Design of a 3 Span
Double Track Reinforced
Concrete Railroad Arch Bridge
C. S. Millard
G. A. Haggander
Millard, Chauncey S.
Design of a three span
double track reinforced
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DESIGN! OF A THREE SPA IT "^^
DOUBLE TRACK REI?T FORCED CO 1^I CRETE
RAILROAD ARCH BRIDGE
A THESIS PRESENTED BY
PRESIDE IT T and FACULTY
ARMOUR I H S T I T U T E OF T E C H ZT L G
for the degree of
BACHELOR OF SCISImCE IIT CIVIL ENGIXEERIITG
HAVING COICPLETED THE PRESCRIBED COURSE OF STUDY
CIVIL E AT G I N E S R I H G
ILLINOIS INSTITUTE OF TECHNOLOGY
PAUL V.GALVIN LIBRARY
35 WEST 33RD STREET
CHICAGO, IL 60616
Chicago, 111., June Ist, 1907.
Dean of Culture Studies.
The bridge, as designed, is a rjroposed structurb for the
C. V. I'T. W. R. R. across the ITorth Branch of the Chicago River,
about four miles above the head of navigation. At the present
time a v/ooden pile trestle carries the railroad across a rat'aer
shalloif; wooded valley. This vallej- of the i;.pper waters of the
ITorth Branch has been under consideration, by various improvenent
associations, as a desirable link in an Outer Park System for the
city of Ciiicago. V/ith ohis fact in nind, the bridge was provided
with two forty foot arches to acconnriodate future driveways.
METHOD OF DESIGiJ.
The method of design used in the aiv^lyses of the arches,
was the graphical system developed by Ivlr. Burton R. Leffler,
Bridge Engineer for the L. S. oc M. S. R. R., in his treatise on
"The Elastic Arch." In indicating the method of design, the cal-
culations for one of the forty foot arches will be explained.
THE FORTY FOOT ARCH.
Two assumptions must be nade to start with, (l) the thick-
ness of the arch ring at the crovm, (2) tlie curve of the intrados.
The thickness of the ring at the cro^vn was assigned to be 2 feet,
the curve of the Intrados v;as struck from 3 centers.
The theory is based on an equal nuraber of horiiiontal di-
visions of the arch ring. The divisions wer^; made on the dotted
ordinates with an ordinate at the center. See Plates I, II and
III. This deteriiiines ^ at the crown, ^.-here dl is the length of
^ a division iieas^jred along the gravity axis, 'which is yet undeter-
mined. Por a trial valae, dl is measured along the mtrados.
I ia talcen at the csnter of each division of the gravity axis, as-
suming a eidth of arch ring of 1 foot. The reiriainder of the arch
ring must be taken so that — is a constant.
Where al' and l' are taken at the springing of the elastic ai"ch.
dl'=2.3o dl= 2 I =11 ^'"T^
h.= depth of the ring at the springing.
h' - 2»5j X 12 _ „ -,
The span of 40 feet is not the true span of the elastic
arch. The span of the elastic arch lies "between the points v:here
the tangents to the arch are fixed in direction. These points
may he located at that portion of the ring where a sudden enlarge-
ment of section takes plstce, or where -r- ceases to he a constant.
In the arc]-- under consideration, this length of span was taken as
The live load was taken as the equivalent uniform load for
Cooper's ElffO with lOOp impact added. The load was considered to
he distributed over a width of 12 feet. Prorri Cooper's Specifica-
tions for Railroad liridges, the equivalent i^niform load for E40
with a span of 32 feet = 7120# per foot. Por EI60 this equals -r
X 7120 = 10680#. With lOO;:^ impact added this equals 213 60# per
foot. In designing the arch., for convenience, a section 1 foot
wide was used. The load on a width of one foot = 1780#. 1800#
per linear foot was the load used. The voltime of the dead load
was scaled from the drawing- The weight of the earth fill was
taken as 100# per cuhic foot, and the w^eif^ht of the arch ring 150#
per cvtic foot. The volv^.e of fill at section = 1.5 x 2 = 3
cuDic feet. The weight of earth = 3 x 100# = 300,-^, The weight
of the arch ring = 2 x 2 x 150 = 600#. The total load at the
croTvn = 900# + 3600^ = 4500#. The length of a division was taken
as 2ft. and thjerefore the live load per section = Z&004h
THE DETERMINATION 0? THE TRuE VALUE OP "H. "
In order to determine the naximun stresses at the various
sections, at least three positions of the live lead rust he con-
sidered. The positions used were 1/2, 3/4 and full loaa. Talc-
ing the position of half load as an example, the load line was
laid off vertically, each, load being nunhered corresponding to its
position along the arch. Assuming H, the horizontal thrust as
40000#, a trial stress diagram, and equilihriun polygon was con-
The next portion of the prohlem, was to locate a line m n,
(See Plates) so that the sijan of the ordinates. called " ordinates,
from tiiis line to the trial equilihriujii pol;v"gcn, would he zero.
All ordinates were '"leasured to the sa^-'-ie scale as t'lat of the arc)'.,
z/s inch = 1 foot. The method of deternjning m rn,can better he
understood \iy referring to the table on Plate I. Column 1 cor-
ta.ins the nurber of the section. Coluinns 2 and 3 give thi.e ordi-
rates to the trial equilil:ri\jn polygon on the right and left sides
of the center; 4 gives the difference betv/'een the right and
left ordinates of the sane section; 5 reives the siirr":;:ation of
Z(^c// ^ ^dz i--3d3 ^ Vc/y)
This siirrr'nation ^var v.sed in +he foi'^r'la for deterriining v w
W^'ere n equals the nurater of equal Fpaces that the arch was di-
vided into. To determine t]xe direction froii in v\^, v vi' v/as laid
off ahove V and v, ir dravm . R, the sum of the ordiniites in col-
umns 2 and 3 of the table = 48.95.
The formi.ila, sn:y gives the distance that n .a, is ahove v, \v
at the niid-ordinate of the polygon.
_R_ = ^8^95 = 2.38
Having dra^vn m m, parallel to v, w, the in ordinates v/ere
measured. These ordinates were recorded in colvjnns 6 and 7.
The y ordinates are given in column S of the tahle. They
were measured from the line joining the springing points o y^ to
the gravity axis of the arch ring. The line k kj^ras ^ext dravm
parallel to o v, and at a distance above it equal to _ ^SL The
line m m,cv.ts the polygon at E and E^. Projecting these points
upv;ard to e and e,, in the line k k^, locates 2 points in the re-
quired pressure curve. The Ic ordinates v/ere measured from Ic k, to
the gravity axis of the arch. The si-^mrmticnsz^T-TTW and^7iy--!>corded
in ccliTirDTS 12 to 14, are the enms of the prodi'^cts of rn, y and k.
"To determine the tme pole distance, the formula
■^— 1^ X Trial Pole Distance is neces;jary. If the true value of H
was as£^a:ned in the first place, tlien^ny =^hy.
True Pole Distance = ||-^- ^ 40000 = 47100#.
To locate the true pole P, P, r v/ac dra,wn paral3 el to in m, ,
P then lies on a horizontal line tlirough r.
In the fundaroental equation ^ ~ T "^ T~~ ^ ~ stress, T the
thriist , A the area, M the moment, c the distance to any fitre and
I the moment of inertia of the section. In this anal^/sis of the
arch the effect of j- has heen oi^iitted.
By experirr.ent it has beer, determined that the effect of t
is to decrease the value of H. The experiments of Prof. Hov;e
h^ve shown that for an arch havinfi; a rise of ■5- the span, the true
value of H is 93-l/2> of the approximate value; for a rise of
1/6, S6;o; and for a rise of l/lO, 69:1. In the arch, designed,
the rise was 3 feet 10 inches, and the span 32 feet, giving a
ratio of o— ^-r* S" interpolating "between the values already men-
tioned the correct percentage for H v;as determined.
True Pole Distance = 72.6;b of the approximate value.
Having determined the trte pole distance, a nev; stress
diagram and equilibrium polygon vras drav/n. e and e, being points
on the required curve, the new polygon must he drawn from one of
these points, and as a ch.eck on the accuracy of the work, it
should pass through the other point. Ilie pret^suro curve having
"been located, it only renair.s to determine the unit stresses
The above figure is a side view of a portion of an arch
ring contained "between two planes perpendicular to the neutral
surface n n' and mahing an angle oc in circuJLar measure, before
strain, between them. A vertical plane ir^idway between the faces
of the arch intersects the neutral surface in the line n n' = As
feet in length, v/hicZi may be calJed the neutral line. The forces
considered all act in this plane.
Let R be the resultant of all external forces acting upon
the section passing through r . Conceive a;pplied at n tvTO opposed
forces +R and -R, each equal and parallel to R. The single force
R is thus replaced hv a couple RR and a force +R acting at n.
The latter may be resolved into components T and IT, tangential and
normal to n n' at n. The force T causes a unifon:i shortening in
all the fibres. The force N is a s}ieari:ig force and jnaiT- be
neglected in de-termining the longitudinal stresses.
The couple RK is principally effected in claanging the
curvature of the arch and its nonent is nost conveniently found ty
"■-.t'ltiplying its horizontal cor^ponent H by tiie vertical distance
from n to R, v;hich 7/e can call t feet. Then ~re liave
M = Ht,in foot po\/nas (j.).
Under the action of this couple the angle oc is changed to or'
and the curvature is increased if R exits the section "oelow n, and
decreased when R exits the section ahove n.
Gall or'- cxT =^o<c and regard M as right handed.
Call distance of any fihre from n n' i^ , this he ing +
ahox'-e and - celoiv. The length of the f ihre hefore flexure
^/il^y- Kcx' after flexurb = A~5 -^ l^^' its change of length is
1/ f<x'- oT ) z^ l^Ac< Calling its cross section a in
sq.uare feet and the unit stress di^e to jY = f i:ounds per square
foot, the stress on the fibre of concrete
and of steel
fa - -^^^/ an^^ /^)
Since f = elo-^j-^ation of fi'cre. -g
since I length of fibre ^ ^
In (3) 71 = 0^ -.There E/= modulus of elasticity of concrete and
E^ = Exodulus of elasticity of steel. A^ -h ^oC can be re-
placed by A3 '.vithout appreciable error. T>ic sujii of all the
stresses d^Jie to flexure on the entire section at n for concrete
or for -Steel
The so-ment of the stress (^3. fj ahoi?.t n on any fihre is(^afv;)
Let ly = ^no^ent of inertia of the concrete of area k, in
feet, and 1^ = moment of inertia of the steel of area A^ in feet
Z^y'^a = Z^ (^l/^cz) for concref-e-
= Z fi/"^^^J for ^tee/
^ ==^/ ^ rL^nJ,)
/ / -^/ -r / /u^
Let f; =■ stress per square inch on an extre^ae fibre of the
concrete -/hose distance from the neutral axis is v, feet. Then
fro:P. /^; /. = ^^~ ^^
and eliminating ^^ hetTveen this e equation and (6) ;7e get
-f - M ^^
and for steel,
Tz - -zr
The direct thrust on tiae arch must now bw found. Let P
the uniform compresaion on concrete of area A, and let np 'oe the
unit compression on steel of area A_2>
Then the total corapresbion on a section = F'f/l / t n /jz)
This is eiual and opposite to T.
^ /jii-n/iz ^^ ^,i-n/]z
The total stress "i^oiild nov,' he the su:r. of the "bending
stresses and the direct thrust. Let s, and s^ he the stress in
pounds per sq'.>are foot on the concrete and steel respect irely at
the upper and lower edges.
-7/ /- /7 Jz
S^ - / 71 ^ A^//^ ]
As an exanple take point 1 on the 80 foot arch.
Live load Thrust = 100,000#
Temperature Thrust = 15,6Q0#
T = 115,600?^
Live load Sending Moment = 133,000 ft. pds .
Temperature " " = -:rO,000 " "
M = 173,000" " "
A, = 3 square feet
A_g= .O-ioo s^i<are leet
n = 15, I; = 2.26, I^= .Ohoo x 1.25
Y^ = l,c, 7^= 1.25
S/ _ 1 15^600 + 175 ,000 x, 1_. 5^^_^
144 5 + 15 X .0435 "2.25 x".0455''x' iTl.o ^
ijounds per square incn
Sg g. lln_,600 + 175^000 x 1.25__ _ .^
144 3 -f- ir; X .0433 -2. 2o* +-'l'5* x' ■.0'4o3''x l"2?j ^
-10,150 ^ . .
+ 5 600 P^^^'-'-^^s per square men.
Let to = rise or fall of tenperature in degrees Panreniieit >
D = span in feet
f = coefl'icient of expansion of concrete.
The lengtheninf: or shortening of the span is Dtf.
This is tne horizontal jnove-ment along the X axis and is
also given by the fon:aiila
which is one of the equations lea-din^ up to the fundarental equa-
Since the end tanfrents are fixed in direction 2/t - O
v/hich. means that H acts alon« the line K K^.
/tl/^ is the sane as^-^^^^^which v/as found in computing the
true pole distance.
The range of te^Tiueratiire vvas 2S'' . If t/ie arch v/as oullt
at 55° the range .?ould be from 25° to 31°. The concret;e is quite
massive and is also covered with earth so tl-iat this ranp;e is un-
As an exa'nple we v/ili fina H for tne 80 foot arch.
I=|- dl = 4 D=64
t =23° E = 144 X 2,000,000
/Z^^ 102.76, f = .0000053
\ X '54 X 23 X .0000056 x 2000000 x 1^4
H = 2 — ; -n-7vT~r--c = 15,o00#.
4 X 102.7 6 '
The direct thrust of the SO foot arch for full load is
125,000 pounds. The direct thi'ust of the 40 foot arch for no
load is 48,000 pounds. The weight of tht concrete and filling
between the points considered in determining the above t-nrusts is
60,600 pounds. Combining t?iese graphically and obtaining the
resultant gave a thrust of 135,000 pounds acting diagonally
thro'ugh the center of the pier base. The vertical component of
this is 17 2,000 pounds which must be taken by tne foundation per
foot of width.
ConaiderinK the arch as 24 feet wide (since the load from
each track wae diBtribbted over Ik^ feet) v/e get a total load on
the foundation of 4,128,000 pounds or 2,064 tons. Thisi load is
taken "by 98 piles giving a load of 21 tons on one pile. Tnit> rcay
soerr exf;esuive, hut the impact \7hich v«afc> included in the thrusta
never reaches the piles, but is absorbed by the inertia of the
fillinr and concrete.
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