THESIS

Analysis of stresses in
' Wash. Ave. Bridge Lansing, Mich.
to Oe ae

1913

 

 
THESIS CN poo a
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3 1293 01591 3

This thesis was contributed by

Joseph Alexander Macdonald

under the aate indicated by the department stam,
to replace the original which was destroyed in the

fire of March 5, 1916.

 

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- THESIS -

An Analysis of the Stresses in the Washingtona
Avenue Concrete Arch Bridge Over the Grand River at the

City of Lensing, iichigan.

B >
- by -
. a LKR
mo oe \ “
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Je 42 idecdoneld, Er. H. tleyer.
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June - 1913.
THESMS

 
The object of this thesis is to theoretically investigate
the stability of snd determine the »aximunm stresses in the arch
ring of the reinforced ccncrete bridge over the Grand River at
washington avenue, Lansing, wichigen.

The aroh ring is of the no hinged ‘olen tyne heving a clair
epen of 120 feet, a rise of 23 Test, and « tot: 1 width of 54 feet
the width of roc dway being 3c fect 15 inches. “he arch ring is
corposed of 1 : 2: 4 Portlerd Cement Concrete with reinforcement
consisting of two 3" x 3" x 3/8" anrles, top and bottom laced.
The reinforeing ribs are nlaced 3 feet c. toa. he erch ring
ig sun orted by zdutmenta consicting cf 1: 3: 7 Cortland Cem-
ent Concrete mixture resting on rock bottor.

The dead load to be ccrried by the erch ring corsiats of
the erch ring, the earth beckfill, end e four inch bricx r<eve-
ment heving © six inch concrete foundetions. At the time of
construction the sven of this bridre w..8 ore of the longest of
its tyne, and we find the deed led to be exces: ive, as is

» - \alown by “able G. a double treck of the Tiechirin United T"reet-
ion Compeny's line reasses over the bridge, which curries the or-
dinery city ears es well <8 the heavy interurben ors of the
bensing - Jackson civision.

Beiny uable to cbtain specifierctions of the iicht-en
United Triection Compery's o: rs for the Live «heel loeds we
seleaoted a standard oir, us fiven by the Dlectric Railwey Jour-
ne} April 22, 1911. nis o.r hee @ total weteht of 80,0°9 F
supported by two truoxs. The tri-"* ..e sp.ced 35 feet o. to oc.

end heve a wheel base of aix:fect six inchos. <A standard uni-

4102497
form live load of 100¢ per squire foot was considered es cover-
ing the bridge excent for the snece ocennied by the ar, end here
considered as a strin 1O feet wide.

In analyzing tne arch we selected c section 3 fer-t wide and
computed tiie "eximun stresses duc to decd ard live loeds. In
our emilysisg we lcou.rned thet the live lord stresses due to the
car were much ererter than the stresses due-to a uniform lod
of 100¢ per squire foot covering the sc:.e urea, ond for this rec-
son we epnited the test to a section directly underneath the c:r
tr:ok. The diatribution of the live axle lo.ds to the creh ring
throveh the earth fillires, is a provlem not very well understood
and its discussion see s to be avoided by the majority cf tech-
nicsal writers. In our analysis we essurned tn..t the :xle load
wes trensmitted over a width of tem feet on the erch fivirg a
concentr..ted losd of 6290 n-unds at each axle of the interurban
cer for an erch ring section three fect wide.

The enalyticel] theory es develoned by “r. C. 4. Veliok weg
the method employed in determinine tne stresses in the aren ring
The funde“sentel fPormulee for ti: elastic theory method are dcs-
cribed and derived on Plite A, the sere being a copy oF treatre-
nt developed by cir. delick.

The éreh ring wis divided into twenty redial sections, the
co-ordinutes for tiie centre of the sceetions b- tine piven in Teble
A, ond & combinetion 07 tre pronert:'es cre round in Tebles B cnd
C. The results obtained from tie tables :entioned ahove «re

then used to ovtein tne veluves of V which «are

We) Hee ay, Gnd ©!

‘pe
tabul: ted i: Table D, end the vleues fiven in this tabie are for
@ vertical loud of 1000# concentreted et the section under
consideration, and also nerlectine the effect of extal thrust.
From the ceélenleticns shown on ?l:.te Bwe finc thet the effect
of exiczl thrust is very smell end for this reason it hes not
been t& -en into consideration.

On Plate 3 the values of H,, V,, Ms end M, were plotted
to scéle and from these the force polygons were conatriucted.
The resections civen in the forec volysrors were then resolved
tnto their respective co: nonents, thus viving the t' ruste and
radial shears at ecch of the sections for & movine load of
10007. The values for tis thrusts, shears, enc moments were
then sc.led érnd tebuleted in Tible E, ana influence lines for
the se:.e were drawn. The result: from Table " were used to
determine the top end vottom fibre stresses, which ere riven
4n Table *, due to e movine losd of 19907, and influence lines
for theese valves were also drawn. In Teble G we tave fiven
the ton and bottom fibre stresses &t cach section for the act-
nal dpad lords, on each srch section, considered us concertrat-
ed loads. These stress::s wre then sum ed un for e&ch section
for the dead loed coverines the entire bridge piving the .alves
x3 found on Plete 9.

The vnosition of the live loed for nroducing mexirum fibre
etress was d: termined by 'e.ns of « slider, which wa constrict-
ed by specine the wheels to the se: e scale «a the horizontal
scale of the influence lines, end this osition we obtained
by placing the slider on t!.e 4: fluerce Jines so that the sum
of th: nroducts of the wheel locds, in un‘te of thovs:nds of

pounds, by their respective stresses as riven by the ordinate
of the influence line, rroduced a m..ximum value, end the resuvlt
thus obtcined was the value of the Sibre stress for the section
under consideration. These vélues we fine tebvleted on “late 9.

The live end dead lozea unit fibre stresa:s were then com-
bined giving the meximun fibre stresses «8 shown by Plite 9.

The analysis for finding the shear at each section wes
e:actly the sa ¢ o8 Tor Tinding the fibre stressaeam. On “late
10 we have tenuleted the srear due to the dead Load concentret-
(on, a8 assed, and ‘n Teble J we heve siven the suwmction of
the shears for erch section produced 7; tie deed lord, cover-
ing the entiro bridge. The vilue of mexirum sheer due to the
live awle losed wes determined v.it': the glider and the in“luence
Lines for rediel shear in t1.6@ sé: 6 menner 2.8 u.8 cribs for
the fibre stresses. The combined mexi w: decd and live loed
ehears ere tebulcted on Plate ll.

The atresses in the steel were dctcr:. ned by the grephice).
method ag shown on Plate 12. ‘The coexfetenttop enc bottom unit
fibre stres=os of the concrete arch were nlotted to scale, as
shown by the horizontal lines, .nd the distence between them
beine equal to the denth of the arch ring. The unit stress in
the concrete st tie steel wee secled from the horizontal 1l‘1re
whose distance fro: the concrete stresses corresponds to the
distéenee of the steel from ti.c o.tre:e ocnerete fibre and the
value of the strese in tiie steel ts ecual to fifteon f the urs-
ured velnue o% the ™ odulue rutio™’) by the stress sezled from
tie diagram. The velue of mentmw ur't stresses in the steel

ie tadnleted in Tehle 9.
Conclusion.

By referring to Plete 9 it wil? be secn that the mexim rv
unit stresses produced by thrust an bencinge sere hish, but thet
the results obteined are vithin the slloweble limits excenrt at
the points (0) end (1) where we find & tonston of 150% ner
squere inch on the bottom 7ibre. The al’owable untt stress
is 50# ner saquere inch.

From the results obtecined in shear «8s is shown by Plste ll
it is found thet the ma-imur ueitt shearing stresses fre excess-
ive, but it is found th.t the she.r bers in the areh ring bring
the unit shesring stresses well within the allowable iimit.

The elloweble un't she.r for conerete alone in 47 # per squere

inch.

nova: Sinee the comnletion of thir thegis we heve obtained
the weiei:it end wheol specing of the heaviest cir thet “eases c-
ver the bridge, from C. 38. Mosgan, Gener }? Suncrintendent of
the .'°..7., and theyare es foltows. The or has e totel
weteht of 84,939 nounds stm-orted by two trvexs eneced 56 fT et
6 inches @. toc.

Stree the live load is bt © @ill vert o* th lord

on tic bridee, end since the ebove ccr is -ractia. lly ths s::e
as the one ve sel cted in vorkire our thes's, ve enneclude that
the results shown in ovr thes’s vonld be changed very sliently

br using the anove er.
 

Theoretical Hpalysis of the Llastic fitch.

——___

 

 

 

 

 

 

 

 

 

 

Fundamenta/ Formulae !—
Let ADBC represent a portion of the lincer arch, and fet (YP Q be any sechon
of the arc ring. Let £ on the linear arch be The center of This sectiore with
Co-ordineles (x.y) measured trom. Let the central angle Af be sublended by the
la geet OM ee ae oe

This sechore of the Arch king will be acted upon by some Resultant Arch Fressure
P which may be resolved into a Radial Shear 5,2 Nortnal Axial Thrast 7, and 2
Berding (lament Te.2 1, These Shears are stna/l trp Ordinary Arches and Areh
Theories neglect them in the Theory of Stresses, just 45 is dora in The Common
Theory of Flectare for Beams. This leaves then, tor consideration, the twe agents
Thrust 6 Bending acting on the section: these both very with the ditlerent
Sections of the arch ring. T shorlens the section DB uniformly an amount
Ta5_, 4 Kise of Temperoiure lengihens the section IL oT) aes th MLae edd a
PE Lea lengthening of the section DB=- BB's2.7.45- ery A negative
Value of M will aause the section DB te deform jn the direction DB” chonging LF
by or emount F¢. P18 meesured fromthe Verlical thra the Crown-TFositive fo
the lett, Degaitve h the kight, hence this change Sé is anegative ene. At xx' then
tee lt ee A Ta, Ma and sine wher MM is negative dg /s negative,
Mahe tae at ae he Now imagine that. fer any arch loeding, starting
AL EL ae he me aie
andthe tarn. Ther MP may be regerded as titmly Fixed fo the fast deformed
Section preceeding and the effect of the de formetiip of MANFR onthe portion of
the arch fo the Right determined. Let us tind the ettect of this deformanore
onthe movement of the point C whose coordinales are(m.n). The change nA,
B38" gwe 7 Thrust & Temperature will produce enegual chenge CC ot C. Hlso
The change 17 Af dae to 17 will produce a change af 6 of CO such that CBC>
EM Ah aa a Ie ie els ah dd
= fxr =-C6'cos $= -BB'coshe— GBP 2% zct# ax —Tax.

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ee a ee ee A a Nye ah Le of this secJior
Oe et pet A ele setay.—TAy The increase in ST la od og
‘nsihve Derding Morment 1 will be AXmm. ees a FE TM cde eh
CdD
(y-r)dp= a aoe The increase ir Kise -r- vieaet Cy ee ae LLEe | Moment
a pr eee Gn = Le or dym= (f7-x) he eas ah ae
The tofa/ inerease tir Se Thrust 7, 2 Kise in temperature of CFE a

me TT la Laer eed aaah aa on this Ssectior alone will be dx =e7ax-

poe 4 ry gale TLL TALES The fote/ increase in Rise-n- due to 2 Thrust 7, a Kise in

ye ke 2 pat 6 a fostive Bending Morment acting on this sector? wore
will be dy = et by - Tay — Mr.s5 . m/7as
AE ET Eds
Lp etxed Areh- Sym metrical- the folowing Conditions are Assumed —
The Koda! Lines at the ringing Kermain CLs ok abt deal tr Direction —- thas, Summing

7 er for a// Sections between Spring ing Foints. a eS ba a ae)
a ns i its are Fixed an Immoveable Distonee apart-/e. are Kigid -

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Plate /2.
DIAGRANS OF FIBRE STRESSES For

DETERMINING STRESSES IM STEEL.
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MICHIGAN STATE UNIV.

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