum Fiber Stresses Produced by the Live Load at Point 1 102

79.    Maximum Fiber Stresses Produced by the Dead Load at

103

Point 7..............................................

80.    Maximum Fiber Stresses Produced by the Live Loads at

io5

Point 7..............................................

81.    Maximum Fiber Stresses Produced at Points 1 and 7 by the

106

106

Dead and Live Loads...................................

82.    Temperature Stresses......................................

83.    Maximum Stresses Produced by Dead Load, Live Load, and

108

no

Changes of Temperature...............................

84.    The Axial Stress..........................................

85.    Assumption that the Steel Resists Entire Bending Moment

Due to Changes of Temperature........................

86.    Third Example..............

87.    The Computation of Hi......

88.    The Computation of Mi and M

CHAPTER IV.

Typical Arches.

hi

hi

PAGB

89.    The Rockville Stone Arch Bridge..................

90.    The Bellefield Stone Arch Bridge..................

91.    The Luxemburg Stone Arch Viaduct...............

92.    Approaches to the Thebes Bridge..................

93.    Vermillion River Plain Concrete Arch Bridge.......

94.    Steel Reinforcement in the Form of Ribs...........

95.    Steel Reinforcement other than in the Form of Ribs

96.    Area of Steel Reinforcement......................

97.    Abstracts from Specifications......................

120

120

121

123

123

124

124

¹²5

APPENDIX.

TABLE I.

129

Physical Properties of Stone and Concrete,

TABLE II.

Data for about Five Hundred Masonry Arch Bridges arranged ACCORDING TO SPAN............................

NOMENCLATURE.

-1

"'A

CL

--

00

.'i.

K---+---O'- -

-----Si---f---»,

--------

i I

Vi'

Fig. a.

i7i = the horizontal thrust at the left support for any loading in general and in special formulas for vertical loads only;

hi = the horizontal thrust at the left support for horizontal loads o ly ;

-£Z”j = the horizontal thrust at the left support for changes of temperature;

Ha = the horizontal thrust at the left support produced by the axial stress ;

M\ = the moment at the left support;

M₂ = the moment at the right support ;

Mx = \he moment at any point having the coordinates # and y;

Fi = the vertical reaction at the left support;

F2 = the vertical reaction at the right support; l = the span of the arch axis ;

/=the rise of the arch axis;

x and y — the coordinates of any point of the arch axis;

0o = — 4>o= one half the total central angle subtended by the axis of the arch;

0 = the angular distance to the left of the crown of any point having the coordinates x and y;

r*

OHII

RADY P AîÆEirf3KIEJ

P=any vertical load ;

Q = any horizontal load ;

a — the abscissa of the point of application of P or Q ;

b = the ordinate of the point of application of P or Q ;

yi, yo, and = ordinates locating the true equilibrium polygon for a vertical load as shown in Fig. a;

Xi, and #2 = abscissas locating the true equilibrium polygon for a horizontal load (see Art. 22);

dsi, ÔS2, etc. = finite lengths into which the axis of the arch is divided; dx=ds cos <£; dy=ds sin 0;

I\, I2, etc. = the moment of inertia of the cross-section of the arch rib for divisions 8si, ds2t etc.

etc.;

di, ^2, etc. —-7—

1 = sign of summation, and when without limits the sum is to be taken from o to /;

X

1 = sum from o to x;

£=the modulus of elasticity;

F=area of arch rib at any section ;

e— coefficient of linear expansion for one degree;

t°— number of degrees change of temperature;

/>=unit stress in extreme fibers of arch rib;

^' = unit stress in steel reinforcement in reinforced-concrete ribs.

Mr V1 yo—7T+TFa >

Hi Hi _( Mi\hi

x° V hJVi*

Mi

m₂

yi = H?

y2

Mi

X2 V₂’

Xl Vi’

Nx = Vx sin cf) + Hx cos <£ = axial or normal stress ; Tx— Vx cos (J)—Hx sin $> = radial shear;

Mx=Mi+ Vix— Hiy— 2P(x— a)+2Q(y— b)\ x>a. . . M2—Mi

* b

+ Ri + IQj-........

Mx = Mi + V\x— hiy + Q(y— b), horizontal load; Mi l—x M2 x Hi l ⁺ ~

Vi =

Mx

Hi

Hi l

7YI

— y+T^~, vertical load. Hi

II

SYMMETRICAL MASONRY ARCHES.

CHAPTER I.

FUNDAMENTAL FORMULAS FOR THE ELASTIC ARCH.

i. Angular Distortion Produced by Bending.—Let Fig. i represent an elastic arch which has been distorted so that the angle <£ has become $ — d(J> at a section having the co-

Fig. i b.

Fig. i a.

tyj "

-e-

hi; W"

B,

b' a'

—X--------

\\

; \'

r\

xs

Fig. i.

ordinates x and y. Let the length of the section at x be taken as ds on the neutral axis, and assume that the dis-tortion is confined to this section and produced by bending alone. Then, according to the common theory of flexure, the distortion of the fibers can be represented by Fig. ia, and the forces producing the distortions by Fig. ib.

In Fig. i a, if cd represents the distortion of a fiber distant n from the neutral axis, cd = +n( — d<p), d(f> and tan d<j> being assumed equal for very small angles.

In Fig. i b, the intensity of the stress producing the distortion cd is pn, which may be taken in terms of the intensity p upon the outer fiber, or

np

The moment of pn about 0 upon the neutral axis of the arch is

n²p

npn=—

and the sum of the moments of all of the intensities is

i    =m -,

ax a

Cl a-i

it

where Ix equals the moment of inertia of the section x, and Mx the bending moment at this section.

Let Ex equal the modulus of elasticity of the material at this section; then, since

Ex =

unit stress unit strain’

ds

ds

. . p Exa

-d(f>

ds '

This expression is not exactly correct, as it assumes the length of all fibers before distortion to be ds, while actually each fiber has a different length. Usually the depth of an arch rib is quite small in comparison with its radius of curvature, so that the error is very small.

p

Substituting this value of p in the expression M x=—Ix and solving for dß,

M xds EXIX '

This represents the change in the angle ß due to the distortion at the section x alone. If the effect of the distortion at all sections from A to B, Fig. i, be represented by Aß, then

Aß = -

%Mx8s oExI x "

If ßo is the total central angle upon the left of the crown and — <f>i that upon the right, then ß₀ — <f>i is the total central angle. The change in this central angle due to the distortions of all sections between o and l (where l is the total span subtending the central angle ßo — ßi) becomes

IMJs o ~EJ~*

Jß₀ = Jßi — I

2. Changes in the Coordinates x and y Produced by Bending only.—Let the distortion at the section x be the same as in the previous article, and assume the point A free to move; then, after the distortion, it would be in some

position as C, Fig. 2. x will be increased by $x and y will be decreased by dy.

Y^ î\ \y/^

jp

Vs*

■x~

Fig. 2.

From Fig. 2,

dy:dv::x:r, or rdy =xdv =xrd<f>-dx:dv:y:r, or rdx =ydv =yrd<j>. dy =xd<f> and dx =yd(f>.

Substituting the value of d<f> from Art. 1,

dy =

Mxds

~EJTX

x and dx =

Mxds rr j

X¹ X

y-

The total change in x and y due to the distortion of all sections between A and B is

Jx = 1

*Mxyds

0 EXIX

and dy = I

*Mxxds

0 B XI x

If now * is assumed to equal l, we may write for the total effect of the distortion at all sections upon the span l

dl = +1

lMxyds 0 Exrx‘

If y is assumed as positive when measured upward

and +C is the value of y when x=l, then, noting that y decreases under the particular distortion assumed,

AC - ~ i"~T-■

o E xI x

3. Changes in x and y Produced by a Direct or Axial Stress.

—A direct or axial stress is one producing a uniform intensity at the section being considered; consequently the distortion of each fiber will be the same over the entire section (the modulus of elasticity Ex being assumed constant for the section).

If Nx is the magnitude of the stress and Fx the area Nx

of the section, =p₀ is the unit stress or intensity upon

■F X

the section x.

*—Na—

In Fig. 3 let a portion of the arch rib ds in length be acted upon by the direct stress Nx, and suppose this stress produces a uniform shortening of the fibers ab ; then

pods

»—S3—5

Fig. 3.

*

If lab for all sections between x and o be represented

E =—>

ab

Ex•

ds

by Js, then

0 Ex

If x=l, and since this distortion is in effect a decreasing ■of the length of the arch axis,

Js =

_ rPods

~o Ex • '

In a similar manner

Ax = - 2

tpodx

and Ay = — 2

*Poày 0 Ex •

Also

Al = - A

PoOX

and

Poày

AC = -1

4. Changes in 5, and y Produced by a Rise of Temperature.

—Let £=the coefficient of expansion for a change of i° in temperature ;

/° =the number of degrees of change in temperature ; &s = the length in which a uniform change of temperature takes place. Then

As = et° Ids,

Ax = et° Idx,

and

Ay^ef Idy. 0

If x = l, then

As = et° Ids,

Al = et° Idx,

and

Ac = et° Idy.

5. The Combination of Bending, Axial Thrust, and Temperature Effects. — Combining the formulas deduced in the previous articles,

IMJs

ö EJ*'

A <j)Q = A (fri — 2

Al = 2-

0 r-I x 0 ■‘~y x 0

Ac= -2

Ipgoy

o EJ, i E,

lM Xxds

+ 2et°dy. 0

In comparing the above equations with those given in the author’s “Treatise on Arches,” it is seen that the signs of the terms containing Mx are of opposite character. If we had assumed the upper fiber extended by the bending, the signs would have been in agreement. The actual sign of the term depends upon Mx, so the disagreement is of no importance as long as the terms are consistently of opposite signs.

6. Neglecting the Axial Stress and Assuming the Modulus of Elasticity as Constant.—* The effect of the axial stress is quite small excepting in arches which are very flat. For fixed arches having a ratio of rise to span of ¹ /i₀ the effect of the axial stress is to reduce the magnitude of the horizontal thrust about 30%, while for a ratio of ²/io this percentage drops to about 10%. Formulas which include the effect of the axial stress become somewhat complex, and as its effect can be found with sufficient accuracy for

* See “A Treatise on Arches,” by Malverd A. Howe. John Wiley & Sons, New York.

practical purposes by another method, we will omit the term containing p₀ in the formulas which follow.

Usually the modulus of elasticity of the material in an arch rib is uniform, so that it will be unnecessary to consider Ex as a variable in our formulas. We will designate the uniform value by E,

The formulas now become

— i ¹

A(pQ-Api =-g-IMxA,

Ac = — yZMxxA +Iet°dy, & o

ds

where A in the second member of each equation

1 X

CHAPTER II.

7. Conditions which must be Satisfied.—(a) The total central angle must remain unchanged, or J(j>— A<j>i=o\

(b)    The length of the span must remain constant, or Al =o ; and

(c)    The relative elevations of the supports must remain unchanged, or Ac = o.

Expressing these conditions in the form of equations, we have from Art. 6

1MXA = o,

IM xyA +et° EIdx = o, o    o

(by

and

— IMxxJ +ef Eldy = o. o    o

. . (4

From (I),

x> a

x> a

Mx =M i +ViX - Hiy - IP(x - a) +lQ(y-b).

o    o

We have three equations (a), (b), and (c), containing in Mx the three unknowns Mi, Vand Hi, and consequently their values can be determined under the assumptions made.

8. Determination of the Horizontal Thrust Hi Produced by Vertical and Horizontal Loads and Changes of Temperature.—

Let two equal vertical loads P and two equal horizontal loads Q be placed upon two points equally distant from the crown. (These may be the vertical and horizontal components of inclined loads.) Then

Li =P,

and

MX=M

Hiy + I mx

— Px —

x>a    x>a    .

ZP(x-a) + ZQ{y~b) i , 0    0    )

where mx = the common moment for symmetrical loads on a simple beam supported at the ends. Substituting the value of Mx in (a) and (6), we obtain

M ilA —HilyJ + ImxJ = o

and

M\Iyd —HiZy²d + ZmxyJ +et° EZdx=o.

Multiplying the first equation by ZyJ and the second by

ZJ, eliminating Mi, and solving for Hi, we obtain

ZyJ

et°EZdx + ZmxyA — ZmrJ-

. . co

y 2 A (ZyJ)²

—YT

which is the general expression for the horizontal thrust produced by two equal and symmetrically placed loads and changes of temperature.

Hereafter all summations between the limits / and o will be designated simply by I, as in the equation for Hi above.

9. The Horizontal Thrust Produced by a Single Vertical Load Placed at any Point upon the Arch.—In this case mx = the common moment due to two equal and symmetrically placed loads, or

x> a

Px — 1P(% —a).

Since the loads are equal and symmetrically placed, the value of Hi for one load must be just one half that for both loads; hence

Hi =

lyA

2mxyA - lmxA- ^

(2yA)²    ’ •

where

x>a

mx =Px - 2P{x-a),

or

Hx-

T 2mxA\y-

lyâ

2A

. .    (2 a)

10. The Horizontal Thrust Produced by a Single Horizontal Load Placed at any Point upon the Arch. — In this case mx= the common moment due to two equal and symmetrically placed loads, or

x>a

mx = IQ(y-b).

Let h\ =the horizontal thrust at the left support due to the load upon the left of the crown, and

h₂ =the horizontal thrust at the left support due to the load upon the right of the crown. Then

Hi —hi +&2 J

but

Q=hi -h₂\

hence

2hx =H i +Q

and

hi i +%Q.

Therefore

lyj

*i=7 Ö +

ImxyJ - ImxJ-

• •    (3)

*> o

where mx = 2Q(y —b) * Also,

.    .    (3a)

*1=7 Q+

ii. The Horizontal Thrust Produced by a Change of Temperature.—We have directly from eq. (1), since Idx=l,

Ht =

et°El

r 2, O'j'i) IfA--

2» *

■ ■    (4)

also,

Ht =

et°El

12. Determination of the Bending Moment at the Left Support Produced by any Single Load and Changes of Temperature.—

From (III),

l —x

Mx=Mr

where

mx=P

*    ax-\~-x-P(x-a) +Q(y-b) x> a.

Taking the two conditions that the angle at the center shall remain unchanged and that the relative elevations of the supports remain constant, we have from (a) and (c)

ZMxA = o

and

— ZMxxA + et°EZdy—o.

Substituting the above value of Mx in these two equations, neglecting the temperature term for the present, we have

/ —x

M\Z

l —x

—MiZ

A +M₂Z^jA —HiZyJ + ZmxA =o, xA —M ₂Z~j A +HiZxyA — ZmxxA =o.

yç2

Multiplying the first equation by Z-jA and the second

X

by 2-jA, they become

MiZ^AZjA+MiZjAZjA-H.ZyAZ^A

x²

+ ZmxAZ-rA ==o,

—Mil

I —X

xAZ^-A —M₂2-jA‘Z~jA + HxZxyAZ^A

— ZmxxA Z~A =o.

Eliminating M₂ by adding these equations, we obtain

Mx — Hx

ZyA (^x

Zx²A

ZxA

^ -f ZmxA (x

Zx²A

ZxA

ZA

ZxA Zx² A ZA ~ ZxA

^ I ZxA Zx²A\ IA\~ŻI ~ ZxA )

(5)

=— and ZyAx =—ZyA. Therefore we have

Since the arch is symmetrical in every particular, ZxA l

ZA

ZyA

ZmxA (x

Zx²A

ZxA

ZA

(7

Zx²A

ZxA

)' '

•    (5a)

For changes of temperature, from (a)

ZMXA= o.

From (III),

Mx = Ml-Hty.

Then

MxZA-HtZyA= o,

or

Mx=Ht

ZyA

ZA

(5&)

13. Formulas which Apply for Vertical Loads only.

ImxA

zyĄy--.yt)    ² w(y Ià

where mx for each load considered has the following value :

Hi-4

Im

(2a)

x>a

mx=Px — IP(x —a).

M\ =H

lyA

ImxA^x

Ix²A

IxA

1A

Ix²A

IxA

)’

•    (5a)

where

. x>a

mx = R\X — IP{x —a),

or the common moment for loads on a simple beam supported at the ends.

Mx=Mi

/ —x

+ M₂j-Hiy+mx, . . . (Ill)

where

x>a

mx=R\X — IP(x —a) . . . ;

V\ =

m₂-m

~ +Ri,

where R = IP

l —a

= the common reaction for loads on a

simple beam supported at the ends. For symmetrical loading

Mx-Hi

lyA ImxA

IA

. .    (50a)

14. A Graphical Determination of mx in (2a) for Vertical

x>a

Loads.—The equation mx =Px - lP(x—a) may be represented graphically as follows: Lay off a load line 2P in length, and with a pole distance of P construct the ordinary equilibrium polygon as indicated in Fig. 4. Since the

---Or--4»---CL-

0:

o. ■i

*---

---5---

____l____

Fig. 4.

loads are equal and symmetrically placed, the reactions are equal and each equal to P. Then in the equilibrium polygon the ordinate ab, before any load is reached, equals x. The moment mx=H(ab) = Px; hence the ordinate ab represents the true value of mx for P = unity.

The ordinate a'b' between the loads equals a and mx = H(a'b') =Pa = R₁x—P(x—a), and as before the ordinate a'b' represents the true value of mx when P=unity.

From the above construction it is evident that when H1 is desired for any single load the graphical construction is quite unnecessary, as mx always equals Px or Pa on the left of the center. Since the equilibrium polygon is symmetrical for each value of mx on the left, there will be a corresponding value upon the right.

In case the values of mx are desired for a combination of loads, the method of procedure is essentially the same as outlined for one load. Lay off a load line equal to

twice the loads for which mx is wanted. Opposite the center of this load line take a pole at any convenient distance H, and construct an equilibrium polygon in the usual manner. The value of mx at any point equals the ordinate of the equilibrium polygon at that point multiplied by the assumed H. In most cases it is more satisfactory to compute the values of mx.

15. A Graphical Determination of Some of the Factors in the Equation for H1 for Vertical Loads.—The expression (2a) in Art. 13 may be written

H\ —

lyAl mx -

ZmxA'

IA

2 lyA^y

lyA

1A

'b

vTfT

39. Thickness of Arch Ring of Stone at the Support.—For

LIVE LOADS.

Dividing both numerator and denominator by 1.34, %

COMPUTATIONS FOR

KEY TO REFERENCES.

PUBLICATIONS

18

vTfT

ImA

VA

ZyA

VA

4---

Fig. 5.

Let ABC, Fig. 5, represent the axis of the arch. Compute lyA

~YJ~ and lay °ff its value upward from A and C. Then

draw DE. The heavy ordinates will be the values of lyA

y~~YÂ~‘

In like manner let A'B'C' represent the equilibrium

x>a

where mx = RiX — 2P(x—a).

A'r

yyi

polygon where the ordinates are indicated in Fig. 6. Then the heavy ordinates represent

Draw D'E' as

mx -

2mxA'

IB'

D'

H22f

2mx/A

H2M

Fig. 6.

16. A Graphical Representation of the Second Term in the Expression for M\ for Vertical Loads.—The second term of (5a) for convenience we will designate as mi, or

lmxA (x

lx² Ai

IxA

m-i =

ImxA

2mxA{x —-J/)

lx²A

1A\ — -

1A

2A

/i 2x²A\ ’ \2 ~ 2x4 )

-

a'

m, J K......rc-

-pr \

¹b'K—

\a‘

-as-

..........

;a"

B"

U-

J_

1 Eh

J H

■:

Fig. 7.

Let the common equilibrium polygon for the given loads be represented by B'AB" in Fig. 7.

We will now prove that when the line A'A" is drawn, so that l(aa')A = o and l(aa')xA =0, the distance A'B'=

m\

When 1 (aa') A = o it at once follows that 1 (ab) A = l(a'b)A. From Fig. 7,

aa' = a'b — ab = a'c +cb — ab.

m 11 —x

-a'c=—²-

aC H L

and cb =~rr

Hence, since ab =

, m₂ * wtj / —# Wj

aa=JTl⁺H~-H’

multiplying through by A,

mx.

~ irA’

. . nt₂x mx l —x aa A — h lA + H l A

also,

(aa')xA —TT JA +

m 1 (/ — x)x mTx

J.

Making l(aa')A — o and l(aa')xA =0 and eliminating jj-between the resulting equations, we obtain

lmxxA — lmxA

lx²A

lx A

lx(l —x)A — 1(1 —x)A

lx² A

lx A

This readily reduces to

y nil =

ImxA [x

Ix²A

IxA

Ix²A

IxA

)’

the second term in the expression for Mi in Art. 13.

From the above demonstration it at once follows that

m₂ =

ImxA (j. —x

Ix²A

IxA

IA

(7

Ix²A

IxA

In Fig. 7

FG =

mi +m₂ ImxA

IA ’

and

A'K =A"K =

ImxA(x —%l)

IA

/i Ix²AV V 2 ~ IxA )

17. A Graphical Representation of Mx for Vertical Loads only.—From (III),

Mx Mil —x M₂x mx

HT⁼⁼H~i~TJrH~il~y^JT{

In Fig. 8 let ABC be the axis of the arch and A'b C the equilibrium polygon for a single load drawn with a

yyi\

pole distance of H1 and located so that A'A" = 7yand

Hi

Then

AA'=A'A"-AA" =

nil TyJ

JTi ~YT]

or

Hi(AA’) =mi-Hl^- = -Mi. In like manner H(CCf) = +M₂.

M\

Let -TT —yi and

n i

M₂

j- =y₂. Then

Hi

Mx

l —x

irryi

x tmx . \

+yrj-y + (jfi=be)

ł/'

= —df + ef — ad + be= —ab.

Therefore Mx=Hi{ab), or the bending moment at any

'a

! if

K—

~~x

L_L_

i®

ic

,0*

2yA

2A

\fa y>

rg

</

i - — —----J— ♦ — —----05* — ~ — - -j -4 — — - — — * —— — - >J

y----£»+-*•-----------t>j    !

P........

es

-'T

le,—

>yt

o)

v.

-?‘T:

■v

Fig. 8.

point equals the ordinate between the axis of the arch and the true equilibrium polygon.

Usually the ordinate ab is so small that no very accurate results can be obtained from a drawing. From the

above demonstration it is evident that

C Ó £? ~

ab =ac —cb =

\mx niil—x m₂ x

HI'

quantities which can be quite accurately determined from a large-scale drawing. However, more satisfactory results will always be obtained by algebraic methods, using graphics merely as a check.

18. The Loads Producing a Maximum Mx and the Ordinates. Locating the True Equilibrium Polygon for a Single Vertical Load.—In Fig. 8 take moments of all the forces upon, the left of b about b, or

—Hiy₀ + Vri<2=o.

M, Vx

y°=~Hi+ma’

which becomes

yo =yi +

Via

nr

Since yi and y₂ are known, Art. 17, the equilibrium polygon is completely located.

Assume that we wish to determine the loading which will produce the maximum positive and negative moments, respectively, at the point K, Fig. 9. Now, since the mo-, ment is proportional to the ordinate between the arch axis and the equilibrium polygon, it is evident that the moment will be zero for any load which has its equilibrium' polygon passing through K. As shown in Fig. 9, the shaded portion of the span loaded will cause one kind of moment and the unshaded portion loaded will produce.

SYMMETRICAL arches fixed at the ends.

²3

the opposite kind. In case the moving load is a uniform load these two moments will be greatest at this point.

For arch ribs which do not have too great a variation in section from the crown the absolute maximum moment

-0

0.

'A

' Vo

Vo

Ai

i.

'c

<-•

-Vi

>-y₂

Fig. 9.

between the crown and the support is between 0.25 and 0.35, the span for uniform moving loads, while the greatest moment of all is at the support.¹

It also appears from examples solved in detail that sensibly the same loading can be used in both cases. The division of the loads is indicated by the sign of Mi, the moment at the support. Loads which produce positive moments at the left support will produce negative moments, at about the three-quarter point of the span.

19. The Effect of the Axial Stress for Vertical Loads only.— The effect of the axial or direct stress is to shorten the arch rib, Art. 3, and may be considered, with a close degree of

approximation, to a certain drop of temperature. Consequently, if we can determine the horizontal thrust produced by axial stress due to any particular loading, we can compute the resulting stresses in the arch rib. We are not concerned with the actual magnitudes of the axial stress at the various points of the rib if we can find the horizontal thrust, The moments and stresses will at once follow by methods outlined for temperature changes.

Formulas which include the effect of the axial stress show that in the expression for Hi the numerator is so slightly affected that the axial stress terms can be neglected without serious error.*

For convenience let N represent the numerator of #1; then the common expression is

Hi =

2Syt{y-y£)

With the effect of the axial stress included this becomes

Hi =

Leti/a=the horizontal thrust due to the axial stress;

dx

+ 2lj=r cos 4>

then

Ha=Hi-Hi'f

Ha Hi' or Hi ~l Hi •

Ha = H

*sy\y-¥z)

JyJ

dx

I. (6)

1J

+ 2 I-pr COS 4>

* “A Treatise on Arches,” by Malverd A. Howe. John Wiley & Sons, New

York.

This value of Ha, which is quickly obtained, is to be treated as the horizontal thrust due to a drop of temperature which would produce a thrust of equal magnitude.

20. Loads which Produce Maximum Values of Tx or Radial

Shear.

Tx = (Vi - IP)cos $ -Hi sin <£.

For loads upon the right of B, Fig. io,

TX = Vi cos (j> —Hx sin 0.

If Si is normal to the radius passing through B, it is evident from the figure that Tx= o, since Vx cos $ = Hi sin <f>. Hence all loads upon the right of P" will produce one kind

p'

p"

\<k

4>

\ (f>

90°

<p

■V---<

Fig. io.

of shear and those upon the left the opposite kind until P> is reached. Since Vx-IP results in a downward force, the loads upon the left of B produce the same kind of shear as those upon the right of P". The fields of loading

producing the. same kind of shear are clearly shown in Fig. io.

2i. Formulas which Apply for Horizontal Loads only.—

From Art. io,

e+

JmxAly —

IyA'

JA

•    •    (3°)

where

mx = JQ{y -b). y>b.

From Art. 12,

hlIJ

Imxd(x

Ix²A

IxA

IA

Ix²A

• (50

JlxA

. where

Mx=Mi

Vi-

mx=Qj + IQ(y-b).

— X    X    b x^>a

I—l~^2y —hiy +Qjx + JQ(y —b),

₊4

y> b.

I-

M₂ -M

The above formulas are for a single horizontal load which produces a thrust at the left support. In practice the reverse may be the case, but the solution of the equations presents no difficulties if care is taken to give mx its proper sign. Of course, when there is a thrust at the left support there will be a pull at the right support.

22. A Graphical Representation of Mx for a Single Horizontal Load.—From (III),

y>b,

Mx =Mi + Vxx -hxy + Q(v -b).

CO

o\

⁹,

cj>

p\a

,cP

LiJ

of "~c*

"TÔ-—6-

.0—-J--

7.________' x.

LÜ

*5"

•a

Pa

f----Xt----

v,

rca-'r-

i---------y

v*

Fig. ii.

For all points between x = o and x=a Mx =Mi + Vxx-hxy,

or

Mx Mi

“ h\ +Vlhx

-y-

Let the equilibrium polygon be constructed as shown

Mi , M₂

Xi =tt~, and x₂=t/—.

V i    V 2

Ftx

On the left of Q, cd=yx    bc=~hT’ hence

Mi

M₂

in Fig. ii, where yi=-^y, y2=^-

ilh

= a6 =cd + c& —y.

hi

For points upon the right of Q we can write

— Mx ~ M2 4- F₂F ^2%

—=b'a'=c'd'+b'c'-y.

Mx

or

_MxJÄ2 V₂x' h₂ h₂

h₂

-r

m₂

c,d'=y₂= 1-, b'c' =

f₂*'

h₂

h₂

h₂

The character of the bending moment is clearly shown in Fig. ii by the shaded areas. The points E, E'} etc., are the points of zero moment.

From Fig. 11,

%1 :y 1 : :*1 + x₀:b,

or

xl I MA h *0 = (6-?.)-=(/>-7^)

If x₀ is computed, it will check the previous work for beating the equilibrium polygon.

23. A Graphical Representation of Mx for Two Equal and Symmetrical Horizontal Loads.—In (3a) mx for the left

co

rct^—

h? Q

Fig. i2.

load (Fig. 12) will evidently equal mx for the load upon the right, but will be opposite in character; therefore

h₁=Q.=h₂ in magnitude. and h₂ will be opposite in direction.

Also, from (5c),

lyA ^EyA M1 —h 1—rr~7 = Q v . .

2 A 2 A

From (III),

x>a

M X=M 1 +Vix-hiy + lQ{y-b),

which becomes, since V\ =0,

y+y-b\ =Q(^-bj

for all points between the loads, and

*=g{

lyA

IA

-y

for all points between the support and a load. This is shown by the shaded ordinates in Fig. 12.

ds

24. Arch Ribs for which A is Constant.—Since A =—, it

is evident that if we so divide the axis in parts of ds 1, ds2, etc., in length, that the quotient of each ds by the moment of inertia of the section of the rib for this distance is constant for all sections, the value of A will be constant. Under this assumption the formulas to be given later can be applied to—

i° Arch ribs of constant cross-section when the axis is divided in equal parts, each ds in length,

2⁰ Parabolic arch ribs for which El cos cj> is constant

* When the span is divided into n parts dx each and x— >H, ~dx, —ox, etc.,

and

when the span is divided into equal parts each dx in length.

os

3° Any arch rib for which -j is constant when the axis

is divided into spaces ds, dsi, ds₂, etc., so that the moment of inertia (usually taken at the center of each division) for each division bears a constant ratio to the length of the division ds.

25. Formulas for Hi and M1 for Vertical Loads when A is

rt »

Constant.—Remembering that    = n, the number of div

isions, we have at once from (2a), Art. 13,

²

( Zmx\

1    Zy\mx — —    )

2    lyiy-yA

Zmx

1    Zmx(y-ya)

2    Zy(y-ya)

(2 b)

where ‘mx=Px — ZP(x —a), x >a, and

* Mi =H\ya —

Also

( l Zx²\ n\2 ~ Zx)

Mi

m₂

= Hiya-

Zmx

Zmx[ x —

n ± (l Zx²\

n\2 ~ Zx)

.    (5d)

n— i „

—r⁹*-

x>a

where mx=R\X— IP{x—a) and 2x = \nl. (See Art. 88.) For any symmetrical loading

. . (5 dd)

n

Hi =\(total load).

26. Formulas for hi and M1 for Horizontal Loads when A is Constant.—From Art. 21,

Imx(y - y g) 1 2y(y~ya) J’

.    -    (3b)

*>-40+

X

where mx = 2Q(y—b), y>b.

2m

Mi =hiya-

(    2x²\

\x~^j

/1    2x²\ ’ *    *

(2 ~ lx )

• • (5*)

b x

where mx = Qt + 2Q(y—b), y>b, and

lx = \nl.

For any symmetrical loading

Mi=hiya. .    .    .

hi=\ (total load).

(See)

27. Formulas for Ht and M1 for Changes of Temperature when A is Constant.—From Art. 11,

Ht =

e1°El

A2y(y-ya)'

From Art. 12,

Mi =Htya.

28. Effect of the Axial Stress when A is a Constant.—From Art. 19,

Ha=H ,/i-

^y(y-ya)

r ,    N    COS 0 ’

2y(y-ya) + 2 FA /

where /fi is the horizontal thrust obtained from formulas which neglect the effect of the axial stress.

29. Determination of Nx, the Normal or Axial Stress, and Tx the Radial Shear at any Point.—In Fig. 13 let Sx be the

<P■

<r.

-'90°.

H* J

<3/

Fig. 13.

stress in the equilibrium polygon in position and magnitude ; then we have at once

Nx = Vx sin (f> +HX cos <j>,

x>a

*>a

where VX = V\—IP and H x =H\ — IQ.

Also,

Tx = Vx cos    sin (f>,

Vx and Hx having the values given above.

30. A Graphical Determination of Nx and Tx for Vertical Loads.—In Fig. 14 let S₂ be the side of the equilibrium

or

Si

h"

CLc’

a.”

"%

CL”

ay

"K

CO

ICO

C/5

j

*$»

or

-Fs*-

sr

Ht

I O»

Fig. 14.

polygon cut by the section where Nx and Tx are desired. From the pole in the force diagram draw a line normal to the section, and at the upper extremity of S₂ drop a perpendicular upon this line, forming a right triangle with 52 as the hypothenuse. The two legs of the triangle will be the magnitudes of Nx and Tx, as indicated in the figure.

31. Fiber Stresses for any Section.—(a) In the case of a steel rib, to which the formulas given above probably more nearly apply than for ribs of any other material, the formula based upon the common theory of flexure may be used. This formula may be written

p - F ± I    p ±MXp,

where p=the stress in the outer fiber;

Nx= the axial stress or the normal component of the resultant stress upon the section being considered ;

.F=the area of the section;

Mx =the bending moment at the section;

z = distance of outer fiber from the neutral axis ;

I = the moment of inertia of the section ;

5=—== the “section modulus.” z

The above formula considers that the modulus of elasticity E is constant throughout the section for all intensities which do not exceed the elastic limit of the steel.

(6) If the arch rib is composed of natural stone vous-soirs, it will be incapable of resisting tension at the joints owing to the uncertainty of the adhesion between the mortar and the stone. Consequently the above formula applies only when the resultant pressure upon any joint lies within the middle third of the joint; that is, the entire joint or section will be in compression.

In case the resultant does not lie within the middle third but does lie within the section we may yet have a perfectly stable structure. Suppose that the resultant cuts the section outside the middle third but not outside the stone, as in Fig. 15.

N* (—(i—->j

<-

Fig. 15.

Let d = distance from edge C. The pressure may be assumed to be uniformly varying from C towards A, so that N x will pass through the center of gravity of the intensities ;

or

then

Nx

₌PÇD₌îf₌lpiiB_xo)i

^    3^ '

As long as is so small that there is no danger of the stone being crushed the arch is stable. It is a recognized fact that this condition exists in a large number of arches now standing.

(c)    Arch ribs constructed of plain concrete are capable of resisting a limited amount of tension, but it is better to treat them the same as if of natural stone. The ring may-crack entirely through and yet be perfectly stable. Small rods of steel distributed laterally and circumferentially near the surfaces of the rib will prevent a considerable number of small cracks which might be produced by change of shape after removing the false work or changes of temperature.

(d)    Reinforced-concrete ribs have circumferential steel rods or bars placed a short distance from the upper and lower surfaces of the rib to resist any tension which may occur. Even in this case the best designers limit the equilibrium polygons for dead and live load to nearly the middle third of the ring, so that there will be no tensile stresses.

The actual distribution of stress on a section of reinforced concrete is at present unknown. Many experiments have been made upon beams reinforced at the bottom, and various formulas advanced to aid in designing such beams, all giving fairly rational results. The elastic theory of the arch assumes that the linear arch is the neutral axis.

of the material arch, and any great departure from the assumed form will affect the stresses; hence, since the experiments upon beams indicate that the neutral axis shifts for different loadings, it is evident that great refinement either in the calculation of stresses or the distribution of stress over a section is entirely out of place.

In the Melan system of reinforcement steel ribs are used spaced about 3 feet on centers. Here the steel may be assumed to resist the bending moments, and the concrete the direct compression. The concrete also prevents the steel ribs from buckling. It is questionable if the above assumption actually obtains. It is well on the side of safety, however.

One of the simplest methods in use merely replaces the steel reinforcement by an equivalent area of concrete and then employs the formula given above.

If the modulus of steel is Es and that of concrete EC}

Ea

then the equivalent area of concrete will be    times the actual area of the steel. The fiber stress in the steel will actually be n times the fiber stress found for concrete in the position the steel occupies.

If a equals the area of the steel and A the area of the concrete, then

F = A +na, uz

I ---h nad²

-b=i—>

1'

ł<

A...

and hence

Nx

Fig. 16.

A +na

±M*JćI

7- + 2 nad²

o

This formula assumes that the concrete resists tensile stresses which it is not capable of doing to any great extent, its tensile strength being somewhere near one tenth the compressive strength.

The above formula may be applied until the maximum safe tensile strength of the concrete or steel is reached, and then the method employed for stone arches when the resultant pressure lies within the ring until the safe compressive strength is reached.

All of the methods are quite approximate for reasons given above, and since the modulus of elasticity of concrete is not constant.

32.    Reliability of the Elastic Theory when Applied to Steel Ribs.—There is but little doubt that the theory is correct for solid steel ribs having a depth which is comparatively small when compared with the radius of curvature, when the loading is applied at isolated points through vertical posts which are unbraced in the plane of the rib. The modulus •of elasticity of steel is quite constant and it is capable of resisting both tension and compression. The deformation of steel either under direct stress or bending follows very closely that found by theory. In truth the theory is probably as exact for steel arch ribs as the common theory of flexure is for steel beams.

33.    Reliability of the Elastic Theory when Applied to Ribs Composed of Natural Stone Voussoirs.—Here we have a material which cannot be trusted in tension ; this is especially true of the joints between the voussoirs. In direct compression the modulus of elasticity is not constant but varies with the load, and then not according to any very definite law. However, within narrow limits it may be considered as constant without serious error. Such being the case we may apply the elastic theory with confidence as long as the equilibrium polygon lies within the middle third of the ring, or when every section or joint is subjected to compressive stresses. We may also consider the theory as applicable when the polygon lies within the ring, provided the compression is not sufficient to crush the stone.

In case the equilibrium polygon passes without the ring at any joint, theoretically a free arch ring would fail. In practice this condition often obtains in stone bridges, yet they do not collapse or show serious signs of failure. It is triie that some joints open slightly, but this appears to have little if any detrimental effect. This apparently proves that the elastic theory cannot be applied under such conditions. It is no fault in theory, but a failure to carry out in practice the assumptions made in applying the theory or basing the application of the theory upon wrong assumptions. For example, the elastic theory assumes a free rib capable of changing shape under various loads, while in practice the great majority of stone bridges have the ring securely clamped beneath the solid spandrel walls and by a mass of concrete backing of varying thickness. Such a structure may be said to become more and more stable under an increasing uniform loading, until the safe crushing strength of the arch stone is reached. This backing exerts a great passive force preventing any upward movement of the arch ring. It is evident, then, that if the ring is stable under the elastic theory assuming a free ring, it will be quite safe when clamped as explained above, and furthermore it does not necessarily follow, because the equilibrium polygon lies without the ring proper

at some joint, that the arch will fail, for the spandrel masonry will prevent a change in shape of the rib to any great extent. The question at once presents itself: What does happen? Probably the masonry readjusts itself until equilibrium exists, the arch joints are compressed unequally, and the friction of the spandrel masonry aids, very materially in reducing the opening or compression of the joints at the extrados-—in fact introducing an effective tension or compression, as the case may be.

Again, in bridges having a considerable depth of side wall above the crown a large portion, if not all, of the ring under the walls might be removed in many cases without complete failure, the wall masonry forming an arch in itself. In conclusion, for the dead and live loads the arch ring which is safe when assumed to be • a free ring will be safe under the usual construction of the spandrels, or if the loads are transmitted to the ring through verticals as in steel structures. All arch rings should be so designed, using a factor of safety of ten for the crushing strength of the stone.

Provision for the stresses produced by changes in temperature was entirely neglected by the old builders, and for that matter by practically all modem builders. A temperature change of but ±40° F., according to the elastic theory, produces a very wide range of stress both of tension and compression. These are a maximum at the supports. If any considerable change of temperature actually occurs and the elastic theory can be correctly applied, the arch ring, if free, should collapse. As stone arch bridges have stood for thousands of years without failure, we must conclude that either the stone does not

change in temperature through anything like the range of change in the air, or the arch ring adjusts itself with the aid of the spandrel masonry so as to resist the temperature stresses without excessive unit stresses, or the theory does not apply. Probably all three conclusions are more or less true. Even in the Northern States it is doubtful if any of the stonework, excepting possibly the more exposed surfaces, has a change of temperature of a great range,— ± 20° F., say. The ring without any doubt adjusts itself to suit new conditions.

To show how small a change would be required in the mortar joints alone to provide for a change of 40° F., take a free rib of granite having a span of 60 ft. and a rise of 8 ft. (measurements taken for the axis). The length of the rib axis is 62.8 ft. The coefficient of expansion for i° F. is 0.0000038. Then the total change in length of the rib is 0.0095 ft.; if there are 42 joints, the change in each joint would be 0.0002 ft. .0024 in., which is too small to be readily detected. Of course the joints do not all distort the same. Again, assume the rib under masonry spandrel walls, and let there be an increase of 40° F. in temperature, and also assume that the rib cannot rise; -then the entire temperature effect must be used up incompressing the ring. The change in length per unit is 4o(.0000038) =0.00015. If the E — 6800000, the stress per square inch is a little over 1000 lbs. This might be increased to 10000 lbs. without the granite being crushed, .even with the dead- and live-load stresses added.

our

Considering our ignorance of the actual temperature • changes and the behavior of the stone under these changes, ¹*t is useless to attempt any theoretical treatment until

knowledge of the subject has been very much increased. The temperature stresses appear to be able to take care of themselves as long as the rib is stable for the dead and live loads.

34. Reliability of the Elastic Theory when Applied to Plain Concrete Ribs.—Here we have a material which is fully as variable in its physical qualities as natural stone. Generally we have no joints to consider and no masonry spandrel backing, but we do have monolithic spandrel side walls clamping the rib, in many instances. As concrete resists tensile stresses but indifferently, it is not safe to permit more than one tenth its safe compressive strength in designing. As this amounts to about 50 lbs. per square inch, it may as well be neglected entirely, and the rib designed for the dead and live loads so that no tension can exist at any section.

The effects of changes of temperature are as uncertain as in stone arches. Having no joints, the ring cannot readily adjust itself, and hence probably resists some tension. As the modulus of elasticity is much less than for natural stone, and the coefficient of expansion but some 60% greater, the theoretical stresses are very much smaller. For free rings no tension exceeding 50 lbs. per square inch should be allowed under any conditions, unless the concrete is reinforced with steel to prevent cracking. At present there appears to be no rational way of determining the amount of steel required so all that can be done is to experiment and follow previous builders where they have been successful. If the rib should crack through, it would not necessarily mean failure, as then the behavior would follow that of a voussoir ring.

35.    Reliability of the Elastic Theory when Applied to Reinforced Concrete Ribs.—Concrete when reinforced with steel is very much more reliable than concrete without the steel. The principal difficulty experienced is the location of the neutral axis of any particular section. The location without any doubt shifts about under the action of different loads. As the elastic theory assumes the arch axis to pass through the neutral axis of each section cf the rib, it is evident that we must assume the axis to lie .at the center of gravity of the section and treat the material according to the common theory of flexure.

While a reinforced rib will safely resist tension by virtue •of the steel, yet the best designers so proportion the arch rib that it is never subjected to tension under dead and live loads. For temperature stresses the compression in the concrete must not exceed about 800 pounds per square inch, including the effect of the dead and live loads. Under this assumption the concrete may crack •on the tension side, and the steel resist all of the tension.

Even when considering the difficulties briefly mentioned above and our almost absolute ignorance of the actual distribution of stress over a reinforced section, we are compelled to accept the elastic theory as our best guide in designing reinforced-concrete ribs.

36.    Reliability of the Elastic Theory : Summary.—For steel ribs it is without doubt quite reliable. For natural stone ■concrete, and reinforced concrete the theory can be used with confidence as long as no tensile stresses occur in the rib. When tensile stresses obtain the theory applied under the usual assumptions is but an approximation.

37. Depth of the Arch Rib.—This must be assumed from the best data available, and then calculations made to see if it will answer under all conditions of loading and changes of temperature. If found necessary, the rib can be modified somewhat without making new calculations by changing the moments of inertia of all sections in the same ratio. The dead- and live-load stresses will remain sensibly unchanged, the change in weight of the rib being very small in comparison with the total dead load. The temperature and axial thrust stresses will be slightly modified. The question of the necessity of a new calculation must be decided by the designer according to his best judgment. In Table II are given the data for a large number of arch ribs to aid in assuming the dimensions of a proposed design.

The articles immediately following give the principal empirical formulas for the dimensions of arch rings, etc.

38. Empirical Formulas for the Thickness of the Ring at the Crown in Stone Arches. — Many formulas have been advanced for the depth of the arch ring at the crown. These are usually based upon the dimensions of arches constructed, and hence they merely indicate that an arch built like one which has been standing some time will probably stand also.

NOMENCLATURE.

£₀= depth of arch ring at the crown, in feet;

R= radius of curvature of intrados at the crown, in feet; /= clear span of arch, in feet;

/ = clear rise of arch, in feet.

Trautwine’s Formulas²—The following formulas apply to circular and elliptical arches:

For first-class cut stone :

to = o. 25VF +0.5/ +0.2. For second-class work :

f₀=o.28iv/^+ 0.5/+0.225. For brickwork or fair rubble :

t₀ =0.333V R +0.5/ +0.267. Low's Formula: f

to =o.i25\/io(/ — f) +2 H,

where H =the surcharge above the extrados at the crown. Rankine's Formulas:

t₀ = Vo.i2R for a single arch ; to = Vo.ijR for an arch in a series.

Perronnet's Formula for circular or elliptical arches:%

to— i +0.035/.

For l=-

Dejardin's Formulas for circular arches :% to = i +0.10 R.

For T

CM w

/₀ = i +0.05 R.

*0 = 1 +0.0357?.

For *-

/ IO

to — I +0.02 7?.

Dejardin's Formula * /or elliptical and basket-handled

arches:

For y- =— /    3

*0 = 1 +0.077?.

Croizette-Desnoyer's Formulas: *

For f >

*0 =0.50 +O.28V 27?.

FOTf = 6-

to =0.50 + 0.20\/ 27?.

For f = — l 12

to =0.50 +0.20\/ 27?.

For elliptical arches use 7? for circle having same rise and span.

German and Russian Practice: *

*o = i +0-035/ +0.0277",

where 77" = the surcharge over the extrados at the crown, including the moving load if any.

Austrian Specifications for large arches of brick and stone: f

/// between \ and f.

For l— 3o metres. . . . *0 = 1.1 m.

For / = 40 For / = 6 5

. .. . *0 = 1.4

.    . . *0 = 2.2

*From paper by E. Sherman Gould, Van Nostrand’s Mag., vol. xxix, p. 450.

t “A Treatise on Arches,” by Malverd A. Howe. Wiley.

For / = 80 metres. . For / = 100 For 1 = 120

¹

“A Treatise on Arches,” by Malverd A. Howe. John Wiley & Sons, New

²

Trautwine’s ‘ Engineer’s Pocket-book.” f Engineering News, June 15, 1905.

% From paper by E. Sherman Gould, Van Nostrand’s Mag., vol. xxix, p. 450.

000

■+-0 -s-k

39. Thickness of Arch Ring of Stone at the Support.—For

semicircular stone arches it is generally assumed that the masonry for 30° from the spring line is self-supporting and consequently has no arch action. If this is so, then the maximum angle which a stone arch ring can be considered to subtend is 6o° each way from the crown. If the loading is so arranged that the equilibrium polygon follows the axis of the ring, then the pressures will vary directly as the secant of the angle </>; consequently the ring thickness 6o° from the crown should be ta=t₀ sec 6o° = 2/₀.

* Croizette-Desnoyer's Formulas for segmental arches:

For7'6

4 = i.4c/₀.

4 i ¹24/0¹

4 = 1.15¹0-

For j = — L 12

4 = I.IO/q.

For basket-handled arches :

when

U|h -ł^ I M U) I H

. . 4 = i.8o/₀;

. . 4 = i.6o/₀;

4 = 1.40/0.

40. Thickness of Abutment. — Trautwine’s rule for all kinds of stone arches is best explained by means of a diagram, Fig. 17. This form of abutment, according to

if

---jjL.-----H-------1-

I«.

__S'.

I,

a- 0.2 R + 0.1/ + 2.0 b^.\h

Fig. 17.

Trautwine, is sufficiently strong to take the thrust due to the dead load before the back filling of earth is in place.

Rankine states that in existing structures the thickness a varies from ^ to £ the radius of the intrados at the crown.

Baker, in “A Treatise on Masonry Construction,” gives a formula, said to represent German and Russian practice, which has the form

a — i +0.04(5/ + 4^),

where h is the distance from the spring line down to the top of the foundation.

41. Thickness of Piers.—In a series of arches it is customary to use several narrow piers and then introduce a. much heavier pier, called an abutment pier. This should be of sufficient strength to resist the thrust from one side without any aid from the arches upon the other side. The thickness will then be the same as if it were an abutment, in reality without earth backing. For the regular piers, various rules have been used. Twice the thickness of the arch ring at the crown plus a fraction of a foot has. been used in very important bridges. Usually piers are: from 2^ to 3 times the thickness of the arch ring at the. crown.

The vertical load upon piers is not very large when measured in tons per square foot, and as far as strength is concerned they could be made considerably smaller than outlined above. The only horizontal thrust to be resisted is the unbalanced thrust produced by the moving load, unless adjacent arches are of different dimensions. With the exception of high bridges the effect of the wind is of no moment.

42. Remarks concerning Empirical Formulas.—The formulas given in the previous articles are based, for the most part, upon actual structures and will without doubt, lead to safe structures if the equivalents in materials and workmanship are held throughout. Apparently the formulas apply to all kinds of stone, as no mention is made of the quality of the materials (excepting Trautwine’s formulas) used. Unquestionably the arch rings were constructed of average materials, probably no better if as good as those used now ; hence the formulas will be of service in assuming dimensions which can be relied upon as

being safe for structures quite similar to those upon which the formulas are based.

43. Albula Railroad Practice * (gage 1 m.).—The following dimensions were used in the construction of a great number of arches on the Albula Railroad.

Base of rail

ł'

im

't~

-a-

t.

0.8

ts

t.

1:40

Fig. i8.

Span.

Key.

Spring. Pier. .

Abutment.

*0 =0-55 te =0.80 b =1.20 a =1.70

0.60

0.90

¹ -35 i .90

m Cn o    o

O O O O

M \n O O O w w co

15

0.80

i. 20

2.10

3-5°

Twenty-six viaducts were built of the spans given below:

i =10 ii 12    14    15    16    20    22    25    27    30    42

Span......

Number of spans....

(All dimensions are in metres.)

= 33    3    7

16    14    15

44. The Dead Load.—Very little is required in the way of discussion in reference to the dead load for steel ribs. The floor and all supports, and even the lateral systems,

* The Engineer, 1904.

can all be designed and the actual weights computed. There remains, then, only the weight of the rib proper to be estimated. The weight of the assumed rib will be sufficiently close for all purposes, as a large error in the weight of the rib will be comparatively small for the entire load. The weight above the rib is usually transmitted to the rib through verticals extending up to the roadway.

In the case of masonry ribs with the spandrels completely filled with earth, sand, gravel, etc., the actual load supported by the rib is not very definite. If the filling is put in in horizontal layers well compacted, the load upon the ring will certainly not exceed the actual weight of the material, and it is very doubtful if such filling creates any considerable horizontal thrust against the rib. If perfectly dry and clean sand or gravel is employed, then there may be horizontal forces acting against the rib. ' These will be very small, however, for segmental arches. This thrust can be found according to the theory of earth pressure.*

The consideration of the horizontal thrust of the spandrel filling is a refinement not warranted in works of this class. The weight of the spandrel filling with pavement, arch rib, etc., should be considered as divided into vertical loads, the horizontal projection of ds being the measure of each division. For computations the load may be assumed to act at the center of the projection of ds.

In case the spandrels are partially filled with concrete its weight may be taken as divided into vertical forces.

* “ Retaining-walls for Earth,” by Malverd A. Howe. John Wiley & Sons, New York.

This is probably not as near the truth as when the fill is made of sand or gravel, but the assumption is on the safe side. Overloading the haunches will cause an upward movement at the crown, and overloading the crown causes the haunches to rise; but when the spandrel filling is partially concrete the passive resistance to an upward movement is very much in excess of its weight ; so also is that of sand or gravel. The arch rib, then, in this type of bridge is anything but a free member, and consequently any great refinement in its design is time wasted. If we can assure ourselves that the rib is safe by adding a few inches to the thickness of the ring, the very small percentage of extra cost need not be considered at all.

When the roadway is supported by longitudinal walls resting upon the rib, the problem is at least as complex as before, for there is no way of knowing how the weights transmitted by the walls are distributed. The only recourse is to treat the material as in the case of sand or gravel filling.

The use of lateral walls or columns to support the roadway places the problem in a shape to be carefully considered theoretically. The actual magnitudes of the loads can be computed and the points of application to the rib are definitely fixed. For long spans this is unquestionably the best and most economical type which can be built. There is an exception to this in very flat arches where the ring occupies the greater portion of the vertical projection of the bridge.

45. Dead-load Equilibrium Polygon Following the Axis of the Arch Rib.—It is assumed that the rib has been dimensioned and that the fill over the crown is known. Compute the weight of the shaded portion in Fig. 19 and call it P Lay off the vertical line DE, and Px from D. Draw DO horizontal and CO parallel to the tangent to the arch axis at b. Then from 0 draw lines parallel to the tangents at c, d, e, etc.; then these lines will cut off on DE the loads P2, P3, etc., for which an equilibrium polygon will pass

an

pi

d'/d

pi

p₂

P3

f.

p,

Fig. 19.

through the points a, b, c, d, etc., and DO will be the horizontal thrust for this loading. A check calculation will show that this is the true horizontal thrust according to the elastic theory, neglecting the effect of the axial stress.

By a similar construction the polygon may be made to pass through the points where the loads are applied to the axis. In either case the bending moments due to the dead load are sensibly zero. This assumes that the loads are reasonably close together.

Filled spandrels can usually be made so that the above conditions are fulfilled by selecting proper filling materials.

CHAPTER III.

EXAMPLES SHOWING THE APPLICATION OF THE FORMULAS, ETC.

46. Preliminary. — In the examples which follow, the ■computations will be given in detail, with suggestions as to methods and checks. In some cases it will be found that the algebraic work necessary to get the data into shape for applying the arch theory requires as much time as the computation of H1 for each load respectively. Some of this work will be found quite unnecessary by many. It is given in one case for the benefit of the few who may use the example as a guide for their first arch calculation.

47. First Example : Data. — Let us assume that the design shall be for a single-track railway bridge with an arch ring of Quincy, Mass., granite, and that the axis of the ring has a span of 60 ft. and a rise of 8 ft. Let the spandrel filling be cinders, sand, or gravel, in such proportions that the total dead load will have its equilibrium polygon following the axis of the ring. Since this is to be a railway bridge, there should be at least 3 ft. of fill between the base of the rail and the arch ring at the crown. This will distribute the moving load which may be assumed at 5000 lbs. per foot of span. If the ties are 8 ft. long, we may assume that the fill will distribute 5000 lbs. over at

S3

least 13 ft. under the ties, or that the moving load will be about 400 lbs. per square foot. 30 lbs. per square foot will cover the weight of the track.

48. Subdivision of the Arch Axis.—This should not be decided upon until the shape of the arch ring is determined. In this case let the ring be of uniform depth throughout; then, in order that A may be constant, the axis should be divided into equal parts. In all summation formulas it is well known that the smaller the divisions are made the more accurate will be the results. In this

10

7.

4,

5,

Sx

----x------>

8y

Hh

ill

A¹

i£=30'

--V-

■-£v

\!

Fig. 20.

particular case ds might be replaced by ds, and the problem solved, as far as Hi, Mi, etc., are concerned, by means of integration.

Ordinarily twenty divisions will give results sufficiently accurate for practical purposes. This number will be used.

.'. R =60.25 ft., and k' = 52.25 ft.

*. <£o =29° 51'.76.

il 50

Sm ^⁰⁼Ä⁼6^5 *

Arc AB =

29.8616 =31.40 ft.

From Fig. 20,

2 7iR 360

Hence £$ = 3.14 ft., and the angle at the center for each division is 2⁰.98616.

49 Computation of x and y.—The values of x and y are computed for the center points of the divisions made above, as shown in detail in Table A.

Table A.

Point.

sin <f>.

cos <j>. R sin 4>- R cos <j>.

3o—Rsin<j>. R cos6 — 52.25.

10

28° 22'. 172 25 22 .996 22 23 .820 19 24 .644 l6 25 .468 13 26 . 292

IO 27 .116

7 27 .940 4 28 .764 i 29 .588

0.47515 0.87991 28.628 .42867 .38102 ■33234 .28275 .23239 .18141 • 12993 .07810 .02606

1-372

4-173

7.044

9-977

12.964

53•⁰¹5

90346 25.827    54-433

92457 22.956    55.705

943x6 20.023    56.625

95919 17.036    57.791

97262 14.001    58.600    15.999

59-247 7.828    59.739

60.065 60.229 60.25

9⁸335 10.930 99T52 99694    4.705

99966    1.570

i.00000

19.070 22.172 25•295 28.430

30.000

0.765

2.183

3- 455

4- 575

5- 541

6- 35° 7.000 7.489

7- 8x5 7-979 8.000

In this particular case we are probably not warranted in using three decimal places in the values of x and y, although the labor is but a very little greater than if but two were used. This is assuming that multiplications are performed by machine or a multiplication-table. After a little practice Crelle’s “Rechentafeln” will be found quite satisfactory for all multiplications and many divisions.

50. Computation of Hi for Unit Loads.—Table B gives in detail the calculations for H1 corresponding to a unit load at each point respectively. Since the arch and the loading are symmetrical, the summations have been made from x=o to x = \l.

In column 2 the positive and negative values of y — ya should sum up the same. As the fourth decimal place has been neglected, the sums differ by 3 in the third decimal place. The method of Art. 14 has been employed in computing mx, which requires the use of but ten multipliers (the values of x) and fifty-five multiplications in the complete determination of Hi for each load.

For the first load each value of y— ya is multiplied by the first value of :r, and therefore, since I(y—ya)= o, Jmx(y —y a) should be zero, and consequently the value of Hi =0 for this load. Using the figures shown in the table, Hi =.000035 for Pi = unity, which is zero for all practical purposes.

The true values of Iy(y—ya) and Imx{y— ya) are twice the numerical values given in the table, but since one expression is in the denominator and the other in the numerator the common factor zero has been neglected.

The method employed in Table B is considerably longer than necessary, but has been used on account of its clearness and because all sums are taken between the same limits.

U.

N3

r-'

TJ

Oe

x=P₃'-

mx(y-ya).

-    6.243 -13.070 —13.102

-    5-213 1-592 7.291

ii.869 I5-3I4 17*610 18.772

-37.628 + 72.448

+ 34 820

2mx(y—ya) ^

y-,

”*4

°-3°75

<sr

Cl CO^Tj-Tł-Tj-Tf-i-Tł-Tt-NNrti-TtrtTtTtTtrl-

HOOOOOOOO

rfNNNNNNNN

ce

61.897

2mx

6.190

fi

! - P,'.

mx{y—y„).

-    6.243 -13.070

-    7.762

-    3.088

•943 4-3x9 7.032 9.072 10-433 ii.121

-3°-i63

+ 42.920

+ 12.757

2mx(y—ya)

o. 1125

Table B.

COMPUTATION OF H, FOR UNIT LOADS.

ar

i-P/.

m z(y-ya)-

y(y-ya).

y-Va-

Values of y.

Point.

Ci cococococococococo

-    6.243

-    4.297

-    2.552

-    1.015

.7,10 I.42O 2.312 2.983

3-430

3-656

rOrOCOfOcOPOtOrOfOfO

hûO'O'û ci ci IOHOO CO CS O CO 10 N O OC tovo TfOO Tf ro CN

covo vO CO h vO M

IT) N (S io es

vO Os m M m ci

O ci O O'û io >o rj- o »o

LO c^vO rf ci COCO K O łowco r^ci O no m ioO

rf co

Cl Cl Cl

38.929

2mx

-14.107 +14. m

13.72°

2mx

— 20.121 + 76.705

—IO.282 +IO.285

3-893

.OO4

+ 56-584

2mx(y—ya)

2 y (y-y a)

0.000035

zL2 • i

765

2.183 3-455 4 575

5- 541

6- 35° 7.000 7.489

7- 8x5

7-979

O M ch rc ^ too t^OO O O

53-I5²

2>

-y

ya= —

10

= 5-3T5

h = - ^wdy-yà)

2 ly(y-ya)

u-j

00

10

P₇-i=P₇'.

mx(y—ya).

CO O N O M rocO io N ^ N O 00 ro    fN

(N O M ro O LO

vO ro <*0 t'» CS vO ci M 1^0

MM    M fO Tj- rt IT;

M Tf*V© 00

-39-79»

+ 191.569

+I5I•771 Imx{y—ya)

it¹? ’ I

■TABLE B.—COMPUTATION OF H, FOR UNIT LOADS—(Continued).

P₆ = i=P₆'.

+5 = 1 =+5 •

P₄ = i=P₄'.

Point.

mx(y — ya)-

mx.

tnx(y — ya).

mx(y—y„).

mx.

cirortt^rtMOOOO ^ Tf t^vo Qn N N C"» pOmOOOnOOOOO

M l-O O O O' O

MMMMM M

h rt w Os

127.801

Imx

12.78

-    6.243 -13.070

-    13.102

-    7.383

2.930

16.551

26.945

34-7⁶4

39-97»

42.616

1-372

4173

7.044

9-977

12.964

I5-99I I5-99I I5-99I x5-991 J5-99¹

-    6.243 -13.070

-    13.102

-    7-3»3

2.930

13.418 2i.844 28.184 32.410 34-549

Tt* t^vO vO vO vO vO vO O O' O' O O O O' o

W W N N N N MMMMMM

co

M rt N 0\

—    6.243 -13.070

—    13. 102

—    7-383 2-255

10.326 16.811 21.690 24-943 26.589

1.372

4-173

7.044

9-977

9-977

9-977

9-977

9-977

9-977

9-977

O h o ro rj- io'O t'-QO Os O

-39.798 + 163.784

+ 123.986

Imx(y — ya)

¹ O955

115-485

lmx

-39-79» + 133-335

+ 93-537 Imx{y—ya)

xMg S£ ‘ooi

-39-79»

+102.614

82.428

lmx

+ 62.816

Imx(y—ya)

¹¹ -55

■8265

10.03

•555

8.243

Im-r

10

Hl

Pri

“ta

r-

tïi

isi

<0

ti

o.

IS

PO

c:

t-'

On

H\.

o.1125 ⁰-3075 0-555 o.8265

¹ 0955

I-34I ¹■5425 1.685 i■7585

14

P — unity.

Load.

TABLE B.—COMPUTATION ÔF Ef, FÔR UNIT LOADS—(Concluded).

13

12

11

+10— i — Pio' ■

Pg = l=Pg'.

Pt-i-Pi'.

Point.

mx(y-ya).

ntx-

mx(y-ya).

mx-

mx(.y-ya).

Yïl x-

—i

Q^ û^

fO O oi po O w PO n 00 O Tj-    o CO PO 10 PO O POO

to M

NO 01 00 PO to M <0 rt NO

01 O M PO On O po PO N Ol

M M

01 Ol N

-39 798

+ 238.820

(v6—&)xiuz

zso'66i+

I-758S

01 PO Tt N rt H O 01 io O

r^* Tt t-^NO O' t''* O'*» On P'0 pOhOOnOnOnOwoit*-

h Tf N O' 01 m O' oi 1/0 00 M M M <N 01 01

pOOOipOOhpooicOh r}* f>* O 00 PO lO PO O pOh

lO h M O) Tf-NO 0100 PON

M ro    vO

01 O h PO On 'O PO PO

M M

t-'* 01

01 PO rf N rf h O 01 to t/0 Tt- t>*vO ONNOnOn POh O OnOnOnO h m oi

oi io O' oi to Lo

M M M Cl 01 01

H l-NO

pOOoipoOhpooiOOO •t N O co PO lO PO O <000 01 o M CO Os to

NO PO PO 01 NÛ 0) 00 to On MM    M PO *t io lO

M 01 rf O

01 PO Tf N h O Ol 01 01 Tf- t^NO o CO M o ON On

M Tt N O'

O H N    toO r^OO On O

146.488

Smx

-39 798

+ 230 465

+190.667

Imx(y—ya)

143-333

Zmx

-39 798 + 214-334

137.107

Smx

+ I74-53⁶

Zmx(y—ya)

14.65

1.685

14-33

iL £1

Im

io

SstS 1

Table B also contains the values of Imx+n, which will be used in computing the values of Mi. Having the values of H\ for unit loads, its value for any other load is simply the product of the load by the values given in Table B.

51. Computation of M\, V1, yx, y₂, and y₀ for Unit Loads.—

The formula for Mx is, Art. 25,

lx²

Mi =Hiya -

ltnxx — ltnx:~ lx

lx²

lx

in which Hi and ya are known from Table B. lx = \nl — ^(20)60 =600. There remains to be found the value of tnx at each point for each load and also the value of lx². Of course lx² can be found by squaring each value of x, but this is rather tiresome, as there are twenty different values. The following method will be found shorter and easier and at the same time a portion of the work in computing mx will be done.

Taking any symmetrical values of that is, the values of ^ for points 1 and i', say,

X² +X1² =X² + (/ — x)² =X² +l² —2 lx +X2 = /² —2 X(l —X)

— I² —2X(\l —X) —lx.

Then, for all points,

til²    ^

l{x² +Xi²) =--2lx(\l —x) —llx = lx²,

2    0    0

til²

— = ^(2o)(6o)² =36000,

— 2 Ix{\l —x)= —2(1498.987) = —2997.974, (Table C.)

(Table C.)

1/2

— llx — — (6o)(i40.48) = —8788.800.

yx2

Ix² = 24213.23 and ^-=40.3554.

The denominator in the equation for M1 now becomes 20(30-40.3554)= -207.1075.

The next step is the determination of mx at each point for each load. This can be done by constructing an equilibrium polygon for each load and scaling the proper ordinates, which leads to 10X20 = 200 separate quantities and then 200 multiplications when mxx is found.

Imxx can be found as follows:

t____

---a—

---->

p=i

p----x-

■**----\l—X‘----4«---il—x

rrix    ¹

—»<—-x'—x--->1

mx~

R₂=-t—

-->

Fig. 21.

From Fig. 21 for load unity,

Ri =

From x = o to x = a,

From x = a to x = l,

■X.

mx^R\X =

2mxx = \l2mx — 2mx(%l —x).

The value of %2mx is given in Table B for two equal and symmetrical loads. This value is equal to 2mx for a single load. This quickly disposes of %l2mx.

The value of 2mx{\l —x) can be found quite easily by remembering that for mx(%l—x) upon the left there will be an mj{\l—x) upon the right but opposite in sign until x = a. For x <a and x = 1 — x,

(mx-}-mx')(%l-x) =RiX(%l-x) -R₂x(%1 -x)    (x<a)

= (Ri -R₂)x($l-x)]

hence

2 (mx +mx')($l -x) = (Ri -R₂) 2 x(^l-x).

For x = a to x = 1 — a,

v/

x = l—a

Now

Therefore

2 mx(\l—x)= 2 \R₂(l — 2x)($l —x) =2R₂($l —x)²\.

x=a

Then

2mx(il-x) = (Rx -R₂)X2ax{\l-x) + ₂R₂’2%l-x)².

x=0

With the above explanations, Table C becomes very simple and gives us all of the coefficients required in treating vertical loads. In col. 21 the values of M1 give also the values of M₂ by merely numbering the points 1', 2',

►4

1, Vi, AND

y.

h*

Ri - R₂.

954

862

766

668

568

466

364

2Ó2

Zb O

9Sl

16

I’m,.

Table B.

13.720

38929

61.897 82.428 100.350

115-485

127.801

I37-I07

143-333

146.488

s/

2x($I — x).

(ł/-*)²-

xVtl-x).

il-x.

ftT

Rv

Point.

39.223

14-934

308.572

508.372

729.210

953-210 1161.645 I335■236 1454-352 l/ic8 087

N O' « O N O lo ©\ Hf u-, 'OOO'O O sQ O sO OOO'O O h HM coo 'fiOH i

Oct^l^O O 'O O'!-* CH CH MCO CH O O'O'W'O CH 00 SO I/O Cf" CH M H

39.223 107.711 i61.638 199.800 220.838 224.000 208.435

I73-59I 119.116 44•635

PO PO'O    N    O    fOfOH N

Tt" M

15

HiVa-

Q 00 T)- O fO tO t^oO *o so o atoioo« w O'Vi't O vo'O Os CO

00 CH M O' co H 0) rt in t-'.OO 00 O'

14

Xmx\l.

O C''*'-' Tf O IOCOH O' 'Ht* SO 00 O'00 I/-) io O 0) 0"0

M    (N O Cf Tj- CO O' 't

MCO ION H O COtH Ofô cfHOO cfO -Cf 00 M P) CO m h CH COCOCOci-rJ-cJ-

13

Column 9 plus

Column 12, 2tnx(łl—x).

12

Column 10 times

Column ii.

142.618 350.182 492.031 569.265 587.606 553-892 477-508 368.023 228.959 77-947

105.199 223.525

255-⁶5S

229.673

173-415

109.696 54.669 18.191 2.080

rfOO MD O N N O' CP rOMOONOOOMCMrf    ^

m Ttt^.O\CMVO O CS LO 00    ^

M M M CS CS CS

O 00 rt CS CS Tt-vO 00 rf 00

fO co CO fO PO fO fO rf Tf

O w CS fOrf Lo'O t^OO O'

PO On t^vo MD t^OO On CS

CSVOWVOHVOHVOCSt^

OOMwcscsPOPOTtrł-

2 I

Hn

a

2286.936

1619.747

1092.585

691.785

401.423

205.423 85 958 24.649

2.465

977

931

883

834

784

733

682

631

578

526

Column 7 times Column 8.

37-419

126.657

236.366

339-592 414.191 444•196 422.839

349-832

226.879

77-947

M N co rj- u->'O t>.00 O' o

Pomt.

M CS PO Tf LO MD t^OO O O

24

Vo-

9.588

9-595

9.614

9-639

9-654

SZ

Vl

O N M O On CN Q\ O ioOnw loO w CO M rf GO CO O O 00 O <N *"» CN i^O\H rovo CO m rf N w N fO h O O On On OnOO 00 t^NO nO

Lo rf ro ro CN m

, V,a

yo^y^+~Er\

1

Vl=M?~M' + Rl'

TABLE C—COMPUTATION OF Mu Vu AND yx—{Concluded).

ZZ

21

20

19

18

17

•14

Mx.

-mi.

Col. (19 — 17)- Col. (15 —20).

_    2x² V-

Imx-.

—ID

2mzx.

Col. (14- 13).

NO CO NO M    W lO N    H    O 00 fO    N Tf    O) Tf O ION

On NO »O00    On O t"-    m    Tf nO M    W N    OnOO loOO nO

On O m    com CN    On fON    O CN    Tf nO 00 On O

NO CN CN M

N Tj- (N H

CN CN CN CN CN CN CO

Tf    Tf vO NO On CN CN CO l/) ro N O' O 00 CN 00 LO CO

N CO CO H IO H \0 CN ON LO O O CN Tf CO H 00 M Tf O

NO O 00 On ^NO to rf M co co co co CN m

NO Tf On M o t^CN LO ON co O H CN CN CO CO CN CN W

Tf LO00 NO ON CN ON LO Tf CO M co hoO co ioOO nO coco co NO no Tf Tf 00 CN nO On »O N t>. Q\ r^.vO m io O CO NO Tf 00 00 Tf NO NO <N NO

On O OO O wcoNCNO co lonO t^OO 00 00 00 t^NO no to Tf co CN M

—    2.673

—    7-583 -12.058 —16.057 -19.548

—    22.496

—    24.896

—    26.708

—    27.921 -28.536 -28.536

—    27.921

—    26.708

—    24.896

—    22.496 -19.548 -16.057 —12.058

—    7-583

—    2.673

268.982

817.688

1364.879

I903-575

2422.894

2910.658

3356.522

3745-i87 4071.031 43l6-693

4472-5⁸7 4528.949 4481.233 43II-538 4018.442 3598.106 3042.105 2348.941 I5i8-o52 554-218

Sx²

+Imx—.

"-ID

2mx

= H,ya-

-2O7.IO75

D= —207.1075.

2 m xX

— 207.1075

-    1.299

-    3-948

-    6.590

-    9.191

—    ii. 699 -14.054 —16.207 —18.083 -19-657 -20.843 -21-595

—    21.868

—    21.637

—    20.818 -19.403

-17-373 —14.689 -11.342

-    7-33°

—    2.676

Mi=Hiya--

Point.

H CN CO Tf I^vû t^OO On

3', etc. The quantities in cols. 22 and 23 reversed give y2 and V₂ respectively. Col. 24 shows how nearly constant y₀ is in this case.

For mathematical accuracy the value of V\ for a load at point i should be unity as given in Table C, which evidently is not the actual condition. When the first point is quite near the support, however, the value of Vi approaches unity very nearly.

In col. 22 the value of y\ for point 1 is not given, since it is not possible to obtain its value directly from the formula Mi+Hi=yi, as Hi is zero. The same is true for point 1'. This will be the condition whenever graphical or algebraic summation methods are used. This difficulty does not occur in integration formulas. Fortunately, the peculiarity of the summation methods is of no practical importance if ds Is not assumed too great. The defect is quite marked where ribs have a much greater depth at the springing than at the crown, and ds is so taken that everywhere ds -4-1 is constant.

52. Depth of Ring and the Dead Load.—An examination of Table II shows that a number of railway bridges have been constructed with spans of about 60 feet with arch rings 3 feet deep. Let this be assumed as the depth of the ring.

The load at point 10 can be found as follows: Divide the vertical projection of the arch as shown in Fig. 22, and carefully scale the distances ab, be, and de. Then the weight of the ring at point 10 is [(bc){de) = (3-oo)(3.i4)]i 70 = 1601 pounds, taking granite at 170 lbs. per cubic foot. Assume the fill to be made of material weighing 95 lbs. per cubic foot, then the weight at 10 is (3.02X3.14)95 ⁼9°5 lbs., say. The weight from the track is (3.14)30=94 lbs.

The total dead load at 10 now is 1601+905 +94 = 2600 pounds. In order that the equilibrium polygon shall pass

Base of rail

10

____-)<r---

-CO-

-CO

10'

"~T-+-

A'

p,0

r.

:/

Fig. 22.

through 10 and 9, Fig. 23, the pole distance must be

2600

2600

tan 2⁰ — 59'.2    0.052

=50000.

2600#

10

S,

H = 50000 ^

CO--'

21)00?

Si

So

S3

S₃

Fig. 2$.

For the polygon to pass through point 8, load 9 must equal [50000 tan 2(2° — 59'.2) = 50000(0.1046)] — 2600 =5250

— 2600 = 2650 pounds. In like manner all loads may be computed, or obtained by drawing strings parallel to the chords: connecting the points of division as indicated in Fig. 23.

COMPUTATION OF DEAD LOAD.

Point.

<f>.

tan <j>.

50000 tan <t>.

Dead Load P.

U£U    Corn-

Table’B. Puted Hi-

M M łO fO Ki KJ

OU Qs'-O o

10

55 880 ii 56 .704 8 57 -528 5 58 -352 2 59 .176

0-574

0.507

0.443

0.382

0.323

0.267 o. 211

0157

0.105 0.052

28700

25350

22150

19100

16150

I335°

10550

7850

5250

2600

OOOOOOOQOO 10O1010OOOÖL0O fOW O OOO 00    vO O

co ro co CS <N cs CN

WWW

0.113 0.308 0-555 0.827 1.096 i-341 1-543 1.685

¹ • 759

362

939

1637

2316 3069 3621 4012 4465 4573

28700

24994

The above table gives the computations necessary for obtaining the proper dead loads and also the corresponding values of Hi. The value of H1 for the entire dead load is in round figures 2(25000) = 50000.

The next step will be the separating of the above dead loads into parts, the ring, filling, and track. The ring and track are fixed, so that their combined weight taken from the total will leave the weight of fill required. The tabular statement on page 68 shows the process in detail.

No great degree of accuracy has been attempted in this table, as a hard rain may change the weight of the fill a considerable amount. The last column gives the average weight per cubic foot of the fill which is necessary to just fulfill the requirement that -the equilibrium polygon coincides with the arch axis. It will be noticed that the weight of the arch ring is very nearly uniform for each section.

The lack of uniformity in the variation of the values given is due to inaccuracies of scaling ab, be, and de from a drawing.

FINAL DEAD LOADS.

Point.

ab.

Fig. 22.

be.

170 lbs. 30 lbs.

_ per cu. per. sq.

ft. ft.

de. Ring. Track.

Ring

and

Track.

Fill.

Area of Fill.

Average Weight of Fill per Cubic Foot.

h « ro    »O'O t^OO 0\ O

IO. IO

8.72

7-45

6.36

5-42

4.62

4.00

3-5²

3.20

3.02

OÓOjOjOoOjOjOjOjOjOo

OOOOOwMtotO-^ O O O La 00Os) 0\Cn 00 O

2.77

2.83

2.91 2.96 3 01 3-°5

3-°9 3 •11 3-13 3-14

1601

1578

1608

1590

1602

1597

1602

1586

1596

1601

83

85

87

89

90

92

93

93

94

94

1684

1663

1695

1679

1692

1689

i695

1679

1690

i695

1666

J537

1355

1271

1108

im

1005

921

960

90S

in t^oo <0 M rt o O 10 CO    w(XK) rt-M O O Cn

(NCNCSMWWMMM

OO 00 OO^sJ ON On On OnLti La La h O 03 00C*> to O

53. Live Load and Loads Producing Maximum Moments.—

The live load is 400 lbs. per linear foot and hence the load at each point is obtained by multiplying de, Fig. 22, by 400. These products are given in Table D.

In order to select the loads which produce maximum moments draw the equilibrium polygons for a load unity at each point respectively, as shown on Plate I. One-half of the polygons are shown. These reversed will be the polygons for loads upon the right of the crown.

By inspection we see that loads 1-8 inclusive produce negative moments at the left support, and the remaining loads produce positive moments.

At the crown loads 1-7 and 7'-i' inclusive produce negative moments, and loads 8-8' inclusive positive moments.

For point 6', between ^ and \ point of the span, loads 1-8' produce negative moments, and loads 7'-i' positive moments.

Table D.

¹

an Nostrand’s Engineering Magazine, vol. xxix, p. 454.

LIVE LOADS.

400 POUNDS PER FOOT.

Pt.

Load.

Loads.

Hy.

Ml

SMi.

El

IVt.

10

10'

9'

8'

7'

6'

5'

4'

3'

2'

1'

o'

1108

1132

1164

1184

1204

1220

1236

I244

I252

1256

1256

1252

I244

1236

1220

1204

1184

I164

II32

II08

II08

224O

3404

4588

579²

7012

8248

9492

10744

12000

13256

14308

15752

16988

18208

19412

20596

21760

22892

24000

00.0 127.4 357-9 657-1 995-¹

1336.5

i657-5

1918.9

2109.6

2208.7 2208.7 2109.6 1918.9 i657-5 1336.5

995-1

657-1

357-9

127.4

00.0

00.0

127.4

485-3

1142.4

2137-5

3474-0

s^-s

7050.4 9160.0

11368.7

13577-4 15687.0 1760s.9

19263.4 20599.9 21595.0 22252.i 22610.0

22737.4 22737.4

-1522.4

-3437-9

—    4462.8 -4636.6

—    4161.0 -3I95-2 -1930.6

—    53¹•²

866.4

2076.2

3020.7

3⁶34-6

3890.0

3768.6

333°-⁶

2670.6 1873.1

1068.6

390-5

3-4

—    1522.4

—    4960.3

—    9423-1

-14059.7 —18220.7

-21415-9

-23346.5

-23877.7 -23011.3 -20935.1

—I79I4-4 —14279.8 —10389.8

—    6621.2

—    3290.6

—    620.0

I253-1 2321.7 2712.2 2715.6

1108.O

1117.3

1119.8

1096.4 1058.3

1002.8 938.1 858.4 770.0 677.0 579-0 482.0

385-6

297.9

2x7.2

145-7

87.6 44.2

14.7 0.0

i108.0 2225.3 3345•i 4441-5 5499.8

6502.6

7440.7

8299.1

9069.1

9746.1 10325. i

10807.1

11192.7 11490.6

11707.8 11853.5

11941.1

11985-3

12000.0

12000.0

If the ring is safe at these three points or even at the spring line and points 6 and 6', it will be safe at all other points.

54. Mi, Vi, and Hi for Live Loads.—These values are

For convenience these values are summed from

obtained by multiplying the values for a load unity given in Table C by the live load. The results are given in Table D.

o to x, as shown.

55. Maximum Moments at Point o Produced by the Live Load.—For loads 1-8 inclusive

Mi = —23878 (Table D).

DAI

RADY rO'.O’TII

u.'i.'AuEJ

AMSETK/I

For loads 9-1' inclusive

M\ = 2716 — (-23878) = +26594.

For a full load

Mi = — 23878 + 26594 = +2716.

For a load up to the crown

Mi = -20935 (Table D).

For load io'-i' inclusive

Mi = +23651 (Table D).

Evidently loading one half the span does not produce maximum moments at point o. The difference between the moment for a load extending from one support up to the crown and the maximum moments will not make any serious difference in the fiber stresses, as the dead load contributes a large portion of these stresses. If temperature effects are considered, the live-load effect becomes almost insignificant.

56. Maximum Moments at the Crown Produced by the Live Load.—For loads 1-7 inclusive

7441

Mi = - 23347 (Table D), V\ =

5132

Hi-

MX=M 1 + V1%-Hiy -1P{x — a).    * = 30 and y = 8.

X

IP(x-a) can be found graphically by means of the ordinary equilibrium polygon. In this instance we will compute its value as shown in the following table.

ŹP(x—a).

Point

P.

il—a.

P(x — a).

i jc8 1132 1164 1184 1204 1220 1236

28.63 25-⁸3 22.96 20.02 17.04 14.00 IO-93

31722

29239

26725

23703

20516

17080

I35°9

Table D

Table C

162494

XP(x—a)

v 1* = 7441 (30) =223230,

H\y = 5132(8) = 4105⁶.

X

IP(x — a) = 162494 ;

Mx= -23347+223230 -41056-162494= -3667.

If this is the moment for loads 1-7, then for loads 1-7 and 7'-i' inclusive, Mx = 2(—3667)= —7334.

If our coefficients are absolutely correct, the moment for loads 7'—1 ' inclusive should be the same as for loads 1-7 inclusive, as assumed. For loads 7'—1 ' inclusive

M x = M i + V\% — Hxy,

M1 = +13105 (Table D), Fi# = 807(30) =24210, Hiy = 5132(8) =41056,

and

M*= +13105+24210-41056= -3741.

This is 3741-3667=74 larger than obtained by above method, an error of about 2%.

Considering that the summation method leads necessarily to approximate results, it will be more consistent when possible to always use the formula for Mx in which

X

2P(x — a) does not appear.

57. Moment at the Crown Produced by Live Loads 1-10 Inclusive.—From Table D,

v/

y2^ = 2254(30) =67620,

M₂ = +23651 2+1^ = 11369(8) =90952.

.'. Mx = +23651 +67620 —90952 = +319.

For a load over all,

21+^ = 2(319) = +638.

58. Moment at the Crown Produced by Loads 8-8' Inclusive.

—We will first compute the moment for loads io', 9', and 8' by the formula

Mx =Mi+ViX—Hiy.

From Table D

il+i=+10545, Vix=> 1447(30) =43410, Hiy = 6237(8) =49896.

Mx= +10545 +43410 -49896 = + 4059-

Check :

I-IO

8-10

From Art. 44, Mx = —3741 for loads 1-7 inclusive. 45, 21+* = + 319 .’. 21+ x = +4060

or practically the same as found above.

The above computations show that the moment at the crown produced by a load covering the half-span is hardly one tenth the maximum moment.

59. Maximum Moment at Point 6' Produced by Live Loads 1-8' Inclusive.—Use the formula

M x = M 2 + V ₂xr— H1 y.

From Table D,

M₂ = + 2715 -(-23347) = +26062, 4559(16) ==72944,

Hiy = 17606(6.35) = 111798.

.\ Mx = +26062 +72944 — 111798= —12792.

60. Maximum Moment at Point 6' Produced by Live Loads. 7'-i' Inclusive.—From Table D,

Mi = +13105, Ui* = 807(44) =35508,

H\y = 5i3²(6-35) =32588,

X

IP(x — a) = 1236(3.07) = 3795,

M x =M 1 + ViX—Hiy — ZP(x — a)

= +13105 +35508 -32588 -3795 = +12230.

61. Moment at Point 6' Produced by Live Loads 1-10 Inclusive.

From Table D,

M₂= +23651,    v₂x'= 2254(16) =36064,

Hiy = 11369(6.35) = 72193.

Mx= +23651 +36064-72193 = -12478,

which is about 2^% less than the maximum moment as found in Art. 59.

62. Moments at all Points Produced by Live Loads 1-8' Inclusive Determined Graphically. — The constructions are given on Plate II. Lay off a load line in the usual way and scale off V1 downward. Horizontally opposite this point, at a distance Hi, take a pole and draw the strings , S₂, S₃, etc. The equilibrium polygon can now be drawn. As check upon the correctness of the polygon the common .1closing line, when transferred to the force polygon, should •cut off the value of R\, the common reaction, on the load line. (In this particular case the check was not perfect, but so close that it was deemed unnecessary to draw a new polygon. The effect will appear later.) The closing line is AB.

Following the methods of Arts. 16 and 17, scale each ordinate of the equilibrium polygon and find the mean ordinate = IB'C' a- 20. At the center of the span scale upward this distance, and through the point just found draw CD parallel to the string S₀ in the force diagram, and scale the ordinates A'B'. Then Mx at any point equals the difference between the ordinate A'B' for that point and the corresponding value of y— ya multiplied by Hi.

The values of Mx+H 1 can be found, also, by drawing the arch axis so that the ya line coincides with the line CD of the equilibrium polygon and scaling the ordinates indicated in the shaded area.

The line CD can also be located b’'r making AC = mi and BD=m₂, where

IM₂ - IH₂ya

and m₂=

mx =

IMi -IHiya

Hi

H₂

The computation of Mx in detail is given in Table E.

Table E.

LIVE LOADS, 1-8' INCLUSIVE.

Point.

A'B'.

See Plate II.

y-y a.

Mz

Mx.

(y—yà) — A'B'.

-5-905 -5°9 -3-48 — 2.01

-S-3IS

-4-55°

-313²

— i.860

590

54

348

•74

.40

.740

.226

r5°

000

174

1.32 2.06 2.60 2.88 a 2.96 2.80

2-43

1.83

¹ °5

.27

•5°

i 035

i. 685 2.174 2.500 2.665 2.665 2.500 2.174 1.685

¹ -°35 . 226

285

375

426

380

295

*35

070

344

635

765

726

-1.28

—    2.05

—    2.78 -3-52 -3-834

.740

— i.860

— 3-¹32 -4-55° 5 • 3X5

540

190

352

030

481

—10387

—    0000 4- 3066

+ 5017

+ 6602

+ 7509

+ 6690 + 5!94 + 2377

-    3345 4- 6198 + 18134 4- 26075

The point of maximum moment is at 6', as stated above, and Mx = —13468. From Art. 59, by computation, Mx = — 12792, showing a difference of 676 or an error of about 5%, corresponding to an ordinate of 0.033 feet. The scale employed was 3 feet to the inch, hence 0.033 feet corresponds to 0.011 of an inch on the drawing. This shows that the greatest care must be employed when graphical methods are applied and all possible checks applied.

63. Maximum Moment at Point 6 Produced by Loads Inclusive. Graphical Determinations.—Plate II .shows the construction, and Table F the computation of Mx in detail. Here again there is a difference in the results obtained by

the two methods. From Art. 60, Mx = +12230, while by graphics Mx= +11520, a difference of 710, or about 6%.

Table F.

LIVE LOADS, r-V INCLUSIVE.

Point.

A'B'.

See Plate II.

y-y «•

Mx

(y-ya)-A'B'.

Mx.

O H Cl co 10O    On

—    2.761

-2.59

—    2.16 -i.71

—    1.26

-5-3IS

-4-55°

-3132

—1.860

-2-554

—1.960

.80

•33

•15

.60

i. 10

.740

.226

.972

.150

O O Os00    %h *b

t-t M

1-55

2.05

2-53

3.00

3-5°

3.28

2.30

•65

—    1.62

-4-45

-7.90

—    9.864

i 035 1.685 2.174 2.500 2.665 2.665 2.500 2.174 1-685

1-035

. 226

.520

1.026

!-3⁶5

1-535

1-574

1.400

1.115

.615

•030

.826

•740

—1.860

— 3-J32 -4-55°

-5135

-1.815

—    2.245 -2.074 -1.390

—    .240 1.318

3- 35°

4- 549

+I3IOS

+10058

+ 4987

+ 4238

+ 93*& + 11520 +10643

+ 7I33

+ 1231

-    6763

—    17188 -23341

6+ Fiber Stresses Produced by Dead and Live Loads.—

From Art. 31,

Nx Mxz

p—f±

For this problem,

F = 3 sq.ft., 2 = 1.5 ft., ¹ I2

NX = (Vi — IP) sin (j>+Hi cos $.

i

= —~bh? = —~ = .

12

Then

►-h|

1-5 X4

EXAMPLES SHOWING APPLICATION OF FORMULAS. U

and

P=Wx±Wx-

Point o.

Dead Load.

From Art. 52, \ the total load = V\ =28700, say 29000, and H\ =50000; then

Nx = 29000(0.498) +50000(0.867) = 14442 +43350 =57792.

£=ł(5779²) +§(o) = r92Ó4, say 19300 comp, for both the upper and lower extreme fibers.

Live Loads.

From Art. 55 . . . ,MX=M\ = —23878 for loads 1-8 incl.

. .. ,MX=M\ = +26594 for loads 9-1' “ 7^ = 8299(0.498) +7050(0.867) =10245 f°r loads 1-8    “

77x = 37oi(o.498)+15687(0.867) = 15443 for loads 9-1'    “

Then

£=§(10245)-§(23878) =3415-15919 =-12500 tension in upper fiber and 3415 +15919 = +19300 compression in the lower fiber for loads 1-8 inclusive.

For loads 9-1' inclusive

£=§(15443) +§(26594) = 5143 + 17729 =22900 compression in the upper fiber and

£ = 5143—17729 = 12600 tension in the lower fiber.

Combined Stresses.

Combining the above results we have for the maximum fiber stresses produced by the dead and live loads the following :

Dead Load...........

L.L. 1-8.............

L.L. 9-1'.............

Maximum compression. . Maximum tension......

Load.

Upper Fiber.

Lower Fiber.

19300 compression 12500 tension 22900 compression 42200

19300 compression 19300

12600 tension 38600

These intensities are pounds per square foot.

For pounds per square inch we have, 293 and 268 as the maximum compression in the upper and lower fibers respectively.

Considering that granite has an ultimate crushing strength of from 13000 to 17000 pounds per square inch, the above fiber stresses are of little consequence if the mortar joints have an equal strength, or even one fourth the strength of the granite. The fiber stresses at other points are obtained in the manner followed for point o. A tabulated statement for points o, 6', and the crown is given below:

FIBER STRESSES.

Load.

Nx.

Mx.

$MX.

p.

Point.

Upper.

Lower.

Dead load.......

L.L. 1—8........

L.L. 9-1'.......

Max. compression. Max. tension.....

57792

10245

15443

-23878

426594

19300

3415

5M8

-I59I9

+17729

+19300 —12500 + 22900 42200

+ 1930C 19300 —12600 38600

Dead load..........

L.L. 1-7 and 7'-i'. • •

L.L. 8-8'..........

Max. compression. . . Max. tension........

50000

10264

12474

16666

- 7482 F 8118

3421

415⁸

— 4988 + 5412

+16666 - 1567

+ 9570

26200

4-16666 4- 8409 - 1254 25100

Crown

Dead load.......

L.L. 1-8'.......

L.L. 7'-o'.......

Max. compression. Max. tension.....

5°5°°

18170

5093

16833

— 12792 +12230

6057

1698

— 8528 + 8154

+16833 - 2471

+ 9852

26700

+16833

+14585

- 6546 3I4°°

6'

6'

6'

6'

6'

In this table all stresses are given in pounds per square

foot.

From the above table we see that there is no tension at the three points considered, and that the maximum compression is well within the safe strength of the material assumed. Also, that the greatest fiber stress is at the supports.

65. Effect of Temperature Changes.—Our knowledge of the effect of changes of temperature upon stone arches is very meager. The coefficients of expansion for different stones are known, but how long it takes for a stone bridge to become of uniform temperature we do not know. Probably all portions of the arch ring are never of the same temperature. The range of the average temperature is probably small. (See Arts. 33 and 34.)

In this case we will assume that the temperature changes 40° above or below the temperature of the arch when built. This is without doubt an excessive range. The horizontal thrust is (Art. 27)

Ht =

et° IE

et? IE

IAy(y-ya) 1.4(113.168)’

where

J =dS -T-I =3.14-^2.25=1.4.

For Quincy granite

e = 0.00000381,

E = 6776000.

Then

H,

(0.00000381) (40) (60) (6776000) (144)

¹58.4

56100.

From Art. 27,

Mi =H<^a = 56100(5.315) =298200,

Mx=M₁-Hty=Ht(ya-y).

The above values of Ht and Mi will have signs depending upon whether the change of temperature is an increase or a decrease.

For falling temperature the upper fibers at the support are in tension, and at the crown in compression.

The following table gives the fiber stresses at the support and the crown.

FIBER STRESSES DUE TO CHANGES OF TEMPERATURE.

POINT O.

Tempera

ture.

Nx.

Mx.

*NX.

ÎMX.

Upper.

P.

Lower.

-400 + 4°°

-48638 + 48638

298200

298200

—16213 + 16213

198800

198800

— 215000 + 215000

+182000 —182000

CROWN.

-4O0 + 4O⁰

— 56x00 + 56100

149600

149600

— 18700 +18700

99700

99700

+ 81000 — 81000

— 118400 + I184OO

Combining the above values with those obtained for the dead load and live load we have

220600

i753°o

For point o:

Maximum compression.

tension.....

Upper Fibers. Lower Fibers.

257200

20820Ö

For the crown:

Maximum compression.

tension.....

107200

65900

1435⁰⁰

IO3OOO

The above values correspond to a maximum compression of 1786 pounds per square inch and a maximum tension of 1446 pounds per square inch. In compression the factor of safety is from 8 to 10, but in tension the ultimate strength of the joints is exceeded. As a large number of railway bridges have been built upon practically the dimensions we assumed and no indications of failure having been found, wTe must conclude that the range of temperature •change assumed in this example is very much too great. Furthermore, it requires a drop in temperature of only four degrees to completely balance the compression produced by the dead load in the upper fibers at the support. Without question, then, our assumptions about the effect of temperature changes are not correct. Until we know more about the subject it is useless to make calculations according to the ordinary assumptions. (See Art. 33-)

66. Effect of the Axial or Direct Stress.—In all of the

work above, the effect of the direct compression or tension has been neglected. If the rib is subjected to a uniform stress, it will be shortened or lengthened according to the character of the stress. All vertical loads produce direct stresses which in effect shorten the rib.

As explained in Art. 19, the horizontal thrust produced by this shortening, when found, will be treated the same as the thrust for a change of temperature.

From Art. 19,

Zy(y-y.a)

Ha=H₁ i-

in which all quantities are known from previous calculations, with the exception of the last term in the denominator. The computation of this term is given in detail below.

Point.

Sx.

cos <t>.

Sx cos <p.

2.77

0.880

2.83

903

2.44

2.56

2.90 2.96 3.⁰¹

925

943

959

2.68

2.79

2.89

3-°5

973

10

3.08 3-11 3-13 3 • T4

983

992

997

999

2.97

303

3 • °9 3.¹²

3 • *4

28.71

= 13.67,    =113.168.

Idx cos 0 = 2(28.71) = 57.42, F = 3, J — 1.4, FJ =4.2, i’&E cos ^    57-42

4.2

FJ

Then

113.168

126.84

= 0.892.    Fa=o.io8Fi = 11% Fi, say.

The value of Hi for the dead load is 50000; then the corresponding axial stress produces a thrust, opposite in character, of 5500. The horizontal thrust produced by a drop of 40° in temperature is 56100; therefore the effect of

5500

= .091 of the stresses due to

56100

this drop of temperature. At joint zero the upper fiber stress due to -40⁰ is 215000 tension. 215000(0.091) = 19600 tension.

the axial stress equals

FIBER STRESSES DUE TO THE AXIAL STRESS.

Point o.

Crown.

Ht.

Ha.

Ht

Upper.

Lower.

Upper.

Lower.

Dead load L.L. i—8. .

9-1'...........

1-7 and 7'-i'. . .

8-8'...........

1-8'...........

7'-i'..........

-40°

50000

7050

15687

10263

12474

17606

5I3²

51600

5500

776

1726

1128

1372

1936

S65

5676

0.091 0.014 0.031 0.020 0.024 0.034 0.001 o. no

—19600

—    220 + 24000

+16600 + 2500 + 5600 + 3600 + 4400 + 6200 +    182

— 20000

+ 7400 +1100 + 2500 + 1600 + 1900 + 2800 +    81

— 8900

—10800

4000

118

+13000

Combining these stresses with the dead- and live-load stresses previously obtained, we have (see Art. 64) :

FINAL STRESSES, INCLUDING EFFECT OF AXIAL STRESS.

Loads.

Upper Fibers.

Maximums.

Point o.

Dead load. . . .

L.L. 1—8......

“ 9-i'.....

+ 19300—19600=—    300

- 12500- 3000= - 15500 + 22900— 6700=+15200

Max. comp.= 12200 “ ten. =18500

Dead load L.L. 1—8.. "    9-1'.

Lower Fibers.

+ 19300+16600= +35900 + 19300+ 2500= +21800 — 12600+ 5600=— 7000

Max. comp.= 57700 ‘ ‘ ten.

Dead load.........

LL. 1-7 and y'-i' . . . ' 8-8'...........

Upper Fibers.

+ 16666+ 7400=+24100 — 1567+ 1600= +

+ 9570+ I900=+H470

Max. comp.= 35600 ‘ ‘ ten.

Crown.

Dead load.....

L L. 1-7 and 7'-i'. .

" 8-8'...

Lower Fibers.

+ 16666— 10800= + 5900 + 8409— 2400= + 6000 - 1254— 2900=— 4200

Max. comp. = 11900 ten.

Temperature.

+ 400 + 400 + 400 ±40°

+ 215000+ 24000= + 191000 ± 182000+ 20000= ± 162000 ± 81000+ 8900= + 72100 + 118400+ 13000= + 105400

Upper fibers at o. Lower “    *0.

Upper :    ■'* crown

Lower '    "    “

These stresses show that the effect of the axial stress is considerable, and also that the fiber stresses at the support are reversed in one case so that the upper fibers are in tension about 128 lbs. per square inch. As this tension is not large and exists for but a short distance, the ring may be considered safe. This assumes that no temperature effects are considered. The maximum compression is 400 lbs. per square inch in the lower fibers at the support.

The effect of the axial stress is to lower the equilibrium polygon at the support and raise it at the crown, or it increases the compression and decreases the tension in the lower fibers and decreases the compression and increases the tension in the upper fibers at the support. While at the crown the reverse is true.

If this arch ring had been assumed free, then the above tension could not have been allowed (see Arts. 31 and 33).

67. A Check upon the Effect of the Axial Stress for Dead Loads. — To show how nearly the results of the above method of considering the axial stress agrees with those obtained by direct calculation, we will briefly compute H1 and Mi for the dead load, and also the fiber stresses at the support (see upper table on page 85).

My = lHiya — Imi = 237049 — 266020 = —29000,

N x = 29000(0.498) +44600(0.867) = 14442 +38668 =53100.

^ ⁼⁼ł(53ioo) ±1(29000) =17700 ± 19300 = 1600 tension in upper fibers = 37000 compression in lower fibers.

From Art. 66 the corresponding stresses are 300 tension and 35900 compression, the results in the table being about 1100 too large numerically. This equals a stress of less than 8 pounds per square inch and for the compression a relative error of 3%+.

COMPUTATION OF Hi and Mi WHEN AXIAL STRESS IS CONSIDERED.

Point.

Common . Hi. Art. 52.

True

Hi.

True

HlVa-

mi for

Load Unity, Table C.

Dead Load, Art. 52-

mi for Dead Load.

O 'O 00^1 0\01 4*. Gi 10 H

1637

2316

3069

3Ó2I

4012

4465

4573

24994

Common Hi (i — 0.108), Art. 66

Table B.

1-377 3.888 6.184 8.234 10.024 11-535 12.767 13.696 I4-3I7 14.634

3350

3200

3°5°

2950

2800

2800

2700

2600

2650

2600

4613

12442 18861 23669 28067 32298

34471

35610

37940

38048

49988

Symmetrical 28700 values

combined 29000 £ D.L. or

say

266020

I mi

say

50000

44600

237049

Vi

Evidently the method employed in Art. 66 is quite accurate enough for practical purposes.

68. Effect of Making Spandrel Filling of Uniform Material Weighing ioo Pounds per Cubic Foot.

COMPUTATION OF Hi.

Point.

Ring and Track.

Fill

ioo Lbs. per Cu. Ft.

Total

Dead

Load.

Common Hi, Load Unity.

Common

Hi.

Hi with Effect of Axial Stress.

o o 00-<r 0\'-o 4^(0 » h

1684

1663

1695

1679

1692

1689

1695

1679

1690

1695

2850

2470

2170

1880

1630

1410

1240

1090

1000

950

4500

4100

3900

3600

33°°

3I0°

2900

2800

2700

2600

335°°

0.0

o. 1125 3°75

5550

8265

1-0955 i.3410 i-54²5 i.6850 I-7585

Table B

St⁰

1998

2727

3396

3889

4319

455°

45 7 ²

271x1

54200(l—O.I08) =48300

54200

67000

COMPUTATION OF

Point.

ntj, Load Unity. Table C.

Dead

Load.

wtj for Dead Load.

10

¹ -377 3.888 6.184 8.234 10.024

¹¹ 535 12.767 13.696 I4-3I7 14.634

Table C

4500

4100

6197

3900

3600

33°° 3100 2900 2800

2700

2600

IS94I

24129

29642

33°79

35759

37024

38349

38656

38048

296824

Mi = lHiya - Inti = 48300(5.315) -296824 = 256715—296824= —40100,

= —0.83 ft.,

Mi

— 40100

yi Hi

Nx = 335°°(°-498) +48300(0.867) = 16683 +41876

48300

= 58600.

p = §(58600) ±§(40100) = 19500 ±26700 = 7200 tension in upper fibers = 46200 compression in bottom fibers.

This shows that the fill over the haunches and near the supports is too heavy for the load upon the crown. The original loading could be made less to an advantage.

At the crown the moment is

Mx=Mi + ViX-Hiy - IP(x-a)

= —40100 + 1005000 —386400 — 566400 = +12100, Nx=Hi sensibly = 48300. p = §(48300) ±§(12100) = 16100 ±8100 = 24200 compression in upper fibers

= 8000

lower fibers.

The equilibrium polygon is Mx + H 1=12100-^-48300 = 0.25 ft. above the neutral axis.

Combining these stresses with the live-load stresses of Art. 66, we have

—    7200 +15200 = 8000 comp, in upper fibers

■ at support

at crown.

—    7200—15500 = 22700 tension in upper fibers + 46200 +21800 =68000 comp, in lower fibers

+ 46200— 7000=39020 + 24200 + 11470=35670 comp, in upper fibers

+ 8000 + 6000 = 14000

lower fibers

If the above tension is considered more than allowable, then the spandrel filling should be made lighter. Since the maximum compression is very much less than the allowable stress for granite, the ring will unquestionably adjust itself by increasing this compression, and not resist much tension, if any (see Art. 31.)

69. The Radial Shear.—From Art. 29,

T x = V*cos (j) —Hx sin <j>.

For point zero, or the support, this becomes Tx = ^1(0.867) -#1(0.498).

For dead load (see Art. 52)

Tx = 28700(0.867) — 50000(0.498) = 24880 — 24900 =0.

For live load over all (see Table D)

T x = 12000(0.867) — 22737(0.498) = 10400 — 11320= —920.

At the crowTn <f> is zero, hence

T X = V X = V\ — IP.

For the dead load Tx=o.

For a live load io'-i' inclusive TX = Vi =2254.

In like manner any other point may be considered. When equilibrium polygons are drawn a glance is sufficient to determine if there is danger of slipping at the joints. Usually the radial shear requires but little attention in stone arches.

70.    Second Example. Data.—For this example we will take a reinforced-concrete rib of the Thacher type.* Clear span 50 ft. and rise 10 ft. The thickness at the crown is taken as 12 inches, and at the spring line 4 feet 6 inches. Plate IV gives all data concerning dimension and reinforcement. The dead weight of the entire structure is assumed at 140 pounds per cubic foot, and the live load 112 pounds per square foot. The first step in the solution of a problem of this type is to obtain all the data shown in Plate IV either by algebraic or graphical methods. In the present instance many of the data were obtained from a carefully constructed drawing as indicated in the figure. The modulus of elasticity of the concrete is assumed to be that of steel, and hence the area of the steel is equivalent to twenty times that area in concrete.

71.    Subdivision of the Arch Axis. — Contrary to the usual custom we will not attempt to so divide the arch ring that ds+1 will be constant, but simply divide the span into twenty equal parts and determine all quantities necessary for points at the centers of these divisions. This is clearly shown in Plate IV.

The moment of inertia at each point is found as shown

* Essentially the arch taken by Professor Cain in “Theory of Concrete Arches and of Vaulted Structures.”

in Table I, page 90. above gives a very complete exposition of the method for dividing the axis so that dsA-I shall be constant.

Prof. Cain in his book referred to

72. Computation of Hi for Unit Loads.—The process is precisely that followed in the first example, only we use the general formula (Art. 13)

2H i —

lmxA ( y —

lyA

1A

lmxB

Tables I and II give the work in detail (see pp. 90, 91). 73. Computation of Mi.—The general formula in this case is (Art. 13)

Mi=Hi

lyA

1A

lmxA[ % —

lx² A

IxA

(*-

lx² A

IxA

)'

lyA

have been found in Tables I and II, so there

remains simply the multiplication of the two factors. The determination of the second term we will take up in detail, as it is well to know a few checks and short methods. Designating this term by mi,

Hi and

1A

lmxAl x —

lx² A

mi =

lx A

lmxA ( x —

lx² A

UlA

lx² A

IxA

lxA(x —

Ź XÔ ~ X <A , b£x 4 ?_b    b //a 4l,\

dx    /dx\²

To find lx²A, let x = —z] then x² =■- J z², where

c /    ' *- * <,

* 4, + éA

-5

/IX

~ d > (<A, b 4 , y --

Ł

TABLE I.—COMPUTATION OF d.

Point No.

W I W

«-ii I «

f«

-12

Ł3 ₂

—+~(h’)²=Ix.    00

12 IO

12

CO I H «O 1^

O -o CC^J O\0i .£» Cm ks m

3.80

2.80 2.08

1-54

1.38

1.30

1.19

I . IO

1.03 i .00

54-87

21.95

9.00 3-⁶5

2.63

2.20 I . 69 1-33 1.09 1.00

4-57

1.83

0-75 0.30 o. 22 o. 18

o. 14 O. II 0.09 0.08

1.90 i .40 i .04

0.77

0.69

0.65

0-59

0-55

0.51

0.50

1-73 1.23 0.87 o. 60

o. 52 0.48

0.42

0.38

0-34

°-33

SyJ _ 1982.4Ó_ là 243.02

2.99

¹-5¹

0.77

0.36

o. 27 0.23 o. 18

o. 14 0.12 o. II

0.598 o. 302 0.154 0.072 0.054 0.046 0.036 0.828 0.024 o. 022

5 ■ I7 2.13 0.90 0-37 o. 27 0.23 o. 18 o. 14 O. II o. IO

3-41 3.21 3.08 2.92 2.81

2-75

2.74 2.70 2.68 2.68

0.66 i. io ¹-5¹ 3-05 3.42 4.66 7-8ç 5.95 10.41 6.98 11.96 7.77 15.22 8.30 19.28 8.71 24.36 8.98 26.80 9.12

0.726

4.606

15-937 96.946 72.662 92.929 126.326 167.929

218.753

244.416

8.1576

[21.51

991.23

243.02

1982.46

SJ

SyJ

TABLE II.—COMPUTATION OF Hi.

Point Noi

JyJ

y-~ or y — 8.1576.

J(y-8.1576)

B.

yd(y— 8.1576)

c.

mx.

mzB.

P₂ = *~P₂'-

m x.

mxB.

X.

0.66 —7.0576 I-5¹ ~5-I076 3.42 -3.4976 7.89 —2.2076 10.41 —1.1776 11.96 —0.3876 15.22 +0.1424 19.28 +0.5524 24.36 +0.8224 26.80 +0.9624

-    4.658

-    7.712

-    11.962 -17.418

-    12.259

-    4.636 + 2.167 +10.650 + 20.034 + 25.792

-    5•I23°

-    23.5256

-    55-7412 -103.6379

-    85.5667

-    36.0192 + 17.9888 + 92.7639 +179.9024 + 235.2259

1-34

1-34

1-34

1-34

1-34

1-34

1-34

1-34

1-34 ¹ • 34

Not necessary to compute, as sum= zero.

1.34—    6.242

4.02 — 31.002 4.02 — 48.087 4.02 — 70.020 4.02 — 49.281 4.02 - 18.637 4.02 +    8.711

4.02 + 42.813 4.02 4- 80.537 4.02 +103.684

1-34

4.02 6.70 9 38 12.06

14-74 17.42 20.10 22.78 25.46

121.51

-58.645 -309.6144 + 58.643 +525.8810

— 223.269 + ²35-745

.002

SB

+ 216.2666

SC

SmxB

+ 12.476 SmxB

10

TABLE II.—COMPUTATION OF H— Concluded.

11

12

13

14

15

16

17

P₃ = i=P₃'.

P₄=i=P₄'.

+5=1= Pb-

P₆=I=P₆'.

X.

mx.

mxB.

mx.

mxB.

mx.

ntxB.

mx.

mxB.

1    1-34

2    A. 02

10

6.70

—    6.242 - 51.002

-    80.145 —116.701

-    82.135

—    31.061 + I4-5I9 + 7¹- 355 + 134.228 + 172.806

6.242 31.002 80. I/IT

1-34 4.02

6,70

9-3⁸ -Ï03T381

9.38 -114.98

9-3⁸ - 43-4⁸6 9.38 + 20.326

9-3⁸ + 99-⁸97 9.38 +187.919 9.38 +241.929

6.242

1-34 -4.02 — 31.002 6.70— 80.145 ■ 38 -163.381

12.06 — 147.844 12.06 — 55.910 12.06 + 26.134 12.06 +128.439 12.06 +241.610 12.06 +311.052

6.242

1-34 -4.02 — 31.002

6.7c— 80.145 9.38 -163.381 12.06 - 147 844 14.74 - 68.335 14.74 + 31.942 14.74 +156.981 14.74 +295.301 14.74 +380.174

1-34 4.02 6.70

9-38

12.06 14.74 17.42 20.10 22.78 25.46

+ 392.908 -347.286

+ 550.071 -439-245

+ 707.235

-484-524

+ 864.398 -496.949

+ 45-⁶22 XmxB

+ 110.826

ImxB

+ 222.711

2mxB

+ 367.449

ImxB

2 Hi — 0.2109

0-5123

1.0298

1.6990

18

19

20

21

22

23

24

25

.s

P7-I-P7'.

+8 = 1 =P₈'.

+9=1 = Pg'.

PlO — I = Pio'.

mx.

mxB.

mx.

mxB.

nix.

mxB.

mx.

m XB.

m C4 co Tf i00 o\

-496.949

-496.949

Ov

O'

\d

O'

O'

O'

-O

17.42+    37-749    17-42+    37-749

17.42+ 185.523 20.X0 + 214.065 17.42+ 348.992 20.10 +402.683 17.42+ 449.297 20.10+ 518.419

17.42 4-    37-749 17-42+    37-749

0.10+ 214.065    20.10+ 214.065

22.78 + 456.374    22.78 +_4Ç+574

22.78+587.542 25.46+ 056.664

+1021.561 - 496-949

+ 1172.916 - 496-949

+ 1295.730 - 496-949

+ 1364.852

- 496.949

+ 524.612

ImxB

+ 675.967

ImxB

+ 798.781

XmxB

+ 867.903

2i7i=2.4257

3-I256

3■6935

4.0131

— = 1.34, or one half of one of the twenty divisions into

dx

which we divided the span of the axis. The first five columns of Table III give the complete determination of lx² A

⁼ 3o.373-

ZxA

1A

(il

Zx²A

IxA

) =243.02(26.8 -30.373) = -868.304.

We now have

tn 1 =

ImxA (x — 30.373)

— 868.304

Cols. 6, 7, 8, and 9 give the deduction of A(x — ^0.373), and in col. 9 the algebraic sum is found to be — 868.308, which should equal the denominator when all work is correct. In this case the difference is 4 in the third decimal place (see cols. 10, 11, and 12).

The next step is the computation of

2

.    ZmxA{x-3 0.373) ₌

— 868.304

I

This may be written

x=aA(x-3O.373)

—nii =K\ 2    -

*=o 648X1.34

mx

1-34

A (x — 30.373)

— 648

X 4*

x'<l£a    — 30-373)

648x1.34

since

mx =R\X for v = o to x=a

and

xf =1 —x.

mx=R₂xf for x = a to x =1,

00

JO

A(x — 30.373).

19.102

39-793

80.961

i65-⁶3S 190.638 186.971 I97-I45 198.063 184.963 131.668 59•844 10.889 60.289 88.383

¹⁰¹-5°5 116.248 109.253 56522 29.003 14-445 586.537

•845

868.308

ą\ą

I I I II I I I I I I+++++++++ + I

VO M <N O' W NO CN 00 NO o O NO 00 CN NO M On w MVO no vo CO Tł* On CN cn COCO CO ro <N cn On ^fOO rf to no

O m vo On TfNO no Th On to m O

MMMM04<N0<<NHMMM

CO M O

243.02

VI

2x*A

x — -rr— or 2x4

*-30-373-

-29.033

-26.353

-23.673

-20.993

-18.3x3

-15-633

-12.953

-IO.273

-    7-593

-    4-913

-    2.233 + 0-447 + 3-I27 + 5-8o7 + 8.487 4-11.167 + 13-847

4-16.527 4-19.207 4-21.887

TABLE III.—COMPUTATION OF Mu

Tzz

A.

1-34

z.

rt CN    OOOO    Tt-    cn    oooo    O 00    O    rf    CN O CO O

coo    N    ro    O    N    Tf    H    N    Tt    h    CO    lO h    CO    to    (N Q\ N

N    o    w    LO00    O    covo    00    M    Tfvo ON <N

h    m    m    CN    CN    04    CN    cocococOThrfrtTl-to

m rJ*vO On

1004.52 2080.78 4275.00 8978.82 10847.22

1T5°5-52 13667.56 16388.00 19926.48 21493.60

1522

1378

1250

1138

1042

962

898

850

818

802

OhNOnhONOO^CO no to Tf 00 Tf On Cl CN COCO

O h io On TfvO

m h m M <N CN

M On

110167.50

2z*A

Symmetrical

values

combined

34

1.34=30.373

121.51

243.02

VI

Ix*J Iz²A 2xA \11A

110167.50

20(243.02)

S-00 N 'S' n'm'nO S' tN? S' ^-no 00 N⁵no' N

h m M O P0 Tf ir>NO t^00 On O 0* <-0 to

M M H M

M CO to N O H fO 1-0 N O H <OIONOnh^<OioNOn

Point.

h N (O't tONO t^-00 On

o *b *«>00 V»5d ViNtt'fo'Ni *m

HH M

's

III.—COMPUTATION OF +7,—<

TABLE

19

18

16

15

14

13

12

11

10

4(^—30.373)^ J(x— 30.373)4^ 648X1.34    ’    648X1.34

r(^l-x) \FzxZ /

Tf M VO 00 TtMOO O to CO to VO H 00 r^M CO NO    00 VO

VONN st HCO O to to NO to 00 O'ONMOOfOfOMONO .

M CN ro st M CO CN NO VO CN CN st N I^nO m rf M O VO m OM

m CN rf00 O'00 00 t^NO tJ-mOhmmmmOOO X

CN co "

VO «oK|

I I I I I II I I I I +++++++++ + I I

TT T N O O to ^foo CO O' CO

M <N CO CO rt* CO H o CN CO Tf IT) lO CO M o

NO CN O' rf CN 00

O H VO t^NO M O' »OX

to CN NO Os CN o to CN O' i⁻ N lO O' ’t

O' *n CO O' to NO no 00

— 27.6402 + 27.6480 +    .0078

^=0

Ri.

X+l-34-    I-34-

868.304-i-1.34 or

4(^—30.373)

xà(x-^l) or xd(x— 26.8).

xJ.

x — il or x— 26.8.

Point.

0\Ni/)fOH ONNiOfCH ONt^iOCOM OnNIO fO w

rOCOCOCOCOCSCSCNCSCSMMMMM

m fOiON On H covot^*ONM co>ONOnh fO ION O' MMMMMCNCNCNCNCNCOCOCOCOCO

648

—0.0296 — .0614

loO rf ON t}-nO i^fO (N io On 00 O O 00 O HMWN^fOOIN

rt-oo Tt-

CN 'O ro On m On

o o o

O N O' O'

CO IO t^NO

MMMM

CN tr Tf CN

oo ^ cn o o o

I I! I I I II I +++++++++

855-711

1567.264

2763-549

4788.584

4523-559

3479.069

2678.271

1730-945

787-357

96.234

23270.543

(H

F+7

1-0.9054

— 2.2454

-1 • 3400

•*IU JO JOJ'BUIUIOUOQ

•6 \OJ 33S

ń

^00-898 = —

(H

r*z

00    M    rj- Os co co    CN CO    IO00 00 NO    N    CO    o    O    t^Os

X O O'    O    to CN m vo    O' co    m    1^00 NO    N    rt    O'    rf-00

CN    Tf    VOVO VO^TfCN    ^    O VO O    rf    CN CO    o    Tf    Tf*

(N    NW t^.NO X    VOX    VO    VO Tf VO NO    CO    TtVO    N    CO

M M CN CO VOVO t^VO VO St rt co H

O NO

NO    X    O    CN r-f-NO    00    O    CN    Tł-TfCN    O CO O    Tj*    CN    OCO'O

st    N    H    rt*    O    fO    N    O    cocoo    N    CO    o    N't    m    t^Tł-

M Tt-vo    O'    cn    rf    O    CN vo

M    M    M    CN    CN CN

vo CN O Tt- CN

CN <N CN M M M

M CN CO Tj- vovO t^X O'

o "b "onoo V-vb "vo'V'cob m

M M

—134.00

+ 134-00

0.00 S(x— 26.8)

LZ

Mi

For

Symmertical Loading.

S'&IS'S

fO to on O h w

m COOO 00

w ^ 't 1C co O ^-00 M CO

MIII+ + + + +

N-:

.9

cc

w

Oit^O OiinfOOiinOiNN^MnO -7-00 PO <N M co N O h T)-vO O NOO H tnito it On'O OCO Oi PO t>.<3

IO PU <3 N IO mH N « -Pt O    M

h co ło^o io it « o poo f^.oo ł^-io po h o o °

•S

MMIII

-30.462 + 50.788

+ 20.326

XM

Lj

<5*    «

.2

jo l±-ih

Mr

8.1576/L.

to NO NO too

MMMMM

OnnO Tt- m O O

No

§

-mi.

fa

—    1-339

—    3.962

—    6.460

—    8.708 -10.445 -11-593 —12.163 —12.126 -11.476 —10.256

—    8.631

—    6.821

—    5-°44

—    3-454

—    2.136

—    1.132

—    0.486

—    0.178

—    o.044

O

♦—)

eo

N

R₂ “B

-1.311 -3-764 -5.726 -6.538 -6-355 -5-463 -3.788 — 1.502 +i.084

+ 3-236

+ 4•498 + 4.762 + 4.301 + 3 • 451 + 2-455 + 1-376

+ 0.489 + 0-137 + 0.0206

Ph

K-

J-H

Ri “A”.

ôo'oo -7-0 O O lOTt-O « aw)i^iOHOO tJ- ŁO^-o M h O W fO'O u-> -<3- m V) «) Ol Ul in O'fo o

O M ■'t'O 00 O P) no PO H OsMC rf M O O O

MMMMM

TABLE

x'-

21

x <(l — a) y 4(:r — 30.373) x' — o 648 X 1.34

See Col. 19.

-52-4534

—    50.1816

—    45.8066

-37-3586

—    28.2446 —19.8636 -11.6556

—    4.0056 + 2.5494 + 6.8124 + 8.5680 + 8.2824 + 6.8814 + 5-1134 + 3-3864

+ 1-7754 + 0.5924 + 0.1564 + 0.0223

“B.”

20

jj(x— 30.37.3Nr 648 X 1.34

See Col. 18.

—    0.0296

—    0.2138

—    o.8388

—    2.6308

—    5.2768

—    8.4558

—    12.4078 —16.9978

—    21.8428 -25.6998

—    27.6402

-27-2538

—    24.9188

—    21.2468 —16.6938

—    ii.1448

—    5-5678

—    2.5158

—    0.8619

“A.”

Point.

h N po rj- i/n\o t^oo oo*o *bioo *N\b *t/W*fo <n m

Dividing both numerator and denominator by 1.34, % becomes z and the denominator 648. Cols. 14 to 26 inclusive show the solution of the above equation in detail. As checks col. 18 should sum zero and col. 19 have an algebraic sum equal to the span, in this case 53.6. The error in each case is 0.007^.

Cols. 26 and 27 give the values of Mi for unit loads.

74. Values of Vi, yi, y2, yo. etc., for Unit Loads.—These quantities are quickly determined as shown in Table IV, which also contains for convenience in future calculations

the values of I of Hi, Vi, V₂, Mi, and M₂.

75.    Values of Hi and Mi for the Dead Load.—Since the Spaa is divided into equal parts, the dead load at each point equals 140 X 2.68 X the ordinate from the intrados to the roadway, nearly; so it is unnecessary to carry the common factor 375.2 through the work. Column 2 of Table V, page 98, contains the ordinates which must be multiplied by 375.2 in order to obtain the dead load assumed at each point.

Tables V and VI give the values of H1 and Mi as found by considering each load separately, and also by considering the loading as a whole. For M1 we have — 40.289 and —40.301. For Hiwe have 46.504 and 46.502, in both cases close agreement.

76.    Location of the Equilibrium Polygon for the Dead Load. —Knowing Hi, V1, and Mi we can graphically locate the polygon. The algebraic determination of Mx+Hi, however, is more accurate and requires hardly any more time. From Arts. 16 and 17,

Mx

H-

ImxA\ i 1A JH~i

CO

im₂.

0

HE^o

+ 0.191

+ 0-873

+ 2.476

+ 5-544 + 10.338 + 16.778 + 24.483 +32.727 +40.464 +46.576

+ 5oi65 + 50.788 + 48.519 +43.856 + 37.611 + 30.992 + 25.392 + 21.665 + 20.326

—    1-339

—    5.066 —10.666 -17.285 -23-53° -28.193

—    30.462 -29.839

—    26.250

—    20.138

—    12.401

—    4-157 + 3-548 + 9.998 + 14-782 + 17.850

+ 19-453

+ 20.135 + 20.326 + 20.326

00 W COW NiOfOO vn O t^vo COCO 1^00 WOO n00 Tf- O lOi-NO N ro O O O' W w NO W00 ^ W co O' O VO NO co t^00 O' co O' h NO    ro covO NO

O m coonnomcjcococow t^OOO conQ

O W co vo vo nO no NO VO no WHMHMHHHMM

H W Tj-NO 00

10

HC^O

P) O PI rt- CO'O (>N fl N CPSO po rf P) O P) O O O O h f<500 oo Ttoo « Nnoo rj-oo oo po h O O O OOOOmpoioOsPOOsiopOmOOOOOO

po ■+ lo'd i^oo o o

Ui

O 00 OOOVO    PO00 PO M Tt t-'.'O 00 O 00 O O

0 Os OsSO H HlOHt^Płl^HlOMHSO OsOsO O Os 00 SO Tj- O SO O Tj-so 00 Os Os Os O' O O

w M P) PO 'S- losd »^00 00 OsOsOsOsOsOsOsOs O O

H H

O O' O' Os

>

HH

SO POP) MpipfPHOO OOOsOOhhhhO

od d o 6 6 6

M H M 1H H W H

Os Os

=.

■Ivj__X

o oo w co oo h w to tooo fofoaw w oo w O On O' N 't o co tovo vo Tf- co tJ-no O lO W O O O

O O O O O O00 t^NO ini-fOM H

•W-vc

'K

w fCI-fOC'NO'Tti- v/~;VO co W Tf\0 'O N rt M CO M Tt 00 CO O O 00 Ti- O CO'O o w TfNO

LOH o m fO fO Tt rf loin lo'O O NO

8 Q\fOiOM

O 1/)PI H

llllll++++++++++++

Point

No.

H w rO’t lO'O t^oo On

O *0 On00    XTi Tt'co'w

W M

vo

co

'lW IOJ SUOrp^HOfBO 3[D31Q

mxA.

t^*0 On O O LO O o

LOW M M QSM ł'*. Tt* QN O

lO'O W 10 O M co to rf vO Tt O OOWWMMO\fOON LOCO 00 LO LO M ro W CO TfvO 00 M w

W 00

50991.42

101982.84

J.

Table I, Col. io.

NO M N Q\W^Û N 00 O O O to rf 00 ^ On W W rooo

O w io On '^■'O -

♦H M M M w W

O H fON

121.51

243.02

TABLE V.—Hx AND Mx FOR DEAD LOAD.

ttii.

Table VI, Col. 6.

My

Dead Load. (a)X(₄).

Mi.

P = P' = i. Table III.

DeadLoad. (l )X(2).

•d

't a'O'O^JNN^I O ^

O w o O O w O 00 m io

OsvO O w N ION « O NIOCI'O O W Tf- 10*0 M W

ro ro ^ rf rt rf

0% Os h O O'iOM Tt'O to CnO O ^'O 0» O O m to ON W On

r}- m

io to IO00 to O w o tooo

MfOtOWM M w w w

MIII + + + + +

—127.288 + 86.999

— 40.289

-    ¹-339

-    3 536

-    4-9l8

-    5.016

~ 3-177 + 0.131

+ 4-171

4- 8.328 + 11.833 + 13.849

-30.462 + 50.788

+ 20.326

•550¹

15396

2.8842 4-53II 6.0315 7•io73 7-6577 8.0518 8.1466

46.5019

ft.

aj

•3

'O

d

O to O to O to to to 00 to iO to to NO rf to On rf h O

m a WiO't fON W W W

zS ■ iS

103.04

H u

P = P' = i. Table II.

0.0576

0.2109

°-5I23 i.0298 i.6990

24257

3-1256

3- ⁶935

4- 0131

16.7675

zyJ 2nixd

For symmetrical loading Mi=Ht~r-j---^—=*379.345 — 419.646== —40.301 for dead load.

vX

Point No.

'm *W VpoVV»00 *t^00 VOs

h m ro -t ioO l^oo On

io+ io'

u*

-1

HH

TABLE

<6)X(7).

------S

-    321.588    g

-    I359-548

-    3085.082    ^

-    5574-63I    *g

-    4499.666    j>

-    1864.924    o

+    927.411    ^

+    4747-983    >

+    9157-541    ^

+11929.832

^(y—8.1576).

Table II, Col. 3.

»n*.

Point No.

53

-16705.439 + 26762.767

+ IOO57.328    ^

VO

Ov

—    4-658

—    7.712

—    II .962 -17.418

—    12.259

—    4.636 + 2.167 + 10.650 + 20.034 + 25.792

i* O^lOi^^NN N O Tt-O W Os O O W Os 00 M

Os MD    O    W N ION (N

O N io M O O W rt i-o MD M w ro fO st rt rf

,,3    5Æ8,<2S,8%g.2ïï

M W W 00 fO O Os CS MIO lOfOOsfONO m COTfrf

H M w w fO to ro fO fO

= 46.504.

RxX —.

1-34

h|’5

1-34

•z

Dead Load.

MD O Tt-00 WOO TtOO

loo so md n* oo oo

^ NO POMD Os w looo *D xoMD O O MD ^ ^

M W CO Tf Lr>MD t^OO Os

OOOvONOOONO

O H oo NlOfOH

° "liOH Os'© O O ro N »O N 00 <3 lO Tf W) KO H w ci (Oł u-)VO

M fOlON Os H

fO io N O'

H H H H

O LO O PO O lo PO IO00 PO LO lO PO MD rf VO Os ^ M O

w ONIO i- PO « W W W

H <N CO ^ IOMD ^00 *Os

M W CO tJ* iosO N-00 Os

IO + io'

Imxd(y—8.i576)_ 10057.328 2yà(y—8.1576) 216.2666

46.5 X 140X2.68= 17400= Hi for dead load.

H1=

Remarks.

The moment Mx at any point equals the ordinate in Col. 6 multiplied by Ht.

TABLE VII.—LOCATION OF EQUILIBRIUM POLYGON FOR DEAD LOAD.

Above or below Axis of Arch.

below

above

below

Mx

Hi

0.844 0.482 o. 126 0.021 0.064 0.047 o. oi5 0.037

O.OII

0.017

0.039

2yA

1Z~&

2m x A

Table II, Col. 2.

2A

m.

2m XA

Hi

2A

i.

■m

IntxJ

2A

mx.

Table VI.

Point

No.

-8.158 -7.058 —5.108 -3-498

—    2.208

—    1.178 -0.388 + 0.142

+ O.552

+ 0.822 + 0.962

— 9.ooo -7-54° ~5-²34 “3-477 -2.144 -1.131 -0-373 + 0.179 + 0.563 + o.805 + 0.923

in hO O O O oo M N in a

NO 'O ro t^vO 'O en en h r^-00

O' O co M O' Ci t^oo vO N m m rj-vO O' in h n cn^j-

Tf CO 01 H

i ii ii M + + + +

^ i^i in io io i/i io io io 10 'OO'O'O'O'O'O'O'O'O'O

O' O' O' O' O' O' O' O' O' O' O'

MMMt-tMMMMMMM

rJ-TtrtrfTtrtrtTfrt-Tj-Tt

rf    CMO    lO    ’O    N    N    (i    O    rt-

O h ci roi- i^oo O' O

«I-

Mx .

Table VII gives the values of in col. 6, showing that

ti i

the polygon nearly coincides with the arch axis.

77. Maximum Fiber Stresses Produced by the Dead Load at Point i.

Moment of inertia = 5.17 = 7.

Area of section in equivalent concrete =3 80+0.20

= 4.00 sq. ft. -=7fi

Dist. outermost fiber of concrete from neut. axis = 1.90 = z.

c.g. of steel above or below neutral axis = 1.73 =z'. .£ = 1.34, y = i.io. sin </> =0.618, cos <£=0.786.

Mx=Mi + ViX-Hiy,

NX = V 1 sin 0 +7/1 cos <£,

TX = V 1 cos <£—77 1 sin 9'>,

Mi = -40.289(375.2) = -15116,

Hi = +46.502(375.2) =    17448,

Vi =51.520(375.2)    =    19330,

= 19193,

Hiy = 17448(1.10)

=    25902.

Vi* = 19330(1.34)

Then

M x = —15116 +25902 —19193= —8400. Prom Table VII we have

Mx = i7448( —0.482) = —8200, a difference of 200 pounds per square foot.

Nx = 19330(0.618) +17448(0.786) =25660,

P= — ±o.367(Mx) =£1ÉÉ£ ±0.367t-8400).

4    4

and

= 9500 comp, in the lower fibers of concrete 3300 comp, in the upper fibers of concrete

The unit stresses in the steel are as follows :

P' =

r nx    }

1    ±0.334Mx \ 20 = (6415 ±2800)20.

and

/. p' = 184300 comp, in lower steel 72300 comp, in upper steel.

The above unit stresses are pounds per square foot. Reducing them to pounds per square inch,

the maximum compression in the concrete is 66 lbs.

steel is 1280 lbs.

These values are quite insignificant when compared with the ultimate strengths of the materials.

78. Maximum Fiber Stresses Produced by the Live Load at Point i.—From Plate III we see that loads 1-7 inclusive produce one kind of stress and loads 8-1 ' inclusive the opposite kind.

A live load of 112 pounds per square foot of roadway is equivalent to about 300 pounds for each division of the span. For loads 1-7 inclusive the fiber stresses are obtained as follows (see Table IV) :

Mi = -30.462(300)

H1=    2.9675(300)

V\ =    6.654(300)

Mx= -9139 +2680 -979 = -7438

and

= -9139

890

=    1996

Vi sin (j) = 2000(0.618) = i±iCOS<£= 890(0.786) =

A* =

1236

700

1936

and

I Q ?6

P = ~~— ±743⁸(°-3⁶7)

= 3214 compression in lower fibers of concrete 2246 tension in upper fibers of concrete.

For the steel we obtain

P' = {^p±743⁸(o-334)} ²⁰

and

= 59400 comp, in lower steel 40000 tension in upper steel. For loads 8-1 ' inclusive we have (see Table IV)

Mi =50.788(300)

Hi = 13.7995(30°)

Vi = 3.346(300)

Mx= +15236 + 1340-4550 = +13330

= +15236 4140

1000

V\ sin </> = iooo(o.6i8) = H\ cos 0 = 4140(0.786) =

620

3²5°

3870

Nx =

P =^-^±13330(0-367 )

4

and

= 5860 compression in upper fibers of concrete 3920 tension in lower fibers of concrete.

For the steel,

pf =[970 ±¹3330(0.334)]²° = io84oo comp, in the upper steel

and

69600 tens, in the lower steel.

79. Maximum Fiber Stresses Produced by the Dead Load at Point 7.

Moment 01 inertia =0.18 (see Table I).

Area of section in equiv. cone. =1.19 +020 = 1.39 sq. ft.

2 = 0.60,

z' = °-43>

x = 17.42,

? = 8.₃.

sin 0=0.208

cos 0 =0.978.

M1 = — 15116

M x=Mi + V\%-H\y - IP(x-a), Nx = (Vi — IP) sin 0 +Hi cos 0,

17448

Vi-

19330,

TX = (V 1 — IP) cos 0—Hi sin 0,

i’P =

15732,

^ = 336728,

Hiy = 144818.

IP(x -a) =

176241,

Mx= +553.

From Table VII,

Mx = i7448( +0.037) = +646.

This indicates a large percentage of error. The error is of no consequence as it amounts to less than 3 pounds per square inch fiber stress. In order that col. 6 of Table VII should be correct much greater accuracy would be required in the previous work. For practical purposes, however, col. 6 is quite accurate enough.

Hi cos 0 = 17064.

(Vi-IP) sin 0 = 749,

Nx = 17813.

P-

Nx

1-39

±MX( 3^) =12820 ±1843,

or

and

p = 14663 compression in upper fibers of concrete p —10977 compression in lower fibers of concrete.

For the steel,

pf =    ±M*(2.4) I 20 = {12820 ±1327)20

or

and

p¹ =282940 compression in upper steel pf = 229860 compression in lower steel.

80. Maximum Fiber Stresses Produced by the Live Loads at Point 7.—From Plate III we find that loads 1-8 inclusive produce positive moments at this point and loads 9-1* negative moments.

Considering first, loads 1-8 inclusive: from Table IV,.

M\ = —29.839(300) = — 8952, (Vi — IP)sm<f>=* 88,.

Hi =

4-53 (3°°) ⁼    1359»

Hi cos (f> = 1329.

Vi =

7.411(300)=    2223,

Nx = 1417.

3⁸725,

ViX— 2223(17.42) =

Hiy= 1359(8.3) = 11280,

IP(x —a) =16884.

Mx = +1609.

1417

1-39

±1610(3^) =io2o±5367.

Then

p = 4347 tension in the lower fibers of concrete

and    p = 6387 compression in the upper fibers of concrete.

For the steel we have

p' = {1020 ±1610(2.4)} 20 = {1020 ±3864120,

or

p' = 56880 tension in the lower steel

and p' =97680 compression in the upper steel

Proceeding in a manner similar to that employed above ■ for loads 8-1' we obtain

p =    8040 compression in lower fibers of concrete

and p =    2640 tension in upper fibers of concrete,

£' = 145280 compression in lower steel

and

p' = 37280 tension in upper steel.

81.    Maximum Fiber Stresses Produced at Points 1 and 7 by the Dead and Live Loads.—Tabulating the above results and combining those producing maximums we have the results given in the table at top of page 107.

The maximum stress in the concrete is 146 pounds compression per square inch and in the steel 2650 pounds compression per square inch, values considerably below the allowable. There is no tension at these points.

82.    Temperature Stresses.—For a change of temperature of ±40° F. the horizontal thrust is 6500 when E = 1500000 and £=0.000006.

MAXIMUM FIBER STRESSES. (pounds per square foot.)

Steel.

Concrete.

Loads, etc.

Point.

Upper.

Lower.

Upper.

Lower.

Dead load..............

Live load 1-7...........

8-1'...........

Maximum compression. ..

tension.......

3920

+ 3300

— 2240 + 5860

9x60

+ 9500

+ 32I4

12714

+ 72300 — 40000 +108400

180700

+184300 + 59400 — 69600

243700

Dead load..............

Live load 1-8............

9-1'...........

Maximum compression.. . ‘ *    tension. ......

+14663 + 6384 — 2640

21047

+ 10977

- 4347

+ 8040

19017

+ 282940 + 97680 - 37280

380620

+ 229860 — 56880 + 145280

375I40

x = 6$oo(y

2yA

lâ

j = 6500(^-8.1576),

NX = H cos <j> — 6500 cos <j>.

For point 1,

Mx = 6500(7.06) =45900, Nx = 6500(0.786) = 5100. For a drop of 40° F.,

or

and

P=~

5IO°

=F 45900(0.367) = -1275F 16850,

p = 18125 tension in upper fibers of concrete P = I5575 compression in lower fibers of concrete.

For the steel,

p' =332100 tension in upper steel

£' = 281100 compression in lower steel.

and

For point 7,

Mx = 6500(0.142) = 923,

Nx = 6500(0.978) =6400.

For a drop of 40° F.,

or

and

or

-923

1-39

■F 6400(3^) = -664=F 21333,

p = 22000 tension in upper fibers of concrete £ = 20700 compression in lower fibers of concrete..

For the steel,

pf = [ — 623^ Ó400(2.4)]20 = — I 2460T 307200

p' = 319700 tension in upper steel

and

p' = 294700 compression in lower steel.

A rise of 40° F. will reverse the above stresses.

83. Maximum Stresses Produced by Dead Load, Live Load, and Changes of Temperature.—Combining the stresses of Art. 81 and 82 we have:

Point i :

£ = 27285 compression j 17065 tension

£ = 28289 compression 10000 tension

upper fibers of concrete ;

lower fibers of concrete ;

pr = 512800 compression 300000 tension

upper steel ;

lower steel.

p' = 524800 compression 166400 tension

Point 7:

p — 43000 compression 10000 tension

p — 397°° compression 14100 tension

pr = 700300 compression 74400 tension

pr = 669800 compression 121700 tension

J upper fibers of concrete ;

lower fibers of concrete ;

I upper steel ;

lower steel.

The allowable compression in the concrete, when temperature is considered, may be assumed at 800X144 = 115200 pounds per square foot, and the tension at 11500 pounds per square foot.

In compression the maximum stresses are considerably less than the allowable, while in tension they are much larger. Yet if the tensile strength of concrete is taken as one tenth the compressive strength, the above stresses are less than the ultimate strength of the mate-rial. If it should happen that a maximum change of temperature and a maximum live load should occur at. the same time, the concrete would probably crack, but. the steel and the compression concrete have ample margin to cover this contingency. It is quite improbable that, a range of ±40° F. ever occurs, so the two sections may be considered safe. The crown should be examined in an actual case. Although the live-load moment will be small, the temperature moment will be considerably larger than at point 7.

84.    The Axial Stress.—Thus far the effect of the axial stress has been neglected. Proceeding in the manner followed in example 1, the value of Ha is found to be about 6.7% of Hi. The effect is seen to be somewhat less than in the previous example. As the rise of the span increases the effect grows less.

85.    Assumption that Steel Resists Entire Bending Moment Due to Changes of Temperature at Point 1.

Max. comp, in upper steel due to D.L. + L.L. =1255 lbs. per sq. in.

Max. comp, in lower steel due to D.L. + L.L. =1700 lbs. per sq. in.

Moment due to ±40°= ±45900 ft.-lbs.

Area of steel = ^[f(2f—f)] =0.70 sq. in.

Dist. c. c. steel = 3.80—0.34 = 3.46 ft.

= 13300 lbs.

Total stress in steel =

459°°

3-46

= 19000 lbs.

Stress per sq. in. =

13300

0.70

Max. comp. = 19030 + 180 + 1700 = 20880 lbs. per sq. in.

Max. tension = —19000 — 180+1225 — 17960 lbs. per sq. in. All well within the elastic limit of the steel.

This shows that even if the ring should crack entirely through at point 1, the steel would safely carry the maximum temperature moment even when combined with the dead- and live-load stresses.

A brief calculation for point 7 and the crown shows that the steel is here stressed well within the elastic limit.

86.    Third Example. — In this example we will take the data used in the second example and show how the computations of Hi and Mi can be quite rapidly made.

87.    The Computation of Hi.—The equation used in the former calculations was

'-JjO

where mx = the common moment for equal and symmetrically placed loads. Assuming unit loads, the following values of mx may be written:

Between the load and the left support

dx

mx=RiX —x =—z, 2

where dx is the length of the division into which the span is divided, or l=ndx.

Between the first load and the center of the span

dx

mx

= RiX— (x —a) =a = &—.

Then

dx

an expression which is very quickly handled numeric-, ally.

Although the general data, such as the values of x, y, I, ds, etc., are given in the second example, we will repeat some of it for convenience.

x.

Vi

7.

GENERAL DATA.

Ss.

4.

yJ.

XyA V IA •

B.

+-^>

1-34

O 10« u->00 t'- O *-* 00 cs

h O 'O O' O'O' h

H KlTt    f'-OO 00 00 O'

5-17 2.13 0.90 0-37 0.27 0.23 o. 18 o. 14 O. II

o. 10

3-41

3.21

3.08

2.92

2.81

0.66

¹-5¹

3-42

7.89

2-75

2.74

2.70 2.68

2.68

10.41

11.96

15.22

19.28

24.36

26.80

0.726

4.606

*5-937 46.946 72.662 92.929 126.326 167.929

218.753

244.416

-7.0576

—    5.1076 -3.4976

—    2.2076

—    i.1776 -0.3876 + 0.1424 + 0.5524 + 0.8224 + 0.9624

—    4-658

—    7.712-

—    II .962--17.418

—    12.259

—    4-636-+ 2.167 + 10.650* + 20.034 + 25.792

121.51

991.23

-58-645 + 58-643.

243.02

1982.46

.002-

IJ

lyj

The values of B in the last column when multiplied by y give the denominator of the expression for Hi.

COMPUTATIONS FOR Hi. (unit loads.)

Point,

c.

z. k.

zB.

x=*a 1 zB. x = o

Il ** II

~*l «

0}

<3

II N II ⁰

to

Cj

H,-

D dx

1c T-

M «    t^OO On O

-    5•I23° i

-    23.5256 3

-    55-7412 5 -103.6379 7

-    85.5667 9

-    36.0192 IX + 17.9888

+ 92-7639

+ 179.9024 + 235-2259 -309.6144 + 525.8810

fO IU N Q\ MMMM

vj Ot Cm

—    4-658-    4.658+4.656    4.656

—    23.136— 27.794+12.368 37.104    9.310    0.0288

—    59.810— 87.604+24.330121.650 34.046 O.IO55

—    121.926—209.530+41.748292.236 82.706 0.2562

—    no.331 —319.861 +54.007486.063166.202 0.5149.

—    50.996-370.857+58.643 645.073 274.216 0.8495 + 28.171—342.686+56.476734.188391.502 1.2129 + I59-75° -182.936 +45.826 687.390404.454 1.5628 + 340-578 +157.642 +25.792 438.464 596.106 1.8468 + 490.048 +647.690

647.680 2.0066

8.3840

+ 216.2666

16.7680

+ 432-5332

IC

The values of H1 are identical with those obtained in the second example. The number of operations is very much reduced and the multiplications simplified. This method is shorter than any algebraic or graphical method advanced up to this time. (See pages 90 and 91.)

EXAMPLES SH O IVIN G APPLICATION OF FORMULAS. 113

88. The Computation of M\ and M₂.—In this case we will employ the formula    =

ImxJ

Im

x (%

)J

AJ

IJ (-\ 2

Ix²J

IxJ

))’

IJ •

This expression contains only known quantities and requires but one division and ten multiplications.

ImTJ

IJ *

dx

ÔX

As before let x =—z and a =—k.

For all points upon the left of the load

l—a 2n—k

2 n—k dx --z—.

mx=RiX,

Ri =

mxJ = J

Upon the right of the load, between x’ = a and x‘ =0

mx=R₂x',

a k

i\₂ —~r =—.

/    2 n

.    . k dx

.'. mxJ =J—z—.

2 n 2

Since J has symmetrical values,

x=*a

dx

I zJ

represents the summation of mxJ from x = o to x=a and x = l —a to x=l.

I mxJ =

Upon the right of the load and until x = l—a, mx=R₂x', and for the two values of mx corresponding to symmetrical values of A this becomes

dx

R₂x' +R₂(l—x,s) =R₂l =k—.

f. x=l/3 1 dx

\k I A \ —,

l x=a J 2

I mxA =

and therefore

f * = a    *=//2 I §x

ImxA — \ 1 zA +k 1 A\—.

U=o

2mxA

I A, the denominator of the expression    is already

known, hence the value of the expression is quickly determined.

ImxA(x —%l),

dx

(x — \l) = (z — n)—, where the values are evidently symmetrical about the center of the span but contrary in sign. Until x =a

mx =

2n—k dx

-z—.

2 n    2

Between x = l—a and x = l

k dx mx =—z—. 2n 2

Then for the symmetrical values of (z—n) which have contrary signs we have for the two values of

2n—k k } dx n—k dx —z—. n 2

2n\Z 2

For x = a to# = o and x'=a to x'=0

I /dx\²    x=a

ImxA{x — %l) ——    J (n—k) I (z—n)zA.

From x=a to x=-l—a or xr = a

mx=R₂x' and mx=R₂(l —x').

For symmetrical points

dx

R₂(l-x’) -R₂xr =2R₂{%1 -x') =—(n — z)—.

Summing the symmetrical values,

ImxA(x—^l) = —

S I ÎB-

For the total summation

lmxA (x —

r x=a    x=i/2 -i /^\²i

=    — k) I (z — ri)zA —k I (z—n)²A    ~-

This expression is somewhat long but very easy to use

Mr

Ix²A

IxA

lx²A

MM

Iz²A,

dx

dx

8x

IxA= — IzA=n—I A, U=n—. 2    2    ' z 2

= (n

nIA

l_ Ix²A 2 IxA

I z²A\ dx

77’

and the denominator becomes

-là.

2z²A\dx n2Al 2

The expression for M\ now becomes

Mx\

2yA

X=l/2    ~| fa

2 zA+k 2 A \ —=r.

Ix = 0    x = a ) 2 A A

*=// 2

± i (n —k) 2 (z — n)zA —k 2 (z —n)²A

-Hr

2A

dx

x=*a

2n\ n —

2z²A

n2A

2A

ImxA

COMPUTATIONS FOR

Point.

z.

zA.

x = a 2 zA. x = o

x = il 2 A. x = a

x = il

k. k 2 A. x — a

2 m XA

2m XA

1-34

2A

10

9    9369    171.21

ii    i3!-5⁶    3°2-77

13    197-86    500.63

15    289.20    789.83

17    414.12    1203.95

19    509.20    1713.15

120.85

119-34

115.92

108.03

97.62

85.66

70.44

51 ■ 16

26.80

M f<) ION O' H

fO lO O'

MMMM

120.85 358.02 579.60 756.2Ï 878.58 942.26 9I5-72 767.40 455-⁶⁰

121.51

363.21

601.89

833-73

1049.79

1245-03

1416.35 1557-23 1659-55

1713-15

Cols. (3 + 6)

O IN    love N-00 0\ Q\

.662

.003

•3i8

•597

.788

.865

.810

.586

.150

.446

dx

= 0.0005514.

2.68

2 IA 2(243.02)

Col. 8 represents the sum of the moments for each load multiplied by the -corresponding value of A, divided by 2A. By ordinary methods the determination of ImxA for one load only requires the scaling of 20 ordinates, 10 additions, and 10 multiplications.

n — k. k.

MMMM Ol Ol VI 'O

COMPUTATION OF

z — n.

-19

z.

ImxA(x—\l)

IA

/1 Zx'O'j '

\2

2xA

{z—n)z. z(z — n)J.

x — a

2 A(z — n)z. (n-k) 2 A(z-n)z.

x — o

-19

12.54

77.01

ii

13

15

17

19

19    -19

12.54 89-55

-75 - 256.50 - 346.05 717.99    —1064.04

-99 -1030.59 -2094.63

—    99 —1184.04 —3278.67

—    9¹ -1385 -02 -4663.69

—    75    —1446.00    —6109.69

—    5¹    -1242.36    -7352.05

509.20 —7861.25

—    238.26

—    1522-35

—    5190.75 -13832-52

-23040.93

-29508.03

-32645-83

-30548.45

—    22056.15

—    7861.25

10

11

12

13

14

— J(z — n)².

x —1/2

- 2 A(z-n)². x = a

X — l/ 2

—k IJ(z-n)². x = a

Cols. (8 + 12).

Col. 13

Multiplied by

— 0.0001034.

O O CO-vj Q\Cn ^ Oo N)

—    361

—    289 -225 —169 —121

81

49

²5

—    238.26

—    436-39

—    769.50

-1333-41

—    1259.61

—    968.76

—    745-78

—    482.00

—    219.24

—    26.80

—    6241.49 -5805.10

-5°35-60 -3702.19

—    2442.58 -1473.82

—    728.04

—    246.04

—    26.80

—    6241.49

-I74I5-3⁰

—    25178.00

-²59i5-33

—    21983.22 —16212.02

—    9464.52

—    3690.60

—    455-⁶⁰

-    6479-75 -18937.65 -30368.75

-39747-8S

-45024.15

-45720-25

-42110.35

-34238-05

-22511.75

-    7861.25

+ 0.670 + 1.958 + 3-I4°

+ 4. IIO + 4.656 + 4-728 + 4-354 + 3-540 + 2.328 + 0.813

ox

2 n {^1

-mu

nZA

= —0.0001034. See Table III, page 93, of the second example.

Col. 14 is the complete value of

Imxi{x— %l)

for each load, 1 to 10

respectively.

Note that cols. 1, 2, 3, 4, 5, and 9 will remain the same as long as «= 20 regardless of the span. The formation of the remaining columns requires but 50 multiplications and 30 additions.

FINAL VALUES OF M. AND M.,.

mi

„W

H1 li •

lmxi lmxi(x — ■$-/)

^ +HS9'

»ml

Ml

m₂.

O O    0\<st -U (0 w

+ o + 0.235 + 0.860 + 2.089 + 4.200 + 6.930 + 9.894 + 12.749 + 15.065 + 16.368

+ 0.662 + 2.003 + 3-3i8 +4-597 + 5.788 + 6.865 + 7.810 + 8.586 +9.150 + 9-446

±0.670 1.958 3.140 4. no 4.656 4.728 4-354 3-540 2.328 ±0.813

1-332

3.961

6.458

8.707

10.444

11-593 12.164 12.I26 II.478 IO.259

0.045

0.178

0.487

1- 132

2- 137

3- 456 5.046 6.822 8-633

-1-332

-3.726

-5-598

—    6.619

—    6.244 -4.663

—    2.270 +0.623

+ 3-587

+ 6.109

+ 0.190 + o.682 +I.602 + 3•068 + 4-793 + 6.438

+ 7•703

+ 8.243 + 7-735

Combining the values found we obtain the values of Mi and M₂ for each load 1 to 10 respectively; for loads 1' to 10' Mi and M₂ simply change places. Compare cols. 6 and 7 with col. 26, page 95.

The values of Vi, y 1, y2, etc., can now be found as in the second example.

The above calculations require but little more time than some of the graphical constructions in common use which only give the equilibrium polygon for one set of loads. Here we can quickly determine the effect of each load and then use those producing maximum results

CHAPTER IV.

TYPICAL ARCHES.

A few typical bridges will be illustrated in this chapter which will clearly show that, as ordinarily constructed,, the arch ring proper is heavily reinforced either by masonry or concrete backing or masonry side walls. Since this masonry does not readily follow the arch ring if it sinks,, the actual dead load is never more than the dead weight of the material above the ring; and since the passive resistance of this masonry against moving upward is large in' case the arch ring has such a tendency, it is evident that any ring which is stable under the elastic theory must be stable in the structures as built. Furthermore, experience teaches that temperature stresses may be ignored in stone arches well backed, as is customary. A recording thermometer placed in the ring of a reinforced-concrete bridge having earth filling indicated that the total range of temperature change did not exceed about 20° F. in some ten or twelve months. All rings without backing should be designed to resist a change of temperature of about ±35° F. In rings like that of the Luxemburg bridge full account of temperature must be considered, the range approaching that for steel.

iï₉

tl.371.2r

Batter to lł

El. 4- 299.2Ś"

Rase of Rail, El.351,0

.'0

§dl.W.+ 324.2 m f /June 2,89 J,

WSm'

May 22,94

-Y----70V

.•18 Stone Coping /    / 18*81 ring C

. mt

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89. The Rockville Stone Arch Bridge.—This is typical of a large number of arches recently con-fks₈r^u«.Ei.+324” structed by the Pennsylvania The arch ring is backed with Portland-cement concrete to such an extent that it is increased in thickness nearly three times near the spring line.

1900.)

90. The Bellefield Stone Arch Bridge, Pittsburg, Pain this bridge the outside spandrels are of solid masonry. Inside there are six longitudinal walls reinforced with three lateral walls. The lateral walls do not support any vertical load, as they stop at the springing of the

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(Eng. News, May 10,

arches between the longitudinal walls. The arch ring is securely held by a backing of concrete and the spandrel walls. (Eng. Record, June 9, 1900.)

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91.    The Luxemburg Stone Arch Viaduct.—This bridge is an excellent illustration of spandrels pierced with lateral arched openings. The elastic theory can be applied with confidence in bridges of this type. (Eng. Record, Oct. 12, 1901.)

92.    Approaches to the Thebes Bridge.—The approaches are composed of eleven plain-concrete arches having a span of 65 feet, and one with a span of 100 feet. The proportions of the 65-foot arches are clearly shown above. Note that the spandrel side walls cover nearly all of the arch ring at the supports, and at the crown two fifths of the ring, practically preventing distortion under moving loads. (R. R. Gazette, Jan. 9, 1903.)

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DETAILS OF ARCHES IN APPROACHES: THEBES BRIDGE

93. Vermillion River Plain-concrete Arch Bridge. — This bridge is composed of three spans. The entire loading above the ring is supported by lateral walls which makes the application of the elastic theory quite rational.

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CONCRETE ARCH BRIDGE OVER SALT FORK OF THE VERMILLION RIVER

The ring was designed without reinforcement, but when built the entire concrete work was reinforced with steel bars. This reinforcement is shown in the Railroad Gazette, Oct. 27, 1905.

94. Steel Reinforcement in the Form of Ribs.— Where the steel reinforcement has been concentrated in concrete

rings the three types shown below have been successfully used. The steel is usually assumed to take the entire bending moment. The two upper types were used

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95.    Steel Reinforcement other than in the Form of Ribs.—

The present method of reinforcement appears to follow the idea of thoroughly distributing the steel in layers a few inches from the upper and lower surfaces of the arch ring. This is accomplished by the use of small rods, wire netting, expanded metal, etc. The majority of reinforced concrete bridges in the United States are reinforced with rolled rods, some plain and some “deformed.”

96.    Area of Steel Reinforcement. — The amount of steel is seldom less than 0.6% or greater than 1% of the

area of the ring at the crown, regardless of the type of reinforcement employed.

97. Abstracts from Specifications.—The following abstracts from specifications will indicate some of the methods employed and assumptions made in the construction of reinforced-concrete bridges.

Conditions of Calculation.—Modulus of elasticity of concrete, 1,400,000 lbs.; modulus of elasticity of steel, 28,000,000 lbs. ; maximum stress per square inch on steel, 10,000 lbs.; maximum compression per square inch on concrete, 500 lbs.; maximum shear per square inch on concrete, 100 lbs.; maximum tension per square inch on concrete, 50 lbs. The above to be exclusive of temperature stresses. The steel ribs, under a stress not exceeding their elastic limit, must be capable of taking the entire bending moment of the arch without aid from the concrete, and have a flange area of not less than one one-hundred-and-fiftieth part of the total area of the arch at crown.

Portland-cement Concrete.—The concrete shall be composed of clean hard broken stone, or gravel with irregular surface, clean sharp sand, and cement, mixed in the proportions hereafter specified. Whenever the amount of work to be done is sufficient to justify it, approved mixing-machines shall be used. The ingredients shall be placed in the machine in a dry state, and in the volumes specified, and be thoroughly mixed, after which clean water shall be added and the mixing continued until the wet mixture is thorough and the mass uniform. No more water shall be used than the concrete will bear without quaking in ramming. The mixing must be made as rapidly as possible, and the batch deposited in the work without delay. If the mixing is done by hand, the cement and sand shall first be thoroughly mixed dry in the proportions specified. The stone, previously drenched with water, shall then be deposited on this mixture. Clean water shall be added and the mass be thoroughly mixed and turned over until each stone is covered with mortar, and the batch shall be deposited without delay, and be thoroughly rammed until all voids are filled. The grades of concrete to be used are as follows: For the arches between skewbacks, one part Portland cement, two parts sand, and four parts broken stone or gravel that will pass through a one-and-one-quarter-inch ring. For the piers, one part Portland cement, three parts sand, and six parts broken stone that will pass through a two-inch ring. For the foundations, abutments, and spandrels, one part Portland cement, four parts sand, and eight parts broken stone or gravel that will pass through a two-inch ring.

Concrete Facing.—Concrete facing will be used and shall be composed of one part Portland cement and two and one-half parts sand, and shall have a thickness of at least one inch on all arch soffits, arch faces, abutments, piers, spandrels, or other exposed surfaces. There must be no definite plane or surface of demarkation between the facing and. the concrete backing. The facing and backing must be deposited in the same layer, and be well rammed in place at the same time. If the arch faces, quoins, or other exposed surfaces are marked to represent masonry, such division-marks shall be made by triangular strips two inches wide and one inch deep fastened to the casing in perfectly straight and parallel lines, and all projecting comers will be beveled to correspond.

Connections.—In connecting concrete already set with new concrete the surface shall be cleaned and roughened, and mopped with a mortar composed of one part Portland cement and one part sand, to cement the parts together.

Arches .*•—The concrete for the arches shall be started simultaneously from both ends of the arch, and be built in longitudinal sections wide enough to inclose at least two steel ribs, and of sufficient width to constitute a day’s work. The concrete shall be deposited in layers, each layer being well rammed in place before the previously deposited layer has had time to partially set. The work shall proceed continuously day and night if necessary to complete each longitudinal section. These sections while being built shall be held in place by substantial timber forms, normal to the centering and parallel to each other, and these forms shall be removed when the section has set sufficiently to admit of it. The sections shall be connected as specified above, and also by steel clamps or rib connections built into the concrete.

Steel Ribs.—Steel ribs shall be imbedded in the concrete of the arch. They shall be spaced at equal distances apart, and be of the number shown on plans. Each rib shall consist of two flat bars of the sizes marked on plans. The bars shall be in lengths of about 30 ft., thoroughly spliced together, and extending into the abutments

* The arch rings are also made in the form of voussoirs so as to symmetrically and uniformly load the falsework to prevent its unsymmetrical or excessive distortion.

as shown. Through the center of each bar shall be driven a line of rivets spaced 8 inches c. to c. with heads projecting about I inch from each face of bar, except through splice-plates, where ordinary heads will be used. The bars shall be in pairs with their centers placed two inches within the inner and outer lines of the arch respectively as shown. All steel must be free from paint and oil, and all scale and rust must be removed before imbedding in the concrete. The tensile strength, limit of elasticity, and ductility shall be determined from a standard test-piece cut from the finished material and turned or planed parallel. The area of cross-section shall not be less than \ square inch. The elongation shall be measured after breaking on an original length of 8 inches. Each melt shall be tested for tension and bending. Test-pieces from finished material prepared as above described shall have an ultimate strength of from 60,000 to 68,000 lbs. per square inch, an elastic limit of not less than one-half of the ultimate, shall elongate not less than 20% in 8 inches, and show a reduction of area at point of fracture of not less than 40%. It must bend cold 180 degrees around a curve whose diameter is equal to the thickness of piece tested without crack or flaw on convex side of bend. In tension tests the fracture must be entirely silky. (Engineering Record, Aug. 3, 1901.)

APPENDIX.

Table I.

PHYSICAL PROPERTIES OF STONE AND CONCRETE.

BUILDING STONES.

<7 = Ultimate crushing strength in pounds per square inch. Test Specimens⁻ Cubes.

R = Cross-bending fiber stress in pounds per square inch.

5 = Ultimate shearing strength in pounds per square inch. e = Expansion per degree F. Determined Wet.

E = Young’s modulus in pounds per square inch. Compression. r = Limits between which E was determined.

W *= Weight in pounds per cubic foot.

Compiled from Tests of Metals and Other Materials, etc. made at Watertown Arsenal, Mass.

GRANITE.

Color, Name, Source, etc.

C R S .    _    .

io² . io². io². 10    ⁸.    10s. to².

Dark.

Light.

Red.

Black.

Light.

Braddock, near Little Rock, Ark. ,

« «    « <    I <    < t    < «

Exeter, Tulare Co., Cal..........

Platte Canon, Colorado..........

Branford, Conn................

Pine Mt., Quarry, Lithonia, Ga. . . Stone“ “    “    “..

Troy, N. H....................

Maine.........................

Mt. Waldo Quarry, Frankfort, Me White Rock Mt., Millbridge, Me. . Cape Ann, Rockport, Mass.......

242

196

226

146

157

277

209

NO O ; <N

tO tO 10

4* « 4*

324 7i

341 7i i-S° is6

461

164

3>

398 57 1-50 161 398 83 10-30 162

Pink.

Pigeon Hill, Rockport. Mass.

Quincy, Mass.............

Milford, Mass.............

262 210 322 199 242 203 173 197 130 238 190

to to to 0\

to

to to

Cn CO

375

337    48    1-50

337    45 10-30 165

400    98    1-50

....    65    10-30

90

to M W . to to

Hbiv) 0 4*.

Pink.

Mottled.

Red.

Light.

Orton ville, Minn........................

Rockville, Stearns Co., Minn.............

Sioux Falls, Minn.......................

Korah Station, Va......................

Broad Rock Quarry, Chesterfield Co., Va. . Snoqualmie District, Wash..............

252

204

181

182

234

154

198

to 10 10 M M M O

397    94

389    60

431

402

i~5°

1-50

TABLE I.—PHYSICAL PROPERTIES OF STONE AND CON CRETE—{Continued).

SANDSTONE.

Color, Name, Source, etc.

C *    $    -    -    ■ w

IO³. IO². IO². IO ⁸.    10s.    I0².

Near Ft. Smith, Sebastian Co., Ark Cabin Creek, Johnson Co., Ark. . . .

Jasper, Ala.......................

Piedmont Quarry, near Oakland, Cal Light Red. St. Vrain Cafion, Col., laminated.    . . .

Red.    Manitou, Col....................

Gray.    Ft. Collins, Col., laminated.......

“    Trinidad, Col...................

Brown.    Portland (Middlesex Quarry), Conn.

Blue.

i 28 185 159 110 US no 117

00

620 613

436

\Ti O'

1-40

1-50

iso

140

140

145

»3

43

lzzzz 1:    Tj:

m 04    o'

Portland, Conn., ist quality

‘ ‘ èridge stone

Cromwell, Conn.......................

Brainard Quarry Co., Conn.............

Near Redfield, Bourbon Co., Kan........

Carreyville, Ky........................

Potomac Red Sandstone Co., Maryland.. .

lé    II    Ił    II    l<

Kibbe, East Longmeadow, Mass.

Soft Saulsbury ' ‘ Hard “    ‘ I

Red Brown, ‘

Maynard, East Longmeadow, Mass.

Worcester '‘    ‘‘    ‘‘ .

Light Drab. Frontinac, Minn..................

“    “ Mautorville, Minn................

“ Yellow Mankato, Minn...................

Kettle River, Pine Co., Minn.......

Ohio............................

Chitwood, Oregon................

Cooper Quarry, Douglas Co., Oregon Coquille River, Coos Co., Oregon. . . Cooper Quarry, Douglas Co., Oregon

Brown.

Salmon.

Blue.

Blue.

Buff.

Buff.

Drab.

Blue.

Mottled.

Cream.

Beaver, Carrol Co., Ark

Batesville, Ark........

Rockwood, Ala........

Bedford, Ind..........

Indiana...............

Oolitic, Bedford, Ind. ..

Johnson, Co. Iowa. Iowa State Quarry

Hutchinson,

Crowley’s, Io Cedar Valley

Iowa

wa

136

139

ISO

96

169

62

107

80

106

• 130

.    190

127

104

8S

140

114

94

98

101

88

96

125

58

87

63

7S

152

M ro O W    «    M

tO MM Os» O A

M 00

to

O'

pooo

S49

S26

Si6

Soi

S 77

S77

S 67

S6i

517

517

686

622

177

177

tO    Ni

00    (0

i~S°

10-20

1-20    134

1-25

10-20    133

1-30

10-20    160

LIMESTONE.

206

to

(1843).

, Iowa

95

(>o

108

MM    tO

to O O    O

471

C\

i-S°

58

389

60

41

71

47

72

36

241

78

So

407

375

Ol (A vj

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1-30

1- 30

2- 30 2-10

TABLE I.—PHYSICAL PROPERTIES OF STONE AND CONCRETE—{Continued).

LIMESTONE.

Color, Name, Source, etc.

io².

io².

io².

io~⁸.

10«.

io².

Light Buff. Ft. Riley, Kansas-Junction City, I"

Mt. Vernon, Ky. . .

Bowling Green, Ky

Hopkinsville, Ky.,

Bowling Green, K'

Spring Ledge, Mt.

Kelly Island, Lake Kasota, Minn. .. .

Wasioja, Dodge Co., Minn......

Carthage, Mo., Quality No. i. . ..

Yammerthal Flint, Buffalo, Ń. Y Isle La Motte, Vt..............

Pink.

is-

Oolitic. . .. Oolitic. . . . y., Oolitic. . Vernon, Ky Erie......

io

33

60

62

78

150

76

122

108

So

40

137

162

237

146

300

13

12

17

12

ii

464

89

40

93

1-20

1-20

464

217

33

10-20

139

17

21

22

219

147

1-40

MARBLE.

Fossiliferous.

Chocolate.

St. Joe, Searcy Co, Ark.....

4 4    4 4    4 4    4 4    4 4

Marble Hill, Ga...........

Tate, Ga., Creole Quarry. . . “    “ Cherokee Quarry.

“    “ Etowah Quarry. .

103

123

US

135

126

141

ii

16

10

13

14

219

202

193

441

82

107

91

55

41

69

18

441

40

91

1-40

1-30

10-30

1-40

1-40

10-30

1—40

10-30

169

168

169

170

168

168

Lee Co., Mass

‘ Kennesaw Quarry

96

13

14

78

Drab.

White.

White.

Blhe.

Faribault, Minn.......................

Tuckahoe, N. Y.......................

Richville, Dekalb, St. L. Co., N. Y.......

Vermont.............................

Rutland, Vt..........................

160

181

178

162

125

90

120 119

139

16

21

iS

562

454

441

634

76

67

136 120

10

19

10

361

66

21

12

312

45

6s

78

10-30

10-30

10-20

10-30

1—40

10-30

1-40

I 70 168

17S

168

Mt. Dark.

Sutherland Falls, Vt.

Roche Harbor, San Juan Co., Wash, Snoqualmie District Wash........

128

162

113

102

90

47

18

23

iS

433

550

75

93

126

1-40

1-40

1-40

CONCRETE.

The physical properties of concrete depend upon so many variable factors that it is useless to attempt to give more than approximate values.

Mr. Edwin Thacher, C.E., has deduced formulas, based upon a large number of experiments made at the Watertown Arsenal, showing the effect of age and composition upon the ultimate strength. Values according to these formulas are given below. For tests of concrete of all kinds reference is made to Tests of Metals and other Materials, etc., made at the Watertown Arsenal, Mass.

Ultimate Strength in Pounds per Square Inch.

Age.

Mixture.

Remarks.

7 Days.

i Month.

3 Months.

6 Months.

I-I-3

I-2-4

i-2*-5

1-3-6

i-3ł~7

1—4—8

1-5—10

I-6-12

1600

1400

13⁰⁰

1200 1100 1000 800 600

2750

2400

2225

2050

1875

I7OO

ISS©

IOOO

3360 2900 2670 2440 2210 1980 iS 20 1060

43OO

37OO

34OO

3IOO

28OO

25OO

I9OO

I3OO

5= 1800 —200*; 7 days 5 = 3100 —350*; i month 5 = 3820 — 460*; 3 months 5 = 4900 — 600*; 6 months

*=parts of sand to one part cement.

5 = ultimate strength for 12-inch cubes.

E= Modulus of Elasticity in Thousands of Pounds per Square Inch.

(Compiled by E. Thacher.)

Age 7 Days.

Age i Month.

Age 3 Months.

Age 6 Months.

Mixture.

IOO

IOO

to

600.

to

IOOO

to

IOO

to

IOO

IOOO.

2000.

600.

to

IOOO

to

IOO

to

IOO

IOOO.

2000.

600.

to

IOOO

to

IOO

to

IOO

1000.

2000.

600.

to

IOOO

to

IOOO.

2000.

1-1-3

1-2-4 I—2ł—5 1-3-6 1—4—8

1-5-1°

1-6-12

2450

2580

2220

i860

2050

2050

1800

1540

1380

1350

2830

2660

2550

2440

2100

I74O

1380

2580

2450

2300

2130

1800

1460

1140

1910

1460

1350

1220

3500

3670

3320

2970

2530

2100 1640

3140

3160

2900

2650

2220

1780

1360

2120

2160

1980

1800

3850

3670

3630

3600

3020

2420

1820

3580

357°

3540

3500

2840

2200

1520

2700

2580

2220

i860

These values are means of tests made upon 12-inch cubes made with four brands of cement respectively. A statement of the data upon which the above tables are based is given in an article by Mr. E. Thacher, entitled Effect of Age and Composition on the Strength and Modulus of Elasticity of Concrete, Cement, May 1902.

EXPANSION OF CONCRETE.

Prof. Pence gives 0.0000055 as the coefficient of expansion for one degree F. for 1-2-4 concrete composed of Lehigh Portland cement and limestone. With the limestone replaced with gravel the coefficient becomes 0.0000054. This makes the coefficient of expansion of concrete and steel essentially the same.

WEIGHT OF CONCRETE.

The weight of concrete will vary somewhat according to the materials used and the methods of mixing. In “ Tests of Metals, etc.,” for 1898, the weights of a large number of 12-inch cubes are given, the proportions varying from 1-1 to 1-4, with 33 and 40 per cent of the stone used as mortar. The mortar was made "‘dry,” “plastic,” and “wet.” The weights per cubic foot ranged from 138.9 to 143.7 pounds. For all ordinary purposes 140 pounds per cubic foot may be used. Some specifications state that concrete shall be taken at 150 pounds per cubic foot.

WEIGHT OF FILLING MATERIAL.

This will vary according to the kind of material and the method of depositing it. For average conditions, when the spandrels are filled with sand or gravel, 100 pounds per cubic foot may be assumed. For gravel deposited in thin layers and rolled, some specifications state that the fill shall be taken as weighing 120 pounds per cubic foot.

Table

II.

MASONRY

ARCHES.

onot 4»- oj to h I Number.

Name.

Place.

Over.

Date.

Engineer.

*0

Span.

Rise.

Thickness of

Arch Ring

at Crown to.

At Springing, is.

Taff Vale Viad.

Queretaro Malaunay Magnolia St.

L. Juniata No. 8

Mass. Ave.

Near

Washington, D. C., U. S. A. Rock Crk.

Créé R.

Taff R.

Brunei

172 6-3 5 Antonio Avana 1840—44 Locke 1894 Brown Brown 1831 Telford 1900-1 Douglas Rennie

P'

50.0    25.0

Cj Curve.

Radius at Crown.

25.0

Width, Face

Crown.

Springing.

Material.

Class of Bridge.

Reference.

14.0

Blue Grit

Ry.

^OO QO-^I OCrv 09 KJ h O O

Chateau Thierry Charles

Starrucca Viad Pont Neuf

Enz

Pont au Change

Court St.

France

Nuremburg, Bavaria Nr. Lanesboro’, Pa.,U.S.A. Paris, France

Wildbad, Germany Paris, France

Rochester, N. Y., U.S.A. Rochester, N. Y.,U. S.A. Bet. Norwood-Bromley, England Dole, France England Gien, France

Marne.

Starrucca Crk. Seine R.

Enz R. Seine R.

Genesee R.

Lon. Croydon Ry.

Doubs R.

Mouse R.

Loire, R. Val.

1786 Perronet 1728

1847 Adams 1578-1604 Cerceau and Marchand 1886 Leibbrand 1639-47

1893

McClintock

Gibbs

1760-64? Gueret 1822 1888-9

Telford

Lethier

HVJ pO W M

50.0 50.0 50.0 50.0 *50. o 50.0 50.0 46. o 39-0

51.0 51.i 51.0 *51 • i

25.0

7-4

12-5

15-0 25.0 6.6 5-9 4.9 17.0 16.0 120. o 21 .9

Si ■ 2 51.2 to 35-2 *52.0 *52.0 52.0

3-1    3.1

2.8    2.8

3.0

7.0

DO

25.0 .124    27.7

38.S -073    5° •⁰

48 . o 25.0 .150 200.

8.8

Aijued’t

y.

A.    March, 1850

B.    June 2,1888

C.    1851-2

H Number,

3¹ • S 8.0

Brick

to

4.0

4-3

2.5

2.3

2-5

3.6

10.7 1.6    3.9

25-6    5-3

17,6|

20

•?!

TO.O 2-5

t¹2.O 2.3

2-3

Ry.

Ry.

H.W.

D.

E.

F. 1852, p. 290 B. Dec. 25, 1902

8.0 &

7.6

37.2.115    19.2 13.8

26.3 -095    24.8 7.0

25.8 .089    72.5

35■9 -°45 25.6. 207

26.7

38.8    .064 34.2 .067 t3°-o

16.0

?Ó4.0 6.0

H.W.

Sandstone

Limestone

Brick

Ry.

H.W.

H.W.

H.W.

H.W.

Aqued’t H.

Woodbury, 1858

F.    1852, p. 278 B. Sept, i, 1888

G.    1891, p. 887

G. 1891, p. 918 F. 1852, p. 276

B. Feb. 2, 1893

C. 1855-6

to    to to to to to

v4    Q\Ui    G> 10

Dinan Viad. Guétin

Stura

Digoin

Roquefavour

Dinan, France Bet. Digoin and bray, France N. of Turin, Italy Montalierie, Italy Digoin, France

Vic. of Marseilles, France

Mains-

Rance R.

Valley

Stura R.

Po R.

Valley

Arc R. and Val.

1845-50

1890-98

Fessard

<*-• (ĄiA Ô <3 oö

52.0

52.0

52.5

52.5 to

42.6

*52.5

52.5

17-5 26 . O

26.2 17-5

14.2

26.2 23-0

3-8

3-6

8.5

H.W.

3-3

3-6

1.

1849

Changed

1890-98

1841-47

Bella

Barbavara

de Montricher

52.5

52-5

52.5

5- i

13-6

3-0

23.0

3-0

3-9

3-9

26.2.114    17-7    3-3

26.2 3°-7

.117

22.1

31-9

tT3 •1 10.0

Ry.

H.W.

Canal

F.    1852, p. 197

G.    VI, 1893

102.7 32.1 3°-7

.042

.093 . 127

Go to G> m O Cn

OCCn M

9.8 Brick(?) Brick

9.8

H.W.

Ry.

Canal

00 Ov O M N CO Tt 10VD    r>.CO

Strasbourgh Sta. Croydon Abattoir St.

Park St. Pathhead

Mill Creek L. Juniata No. 13

Paris, France Near Croydon, England Paris, France Nemours, France Stirling, Scotland Moret, France Moulins, France Hartford, Conn., U.S.A. Pathhead, Scotland,U.S.A.

W. of Bird-in-Hand, Pa., i mi. W. of Tyrone, Pa., U.S.A.

Station Platform Lon. & C. Ry.

Loing R.

Forth R.

Loing R.

Allier R.

Tyne R

Mill Creek L. Juniata R.

1805

1400t

1771

1758-60

1898

1830

1889-90

1892-93

Gibbs

Perronet

Perronet

Regemortes

Graves

Telford

Brown

Brown

S2.5

49.2

16.4

*52.7

53-0

53.0

53-0

53-0

53-3

53- 9 54.0

54-0 & 50.0

54- 0 54.0

26.3

26.3

24.6    .134 8.2

71.6    .041

Canal

G. 1888, p. 363 G. 1899

F. 1852, p. 296 F. 1852, p. 296 F. 1852, p. 286

A. 1855, P- 65

.6

3-0

71 -3

99.1

39-2

61.2

.042

.032

.071

.072

42

?20 . O

8.2

6.4

7-3

3-3

4-3

34-0 70. o 24. o

12.8

25.0

13-5

13.5

2.7

3-2

2.7

3-2

DO

33.7 .081 33-7 -095

32.0

49.4

4.0

Mill’e Grit Brick

Freestone

Freestone

Brick

H.W. H.W. Ry. H.W. ?

H.W. ? H.W. H.W. H.W.

Ry.

Ry.

F. 1852, p. 296 C. 1855-6 I.

F. 1852, p. 286

I.

J.

F. 1852, p. 276 B. Jan. 12, 1899 F. 1852, p. 195

E.

E.

* Maximum.

t About.

Remarks.—1. Piers 14' octagon. On curve. Skew 40⁰.    1320' Radius. 2. Max. H. =95'

Av. = 75'-8o'.    3. Found, upon piles. 4. 4 tracks. Intrados to base of rail, 5'.6.    5. Middle

Div. Penn. Ry. 7. Skew 17⁰. Cost $132,000.    11. 2 tracks. Max. H. = no'. N. Y., L. E..

& W. 12. Repaired 1886.    13. Middle 3d joints at crown and springing filled with lead-

18. Ribbed skew. 20. H. = 134.5'. Piers hollow. 21. Approaches to metal spans. 22. Max.

23. Max. H. = 39'.4. Max. H. = 271'.o.

H. = 130'.5. Nat. Route No. 176.

Genoa. 27. 3 tiers of arches.    .

36. Max. H. = 75'.o. The 54spans are under roadway. 38. 4 tracks. Mid. Div. Penn. Ry. ribbed skew.

24. Route Turin-Milan. 29. Ribbed skew.

37. Three

25. Turin-35. Stone facing-tracks—1° curve"

TABLE II.—DATA FOR ABOUT 500 ARCH BRIDGES

MASONRY

ARRANGED ACCORDING TO SPAN—{Continued).

ARCHES.

•49HuinN|    5 in'?.    Jqvç «ç ^

Name.

Place.

Over.

Date.

Engineer.

Span.

Rise.

Thickness of

Arch Ring

at Crown 1₀.

Kennet

Monocacy Viad.

Nashawtuc

Ouctoine

Big Conestoga

Peas

Oder

Bachthal

Dauphin

Cannes

Spreethal No. 28

Washington Nôtre Dame

Gravant

Zempoala Aq. Monford

Johnstown

Carrolton Viad. Jamaica St.

Near Caversham, England Kennet R. Monocacy, Penn., U.S.A.

Concord, Mass., U.S.A.

E. of Lancaster, Pa.,U.S.A.

Bet. Berwick and Edinburgh, Scotland Kunnersdorf, Saxony Dermitz, Saxony

France

Löbau, Saxony Königstein, Saxony

Saxony

Neuneck, Germany 26.5 m^Pittsburg, Pa.,

New York, N. Y., U.S.A.

Paris, France

Chateau Thierry, France

Pontlieu, France France

7 m. south of Huauchin-ango, Mexico England

Johnstown, Penn., U.S.A.

Llanrwst, Wales

Nr. Baltimore, Md., U.S.A.

Glasgow, Scotland

Sudbury R. Rieneros R.

Deep Dingle

Oder R.

Bach R. Romanche R. Beauvoir Ravine Spree R. ?

Spree R. Glatt R. Raccoon Crk.

Harlem R.

Seine R.

Marne R.

Huisine R. ? Yonne R.

Valley

Severn R.

Conemaugh R.

Conway R. Patapsco R. Clyde R.

At Springing, ts.

1883

1770-90

1844- 5

1842-4

1843-47

1845- 46

1845-46

1886

1887-88

1886-89

1507

1765

1773

1760

1553-7°

1790-92

1888

1634-36

1833-35

1833-36

Fisk

Wheeler

Garipuy

Brown

Henderson

Potie

Cunib

32

Leibbrand

Hutton

Jaconde

Pérronet

Voglie

Adwine

Tembleque

Telford

Brown

Inigo Jones

Latrobe

Telford

15

68

*54 -o 54-0 54-0

54- 5 *54-5 *55-0

*55-8

55.8

55- 8 35-8 55-8 37-9

*55-8

*55-8

C-C55.8

36.0

36.0

56.7

t03i. 2 57-5 51.1 57-5 57-5 to 53-9 *58.0

9.0

6.0

17 . i 27-3

2.6

2.5

I . 2 2.1

DtaOta Curve.

Radius at Crown.

to

to Face at Crown. Thickness of Piers at Springing.

Material.

Class of Bridge.

Reference.

Nupber,

27.9

27.9    3.0

27-9    3-1

27.9

18.9

27.9

C-C9.8 Ji .0

28.0    3.0

14.0    2.0

28.4 3.3 15-6

21 -3 29. o

4-3

50.0

Ó3-75

27-3

.050

.019

?24.0

Brick

Ry.

25-0

12.8

Granite

Canal I. H.W.

K. Dec. 20, 1895

vVm. Wheeler F. 1852, p. 280

Ry.

E.

H.W. L.

*>5

■67

■’68

69

70

71

■72

Brives

Tournelle

Marie

Aelius

Sèvres

Rahway Ave.

Washington

Ingersheim

-73

-.74

Trenton L. S. & M.

S. Ry.

France Paris, France

Paris, France

Rome, Italy Near Paris, France

Elizabeth, N. J., U.S.A. New York, N. Y., U.S.A. France

Near Trenton, N.J., U.S.A. Elyria, Ohio, U.S.A.

Loire R. Seine R.

Seine R.

Tiber R. Seine R.

Rahway Ave. Harlem R. Tech R.

Delaware R.

W. Br. Black R.

1772

1630-36

1635-58

Grangent

Marie

Marie

136

1820

i886—89

1773

1902

Hadrian

Beaupre

Brown

Hutton

Clinchamp

Brown

18

58.0

30.0

38.0

40.0

*58.0

58.3 58.S 57-8 55-5 52.0

*58.6 38.7 to 44.8 38.7 to 44.8 59-0

59.0

16.4 59.2 597

59-7 to 50.1 60.0 60. o

22.3 20. o

14.5

14.5

17.0

29.2 10.8

10.3 9-7

3°

2.7

2.7

2.7

i -5

2.7

2.5

2.3

2.5

tu $ ta ta tata £ ta £ £

27.88 27.88

27.9

27.9

27.9

28. o

55-8

28.4

41-5

42.6 45-8

29.O

36.2 21 .0

29.2 43-7

. 108 .III

. 108

.036

. 187

.096

19.7

19.7

13- i 107.6

80.8

.089

.093

.075

.135

.085

.037

7-4

9.2

4-9

9.5

iS-8

+ 13.2

77-4

35-2 35-2

4-7

Brick

Stone

12.8

14.4

12.8

12.8

24.0    11.0

48.0 48. o 14.0

40. o

6.0

Sandstone

10.0 10.0

Granite

26

29.8 22.4 29.8

22.4

29-5

29-5

8.2

9-5

29.8

ii.7

12.0

30.0

3-2

5-4

45-

29.

.069 . 281

29. i 53-3

ii .7

12.8

4-3

3-3

3-2

2. o

3-2

3-2

4-5

3-3 2 . o

3-3

29.

29.

29.

5°-

29.

43-

30 •

77-7

t33-5 42.6

ii -7

+ 23.0 Travert’e

12. i

62.2 80.8

52.0

22.2

8.3

8.0

Berea

sandstone

Ry.

Ry.

Ry.

Ry.

Ry.

H.W.

Ry.

H.W.

H.W.

H.W.

Aqued't

Ry.

H.W.

Ry.

H.W.

H.W.

H.W.

H.W.

H.W.

Ry.

H.W.

Ry.

Ry.

F. 1852, p. 229 F. 1852, p. 226 F. 1852, p. 292 F. 1852, p. 292 F. 1852, p. 222 F. 1852, p. 22s

F.    1852, p. 228

G.    1891-1, p. 920

D.

Washington Bridge by Hutton F. 1852, p. 274

F. 1852, p. 280

F. 1852, p. 282 F. 1852, p. 280

B. July 7,1888, p. 2

F. 1852, p. 284

B. July 20, 1889

E.

L.

I. , F. 1852, p. 237 F. 1852, p. 290

F. 1832, p. 280 F. 1852, p. 276

F. 1832, p. 276

M.    Feb. 18, 1899

A.    April, 1847 F. 1832, p. 288 D.

See No. 54 F. 1852, p. 282

B.    Jan. 30, T9C2

N.    June 8, 1899

O' O O O' o

v* N H 9    '999 h 9^

Remarks.—39- Skew. S. E. Ry. Co.

Fitchburg, Mass. 43. Penn. Ry.

Pile found. H. = s₉, L. =723.    49.

'51. Saxony-Silesia, H. =66.9.

-54. Apnroach to metal spans.

Yangents. Max. span is highest.

* Maximum.

40. Chesapeake & Ohio Canal. 41. Granite from 44. Max. H. = 124 -0.    45- Löbau-Zitlau, H.=Ó2.3.

Saxony-Silesia, H.=9S'.o.    50. Prague-Dresden, H. = ₃₃.6.

I²,' Sheet-lead Hinges,_ 3.    S3- 2 tracks. Rail 2₇'.s above key.

36. See No. 9._ qS9. Waterway 8i"X 12". H. = i2₄'.o. On two 61. Skew 55 . Ribbed. Stood through Johnstown Flood.

46.

t About.

62 On Ai° curve. Arches Cyl. H.=66'.o.    64. ist Bridge by Mylne. 1768-72 called New Jamaica

St' Bridge. Old Bromielaw Bridge. 68. Originally 8 arches 5 now buned. To give access to St Ange Castle. 69. Paris-Versailles. 7°- five tracks. Ribbed arch Skew 45 44- Pa. Ry-N. Y. Div. 73. Two abut, piers, 22'.o. Skew 71° 30 .    74. Twirl arches 4.2 apart. 1 o

tracks. L. S. & M. S. Rv.

75

76

77

78

79

:8o

181

, :82 -83

.’84

• 85

.86

•87

.'88

■89

<>1

■92

•93

•94

95

96

97

Name.

Dee

S. approach Voyne Viad.

Muddy Crk.

W. Jersey St. Kennet

Staines

Ballater

Stirling

Richmond

Alness

Anker

Dutton Viad. Holy Cross (old) Kingston

Conemaugh

Ben’s Creek L. Juniata No. 7

Big Chiques Big Viaduct L. Conemaugh No. 6

98 Chestnut St.

99

100

301

302 ÏO3

Congleton Viad. Tweed

*04

105 Görlitz

306

307

W. Grand St.

TABLE II.—DATA FOR ABOUT 500 ARCH BRIDGES

MASONRY

ARRANGED ACCORDING TO SPAN—{Continued). ARCHES.

Place.

Over.

Date.

Engineer.

No. of Spans.

<.

Span. Rise.

Arch Ring at Crown tn.

At Springing, ta.

Minneapolis. Minn. U.S.A. Mississippi R.

Val. Llangallen, Wales Drogheda Ireland

Addystone O. U.S.A.

Elizabeth N. J. U.S.A. Near Caversham. England

Staines, England

Ballater, Scotland Stirling, Scotland

Richmond, England Alness, Scotland Warfield, England England

England

Feldkirch, Austria Kingston England

Saumur, France W. ot Ben's Crk, Penn., U.S.A.

Lilly-Portage, Pa., U.S.A. i m. E. Schoen berger' s Penn., U.S.A.

Penn., U.S.A.

Viad. Sta., Penn.. U.S.A. E. of L. Conemaugh, Pa., U.S.A.

Philadelphia, Pa., U.S.A. Bewdley England

Congleton England Ratisbon, Germany Berwick England Charmes, France

Kew England

Near Görlitz, Silesia

Dole France

Dee R.

Boyne R.

Muddy Crk.

W. Jersey St. Kennet R.

Thames R.

Dee R. Forth R.

1851-67

+ 1895

189 -+ 1840

1796

1809

1829-32

Macneill

Kittredge

Brown

Brunei

Sanby

Telford

Stevenson

to

Thames R.

Weaver Val. Ill. R. Thames R.

Arm of Loire R. Conemaugh R.

Conemaugh R.

L. Juniata R.

Big Chiques Crk. L. Conemaugh R. L. Conemaugh R.

Schuylkill R. Severn R.

Dane R. & H.W. Danube R. Tweed R.

Moselle R.

Thames R.

Neisse R. & Val.

Doubs R.

Elizabeth, N. J U.S.A. W. Grand St.

1774-77

1816

1846

Payne & Couse

Telford

Grainger

Stephenson

13th cen.?

1825-28 Lapidge

1756-64

1896

1896

1889

1884

1889

1889-90

1861-66

1797-9

1839-

1135

1847-50

1740

1844-47

C/t

Voglio& Cessart Brown

Brown

Brown

Brown

Brown

Brown

Kneass

Telford

Stevenson

Stevenson

1760-64 Gueret 189- Brown

* Maximum.

c*>

M tO H 4^.

O OOCa to

00 Q\ Q\ Co

Remarks.— 76. H. —147'.6. Intrados to rail = 6'.i. Shrewsbury-Chester, stone facing. 78. Approach to metal spans Av. H.=9°'.    78. Big 4 Ry. 79. Pa. Ry., N. Y. Div. Skew 6o°.

Ribbed    80 Great Western Ry. Co. 81. Closed 1797 on account of poor foundation.

84. Cieai headway above L.W. = 25'.o.    86. Leeds-Thirsk. H.=9o'.o.    87. 816' long.

Grand June. Ry. H —    89. Replaced. 90. Found, upon blue clay. Kingston-Ham

•wick. 91. A. 1856, p. 376. Found, upon piles. 92. Pa. Ry. Pitts. Div. Four tracks.

88.

pton. 03. Pa-

60.0

57-0

54°

60.0

60.0

15-0

14-3

13-5

30.0

30.0

D Curve.

Radius at Crown.

M|°

Width, Face to Face at Crown. Thickness of Piers at Springing.

40.0    8.0

Material. Class of Bridge.

Reference.

Number.

H.W. N. Nov. 23,1895    75

60.0

55.0

do .0 60.0 18.0 do. o 52.0 do .0 60.0 58.0 53-5 *60.0 60.0 60. o 60.0 30.0 do. o do.o do .0 56.0 52.0 *60. o 60.0

60.0

do.o

do.o do. o do.o

do.o do.o 52.0 do. 8 do. 8

61.5

di.8 to 34 -1 di.8 to

38.4

61.8 41.2 30-9

24.8 di.8 to

51-2 62.0

1.5

1.5

13-5! 2.8 3-5 12.5 2.8 3-5 10.3 2.8 3-5

20.0

3°.o

30.0

19.0

18.3.

16.5

21 .O 20.0

20.0 15-0

15.0

30.0 20.0

18.0 18.0 16.0 20.0 3°-4

30.8 3°-9

17.1

30.9

19.2 30-9 20.6 15-5 12.4 19.2

DDD

30.0

30.0

27.8

131

Brick

Ry.

Ry.

76

4.8

3-0    3 -°

3-0    3-0

3-0    3-°

2.8

2.8

2.7

3-0

2.5

2.0

3-2

4-3

2.7

4-3

9.0¹    3.6

2.7

3.0

3-6

S    dd d£d do dm

DM

43-5 •    3°.°

37.5 .080 52.1 .067

40.0 .070 38.6 .073 39-8 .073

30-o

55-4

32.5

32.5

37.5

37-5

30.0

32.5

51.0

?3Ó.o

?3² • 8

25.0

31 -o

30.0 ?i i .0 ?25.0

10.0

Berea

sandstone

Ry.

F. 1852, p. 156 A. July 1851, p.384 77

Blue

78

.087

.092

.092 . 080

.075

.090 . 092

44-7

83-5

8.0

9.0

Brick

Brick

Ry.

D.

79

80

H^V.

Granite Greenst’e from near Stirling

Brick

BrickRing

fio .0 9-3 8.7 12.8

36.6

11.0

36.5

f4i .0

Brick faced with stone

7.0

112. o 8.0 7.0

BrickRing

H.W.

H.W.

H.W.

Ry.

Ry.

H.W.

H.W.

H.W.

Ry.

Ry.

Ry.

Ry.

Ry.

Ry.

K. Dec. 20, 1895 K. Sept. 13,1895    81

C. 1855-56 C. 1855-56

K. July 12, 1895 F. 1852, p. 288 F. 1852, p. 169 F. 1852, p. 188

A. 1837, 8, p. 125 O. June 1898 A. Dec. 1842

K.

L.

D.

E.

E.

E.

E.

E.

82

83

84

85

87

88

89

90

July 26, 1895 F, 1852, p. 276

91

92

93

94

95

96

97

34-0

34-0

30.4

30.8

3°-9

3°-9

.074

.059

• 105

.139

.087

28. o

31.0 25 • 6

8.0

20.3 5-2

17 . i

8.5

Brick

Brick

Brick

H.W.

HTV.

ilw:

I.

98

3°-9 20.6 15-5 12.4

44-7

57-9

.096

.062    51.0

Red Gran.

it.5 to 10.7

Ry.

Ry.

I. L.'F. 1852 p.284 99

A. 1839, P- 444 F. 1852, p. 274 F. 1852, p. 155 F. 1852, p. 276

F. 1852, p. 282

F. 1852, p. 215

F. 1852, p. 280

D.

100

101

102

103

104

105

106

107

Ry. Pitts. Div. 94- Ribbed. Skev'45^    pfeS Ä 5° 3₃'

°n 2° S-57^^; S

tł\⁹⁶on curve⁶' bX-SL- 107. Pa. Ry.. N. Y. Div. Skew 57° 4*'• Ribbed. Four tracks

curve.

IO

MASONRY

ARCHES.

Number.

Name.

Place.

Over.

Date.

Engineer.

G

P..

CO

Span.

Rise.

2;

Thickness of

Arch Ring

at Crown /o.

At Springing, is.

Curve.

Radius at Crown.

to.

\ V HUH, 1 [VI ,

to Face at Crown. Thickness of Piers at Springing.

Material.

Class of Bridge.

Reference.

Number.

Holy Cross (new) St. Angelo Barton Aq.

Actius

Neuf

121

123

125

127

Rock River Coldstream L. Conemaugh No. 3 L. Conemaugh No. 2 Stockport Viad.

Ri vanna Aq. Teviot-Tweed Houghton River Conon

Feldkirch, Austria Rome, Italy Worsley, England Athlone, Ireland Mirepoix, France Frouard, France Ferté, France Montélimar, France Rome, Italy

Paris, France

Watertown, Wis., U.S.A. Coldstream, Scotland Summerhill, Penn., U.S.A. Penn., U.S.A.

Stockport, England Carlisle, Scotland U.S.A.

Near Kelso, Scotland England

Ill R.

Tiber R. Irwell R. Shannon R. Lers R. Moselle R. Marne R. Roubion R.

Seine R.

Rock R.

Tweed R.

L. Con. R. H.W. L. Conemaugh R.

Eden R.

Teviot R.

Conon R.

1898

135

1760-

1844

177 6-90 1788

1806

138

1578-1604

1902-03

1771-

1887

Hadrian Brinkley Rhodes Garipuy Le Creuix Pitrou Voglie M. Rusticus

Cerceau & Marchand Loweth Smeaton Brown Brown

1794-95

1809

Smirke

Eilet

Elliot

Haskoll

Telford

^ i/y⁴-* n m

128

Boberthal

129

130

131

132

133

134

Cher

Scrivia

Cinq-Mars

135

136

137

Chante-Perdrix Landwasser Raritan River

V.

Near Bunzlau, Silesia

France

Italy

France

Val-Benoist, Belgium Furand

Auzon, France France

New JIrunswick, N. J.,

Bober R. & Val.

Cher R.

Scrivia R.

Loire R.

Furand, R. Vienne R.

Landwasser R. Raritan R.

1850

1845- 46 1832 1834

1846- 47

1901

1902-3

Beaudemoulin

Ferraris

Bailloud

Montluisant

Beaudemoulin

Lamothe

Brown

62.3 *62.3 *63.0 63.0 639 63.9 639

64. o 64.0 to 25-4 64.0 to 45-3 64. o 64.0 64.0 64. o 65.0 65.0

65.0

*65.0

65.0

65.0

55-0

45-0

65 • 6

less 65.6 65.6 65 • 6 65 ■ 6 65 • 6 65 • 6

65.6

65.6

66.0

15.6

31 • 5

31 -5

16.0 16.0 32.5 21.0 iS •0 17.0 32.5 21.8

21.9 13- i 21.6 8.8

32.8

21.9

32- 8 32.8

33- 0

?5°.8

2.7

2.8

3-8

2.8

2.8

3-0

3-3

3-9

3-9

3-3

3-3

3-3

3.0

3.0

2.7

7-3

2.8

2.8

4.4

SS • 4 SS • 7 59-7

50.6 32.0

12.7 32.0

22.7 39-3

40.0 40. o

32.S

42.7

32. S

36.4

47.6 47-5 47-7 65 • S 32.8 47-5

32.8

33-0

.096 . 092 . 062 . 084

.150

• 141

?42.0

+25.6 ii.7 12.8

Limestone

H.W.

H.W.

H.W.

.076

.067

.086

. 066

.086

.082

22.4

19.4 50.8 24.s

72. S 13-8 28.3    8.0

23.0

Ring sand [stone

32.0 t36.o

+23.0

20.0 8.0 to

6.5

7.0

.069 .082 . 082

.054

. IOI

.069

.091

29 • S 28.9

26.2

26.2

8.5

55.°

8.5

13-1 ii -5

8-5

il -5 8.0

Brick

Freestone

Brick ring

Freestone

Limestone

H.W.

H.W.

Ry.

Ry.

Ry.

ILW.

Canal

H.W.

H.W.

Ry.

Ry.

Ry.

Ry.

Ry.

H.W.

Ry.

Ry.

Ry.

Ry.

O. June 1898 L. Q.

R. Dec. 1888

A.    1844, p. 444 F. 1852, p. 282 F. 1852, p. 284 F. 1852, p. 276 F. 1852, p. 286 F. 1852, p. 274

F. 1852, p. 276

B.    Mar. 26, 1903

108.

109.

110. hi 112

115

14

US 116

117

L.

E.

E.

F. 1852, p. 158. J

C. 1855-6

Pub. W’ks.U. S.,’41

L.

I.

L. J. F. i852,p. 286

11& II9-

120

121 122-12 S

124

125

126 ¹ 27

F. 1852, p. 212

F. 1852, p. 294 F. 1852, p. 296 F. 1852, p. 294

I2&

J.

J. F. 1852, p. 290 J. F. 1852, p. 294 G. ist Tri., 1901 Engineer, April, ’04 N. Oct. io, 1903

129'

130

131 13*

I3J

134

135

136-

!37

138

Kew

Kew, England

Thames R.

1789

Paine

139

Bow

140

141

Montignac

142

143

144

Brig o’Balgownie

145

146

Bellecour Viad. d'Arles

Stratford, England Near York, England France

Lancaster, England Old Aberdeen, Scotland Horbury, England

Lyon, France Near d’Arles, France

Lea R.

1835-39

Ouse R.

Vézère R.

1766-72

Walker & Burgess

J. & B. Greene

Tardif

56.0

51 o

12.0 66 o 55.0 45-8 66. o

N> K) M A bi CO

3-2

3-3

28. o 25-5 39-0

.114

. 084

24.0

Don R. Aire R.

1281

1775

Bishop Cheyne Clinchamp

13-8    2.s

4.0

81.0

.031

to

H.W.

K. June 14,1895

138

Valley

1789-1810

Ot h Ol 09    09

66. o

66.1 to 42.6 66.9 67. o

67.9

5 5-4 68.2

68.9

19-3    3-5

42.5

29.6

10.o

Granite

ring

Brick interior

H.W.

Ry.

A. Oct. 1837.p. 14; A. April 1839. S

139

21.3

17.i

S.

140

MM    -4    09

09 +*•    M    09

° 4*_O_<-n

33 -5

13 • i

Canal

H.W.

57-S

53-3

■ III

ii.7

r8.

Ry.

F. 1852, p. 280

F. 1852, p. 286

F. 1852, p. 282

F. 1852, p. 284 F. 1852, p. 129

L.

141

142

M3

144

145

146'

* Maximum

Remarks.—-108. Replaced No. 89. no. Removed for Manch. Ship. Canal. in. Gravel foundation. Coffer-dams used. H. = ₉8'.₅.    116. See Nos. 68 and 109.    117. Repaired 1840-

R^arqle* b^o    tbe circulaT See No. 12.    118. C. M. & St P. Ry. 120. Pitts. Div.

Pa. Ry. Skew 6° . Ribbed. On piles.    121. Pitts. Div. Pa. Rv. Stone parapet. 122 Manchester-Birmingham.    H. -105..9.    123. Intrados has five centres. 124. James River and

Kanawha Canal.    128. Berhn-Breslau. Intrados of each at same elevation; 75 ft. high

t About.

129. Tours-Bordeaux. Skew 34⁰ 30'.    130. Turin-Genoa. 131. Tours-Nantes. 134- Lours-

Bordeaux. 135- Cost 1,017.300 ?•    136. Thusis-Engadine. i37- Penn. Ry. 139- Replaced

old bridge of 1100-1118. Slight skew. Foundation on gravel. 140. Great North of England Ky. On piles.    142. Bottom of canal to intrados= 8'.5.    146. Avignon-Marseilles. H.= 27 .9. rile

foundation.

I42

APPENDIX.

APPENDIX.

143

TABLE II.—DATA FOR ABOUT 500 ARCH BRIDGES

ARRANGED ACCORDING TO SPAN—{Continued).

MASONRY

ARCHES.

jo

Name.

Place.

Over.

Date.

Engineer.

No. of Spans.

Span.

Rise.

Thickness of

Arch Ring

at Crown to.

At Spnng-

mg, ta.

Curve.

Radius at Crown.

ftp

Width, Face to Face at Crown. Thickness of Piers at Springing.

Material.

Class of Bridge.

Reference.

Number.

147    Central Ave.

148    Bord

149    Karlsbürcke

150    Potarch

Indianapolis, Ind., U.S.A. Fall Crk.

Oeil R. ( ?) Moldau Dee R.

451 Lockwood

Prague, Austria Wales( ?)

Nr. Huddersfield, England Sheffield R. Val.

1899

1764 Leclerc

1357-1507

1813 Telford

1846-49 Hawkshaw

16

152    Wellesley

153    Whamcliffe Viad.

154    Crum Elbow

Limerick, Ireland Brent-Knoll, Enlgand

Shannon R. Brent R.

155

Hyde-Pk -on-Hudson, N. Crum Elbow Crk.

Y., U .0.A.

156

157

158

159

160

Wissahickon Shock’s Mills Rockville

Mazares

161

162

163

164

165

Homps Spoleto Aq. Colorado St.

166

167

North

Pa., U.S.A.    Lea R-

Shock’s Mills, Pa., U.S.A. Susquehanna R. Rockville, Penn., U.S.A. Susquehanna R.

Fran ce ( ?)

Pavia, Italy Swatara, Penn., U.S.A.

Spoleto, Italy St. Paul, Minn., U.S.A.

Helmsdale, Scotland! ?)

Luxemburg, Germany

Edinburgh, Scotland

Lère R.

Ticino R.

Aude R.

Valley! ?)

Helmsdale R. ! ?) Petrusse R.

WaverleyRy. Sta.

1827 Nimmo 1836-37 Brunei

1898 Morris

Braithwaite 1881-2 Buchholz 1903- Brown 1901 Brown

¹7 8 7    Pertichamp

14th Cent. Under Visconti Osborn 1785 Ducros 741 Theodelapius 1889 Rundlett 1816 Telford 1899-1903 Sejoume

1763t

32

28

48

10

69.    o 69.0 69.2

69.5

70.    o 60. o 70. o 45-0 30-0 70. o 70. o

70. o

70. o 70.0 70.0 70.0

70.3

70.4 70. o

70.2

70.3

70.5

70.7

70.8

14.0

15-0

25 • 6

2 .O 2 .O

25.8 2.5

8.0    2.6

15-0 ti-5 9.0 J 2 . o 17-5    3-0

7.0 2.5

17-5    3-8

23.0 3-0 20.0 3.5 20.0 3.5

35-2

64.0 3.9 25.0 3-5 9-4 4-3 35-2

ii .0 3.5 25.0

35-4 2.9

ci hi

36.6

50.0 5° •⁰

6.0 Limestone H.W. and H. W. Klausman 6.0 RingOö- E. Ry.

[litic

147

.068    20.0    10.o

28.0    4.5 Sandstone

Ry.

tl -5 t3 • 6

2.5

3-5

3-5

t4 • 6

hi hi

15.0

40.6 40.6

35-2

85-3

37-0 68 2 35-2 62. o

. 100

43.0

35-0

10. o

Brick, stone fac’g

H.W.

Ry.

H.W.

F. 1852, p. 280 P. 1896, p. 126 F. 1852, p. 288

C. 1850

A. April, 1851, p. 21S

C. 1855-6. S, Q.

A.    1837-8, p. 126; F. 1852, p. 181

B.    Feb. 16, 1899

148

149

150

151

152

>53

154

30.0

28.0    9.5

.086 28.0 8.0 .086 50.0 8.0

.046

.093    22.5

.063

.056 f5o.o

Brick ring Talcose si.

16.0

18.7 8.5

ii . 7

11.7

Stone

ring

Brick

Brick

Freestone

Brick

Limestone

21.0    14.0

Ry.

Ry.

Ry.

Ry.

H.W.

Ry.

S.

B. May 2, 1902 T. March ii, 1904 B. March io, 1900 T. Oct. 25, 1901 F. 1852, p. 284 F. 1852, p. 276

155

156

157

158

I.

168

169

170

171

North Loch Schuylkill Black Rock Tunnel London !old)

172

Brunswick

Nr. Edinburgh, Scotland N. Loch Valley Philadelphia, Pa., U.S.A. Schuylkill R.

London, England

N. Brunswick, N. J.,U.S.A. Raritan R.

Thames R.

1836

II76-1209

1758

1902

Mylne

Robinson Peter of Cole-[church

19

71.0

Brown

473

174

175

176

Aulne

Boston Ave. Żeniec

Schmiedtobel

177    Po

178

179    Teviot-Tweed

180    Conon Viad.

181    Staines

182

France

Medford, Mass., U.S.A. Austria

Near Klösterle, Austria

Near Valenza, Italy Dresden, Saxony Kelso, Scotland Conon, Scotland

Staines, England

Fucecchio, Italy

Mystic R.

Schmiedtobel

Po R.

Elbe R. Tweed R. Conon R.

Thames R.

Amo R.

1900

1882?

1850

1179—1260

1799-1803

1832

1869

Arnoux

Bailey

Huss

Rovere

Fotius

Rennie

Mitchell

Rennie

21

18

*72.0 72.0 72.0 9 to 20 72.0 72.0 66. o 56.0 51.0 72.2 72.2 72.2 72.2 39-4 72.2 72.5 73-0

73-0

35-5

36.1 4-1 19.7

ii . 2    3.8

36.3 6.4 21.0

3.0

35-5

2.8

3-3

3-3

3-2

3-0

2.5

4-3

7-5

4.0

36.0

47-5

47-5

36.0 33-0 28. o 25-5 31 • i

49.8

36. I

19.7

63.9

36.3

.078 42.3 . 042

54.5 13.5 Craigleith stone

.059    18.3    8.0

40t 25' to

[34'

.092 55.0 .100    55.0

•114 55.0

.118 55.0 26.6 .050 56.0

.114

J.    F. 1852, p. 284

J- 0.

N. Nov. 23, 1889 F. 1852, p. 288 H.W. N. Oct. 12, 1901 N. Mar. 1, 1902

K.    Oct. 1899, pp.

423, 491

Aqued’t

H.W.

H.W.

H.W. I.

Ry. F. 1852, p. 292

159

160

161

162

163

164

165

166

167

9.0

8.0

[17.i

9.84&

.059 32.8 .180 37.3    29.9

12 6. o

168

169

170

171

Penn. Ry. B. Jan. 30, 1902,p.172 4 tracks

Ry.

I.

H.W. L.

86

Granite

Brick

Ry.

H.W.

Ry.

Ry.

G. ist Tri. 1901 Win. G. Taylor B. Dec. 7, 1893 G. 1888, XVI, p.

173

174

175

176

575

Ry. F. 1852, p. 296 H.W. F. 1852, p. 274 H.W. L.

Ry.

177

178

74.0 66. o 74.0

93    30

78. o

.038 34.0    8.3

Granite

H.W. C.

8.3

K. Sept. 20, 1867, pp. 206, 257 S. K. Sept. 1895. P- 336 Jour. F. Inst., Feb. 1870, p. 101

13,

179

180

181

182

* Maximum.

Remarks.—147. Cost about $39,000. Pile foundation 3'center to center iao Partiallv

ribbedyedHfl=id22'o⁹⁰- ₁,I25I’’BHndmerle^’& tSheffield Ry' 7o' and «'"arches otf’skew an£ Aó Coct    Tr„⁵ p Bell-mouthed type. 153- Great Western Ry. of England.

156. Cost «375,000.    157 Penn. Ry., two tracks. 158. Penn. Rv. four tracks 6° curve at

16? Lebanon Valle^Rv    i^Nnw’brid®e- p?OVeI?d’ Roofsupported by marble columns

• ^eDanon valley Ky,    163. Now m use. Piers of stone. H. = 29₂'.. at springing. 164

t About.

Skew 60°. Pile foundation. i66.‘ Four lateral arches in each spandrel. 167. Replaced bv steel 1898-9.    168. H. = 65'.o to top of parapet. 169. To Mt. Carbon, W. Va. 170. Phila.

& Reading Ry. 171. Replaced in 1824-31.    172. 72'span. Stepped Skew 63° 10'. Backing

of masonrv 24' to 28' above springing, then earth fill. 173- Cost 2^165,000 f.

$17 100 Skew 16°.    175. Austrian State Ry. 176. Austrian State Ry.

to Lake’Maggiore. 179. Below Elliot’s Bridge. 180. Highland Ry. one track.

174. Cost 177. Alessandria

APPENDIX.

HS

TABLE II.—DATA FOR ABOUT 530 ARCH BRIDGES

ARRANGED ACCORDING TO SPAN—(Continued).

MASONRY

ARCHES.

Number.

Name.

Place.

Over.

Date.

Engineer.

No. of Spans.

Span.

Rise.

Arch Ring at Crown t₀.

At Springing, is.

CO CO 00 Cn 2. W

i86

187

188

189

190

191

192

193

194

195

196

197

198

Scotland    Earn R.

N. Brunswick,N.J..U.S.A. Raritan R.

Allantown Englandf ?) Whitadder R.

Westminster (old) Westminster, England Thames R.

St. Maxence, France Navilly, France Roanne, France

Compiègne, France

Semur, France Paris, France

Rome, Italy Perth, Scotland Read ville, Mass., U.S.A.

Italy

Orleans, France Glasgow, Scotland

Albany St. Whitadder

Pont Royal Cestius

Hyde Park

Crown St.

or Hutchenson

Oise R. Doubs R. Loire R.

Oise R.

Armançon R. Seine R.

Tiber R.

Tay R.

Hyde Park Ave. Taro R.

Loire R.

Clyde R.

1781-1821 Rennie

1892 Dean and Westbrook

1842 Stevenson

1738-50 Labelye

1774-85 Perronet 1780 Gauthey 1789-1809 Vareigne and Vimar

1783 Lahite

1780 Dumorey 1685 Mansard

ist c. B.c. Under Cestius 17 60-71 Smeaton 1897-98 Curtis 1816-20 Cocconcelli

1829-33 Stevenson

9ł=-L W M W W V! W MW

ÖS M

75.0

*75.0

18.5

15.0

2.4    2.4

Ota Curve.

Radius at Crown.

54-4

.044

Width, Face to Face at Crown. Thickness of Piers at Springing.

40.9    18.6

?35 ■ o io.o

Material.

Class of Bridge.

Reference.

199

Molle

Tiber R.

200

Fabricius

201

202

203

204

205

Annan High Bridge

Conewago Schuylkill Falls

206

207

208

209

210

Conemaugh Posen Viad. Vittorio Painsville Viad.

211

Pont du Gard

Near Rome, Italy

Rome, Italy

Scotland

Near Johnstown,

New York, N. Y.,

W. of Conew’o, Pa., U.S.A. Conewago Crk. Philadelphia, Pa., U.S.A. E. Pk. Drive

Viad. Station, Pa., U.S.A. Conemaugh R. Posen, Germany Turin, Italy

Near Painsville.O.,U. S.A.

Trilport, France

14 m. from Nismes, France Gardon R.

Tiber R.

Avon R.

Scotland Annan R. U.S.A. Harlem R.

Po R.

Marne R.

1100 b.c. Scaurus JÓ2 b.c. Fabricius

1820

Telford

m 64 61

75.0

4=76.0

76.0

76.7

76.7

76.7    to 70.3

76.8

76.8    to 68.2 76.8

4=77.0 78. o 78.7 79.0 79.0 74-5 • 65.0 79-3 to 51.0 80. o

70.5

80.0

ii .5

38.0

6.4

25 ■ 6

26.6

25.6 19-3

38.4

25-7

24.6

38.4

14-

21.6

26.3

13.4 11.8

8.7

40. o

39-8

20.0

2-5    3-0

7.6    14.0

38.0

. 200

f30.o    10.0

4.8

4-3

3-2

4-3

3-2

3-2

4.8

4.8

3-0 I . 2

4.0

118.2 53-3 63-9

53-3

38.4

38.4

. 041 . 080 .050

. 080

.084

■ 125

46.9 18.1-12.8

16.0 35 •¹    13-3

76.7

.<>

32.0 12.8

55-4 14-9

Brick stone fac’g Soft red sandstonel Portland stone

H.W.

H.W.

H.W.

H.W.

H.W.

H.W.

H.W.

S. 1839 B. April P- 373

A. March, 1844, p. 128

K. 1895, p. 306; L. F.

S. F. 1852, p. 282 F. 1852, p. 284 F. 1852, p. 284

16, 1892,

183

184

185

186

187

188

189

3-0

60.

46.

.049

.026

30-9 + 26.0 165.0

H.W.

H.W.

3-5    4-5

3-5

3-5

6.0

6.0

64.9

64.7

65.0

40.0

39-8

50-0

■ 054 .054 .054

.150

.151

38.0

15. i

Brick

Ry.

F. 1852, p. 278

F. 1852, p. 282 F. 1852, p. 276

Q. F. 1852, p. 274 L.

T. Aug. 12, 1898 F. 1852, p. 288

190

191

192

193

194

195

196

Sandstone

+ 29.0 51.2 t32.o

27. o

Peperino, tufa and traverti’e

Ry.

H.W.

H.W.

H.W.

I.

197

C. 1855-6

198

(N ro rf iovC

MMMMMM N N N N fl N

Oise

Prague Austria York, England Near Montlouis, France Tablonica Austria Baiersbonn, Germany Near Pontoise, France

Moldau R. Ouse R. Loire R.

Forbach R. Oise R.

1820 Telford 1837-42 Jervis

1891-92 Brown 1890 Nichols

1833 Penn. Ry.

1810 Pertinchamp

1758-64 Peronnet

Bet. 27 Under Agrippa

B.C.-14A.D ist tier 2d tier top tier

16th cent.

1845 Morandière

1890 Leibbrand 1843 De Breville and Couche

80. o 80. o 50.0 80. o 80. o

20. o 40. o 25.0 40. o 26.0

3-0

2.5

2.5

M tO ^ t+i

m fÔ6< WVÔCJÔ ^ 61

,    M «O    M

80. o 80.0 4=8o . o 80.0 80.4 to

76.7

80.5

63.0

51.0

40. o 16.0

40. o 28.8 27.7 40.3

31-5

25-5

3-5

3-0

3-0

4-7

3-0

4.8

5-3

5-0

5-0

3-5

3-0

3-5

3°

5°.o 40. o 25.0 40. o 43-8

40. o 58.0

40. o 42. o

40-3

31.5

25.5

.060

.062

.088 . 069

.075

. 081

.075

20.0 12 I .0

25-0

30.0

90. O 32.O

20.8

Aqued’t

Q. Cresy’s Enc. C. 199 E.

F. 1852, p. 274 M. No.1207 F. 1852, p. 288

F. 1852, p. 288 Johnson's Ency.

200

201

202

203

12.0

10. o 16.0

Sandstone

Brick

Granite

Freestone

Ry.

E.

Ry •

Ry.

Ry.

H.W.

Aqued’t

and

H.W.

B. May 24, 1894;

Jan. 24, 1891 Q. Am. Sup.

204

205

.8

*80.9 •*81. o 81.2 82. o 82. o 82. o

7-

7-

15-0 11.8

26.3

23.3

3-6

5-2

Granite

Ry.

H.W.

78.

28.2    10.7

Ry.

9-

ii.7

2.0 4.6

77-

?2 I . 7 25-4

IO . I

Ry.

H.W.

Ry.

I.

Ü.

B. May 2, 1902 Q. F. 1852, p. 280

Johnson’s Ency.

P. Oct. 1896,p. 122

206

207

208

209

210

211

K. May 10, 1878

K. Dec. 22, 1871 B. Dec. 7, 1893 G. ist T. 1901 F. 1852, p. 294

0.

212

213

214

215

216

217

* Maximum.

Remarks. 183. On piles. 184. Skew. 186. First lise of modem caisson. Replaced by * cast-iron bridge. 187. Radial joints in spandrels. 193. Continuation of Fabricius Bridge. 195-, Skew 61 and 77° 52'. N. Y., N. H. & H. Ry. 200. 13' arch in pier. 201. Glasgow-Carlisle. 203. H. = ioo'.o. Parapet 116' above water. 204. Phila. Div. Pa. Ry. 6° curve. Two tracks. 205. Phila. & Reading Ry. 206. Pitts. Div. Pa. Ry. Destroyed by Johnstown Flood, 1889.    208. Commenced by French 1810. Completed by King Victor

t About.

Emmanuel. 209. Lake Shore & M. S. R.R. 210. First bridge entirely designed by Peronnet. 211. Fifth century, ends destroyed. Repaired 1743 and piers prolonged for new bridge. H.— 160'.o. 212. Between Karlin and Bubua. Viaduct has 87 arches. 214. Orleans-Tours. Damaged in War 1870-71.    215. Austrian State Ry. 216. Three-lead hinges. Cost 18,260 f.

217. Skew 76°. Ch. de fer du Nord.

Name.

Place.

Over.

Date.

218

219

220

221

222

223

224

225

226

227

228

229

230

Crueize Viad. Stulz Viad.

Mussy Viad.

Pont Royal

Cart

Big Walnut Cognet

Near Marvejois, France

231

232

233

Lea Cut

234

Salarius

235

Fouchards

236

Pont de Pierre

237

238

239    Dee Viad.

240    Dunkeld

241    Dean

242    Licking Aq.

243

244

245, Enz 246 Jena

247

248

249

250

251

Alcantara Louis XVI! ?) Spey Trinity

252

253

St. Ed me

2-4

Vecchio

25

Mussy, France

Paris, France Moret, France Elkader, Iowa, U.S.A. Paisley, Scotland U.S.A.

Sisteron, France Hautes Alpes, France Maligny

Darlaston, England Coatsville, Pa., U.S.A.

Blois, France

Bordeaux, France

Lea Cut, England

Narses, Italy

Samur, France

Grenoble, France

La Voulte, France Albois( ?), France Bet. Rhos-y-Medre and Chirk, Wales(?) Dunkeld, Scotland

Near Edinburgh, Scotland

Castellane, France Romans, France

Near Hofen, Germany

Paris, France

Stonleigh, Enlgand Toledo, Spain France

Fochabers, Scotland Florence, Italy

Pontoise, France Nogent-on-Seine, France

Florence, Italy

Neuvillr, Prance

Crueize R. Stulz Gorge

Mussy R.

Seine R. Loing R. Turkey R. Cart R.

1892-6

Durance R. Drąc R. Serin R.

W. B. Brandywine Loire R.

Garonne R.

Lea Cut R.

Teverone R.

Thouet R.

Isère

Allier R. Aveyron R. Dee R.

Tay R.

Licking R. jVerdon R. Isère

Enz R.

Seine R.

Avon R( ?) Tagus R.

Spey R. Arno R.

Oise R.! ?) Seine R.

Arno R.

Ain R.

1685

1771

1888

1839

1902

1500

1605

1902

1723

1813-22

Rebuilt 6th cent.

1839

1770

ti849

1809

1404

1885

1806-12

MASONRY

Engineer.

in

'S

Geoffroy, Morris 18 and Pouthier Mansard Peronnet Tschirgi Locke Graham

Werbruge

Brown

Gabriel

Deschamps

Trudaine or Voglie Picot

1781-1821

997

1791

1569

1772

1766-69

1177

1775

Boesnier

Telford

Telford Fisk

Leibbrand

Lamandé

Rennie Romans! ?) Perronet Bum

Ammanati

Peronnet

Peronnet

Gaddi

Aubry

* Maximum.

17

19

rt

Remarks.—-218. H. = 207'.6 Midland Ry. 219. Thusis-Engadine. 220. B. Nov. 8 1804, p. 388. Pans, Lyon. 224 Glasgow & Paisley Joint Ry. 225. B. & O. Ry., Newark Div., two tracks. 230. Over Wilmington & Northern Ry. and deep ravine. 234 Blown up in

ARCHES.

Span.

Rise.

Thickness of Arch Ring at Crown to.

be

■g*

£2

.5

Curve.

Radius at Crown.

Width, Face to Face at Crown. Thickness of Piers at Springing.

Material.

Class of Bridge.

Reference.

Number

*82. o 82.0

t4i .0 41. o

T4- ³

3-3

fS.2

4.9

3C 41.0

.102 ?2Ó.2    16.4

8.5

Ry.

Ry.

82.2

41 . I

4-6

02    41.i

*82.3

83.1

84.0

85.0

85.0 85 • 2 *85.3 85-3

86.    o 86.0 78.0

86.3 to 54-3 86.9 to 65-9

87.    o

87.8

88.4

88.6

75-4

*89.5

*89.5

*90. o

90.0 t< 22.0 90. o 90. o 90-3 91.1 t< 70.2

*91.8

97. o *93-o 94.0 *95 • ° 95-8 t< 87.6 95-9 95-9

*95-9 96. o

.112 26.516.4-23.0

Granite

Ry.

R. March, 1891 Engint er, April 8, • 357, 366 30, 1897,

27.9

18.0

57 • 5 42.6 42.6 13-5 43-0 39° 30-9

28.9

16.0

43-9

7-4

22.1

20.7

28.8 33-°

30-o

30.0

15.0

29.9

23-9

C-CÇ.2

10.8

13.0

46.5

9.8

16.0

14.8

28.8

19.2

26.6

4-3

3-0

3-3

2.7

4.6

3-0

3-5

6.9

3-9

3-9

3-8

3-9

3-5

3-7

3-0

3-0

2.8

4-3

4-3

3-3

4-7

+4-6

3-7

3-2

5-3

5-3

5-3

4-3

4.0

3-3

4.9

t8.o

3-2

C 153.4

C 45-5

t6o.o .028 41.6 .066 30.0

8.0

H.W.

04, pp K. April

P- 575

L.

Limestone

H.W.

Ry.

32.0    10.o

Ry.

C2

Pi Pi Pi O

PI T) Pl^ pi

42.6 42.6 75-2

43.0

39-0

57-5

43-9

84.2 114.8

52.2

48.7

48.8 76. o

73.5

55 • i

119.4

102.0

87.9

CO v| On Ca O La

71 . i

.108 ii.8 .070    21.3

.047 ?26,5

Freestone

H.W.

H.W.

H.W.

Ry.

. 120

22.4-159 49.2    13.8

H.W.

30.0

27.8

10. i

.034 32.8    16.4

12.8

.070    38.4    13.9

.062    27.0 16.0—

14.0

.061

• 037

.058

.077    19.9    28.8

.041    10.7

.046 46.4    9.8

.052    16.0

21.5

33.8    26.3

.032    41.s

.066 32.0

9.8

.062j103 .o 23. s . 060I

19.2

Brick and stone Brick, stone trim

Fine sjtone

Aqued’t

Sandstone

Freestone

White

marble

H.W.

H.W.

Ry.

H.W.

H.W.

H.W.

H.W.

H.W.

H.W.

H.W.

H.W.

F. 1852, p. 280

B.    April' ii, 1891 A. 1839, p. 313 T. Aug. 22, 1902 F. 1832, p. 282 F. 1852, p. 276 J. F. 1852, p. 282

C.    1835-56 T. Nov. 21,

p. 898 F. 1832, p. 276 P. 1886, p. 134 J. F. 1852, p. 288

C. 1833-36

M. Feb. 18, 1899.

p. 19346 F. 1852, p. 282

F. 1852, p. 292

F. 1832, p. 282 F. 1852, p. 280 A. Oct. ’48, p. 317

L. F. 1852, p. 286

221

H.W.( ?) H.W.

!9°2,

229

230

231

232

233

234

23s

236

237

238

239

240

I.

I.

F. 1832, p. 274

F.    1852, p. 282

K. 1892, p. 560;

G.    11891, p. 920

C. 1833-36; H. F. 1852, p. 286

P. 1896, p. 130 Woodbury, 1858

A. April, 1847, p. 104

F. 1832, p. 280 F. 1852, p. 280; V.

July 17, 1897 J. F. 1852, p. 276; P. 1896, p. 129 F. 1832, p. 282

S.

t About.

1867.    239. Shrewsbury & Chester Ry. 242. Chesapeake & Ohio Canal,

"hinges.” 247. On piles. 254. Covered. Rebuilt about 1350.

241

242

243

244

24s

246

247

248

249

250

251

252

253

254

255

245. Three lead

MASONRY

ARCHES.

Number.

Name.

Place.

Over.

Date.

Engineer.

No. of Spans. »

Span.

Rise.

256 Dean

257

258 Fleischbrücke

Edinburgh, Scotland

Drôme, France Nuremberg, Bavaria

I Thickness of I Arch Ring at Crown tn.

ic

.a

Curve.

Radius at Crown.

l±.

Width, Face to Face at Crown. Thickness of Piers at Springing.

Material.

Class of Bridge.

Reference.

Number.

1831

Telford

1774

1599

Bouchet

Carln

259

.260

Imnau

261

.262

Rialto

.263 Margherita

Near Imnau Charrey, France Near Chalonnes, France Venice, Italy

Rome, Italy

Eyach R. Saône R. Loire R. Grand Canal

Tiber R.

1896

1888

1864-5

1588-91

1891

Liebbrand Mocquery Morandiere Ant. da Ponte

Vescovali

Ul

96. o 90.0 *96.0 97.0

C-C98.4

*98.4

98.4

98.5

*99-°

16.7

3°

27.7 13-0

3-°

2.3

6.4

4.0

C-C9.8

12.3

+ 24.6 23.0

16.5

3-8

4-9

t5-°

f6. o

.26

04

-265

Pont du Jour

Carbonne, France Paris, France

Garonne R. Seine R.

1770

1864

Saget

Bassompierre

Ul GJ

Alcantara Aq. Bishop Aukland Etherow River

269    Blackfriars (old)

270    Alcantara

271    Wellington

272    Rutherglen

.266

267

268

■²73

Near Lisbon, Portugal England

London England

Alcantara, Portugal Leeds, England

Bet. Glasgow and Rutherglen, Scotland

Minneapolis, Minn., U.S.A.

Wear R. Ethei*ow R. Thames R.

Tagus R. Aire R.

Clyde R.

Mississippi R.

1731-75

1388

1760—70

loot

1816-19

189s

w IO CO co

99.1

99.2 IS-8

40.5

31.2

7-9

Hoskoll Robt. Mylne

Trajan Jno. Rennie

Crouch & Hogg

1882-93 Smith

M Ul

*100.0 100.0 *100 .0 *100.0

*100.0

100.0

100.0 90.0

100.0

88.0 22.0 25.0 43-0

50.0

15 -°

12.6 ii .7

39-7

3-7

5-3

1.8

4.0

5-°

4.0

4.0

4.0

3-0

1.8

4.0

7.0

4.0

4.0

D    bio Ai Citait) C)    -atrj

77-5

48.7

74-5

80.0

■ 030 . 062 .086 .050

41 . o

H.W.

128.4

104.5

73 • S

7-9

62.5

50.0 90.8

97.6

91.4

.012

.036

.050

. 064

.041

■ 044

53-3

?i3 • o 18.2 ?26.2 64. O

?67-5

17. i

H.W.

25.6    22.6

101.7

29 S

?45-o

Beton

Limestone

Marble

Rezzato & travertine stone

Stone fro Châ Haudon

teau

Granite

Brown

sandstone

tso.o 13.5

Granite

H.W.

H.W.

Ry.

H.W.

H.W.

m H.W.

Ry.

Aqued’t

H.W.

H/W.

H.W.

H.W.

H.W.

F. 1852, p. 192

F.    1852, p. 282

P.    1896, p. 132; F.

1852, p. 276

G.    ’98, 2d Tri.

G. VI, 1896, p. 737 K. Oct. 18, 1867

Q.    P. 1896, p. 122 ; F. 1852, p. 276

R.    June, 1892, p.

260

F. 1852, p. 280 K. Feb. 8& Jan. 25, 1867

P. 1896, p. 137

256

257

258

259

260

261

262

263

264

265

I.

r.

L. P. 1896, p. 136; F. 1852, p. 280 L.

A. 1844, p. 128 and

246

266

267

268

269

270

271

274    Elster Viad.

275    Göltzsh

Bet. Reichenbach and Plauen, Saxony Bet. Reichenbach and Plauen, Saxony

Elster R. & V. (two tiers) Göltzsh R. & V. (four tiers)

1846—50

1846-5-

Wilke Wilke ( ?)

00

276 Lempde

277;

278    Montlyon

279    Pont au Double

280    Guillotière

281

P. de la Concorde

Rouen, France France

Paris, France

Lyons, France

Paris, France

Alagnon Seine R. Durance R.

Seine R.

Rhone R.

Seine R.

178s

1805

1847

1265

1787-92

Mauriset

Lamandé

Delbergue-Cor-

mon

De Lagalisserie

Ass. des frères Dupont Perronet

H H (O to O O'Gj O

00

80. o

71.4

43 • 9 40, o

100.3-23.2 100.3 92.9 46.8 44.6 41.8 39.0

101.2 *101.7 101.7

101.8

40. o 15-0

13.0

5-3

46.4

23.4

32.0

13-7

32.0

9.8

2-7

2.7

2-5

2-7

3-7

3-7

1.5

1-5

1.5

1-5

4-5

5-3

40.0

46.8

23.4

23-4

23.4

23-4

95-7

77-8

. 067

.080 . 064 . 064 . 064 .064

.047

28.0 7.0-14.0

Limestone

Engineer, Aug. 23, 1895. p. 182 West. Soc. Eng.

Vol. 8, 1903, p. 421

Jour.

26.1

?2Ó. I

Brick mostly Brick mostly Rings of brick

H.W.

F. 1852, p. 209

F. 1852, p. 199 Q. Am. Sup.

272

²73

282

283

284

Gère Avignon

Munich, Bavaria Vienna, Austria Avignon, France

Isar R.( ?) Rhone R.

1814 1781 1177-87

Wiebeking

Vimar

Benezet

to

102.3-26.2 102.3

92.7

831

102.3 102.7 *102.9

38.4

136.2 62.9

• 039

• 034

52.2

34 -1

Millstone

grit

H.W.

H.W.

9.8

8.7

3-7

6.4

17. i

28.2 51 • 5

3-4

3-2

4-3

5- 2

2.4

148. o

127.5

138.2 82.5 57-4

51 •

.025 .027 .023 .050 . 090

• 047

51.1    9.6

42.6 35-6

15.4 22.8

9.6

Freestone

Freestone

Freestone

H.W.

H.W.

H.W.

F. 1852, p. 284 H.

F. 1852, p. 286 J. F. 1852, p. 296 F. 1852, p. 274 J. F. 1852, p. 284

2₇6

²77

²78

²79

280

²8r

J. F. 1852, p. 288 F. 1852, p. 284 J. L. P. 1896,

F. 1852,

p. 274

282

²»3

284

* Maximum.

„„„ Remarks—256    ₉6'.o arches are under sidewalks. 257. 90\o arches are under roadway.

259. Ihree granite hinges. 261. Granite piers on concrete foundation. Two tracks. 265. Parapets, etc., Jura marble. 266. H. = 23o'.o. Highest single tier of stone arches in the world.

f About.

269. Replaced by cast iron, 1865. neapolis Union Ry. Two tracks. 3Ó4'.o.    284. In ruins.

270. H. = 2io'.o.    271. Coffer-dams employed. 273. Min-

274. Saxony-Bavaria. 275. Saxony-Bavaria. H. =»

MASONRY

ARCHES.

Name.

Place.

Over.

Date.

Engineer.

No, of Spans,

Span.

Rise.

Thickness of Arch Ring at Crown ip.

At Springing ts-

Curve.

Radius at Crown.

Width, Face to Face at Crown. Thickness of Piers at Springing.

Material.

Class of Bridge.

Reference.

Nuinber.

285

Port de Piles, France

Creuse R.

1846-47 Bayeux

103.8

40.5

4-3    4-3

70.8 .060 30.0    19.2

Freestone Ry.

286 Hérault

287

Route of Nice, France Prague, Bohemia

Hérault R.( ?) Moldau R.

Grangent 1878 Reiter

288

280 Wissahickon

Marbach, Germany Philadelphia, Pa., U.S.A.

Murr R.

Wissahickon Crk.

1887

1897

Leibbrand Gen. Thayer

98.4 104.4 105.0 99.8 94-5 89-3 105. o 105.0

iS-4

16.2 15-3 14-5

13.7

10.2 11. o

2.7

4-0 S•3

3-9    4-9

3-0    4-5

90.5 .029    22.4

39•8    13 -1

Granite

H.W.

C. 1851-52; J. F.

1852 p. 276 F. 1852, p. 294 F. 1852, p. 280 K. May io, 1878. p.

285:

286

C 140.2 .028 18.4 C( ?) 118. i .025 ?35.0

Cousho-

hocken

stone

H.W.

H.W.

3S9

287

G. 1891. I, p. 922 B. Sept. 9,1897, p. 162

28»

289-

290    Potomac Aq.

291    Ponthaut

292

Washington, D. C., U.S.A. Potomac R.

Bonne R. Loire R.

Germany Orleans France

293

294

295

296

297

298 Lodi St.

299

300    2d Worochta

Hartford, Conn., U. S. A. Connecticut R. Baiersbronn, Germany    Murg. R.

Wurtemberg, Germany    Nagold R.

Winstone, England    Tees R.

Sault, France    Rhone R. Br.

W. Br. Black R.

Garonne R.

1793

1750-60 Hupeau

Graves Leibbrand

1903 1889 1882

1762 Robinson 1825-27 Montluisant

301

302

303

304    Grand-Maître

305    Cresheim

306    Napoleon Tongueland

3° 7

308

Elyria, Ohio, U.S.A.

Toulouse, France

Worochta, Austria

St. Esprit, France

Nantes, France Mantes, France

Fontainebleau, France

Fairmont Park, Philadel- Cresheim Crk.

phia, Penn., U.S.A.

Paris, France Near Kirkendbright, Scotland

Hartford, Conn., U.S.A. Connecticut, R.

Pruth R.

Rhone R.

Loire R. Seine R.

Fontainebleau V.

1894 Jackson and Bunce

1543-1632 Souffron

1892-93 Huss

1265-1309 Ass. des frères Dupont

i7S7-⁶S Hupeau 1869 Belgrand 1892 Webster

*105.o 106.3 106.5-98.0 108.0 108.2 108.8 108.8 in -.5 91.8 112 . o

ii3•o-

44.8 II3•5-

32.8

S3 • i

28.8 16.0 27. o

10.8 10.8

31-9

29.S

I9S

Dee R.

r8o6

Telford

1904- Graves

19

21

Ma ny

114.i-

81.0

38.4

56.8

44.8

115 • 2 *U5-4

US -8 t42-5 116. o

116. o 118.0 Small 68.0

34-4

34-0

19-3

21.2

14.8

38.0

21. i

6.9

2.0

3-3

4.6

4-3

hi hi

53.1. 108 29.5 83.9 .083 49.0 19.2-18. i ti8. o

Granite

3 • S 4-3

3-7

4-3

6.7

21.7?

ti6.4

114.2 .057    23.0 22.2

81.0 .037

38.0

76.8.049 64.0    26.6

56.8 .076    14.7

70.4.084 17.6 27.8

Elyria

sandstone

Brick

Aqued’t A. 1837,8, p. 148 F. 1852, p. 284 H.W. L. F. 1852, p. 276

H.W.    N.    Dec. 26, 1903

H.W.    G.    ist Tri., 1901

H.W. G. 1891, i,p. 903

F. 1852, p. 288

H.W. C. H. Snow, City Engineer, Elyria, O. H.W.    F.    1852, p. 276

Ry.    B.    Dec. 7, 1893, p.

448

H.W. F. 1852, p. 274

290’

291

292-

293

294

L.

295

296-

297

298

299.

3°o-

301:

6.4

6.4

Beton

3•S 4-5

10.0

BufE

sandstone

o.

H.W. I.

Aqued’t K. Oct. 1869, p. 275

Sewer B. Aug. 31,1893, P-170

30 2-305

304

305

4.0

3-6

C 64.8 .056 24.0

Ry. I.

H.W. L. F. 1852, p. 286

30&

307

102.0 15.0-40.0

Granite

309 Waterloo (new) 3to Devil’s Br.

3TI Têtes

312

313    Vingeanne Val.

314

3rs Maidenhead

London, England Near Lucca, Italy France

Bourbonnais, France Near Oisilly, France Rumillv, France Maidenhead, England

Thames R.

Serchio R.

i8t7

iooof

Rennie

74.0 81 .0 108. o 1150

I 19. o 120.0

120.5

Durance R.

Vingeanne Val. Cheran R. Thames R.

1732

Hanriana Vaudra y

22.9 2 S ■ I 27.0

28.8

29.8 34-6

60.3

H.W. and T. Feb. 19,1904, p. El. Ry.    123

N. Dec. 26, 1903 N. Dec. 31,1904,p.

308-

4 • S 10.0

44.0 20.0

Granite

H.W.

t4-S

C2 60.3 .074 12.0

765

1785

j 1832-38

Garella

Brunei

123.6 124.0 127.0

127.6 128.0-

21.0

6l .

6.9

46. o

63-

24-3

Limestone

sandstone

H.W. C.

S. K. Feb. 22, 1895, p. 236; F. 1852

309

310

4-7

2.7    3-6

S•3    7-S

C2    61.8

c 255.7 E

C2 63.8 E 169.0

.005 iS-9 .010

Granité

H.W.

Ry.

. 086

•031

Go* N) M

to

...

QUO

Freestone

Brick

Ry.

H.W.

Ry.

F. 1852, p. 278

B. Dec. 7, 1893

J.    F. 1852, p. 284 P. W’ks.G.B., ’46;

K.    Oct. 25, 1895

i.

311

312

313

314

315

* Maximum.

t About.

Remarks.—285. Tours-Bordeaux. Two tracks. 69° 26'. Ten 4' ribs. 293. See No. 308.

299. Stone trimmings. 300. Austrian State Ry.

288. Three lead. ‘ hinges.’ 294. Three lead “hinges/’

301. Small arches in piers.

280. Skew

Cost

23,890 f. 304. Paris

water-supply, arches. 313. E. Ry. of France.

308. Pneumatic foundations. Cost (est.) $1,600,000. 315. Great W. Ry.

310. Four small side

MASONRY

ARCHES.

Numbeh

Name.

Place.

Over.

Date.

■ Engineer.

r.

Span. Rise.

Thickness of Arch Ring at Crown ip.

At Springing, ts.

316

.317

318

Echo Br.

319

320

321 .322

North Ave.

.323

ist Worochta

Neuilly, France

Mantes, France

Newton Upper Falls, Mass.

U.S.A.

Elyria, Ohio Aberdeen, Scotland Wan Hsien, China Baltimore, Md., U.S.A.

Worochta, Austria

Seine R.

Seine R.

Charles R.

Black R.

Den Bum Rill

Gorge-Jones’ F’ 11s

Pruth R.

1768-74 iPerronet

1757-65 Hupeau under Perronet 1876 Fitzgerald

*128.2    32.0    5.3

1801 +

Telford

1893-95 Smith 1892-93 Huss

128.2    38.5

11 5 • 4 34-9 129.0    42.3

34-°

37-°

37.0 129.0 130.0 130.0

6.4

G ta Curve-.'

Radius at Crown.

7cJ°

Width, Face to Face at Crown. Thickness of Piers at Springing.

Material.

Class of Bridge.

Reference.

Number.

5.0    6.0

IO to

s, i i

.021

.071

.074

47.9    14.

35-6    25.

Freestone

18.0

Granite

H.W.

H.W.

H.W.

Aqued’t

J. P. 1896; F.

1852. p. 280 V. July i 7, 1897 1852, p. 276 ston Water W’ks

3 X«¹

3ir

318"

.324

Boucicault

Verjux, France

Saône R.

1888-90 Jozoti

131 • ²

36.1

26.2

13¹ • ²

18. 5

14.0 29. o 65.0

26.0 5.0 8.4 32.8 4.6 7.2

16.4

3-4 t4-9

hi

18.5

27. o 43-°

100.0

14-7

Granite

Fteley

. 298.

16.

BrickRing

H.W. H.W. H.W. & E. R.R.

Ry.

P- 477

June 19,1902 July 6,1893, p. 7

3i<*

320

32*

322-

B. Dec. 7, 1893, p.

448

3231

177.0

.0x9

?2Ó . O

Villebois

stone

H.W.

.325

Moret Viad.

326

327

St. Martin

328

329

Vizille

331

Moret, France

Scrivia, Italy

Toledo, Spain Villeneuve, France Near Grenoble, France Near Bludenz, Austria Verdun, France

Loing Val. Scrivia R.

1847-49 1850 +

Ranco

30

131 . 2 32.8 131.²

16.4

16.4

43-7

5-9

Tagus R.

Lot R.

Romanche R. Gorge Doubs R.

33 2

333

Castalet Albula R. Viad.

Sales

Al. R. Gorge

1203

1732 1766 1884 1895-97' Jozon

Séjourné

Bouchet

Huss

i9°3

*132.0

*132.6

133- 8

134- 5 134-5 126.3 134-5 137-8

66.3 + 38.2 42.6 3°- i 27.9

5-3

5-6

3-9

3-9

10.2

S? hl£h3 hi

139-4 16.4 86.9

66.3 115 • o

.019 . 068

29.5

8.

Ry.

Brick

. 080 .067

19.6 32.3

19.7

13.1

Ry., Turin to Genoa H.W. H.W.

Limestone

Ry.

H.W.

8.5

Ry.

335

337

Br. C33

St. Sauveur

Pont-y-tu-prydd

Alma

Bellows Falls, Vt., U.S.A.

France

Nr. Newbridge, S. Wales. Paris, France

Connecticut R.

1899 Cheever

98.4

140.0

20. o

4.0

4.0

Taff R. Seine R.

1755

1855

, Edwards Darcel

140.0

140.0

I4I-4

35-° 28.2

4-9

<0 G

132.6

.030

27.0

15-8

16.4

Freestone

Millstone

grit

Ry.

H.W.

H.W.

338

Near Narni, Italy

Bet. 27 B.C.-14A.D

339

Putney Road

Putney, England

Thames R.

+ 1882

Bazalgettc

-340

341

Outer Maximilian Verone

Munich, Bavaria

Near Vieux-Château, Italy

Isar R. Adige R.

1904

1354

Under Scala

Pont-du-Cèret

Moulins, France Near Perpignan, France Turin, Italy

Allier R.

Tech R.

Dora Riparia R.

x 705-1710

Mansard

1336

1834

Mosca

* Maximum.

Remarks.—316. In design R. = i6o'.o.    317. Destroyed in War 1870.    318. Sudbury Aqueduct for Boston. H. = 79,-o.    321. Slightly pointed. 322. Skew 55⁰. Ribbed. 323. Austrian.

State Ry. 324. Radius at spr. = 75'.5. Paris-Lyon. Two tracks. Approach to metal spans crossing fiver. Curve about o° 52'.    330. H. ==x6s'.o. Slight curve. 331. Extrados arc of circle 144'.3.

126.0

142.0

135-0

114.    o 75-0

144-0 129.0 112. o 144-3 146.0

87.4

•33 o

147- I

115.    I 147.6

148.0

25 - ²

19-3 16.3 13-0

35-8 21.3 17 - i

73-8

18.0

4.9

4-5

4-3

4.2

5-3

4.6

4.9

H.W.

B. May 18, 1893 G. 1892, p.445 ; P.

1892, p. 50 F. 1852, p. 117

F. 1852, p. 296

P. 1896, p. 130 F. 1852, p. 276

F.    1852, p. 280

G.    1888, p. 575

G. 1897, 4°, p. 179

B. Feb.27,1902 Engineer, April 8 and Mar. 4, 1904, pp. 228 and 355 B. June 21, 1900, p.

402, and Blues P. 1896, p. 140 0- L.

0- J-

A. 1856, p. 376

324t

325;

32fr

32 T

328

32<>-

330

33*

3322

33J-

334£

3 3 St 336« 33T

L.

33&:

5-5

5-3

5-²

13-1

144.0

136.0

127.0

90-5

73-

i 60.

.031

.032

.033

• 059

• 031

47.0

71-5

22.4

18.0

nd

19.0

20.

36.2

and

12.8

40. o

22.

36.

Granite

ring

H.W.

Limestone

Freestone

Granite

H.W.

H.W.

H.W.

H.W.

K. May 17, 1895 K. July 23, 1886, P- 85

B. Oct. 27, 1904 J. F. 1852, p. 274

339'

34°’

34*

F. r852, p. 2.76

B. Dec. 7, 1893 F. 1852, p. 274 I. U. 1846, p. 27 F. 1852, p. 290:

342-

34S

344

t About.

and 15o'.9 radius.    333. Thusis-Engandine. H. = 282'.    335. H. = 21 s'.o.    337- Rubble, grouted.

Foundation on piles. 338. Probably the most magnificent bridge built by the Romans xn Italy.    340. Three metal hinges. Failed by hinges slipping, June 27, 1904.    342. Failed:

1710-    343- Mostly brick. Stone ring.

MASONRY

ARCHES.

Nuttiber.

Name.

Place.

Over.

Date.

Engineer.

No. of Spans.

Span.

Rise.

rC

Arch Ring

at Crown t₀.

At Springing, ts.

545

346

Bellefield

347

348

Claix

349

350

Gloucester

351

Vieille Brioud*

352

London

Nr. Kleinwolmsdorf!, Sax. Pittsburgh, Penn., U.S.A.

Near Grenoble, France Elyria, Ohio, U.S.A. Gloucester, England Berne, Switzerland Brioude. France

London, England

Roeder R.

St. Pierre Hollow

Drąc R. Black R. Severn R. Aar R. Allier R.

Thames R.

1896-

i6it

1886 1827 + 1204 1454

1821-30

Rust

Kinney

Telford

Greiner and Es-

tone

Rennie

148.6 150-0 3

49-5

36.6

54-4 27 . o 54-0

150.2 150.0 iS°-°

15° - 5 iso-9 75-5

5-6

4.0

6.0

Ci Curve.

Radius at Crown.

Width, Face to Face at Crown. Thickness of Piers at Springing.

Material.

Class of Bridge.

Reference.

80. I .070 t 26 . o 82.0

Gray

sandstone

Ry.

H.W.

3-2

3-8

4-3

4-5

Sandstone

75-5 .057

24.7

H.W.

H.W.

H.W.

H.W.

H.W.

353

354

355

356

Jainma

Main St Tyne

Wear Viad. Victoria

357

358

359

Ny deck

Near Tournon Jamma, Austria

Wheeling, W. Va., U.S.A. Near Newcastle, Eng. Sunderland Lambton

Gignac, France

Near Lavaur, France Berne, Switzerland

Low

Doux R. Pruth R.

Wheeling Crk. Tyne R.

Wear R.

Wear R & Val.

Hérault R.

Agout R. Aar R.

1545

1892-3

1892

1777-93

1775

1840-44

Huss

Hoge & White

Garipuy

Saget Mülle r

152.0 37.7

140.0

130.0

156.7    65.0 157.4

29.5

159.0 28.4 159-9    79-9

144.C    72.0

99.8 49.9 20.2

160.0I 44.0 83.1 41-6

60.5    65.0

160.7

10.1

4.8

4.6

4-5

5-6

On CN CN 4^ -£■    4^

O' O' O'Cn

La

.0

9.0

8.5

8.5

4.6

4.6

4.6

162.0 .029

56.1    24.0

—22.0

Granite

H.W.

78.9 .035

125.4 .036

79.9    .058 72.0 .064

49.9    .092

117 • 7 41.6

103-4

.055

.156

.095

16.0

14.7

H.W.

Ry.

F. 1852, p. 294 B. June 22,1899,p. 346

391

H, F. 1852, p. 276 347 B. May 31, 1890 H, F. 1852, p. 290 349 B. Dec. 19, 1895 A. 1844, p. 247; F. 351 1852, p. 274 A. 1847, p. 106 F. 1852, p. 290 P. 1896, p. 128

F. 1852, p. 276    353

Dec. 7, 1893, P- 354

345

348

350

352

48.0

25.8

H.W.

38.4

39-8

21.5

8&

23-

21 • 5

25.6

Soft

Sandstone 1+ miles from Br. Freestone

Ry.

H.W.

H.W.

H.W.

448

355

Blues

F. 1852, p. 178 A. 1837-38.p.57    356

C. 1855-56

H. J. F. 1852,p. 284 357

360

361

362

Antoinette

Ballochmoyle

Vieille Brioude

363

364

Gour Noir

365

366

367

Grosvenor

Lavaur

368

.369

.370

Near Ballochmoyle, Scot.

Brioude, France

Near Coppel, Germany

4 k. from Uzerche, France

Turin, Italy Chester, England Near Lavaur, France

Bogenhausen, Bavaria

Germany

Jaremcze, Austria

Ayr R.

Allier R.

Schwaendenholz Ravine & Brook Vézère R.

Dora Riparia R. Dee R.

Agout R.

Isar R.

Gutach R.

Pruth R.

Séjourné

Millar

Romans

vO M

164.0

180.0    90.0

50.0    25.0

183.7 f6o.o

6.0?

6.0?

1901

1888—9

1833

1832-3

1888

1901-02

1901

1892-3

Daigrement

Hartley

Hartley

Séjourné

Fischer, Archt.

Huss

4-5

5-3

90.0

25.0

91.8

.067 . 180 .058

28.0

16.0

187.0 55.8 5.9 196.8 52.8 5.6

200.0 200.0

201.7 c.-c. 209.9

210 . O

42.0 42. o

4.0

4-5

90.2 c.-c.

21.4

52.5

3-4

6.6

Ry.

H. F. 1852, p. 282 O. Nov. ’97, P- 322 B. Dec. 19, 1895 F. 1852, p. 294

B.    Feb. 27, 1902

C.    1851-52

358

359

360'

361

8.5

14-4

13-8

118. i

.047

7.0

9-2

140.0

140.0

.028

.032

35-5

60.6

13.7

Volcanic

rock

Sandstone

Granite

ring

Sandstone

Limestone

Sandstone

H.W.

Ry.

Ry.

H.W.

H.W.

H/W.

Ry.

A.    June, 44. P- 247

B.    Dec. 26,1901, p.

487

G. 1892, p. 545

362

363

364

H.

371

Cabin John

372

373

Trezzo

374

375

Plauen

Washington, D. C., U.S.A. Cabin John Crk. Italy

Near Trezzo, Italy

Luxemburg, Germany Plauen, Saxony

Adda R. Adda R.

Petrusse R Valley

Remarks.—345. Saxony-Silesia. Cut-stone ring, ’bridge over Aar near Nydeck castle.

Durham Tunc. Ry. H. = isi' about.

362. Fell 1822. See No. 351.

1857-64 Meigs

213.0 39-4 26.2 220.0

59-0

57-3

1903

1380

1899-03

1905

Under Barnabo Visconti Séjourné Leibold

t23°.o

251.0

277-7

295.3

43-3

87.8

101. 56.

15

6.9

4.2

4-9

4.0

4-7

4.9

10.2

14.7

Ry.

p. 225

.x. G. June, 1891 ; B. Oct.T9i,Dec.7,’93 R. 1889, p. 584 N. Oct. 4, 1902 B. Sept. 18, 1902 B. Jan. 18, 1902 B. Dec. 26, 1901 B. Dec. 7,    1893,

p- 447

A.

365

366

367

368

369

370

6.2

7.2

4.0

7.1

6.6

134-3

246.0

133-6

.031

.020

.030

344.5 .044

20.3

17-3

18. i

52.5

Granite

ring

Granite

Aq. and H.W.

H.W.

Hard slate

H.W.

H.W.

K. April 19, 1867; N. July 29, 1899 N. Oct. 17, 1903 B. Dec. 7, 1893:

C. 1855

B. Feb. 27,1902 B. Aug. 17, 1905, p. 156

371

372

373

374

375

* Maximum.

350. First stone

_    356.

361. Glasgow and S. W. Ry. = 124'.5.    364. Limoges-Brive.

348. Rock foundation.

353. Rock foundation. 354. Austrian State Ry. 358. H. says five arches.

363. On curve P— 2660'. Clear H.

+ About.

374-

DAM

P(

-i 1

SKIE]

PLAIN CON

CRETE ARCHES.

Number.

Name.

Place.

Over.

Date.

Engineer.

No. of Spans.

Span.

Rise.

Thickness at Crown.

At Spring-

ing.

Curve.

Radius at Crown.

Thickness at Crown = tg.

EL.

Width, Face to Face at Crown. Thickness of Piers at Springing.

Class of Bridge.

Reference.

Number

rO

Fem St.

Casey R. Bridge No. 41

Cannington Viad. Ewarton Br. Lochnanuamh Viad.

Finnan Viad.

O'-bn 4k C/j

Bridge No. 242

00 a

Bridge No. 163

Ashtabula Br.

to    to

4k    Gj

to to O'

St. Ana Viad.

Morar Viad.

W. Hartford, Conn., U.S.A Trout Brook

Las Marias, Porto Rico Casey R. Sharpsville, Penn., U.S A. Pine Run Cheltenham, Mo., U.S.A. Des Peres R. Bet. Manati and Aales, Quebrada R.(?) Porto Rico

Mansfield, Ohio, U.S.A.

Bet. Santiago and El Caney, Cuba Cannington, England Jamaica, W. I.

Scotland

Scotland

Scotland

Washington, D.C., U.S.A.

Northampton, Pa., U.S.A.

Adjuntas, Porto Rico Salt River, Ariz., U.S.A.

W. of Cincinnati, O.,

U. S. A.

Thebes, Ill., U. S.A.

Concord, Mass., U.S.A.

W.of Cincinnati, O., U.S.A.

Ehingen, Wurtemberg

Ashtabula, Ohio, U.S.A.

Near Rechtenstein, Wurtemberg

Plano, Ill., U.S.A.

Near San Leandro, Cal.,

U.S.A.

Riverside, Cal., U.S.A.

Scotland

San Juan R.

Ravine

Amabol Bum

Finnan Valley

Broad Branch HokendauquaCrk.

and Highway Small stream Dam spillway Tanner’s Crk.

Bank of Mississippi R.

Assabet R.

Tanner’s Crk. Danube R.

Ashtabula R. Danube R.

Big Rock Crk. S. Leandro Crk.

Santa Ana R.

Morar R. & H.W.

1902- 3 Crawford

1899(1) Buel

1900    Geer 1904(1) Purd on

899( 1) Buel

i9°4(?) Keith 1902 Rockenbach

1900-02 Pain 1881-82 Bell 1899t Simpson and Wilson

1899t Simpson and Wilson

1899t Simpson and Wilson

1901    Douglas

1900    Thompson

1899(1) Buel 1905-

1903- 4 Kittridge

Noble and Mo-jeski

1901    Worcester

1903-04 Kittridge 1898

1902

1904

1893

1903-4

1901

1902-04

1898—9

26.0 10.0 30.0 30.0 36.0 39-4

40.0

40.0

Beckwith

Braun

Breckenridge County Surveyor Hawgood

Simpson and Wilson

10

21

.    gâjlgjlgr

So .0 50.0 50.0

50.0

50.0

50.3

51.8

34-0

55-0

59-0 60. o 40. o 65.0

66.0

68.0 69.0 66. o 74.0 74-4

75-0

81.3

86. c

38.5

90. o

5-0

3.0

i S •⁰

IS.o 18.0 9.8

7 • S ii -S

16.0

22.2

25.0

t7 -° 13.5

il -3 11 . o

OJ K) u N O O'

o\ o o

17.0

37-0

8.2

26. o

43-0

24.0

2.0

1.3

2.5

2.0 i 3

2.5    2.5

1.5

2.5

2.0

2-5

cs cs ot (J0<0

19.5    2.0

S-S 1-3

15-0    2.5

15-0 2.5 18.0

• 103

. 236 . 167 . 167

T30 - ° 4-0

7S • 5

87.0 s.2

H.W. N. April 25,1903 H.W. Cem., Jan. 1902 T. Nov. 16, 1900 B. Nov. 3, 1904 Cem., Jan. 1902, p.

Ry.

Ry-

H.W.

.8

2.5

3.0

ti.8 tó.o

3-5

2.8

1.5

ti -5

2.7

2.3

3-3

3-5    6.0

2.3    3.0

t6. s

2. i

3-0

3.0

3-5

3-0

til

30.0 1.5

2.5

25.2    2.0

25.0    2.5

+ 1.8 31 • S 3-5 28.0 2.8

i -S

I -s

2.7

20.0    2.3

32-5    3-3

64-0    3.5

2-3

37.0 6.s

2. i

.050

.079

. 100

.hi . 100

.115

.102

■ 055

. 176

40. o

20.0 8.0

16.0

16.0 6.0

26. o 42. o

10.3 6.s

T33-0 9-3

28.0 12.0

t3S-û

33.0 12.5 24.6 6.6

I4S-0

H.W.

H.W.

328

N. Feb. 18, 1905 B. Jun. 13,1903, p.

Light Ry.

Ry.

Ry.

Ry.

Ry.

H.W.

Ry.

H.W.

H.W.

Ry.

Ry.-

H.W.

HAY.

Ry.

549

N. Oct. 21, 1905 B. July 27, 1893 B. Feb. 9, 1899, p.

85

B. Feb 9, 1899, p.

85

B. Feb. 9, 1899, p.

8s

N. Jun. 8, 1901, p.

541

Cem., Jan. 1902 N. Oct.

N. Mar. 5, 1904, p.

14, i9°5

292

T. Jan. 9,’03, p. 21 ; B. Nov.20,1902 Municip. Engineerring, March,1902 N. Mar. 5, 1904 B. Jan. 9, 1902, p.

35

OOO **4Qs Cn 4k OJ to

15'-

20'

43-0 3.0 61.5    3-0

43-S 3-S

3-0

.070 . 048

. 081

44.0

fso.o

H^Y.

Ry.

Ry.

T. Jan. 27, 1905 Y. 1898

N. Jan. 2, '04, p. 18 B. Aug. 27,1903, p. 174

N. Sept. 9, 1905, p. 284

B. Feb. 9, 1899, p.

85

o    M    IN    PO ■»*    too    00    &    O M    NPO    Tti/>vO

M    M    M    MM    MMM    M    M    M N    Wtt    CN CN    CN    CS

OJ    OJ Oj to to

to    -H    O    O 00

Near Imnau, Germany Pittsburg, Penn., U.S.A.

Near Tarvis, Austria

Thebes, Ill., U.S A.

Near Mechanicsville, N. Y., U.S.A.

Eyach R.

SiLver Lake

Schlitza R.

Bank of Mississippi R. Anthony Kill s

1896

1905

t¹903

1902

Leibbrand

Brown

50.0 20. o 98.4 100.0 80. o 100.0

9.8 SO .0 40. o 10. o

1.5

4.0

4.0

Noble and Mo-jeski

100. o

100.0

50.0

3-6

2-3

2-3

So.o

4-5

«    cs

<0    <0

5° • ° 40. o

50.0

1-5

4.0

3-5

. 080 .088

. 090

8.2

54-0 12.0

28. o

H.W.

Ry.

H.W.

Ry.

El. Ry.

G. 2 Tri., 1898 N. May 6,

528

Engineer. April 22, 1904, p. 424 an. 9,’ö2,p. 21 ; ov. 20,1902 B. Nov. s, 1903, p. 408

1905, p.

T.

B.

00 Ov O    m N

<N IN    fO PC fO

* Maximum.

Remarks, i. Cost $4050.    3. Skew, is⁰ o'. Penn. Rv. 4. 1:6 “chats” St I &■ «8 F

6. Three cast-iron hinges. 7. Contract price, $31,000.    9. On curve ,980'R Jamaica

12. On curve 1200' 7?    L=i2₄8'; H, = ioo' 13 Pebble-faced. Cosi

?fÆJ’ Ex-metal used m radial planes. 15. 1600'above seadevel. 12 fill over key. 17. Big 4 Ry., Chicago Div. 18. Approach to

Ry. _

Govt. Rys.

14. Three tracks. C. R.R. 16. Very flat arches ; about

t About.

Thebes Bridge. 20. “Big 4” Ry., Chicago Div. 21. Cost $21,000.    22. L. S. & M. S. Ry.

Four tracks. 23. Three lead “hinges.” 24. C. B. & Q. Ry. Two tracks. 25. Skew, io° Cost $25,840.    26. One track. S. P., L. A. & S. L. Ry. 27. Mallaig Ex. of W. Highland Ry.

28. Three granite^‘hinges.” 29. Penn. Ry. Four tracks, 5° curve. 30. Three steel “hinges.”

31. Approach to Thebes Bridge.

TABLE IL—DATA FOR ABOUT 500 ARCH BRIDGES

PLAIN CON

ARRANGED ACCORDING TO SPAN—{Continued).

CRETE ARCHES.

Number.

Name.

Place.

Over.

Date.

Engineer.

No. of Spans.

Span. Rise.

Thickness at Crown.

At Springing.

Curve.

Radius R at Crown.

Thickness at Crown = to.

Width, Face to Face at Crown. Thickness of Piers at Springing.

Class of Bridge.

Reference.

Number.

to

Danville Arch

2 miles from Danville, Vermillion R. Ill., U.S.A.

Near Mittenberg, Germany Main R.

1905

“Big 4“

100. o

40.0    4.0

51-3 4-0

.088

42.0    15.0

Ry.

N. Mar. 3, 1906, p. 33

338

1898-99 Fleischman and Bosch

80. o 112 . o

107.3

Grand Maître

16th St. Borrowdale

Fontainebleu Forest, Fra’e Valley Kirchheim, Wurtemberg Neckaar R. Washington, D. C., U.S.A. Piney Branch Scotland    Bor’dale Bum

O' O w 04    tO

tO    rf rj-    rf

Coulouvrenière Geneva, Switzerland    Rhone R.

Big Muddy    Near Grand Tower, Ill., Big Muddy R.

U.S.A.

Inzigkofen, Wurtemberg Danube R. London, England    Thames R.

Washington, D. C., U.S.A. Rock Creek

Inzigkofen

Vauxhall

Conn. Ave. Br.

1869 Belgrand 1898t

1905 Douglas 1898-99 Simpson and Wilson

1895    Butticaz 1901-03 Parkhurst

1896    Leibbrand 1899 Binnie

■t*.

GJ to to

102.3

*115.8

124.6

125.0

127-5

30.0

16.4

17.5

14.8

18.3

13-8

19.0

ti9-3

19.0

39-0

22.5

3-6

2-5

2.5

2.3

1.3

5.0

4.0

20. o

131 • 2

140.0

18.2

30.0

3-0

7.0

2.8

3-0

Par

41.7    3-6

2.5

2.5

2.3

ti -3

5.0

4.0

3-0 120 . O

127-3

I167.0

3-0

7.0

.086

23.0 10.2

ti8. o 25.0

.024

.042    50.6

H.W. B. July 25,1901,p. 34

Aqued’t

H.W.

H.W.

Ry.

H.W.

Ry.

61

K. Oct. ’69, p. 275 B. Mar. 9, 1900 B. Nov. 16, 1905 B. Feb. 9, 1899, p.

85

Y. 1898 B. N

35

36

37

38

39

mO    t'-

■'t    '<*•

Munderkingen, Wurtem- Danube R. berg

Near Oviédo, Spain    Nalon R.

Neckarhausen, Germany Neckar R.

Ulm, Germany    Ry. Yards

1889-1906 Morison & Biddle Douglas Leibbrand

1893

Proposed

1903t Leibbrand

1905t

cn to m m

141 .0

144.6

130.6 150.0

82.0

164:0

14.4

fi 8.6 120 . O

75-0

41.0 16.4

165.0 165.0

215.0

18.8

13-5

2.3

3-9

3-9

5-0

3-3

3-3

3-7

2.8

3-9

3-9

3-6

3-7

3-7

75.0

41.0

2.3

3-9

3-9

5-0

3-3

3-3

3-7

2.8

. 067

.080

fi2.5

+84.0

52.0

20.0

t20.2

fi7.o

15.8

+46.0

ov. 12,1903,p.

423

B. April 22,1897

N. Feb. 25, 1899 N. July 8, 1905, P-30

B. June i, 1905 H.W. G. 3 Tri., 1897, P-356

H.W. B. Sept. 26,1901 H.W. Engineer, Dec. 30, 1904, p. 650

H.W. B. March 15, 1906

H.W.

H.W.

40

41

42

43

44

45

46

47

* Maximum.

Remarks.—34. Three lead “hinges.” 35. Paris water-supply from Vanne. 36- Three lead “hinges.” 38. Mallaig Ex. of W. Highland Ry. 39- Three “hinges.” 40. Two tracks. Ill. Cent. Ry. 41. Three cast-iron “hinges.” 42. Three “hinges.” 44. Three steel “hinges."

t About.

45. Three “hinges.” 46. Three cast-iron and steel “hinges.” centre of hinges i87'.o; rise centre to centre of hinges 18'.7; cost $45,000.

47. Three “hinges;” centre to

TABLE II.—DATA FOR ABOUT 500 ARCH BRIDGES

REINFORCED CON

ARRANGED ACCORDING TO SPAN—{Continued).

CRETE ARCHES.

Number.

Name.

Place.

Over.

Date.

Engineer.

p.

'S

Span.

Rise.

Thickness at Crown.

At Springing.

Curve.

Radius R at Crown.

Kind of Steel.

Per Cent Steel at the Crown.

I Width, Face to Face at Crown Thickness of Piers at Springing.

Class of Bridge.

Reference.

Number.

Ridgewood Ave.

Linwood Ave.

12

S.

Jefferson St. McKinley Arch

San Miguel

Elkhart, Ind., U.S.A. La Rue H. W. Vulcanite, N. J., U.S.A. Highway Rock Rapids, la., U.S.A. Ravine Ridgewood, N. J., U.S.A. Br. of Saddle C. Marion Co., Ind., U.S.A.

Waldwick, N. J., U.S.A. Stream Mahwah, N. J., U.S.A. Stream Crystal Lake, N. J., U.S.A. Stream Delaware Co., Penn.,U.S.A Stream Wayne Township, N. J., Stream U.S.A.

Ridgewood, N. J., U.S.A. Saddle R.

W. Edwardsville, Kan., Mission Creek U.S.A.

Indian Creek, Ill., U.S.A. Indian Creek Ocnomowoc, Wis., U.S.A. Lake Battle Creek, Mich., U.S.A. Kalamazoo R. St. Louis, Mo., U.S.A. Des Peres R. Sorsogan, Philippines Stream Manila, Philippines    Estero S. Miguel

Albion, Mich., U.S.A.

1903

1905-6

1894

1897 1899 1898?

1898 1898 1905 ? 1896

1895

1904?

1903

1899

1899

1902

1905

1905

1898?

Osgood M. A. C. Co. M. A. C. Co.

M. A. C. Co. M. A. C. Co. M. A. C. Co.

M. A. C. Co.

M. A. C. Co. Walter

30-0

30.0

30-0

30-0

32.0

32.0

32.0

32.0

33-0

35-0

9.0

to-o

3-3

6.4

6.6

2.5

39.0

f" Jt

4" o

• S3

Ry.

3-°

3-2

3-2

6.4

16.0 3 • S

40.0    8.0

4O.O +20.0

0.9

0.9

0.9

0.9

0.5

x .0 2.2

1.8

2. o 2.0 3-3

0.9

2. o

10.0

1.02

.80

32.5

17-5 26. o

Ry.

B. July 14 1904 N. Sept. 9, 1905

.85

• 93

. 80

hi

.82

24. o 22.0 26.0

25.0

i"J

.66

.70

20. o 38.0

20

21

22

Como Park Como Park Mount St.

23

24

25

Salem St.

Florida Keys Viad. Lamington Br.

26

St. Paul, Minn., U.S.A. Rapid Transit Ry. St. Paul, Minn., U.S.A. Rapid Transit Ry. Atlantic Highlands, N. J., Grand Ave.

U.S.A.

Carbondale, Penn., U.S.A. Lackawanna R. Florida Keys, U.S.A. Salt Water Marysborough, Queens- Mary R.

Beargrass Creek

land

Louisville, Ky., U.S.A.

1904

1904

1895-96

1896

1905-

1896

1897

Smith

Hall

Reiseger Phillips Stevens White & Co. Keepers & Thacher Wilson Wilson M. A. C. Co.

M. A. C. Co.

Brady

40. o 42. o 42. o 45.0 45.0 45-9

46.7

SO.o

So.o

50.0

50.0

50.0

50.0

20.

6.7

2.5

o.S

ił" J

6.

6-3

0.8

0.8

2.5

2.5

3-0

i" J

Var.

43.0

33-Q

+ •73 + 99 i. 11

32.3

42. o 66.0 45-0 +33-0

34-0 + 28.0

+ 17.0 fx 7 • o 25.0

H.W. N. July 8, 1905 H.W. B. Sept. 21, 1899

Foot-bri’e N. Dec. 3, 1904 Foot-bri’e B. April 6, 1905

N. August 22, 1896

H.W.

1900

M n T+ \r>\0    00 O' o

O OCW

20

21

22

8.

2.8

25-

4-

3-7

25-0

i" J

i • 25 .76

.71

55.0

130

22.7

H.W.

ILW.

X.

27

28

Mich. Cent. Ry.

Decatur Township, Ind., Goose Creek

Southern Bvd.

U.S.A.

Detroit, Mich., U.S.A.

Keepers & Thacher Nelson

50.0 XI.

50.0

S-o

41.7

3"Xi"

+6o.o

H.W.

H.W.

29

30

31

32

33

34

Arch St. Jackson St. Sixth Ave. Goat Island

Hyde Park, N. Y., U.S.A. Crum Elbow C.

Plainwell, Mich., U.S.A.

Paterson, N. J., U.S.A.

Kalamazoo R. Passaic R.

Jackson St. Carbondale, Penn., U.S.A. Lackawanna R. Niagara Falls, N. Y., Niagara R. U.S.A.

Newark, N. J. Carbondale, P

U.S.A.

1895-6

1897

1903

1903

1904 1896

1900-1

Keepers & Thacher M. A. C. Co.

50.3    9-

i -3

7-3

38. 2

3-75

109.9

Ry.

33

36

37

38

39

40

41

Castle Eichom Eighth Ave. Franklin Br. Montgomery St.

Clifton, N. J., U.S.A.    Passaic    R.

Route Neutra, Hungary Route Nymphenburg,

Wurtemberg Mähren, Austria    Ravine

Carbondale, Penn., U.S.A. Lackawanna R. St. Louis, Mo., U.S.A.    Des Peres R.

Jersey City, N. J., U.S.A. Street

1903

1892

1898

1896

1897-98

1895-96

Courtwright Schwiers Osgood M. A. C. Co. Buck (State) Waldo (Con.)

Schwiers

Venier M. A. C. Co. Dean

M. A. C. Co.

33.0 26. o 54-0 54-2 54-3

54- 6

55- O

s°.5

+55.°

55-8

56.7

7-

7-

8.

10.

5-

10.

9-

3-

3-

5-

6.

0.7

0.9

2. o 0.8 x .0

1.9

I . o

0.9

I . o

2-3

1.7

4.2

3-0

2.2 3-2

3.6

i ■ 23 i. 00

48.4

32.8

48.0

ił" o

3-6°

1-37 x . 12

• 63

.66

2.40

1.02 I . 40 1.02

17.0

+ 24.0 +43-3 +32.0 48.0 +40.0

+ 30.0

+ 197 +32.8

21.3 49.0 32.0 83-3

H.W.

6.0

8.0

H.W.

H.W.

Ry.

H.W.

H.W.

H.W.

H.W.

H.W.

H.W.

H.W.

H.W.

H.W.

B. Oct. 19, 1905 N. Nov. 17, 1900 B. Feb. 14, 1901 K. & T. Blues

Cement, Nov. 1901

N. Sept. 28, 1895 T. March 3, 1899 B. Nov. 10, ’ 98; X.

B. May 12, 1904 N. Sept. 10, 1904 N. August 6, 1904

B. Dec. 6, 1900

23

24

25

26

27

28

29

30

31

32

33

34

N. Sept. 10, 1904 G. ist Tri., 1904 G. ist Tri., 1904

Öst. Monat.Baud’st

X.

X.

X.

33

36

37

38

39

40

41

Remarks.—1. L. S. & M. S. Ry. I beams, 36" centre to centre, type. 6. Nine 5" 9.75-lb. I beams centre to centre. 0 XT;— ■"---

t About.    + J = Johnson bars.    O = Round bars,

tre to centre. 23. Nineteen 7" 15-lb. I beams; 36" centre to centre. 24. Arches vary in s but all same type. 25. 41.25-lb. rails; 2' centre to centre. 27. Melan type.

4" X 4" X Y' angles; 22$" centre to centre. 29. Five 7" is-lb. I beams, 36" centre to centre ; fivr 5" 0.75-lb. I beams, 36" centre to centre. 30. 4"-6” channels, 1.9' centre to centre. 32- Ce R. R. ôf N. T-, two tracks. 33. Sixteen 7" 15-lb. I beams; 36" centre to centre. 34- Thacher type. 35. See No. 31.    36. Wiinch type. 37- Monier type. 38. Melan type; four I

beams. 39. Sixteen 7" 15-lb. I beams, 36" centre to centre. 4°- Eleven 8" 18-lb. I beams; 36" centre to centre. 41. Twenty-one 7'’ 15-lb. I beams 36" centre to centre; two elevated tracks.

* Maximum.

2. C. R. R. of N. J.; two tracks. 3. Six 4" 7.5-lb 5" 9.75-lb. I beams; 36" centre to centre. 5- Melan centre to centre. 7. Nine 5" 9.75-lb. I beams; 31'“ centre to centre. 9. Skew. Phila. R. T.

xi. Seven 5" 9.75-lb. I beams 36" C. I. & St. L. Ry., Short Line; two tracks. 15. Melan type; twenty-one 6" I beams. 17. Various type. 20. 5" 9.75-lb. I be""'-⁻

4. Nine 5

v.,j * „___ns; 34" centre __

8. Nine 5" 9.75-lb. I beams; 36    _—

10. Nine 5" 9.75-lb. I beams; 35" centre to centre.

..cuo ™    12. U. P. Ry. ; two tracks.

14. Flat bars and expanded metal. : „    ..    .    .

sizes and shapes of bars. 19. Thacher type. 20. 5" 9.75-lb. I beams; 38" centre to centre inst. Four angles, 2"X2"Xi"; 38" centre to centre ( ?).    22. Eight 6" 12.25-lb. I beams; 36" cen-

Co.

centre to centre

13-

28.

pan

Feu,

REINFORCED CON

CRETE ARCHES.

Number.

Name.

Place.

Over.

Date.

Engineer.

No. of Spans.

Span.

Rise.

Thickness at Crown.

At Springing.

Curve.

Radius R at Crown.

Kind of Steel.

Steel at the Crown.

Width, Face to Face at Crown.

Thickness of Piers at Springing.

Class of Bridge.

Reference.

42

43

44

45

46

Herkimer Yiad.

47

48

49

50

51

Bloomfield Ave.

52

53

54

56

57

58

59

Cedar R. Meridian St. Illinois St. Wealthy Ave.

60

61

62

Hamilton St.

63

64

65

66

Rock Creek Soissons

Troy, N. Y., U.S.A.

Route Ebhausen, Wurtemberg

Vigneux, France Italy

Herkimer, N. Y., U.S.A.

Jacksonville, Fla., U.S.A.

Auch, France Military Road, San Juan, Ponce, Porto Rico Newark, N. J., U.S.A. Cincinnati, Ohio, U.S.A.

Trinidad, Col., U.S.A. Copenhagen, Denmark Route Painpardu, Belgium La Salle, Ill., U.S.A. Waterloo, la., U.S.A. Indianapolis, Ind., U.S.A. Indianapolis, Ind., U.S.A. Grand Rapids, Mich., U.S.A.

Wabash, Ind., U.S.A.

Hartford, Conn., U.S.A. Hyde Park, N. Y., U.S.A. Polasky, Cal., U.S.A. Route Bade, Austria Washington, D. C., U.S.A. Soissons, France

Wynant’s Kill

1897 Kenney 1891

Dora R.

W. Canada Crk.

McCoy’s C. & R.R.

Gers R. Guayo R.

Park drive Park drive

Purgatorie R. Railway

Gorge Cedar R. Fall Creek Fall Creek Grand R.

Creek

Park R.

Crum Elbow C. S. Joaquin R.

Rock Creek L’Aisne

1900

1902

1902-3 Osborn E. Co.

1903-4

1899

1900-01 Thacher

1904 Reynolds 1894-95 M. A. C. Co.

1905 Hibbard 1879

1899

1905 Strauss 1902-3 Z

1900 1900 190

Jeup

Jeup

Anderson

1905

1898

1897

1905

1900

1901—

1902

M. A. C. Co. M. A. C. Co. Leonard

Beach

Riboud

ir

10

65.0

65.6

65 • 6

65.6

66.0

62.0

*68.0

68.9 70. o

70.0

70.0

70.0

71.7

71.8 72.0 72.0 74.0 74.0

75.0

75-0

75-0 75-0 75-0 77-4 80. o 80.7

8-5

8.2

I . o

0.7

1.23    27.0

ti3- J

H.W.

H.W.

X.

G. ist Tri., 1904

42-

43

14.8

6.6

14.0

12.0

7.0

6.6 2-7. o 1-7-5 8.5 10.o

7.0

8.5

9.2

7-5

7-2

9-5

9-5

14.0

18.0

7-5

14.7 ti i . o 7-7 15-0

7-9

1.8    4.5

1.8    4.5

1-5

i .0

1.3

4.0

i. 2 0.9

1.3

t2 . O I . 2

3-0 i . 2

t2 .O

2.7

40-5 4Ó. o

106.3

77.2

ił" Tt

ii" T

I -3

1.3

1.5

1.3

1-3

1.5

1.5

1.0

.96

.96

ti3-1 +46.6

8.0

52.0 t7-o

20. o

H.W. H.W. E. Ry.

H.W. & E. Ry. H.W. H.W.

G. ist Tri., 1904 G. ist Tri., 1904 N. Feb. 22,1903,p.

240

T. July 3, 1903, p.

428

G. ist Tri., 1904 N. Aug. 3, 1901, p.

44

45

46

47

48.

1.8

1.8

3-3

4-5

fi .9

9.0

i Xi" Ts 2i"Xf"

3C

Par

SC

97-5

62.5

i"Xii" 1" X 3"

i" J

+65.0 t.8o 32.s

2.67 fÓ5.0 7.0

t39-4

+5.0

+46.0

1.3 T70.0 8.0 1.3 t6o.o 8.0 +39.0

+32.0

H.W.

H.W.

98

49

H.W.

Foot-bri’e

H.W.

Foot-bri’e

H.W.

H.W.

H.W.

H.W.

H.W.

N. Aug. 12,1905,p.

M.    A. C. Co., B.

Oct. 3,1895

N.    Feb. 10, 1906 B. July 21, 1898 G. ist Tri., 1904 B. Sept. 21, 1905 N. Feb. 13, 1904 B. April ii, 1901 -B. April 11,1901 B. Mar. 22,1906, p.

321

B. Mar. 13, 1906 N. Dec. 2, 1905

50-

5i

52-

53

54

56

58

59

60'

i. 17 .69 i. 12

.90

49-3 20. o 19-5 t39-4 27. o 45-9

5-0

4.9

H.W. H.W. H.W. H.W. H.W. H.W. &

X.

X.

61

Ry.

N. Feb. 24, 1906 G. ist Tri., 1904 B. Aug. 14, 1902 G. ist Tri.,

G. 3d Tri., 1903, p.

1904

62-

63

64

65

67

68

69

De l’Empereur Fabriano Viad.

70

71

72

Seeley St

73

Haider

Sarajero, Bosnie Italy

Rt. Payerbach, Austria Brooklyn, N. Y., U.S.A. Austria

Gr’d Rapids, Mich., U.S.A

Lenne R.

Prospect Ave. Białka R. ?

Grand R.

1904

1897

1905

1900

1903-4 Foot 1894

1903-4 Anderson

74

75

Main St. West St.

76

77

78

N. Sixth Ave.

Dayton, Ohio, U.S.A. Paterson, N. J., U.S.A.

Yorktown, Ind., U.S.A.

Papigus, Italy

Des Moines, la., U.S.A.

Great Miai Passaic R.

mi R.

Stream Nera R.

Des Moines R.

1902-3 Turner 1897-8 M. A. C. Co.

1905 ? Luten

1901-2? Z

80.3

79.6 82.0

60.7

83.2 84.9

3°.2

85.3

85.3

86.3

87.0 83.0 79.0 *88.0 89.0

88.5

95-0 95-2 too .0

8.2

7-9

8-3

26.9 13-1

i .0 i . o i. 6

1.    6

i .0

2.    o

9.0 9.0 2.2 i. 6

tg.o

3-3

H.W.

47

5-9

1.5

8.5 20.6

t3-5

4.8 fio. o

+1.5

8.0 11 . o

9-5

9-5

11 . i

2.8

H" J

ii"T

t.76

1.5

3.0

1.16 i. 22

t37 • 6 t32.o

ti8.o

53.1

64.3

H.W.

H.W.

9.0

H.W.

H.W.

H.W.

H.W.

1.3

1.3

5-5

5-5

2.3

1.8

3-3

i" 0

1-37

1.37

54-0

50.8

8.8

8.5

H.W.

H.W.

G. 4thTri., 1905,p.

G. ist Tri.,

N. Dec. 9, 1905, p.

645

G. ist Tri., 1904 B. Dec.

G. ist B. Dec. i, 1904,p. 489

B. May 19, 1904

1904

fi8.o

. 31, Tri.,

1903

1904

67

68

69

7 °'

71

72-

73

X.

74

r .9

4.0

i .9

4.2

4-3

68.8

79.2

93-6

42.7

10. o

H.W.

B. March 16, 1899 B. May 11, 1905 G. 4th Tri Cernent, July, 1902

i-, 1905

75

76

77

78

* Maximum.

f About.

î T = Thacher bars.

0 = Round bars.

J=Johnson bars.

Remarks Piketty type, and Thacher bars, tracks, rails.

42. Nine 8" 18-lb. I beams; 36" centre to centre. 43. Monier type. 44 45- Hennebique type. 46. U. & M. V. Ry. Two tracks. 47. Melan ribs 48. Bonna type. 49- Thacher type. 50. Melan type. Two E. Rv. 51. Eleven 9" 2i-lb. I beams; 36" centre to centre. S3- Five 18.8 lb. (per foot! 54- Hennebique type. 55. Two nbs. In Deer Park. 36. Thacher type. 57. 10" ²5-lb. I beams; 36 centre to centre. 38. 10" 25-lb. I beams; 36" centre to centre. 60. Kahn

bars    61. Seventeen 9" 21-lb. I beams; 36" centre to centre. 62. Seven 9" 18-lb. I beams.

63. Spandrel wall tied to ring. 64. Hennebique type. 65. Four 3" X 3" X 6-lb. angles; 33, centre to centre. 66. Hennebique type. 67. Large arch has three hmges. 68. Wunch type. 69. Total length = 3 54'-    70. Melantype. 71. Skew. 72. Monier type. 74- Four

angles 2*,'X 2*" XAs"' 36" centre to centre. 75- Seventeen 10" 25-lb. I beams. 35i centre to centre. 77. Five ribs; four angles latticed. 78. Melantype; four angles latticed.

TABLE II.—DATA FOR ABOUT 500 ARCH BRIDGES

ARRANGED ACCORDING TO SPAN—(Continued).

REINFORCED CON

CRETE ARCHES.

Number.

Name.

Place.

Over.

Date.

Engineer.

No. of Spans.

Span.

Rise.

Thickness at Crown.

At Springing.

Curve.

Radius R at Crown.

Kind of Steel.

Per Cent Steel at the Crown.

Width, Face to Face at Crown. Thickness of Piers at Springing.

Class of Bridge.

Reference.

Number.

79

80

Icy Glen François-Joseph

81

82

Green Island

Stockbridge, Mass., U.S.A. Buda-Pesth( ?), Austria-Hungary Laibach, Austria Niagara F’lls, N.Y., U.S.A.

Housatonic R. Danube R. ?

Laibach R. Niagara R.

189s

1900

1900-1

1900-1

M. A. C. Co.

Melan

Buck (Con.) Waldo (State)

100. o 108.2

108.3

I 10. o

10. o 14.4

14.6 ii -5

0.8

1-7

3-2

2.5 + 2.2

5-9

123.c

6"XÎ"

2.30    7 ■ S

+45-9

+5°-°

.63    40.0 13.s

83

Third Street

Dayton, Ohio, U.S.A.

Great Miami R.

1904-

Turner

Wayne St.

Peru, Ind., U.S.A.

Wabash R.

1905

Luten

IO3.5

110 . O 100.0 90.0 80. o 100

10. o

14-3 13-3 il -3

9-7

3-3

6.3

.66

.69

.69

.91

+62.0

Foot-bn’e    X.    B. Nov. 7, 95

H.W.    G.    ist Tri., 1904

H.W.    B.    July 16, 1903

H.W.    B.    Dec. 6, 1900

N. Feb. 16, 1901, p. 146

O' 0 M «

t^OO 00 00

32 ?

ii .0 10.o 9.0

H.W.

H.W.

T. Mar. 4, 1904, p.

154

B. Mar. 29,1906, p.

95

85

87

Lake Park

Jacaquas R.

Washington Ave.,So. Y-Bridge

Portugal

Milwaukee, Wis., U.S.A. Yellowstone Nat. Park, U.S.A.

Military Road, San Juan-Ponce, Porto Rico

Lansing, Mich., U.S.A. Zanesville, Ohio, U.S.A.

Pena R.

Ravine

Yellowstone R.

Jacaquas R.

Grand R. Muskingum R.

1901

1905

1903

1900-1

1902

1900-2

Turneaure

Chittenden

Jackson

Collar

Landor

85

75

.s

II4-

118.0

120.0

120.0

100. o 120.0 122.0

81.0

14.4

18.0

15.0

S-o 2. o

5.0

4.0

"X3"K

12.0    2.3

u -3

1.9

347

00    00

4k    Gj

3C-

226.0

167.2

₄"Xi"

ii .8

14-0 ti-39 17-5

.63    20.0 12.o

. 80

Tramway G. ist Tri., 1904 N. Nov. 25,1905 B. Jan. 14, 1904, p.

25

N. Aug. 3, 1901 B. Aug. i, 1901, p.

H.W.

H.W.

H.W.

U» 00

00 00 00    00

14-5

ii.5

14-5

10.9

2.5

1-5

3"&S"

Xi"

54-0

43-0

H.W. Cement, Mar. 1902 H.W. N. Mar. 1, 1902, p. El. Ry.

194

91

92

Kansas Ave.

Route Wildegg, Switz Topeka, Kan., U.S.A.

Kansas R.

1890

1896-98

M. A. C. Co.

99.0

122.0

+25 - °

6.3

ii .4 18.9

+0.8

93

Park Ave. Schwimmschulbrücke

95

97

St. Pierre Hollow

99

100

101

Gruenwald

Newark, N. J., U.S.A. Steyr

Playa-del-Rey, Cal.,U.S.A. Route Waidhofen, Austria Schenley Park, Pittsburg, Penn., U.S.A. Chatellerault, France

Route Bormida, Italy Decize, France Munich, Bavaria

Park

Stream

190s

1897+

Reynolds

St. Pierre Hollow Vienne R.

1906

Proposed

De Palo

110. o 97-5 132.0

138.4

16.3

14.6

16.2

9.4

Keepers and Thacher

146.0

144-3

150.0

18.0

Loire R. Isar R.

1900

1902

1904

Mörsch

164.0

131- 2 167.3 183.7

230.0

29.8

15.8

13.2 16.7

15.3

42.0

1.8

1.7

2.0

2.0

1.8

1.4

1.9

2.5

5-0

2.3

8.0

+3.0 +2.6

2.5

CO 00 PO 10 lO

+ 12.8 3Ó.0

+ 72.0 19.7

H.W.

H.W.

H.W.

H.W.

G. ist Tri., 1904 M. A. C. Co. B. April 2, 1896

4 angles

i.i"0

.04 84.0

H.W.

H.W.

N. Aug. 12,1905 Z. Oe. Ing. u. Arch.

Ver., Dec. 23/98 N. Mar. 31, 1906 G. ist Tri., 1904 K. & T. Blues

fi9-o

H.W.

34-4 .10 26.3

H.W.

H.W.

H.W.

G. ist Tri.. 1904

B. April 10,1902 G. 4th Tri., 1905 B. Feb. 20, 1905

000 o 000    00

M o O 00 0\Crt 4k Gj

* Maximum.

Remarks.—79. Four 7" 15-lb. I beams; 28" centre to centre. 80. Three “hinges.” Lattice ribs. 81. Three “hinges.” Fourteen lattice ribs. 83. Four angles, 2$-"X2$"XxV'. 34" centre to centre. 85. Hennebique type. 87. Four 2Y'X 3"XY' angles; 30" centre to centre. 88. Thacher type. 89. Melan type. 90. Thacher type. In plan, Y-shaped.

+ About.

91. Monier type. 3 " X 3" X i"; 36" 96. Monier type, type.

t O = Round bars.

K = Khan bars.

92. Four angles, 3"X3"X7.2 lbs. Twelve ribs.

Thacher bars. 94. Melan type. Four angles, 3"X3"X7.2 lbs.

centre to centre.

97. Twenty-seven ribs, ioi. Three steel “hinges.”

93. Four angles 95. Foot-bridge 98. Hennebique

KEY TO REFERENCES.

Symbol.

Name of Periodical or Book, etc.

Address.

^'-HHWO^Wöntd>

Civil Engineers’ and Architects’ Journal . . . Engineering News and Am. Ry. Journal. . .

Weale’s Bridges.........................

Penn. Ry. Co.’s Blues...................

Wm. H. Brown, Ch. Eng’r Penn. Ry. Co.. . Construction des Viaducs, Tony Fontenay. .

Annales des Ponts et Chausses............

Mahan’s Civil Engineering...............

Masonry Construction by Baker..........

Spon’s Dictionary of Engineering.........

Engineering............................

Edinburgh Encyclopaedia, çth ed.........

Scientific American Supplement...........

Engineering Record.....................

Engineering Magazine...................

Journal of the Association of Engineering

Societie'.............................

Encyclopædia Britannica, çth ed..........

Railway and Engineering Journal.........

Cresy’ Bridges.........................

Railway Gazette........................

Murray’s Handbook of Northern Italy. . . .

Le Genie Civil..........................

Messrs. Keepers & Thacher.............

The Melan Arch Construction Co.........

Transactions of Am. Soc. C. E............

Concrete-steel Engineering Co............

New York City, U. S. A.

Philadelphia, Pa., U. S. A.

Philadelphia, Pa., U. S. A.

Paris, France.

Paris, France.

John Wiley & Sons, New York City, U. S. A. John Wiley & Sons, New York City, U. S. A.

London, England.

New York City, U. S. A. New York City, U. S. A. New York City, U. S. A.

New York City, U. S. A.

John Wiley & Sons, New York City, U. S. A. New York City, U. S. A.

Paris, France.

Paterson, N. J., U. S. A.

New York City, U. S. A.

New York City, U. S. A.

New York City, U. S. A.

BIBLIOTEKA POLITECHNICZNA

KRAKÓW

Loads Dead 3350 Live 1108

3200

3050

2950

2800

2800

2700

2600

2650

2600

1132

1164

1184

1204 -

1220

1236

1244

1252

1256

2 D. L. 1-10 inc. = 29000 2 L. L. 1-8'inc. = 15752

2 D. L. 1-l inc. —• 58OO0 SL.L. 7-1'inc. =8243

\0-~-

~Q~

i£

Va

a------Ąi------_0_4_------As

-8Ó

X_J!_

------A⁸

A⁹'

A¹-⁰-

?r

10

I----s, !

. Q'.....-s„ O'“-

—iß--

Ns

- S₃

Dead Load Diagram

ss.

->

s*

s,

A !i

S₄.

S₅

Hr 50000

Sc

S₆

----A.----

Ss,

S-

-9-

S,

Ss

s₉

ffl,

SAB

s₉

S>!^|

-Sjo

GRANITE ARCH

SPAN 60' RISE 8'

----6

-i--------6-

--6--------i-------6—

-ó—

Ó-

^Sn

r

.

’ j NSłr

M

m

ÉÜÉ

■ii

1'

.

1

... . r

■(■•«.-:■» .;    . i

BOÎ1A 3TB T p

'* : .

....... L-......

*

]Ą

;••

•Ovï". .'

.....~r~ '■--

*■' v> •:■;

. .. ,>

i w

3W

1

.

V'v

a

,

■

V

ft ; -    ?

11

.-- -***. <«■ -• ■'■•>-.*■•■ •»*—

:

V ;

■

T

.

itmr

■lii

s₈

This load diagram for the upper equilibrium polygon. For loads 1 to 8' inclusive Hi=17606

v,= urn

Point of greatest moment at 6'

LIVE LOADS ONLY..

St

,ei

sr

sr

sr

sr

sr

sr

sr

Sr

Sio

s-

Sir

Sir

Sir

S-

So

Mi

line

S»

o ijd

7'

Ri >Vi

So

ring„

j§»

Sio

J y/////////////////,-

Sn

Sis

Center

line

of

arch

Closing • ,-vline

SBC'

S₈

s,

Sl4

So

[[| 1 LLlus^

S₆

3"

2'

10

10'

9'

8'

7'

6'

This load diagram for the lower equilibrium polygon. For loads 7'to 1'inclusive

Hi=5132 V, = 807

' Point of greatest moment at 6'

LIVE LOADS ONLY.

Si

■s,

S₃

s₄

s₅

So

s,

-Ss

il

#t P

F

m

ü

«f

::

■ -

1 ;

%

.il

i

iih;.    ' ‘‘‘i '

i

'

■:wtS

M

'\Æ

I M

’■T-

m

?

v.

C- ■■

m

m

S, :

’ n

!

i

; ::>■

«

m

Curve

Vo

Line S y A

o

_L_

Positive

A*

'3

10

lQi

9'

8'

7'

6'

51

il

8'

2'

f Xi

—=2.0-

T

oo

''""■Oiegative

'o

GO

O»

-2 1,2

<35

<=>

CONCRETE-STEEL ARCH SPAN 53!6 RISE 9.'I2

'alues of [Vj

.

H

:

.

m

i

m

m

■

«• •

«

&

•' .

1

ACTUAL LOADS ■= QU ANTITIES BELOW xf(2.68 x 1,40)=375.2 =ßl q D.L. ç.11-50_q9.55 u_^7.30

9

5.63

I

9

4.40

^3.55

K2.93

o

i

QJ~

-62.45

9

p.

PLATE IV

9

2.18

P

9

2.03

Q

/K

s

->r

1

9

p

L.L. = 112 per o ' j D.L.= 1409er cu. ft.

P

\

p,

P

•fié*—V—"T

Ç3..U_____

_Aï***S—<j>

ł

\

A-i:i9 1 = 0.18

h =l'.10 1=0.14

â-i:o8 1 = 0.11

a=i:oo 1 = 0.10

ii

.

h= (l'.OO)

9

v«R

©

P

/i = 1(38 1 = 0.27

9:

- \

\

h - l.'3ü 1 = 0.23

(12:45)

6

I

I

,A'

h =1.54 1 = 0.37

p,

\

\

»

\

CROSS-SECTION OF RIB

i

__04ï___

T

1

Steel

►si

^y

Àf/

.A'

h = 2,'os

= 0 01

\

\

\

%///■

'm

Concrete

1

0.17

I

x

✓

■V

\

\

\

\

T

.

9 IX

9

&

<r

X

/

8

%

I

oy^>

à—-1:8-

/

y

I

ft-2.'80 1 = 2.13

œ-1.34    / j æ=4.0*

i

1

2

Ail dimensions [scaled from drawing excepting those enclosed ( )

I as= 6.70

3

A=a8o

I * 5.17

K

2:68

->>

a? =9.38

as = 12.06

5_r.⁶_-?„(>

(9

4

5

N

\

\

\

\

\

\

\

\

2:0-r*

L+ 20 |₂=J9 0.2 (1-0.17/ = !»

/

- Clear Rise =(10.'00)

•» Rise of Axis =9(12 =/

a-= 14.74 y= 7.77

6

• rf^OrïMZÂ

1

:\jl±

r

Z K I /

I

\

\

—O-

\

\

\

\

\

as=17.42

=20.10 . = 8.71

7

8

i Clear Span = (25.'00) \    ____

I Span of Axis = 25.'00 +1 8 = 26.'80 — ^ I

\

\

-o-

X=22.78 8.98

o

9>

X -25.46 V” 9.12

■ 10

T

-----^

99

N

y

X.K

cc 0c

Re-61'.62

\

T

\

:

§

■ r.....................................—

,

■;> Y’.-'

........ :    ..B,.:;—*,...

.4

<

)

!

I'

■i......

!

»

j    f*.

:

■ ■

-...........

.

I

{

.

;

; :

ti

.....> :

I

I

!

-I

INDEX.

PAGE

Abutments, Albula Railway............

thickness of...............

Albula Railway, bridge dimensions......

Arch axis, subdivision of...............

Arch bridges, typical..................

Arch ring, with constant à.............

depth at support............

depth at crown.............

fixed, general formulas.......

Austrian specifications.................

Axial stress, change in coordinates......

constant J................

Example i................

Example 2................

horizontal thrust, Ha.......

vertical loads..............

Baker, thickness of abutments..........

Bellefield bridge.......................

Bending moment, taken entirely by steel.

Concrete, coefficient of expansion........

in compression...............

in tension...................

physical properties...........

plain.......................

reinforced...................

Coordinates x and y...................

Croizette-Besnoyer’s formulas...........

Bead load computations...............

Becimals, number.....................

Dejardin’s formulas...................

Bistortion, angular, produced by bending. Elasticity, modulus....................

49

47

49

54, 88

. 119

29

46

43

45

32

81

110

24

23

47

. . 120 36, no

132

36, 37

37

131

35, 41 35, 42

3, 5

46

65

55

44, 45

7, 8, 34, 36

PAGE

Elastic theory, reliability................

Empirical formulas, remarks..............

Equilibrium polygon, following arch axis. . .

dead load, Example 2,

37, 42

₄&

51

Example, first. .

second, third. .

96.

53

88.

hi

Fiber stresses, concrete ribs.

dead and live loads, Example 1.

Example 2

35.

76

final in Example 1........

general formulas...........

maximum, Example 2......

polygon outside middle third, reinforced concrete.........

102, 103, 105, 106, 107

83.

33

108, 109.

34

stone ribs.

36-

temperature changes, Example 1

Fill, above key, railroad bridges..............

weight................................

Formulas, general..........................

34

8o-

53

general, constant A...............

Garman practice, depth of key..............

Granite, physical properties................

Horizontal loads, constant A, general formulas.

general formulas...........

horizontal thrust..........

...... 133

7, 8, 15, 26. . .. . 30, 31

45

129.

31

26.

moments Mx. symmetrical. Horizontal thrust, axial stress.

27.

28.

constant A, vertical loads. . , horizontal loads.

24

20.

dead load, Example 1 Example 2

general formulas.....

horizontal loads.....

31

67

96-

10

Keystone, Albula Railway.....

depth of, Example 1

depth.............

empirical formulas. .

live load, Example 1..........

temperature changes..........

unit loads, Example 1.......

Example 2.........

Example 3.........

vertical loads, general formulas.

6cp

12

5&

89

49?

65

43

43, 44,. 45,. 4fi

Limestone, physical properties........................

Live load, distribution............................. ..

Luxemburg bridge..................................

m„ graphical representation..........................

Mi, live load, Example 1. ............................

unit loads, Example 1............................

Example 2............................

Example 3............................

on xj graphical determination....................... . . .

Mx, graphical representation.........................

Marble, physical properties...........................

Maximum moments, crown, live load, Example 1.......

graphical determination, Example 1

live load, Example 1..............

point O, live load, Example 1......

point 6', live load, Example 1......

PAGE

I3O

53

121

17, 18

69

60

89

”3

Moment Mv general......

vertical loads.

Moment, maximum Mx, vertical loads.

Neutral axis........................

Pence, coefficient of expansion.......

Peronnet’s formulas................

Piers, Albula Railway..............

thickness....................

Rankine’s formulas.................

Reinforcement, steel, area...........

steel-ribs............

I3I

44

49

48

steel-rods, etc. Reliability, elastic theory. . . .

Rockville bridge............

Sandstone, physical properties.

Second example, data........

Shear Tx...................

Spandrel filling.............

44, 47 . 124

• 123

. 124

41

120

130

88

25, 32, 33, 87

Spandrel masonry.

horizontal thrust, uniform weight. . weight.......

50

50

85

50

weight.............

walls..............

Specifications, abstracts...............

Stone, physical properties.............

Ex , computation....................

Table I, physical properties of stone, etc. Table II. arch data...................

38, 39, 40

50

119

125

129

60

129

Temperature.

PAGE

horizontal thrust............

range......................

reinforced concrete. . ..,.....

stresses, Example 1.........

Example 2.........

Thacher, properties of concrete...........

Thebes bridge..........................

Third example..........................

Track, weight...........................

Trautwine’s formulas....................

Unit loads, values of V₁, yv y₂, etc........

Vermillion River bridge..................

Vertical loads, constant J, general formulas.

general formulas...........

horizontal thrust...........

Mj, graphically............

Mx, graphically............

Vu live load, Example 1.................

unit loads, Example 1................

Example 2................

yu y₂, etc., unit loads, Example 1.........

6, 31, 35, 39, 40, 41

118

42

79

106

131

121

54

44, 47 96, 97

123

30

15

17, 18

16

69

60, 69

97

60, 65, 97

SHORT-TITLE CATALOGUE

OF THE

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OF

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Limited.

JOHN WILEY &

New York.

London: CHAPMAN & HALL,

ARRANGED UNDER SUBJECTS.

Descriptive circulars sent on application. Books marked with an asterisk (*) are sold at net prices only, a double asterisk (**) books sold under the rules of the American Publishers’ Association at net prices subject to an extra charge for postage. All books are bound in cloth unless otherwise stated.

AGRICULTURE.

Armsby’s Manual of Cattle-feeding...........................

Principles of Animal Nutrition..........................

Budd and Hansen’s American Horticultural Manual:

Part I. Propagation, Culture, and Improvement............

Part II. Systematic Pomology...........................

Downing’s Fruits and Fruit-trees of America.................

Elliott’s Engineering for Land Drainage......................

Practical Farm Drainage...............................

Green’s Principles of American Forestry......................

Grotenfelt’s Principles of Modern Dairy Practice. (Woll.). . . . .

Kemp’s Landscape Gardening...............................

Maynard’s Landscape Gardening as Applied to Home Decoration. * McKay and Larsen’s Principles and Practice of Butter-making

Sanderson’s Insects Injurious to Staple Crops..................

Insects Injurious to Garden Crops. (In preparation.)

Insects Injuring Fruits. (In preparation.)

Stockbridge’s Rocks and Soils...............................

Winton’s Microscopy of Vegetable Foods....................

Woll’s Handbook for Farmers and Dairymen..................

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Bashore’s Sanitation of a Country House.................

Berg’s Buildings and Structures of American Railroads.....

Birkmire’s Planning and Construction of American Theatres.

Architectural Iron and Steel........................

Compound Riveted Girders as Applied in Buildings.....

Planning and Construction of High Office Buildings. . . .

Skeleton Construction in Buildings..................

Brigg’s Modern American School Buildings. . . ............

Carpenter’s Heating and Ventilating of Buildings........

Freitag’s Architectural Engineering......................

Fireproofing of Steel Buildings......................

French and Ives’s Stereotomy..........................

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Chase’s Screw Propellers and Marine Propulsion. . . .

Cloke’s Gunner’s Examiner.....................

Craig’s Azimuth................................

Crehore and Squier’s Polarizing Photo-chronograph.

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Treatise on the Military Law of United States.

De Brack’s Cavalry Outposts Duties. (Carr.)..

Dietz’s Soldier’s First Aid Handbook.........

*    Dredge’s Modern French Artillery.........

Durand’s Resistance and Propulsion of Ships. .

*    Dyer’s Handbook of Light Artillery.........

Eissler’s Modern High Explosives............

*    Fiebeger’s Text-book on Field Fortification..

Hamilton’s The Gunner’s Catechism.........

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Ingalls’s Handbook of Problems in Direct Fire.

Ballistic Tables......................

*    Lyons’s Treatise on Electromagnetic Phenomena. Vols. I. and II. .8vo, each,

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Manual for Courts-martial.........

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Theatre Fires and Panics..............

♦Greene’s Structural Mechanics............

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Holly’s Carpenters’ and Joiners’ Handbook.........

Johnson’s Statics by Algebraic and Graphic Methods.

75

Non-metallic Minerals : Their Occurrence and Uses

Monckton’s Stair-building..........................

Patton’s Practical Treatise on Foundations.............

Peabody’s Naval Architecture...............................

Rice’s Concrete-block Manufacture.........................

Richey’s Handbook for Superintendents of Construction.........

Sabin’s Industrial and Artistic Technology of Paints and Varnish.

Siebert and Biggin’s Modern Stone-cutting and Masonry........

Snow’s Principal Species of Wood...........................

Sondericker’s Graphic Statics with Applications to Trusses, Beams, and Arches.

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Law of- Operations Preliminary to Construction in Engineering and Architecture.

Towne’s Locks and Builders’ Hardware...........

Wait’s Engineering and Architectural Jurisprudence

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Wood’s Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. . 8vo, 4 00 Worcester and Atkinson’s Small Hospitals, Establishment and Maintenance, Suggestions for Hospital Architecture, with Plans for a Small Hospital.

The World’s Columbian Exposition of 1893.

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Nixon’s Adjutants’ Manual..............

Peabody’s Naval Architecture...........

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Powell’s Army Officer’s Examiner.......

Sharpe’s Art of Subsisting Armies in War.

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Winthrop’s Abridgment of Military Law........

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Young’s Simple Elements of Navigation.........

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Furman’s Manual of Practical Assaying.

Lodge’s Notes on Assaying and Metallurgical Laboratory Experiments. .. .8vo, 3 00

Low’s Technical Methods of Ore Analysis.....................

Miller’s Manual of Assaying.................................

Minet’s Production of Aluminum and its Industrial Use. (Waldo.). O’Driscoll’s Notes on the Treatment of Gold Ores..............

8vo, 3 00

Ricketts and Miller’s Notes on Assaying............

Robine and Lenglen’s Cyanide Industry. (Le Clerc.).

Ulke’s Modern Electrolytic Copper Refining.........

Wilson’s Cyanide Processes.......................

Chlorination Process.........................

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■Comstock’s Field Astronomy for Engineers................

Craig’s Azimuth........................................

Doolittle’s Treatise on Practical Astronomy................

Gore’s Elements of Geodesy.............................

Hayford’s Text-book of Geodetic Astronomy...............

Merriman’s Elements of Precise Surveying and Geodesy.....

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Allen’s Tables for Iron Analysis...........................

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Austen’s Notes for Chemical Students......................

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Duhem’s Thermodynamics and Chemistry. (Burgess.)............

Eissler’s Modern High Explosives...............................

Effront’s Enzymes and their Applications. (Prescott.)............

Erdmann’s Introduction to Chemical Preparations. (Dunlap.).....

Fletcher’s Practical Instructions in Quantitative Assaying with the Blowpipe.

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Furman’s Manual of Practical Assaying......................

*    Getman’s Exercises in Physical Chemistry...................

Gill’s Gas and Fuel Analysis for Engineers....................

Grotenfelt’s Principles of Modern Dairy Practice. (Woll.)......

Groth’s Introduction to Chemical Crystallography (Marshall) . . .

Hammarsten’s Text-book of Physiological Chemistry. (Mandel.)

Helm’s Principles of Mathematical Chemistry. (Morgan.).

Hering’s Ready Reference Tables (Conversion Factors). . . .

Hind’s Inorganic Chemistry...........................

Laboratory Manual for Students...................

Holleman’s Text-book of Inorganic Chemistry. (Cooper.).

Text-book of Organic Chemistry. (Walker and Mott.),

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Hopkins’s Oil-chemists’ Handbook......................

Jackson’s Directions for Laboratory Work in Physiological Chemistry. ,8vo,    1 25

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Keep’s Cast Iron............................

Ladd’s Manual of Quantitative Chemical Analysis

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Chemistry. (Tingle.)

Control.....................................

Löb’s Electrochemistry of Organic Compounds. (Lorenz.).

8vo, . 8vo,

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8vo,

Low’s Technical Method of Ore Analysis

Lunge’s Techno-chemical Analysis. (Cohn.)..................

*    McKay and Larsen’s Principles and Practice of Butter-making

Mandel’s Handbook for Bio-chemical Laboratory..............

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8vo,

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Examination of Water. (Chemical and Bacteriological.)

Matthew’s The Textile Fibres............................

Meyer’s Determination of Radicles in Carbon Compounds. (Tingle.). . i2mo,

i2mo,

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Morgan’s An Outline of the Theory of Solutions and its Results

Elements of Physical Chemistry........................

* Physical Chemistry for Electrical Engineers............

Morse’s Calculations used in Cane-sugar Factories.

Mulliken’s General Method for the Identification of Fure Organic Compounds.

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Prescott and Winslow’s Elements of Water Eacteriology, with Special Refer

Ricketts and Miller’s Notes on Assaying........................

Rideal’s Sewage and the Bacterial Purification of Sewage.........

Disinfection and the Preservation of Food..................

Riggs’s Elementary Manual for the Chemical Laboratory.........

Robine and Lenglen’s Cyanide Industry. (Le Clerc.)............

Rostoski’s Serum Diagnosis. (Bolduan.).......................

Ruddiman’s Incompatibilities in Prescriptions...................

*    Whys in Pharmacy.....................................

Sabin’s Industrial and Artistic Technology of Paints and Varnish. .. Salkowski’s Physiological and Pathological Chemistry. (OrndorfL). Schimpfs Text-book of Volumetric Analysis.....................

Essentials of Volumetric Analysis..........................

*    Qualitative Chemical Analysis............................

Smith’s Lecture Notes on Chemistry for Dental Students.........

duan.) .................■.......................

Wehrenfennig's Analysis and Softening of Boiler Feed-Water Wells’s Laboratory Guide in Qualitative Chemical Analysis.. . .

Short Course in Inorganic Qualitative Chemical Analysis for Engineering

Students....................

Text-book of Chemical Arithmetic . .

Whipple’s Microscopy of Drinking-water.

Wilson’s Cyanide Processes............

Chlorination Process..............

Winton’s Microscopy of Vegetable Foods.

Wulling’s Elementary Course in Inorganic, Pharmaceutical, and Medical Chemistry.

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Poole’s Calorific Power of Fuels..............................

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Spencer’s Handbook for Chemists of Beet-sugar Houses.....i6mo, morocco. 3 00

Handbook for Cane Sugar Manufacturers.

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Stockbridge’s Rocks and Soils.............................,,

*    Tillman’s Elementary Lessons in Heat.....................

*    Descriptive General Chemistry........................

Treadwell’s Qualitative Analysis. (Hall.)...................

Quantitative Analysis. (Hall.).........................

Turneaure and Russell’s Public Water-supplies...............

Van Deventer’s Physical Chemistry for Beginners. (Boltwood.)

*    Walke’s Lectures on Explosives......................

Ware’s Beet-sugar Manufacture and Refining............

Washington’s Manual of the Chemical Analysis of Rocks.

Wassermann’s Immune Sera : Hæmolysins, Cytotoxins, and Precipitir.s. (Bol

i2mo, i co . 8vo, 4 00 .,8vo, i 50

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CIVIL ENGINEERING.

BRIDGES AND ROOFS. HYDRAULICS. MATERIALS OF ENGINEERING. RAILWAY ENGINEERING.

Baker’s Engineers’ Surveying Instruments. Bixby’s Graphical Computing Table......

i2mo, 3 oo

Paper igi X 24! inches.    25

** Burr’s Ancient and Modern Engineering and the Isthmian Cana.. (Postage,

27 cents additional.)............

Comstock’s Field Astronomy for Engineers.

Davis’s Elevation and Stadia Tables.......

Elliott’s Engineering for Land Drainage.................

Practical Farm Drainage...........................

*Fiebeger’s Treatise on Civil Engineering..............

Flemer’s Phototopographic Methods and Instruments......

Folwell’s Sewerage. (Designing and Maintenance.)........

Freitag’s Architectural Engineering. 2d Edition, Rewritten

French and Ives’s Stereotomy...........................

Goodhue's Municipal Improvements.....................

Goodrich’s Economic Disposal of Towns’ Refuse...........

Gore’s Elements of Geodesy............................

Hayford’s Text-book of Geodetic Astronomy..............

Hering’s Ready Reference Tables (Conversion Factors).....

Howe’s Retaining Walls for Earth......................

* Ives’s Adjustments of the Engineer’s Transit and Level.....

Johnson’s (J. B.) Theory and Practice of Surveying........

Johnson’s (L. J.) Statics by Algebraic and Graphic Methods.

. .8vo, 3 50 ..8vo, 2 50 ..8vo, i 00 limo, i 50

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Laplace’s Philosophical Essay on Probabilities. (Truscott and Emory.). i2mo, 2 00

Mahan’s Treatise on Civil Engineering. (1873.) (Wood.).

* Descriptive Geometry.............................

Merriman’s Elements of Precise Surveying and Geodesy. . .

Merriman and Brooks’s Handbook for Surveyors.........

Nugent’s Plane Surveying.............................

Ogden’s Sewer Design.................................

Parsons’s Disposal of Municipal Refuse.................

Patton’s Treatise on Civil Engineering..................

Reed’s Topographical Drawing and Sketching............

Rideal’s Sewage and the Bacterial Purification of Sewage. .

Siebert and Biggin’s Modern Stone-cutting and Masonry. ..

Smith’s Manual of Topographical Drawing. (McMillan.). .

Sondericker’s Graphic Statics, with Applications to Trusses, Beams, and Arches.

..........8vo, 5 00

..........8vo i 50

..........8vo, 2 50

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8vo, 2 00

Taylor and Thompson’s Treatise on Concrete, Plain and Reinforced.....8vo, 5 00

* Trautwine’s Civil Engineer’s Pocket-book........

Venable’s Garbage Crematories in America.......

Wait’s Engineering and Architectural Jurisprudence.

i6mo, morocco, 5 00

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..........8vo,    6    00

Sheep, 6 50

Law of Operations Preliminary to Construction in Engineering and Archi-

tecture.

. 8vo, 5 00 Sheep, 5 50 . 8vo, 3 00 . .8vo, 0 50

Law of Contracts........................

Warren’s Stereotomy—Problems in Stone-cutting.

Webb’s Problems in the Use and Adjustment of Engineering Instruments.

i6mo, morocco, 1 25 8vo, 3 50

Wilson’s Topographic Surveying.

BRIDGES AND ROOFS.

Boiler’;. Practical Treatise on the Construction of Iron Highway Bridges. .8vo, 2 00 * Thames River Bridge...................................4to, paper, 5 00

Burr’s Course on the Stresses in Bridges and Roof Trusses, Arched Ribs, and Suspension Bridges.

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Burr and Falk’s Influence Lines for Bridge and Roof Computations. .. .8vo, 3 00

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Small 4to, 10 00

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......8vo, i 25

......8vo, 2 50

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Foster’s Treatise on Wboden Trestle Bridges.....

Fowler’s Ordinary Foundations...............

Greene’s Roof Trusses.......................

Bridge Trusses.................

Arches in Wood, Iron, and Stone. Howe’s Treatise on Arches..........

Design of Simple Roof-trusses in Wood and Steel.

Johnson, Bryan, and Turneaure’s Theory and Practice in the Designing of

Small 4to, 10 00

Modern Framed Structures.

Merriman and Jacoby’s Text-book on Roofs and Bridges:

Part I. Stresses in Simple Trusses...............

Part II. Graphic Statics........................

Part III. Bridge Design........................

Part IV. Higher Structures...................

Morison’s Memphis Bridge...........................

8vo, 2 50 8vo, 2 50 8vo, 2 50 8vo, 2 50 . 4to, 10 00

Waddell’s De Pontibus, a Pocket-book for Bridge Engineers. ,i6mo, moroccd, 2 00

* Specifications for Steel Bridges......................

Wright’s Designing of Draw-spans. Two parts in one volume.

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HYDRAULICS.

Bazin’s Experiments upon the Contraction of the Liquid Vein Issuing from

..........8 vo,

..........8 vo,

..........8vo,

...... • • .paper,

..........8vo,

i6mo, morocco,

.........i2mo,

........'.8vo,

..........8vo,

.........i2mo,

an Orifice. (Trautwine.)......................

Bovey’s Treatise on Hydraulics..........................

Church’s Mechanics of Engineering......................

Diagrams of Mean Velocity of Water in Open Channels

Hydraulic Motors.................................

Coffin’s Graphical Solution of Hydraulic Problems.........

Flather’s Dynamometers, and the Measurement of Power. . .

Folwell’s Water-supply Engineering.....................

Frizell’s Water-power..................................

Fuertes’s Water and Public Health......................

Water-filtration Works............................

i2mo,

to H m ^ ÙJ to to w ON CA to

Ganguillet and Kutter’s General Formula for the Uniform Flow of Water in

8vo, 8vo, 8vo,

Herschel’s 115 Experiments on the Carrying Capacity of Large, Riveted, Metal Conduits.

Rivers and Other Channels. (Hering and Trautwine.).

Hazen’s Filtration of Public Water-supply...................

Hazlehurst’s Towers and Tanks for Water-works.............

8vo,

Mason’s Water-supply. (Considered Principally from a Sanitary Standpoint.)

8vo,

8vo,

8vo,

Merriman’s Treatise on Hydraulics.......

* Michie’s Elements of Analytical Mechanic*.

Schuyler’s Reservoirs for Irrigation, Water-power, and Domestic Water-

Large 8vo,

** Thomas and Watt’s Improvement of Rivers. (Post., 44c. additional.).4to,

......8vo,

......4to,

......4to,

......8vo,

Small 8vo,

......8vo,

......8vo,

......8vo,

supply.

Turneaure and Russell’s Public Water-supplies.............

Wegmann’s Design and Construction of Dams..............

Water-supply of the City of New York from 1658 to 1895.

Williams and Hazen’s Hydraulic Tables...................

Wilson’s Irrigation Engineering...........................

Wolff’s Windmill as a Prime Mover. . ....................

Wood’s Turbines...................................

Elements of Analytical Mechanics....................

§ § §    8 £ § § § 8> 8>    § 8 §>    §    §88    §o§o8v>8§8>o

in t inoininoMtinfi")

to

MATERIALS OF ENGINEERING.

Baker’s Treatise on Masonry Construction.

Roads and Pavements..............

Black’s United States Public Works......

.......8 vo,

......8vo,

Oblong 4to, .......8vo,

Burr’s Elasticity and Resistance of the Materials of Engineering. . . . . . 8vo, Byrne’s Highway Construction.

Inspection of the Materials and Workmanship Employed in Construction.

* Bovey’s Strength of Materials and Theory of Structures. .

8vo,

in in in i> i** in

o Ol Ol o o o o o o o o o

Church’s Mechanics of Engineering...........................

Du Bois’s Mechanics of Engineering. Vol. I...................

*Eckei’s Cements, Limes, and Plasters........................

Johnson’s Materials of Construction..........................

Fowler’s Ordinary Foundations..............................

* Greene’s Structural Mechanics..............................

Keep’s Cast Iron..........................................

Lanza’s Applied Mechanics.................................

Marten’s Handbook on Testing Materials. (Henning.) 2 vols. .

Maurer’s Technical Mechanics..............................

Merrill’s Stones for Building and Decoration..................

Merriman’s Mechanics of Materials..........................

Strength of Materials..................................

Metcalf’s Steel. A Manual for Steel-users....................

Patton’s Practical Treatise on Foundations....................

Richardson’s Modern Asphalt Pavements....................

Richey’s Handbook for Superintendents of Construction..........

Rockwell’s Roads and Pavements in France...........

Sabin’s Industrial and Artistic Technology of Paints and Varnish.

Smith’s Materials of Machines...............................

Snow’s Principal Species of Wood...........................

•Spalding’s Hydraulic Cement................................

Text-book on Roads and Pavements......................

Taylor and Thompson’s Treatise on Concrete, Plain and Reinforced.

Thurston’s Materials of Engineering. 3 Parts...................

Part I. Non-metallic Materials of Engineering and Metallurgy Part II. Iron and Steel..................................

•i6mo,

......8 vo,

.Small 4to,

......8vo,

■Large 8vo,

......8vo,

......8vo,

......8vo,

......8vo,

......8vo,

......8vo,

..... 8vo,

......8vo,

.....i2mo,

.....nmo,

.......8vo,

.......8vo,

i6mo, mor.,

.....nmo,

........8vo,

.....nmo,

.......8vo,

.....nmo,

.....i2mo,

8vo, 8vo, 8vo, 8vo,

Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their

8vo, 8vo, 8vo,

Waddell’s De Pontibus. (A Pocket-book for Bridge Engineers.). . i6mo, msr., Specifications for Steel Bridges.

Wood’s (De V.) Treatise on the Resistance of Materials, and an Appendix on

8vo, 8vo,

Wood’s (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and

8vo,

Constituents.

Thurston’s Text-book of the Materials of Construction,

Tillson’s Street Pavements and Paving Materials.

the Preservation of Timber.

Wood’s (De V.) Elements of Analytical Mechanics.

Steel.

RAILWAY ENGINEERING.

nmo,

Oj NJ 00 C/I KJ to Gj m Oj h ^ Cm (a KJ h (/i Ui ^ 'J 'J M M U O' O' "4 O' U

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M KJ A Ol M

A UM

3x5 inches, morocco, 1 25

...............4to,    5    00

.....16mo, morocco, 1 50

.....i6mo, morocco, 2 50

.....i6mo, morocco, 1 50

...............8vo,    i    50

Andrew’s Handbook for Street Railway Engineers.. . .

Berg’s Buildings and Structures of American Railroads

Brook’s Handbook of Street Railroad Location.......

Butt’s Civil Engineer’s Field-book..................

Crandall’s Transition Curve........................

Railway and Other Earthwork Tables...........

Dawson’s “Engineering” and Electric Traction Pocket-book. . i6mo, morocco, 5 00

Dredge’s History of the Pennsylvania Railroad: (1879)

* Drinker’s Tunnelling, Explosive Compounds, and Rock Drills.4to, half mor., 25 00

Fisher’s Table of Cubic Yards

Cardboard, 25

Godwin’s Railroad Engineers’ Field-book and Explorers’ Guide. . . i6mo, mor., 2 50 Howard’s Transition Curve Field-book.

i6mo, morocco, 1 50

Hudson’s Tables for Calculating the Cubic Contents of Excavations and Embankments.

..........8vo,    i    00

.........i6mo,    i    00

i6mo, morocco, 3 00 i6mo, morocco, 3 00 i6mo, morocco, 3 00 iémo, morocco, 1 50 ..........8vo, i 50

Molitor and Beard’s Manual for Resident Engineers.

Nagle’s Field Manual for Railroad Engineers......

Philbrick’s Field Manual for Engineers...........

Searles’s Field Engineering.....................

Railroad Spiral............................

Taylor's Prismoidal Formulas and Earthwork.....

* Trautwine’s Method of Calculating the Cube Contents of Excavations and

Embankments by the Aid of Diagrams..................

The Field Practice of Laying Out Circular Curves for Railroads.

8vo, 2 00

i2mo, morocco, 2 50

........Paper,    25

i6mo, morocco, 5 00

Cross-section Sheet......

Webb’s Railroad Construction.

Wellington’s Economic Theory of the Location of Railways.......Small 8vo, 5 00

DRAWING.

Barr’s Kinematics of Machinery. * Bartlett’s Mechanical Drawing.

Abridged Ed.

8vo, 2 50 8vo, 3 00 8vo, i 50

Coolidge’s Manual of Drawing.

Coolidge and Freeman’s Elements of General Drafting for Mechanical Engi-

Oblong 4to, 2 50

8vo,paper 1 00

neers.

Durley’s Kinematics of Machines............................

Emch’s Introduction to Projective Geometry and its Applications.

Hill’s Text-book on Shades and Shadows, and Perspective.......

Jamison’s Elements of Mechanical Drawing...................

Advanced Mechanical Drawing.........................

Jones’s Machine Design:

Part I. Kinematics of Machinery.......................

Part II. Form, Strength, and Proportions of Parts........

MacCord’s Elements of Descriptive Geometry.................

Kinematics ; or, Practical Mechanism....................

Mechanical Drawing..................................

Veloeity Diagrams...................................

MacLeod’s Descriptive Geometry............................

* Mahan’s Descriptive Geometry and Stone-cutting.............

Industrial Drawing. (Thompson.)......................

Moyer’s Descriptive Geometry...............................

Reed’s Topographical Drawing and Sketching................

Reid’s Course in Mechanical Drawing........................

.,....8vo, i 50

......8vo, 3 00

......8vo, 3 00

......8vo, 5 00

......4to, 4 00

......8vo, i 50

Small 8vo, 1 50

......8vo, i 50

......8vo, 3 50

......8vo, 2 00

. 4to, s 00 8vo, 2 00

Text-book of Mechanical Drawing and Elementary Machine Design.8vo, 3 00 Robinson’s Principles of Mechanism.

Schwamb and Merrill’s Elements of Mechanism.

Smith’s (R. S.) Manual of Topographical Drawing. (McMillan.)

Smith (A. W.) and Marx’s Machine Design..................

Warren’s Elements of Plane and Solid Free-hand Geometrical Drawing. i2mo,

Drafting Instruments and Operations.........

Manual of Elementary Projection Drawing. ..

Manual of Elementary Problems in the Linear Perspective of Form and Shadow

8vo, 3 00 8vo, 3 00 8vo, 2 50

8vo, 3 00

i 00

i2mo, i 25

i2mo, i 50

Plane Problems in Elementary Geometry

i2mo, i 00 i2mo, i 25

Warren’s Primary Geometry.................................

Elements of Descriptive Geometry, Shadows, and Perspective.

General Problems of Shades and Shadows.................

Elements of Machine Construction and Drawing............

Problems, Theorems, and Examples in Descriptive Geometry. Weisbach’s Kinematics and Power of Transmission.

Klein.)....................................

Whelpley’s Practical Instruction in the Art of Letter Engraving.

Wilson’s (H. M.) Topographic Surveying.....................

Wilson’s (V. T.) Free-hand Perspective......................

Wilson’s (V. T.) Free-hand Lettering.......................

Woolf’s Elementary Course in Descriptive Geometry..........

(Hermann and

i2mo, 75 • -8vo,    3 50

.,8vo, 3 00 . . 8vo, .7 5a . .8vo, 2 50

8vo, 5 00

ELECTRICITY AND PHYSICS.

Anthony and Brackett’s Text-book of Physics. (Magie.)

Small 8vo, 3 00

Anthony’s Lecture-notes on the Theory of Electrical Measurements. . . . i2mo, 1 00 Benjamin’s History of Electricity.

Voltaic Cell...............

Classen’s Quantitative Chemical Analysis by Electrolysis. (Boltwood.).8vo, 3 00 Crehore and Squier’s Polarizing Photo-chronograph.

Dawson’s “Engineering” and Electric Traction Pocket-book. i6mo, morocco, 5 oa Dolezalek’s Theory of the Lead Accumulator (Storage Battery). (Von

Ende.)....................................

Duhem’s Thermodynamics and Chemistry. (Burgess.). .

Flather’s Dynamometers, and the Measurement of Power.

Gilbert’s De Magnete. (Mottelay.)...................

Hanchett’s Alternating Currents Explained.............

Hering’s Ready Reference Tables (Conversion Factors). .

Holman’s Precision of Measurements.................

. . ,8vo, 3 00 . . 8vo, 3 00

8vo, 3 00

i2mo, 2 50 . .8vo, 4 00 i2mo, 3 00 . .8vo, 2 5a

.........i2mo, i 00

i6mo, morocco, 2 5a ........t .8vo, 2 00

Telescopic Mirror-scale Method, Adjustments, and Tests. . . .Large 8vo, Kinzbrunner’s Testing of Continuous-current Machines.

Landauer’s Spectrum Analysis. (Tingle.)............

75

8vo, 2 00 8vo, 3 oa

Le Chatelier's High-temperature Measurements. (Boudouard—Burgess.) nmo, 3 oa Lob’s Electrochemistry of Organic Compounds. (Lorenz.).

8vo, 3 00

*    Lyons’s Treatise on Electromagnetic Phenomena. Vols. I. and II. 8vo, each, 6 00

*    Michie’s Elements of Wave Motion Relating to Sound and Light. .

Niaudet’s Elementary Treatise on Electric Batteries. (Fishback.). .

8vo, 4 oa

i2mo, 2 sa

*    Parshall and Hobart’s Electric Machine Design

Ryan, Norris, and Hoxie’s Electrical Machinery. Vol. I

Thurston’s Stationary Steam-engines.................

* Tillman’s Elementary Lessons in Heat..............

Tory and Pitcher’s Manual of Laboratory Physics.....

Ulke’s Modern Electrolytic Copper Refining...........

LAW.

*    Davis’s Elements of Law....................

*    Treatise on the Military Law of United States.

Manual for Courts-martial......................

Wait’s Engineering and Architectural Jurisprudence.

4to, half morocco, 12 50

...........8vo,    2    5a

...........8vo,    7    oa

Sheep, 7 50 i6mo, morocco, 1 50

...........8vo, 6 00

Sheep, 6 50

Law of Operations Preliminary to Construction in Engineering and Archi-

tecture.

Law of Contracts................

Winthrop’s Abridgment of Military Law.

. .8vo 5 00 Sheep, s 5a . .8vo, 3 00 . i2mo, 2 5a

MANUFACTURES.

Bernadou’s Smokeless Powder—Nitro-cellulose and Theory of the Cellulose

Molecule.....

Bolland’s Iron Founder.

i2tno, 2 i2mo, 2

“The Iron Founder,” Supplement.

Encyclopedia of Founding and Dictionary of Foundry Terms Used in the

Practice of Moulding......................

* Eckel’s Cements, Limes, and Plasters..............

Eissler’s Modern High Explosives...................

Effront’s Enzymes and their Applications. (Prescott.).

Fitzgerald’s Boston Machinist......................

Ford’s Boiler Making for Boiler Makers..............

Hopkin’s Oil-chemists’ Handbook..................

Keep’s Cast Iron.................................

nmo, 2

X2mo, 3 ..8vo, 6 ..8vo, 4 ..8vo, 3

i2mo, i i8mo, i •-8vo, 3 ,.8vo, 2

Leach’s The Inspection and Analysis of Food with Special Reference to State

Control............................................

* McKay and Larsen’s Principles and Practice of Butter-making

Matthews’s The Textile Fibres................................

Metcalf’s Steel. A Manual for Steel-users.....................

Large 8vo, 7

......8vo, i

......8vo, 3

i2mo, 2

Metcalfe’s Cost of Manufactures—And the Administration of Workshops.8vo, 5

Meyer’s Modern Locomotive Construction........

Morse’s Calculations used in Cane-sugar Factories.

* Reisig’s Guide to Piece-dyeing................

Ricç’s Concrete-block Manufacture.............

Sabin’s Industrial and Artistic Technology of Paints and Varnish.

Smith’s Press-working of Metals....................

Spalding’s Hydraulic Cement........................

Spencer’s Handbook for Chemists of Beet-sugar Houses.

Handbook for Cane Sugar Manufacturers.........

8vo, 3 8vo, 3

.........i2mo,    2

i6mo, morocco, 3 i6mo, morocco, 3

tion..........................

* Walke’s Lectures on Explosives...........

Ware’s Beet-sugar Manufacture and Refining.

West’s American Foundry Practice..........

Moulder’s Text-book..................

Wolff’s Windmill as a Prime Mover.......

Wood’s Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. .8vo, 4

I2H10, 2

i2mo, 2 . 8vo, 3

o 9 o 9 9 9 ç 080899990999090    08888889    333

MATHEMATICS.

Baker’s Elliptic Functions..................................

*    Bass’s Elements of Differential Calculus.....................

Briggs’s Elements of Plane Analytic Geometry.................

Compton’s Manual of Logarithmic Computations...............

Davis’s Introduction to the Logic of Algebra..................

*    Dickson’s College Algebra................................

*    Introduction to the Theory of Algebraic Equations.........

Emch’s Introduction to Projective Geometry and its Applications.

Halsted’s Elements of Geometry.............................

Elementary Synthetic Geometry.........................

Rational Geometry....................................

*    Johnson’s (J. B.) Three-place Logarithmic Tables: Vest-pocket size.paper, 15

100 copies for 5 00

Mounted on heavy cardboard, 8X10 inches, 25 xo copies for 2 00

nmo, i 75

Johnson’s (W. W.) Elementary Treatise on Differential Calculus. .Small 8vo, 3 00 Elementary Treatise on the Integral Calculus

Small 8vo, 1 50

Johnson’s (W. W.) Curve Tracing in Cartesian Co-ordinates Johnson’s (W. W.) Treatise on Ordinary and Partial Differential Equations.

Small 8vo, 3 50

Johnson’s (W. W.) Theory of Errors and the Method of Least Squares. i2mo, 1 50

*    Johnson’s (W. W.) Theoretical Mechanics. . .

Laplace’s Philosophical Essay on Probabilities. (Truscott and Emory.). umo, 2 00

*    Ludlow and Bass. Elements of Trigonometry and Logarithmic and Other

Tables...............................

Trigonometry and Tables published separately

*    Ludlow’s Logarithmic and Trigonometric Tables.

Manning’s Irrational Numbers and their Representation by Sequences and Series

i2mo, 3 00

. . 8vo, 3 00 Each, 2 00

8vo, i 00

Mathematical Monographs. Edited by Mansfield Merriman and Robert S. Woodward. . . .

No. i. History of Modern Mathematics, by David Eugene Smith. No. 2. Synthetic Projective Geometry, by George Bruce Halsted. No. 3. Determinants, by Laenas Gifford Weld. No. 4. Hyperbolic Functions, by James McMahon. No. 5. Harmonic Functions, by William E. Byerly. No. 6. Grassmann’s Space Analysis, by Edward W. Hyde. No. 7. Probability and Theory of Errors, by Robert S. Woodward. No. 8. Vector Analysis and Quaternions, by Alexander Macfarlane. No. 9. Differential Equations, by William Woolsey Johnson. No. 10. The Solution of Equations, by Mansfield Merriman. No. 11. Functions of a Complex Variable, by Thomas S. Fiske.

Maurer’s Technical Mechanics....................................

i2mo, i 25

. . . . Octavo, each 1 00

8vo, 4 00 . .8vo, 2 00

Merriman’s Method of Least Squares. .

Rice and Johnson’s Elementary Treatise on the Differential Calculus.. Sm. 8vo, 3 00

Small 8vo, 2 50 ......8vo, 2 00

Differential and Integral Calculus. 2 vols, in one.

Wood’s Elements of Co-ordinate Geometry..........

Trigonometry; Analytical, Plane, and Spherical .

i2mo, i 00

MECHANICAL ENGINEERING.

MATERIALS OF ENGINEERING, STEAM-ENGINES AND BOILERS.

Bacon’s Forge Practice..............

Baldwin’s Steam Heating for Buildings.

Barr’s Kinematics of Machinery.......

* Bartlett’s Mechanical Drawing.......

Abridged Ed.

Benjamin’s Wrinkles and Recipes.....

Carpenter’s Experimental Engineering. Heating and Ventilating Buildings.

i2mo, i2mo, ..8vo, ..8vo, ..8vo, i2mo, ..8vo, . . 8vo,

OOOUlOOlÜlUl

00000000

Cary’s Smoke Suppression in Plants using Bituminous Coal. (In Prepara

tion.)

.Small 8vo, 8vo, paper,

Clerk’s Gas and Oil Engine. ..

Coolidge’s Manual of Drawing Coolidge and Freeman’s Elements of General Drafting for Mechanical Engineers ...................................

Cromwell’s Treatise on Toothed Gearing..............

Treatise on Belts and Pulleys....................

Durley’s Kinematics of Machines.....................

Flather’s Dynamometers and the Measurement of Power

Rope Driving.................................

Gill’s Gas and Fuel Analysis for Engineers............

Hall’s Car Lubrication.............................

Hering’s Ready Reference Tables (Conversion Factors).

Oblong 4to,

.........i2mo,

.........i2mo,

..........8vo,

.........i2mo,

.........i2mo,

.........i2mo,

.........i2mo,

i6mo, morocco,

M A

OlOKiOOOUlUlC/l    OO

OOÜIOOOOOO    OO

tOMM (O OJ ^ MMK)

Hutton’s The Gas Engine..........................

Jamison’s Mechanical Drawing.....................

Jones’s Machine Design:

Part I. Kinematics of Machinery...............

Part II. Form, Strength, and Proportions of Parts.

Kent’s Mechanical Engineers’ Pocket-book...........

Kerr’s Power and Power Transmission...............

Leonard’s Machine Shop, Tools, and Methods.........

8vo, 5 oo 8vo, 2 50

MacCord’s Kinematics; or, Practical Mechanism.

Mechanical Drawing........................

Velocity Diagrams..........................

MacFarland’s Standard Reduction Factors for Gases.

Mahan’s Industrial Drawing. (Thompson.)........

Poole’s Calorific Power of Fuels..................

8vo, 5 00

• 4to, 4 00 8vo, i 50 8vo, i 50 8vo, 3 50 8vo, 3 00 8vo, 2 00

Text-book of Mechanical Drawing and Elementary Machine Design. 8vo, 3 00

Reid’s Course in Mechanical Drawing.

Richard’s Compressed Air...................

Robinson’s Principles of Mechanism..........

Schwamb and Merrill’s Elements of Mechanism.

Smith’s (O.) Press-working of Metals.....

Smith (A. W.) and Marx’s Machine Design.

Thurston’s Treatise on Friction and Lost Work in Machinery and Mill Work.

i2mo, i 50 ..8vo, 3 00 .,8vo, 3 00 . . 8vo, 3 00 . .8vo, 3 00

8vo, 3 00

Animal as a Machine and Prime Motor, and the Laws of Energetics. i2mo, 1 00 Warren’s Elements of Machine Construction and Drawing.

Weisbach’s Kinematics and the Power of Transmission. (Herrmann— Klein.).

8vo, 7 50

8vo, 5 00

Machinery of Transmission and Governors. (Herrmann—Klein.). .8vo, 5 00 Wolff’s Windmill as a Prime Mover.

Wood’s Turbines.................

8vo, 3 00 8vo, 2 50

MATERIALS OP ENGINEERING.

*    Bovey’s Strength of Materials and Theory of Structures.

Burr’s Elasticity and Resistance of the Materials of Engineering. 6th Edition.

Reset............................................

Church’s Mechanics of Engineering..........................

*    Greene’s Structural Mechanics............................

Johnson’s Materials of Construction.........................

Keep’s Cast Iron........................................

Lanza’s Applied Mechanics..................................

Martens’s Handbook on Testing Materials. (Henning.)........

Maurer’s Technical Mechanics..............................

Merriman’s Mechanics of Materials..........................

Strength of Materials.................................

Metcalf’s Steel. A manual for Steel-users...................

Sabin’s Industrial and Artistic Technology of Paints and Varnish.

Smith’s Materials of Machines..............................

Thurston’s Materials of Engineering........................

Part II. Iron and Steel................................

Part III. A Treatise on Brasses, Bronzes, and Other Alloys and their

8vo, 7 50

8vo, 7 50 8vo, 6 00 8vo, 2 50 8vo, 6 00 8vo, 2 50 8vo, 7 50 8vo, 7 50 8vo, 4 00 8vo, 5 00

i2mo, i 00

.....i2mo, 2 00

.......8vo,    3    00

.....i2mo, i 00

3vols.,8vo, 8 00 .......8vo,    3    50

Constituents.......................

Text-book of the Materials of Construction.

Wood’s (De V ) Treatise on the Resistance of Materials and an Appendix on

8vo, 2 50 8vo, 5 00

the Preservation of Timber......

Elements of Analytical Mechanics.

.8vo, 2 00 8vo, 3 00

Wood’s (M. P.) Rustless Coatings: Corrosion and Electrolysis of Iron and Steel,

STEAM-ENGINES AND BOILERS.

Berry’s Temperature-entropy Diagram.......................

Carnot’s Reflections on the Motive Power of Heat. (Thurston.).

Dawson’s “Engineering” and Electric Traction Pocket-book. . . ,i6mo,mor., 5

i8mo, i ..8vo, 2

Ford’s Boiler Making for Boiler Makers..................

Goss’s Locomotive Sparks..............................

Hemenway’s Indicator Practice and Steam-engine Economy

Hutton’s Mechanical Engineering of Power Plants.........

Heat and Heat-engines............................

Kent’s Steam boiler Economy...........................

Kneass’s Practice and Theory of the Injector..............

MacCord’s Slide-valves.................................

Meyer’s Modern Locomotive Construction......................

Peabody’s Manual of the Steam-engine Indicator................

Tables of the Properties of Saturated Steam and Other Vapors Thermodynamics of the Steam-engine and Other Heat-engines

Valve-gears for Steam-engines............................

Peabody and Miller’s Steam-boilers.............................

Pray’s Twenty Years with the Indicator........................

i2mo, i

i2mo, i

i2mo, 2 • 8vo, 5 ..8vo. 5 ..8vo, 4 ..8vo, i ..8vo, 2 . 4to, 10 i2mo, i

.....8vo, i

......8vo, 5

.....8vo, 2

......8vo, 4

Large 8vo, 2

Pupin’s Thermodynamics of Reversible Cycles in Gases and Saturated Vapors.

(Osterberg.)..............................

Reagan’s Locomotives : Simple Compound, and Electric Rontgen’s Principles of Thermodynamics. (Du Bois.). .

Sinclair’s Locomotive Engine Running and Management Smart’s Handbook of Engineering Laboratory Practice. .

Snow’s Steam-boiler Practice........................

Spangler’s Valve-gears.....................................

Notes on Thermodynamics.............................

Spangler, Greene, and Marshall’s Elements of Steam-engineering Thomas’s Steam-turbines..................................

nmo, i i2tno, 2 ..8vo, 5 i2mo, 2 i2mo, 2 ■8vo, 3 ..8vo, 2

Thurston’s Handy Tables.......................

Manual of the Steam-engine................

Part I. History, Structure, and Theory......

Part II. Design, Construction, and Operation.

.....i2mo, i

Handbook of Engine and Boiler Trials, and the Use of the Indicator and

8vo, s ,8vo, 2

the Prony Brake.....................................

Stationary Steam-engines................................

Steam-boiler Explosions in Theory and in Practice...........

Manual of Steam-boilers, their Designs, Construction, and Operation Wehrenfenning’s Analysis and Softening of Boiler Feed-water (Patterson) 8vo, 4 Weisbach’s Heat, Steam, and Steam-engines. (Du Bois.)

Whitham’s Steam-engine Design......................

Wood’s Thermodynamics, Heat Motors, and Refrigerating Machines. . .8vo, 4

i2tno, i . .8vo. 5

8vo, 5 8vo, 5

!?&ooooooo£oo®oo&o!n Ä S § o n o w o o S S o o o § S, S> 8 § § 8 $

MECHANICS AND MACHINERY.

Barr’s Kinematics of Machinery......................

* Bovey’s Strength of Materials and Theory of Structures

Chase’s The Art of Pattern-making...................

Church’s Mechanics of Engineering..................

Notes and Examples in Mechanics................

Compton’s First Lessons in Metal-working............

Compton and De Groodt’s The Speed Lathe............

..8vo, 2 50 . .8vo, 7 50 i2mo, 2 50 ..8vo, 6 00 ..8vo, 2 00 . i2mo, i 50 i2mo, i 50

i2mo, i 50

Cromwell’s Treatise on Toothed Gearing Treatise on Belts and Pulleys......

i2mo, 2 00

Dana’s Text-book of Elementary Mechanics for Colleges and Schools. .i2mo, 1 50 Dingey’s Machinery Pattern Making Dredge’s Record of the Transportation Exhibits Building of the World’s

Columbian Exposition of 1893.......

Du Bois’s Elementary Principles of Mechanics: Vol. I. Kinematics.................

4to half morocco, 5 00

.....8vo, 3 50

.....8vo, 4 00

Small 4to, 7 50 Small 4to, 10 00 . 8vo, 4 00 i6mo, i 00 i2mo, 3 00 i2mo, 2 00 . 8to, 2 00 . .8vo, 2 50 i2mo, i 00 i8mo, 75

Vol. II. Statics.................

Mechanics of Engineering. Vol. I.

Vol. II.

Dur ley’s Kinematics of Machines.................................

Fitzgerald’s Boston Machinist....................................

Flather’s Dynamometers, and the Measurement of Power............

Rope Driving..............................................

Goss’s Locomotive Sparks.......................................

* Greene’s Structural Mechanics...................................

Hall’s Car Lubrication..........................................

Holly’s Art of Saw Filing........................................

James’s Kinematics of a Point and the Rational Mechanics of a Particle.

Small 8vo, 2 00

i2mo, 3 00 . .8vo, 2 00

8vo, i 50 .8vo, 3 00 .8vo 2 00

* Johnson’s (W. W.) Theoretical Mechanics..............

Johnson’s (L. J.) Statics by Graphic and Algebraic Methods. Jones’s Machine Design:

Part I. Kinematics of Machinery.................

Part II. Form, Strength, and Proportions of Parts. . .

Kerr’s Power and Power Transmission..................

• 8vo, 7 50 8vo, 4 00

*    Lorenz’s Modern Refrigerating Machinery. (Pope, Haven, and Dean.) .8vo,    4 00

MacCord’s Kinematics ; or, Practical Mechanism. .

Velocity Diagrams.........................

*    Martin’s Text Book on Mechanics, Vol. I, Statics.

Maurer’s Technical Mechanics..................

Merriman’s Mechanics of Materials..............

Elements of Mechanics.....................

Lanza’s Applied Mechanics............................

Leonard’s Machine Shop, Tools, and Methods.............

.8vo, 5 00 8vo, i 50

i2mo, i 25 . .8vo, 4 00 .,8vo, 5 00

*    Michie’s Elements of Analytical Mechanics..........

*    Parshall and Hobart’s Electric Machine Design.........

Reagan’s Locomotives: Simple, Compound, and Electric.

Reid’s Course in Mechanical Drawing.................

Text-book of Mechanical Drawing and Elementary Machine Design.8vo, 3 00

i2mo, i 50 . .8vo, 3 00 . .8vo, 2 50 . .8vo, 3 00

Richards’s Compressed Air.........'.........................

Robinson’s Principles of Mechanism.........................

Ryan, Norris, and Hoxie’s Electrical Machinery. Vol. I........

Schwamb and Merrill’s Elements of Mechanism...............

i2mo, 2 00 . . 8vo, 3 00 i2mo, i 00 . 8vo, 3 00 . .8vo, 3 00

Sinclair’s Locomotive-engine Running and Management........

Smith’s (O.) Press-working of Metals........................

Smith’s (A. W.) Materials of Machines........................

Smith (A. W.) and Marx’s Machine Design...................

Spangler, Greene, and Marshall’s Elements of Steam-engineering.

8vo, 3 00

Animal as a Machine and Prime Motor, and the Lawc of Energetics. i2mo, 1 00

8vo, 7 50

Weisbach’s Kinematics and Power of Transmission. (Herrmann—Klein.).8vo, 5 00 Machinery of Transmission and Governors.    (Herrmann—Klein.).8vo, 5 00

Wood’s Elements of Analytical Mechanics...........................8vo, 3 00

Principles of Elementary Mechanics...

Turbines...........................

The World’s Columbian Exposition of 1893

Thurston’s Treatise on Friction and Lost Work in Machinery and Mill Work.

Warren’s Elements of Machine Construction and Drawing.

X2mo, i 25 ..8vo, 2 50 .-4to, i 00

■Steel’s Treatise on the Diseases of the Dog............

. ,8vo, 3 S® . .4to, i 00 i2mo, i 00 i2mo, i 50

The World’s Columbian Exposition of 1893...........

Von Behring’s Suppression of Tuberculosis. (Bolduan.).

Winslow’s Elements of Applied Microscopy...........

Worcester and Atkinson. Small Hospitals, Establishment and Maintenance;

Suggestions for Hospital Architecture : Plans for Small Hospital. 12mo, 1 25

HEBREW AND CHALDEE TEXT-BOOKS.

i2mo, i 25 . 8vo, 2 00

Green’s Elementary Hebrew Grammar.

Hebrew Chrestomathy..........

Gesenius’s Hebrew and Chaldee Lexicon to the Old Testament Scriptures.

Small 4to, half morocco, 5 00 ..................8vo, 2 25

(Tregelles.). ..

Letteris’s Hebrew Bible.

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