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Class Q>ZO • Q>







TRANSACTIONS



AMERICAN SOCIETY



CIYIL ENGINEERS.



(INSTITUTED 1852.)



VOL. LI.
DECEMBER, 1903.



Edited by the Ssci-etary, under the direction of the Committee on Publications.

Reprints from this pubUcation, which is copyrighted, may be made on condition that
the full title of Paper, name of Author and page reference are given.



NEW YORK :

PUBLISHED BY THE SOCIETY,



1903.



"^Gg^.S



Entered according to Act of Congress, by the American Society of Civil Engineers,
in the Office of the Librarian of Congress, at Washington.



Note. — This Society is not responsible, as a body, for the facts and opinions advanced
in any of its publications.



COIS^TENTS.



PAPERS.

No. PAGE.

956 DEFLECTIONS OF BEAMS WITH VARIABLE MOMENTS OF INERTIA.

By C. W. Hudson 1

Discussion on Paper No. 956 :

By Irveng p. Church 18

Mansfield Merriman 30

C. H. LiNDENBERGER 21

C. W. Hudson 33

957 AN INVESTIGATION OF THE PROPERTIES OF BRICK, UNDER

DIFFERENT PHYSICAL CONDITIONS.

By Sherman M. Turrill 35

Discussion on Paper No. 957 :

By E. J. McCaustland 65

Sherman M. Turrill C6

958 TIMBER TESTS. AN INFORMAL DISCUSSION.

By W. K. Hatt 67

Hermann von Schrenk 83

Gaetano Lanza 86

A. L. Johnson 90

S. Bent Russell 97

W. K . Hatt 98

959 LOADINGS FOR RAILROAD BRIDGES. AN INFORMAL DISCUSSION.

By Henry W. Hodge 105

J. W. Schaub 109

Emil Swensson 109

Theodore Cooper 109

A. J. Himes 112

Henry W. Hodge 112

960 IMPERVIOUS CONCRETE. AN INFORMAL DISCUSSION.

By R. W. Lesley 114

J. James R. Croes 1 21

J. W. Schaub 123

B. R. Green 124

Oscar Lowinson 125

Edward Cunningham 126

W. K. Hatt 128

Theodore Belzner 130

Sanford E. Thompson 130

William B, Fuller 133



isees



IV

No. PAGE.

961 AUTOMATIC MODULES FOR REGULATING THE SPEED OF FILTRA=

TION.
By Charles Anthony, Jr 136

Discussion on Paper No. 961 :

By John H. Gregoey 145

W. R. COPELAJJD 152

John W. Hill 156

Charles Anthony, Jr 158

962 TESTS OF THE EFFICIENCY OF HOISTING TACKLE.

By S. P. Mitchell 161



963 THEORY OF CENTRIFUGAL PUMPS AND FANS: ANALYSIS OF
THEIR ACTION, WITH SUGGESTIONS FOR DESIGNS.

By Elmo G. Harris lo6

Discussion on Paper No. 963 :

By William Mayo Venable 224

E. T. Adams 231

Allen Hazen 231

Joseph- Mayer 232

Theodore Horton 244

Elmo G. Harris 250



961 AN EXPERIMENTAL STUDY OF THE RESISTANCES TO THE FLOW
OF WATER IN PIPES.

By Augustus V, Saph and Ernest W. Schoder 2.53

Discussion on Paper No. 9(54 :

By A. Flamant 313

Hiram F. Mills 314

Edgar C. Thrupp 315

Allen Hazen 316

E. G. CoKER 322

(iEORGE H. Fenkell 323

Gardner S. Williams 326

Augustus V. Saph and Ernest W. Schoder 328



965 THE LEVEE THEORY ON|THE MISSISSIPPI RIVER. AN INFORMAL

DISCUSSION.

By B. M. Harrod 3.31

L. W. Brown 344

J. A. Ockerson 3.5G

Lewis M. Haupt 359

B. F. Thomas 380

Henry B. Richardson 385

T. G. Dabney 389

B. M. Harrod 400

966 SEWAGE PURIFICATION. AN INFORMAL DISCUSSION.

By Rudolph Hering 415

George W. Rafter 416

L. J. Le Cokte 422



No. PAGE.

967 SOME RAILWAY CONSTRUCTION IN OKLAHOMA.

By A. Q. Allan 424

Discussion on Paper No. 967:

By Samuel H. Lea 436

Emile Low 4.37

H. F. Dunham 4-39

J. P. Snow 440

A. G. Allan 440

968 THE FATIGUE OF CEMENT PRODUCTS.

By J. L. Van Ornum 443

Discussion on Paper No. 968:

By E. R. Buckley 446

L. F. Bellinger 447

A. L. Johnson 448

H. F. Dunham 449

J. L. Van Ornum 450



MEMOIRS OF DECEASED MEMBERS.



Fredertck Winn Bond, M. Am. Soe. C. E 452

John Butler Johnson. M. Am. Soc. C. E 454

William Shattuck Lincoln, M. Am. Soc. C. E 457

George Henry Mendell, M. Am. Soc. C. E 459

Martinius Stixrud, M. Am. Soc. C. E 463



PLATES



plate. paper, page.
I. Oven for Reheating Brick: and Instrument for Determining the

Elastic Properties of Brick 957 39

IL Curves Showing Elastic Properties of Brick 957 51

III. Brick Tested for Compression 957 53

IV. Brick Tested for Compression 957 55

V. Brick Tested for Compression 957 57

VI. Brick Tested for Tension, Bending, Shear and Torsion 957 .59

Vn. Piezometric Fittings on small pipes 964 261

VIII. Diagram showing Effect of Temperature on Loss of Head 964 291

IX. Diagram showing Temperature Effect in Brass Pipe 964 293

X. Logarithmic Plotting of Flow in Brass and Galvanized Iron Pipes. 964 299

XL Logarithmic Plotting of Flow in Pipes of Various Sizes 964 307

XIL Single and Double Railway Arch Culverts 967 425

XIIL Large Railway Arch Culvert 967 429

XIV. Box Culvert, and Reservoir Spillway 967 431

XV. Arch Culvert on the line of the Wheeling Bridge and Terminal

Company 967 441



AMERICAN SOCIETY OF CIVIL ENGINEEES,

INSTITUTED 185 2.



TRANSACTIONS.



Paper No. 956.



DEFLECTIONS OF BEAMS
WITH VARIABLE MOMENTS OF INERTIA.

By C. W. Hudson, M. Am. Soc. C. E.



With Discussion by Messes. Irving P. Chuech, Mansfield Meeeiman,

C. H. LiNDENBEEGEE AND C. W. HUDSON.



The determination of the deflections for the special types of open-
webbed, elastic frameworks often met in the practice of the bridge
engineer has received much attention, and very exact solutions are
made for these jaroblemsby both graphical and analytical methods.

The determination of the deflections for solid-webbed girders, which
is of equal theoretical interest and often of great practical imjaortance,
however, has been confined chiefly to girders with a constant moment
of inertia, and such determinations have usually been made by means
of the equation of the elastic curve. It is the purpose of this pajjer to
give a more general method for determining the deflections of solid-
webbed girders under flexure, and to illustrate its application to the
forms of girders most frequently met in practical bridge construction.
The plate-girder drawbridge, owing to its frequent use, is one of the
most important structures with which the bridge engineer has to deal.
Application of the general formula to this case has been made, with the
object of giving a simple method of making a better design for the
girders and end-lifting machinery for these bridges.
* Presented at the Meeting of May 6th, 1903.



2 DEFLECTIOKS OF BEAMS.

The writer believes that the determination of the deflection of girders
with solid webs is effected best by means of the method of work. The
general formula is derived, it will be noticed, in a manner very similar
to that commonly used in obtaining the general formiila for the deflec-
tions of articulate structures.

Let us suppose a beam of any shape in equilibrium under the action
of a given loading, and that the following derivation is made under the
usual conditions imposed in the development of the fundamental
formulas for flexure, and under the supposition that, where the
forces producing bending have components in the direction of the
length of the beam, the efi"ect of such components in producing dis-
tortion may be neglected.

Let M = the bending moment at any section of the beam due to the
given loading;
S = the unit stress due to flexure on any fiber of the cross-
section, the fiber being distant _y from the neutral axis,
and
/== the moment of inertia of the cross-section for which the
other qiiantities are taken.

31 y
Then, from the fundamental relation for flexure, we have S^= j .

Under the action of flexure, the fibers on one side of the neutral axis

of a beam are lengthened, and on the other side they are shortened.

Let dx (an infinitesimal) be a portion of the length of the beam, and

A be the change in length of any fiber distant if from the neutral axis,

due to flexure.

S' Z(
From Hooke's Law, we have A = -^ , in general, and for a fiber of

a length d x, A = — =^, in which £'is the coeflBcient of elasticity of the
material.

Since 8 = -—,

X=^dx.

Now, simultaneously with the given loading, sujipose a force of
unity to be applied to the beam at any point in any desired direction.
Let TO be the bending moment at any section of the beam thereby pro-
duced. The stress on any fiber (the same fiber for which A is taken) of



DEFLECTIONS OF BEAMS.



any section due to the assumed load of unity = -y^, and the work done

by this stress, assuming it to be gradually produced, acting through

,1 -,• , 1 • T . iny , M^i my , Mmy^dx

the distance A, is equal to -— ^ X A = — -V X -^uix= — _ .^^ ,„ — ,

Letting C d A represent' the area of the entire cross-section, the
work done on an infinitesimal portion of the length of the beam

Mm / ^ 2 , a\ J 1 ■ r- 2 7 i r XI • 1 Mmdx

''^ o^PT^ ( f*^ d A\ d X, and since | y d A = 1, this work = ^ ^ ,,

and for the internal work throughout the length of the beam due to
flexure we have:

'Jim dx



W.



-L



1)



2EI

Again, let /i be the deflection, due to the given loading, of the
point at which the force of unity is applied, measured along the line
of action of the force, unity, then the work done by the force of unity
gradually applied is :



W=-^A



(2)



Making the total work of the internal stresses due to flexure (1)
equal to the total work of the external forces (2), we have:






Mm dx



2 El

M VI d X*
LI ■



(3)



which is the general value for the deflection of any point in a beam.

As an illustration of the general ajaplication of this formula, sup-
pose we find the horizontal deflection of the upper corner of a simple
I-beam span, loaded at the center with the weight P. Fig. 1 will
make the j^roblem more clear.

P



Unity-






" 'A



1-

FlG.l.



.Unity



* This formula was first derived by Professor Fraenkel.



4 DEFLECTIONS OF BEAMS.

Taking each half of the beam separately, and taking the ends of the
neutral axis of the beam as the origin of x, we have :

' Mmdx r—Mm.dx „ ,, , n, i t»
-=-^^ — for the left half
hi 1



_ /•' Mmdx _ rL

-J o ~ET- -J ;

_|_ / "2" — ' for the right half,
J o E J



in which.



p

31= + 75- X X, for both right and left halves of the beam,

m = + ^ X X, for the left half of the beam,

h h

and m =- ^ X .r -|- -x, for the right half of the beam.

Substituting these values for M and m in the general equation, we
have:

2 P h X d X



r- phx^dx /"t Phx'd x r-.



4: EI



=/:-



T-Phxdx Phl-



4: EI —d2EI-

If m make P = 29 000, I = 360 ins. and /= 1 140 (inches ^"'), this
being the value for the moment of inertia of a 20-in. I-beam (192 lbs
per yard], then:

. 29 000 X 20 X (360)' 1 .

^ = 32 X 29 000 000x1140- 14 '"•' ''^^''^^-

Iq the problems of practice it is usually the vertical deflection of
some j)oint in a horizontal beam under vertical loading that is desired.
The following five problems are deemed sufficient to show the applica-
tion of the general formula to finding such deflections. The girders in
the following problems have four values for the moment of inertia; the
resulting expressions for the deflections, however, are of such form
that expressions for the value of the deflection for a girder with a
greater or less number of values of the moment of inertia can be
written by inspection. For girders with inclined flanges, values of the
moment of inertia can be determined at stated intervals, and these
values considered constant for the intervals taken.

Problem No. 1. — Find the vertical deflection at the center of a plate-
girder span with three cover-plates, the girder being loaded uniformly



DEFLECTIONS OF BEAMS.



tlii'ouglioiit its length, /, with a k-
vertical load of ir ijounds per unit j
of length. Fig. 2 will make the p
problem more clear.

As the moment of inertia of the
girder has four values, the second
Ijart of the general equation.



- *(



w Pi mnds per linear unit



J



/Mm dx f-



EI
will have four parts :



31 m d X
El^'











_


1 Unity

~1 ^




1
1


r

h-


1

— t


h


1
1


h


' T '
1 -'-J 1


h


-1 \

1 /. 1 h

-\ \ -


1


— 1=
1
1




1


^^


1 1
I 1






< — b


- — ->i




1




1 1




U— 6—- *


e


-c—





-i)




1 I





c


l(




. _








J^-








Fig. 3.



■ — x^) d X



. _ w r'^ [I X- — X'' ) d X w r ( I X- — X

~2eJ o Ti ^ 2eJ a I,

j_

,0 r' {l.-c' - x')dx w r^ jlx'-x'

+ 2eJ b l, "^ 2EJ c h



2E l37i'~rTi'^



8/,



473 +



4:1,

16 c'



+ 1



) d X
c'-b^)



SL



24/,



wl'



+



w I



384 EI^ ' 6 E



(



64 7,






+






8E\1, /, "^ X~X"^"J,"~^/



which is the expression for the value of the desired deflection.

If Ii = lo = T^^ I^, that is, if the moment of inertia of the beam be
constant, the expression becomes:

^ 380;7'

which is the ^Vell- known formula for the deflection at the center
of a simple beam of constant moment of inertia, under uniform
loading.

Problem No. 2.— Find the
vertical deflection at the center
of a i)late-girder span with three
cover-ialates, due to a vertical
load, P, at any jsoint in the
span.

The load, P, will be placed T ^ j^

on the first cover-plate, and the Fig. 3.




6 DEFLECTIONS OF BEAMS.

expression found for the value of the deflection. The expression
for the deflection for any desired j^osition of P can then be written by
inspection.

y*i Mm d, X

..« P (1 — k) .r d X rki P [1^ k) x^ d x



A =



ra P [\ — k) X- d X ri^-i
Jo 2^ A +J«



2 JBJI2

•b P k {I X — x") dx re p k [I X — x^) d X



/'!} p k [I X — X-) a X r c

/^r P k [l X — X-) dx re P k x^ dx r
^ 2^^ +y, 2^/, V.

nc Pkx^dx r^ Pkx? d X

+J , 2EI, J l-^Wir



2 EL



P ( 1 - k)
6E






P_k_ ( l^ __i^_Llf_i£.,_l__ f_^\
"•" 4 ^ V /, li h /i 4 /, I, )

^E \ I, I-i h ' I^ ^ 9>I^ I, )



Pk

h^ c' c' . P



'^ 6E\ I^ l, 1.2 h ' ii l^^SlJ

~ 6 ^ \ /i lJ 12E I,

, Pklf H' }? 6' c' \ , Pk l^



/ b- b- c- C \



which is the expression for the value of the desired deflection.

For a load, P, on the second cover-plate, by inspection, can be
written :

P Ti'' f_



6E \ /, ~ 7, + 1.2 ~ h)



VI EI.^

Pkl / r c^\ P k f'

+ 4:E\7:^ i7/ + lG^//

If we make /j = /^ = /j = /n in either of the above values for J,
we have :

and if A- = — , that is, for P in the center of the girder,

~4:SEI'



DEFLECTIONS OF BEAMS.



Problem No. -5.— Find the
vertical deflection at tlie loaded
point of a plate-girder span with
three cover-plates, due to a
vertical load, P, at any jjoint
in the span.

P will be placed on the first
cover-plate, as in Problem No. 2.

J — - r M m d X
J o ~^~I~




J kl



P (1 — kf X- d X
E I^

PTc' [l — xfdx
EL,



+



+



r^"' p (1 — ^f x^ d x

J a E I2

r Pl^ {I — xf d X
»' h



-{-f^ P k^ {I— xf d X f
•^ c EI, +-^ o



EL



P k^ X- d X



EL^



+



r

«/ a



PT^oi? dx



EL,

p a — kf



3 E

p k^ t'






+
+

kl



f



b



P J^o^d X
EL,



+



/;



P k^ x^ d X

—ejt'



i)



E



/ K I C



+



2 7,



i)

P k^ I /b^ _ Jrjr , _c^ _ _^ , J^ ^\

- {l, "■ L, -^ L, /s "^ 4 1, I, )

vx r + /3 ~ /3 + 8 /, - /, ;



E

P k^



-r- +



SE \ I, h ' h L, ^ SL,

p k' ^y_ ^ Jy^ _ —^



i!: + -^_
' ^ h



f-



TeKJi ' 1-2 - ^3 ^3

pi'T^ f 1 2fc ^-2 N P(i



8
2 A-



+



2,E

PT^f / b^
E \L,

Pk^l






E
2Pk''-



\L~~L,



+



+



c c \



C"

&3



'L r K' / a" a" . 0" ^^ a. ^^ — \

which is the expression for the value of the desired deflection.



DEFLECTIONS OF BEAMS.



If we make 1^ = 1.^ —
_ P f k'' (1 — kf,



A = —"^l



Py_.
4S>E'I

Problem No. '4.— Find the
vertical deflections of the ends
of a plate-girder sjjan with three
cover-plates, sujiported at the
center, only, and loaded with a
uniform vertical load, iv, per unit
of length.

M m d X



Zj = /^, then
and, in addition, if we make A: = -^' then



mimwi,



rW pounds per linear unit



Unity



Unity



EI



r

*J o

y " «' x^ d X I" w x^ d X r*




w



2 EI.

I'



which is the expression for the value of the desired deflection.

If I^ = I, = I.^ = 4 then,
_ w l^
~ SEP

Problem JVo. 5.— Find the
vertical deflection of the ends of
a jjlate-girder span with three
cover-plates, supi^orted at the
center, only, and loaded at each k- — 6-
end with a load, P. t"

k

= /'

*^ O



A =



Mm d X
EI

P x^ d X




+



f^ Px^dx r_p^^_dx f

J a EL ^ J b Els -^ '



EI^ ' J a EI,
.,3 z,3 „3 „3



^ Px'dx

: EI^



= /

- JL (iL j^^ _fL X. "^ ^' ^' _^' ^
zeKi,^ I, h. I, ~h^x~ rj

3^7, +3 ^V/, L'^' I. ~T^^ I^ I J
which is the expression for the value of the desired deflection.



DEFLECTIONS OF BEAMS.

If /^ = /, = I.^ = I^, then,

P f



By substituting for A, in tlie above exiiression, the distance it is
desired to raise the end of a plate-girder draw, and solving for P, the
uplift necessary to raise the end the desired amount is obtained.

In' applying the results of the preceding problems, a deduction of
2 or 3 ft. from the over-all lengths of the cover-plates should be made,
for the reason that enough rivets to fully develop the value of a cover-
plate to resist stress are not generally contained in a less distance than
the first 2 or 3 ft. at the end of the cover-plate.

By the aid of the results of Problems Nos. 4 and 5, the reactions for
center-bearing plate-girder draw spans, with equal arms under uniform
loading, can be obtained.

Let iij = i?3 be the reactions at the ends of a two-span plate-girder
with three cover-plates. From the result of Problem No. 5, we have, for
the vertical motion, J, due to a force, R^ = i?.,, at the ends, the fol-
lowing :

_ R^ f R, /r? g' b\ __ &^ , c^ _ _^\

3 ^/, "^ 3 ^ Wi h I2 ~^^ h h)

If, at the same time, we make the value for A given by the result of
Problem No. i equal to the value for A above, and solve the equation
for i?j, we have:



10 I*
8T



3i, + 3 Va /,"'"/, Z+h~lJ

_ 3^ T, T,'~%'^ T^~ h^ I^~7,

~ ' 8 P rr^ a' b^ b'^ , c' c"

which is the expression for the value of the end reactions for a plate-
girder of two eqiial spans having four values for the moment of inertia,
and loaded uniformly throughout its length by w pounds per unit of

length.

3
If I^ = I, = I.^ = I^, then i?! = i?3 = -g- w ^.

It would not be a difficult matter to derive values for the reactions
for a more general case of loading and span lengths; but the ordinary



10



DEFLECTIONS OF BEAMS,



plate-girder draw has equal arms, and the only place, under the gen-
eral practice in designing, where it is necessary to consider the effect
of continuous loading is near and over the center support. The
maximum values for shear and moment at the center support are given
under conditions closely approximating continuous uniform load, and
this is the problem we have solved.

In order to see how much the ordinary method (using a constant
moment of inertia), of computing the moments over the center supports
of plate-girder draws differs from this more exact method, the com-
parison will be made for two widely different cases.

Case /.— A very light, single-track, center-bearing, plate-girder
draw, 72^ ins. deejj, from out to out of flange-angles.



Lena:th


Effective








over all.
1 web plate, 72 x f ins. 138 ft.


length.

and 136 ft.




Y~-\


r


4 flange-angles, 5x5 '• 138 "


" 136 "








(55 lbs. per yard.)










2 cover-plates, 12 X | " 98 "


96 "




.2




2 " " 12 X 1 " 22 "


20 "




5




2 " " 12 X 1 " 10 "


8"




i-




/i = 38 284 ins.* = 1.85 ft.* and


a = 20 ft.








I.^= 58 335 " = 2.81 "
I^= 79 081 " =3.81 "


b = 58 "
c = 64 "




L_j


L






/, = 100 526 " =4.85 "


Z = 68 "




Fig. 7.


-^ = 4 324.3, 4^ = 2 847.0,4^ =

h ly Ix


8 648.6,




= 5 694.0,


-y- = 69 434.9, 4- = 51 ^10.5, ~ =

l-l J.i Jo


4 027 206.3,


b*


= 2 970 209.0,


e' r^ ,■'
-jr-= 68 804.2, - = 54 050.3, \-
h Ji h


4 403 468.8,


^4


= 3 459 219.8,


f Z*
-y-= 64 831.3, 4- =

^4 A


4 408 531.1,






-)- = 207 394.7 — - 108 107.8 + =


12 847 854.8


= 6 435 122.8


I' a' a' b'
E - '-«■ ^^ ^' ^^ ^^ "


I. ^ J,





^4




8 ^i ,/ ^,i ^;.
^4 "^ A J. ^ I.


b' c'
h "^ h





C*
^4


3 12 847 854.8 — 6 435 132.8
~ 8 '^ 207 394.7 — 108 107.8 "


= -|- w 64.59 -




= 24.22 w.





DEFLECTIONS OF BEAMS.



11



The moment over the center support for this case is:
il/= 68 10 (24.22 — 34.00) = — 665.04 ic foot-pounds.
The moment over the center support for constant moment of inertia



31^-



— wV'^ — 578 »' foot-jDounds,

which shows that this more exact method gives a moment over the
center support for this case 15% greater than the usual method.

Case II. — A very heavy, double-track, center-bearing, jilate-girder
draw, 101| ins. deep, from out to out of flange angles.



1 web pLate, lUl X
4 side flange-plates, 18 x f
4 flange angles, 8x6

(125 lbs. per yard.

2 cover-plates, 20 X \
2 " " 20 X f
2 " " 20 X f
2 " " 20 X f

/j = 353 750 ins
/, = 434 500 "
ig = 517 600 "



Effective
length.

ins., 159 ft. and 156 ft.



Length
over all.



159
159

159
84
24
16



156
156

156
82
22
14



= 17.06 ft.* and rt = 37 ft.
= 20.96 " " & = 67 "
= 24.96 " " c = 71 "



I^ = 603 100



= 29.08



I = 78



-^ = 2 969.1, ^ =
A -'2

4- = U 349.4, 4- =

1.-, lo



2 416.7, ~ = 109 857.3, -f
12 049.9, -^ = 961 408.5,-^



-^ =14 339.3, — = 12 307.8, 4" = 1 018 096.2,



A



1.

+



= 16 318.8,



I'



1 272 869.9,



47 976.6 — = 26 774.4 + = 3 362 231.9



1 770 606.9



„ 3 3 362 231.9



1770 606.9 3 „^ ^„ ,,^ ,^

= - w 75.07 = 28.15 w.



47 976.6—26 774.4
The moment over the center support, for this case, is
31= 78 IV (28.15 — 39.00) = — 846.30 iv foot-pounds.



12 DEFLECTIONS OF BEAMS.

The moment over the center support, for constant moment of
inertia, is:

M r= — 10 p =^ 760.5 w foot-pounds.

o

Which shows that this more exact method gives a moment over the
center support for this case 11%" greater than the usual method.

The two i^late-girder drawbridges selected for investigation may be
said to represent fairly the results obtained by the method commonly
used in designing such structures. In view of the restilts of the
investigation for these two cases, it is fair to assume that many center-
bearing plate-girder drawbridges are stressed considerably higher, for
a short distance over the center support, than their designers intended
they should be. This condition of affairs, it is believed, is quite gen-
erally known, but, as far as the writer knows, has not influenced
designers. The additional material required for a more correct design
is a very small percentage of the whole weight, and, considering the
increased strength it would give, the addition should always be made.

The objection may be made that it is hardly necessary to go to any
great refinement in designing plate-girder drawbridges, as it is not
certain that the end machinery will act as designed, and, further, even
though the condition of the supports be as assumed, until account is
taken of the effect of the shearing stresses in producing distortion,
the true reactions cannot be obtained. The fact that the end supports
may be out of their proper j)osition, and in such a manner as to cause
increased bending moments over the center support, is an argument in
favor of the most refined methods in designing the structure. It is not
customary to consider the effect of the shearing stresses in i:)roducing
distortion, in designing plate-girder drawbridges; the method used in
investigating Case I and Case II is simply a more refined application
of the usual method (which considers only flexural stresses). In order
to determine the effect of the shearing stresses in producing deflection,
the following investigation is made:

Let us take any portion of a beam in 1

equilibrium under the action of trans- ^^ — tF - j

verse loading, as shown in Fig. 9. If P =^ j "

be the resultant in position, direction " ^ >

and amount of all the forces to the right 'S ^ „

'^ I Fig. 9.

of the section ?« n, then it is clear that I"-



DEFLECTIONS OF BEAMS. 13

the poi'tion of the beam under consideration wonld be kept in equi-
librium by the couj^le, Fa, and the shear, S.

In general, no matter what may be the actual distribution of the
stresses on any section of a beam under a given loading, these stresses
may be resolved into an eqiiivalent couple and shear. The effect of the
couple in doing work has been determined in accordance with the
common theory of flexure. Going back to (1), and rewriting the
■expression for the work of the intei'nal stresses, we have
'^ Mm dx



- for the work of the flexui'al stresses )



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