.~.

..

COMBINFD BENDING AND TORSION

OF

PLA'fE GIRDERS

Cooperative Investigation By

Lehigh University

and

Swarthmore College

Sponsored By

Pennsylvania Department of Highways

and

Bureau of Publi c Roads

Conducted 'By

Gerald G. !tubo Samuel T. Carpen'cer

Bruce G. Johnston William J. Eney

Final Report

Fritz Laboratory Project Noo 2158

(Not for Puhlication)

'-

I

II

TABLE OF CONTENTS

S~'no'psis. •

Introduct,:i.on '. •

. . 1

2

IV

Combined Bending aJld To~C'siol1 Theory - -Summary

E:{perimental Program

5

10

v /I.nalytical St.udy 0 f Results • Co 0 l;I 27

."

....

AppendiJt A

Appendix B

Appendix C

Conclusions

Bibli 0 gra.phY

AcK.'1owledgment s

Illustrations

?

o ., •. () 0 ;).

44

45

'.

"COMBINED BENDING AND TORSION OF PLATE GIRDERS"

CHtlPTER I SYNOPSIS

An extensive investigation of plate girder behavior under torsion

loading was undertaken by the FrItz Engineering Laboratory of Lehigh

Universl ty. This report covers the third phase of the project, the

experimental test program of which 'Was carried out at Swarthr.lore College.

Full-size plate girder specima~s, approximately 4 feet deep and 38

feet long, of bolted and riveted construction were subjected to combined

bending and torsion loads. A concentrated vertical load was ap~lied qy a

testing machine at midspan while the specimen was simply supported, free

to warp but restrained against twi sting at the ends. The behavior of the

specimens under various combinations of lateral eccentricity and height

of load api'lication was observed. The transverse and longitudinal

variation of the hending and shear stresses, the vertical and lateral

di splacements of the section, WId the angule:.r distortion of both web

and noogs were Dieasured by SR-4 gages, dial gages and level bars

respectively. The experimentally-determined stresses and di stortions

were compured 'with corresponding computed values obtrdned from repre­

sentative theories.

The results in general confirmed expectations in that the linear

theolj' , admittedly approximate, gave values on the low side, espe_ci;l11y

in the case of the critical flange normal stress. On the other hand

the values based on the non-linear theories, rigorously incorporating

the effect of deflections, checked satisfactorily with measured values.

1

CH~TER II INTRUDUCTluN

A. Program

A comprehensive investigation of the behavior of bunt-up

structural members when subjected to loading conditions involving

torsion was undertaken by Lehigh University at the Fritz Engineering

Laboratory. The complete program was divided into three phases•

. The first phase covered the basic uniform torsion case of a plate

gl~.'der subjected to a shaft loading without end restraints. Both

Sllallow and deep girders were tested hy Chang and Johnston at Lehigh

University as part of project No. 211.(10)

In'the second phase a series of shallow plate girder specimens

were placed under non-unifonn torsion by means of a shaft loading w1.th

end restraint. These tests were conducted by Kubo, Johnston, and Elley

at Lehigh Universi ty under proj ect No. 215A. (11)

The third phase of this program is based on combined bending and

torsion loading tests of plate girders of average proportions perfonned

at Swarthmore College by I{ubo, Carpenter, Johnston, and Eney. This

project, designated No. 215B, is SUlJJlIIarized in this report.

B. Sponsorship

All three phases of this program have been SUPi;orted by gr~ts from

the Department of Highways of the Commonwealth of Pennsylvania in

cooperation with the Federal Bureau of Public Roads. Both Lehigh

tJniversity and Swarthmore College have made available thei r testing

facUi ties without charge.

Note: References designated by a superscript number are summarizedin the Bibliography, AppendiX A

2.

C. Classification of Torsion

A member subjected to twist is under unifo11ll torsion if there is

no restraint of the warping tendency. This condition is also knOlolll

as the St. Venant or pure torsion. A member of constant torsional

rigidity under such conditions mil have only torsional shear stresses

and will exhi'lJit a constant unit angle of twist. Among the common cases

of uniform torsion are the following:

1. Cylindrical member without end restraint under shaft loading

2. Cylindrioal member with end restraint under shaft loading

3. Non-circular section without end restraint under shaft loading.

When the warping tendency of non-circular sections is restrained,

ei ther by physical means or by virtue of sy:nnnetrr, then nomel and

shear stresses due to flange hending are produced in addition to the

torsional shear stresses. The unit ungle of twist is no longer co~stant

along the span. The angular distortions are reduced in magnitude com­

pared to the uniform torsion case. Representative cases of non-uniform

torsion are the following;

1. Open section, restrained at one end, subj ected to & shaft torque

2. Cantilever beam acted upon by an eccentrically applied load

3. Simply supported beam with a transverse load applied eccentrically

at some intermediate point•. This combined bending and torsion

loading is the one most commonly met in practice.

D. Objectives

This third phase of the non-uniform torsion project was designed

to investigate the behavior of full-size plate girders under combined

bending and torsion loading. Although formulas based on superposition

of effects have been widely used to compute stresses and distortions,

it 1s recognized that they are ba.sed on assumptions llhich are not

strictly correct. The spec!fie obJ ect!ves of this investigation were

as follows;

1. To determine the resultinR distribution and variation of the nonnal

and shear stresses in the web and nange of plate girders and to

check against theoretical computed values

2. To measure the angular and linear distortions and to compare the

results with computed values obtained from the available theories

3. To investigate the non-linearity of stress and distortion due to

the influence of the deflections on the bending and torsional moments

4. To compare the efficacy of the available theories in predicting

the characteristic behavior of plate girders

5. To examine the effect of section properties on the resuits, both

experimentally and analytically.

E. Scope

A lim! ted yet representative group of full-size plate girders of

bolted and riveted construction were included in this investigation~

The three specimens uaed were of average depth with bending· and

torsional section properties kept constant over the Sp8ll length of

36 1 8". A single, concentrated, eccentric load applied vertically at

the center-line was the loading used to produce the comhined hending

and torsion condition.

The magni tude of the applied load was lim! ted so that the maximum

combined normal and shear stresses were well below the elastic linlit.

In thi13 way the basic section of web plate and flange angles could be

retested, after predetermined modifications, as another specimen.

4

CHJu-TF:R III CONBINFD BE.ND!NG AND TORSION TiiIGRY - St'HMARY

A. ADalytical DeveloQlllcnt -- Gener..iih

A con:.liion case of a structur~.l lliCllihcr subj ected to cOrI:bined bending

and torsion load1n"i is a hetJ!l or gir-;le,uncit;>r the action of c system

of eccentrlcallr applied tr;.:J1;wer3e forces. A;"3 a clor::e approximation,

the shear center a"10 a shnft to!"CjU!3 of a mClt71Itu6e eqlJsl to t',e product

readily predi etahle if ttl? i:-.flucnce ef tbe lutsral di ~pl[-:.ce!Tif.mt is

neglected.

Severf..l analytical s01ut.lons ,of' this p:t'o}:'lcm hc.l'lei'€en pre,posed.

'As .i s usually the case, the degree of precision :iri:pro'les as the number

of aSGumptions are reduced. Ho·....ever, the complexity of the der:1 vation

also locreas,e~,. 10 genc!"Cil the metho~~::; are basicr"lly, linear or non-linear.

The linear theory 1s based on the assurnpti<;Jn of s:ilull de(lectt0ns which '

CEiIl be neglected. In the non-'linear tileorie(;, deflections are con~~idered

too large to be neglected.

Rep re.:sentati ve of the various available methods ,and selected for

comparatiie purposes in th! s report were the, following:

(1) Bethlehem Steel Co. Formulae

(2) Timosh'enko Solution

(.3) Sourochnikoff3 s Method

(4) Petterssono's Theory.

.~. R. BCJdlich~ St.eel Co. Formulae -- Summ~ry

I

The basic Timoshenko solution has heen summarized and applied to

several typicd cases of unifom and non-uni fom torsion. (1) The stress

and rotation functions are expressed in terrr.s of the cri ticul v~ues.

The comhined stress values are obtained by superimposing the vertical

and torsion bending effects only. These formulo.e are included in this

study since they represent the simplest available solution.

c. Timoshenko Solution -- SUlIlDlary

The basic equilihrium equations for an I-heum of bisynunetric<.J. cross

section suhjected to non-unifonn torsion was solved hy Tlmoshenko. (2)

Essentially it is the linear, smll1l denection theory. For the cOI:';bined

bending and torsion case, the bending and torsional stresses are super-

imposed to obtain the resultant stresses. In the case of an eccentrically

applied vertical load, the lateral, vertic<.l and torsions.! bending stresses

are considered. However, the lateral bending and torsional stressps are

based on the initlul eccentricity and do not include the effect of the

induced displacements.

This solution has been extended to cover a large nUIllher of different

combinations of loading and end conditions and aptien:rs in various

forms.(3)(4)(5)(6)

D. Sourochnikoffos Method -- Summar.y.,This non-linear solution(?) is an extension of the basic linear

theory in that it takes into consideration the influence of the

deflections on the bending and torsional moments as well as the angular

distortions.

The effect of a pure torque acting at the midspan of a f imply

supported beam of bisymmetrical section is first investigated. Four

typical variations of torque along the span are considered. It was

found that the ratio of the angle of twist to the maximum angle of twist

at midspan was ner...rly pa.rabolic in all four cases. Since the torque

curve in the case of combined bending and torsion falls between the cases

previously studied, it is assumed that the angle of twist curve CM be

represented by a second degree parabola. The bending moment in the

lateral direction due to this assumed rotation leads to the lateral

deflection by means of the flexural relationship. The actual torque at

any point can now be evaluated. 4 simplifying assumption that the

adjusted torque is constant yields an expression for the maximum angle

of twist under combined hending and torsion alone. The maximum normal'

flange fiher stress at midspan is obtained by superimposing the vertical

bending, lateral hending and torsion bending effects. The total stress

is not proportional to the load, since the latter tlro components are

functions of the maximum angle of twist which in turn is not proportional

to the load•

•

7

., E. PetterssonQs Theory -- SwmIl.a17

The combined bending and torsion case represented by a simply

supported I-beam of bisymmetrical cross-section loaded at midspan by

an ohlique eccentric concentrated load has been solved by Pettersson. (8) (9)

Since this is a stress problem where the law of superposition is not

strictly' applicable, denections are taken into account. The differ­

ential equation for the angle of rotation has been deduced and solved

by means of infinite power series. The angle of rotation and its second

derivative, Wlch are used to obtain the combined stress, have been

transfomed so as to obtain a direct expression of the effect produced

by the risk of lateral buckling. The maximum an~le of twist in com­

bined bending and torsion is expressed in terms of the corresponding

distortion produced by la twisting moment and a factor 'Which is equal

to ~ty,when the concentrated load is infinitely small, and which

increases indefinitely as the applied load approaches the critical value

in lateral buckling.

F. Experimental Work -- R@view

In contrast to the number of analytical solutions that have heen-, -

published on the suhject of combined bending and torsion of I-Beams,

the number of reports on actual laboratory tests is qUite 'limited.

The experiniental investigations that have been reported to date have

utilized either rolled sections or welded I-BEans.

In a series of eXperiments conducted a:t tiie Royal Inst,itute of

Technology, two specimens of bisymmetrical and monosymmetrical cross­

sections were ,.n. turn. ~bj ected to an eccentric vertical load at the

center of a simply supported span. (9) The sections were of welded

,,-construction approJd,.mately' 10 inches deep, and 20 feet long with a

gL (=.!.) ratio of 4.7 and 6.4 respectively. Weights were suspendedL

from the beam specimen by means of a yoke. The theoretical. end condi-

tions were carefully maintained by means of vertical hinges, single-

direction axial ball!>~_~JlgS and a system of pulleys and weights.

The strains were measured by electric resistance type gages at

selected points, 'While the horizontal and vertical displacements were

measured by means of a theodolite sighted on drawing pins attached to

the fianges.

The comparison of observed results and calculated values based

on the Pettersson Theory was exceptionally good. Direct comparison

with the predicted values based on other theories were not included.

q

CHAPTER IV - ~'XPERINENTAL PROGRAM

A. General

In the in1tial series of tests conducted at Lehigh Un!verst ty

to investigate the unifol1I1 torsion behavior of built-up stroctural

mem~ers,(lO) a wide range of spec1men~ were tested. Included in this

group were a series of shallow as well as relatively deE"p ;'lhte girders

of bolted, riveted and welded construction, 14!" and 48~n back to back

of angles res~ectively. The shallow sf/ecimen.::; were used in the second

phase of the ex?erimentu.t progrum at Lehigh UntversityCll) which W~lS,

devoted to the study of non-unifonn torsion as exemplified by shaft

torque loading applied at mid~p1:lI1 wi th the concomit<mt restr-,,-int of

warping. The deeper girder,;; were not -1;ested at that time with t~e

available equipment bechUse of their linli ted lcngth-to-depth ratio.

It was felt that the dedred ty~·lc;..u. non-unlfornJ torsion behavior

could not be' produced in so short a spun.

When it wus decided to extend the progr<.iJr. to cover the cOJIobined

bending and torsion loading of pl.:lte girders, eCOnOl:lY dictated the

selection' of sections wich were siJ:J.:H;...r to tho:3e for which torsional

constants had heen obtained experimentally in the first phfl~'e. Due

to the limitation of time, the basic torsional const~nt te~t could

not ~e conducted on the ne'" specimens. Therefore, the pr€.'viously-

obtained experimental values were ada~ted for u~e in this report with

rea.sonahle assurance as to their a~'plicubility.

ro

-,

B. Test Specimens - Fabrication IJIld Properties

All significunt details of specimen T6A-4 und T6B-l of the

uni form torsion tests(lO) were duplic::lted in the design of the new

specimen. In order to obtain the maxiu,um possihle reuse of material,

it was decided to start with high tensile boltti /;is connectors and

to achieve def.ired variations in torsional properties hy chaneing

the number of cover plates. Stresses were to be kept wi thin

reusona~le limits.

Features of all three sections finully selected for this test

program are summarized in Fig. 4:1. All sections are similar except

for the numher of cover plates or the mode of fabrication. Fa~rica­

tion detuils of a represent£.tive specimen, T12-R, are shO\J!l in Fig.4::..,

For practical reasons smne lliodifications in fahrication details

were introduced in the new s!iecimens. The basic pitch of holes for

connectors was made the SBIDe for hath the horizontal and vertical

legs of the flan~e ungles. The spacing of the intermediate stiffeners

was increased slightly in order to maint~in reusonutly uniform spacing

along the span. i'rovisions were also mco.de for the contem!,lsted

addition of another set of stiffeners in each panel. These additional

holes were not utilized since time did not permit the study of the

effect of stiffener spacing.

I J

Since this program was an extension 01' two previous

investigations, (10) (11) the numericc.J. sequence of specimen-

designl:l.tion 'WaS continued. It is to be noted thCl.t specimen T6A-4

and T6B-l of ~he first ~huse are simil'.~r to the ne'" specimens

TIO-B and Tl2-R respectively. A specimen corresponding to TIl-P

without cover plates 'Was not tested in the first phat3e. The letter

designation following the specimen number refers to the mode of

fabrication, B for bolted and R for riveted construction.

Shop fabrici...i.tion procedures, as well as subsequent larorator,y

'adjustments, 'Were carefully standurdized in order to obtain s}lecimens

structurally equivalent to those previcuf.J.y tested. The fabrication

of the specimen and the auxiliary support frmne:: \lb.S undertaken

by the Bethlehem Steel Company at, their PottstOlm, Pennsylvania plant.

The high-strength bolts were tightened by means of a calibrated

torque 'Wl"ench to a uniform value of 300 lh.-ft. torque. No case-"

hardened 'Washers hud been used in the belted specimens of the first

phase. It 'Was recognized that such wushers 'Would insure rr.ore uniform

resultant bolt tension and that the ~lt tension could ulso have been

increased so as to conro~ to current specifications by tightening

the nuts 'With a pneumutic gun. However, it WE.S felt that such

modifications 'Would introduce an indeterminCJ.te change in the torsion

constant of the new specimen. Consequently the roltlug ~rocel:ure

used on the originiil s~ec1men, T6A-4, \01...5 adopted for both specimens

T10-B and Tll-B.

•

The subsequent modification of specimen TlQ-:B to Tl2-R involved

the replacement of the bolts by rivets. In order to save the cost

of transporting the 7-ton specimen to the shop und bacit, r1 vcting

hy meuns of a pneumatic hummer at the Swarthlilore College laooratory'

was considered. nowever, the ori~inal specimen, T6B-l, ,held been

riveted by mecms of a bull riveter under standard shop conditions.

Therefore, for the reason previously cited, the bolted 2late girder

specimen TlO-B was shipped back to the fabric~ting shop at Pottstown

for riveUng and returned as specimen Tl2-R.

The torsion constant K of a plate gi rder can ~e detern:ined by

both analytical and experimental means. The values used in this

report are summarized in Fig. 4:.3. The system of designation is

pattemed after the one used in the non-uni form torsion report. (11)

The experimental value, Test (1C).tV is bu.sed on the re~ults of the

unifonn torsion'tests conducted on sin:il~r specin.ens in the first

phase. The torsion.u rigidity KG is equiValent to the slope of the

straight-line portion of the torque va. unit angle of twist curve.

The constant (Ks)D is the lower limit v.:.a.lue based on the assump­

tion that all elements of the cross-section twist through the srune

angle hut separately. Due to the lack of cover plates in specimen

Tll-B, there is only limited unity of action between the elements.

The separate-action constant was taken to be the re,present;;;tive value.

No experimental value was availahle for this specimen.

As the upper 11m!t the torsion constunt of a bullt-up section

would approach that of an equivalent solid section of the sume

external dimensions •

13

.'Since the elements of a bolted or ri.veted plute gi rOer are

actually connected alon~ well-defined .g~;e lines and intp.rpl~oe'\

friction does exLt, the so called integri;:.l-~ction constant K1

must full somewhere between the two extrelte "cUuer,. Basfd on the

ooncept r-ro.posed hy de Vries, (l?) 1.t i~; assumed tha.t the core portion

in each flmlge acts as a solid section whilE'! the r~.l\;:tinder of the

section acts as sep~rr.~te rect<:mgles when the section is twi sted.

The cOffijJuted value (K1 ) B is the integr::i.l-action constant

proposed for design in which the hump and end effects, which tend

to neutralize each other, have been n.eglected. The (Kr )c value

incorporutes both corrections.

•

C. Test set-up

The unusual magnitude of the space and force requirements of

the full-size plate girder tests necessitated careful design of the

handling, loading and support arrangements. ;;)everal possible proposals

were studied from the standpoint of prO-ct! ~ability as well as efficiency.

The first prohlem was the choice of the loading system. It was

recognized that a system of suspended dead loads would hest f'1 t tIle

assumed conditions of the theory. However, such a syste.m of weig11ts

would remain posed like the Sl,'Ord of Damocles throughout the loo.ding

and unloading process. The ever-present possibility of' an unfortunate

accident was deemed sufficient grounds for the eliminntion of suspended

weights in favor of the use of a un! versal testing machine for loading.

Since the specimen weighed approximately seven tons, an overhead

crane of sufficient capacity and muneuverabllity would be required to

place it in the p!"\.lper position for testing. The ma.zni tude of the

forces that were contemplated were such that a unl versal testing

machine of at least 100,000 pounds capacity in compression w~s indicated.

The requirement that t."le specimen be supported at hoth ends meant that

the effective bed of the testing machine had to be ut lea~t 38 feet ·long.

All three major specj.fications could be met by the available

facili ties in the Civil Engineering Laboratory at SwarlhmoreCollege.

Arrangements were me.de in the summer of 1950 by Dr. Bruce JChnston,

at that time Director of the Frl tz Engineering Laboratory at Lehigh

University, with Professor Samuel Carpenter, Chatman of the Civil

Engineering Department at Swarthmore College, to handle the test

progr;~ as a cooperative effort by both institutions•

15

•

Several possible loading putterl1S werecon ..idereci. However,

it was decided to concentrate on the fund~ent&l case of a con­

centrated verticul loud applied with an initial eccentricity at the

centerline. A schematic sketch of the test set-up for a typico.J.

combined bending and torsion test is sho'\.'l1 in Fig. 4:4.

In addition to the requirement of load capacity, it wus

necessary that the point of application of the concentrated load

be adjustable. The eccentricity with respect to the longitudinal

axis wan to vary from five to t,,:enty inchee.. The applied load. wus

to be set at t",'O levels; first, at th~ top of the flange and then

at mid-height. Further, the load was to remain essenthl1ly' vertical

during t.l1e ensuing distortion of the specimen. Based on practical

considera.tions the use of a single all-purpose loadinr; hracket was

abundoned in favor of two separute brackets, one for each level.

The first eccentric loading system, shown schematically in Fig.

4: 5, was desj gned to Apply u vertical eccentric load at the level

of the top flange. A 't racket was bolted on one side of the plate

girder specimen. A strap plate served both a tie and as a support

for a serles of grooved plates pli;i.ced at interv.ils of five inches.

A surface-hardened knife edge was tack-welded to a steel block which

in tum was welded to a rigid flat plate. A nest of three-inch

diameter steel rollers, held together ~ spacer burs at hath ends so

as to be free to roll as a unit in ei ther direction nOnT!al to their

common axis, was placed under another smooth steel plate which was

bolted to the underside of the testing machine he~~ to serve as a

bearing plate•

IG

•

The roller nest w<J.s desiGIled to j.lenni t the l,~teral mOVel11pnt

of the low~r plate relati~e to the upper pl~te, therehy reflecting

the movement 0 f the girder specimen wi t!'1out altering the initiul

parallel alignment. Thus the applied load distributed O'ITf."r a t.1.de

arei::l. of the lower pl-:.te for stability, 'WOuld be tr;,nsmitted to the

girder as a concentrated, verticQ]. lOf:.d.

Since the loading device had to he kei>t ut the centerllne of

the testing machine, the p;1 rder stiec1men wa:: shi fted l:ltert:.lly as a

unit a specified distl:illce when uctifferent eccentr1 city Vl'" to he

introduced.

To apply a concentr~ted vertical eccentric loaQ at n.io-heigf1t

of the girder specimpn, a second loading system, shot,n in Fi~. 4:6,

was devised. A new rracket with a line of horizontally-Rpaced holes

replaced the former bracket at 1llidspan. These reamed holes were

located so that they would ~e at the proper distance from the

longitudinal axis of the specimen.

A bracket, made uf of a tee-section, was attach€'d to t~e under­

side of the testing machine heud. A loading post, fubric..:.ted from a

long length of 4 x 4 inch square steel tubinG, WuS connected at its

upper and low;r end3to the re3r-ecti~e brac;cet~ hy a t-,~1r of 1.25 inch

round pins inserted throt.:gh the D1J..tching hole~;. When the girder was

properly aligned, the loading post WOL':ld he vertic.~ and ut t:le·

c.el3ired eccentricity.

It was recognized that since the specimen would twist and te

displaced vertically and horizontally by the load, the post would not

remain vertical under this systelli. However a mitigating factor was

17

•

the rela.tively long Ipngth of 6 feet hetween pins. The luterul

displacement of the load point ",as measured dirp.ctly by meC:illS of a

suspended .t>lumb bob. The corre8~nding change in the direction of

the applied load from the vertical ·was SClall.

The requirements which governed the desIgn of the end support

arrangement were ilS follows:

a. the specimen should he simply-supported at both ends for ~ending

in the vertical plane

b. there should be no twisting of the section at e1ther end

c. there should be no restraint of the warping tendency at the ends.

After investiguting severi.l1 altemative arroogemcnt s, the

system shown in Fig. 4: 7 wus adopted for the east end. A slightly

modified arrangement WtiS used at the west end of the g-irder specimen.

Under each end of the girder was placed a welded frume from which the

necessary' support features were projected. The structural fr:-·tle was

designed as a rigid knee. At each end of the span a 3-inoh' diameter

roller was pos!tioned hetween two base-plates fllaced under and normal

to the girder. Under cert;:.in loading conditions it was found that

friction alone could not sup~ly the requisite restruintagainst lateral

movement of the ends. A collar was adcied to the roller rr.! shrink

fi tting a ring of rectangular cross section at the Irliddle. Matching

grooves were machined on the contact face of the base plates. The

roller was free to roll but was positively restrained from shifting

sideways.

Due to the application of the .eccentri c load at mids}1!.1!l, the

girder will twist about the points of verticul f;U~port. It would

•

have been. desira.ble to ha.ve the specimen twist about its longitudinal

axis at mid-height. ° However, the fabrioationof a suital,le support

fixture.~ adequate for the loads to he transmitted, would have been

so oore;licated that this idea was 'bypassed in favor of a simpler

arraoJgement. A knife-edge and matching grooved plate was placed

pa:'allel and 'directly under the girder lleb between the lower ~se

;Jlate and the top of the support frame. The contact surface of the

knife-edge would be the point about ~lich the.girder would tend to

ro1!ate.

The restraint against thi s rotation wus provided by a ptiir of

° tie rods. The upper and lower rods extended out horizontal~y from

the girder and were connected to vertical posts of the support frane

by suitable fixtures. A thrust baa-rin,; was placed und.er the locking

nut so that it could be readily turned on the threaded rod even under

pressure. Thus the end crOss sections could be broup,ht ,back to itso

original vertic~ position after the load had been applied•. A plumb

'hol:' suspended from the top flange post a scale attaohed to the hottom

nange served as an inQ1cator of the magnitude of rotation at that end.

The resisting torque at each end could be evaluated from statics

if the stress in each tie rod were known. A pair of electric strain

gages was cemented to the machined throat section of the rods used at

the east end. Calibration constants had been obtained from stW'ldard

tension tests for each rod.

Since two calitr&.ted ring dynamometers were aVailahle, they were

inserted into the tie rods used at the west end. The di~ readings

•

were a direct measure of the change in the linear dimension calJsed

by the stress transmitted by' the dynamometer.

D. Anstruntentat10n

The behavior" of the girder speeimen under loud was to he

evaluated qllantitatively in tetms of strains and distortions. Since

there were tw symmetriccl. hulr spans, it lrclB decided to use the west

half of the specimen for a comt>rehens1va strain survey and the east

half to study the llnes.r and angulJ.r movements.

An extendve system of electri c strain g<igC5 was required to

determine the normal and sheu.r stress distribution along the spcm as·

well as around the section. Approximately 100 S&-4 gages of various

types were cemented to the specimen at selected stations. Nonnalu • .

stresses in a given direction were measured by an un1-axial gage,

shear stress combined with one nonnnl stress by a bi-axial gage and

principal stresses and their orientation at Co give-.n point 'by a tri-

axial gage.

Since all three specimens were modifications of the SaIne basic

section, the strain gages used on one were usually availa'hle for use

in the others. However, some replacements and add!tionf'i were mude

necessary by accidents in handling or changes in section.

The major stationS were spaced 5' -6" apart along the half span.

The variation in normai and shear stresses at these stations across

the nang~ and the web could be defined from t."le strain readings. For

additional flange stress values along the axis, intermediate stations

were established•

..

The technique of cementing gages E.nd. wiring le:-.ds follol(ed

standard la1:orc1.tory practice. U'W1tchinB units l(ere used to ~ring

the numerous leuds into the portable strain indlcCl.tors. Systematic

checking procedures were 1nstituted to min11llize instrumentul as 'W'p.ll

as human errors.

The varial,le nature of the anticipa.ted distortions were such

that a comprehensiV9 netvork of linear and angular measurements waS

also required. The relative transverse and verticc.:J. displacements

of points along t."le longitudinul axis as vell ~w the angle of' t\d.st

of the l(eb and nange \lere deaired. In line with the pr€v1oudy

designated p.:.:tttem, the. system of dial gages and level rar sup~orts,

eventually selected, l(£ll~ placed over the east half of the girder specimen.

The angle of Mst of both web and flange at selected stations

was measured by a level bar wi th a 2-1nch gage length which could be

set up on special sup~orts tack welded to the specimen. The arrange­

ment and stationing of the level bar supports are shot.J1l schemo.tically

in Fig. 4:8. Since the angle of Mst was var1o.l-le along the span

due to restraint of warping, as m.my stations as practical-le were

estarlished.

Details of the adjustahle type of level biir, improved l:'y .Mr. Kasten

of the Swarthmo~ College lngineerlng Lo.bor!J.tory staff, are sketched in

Fig. 4;9. A micrometer screw was used to hring the har ba.ck to level

as indicated by a spirit bubble after euch load increment. The

corresponding c:1~lnge in the Ames di<:J. guges set at one of the gage

points gave the desired measure of the ongul~r distortion.

2/

The runge of the instrumE'nt was increased apprec3 al:'!ly by means

of a filler block of known thickness which could he inserted or

removed from ~etween the fixed points and the muln hare This adju~t­

ment was equivalent to a. renet of the Ames dial ry an amount equal

to the thickness. The vertic.:J. poE':itioning of the dial gc.ge ~~d the

micrometer screw was also adjustahle, which elimina.ted the need for

resetting the adjustable plate which held the level har support in

placE..

Th's lateral and vertical components of displacements of the

stations along the longitudinal axis were to 1'-e measured with respect

to vertic:;.l and horizontal reference plmles respectively join:l.ng

corresponding' points at the ends of the specimen. The net displace­

ment could be obtained by subtracting from the gross value a correction

for the yielding of the supports, if any.

The universal testing machine at Swarthmore College was equipped

vdth a specially-designed, welded platform fraIle made up of heavy­

rolled beams which made it possiHe to conduct flexure test:; of long­

span beams. In the conventional arrangement, load applied by hydr<::tulic

pressure to the loading ram under the platfol1ll is transferred by the

lower crosshead to the tension screws, thence to the specimen ~rway

of the upper crossheud and the measuring capsule. The upwurd-acting

reaction under the platfom is balanced by the sum of the downward­

acting beam reactions on top of the platform. No unballlnced external

forces are introduced under the base of the testing machine frame since

the forces constitute a closed system. The full capacity of 600,000

pounds would therefore be available for loading.

The anticipated elastic deformation of the platform as a cantj­

levered frame under the be3Ill reaction$, although small, proved to be

a troublesome factor in the design of a. distortion-measu!'ing system.

After sever~ schemes to compensate for this factor were tried out

on a vertical ~endjng test of a specimen, it Wt:l.S decided thti.t ~

different arrangement should r.e ex~lored.

The platfonn of the testing macMne was blocked up at both ends

\1y steel shims to eliminate the elastic deformations ·and the attp.ndant

support settlement at the ends of the specimen. Under load the beam

end reactions 'WOuld be transferred directly out of the system to the

floor of the platform pit. The u:t>w~:.rd-ucting reaction unde~ the

platform, no longer balanced by the bel.:In reactions, would become an

uplift force on the testing machine frame. Under this arrangement,

the maximum load wuld l'-e limited by the capacity of the anchor bolts

and the weight of the frame itself. The safe load was approximately

100 kips. Since the maxi.mum load contem:t>l..ted in tht s progrClJll was

also 100 kips, this arrangerr.ent was adopted for use.

Now the reference plane for vertical displ~ca~ents could be the

stahle floor of the laboratory. A grid-work of light ch~nel sections,

tack-welded together, was placed under the east hulf of the girder

specimen straddling the testing machine pit.

To measure the transverGe displacement of points along the span

under load, both the horizontl:l.1 .md verticcl components 'WOuld have to

be obtained separately. Another complication was the fact that a dial

measuring the horizontal component 'WOuld theoretic:uly have to shift

vertically to comj,Jensllte for the latter component of displacement.

23

This fE:.ctor could he neutralized 'by' fixing the dhl at El considerable

distar.ce away f'rom the specimen ond connecting the plungp.r of the dial

to tae specimen with a taut length of fine flexir.le wire. The change

in direction of the wire would be so smail that the change in dial

reading could he interpreted as the desired component of displacement.

A set of three dial gages was used at each station. Two gages

were placed below the specimen symmetrically about the web. The gages

were connected directly to outriggers tack-welded to the web at mid­

heigh~. The gage supportf:' on each side were spaced along 50 f·teel angle­

section which could ~e shifted parElllel to the girder on top ~f thA

dial support frame. When properly aligned, the dial guge3 were directly

below the point of attachment or the connecting wirp. to the outrigger~.

The wire was kept taut by means of a suspended weight attached to the

lower end of the plunger. These dial readings could be comh.ined to

give the vertical defleation as well as another independent va.lue of

the angle of twist at the station. Fig. 4:8 shows schematically how

the level bar and dial gage systems were combined on the south side

of the web.

The third gage of each set was placed on a horizontal line directly

opposite the mid-hei;Jht or the specimen at each stution. A vertical

rod was cantilevered from the dial suP;.ort frame. A smUl plute was

post tioned on the rod and held by a perl. r of locking nuts. The dial

gage was clunpted to ·the plate in a horizontal po.=,ition. A nexible

wire oonnected the forward end of the plunger to the specimen. The

oppoai te end or the plunger WliS connected to a light weight by a. wire

riding over a pulley to keep the wire system 'tuut. Since the forces

involved were very small, the dial support could be considered fixed

in space. The change in dial reading gave the later~ displacement

of the specimen.

Fig. 4:10 shoW's the battery or north side dii:U gages set up on

the suplJOrt frame. An overall vieW' of the test set-up is pictured

in Fig. 4:11.

•

E. Test Schedule

Verticul and lateral bending tests of each spec1men were con­

templated in order to oht&:ln vuues of the flexurul constants; however,

the laree number of combined l:'ending and torfdon tests that h(';.d been

planned made for a very tight schedule. It wu.s necessary at times

to pass up a lateral bending test or to conduct a purtial test hy

taking only a limi ted number of readings. Fig. 4:12 gives the summary

of the test program as conducted.

Major problems in handling of the girder specimen were encountered

in the course of the investigation. .To move the specimen into the

laboratory and then into the testing machine required a ten-ton over­

head crane and careful maneuvering as indicated in Fig. 4:13 •

.'After a combined bending and torsion test for a glven eccentricity

had been completed, the next eccentricity was introduced by shifting

both end SUPfiort frames the proper amount. This movement was accomplished

by jacking against a. h.loclt fixed to the floor at . each end or taking up

on a tension rod depending on the direction desired. The eccentric

load had been removed ~o that only the friction between the base of

the end support frarae and the top of the pla.tform had to be ov€'rcome.

When the girder specimen was to be tested in lateral hending, it

had to be tumed on its side. A apec! al bracket with a large diameter

rod as a hondle was bolted to each end of' the girder. The end support

was removed after the specimen was jacked up. A ball bearing race

supported on a grillage was slipped into place. \-Yhen the jacks were

lowered~ the specimen could be turned quite readily. Fig. 4:14 shows

the girder in the process of being turned on its side•

CHAPTER V ANALYTIChL STUDY Of RESULTS

A. Preliminary Tests

A short segment of each main member oJ the plate girder specimen

had been reserved for tansile tests. Flat coupons were prepe.red and

tested for the usual tensile properties. The results are summarized

in tabular form in Fig. 5:1. As noted, the material sa.tisfied the

standard specifications of the A. 3;T .M. Designation A7-46 for structural

steel. A common set of elasticity values was adopted and used i~ all

of the subsequent computations so that the results would 'be comparable.

There are several ways in which the moment of inertia of a plate

girder section CWl be computed. To indicate the range of variation

three values of Ix were computed for each specimen. The computed gross

Ix "r8S based on the typical solid section. The net Ix, used in this

report, was obtained on the basis of a section with holes deducted from

both top and bottom flanges. A weighted value of Ix was computed by

proportion, considering the equiValent length of holes and the typical

rivet pitch.

Two sets of experimental values were obtained from. sach vertical

bending test; namely, denections and flange normal stresses along the

speno These experimental values were plotted at the corresponding

points. A value of Ix was found by trial which yielded a curve matchililg

~e plotted points quite closely. The two values of I:l! were averaged

to obtain the effect!ve Ix.

Typical results for the ri~eted specimen ll T12-R, are reproduced

in Figs. ;: 2 and ;: 3, based on the test w:l.th thp. concentrated load at

the centerline. As an independent check, another verticc:.l llending test

wi.th concentra.ted loads at two symmetrical points was also conducted

. for this specimen. A similar set of curves llCS prepared ror each of the

three specimenso The bending stress variation for the bolted specimen,

TIO-B, 1s shown in Fig. 5~4.

The vertical bending· properties are su:mm.arized in Fig. 5: 5. It

is of' interest to note the following results which are based on a

limited number of tests and are not represented to be comprehensive:

1. The values of effective Ix based on measured deflections and

normal stresses checked quite well for each specimen, the

maximum difference being 6%

2. The two vertical bending tests for the riveted specimen,

Tl2-R, yielded almost identical values for the effective Ix

3. The average effective value of Ix for the riveted specimen,

T12-R, was approximately 5% higher than that for the bolted

version, TID-B.

The moment of' inertia, 1y , about the weak axis was obtained in a

manner similar to that outlined for Ix. A typical test curve depicting

the variation in flexural stress at the f"..xtrame edges I along the spew

is show.n in Figo 5g6.

A typical comparison of measured and computed values of deflection

for the bolted specimen without cover plates is sho1-m in Figo 5~7.

The lateral bending properti.es are summarL~ed in Figo 5~8.

The lateral bending te3t for the riveted spl8cimen ll Tl2-R, "ras not

conducted due to the lack of time. However ll it appears reasonahle

28

to assume that the difference between the ly test values for the

riveted and bolted specimens of the same mElke-up should be less

pronounced than in the previous ca.se for Ix. The correspondin~ va!Up.s

for Specimen T10-B were adopted for use in the 1'12-R computations.

Since sections similar to those previously tested in pure torsion

by Chang and Johnston{lO) were being used in this investigation of

combined handing and tox-sion p no additional calibration tests were

perfomedo The experimental and computed values of the torsion

eop.stHnib adopted for uee in this :report are summarlz<3d in Figo 4:30

Later this operat~on 'Was discontinued 'When it

•

Bo Combined Bending~ Torsion Tests

The plate girder specimen was carefully·aligned in the testing

machine so that the load would be applied with the proper eccentricity.

Then the instrumentation for both strain und di stortion mea.surements

'Was iDf~talled 0 A couple of dry runs were made with certain cr!ti cal

gages under constant observation to detect any unpredicted behavior.

A small in!tie.l load was apVlied to take up any sla.ck that tight

be in the system. Initial readings were taken of all gages. The design

load was applied in increments and then removed. A complete set of

readings was usually taken at the end of each increment. The maximum

load was lilLited to a value such that all stresses were well within

the elastic range.

In the first test the end tie-rods were adjusted after each

increment of load to bring the end section back to its original

vertical p05ition~ as indicated by the suspended plumb bob, hefore

readings were taken 0

I

was noted that the end rotation 'Wus quite small.

'Wherever possihle» the test data wa.s plotted againf:t the load

increments and the value corresponding to the maximum load was deter-

mined graph! cally. Thl s procedure eliminated the poss!hili ty of a

single erratic reading being used as a representative test value.

The 5&-4 strain gage readings were converted to stresses in t.l1e

prescribed lil811raer$ using the previously adopted elasMc constants.

ThiS level bar riladings were reduced to angles of twist by

dividing the change in dial reading by the gage length. The vertical

dial gage readings were combined to give both the vertical displaC~lent

30

and the angle of twist. The unit angle of twist at points lrldwu.y

between stations was determined by d1vidlng the difference in the

angle of twist by the distance 't"-etween the stations.

It was recognized that since the nange does not twist as much

as the 'Wsb at any given station, those two values must not be com'Mned.

ThiS angle of twist of the web form ·from both the level bar and dial

gage :readings checked quite welL The average value was used in the

comparative curves.

In the computations for stress and distortions under combined

hending and torsion, values of section properties and elastic constants

were required. Experimental as well as computed values were usually

availa'ble. For comparative purposes, it 'Was decided to group the

values into logi.cal combinations which are sUl:lIIlarized in Fig. 5: 9.

All of the test values were grouped together and used in combination "AI1.

Approximate values that 'WOuld nomally be used in desi~n computations

'Wsrs desiglllated as Combination RBn. More exact, computed values incor­

porating certain refinement s were used in Combination ViC". The minimum

values of' the torsional and flexural constants were grouped as

Combination aDI'i.

It had been established in the second phase on non-uniform torsion

of plate girders(ll) that the theory for rolled hearns could be extpnded

to built-up sections if the proper constants were introduced. Of the

various available analytical solutions covering t.~e cOIJh1ned hending

and torsion of beams, four representative ones were selected for com-

parative purposes. These were des1~ated the Bethlehem .:teel Co.

Formulae(l), the Timoshenko solution(2), the Sourochnikoff method(7),

.31

and the Pettersson theory(8). For convenif>..Dce the compututions vere

differentiated Py a letter corresponding to the Cirst letter in the

name. For example, all computations hased on the Timoshenko equations

are preceded by ":.he letter T. A fifth method, utili zin~ actual measured

distOrtions and equilibrium condi tionfl, was de~ignated comput~ttion "H".Finally the computations were grouped into series according to t.l1e

method, t,~e end conditionsl' and combination of constc.nts used. Each

set of computed values 'WOuld be represEnted ~y a method and serles

designation such as T2.

Tha notation used is summarized in Fig. 5:9. For comparative

purposes the results of the analytical study are presented in terms of

either Series 1, 2 or 4.

32

c. .Comparison of Analytical Methods

Since one of the objectives of this investigo:.tion vas to compare

the efficacy of the available theories in the predict1.on of the

charo.cteristic behavior of plate girders» comput".t:'ons were carried out

hy each of the four methods wherever possibls. The same combination

of constants was used in anyone series.

The three components of flange stress are the verticiU bending»

lateral hendjng and torsion~ hending stresses. These are all addi. tive

at the south side of the top flange for specimens eccentrlc:,;.lly loaded

on the north side.

A breakdown of' the components as determined by. the four methods

for two different combinations of constants for a given specimen and

loadi.ng is tabult.ted in Fig. 5:10. The vertic&l bending stress remains

the saIne» regardless of the method of computation. The lateral hending

stress» ~en included» varies over a limited range •. The largest and

most sensitive component is the torsional ~ending stress.

The variation in the three components with 10lld resulting from the

use of the Pettersson fC'I'lIlula for a. given specimen is shown graphically

in Fig. 5:11. The vertical bending stress viJ.ries linearly with the

applied load» while the lateral and torsional bending components increase

at an increasing rate with loud. The relative magnitude of the components

will of course vary as the load and initial eccentric!ty are ch;:...nged.

Wi t.1t greater eccentri c1 ties the torsional and l,-,tcrcl bending stresses

will increase at the expense of the verticul bending component.

33

" The manner in 'Which the computea mwciD'lur. comhined flange stress

\ varies 'Witn load for a typi cd specimen andlo&.ding 1 s shown gra~hic[,l1y

in Fig. 5:12 ~ The simp],i ried Bethlehetli Steel Co. Formula yie1d~ a

strui ght line vanation wi th loud. The Sourochnikofr and Petters son

methods exhiHted t~e charactElristic amplific:..tion of 'stress due to the

inf).uence of deflections, th~ latter usuC1l1y <%1 vin?. slightly higher

values than the former. The Timoshenko solution, though approximatE',

Incluqes a latera+ hending t?rm 'Whlqh resulted in ~~lues intermediate

bet\"een, the maximum and tQe minimum. For the majori ty of cases investi-

The increase in the effect!ve torsioniu am of an inith,lly eccentric

load appli,ed ahOve rc.idhelght of u bisymmetric::.:,l section due to twist

results in a ~on-Hnear variation of tile ~gle ~f' twist ldth load. The

deflection methods of Souroc.hnikoff 'and Petters50n re.cognize this

behav~or and show an ~ncreasing rate of; incre,ase of the an~lar distor...

t~o.l1 ~th ],oad. Th~ aPJ?ro:lq,~~te ~et-hlehe.ro Steel Co~ and Tim~shenk()

soluti ons are not sellsi tive to thi s effect and a straight line variation

results. Fig. 5:13 shows gr-dphically hoW' the computed v~ues of the

centerline angle of twist vary as a function of the magnitude of the

eccentric load for a given specimen.

The superposition of the computed vertical bending and the torsional

bending effects with or 'Without the lateral l"cnd1ng component under

, \ combined hending and torsion loading yields resultant normal stresses

''Which differ 1n value at each of the four .nange comers. "!ik the

vertical eccentric load applied on the north side, the component stresses

34

: . '. .~; .

at the south side of the top nange are all cOlr.pressive and ther£fore

addi tive while one or more of the components a.re subtractive at each

of the other cornerS9 Fig. 5;14 shows how thea comhined flange stress·

at each comer, computed by the Timoshenko fozmula, varies along the

span for a typical 'sp~cimen and load.

Another factor which was investigated was the effect of the

variation of the eccentricity of the vertical load on the computed

maximum ne:nge stress.· In order to establish a basis for compari~on,

the applied load was modified so that the combination of load and

eccentric!ty produced the SSffie computed maximum stress by une of the

Eethlehem Steel Co. formulae. Then the corresponding stress 'W'S.s com-

puted by each of the other three methods. The resulting values

are summarized in Fig. 5:15. Under these conditions of decreasing load

'Wi th increasing eccentrici ty, the combined nanga stress computed by

the Timoshanko, Pettersson p and Sourochnikoff methods sh~~ed a gradual

decline in magnitude.

Both of the non-linear solutions of the comhined hending and

torsion problem take into account the effect of the change in height

of eccentric load application. As an example the Pettersson solution

for the maximum comhined stress when a 50 kip load is applied w:fth an

initial eccentricity of 10 inches first at midheight (a =0) and

then at the top of nanga (a = 27.5 V1 ) is depicted in Fig. 5·16. The

nange stress is increased by' approximately 8% as a result of the

increase in the height of load application in this particular case.

The angular distortions would also increase due to this change.

35

D. Comparison 2! Computed and Experimental Values

Using vadous comhinations of' constants in the sp.lected analytical

methods, the important stress and distortion functionE were computed.

Among these were the variation along the span of the normal stress at

the flan~e corners, t..~e web angle of twist, and the combined shear stress

in the flange and web. These computed values were compared graphically

'With the results obtained from the experimental data. It is to. ~e noted

that the amount of calculations involved increased rapidly as refinements

are introduced into the theory.

"With each of the methods used, it was possible to obtain a computed

value for the combined normal stress at midspan. These values are

plotted at the centerline and identified by a distinctive symbol. However,

only the Timoshenko theory could he used directly to compute values along

the span. For comparative purposes a modified Pettersson curve was

obtained by proportion. Values at intermediate points were computed by

multiplying the TiILoshenko value at that point by the ratio of the

Pettersson to Timoshp..nko stress at midspan.

The experimental values were reduced from the test data obtained

from SR-4 strain gages stationed along the span. A t~,pica1 comparison

of experimental versus computed flange nomal stress is shown graphically

in F~g~ 5:17 for the bolted specimen T10-B. There was some scatter in

the experimentally determined points. In general the modified fettersson

curve afforded a closer approximation to test values the~ the Timoshenko

solution, which was consistently low in the region of high values. The

approximate Bethlehp~ Steel Co. solution was also off appreciahly on the

36

unsafe side at midspan. The Sourochnikoff curve would have fallen

slightly under the i'ett.ersson curve, based on pr~v1oul'l studies. After

making allowc:nees for experimental variations, it appears tha.t the

approximate methods, although simpler to apply, give combined st!'esses

which are incorrect on the unsafe side.

Computation "H", developed in connection with the investig""tion

of non·,uniform torsion of plate girders, (11) provides another method

of evaluc.ting the veracity of the measured strains ana the computed

stresses. This method, based on measured angles of tidst along the

span, theoretically adjusts for possible deviations from assumed test

conditions.

The experimental data froID a typical riveted specin:en test are

plotted against the Timoshenko and Computation nHn solutions in Fig. 5:18.

As in the previous case, the Timoshenko and Bethlehem Steel Co. solut:lons

are mdervalued. The Computation "H" curve £1ts the plot of mea3ured

values quite \lell.

37

•

The resultunt shear stress in the nanga is computed hy super-

imposing the torsion61 and trunsverse shear components. Both a.re

based on certain assumptions regarding the dlstributioD of forces

across the flange. The procedure for comMned bending ond torsion

loading is essentially the 56.!Ue as for the non":unifoz,n torsion case. (11)

The torsional ~hear stress, based OIl the de Vries asswnption, varies,

directly with '(,he effective thickness of the flange. The transverse

shear stress is assumed to vary parabolically across the flange.

Two longi tudina! gage lines were selected for study: the flange

centerline and a line between the r1vet line and the edge of the cove"r

plates. The Timoshenko formula was used to compute the variation along

the span hased on certain design constants. The curve for the flange

centerline is shown solid, 'While that for the outside gage line is dashed.

The AX-5 strain gage data were converted into normal and shear

stresses by the conventional relationships. The resultent shear stress

was plotted as circled points at the respective stutions. Typical

results for the bolted and riveted specimen are shown in Figs. 5:19 and

5:20.

In spite of the relatively low rllIlge of' measured stress and the

approximations made in the theoretical computations, the check of

computed and experimental values was reasonably good. It 'WaS con-

eluded that the maximum shear stresses in the nange along the center-

line can he predicted adequately hy the Timoshenko fONula•

38

The resultant shear stress in the web is made up of two parts;f

namely, t!1e torsiontU and the trcmsV'erse shear stresses. The first com-

ponent is essentially a shear force per unit length which tollo\ls a tip-

to-tail procession around the section. The torsional shear stress varies

directly as the torsional shear torque which in tum vuries 'With the

un1 t angle of twist along the SpQ.ll. The transverse shear stress due to

vert!cal bending is assumed to be un! formly dl stri~uted over the \leb

area and proportional to the vertical shear.

For the specirr.ens as loaded, the two shear stresses are addi tive

on the north side of the web and subtractive on the south side. The

Timoshenko method, based on certain design constants, was used to com-

pute the torsional shear stress. The combined shear stress \las then

. determined and plotted a.long the span.

The AX-5 strain gage readings vere reduced to bending and shear

stresses. The shear stresses were plotted at the respective statj ons

. as circled points. Fig. 5:21 shows the maximum combined shear stresses

.39

from a typical load test While the minimum values are depicted in Fig. 5:22.

Here again the check of experimental and con:puted v~lues was good

in view of the relatively loW' rr.mge of stress. Even in the case sho'Wll

in Fig. 5: 22 Where the sign of the shear stress underwent a reversal,

the theoretical variation was reasonably well matched.

CHAPTER VI CONCLUSIONS

A. Evaluation g.f.~ Program

An extendve e:xperimentw. investigation of built-up structural

members suhjected to torsion loading was laid out by' Dr. Bruce Johnston

while Director of the Fritz Engineering Lal:oratory.. The test specireens

ranged from a series of flat plates to full-size plate ~irders, which

were fabri cated as 001ted, riveted or welded uni ts. The loading condi­

tions included pure torsion, non-uniform torsion and finally combined

bending and torsion.

For rolled beams and equivalent welded sections, a small-scale

model of the proper proportions could have been tested with reasona'hle

assurance that correlation between 'prototype and model would be main-."

tained. However, for built-up sections held together by connectors the

.• scale effect would be difficult to control and evaluate. In order to

by-pass this factor, full size specimens utilizing conventional modes

of fabrication were used in this investigationo

On the other hand the use of full-size specimens in place of models

introduces certain requirement s and lim!tations. Larger capac!ty units

must be provided for handling and testingo Material and oFerating costs

are increasedo The longer period per test reduces the number of set-ups

that can he 1nvestigs.ted.

The calibration tests of the plate girder specimens, conducted for

the purpose of evaluating section constants, gave a good indication of

resul ts that could be expected later in the combined bending and torsion

tests. The vertical bending test, for example, after several

•

40

modifications, yielded consistent results between computed and

experimental values. The preliminary tests also served ~o check out

the individual SR-4 strain gages as well as the d1al gage method of

measuring displacements. In general p the experimental data obtained

from the combined bending and torsion tests were adequate and of

acceptahle quality.

B. SUmmarY of Test Results

Comparison of observed and predicted behavior of the plate girder

specimens led to the follo\rlng conclusions:

1. The mode of fabrication has an influence on the effective value

of section properties.

For example, based on the vertical bending tests, the

affectiva value of Ix for the riveted spec1roE'll lias approximately

5% higher than that for the bolted version of identical section.

A more sensitive section property is the torsion constant of a

plate girder. From the pure torsion perfonned by Chang mid

Johnston, (10) the torsion constant for the riveted girder lias

found to he about 18% higher than that for the corresponding

bolted specimen. It must 1:'e pointed out that these differences

",ere experimentally determ~ned for certain specimens as fabri­

cated. Since fabrica.tion techniques, especially in boltEld

construction, can vary over a wide range, no single value

would 1'e applicable in every case.

41

2. The linear theory of combined bending 6J1d torsion can 1:>e used

to predict the flange and web sheur stresses.

llhen applied to the plate gil'ders tested, the comparison

of computed and measured shear stresses was favorable in spite

of certain assumptions made regarding integral action behavior.

It is to be noted that the shear condi tioD was not crt tical in

these specimens and the shear stresses were reIat!vely low in

magnitude.

3. The linear theory of combined bending and torsion 11 elas under­

valued predictions of flange normal stresses.

The magIll tude of the measured normal stresses in the flange

was generally larger than that computed by the apPl"Oxireate methods.

This discrepancy on the unsafe side was in line with expectatlons.

Both the Bethlehem Steel Coo and Timoshenko methods neglect the

amplifying effect of deflections. These approxin:.ate methods are

relatively simple to apply. However, it is to be expected that

the difference between actual and computed values of maximum

combined normal stress will increase as the section becomes

deeper and more slender.

4. Analytical methods ba.sed on the non-linear theory of combined

bending and torsion can predlct the normal flange bending stress

condition.

The computed values of the normal stress in the flange of

the plate girders tested checked reasonably well with ~easured

42

values at corresponding points. This improved cheek was ex­

pected since certain provisions for the effect of deflections

have been incorporated in both the Souroehnikoff and Pettersson

solutions. For the particular combination of section properties

and loading conditions used in this investigation, computed

stresses by the two non-linear methods did not differ appreciably.

43

APiENDIX A

BIDLtOGRAPHY

(1) Bethlehem Steel Co. Booklet 5-570 "Torsional Stresses inStructural Beams." Bethlehem, Pennsylvania.. 1950.

(2) Timoshenko, s. Dull. Po1ytech. Inst., St. Petersburg. Vol. 4,1905.

0) Timoshenko, S. "Theory of Elastic Stl:lbi1i ty. Q McGraw-Hill BookCo., Inc. New York. 1936.

(4) Timoshenko, S. "Theory of Bending Torsion and Bending of T'nin­walled Members of Open Cross Sections. 91 Journal. of theFranklin lnsti tute, Vol. 239, april, 1945.

44

(5) Lyse, I. and Johnston, B.Trans. A.S.C.E.

"Structural Beams in Torsion. ft

VoL 101, 1936.

(6) Goldberg, J. "Torsion of I-type and H-type Beams. n Proc. A.S.C.F..Separate No. 145. August, 1952.

(7) Sourochnikoff, B. "Strength of I-beams in Combined Bending andTorsion." Proc. A.S.C.E. Separate No. 33. September,1950.

(8) Pettersson, O. "Combined Bending and Torsion of Simply SupportedBeams of Bisymmetrical Cross Section. n Bull. No.3,Division of Building Statics cmd Structural Fngineering,Royal Institute of Technology, Stook.'J.olru, Sweden.

(9) Pettersson, O. "Combined Bending and Torsion of I-beams ofMonosymmetric&1 Cross Section." &'11. No. 10, Divisionof Building Stat! os and Structural Engineering, RoyalInstitute of Technology, Stockholm, Sweden.

(10) Chang, F. and Johnston, B. "Torsion of Plate Girders." Trans. A. S.C.E. Vol. 118, 1953.

(11) Kubo, G. "Non-Uniform Torsion of Plate Girders. tt Proc. Sep. 449.A.S.C.r. 1954.

(12) de Vries, K. ItStrength of Beams a.s Determined by Lateral Buckling."Trans. A.S.C.E. Vol. 101, 1936. .

AP}?FBDIX B

ACKNOl>lLEOOMENTS

The investigation of plate girder benavlor under combined bending

and torsion loading was suggested by Dr. Broce G. Johnston, 'While Director

of the Fritz Engineering Laboratory of Lehigh University, as a logical

extension of the unifoIm and non-uniform torsion study.

The test program was carried out at S'Warlhmore College m. th the

cooperation and guidance of !'rofessor Samuel Carpenter, Chairm.m of the

Department of Civil Engineering. Mr. Ed K.asten and other members of the

laboratory stuff provided valuable technical assistance in many phases

of the experimental study.

Professor william J. Eney, Head of the Department of Civil Engineering,

Lehigh University, and present Director of the Fritz l!ilgineering Lar.oratory,

has lent invaluable aid as staff coordinator in aU a.dministrative and

fiscal matters and in the prepi:i.ration of the final. report.

Consultant on various technical aspects of· the P1'Oj ect has been

Mr. Karl de Vries of the Bethlehem Steel Company.

The Pennsylvania Department of Highways and the Bureau of Puhlic

Roads have been most cooperative as co-sponsors of all three phases of

this project. The sponsors were retJresented by Messrs. H. G. VanRiper

and "I. R\ He1"lIlan of the Department of Highways and l-1essrs. Raymond Archihald,E..L..ECIGIr::t:J.Q

...Jeub Ed oksRTh; and Neil Van Eanam of the Buretl.u or fuhlic Roads.

The experimental phase, the analytical study, and the cOIlpilation of

the report have been supervised by Dr. Gerald G. KUho acting as Proj eet

Director.

45

•

ILLUSTRATIONS

46

-----------_._--------------_._._-------_._----.__._----_._-~--_._----_..:.-_..,;,:------...:. --...::.:--.:_----=-

47

-----------------------------------

T,12-R

I

I

I,'~48 111 ­: 2

TIO-B

Sketch

Mark

Fig. 4:1SUMMARY OF TEST SPECIMllES

COMBINED BENDING AND TORSION TESrS

Phase 3 Series B Deep GirdersI-------,.--------------T--- I

I TII-B _II'

f--------l----------ll----~----+-I------:---------i

~~~Sl'-- ----1----- Ett=:;;~SI

II

!

\

i

)0 •

- ,--'·1'

I. I

I

I I ~-:.---1---- ~.:~~

-L II--__---,-_,-....-.--.J..-----.Bo--l-t-ed--- _ _.~lt:~__I___'__Ri_-v_e_t_e_d_--_-J

l,

Notes:\'

All specimens were 38' 0" lo~g

All specimens were 48~n b. to b..of angles'

All specimens were made up of the' same materialI-Web Plate 48 x in .4-Flange angles 6 x 6 x 5/8

Cover Plates 14 x 5/8 (where used). ,

Holes for connectors were spaced at 2!" c. to c. except at stiffeners

Stiffeners were spaced 3' 8" apart

Combined Bending and Torsion of Plate GirdersTesting Program conducted .at Swarthmore CollegeLehigh Unlvers1ty - Fritz Engineering Laboratory Project 215B

I

I

/'-----.-------- -------------.---.-----.---.---~---_----.--------;..J-i -,

",

NOTES,

DIMENSIONCODE

A: 4 AT 2~'= 10"

B: 6 AT 2~"= 1'-3"

c= 3~"

D =34"E= 2~"

F= 3~"

G=24 "

H=3~"

J =2~"

K= 2~"

L=4i"M= 5~"

N =4~"

R=~"

DETAilMATERIAL

SI, L5X3~X~

52: L6X3~ X~

P I: PL. 7~ X~

P2· PL.3X~

G G

F L

--------+--;n

'"~.. in

;" N0:

~,

t;;; ",,,,

""N '" I'"'" !

c c c c c c C C 8 F G G F

1'-10" 1'-10" 1'-10" 2'-O~"

3'- 8"· 3'-IO~"

SYM. ABT.

3'-8"

18'-4"

NOTE: BOTTOM FLANGE RIVETING SAME AS TOP FLANGE

" . n

J1 l~ 1"'~

_ c

lr - fr.:T - Or" "

c c c c c c c c c c c~ ~ c c

3"8"3'-5~ "

8"

38'~ 0" O. TO O.

NOTES' MATERIAL, STRUCTURAL STEEL ASTM A1'"46

HOLES: 15/16" SUS-PUNCHEO 110 REAMED

RIVETS, 7/80

' SOLTS, 7/80

' <HIGH-STRENGTH) H. H.1Io N.

COMSINED SENDING 110 TORSION TESTLEHIGH UNIV. - FRITZ LAS. PROJECT 215 S

TESTS CONDUCTED AT SWARTHMORE COLLEGE

PLATE GIRDER SPECIMEN TI2-R

TlO-S SIMILAR TO TI2-R BUT SOLTEDTII-S NO COVER PLATES-BOLTED

~L.~L

.. -- --- ---,--------_.- ------_._._-- - ---

Fit> 4;3SUMMARY OF TORSIONAL PROPERTIES

i----·---~-- -----_.. -.------- .

i IIIII

Torsion Constant K (in. 4) I

~:t:~~e~:~~e~e~~:~··-·-I-~·~~~~T;~~-·~ ~-~~C~~~~~-=L -~;~~;- ~~~l: Lehigh Un!versi ty Mark T6A-4 . I T6B-1 I -- Ir--·-----·--'---r-·-·-·-..--·-·----.. - ..I---·-------,

I II I lI -T---II ' 'I I 'II I II Sketch 48~" II -L__ ~Lt I iI-I . __. __ ._._... l- Bo~~~~_____ I Riveted t Bolted I

.1 A. Experimental Value ·-·-------·------·-··-·r---- II (Bused on Test) , I I

I 1. Test (K)A i 59.60 73.10 i -- I.--.----.----1---·---....-·-·-·-----·-·-1-..------....·---·-·..-----l.· ---' -' -..-- ~.--~

B. Integr~Action Constant . I(Computed) ,

·'

5.8910.2810.28

Experimental constant based on uniform torsion test

- Integral-action constant, incorporating edge and hump effects,

- Separate-action constant, assuming elements act separately

- Integral-action constant, neglecting edge and hump effects

Test (K) A

(KI)B

(KI)C

(KS)D

1. (KI)B 62.43 62.43

I 2. (KI)C . I 66.74 I 66.74 ! --

IC. -;:::::;e~~~~~-~:~;r·-..-----------·_.. -·--.. -'---'---..·-..--·r-·---·-----·I (Computed) I I

I Ii 1. (KS)D . I1---·---_·_-_·_.._- ---- -..- , - _--.- - _..- - - -- ..---- -" " '--" _!I '

I

III

•

IComputed values based on nominal dimensions 1I

Uniform torsion tests conducted at Lehigh University by Chang and Johnston (10) II!

on Phase I specimens

Combined Bending and Torsion of Plate Girders II

Lehigh University - Fritz Engineering Laboratory Project 2l5B

_' .._ ._ _ __.. __.. .. _. __.._., __ . . . ..J

..

EccentrioVertical Loadapplied throughbracket

BracketBorthSide

EAST !liD

Tie Rods wi tbSR-4 gages

WW PID

Tie Rods withRing DynlllROllleter

'----- MOVABLELOADINGDEVI CE

TESTI NG MACHINE

'-------- BRACKET

.L--"'T"""--++--1--T+--"'T"""----'LTEST IN GMACHINE

HEAD

•-.,. 00I

'"0

0

-N0.,-.,.

-.,.0I

-N

GIRDER

STRAPPLATE----~

FIG. 4:5

ECCENTRIC LOADING SYSTEM I

CONCENTRATED VERTICAL LOAD AT MIDSPAN

ACTING AT TOP OF FLANGE

SECTION "Cr

Combined Bendinr, and Torsion of Plate GirdersTestinr program conducted at Swarthmore CollegeLehi~h University - Fritz Engineerinp, Laboratory Project 2l5B

Fig. 4:4SCHI!HATIC SUTCH TEST SEl-UP

Combined Bending end Torsion ot Plate GirdersTesting program conduoted at Swarthmore CollegeLehigh University - Frita !l:lgineerlng LaboratoJ7 Projeot 21'B

'"\

o

51

BRACKET

R PIN

TESTING MACHINE HEAD \

/~-+~~

\~~

\mv~OADING POST

\ I I PINI II II

GIRDER t.. e=20"VARIABLE I

+- +-"I 51'

y-0

10" I. :\ I

II-<0

~• r;: -~

I I

1°? 1\I II00 I II, •

= 00 III

-~ I

'['0 00 i\ I II I

U :I 0<1>(\J00 hi

I I/PIN'" 00 ~ I-

- 1-. ..... • -~"r ~

0·0,;.

00 ~= cbo-~ ; ,

\1HOLE FO0 ~'0

I GO(\J

le0I, . BRACKET(l)0

•== ~ /I

L

=

FIG. 4:6

ECCENTRIC LOADING SYSTEM II

..

CONCENTRATED VERTICAL LOAD AT MIDSPAN

ACTING AT MIDHEIGHT

SECTION "C II2

. "I

Com"ined' Bending and Torsion of Plate Gi reiersTesting Program ~onducted at Sw~rtnmore CollegeLehigh University - Fritz Engineering LarQ:r'[,tory Proj ect 2l5B

52

J)

1 '

...

12 'IF 27

(LOAD)P (AT MIDSPAN)

f..I

I

NORTHSIDE

GIRDERSPECIMEN

// TIE ROO WITH

SR-4 AGES

-.......­. ...

eGIRDER-

SOUTHSIDE

...

:'lBASE PLATES

KNIFE -'--~-LJ--""" (GROOV 0) .EDGE ,--

\,--_"::=~~---I

TIE 00 WITHS -4 GAGES

3"t ROLLERWITH COLLAR~

SUPPORT/ FAME

(wELDED)

PLAN V, EWPARTIAL

N

....N

THRUSTt3EARING

KEYIN SLOT

'11=1

F.lG.4:7

GIRDER SUPPORT FRAME -AT EAST END

(FRAME AT WE ST END 51 MI LAR EXCEPT TH TTIE RODS HAVE RING DYNAMOMETER)

TOP OFTESTINGMACHINE

8EO

SECTION "0"

Combined Bending and Torsion' of Plate Gird rsTesting program conducted at Swarthmore CO~lege

Lehigh University - Fritz Engineering L boratorJ Project 215B

'--- ~_-.....---------------_::_-------_:__----J.

53_____---'"~ _4.-'---~--

1

i

@ p

RE$TRAINEDEND

L;2 c 220 II (EAST HALF) FREE-~._~ .... END

..!

. ISTATiON l

II

1

NORTH--/

SIDE /',

WIRE

0'4- AMESIAL

FIG. 4:8 .

/2J

/V LEVEL BAR SUPPORT-- ,/'

(?)

Sf ~I ~I I1 1'-2 311

,3'- 8" 3'- 8" 3'- a" 3 '- 8" 12'-5 LI'. 4

<t) -@ @) ®

ILJliL 2- / ~-~~t 6/1 '=#= ~ , ~F--t---'1

I dI, ~ tIt- ,~t;1I I l' --

IL/<-- -:--- ----------:-- -- --- -----: -- -1- - ,~

. , .! ---_._---II

II

TEST SET-UP -OISTO

MEASUREMENT O~ AN'"'LE OF TWIST, VERTICAL AND LATERA DISPLACEMENT

SCHEMATIC C"KETCH

I Combined Bending and Torsion of Plate GirdersI Testing program conducted at Swarthmore College.L L_e_h_ig_h_tJn_i_v_e_rs_i_t~_1--_Fri._t_z_Eil_gi_nee-e_r_in_g_L_a-..,:oo.;....,-.ra_t_o_ry_P_ro:t_2l5B

54

®II

( PLAN VIEW i;

I,

II

-I ::>1I

0 0 C© III

\ II .I

STUD!BOLTS

II..

NUT WITHSHOULDER

SPIRIT- LEVEL

UBBLE

IR VA L FILLE f\ - }t---+-:i----i

FIXED POINT

F= 20"

MICROMETERSCREW

fIG.4:9

EVEL BAR WITH SUPPO T

OVEO AOJU~TA L£ TYPE

POStTlONS OF DIAL, MIC OMETER SC EW, AND

'fIXED POINTS ARE ADJ STA LE

Combined B nding and Torsion of Plate GirdersTesting program conducted at S~rthmore CollegeLehigh University - Fritz Engineering Laboratory Project 2l5B

PLATE GIRDER --­SPECIMEN

~-­.,

..

PIG. 10 DUL G4GB SUPPORT FHA.LOADI.G BEAD OB BRACKft

PIG. 11 OVERALL VIEW OP TEST SET-UP

Fig. 4:12 COMBINED BENDING AND TORSION OF PLATE GIRDERS

SUMMARY OF TEST PROGRAM

Specimen and Test Bo.

56 .

..-~ ,

T10-B

. I eI

T1I-B T12-R

Sketch T"~: II4 8 ~" -- .-_._-- . ~- .•

J_..~L -.JL ~L

.j-

A.

Fabrication

1. VertioalBending

.Bolted

. 101

Bolted

202

Riveted .

301 A &B

2. LateralPreliminary Bending

Tests :3. Pur. Torsicn

114

T6A-4Chang

201'f6B-1Chang

,", .,I

I.! .

, ;..;

; 1.. ·i

"

B.

Compined

Bending

e= 51t 102 20)Loading

FlO" 103 al4

I e=15" 104 (Partial)', ---~

e=20" 105 206,

i=27.5~·

- "

306

Torsion

Testse=10" --- ---

Loadinge=15" -- ---II

i=O e=20" -- ---,.,

401

402 &: 601

40)

Phase,;},·: Tests at Lehigh University -- by' Change (Pure Torsion)

Phase. 2: n n n " -- by Kubo (Non-Uniform Torsion)

I,:\'-.~

Phas~ 3: n n Swarthmore College .'-"'" by Kubo (Combined Bending and Torsion)

FIG. 4:13MANEUVERING SPECIMEN

INTOLAEORATORY

--FIG. 4:14

roTATING SPECIMm FROMVERTICAL TO HORIZONTAL POSITION

57

.-

Fig. 5;1

SUMMARY OF TENSION TESTS OF COUPONS---~.--_ .. - --- T

Specimen Modulus Ultimate % HedU~1on ICoupon ~asticity Stress iElongationGroup Mark Location E (ksi) (ksi) in 2 in. of Area

I

TB-2 Flange Angles 2$,200 62.90 49 56 II TB-1 n n 31,700 62.90 47 56

Avg. 29,950 62.90 48 '56

TA-1 Weh Plate 30,600 58.70 46 57TA-2 n t1 31,800 58.70 46 - 58TA-3 n n 30,050 58.40 -- 58IITA-4 n II 29,900 58.40 -- 58

Avg. 30,590 58.55 46 58

TC-1 Cover Plate 31,600 69.40 45 56III TC-2 n n 30,400 66.60 46 55

Avl!.. 31,000 68.00 45.5 55.5

TC-3 Cover Plate32,700 65.60 44 64 I

at 45°IV TC-4 I'i tl 31,400 67.70 43 53

Avg. 32,050 66.65 43.5 58.5 I

"-

TSB-1 stiffener Angles 32,100 64.00 42 60

V TSB-2 " " 33,000 63.50 40 61.5

Avg. 32,~50 63.75 41 61

Avg. of All Specimens: 31,110 63.07 45 58Avg. of Groups I,ll, & 111= 30,510 63.15 46.5 56.5

Avg. of Groups I, II, III & n 30,900 64.03 45.8 57.0

ASl'M - 1.. 7 - 46 Specifications -- 60 to 72 22% --

Combined Bending and Torsion of Plate GirdersTensile Tests conducted at Lehigh Un!versttyLehigh Un!verstty - Fritz Ehgineering Laboratory Proj ect 2~5B

:..'

.i

...., '.

Fig. - 5:2 'VERfICaL DEFLECTION !LONG SPAN- EXPERIMENTAt va. COMPUTED

VERrICAL BENDIN'G TEST. ~ CQN(;MRATED LOAD AT CmTERLINE

"," ..,"

p

(40,800)

(35',IGO)

(37, 000)

Notes'~

o Expo Value - SR-4 Gages

. -:~~ CO~lputed V'alues--- <2>: Based on net Ix-'--WB d Iase on,gross x

- - -')I(. Based 'on effective IxVerticalRending

T12-R-301A

e = 08 P = .100

Riveted

~."

? = ,'E 2. 0 ' ---,...------,-----.-----Jl"""""1I- i I

IIII I ' I,'-r T --..-- --r' -,-I ' I

I,' I'~"/I'I !.../

I I ! ........

'1 - I j' i I !J ........ :. , I6 --------.. ·--.. -i----:-- . ------:-----------i------ i~--- .-'-.--- --T ----~---(

r- I I I ' ~yY../., i. i 'I'• - II i I 'II ../../ -, , ! ..:" i

~ j" -. j . I ' .~I. '~'. ../1'../' ' ' " - ';,1', ,1'" ----'-----,.\1'.1,-,'-'-----.,-,-.,11

1

. '-J 4- ----i-.-:...· ---.-.~----~--~. -,--,-~--f---:--.,~-- --f - -o I ! I . '/'I I I ........ 'I I- I 1 1

! I, ' • ../ I i ~ II I ~~I I I I ;I : ; ~",I i r i'-+--- -"-~-----:---~ I-~--~ ----'-,~r._-------~--,-------+---.....,----,.-:- ...j ~...._-------I

. '. i~j I I' Ii '_~',I,'~ I '1 !' ;,'

'j I I; , 1 : ' '

@ ,@', ®STATtoN·

Fj~. 5;.3 vERTICAL BENDING ST:lliSS ALONG SPAN EXPEF1ME;NT 1;L vs 0 C0NFUTED

VERTICAL BEJ.'lDING, TEST - Cc.iNCENTRATED LOAD AT CENTEPLINE

t' ','

_rL/2. '=, 220 II

I

, .'

iIII

Gages~ 311-4Value

STJ.:1'ION

Notes:,T10-B-IOI

o

8

z ----

b

,Fig. 5:4,- '.

. VERTICAL: BEi-JDINGTE.:',f - CONCEr~TIbTED LOiill l;,T CH;TEiGINE

- ~'.

,------------- ---- -- -

Fig. 5: 5

SUMMARY OF VERTICAL BENDING PROPERTIES

Bolted

202

~L

19,1,30

16,400

18,300

II

TII-B

--- .. -- ----- --- ,- - - - --- - ----·-iI

I

I1------------1

IiI-~---,

37,500

.39,200

Tl2-R

38,200

(in.4)

39,000

35,160

37,000IiI

____________J .__ L -._,

40,800

35,160

36,150

II

__ 1. __ •.. _

Sketch

2. Net

1. Gross

3. Wetghted Ix

I1. Deflection Curve I 36,600

!2. Normal Stress Curve I 35,700

1-1 _-

I.3. Avg. Effective Ix I

I

liTI4Br I

~L~!I !

Bolted i Riveted I--------- -------r-------------------,---------T-~- ---- --1-A. Computed Values I J

I !I I, 40,800 II, I35,160i :

39,000 ! 39,000 I 39,000 !----------- ---------.- --'--. -----------j------.-----.---.--+ -----...--- -------- t

i :I iB. Test Values

TlO-B_________1. -----

Swarthmore T~st ~~_. jl- l_0_l -I-__3_0_1_.8._~ 30_1_BII

Ii!

Lehigh Mark

i Moment of Inertia Ix

i--------------------T-------------- -i Specimen!

III

Il-..--

.'

Notes:

Test 101Test 301A ­Test 301B ­

Test 202 -

Concentrated load at centerlineConcentrated load at centerlineConcentrnted loads at two symmetrical

Concentrated load at centerlinepoints

Combined Bending and Torsion of Plate GirdersTesting Program conducted at SW'arthmore CollegeLehigh University - Frltz Engineering Labor!~tory Project 2.l5B

I

I__ .__.1

r .....

- - -,::...::=-==~=---- -it

(780)

I y (11'1:4-)

(004)

( G4-0)

1.. _.. - j

I

~~~ ......' --"- - _.-

I. !

I yI yI y

I I..+_-_...--.. _.. ·1

! \

~= 220"

Exp. Value - "SR-4 Gages

~'-.

Computed YaluesBased on netBased on grossBased on effective

; --

Notes:o

-----~--S:ij

-----~

........~_.. ,1._ .......•......!

Bending

e =0"P -= 6 kipBolted

TIO-B-1l4

2. .. ----

8

.~,V) 4

G

STATION

Fig. 5:6 . LATERAL BENDING STRESS l.LO"NG Sf'.tt'll - EXPEEIHENTLL v~". CCJHi'lrn.D

.LATERAL BENDING TEsr CONCENTR.ATED LOAD h'r CENTF.ItLINE

"',-.'

>.i:

....,.. ,,,'."'":~t

.'

- --_.---_.._---_._-~ _ ~-_ ..~----- .._.__._.- .... _- --,_._----,.......,..--------- ""'--- '"------_ ....-

,....,.---------------------_.__.._---_.

.501lL,.' "="'.~....._""""====_:.===_=::o:...~=~~.~___

( 1(0)

VERTICAL' DEfLECTIONS ALONG SPlu"I - EXPERIr'1ENT~;..l. V.s. Cm1PUTED

.L.ATl:fVL BEnDING TEciT - ceJNCFJ-iTRhTED 101\0 AT ·'JENTF9J..ntE

Fig. 5;8

SULviMARY Of LATERAL BENDTNG PROPERTIES

IIiI

iII

---'!

iIJ

209

163

170

156

150

IIII,

1I

I

II. _._. __ . _...._ .1. -----_ .. __ .... _.

**

*

604

780

(in. 4)

I- ,

673

Moment of Inertia I y

Concentrated load at centerlineConcentrated -iC>ad at cente~line

Test 111._Test 201

1. Deflection CurveI

- i - 2. Nonnal Stress CurveI ,! .3. Avg. Effective I yI1-----· ----_.- -" -- - --- --- -.... -. -.j Notes:IIIi

.-.. ! ,,

- ,

"

i

[~=~=~=~~-:._~=~~_~'-~--.~-.--- .-~~--~ .i__=_=~.~-~~~~-~~_~~~~=~~ __~=~~~~i~:~;~; •.-~.~-~_~~_-_~ •.~= •.~_-_=_~~_~~--_~-~---'Lehigh Mark I T10-B _I T12-R I T11-B

i------- ---- 1\-------- -- . ----- -I .---- ---.-------- --.

j ,~SwarthmoreTestNo. f 114 1---- - 1- 2Cl'11. -- -- ------------- -----. .- -- --------._----.----j-------.-------------- ---...-.- '-r ..----.-- ---------..--- --. --+-_.. --.-----.--_.- -------.-.

I-Ii I !

I i- . i

I I III, I II~:~_.,.-"-:.."-"" Sketch III ~ L I ~ L ~ L i

II I

i Bol ted I Riveted Bol ted Ii J . .. .-----~ __.. .:.. ._ ... .. J

I------------------~---.. --.--. ---- -_....- -j--------...-----------------.-- I . I

i I 'I I Ir IIC. Computed I II 1. Gross I 780 Ij 2. Net I 604,

: I Ii 3. Weighted I y IIIi D. Test Values i

II !I 706 1

I 640I

j

i

II

Ij

IL .._

* No experimental values are available. Corresponding values from TIO-Bwere used in the computations.

Combined Bending and Torsion of Plate GirdersTesting program conducted at Swarthmore CollegoLehigh University - Fritz Engineering Laboratory Project 215B

,,\ -

•

66

Fig. 519SUMMARY !IF NCiTATION

<..

Method or Computation &1dComb.

Series ofB T 6 P H Condition Constants

1 Bl T1 Sl Pl - Ide81 B

2 B2 T2 S2 P2 -- Ideal A

3 B3 T3 63 PJ -- Actual A

4 B4 T4 64 P4 -- Ideal D/

'} B5 T5 65 P5 -- Ideal C

6 -- -- -- -- H6 Actual A"- _.

Notes:

B - B. 6. Co. Fonnula (Approximate - Linear theory)

Method T - Timoshenko " ( II " II )

of S - Sourochnikoff II (Deflection - Non-linear theory)

Computation P - Pettersson " ( n " II )

H - Based on measured distortions

A - Test values of (K)A Test Ix Text I yCombination

B - Computed Design (KI)B Gross Ix Gross Iyof

C - Computed Theoretical (Kr)c Weighted Ix Weighted IyConstants

D - Computed (KS)D Net Ix Net Iy

Combined Bending and Torsion of Plate GirdersTesting Program conducted at Swarthmore CollegeLehigh University - Fritz Engineering Laboratory Project 215B

Fig. 5110

COMPARISON OF STRESS COI'iPuNmTS

MAXIMUH COMDINFD NUR'b.L STRES5 ar CENTF.RLINE

SUMM~ OF CONk'UTFD VALUES

67

Computation

Series Method

Specimen TIO-B-I03

Bolted P = 50 kip

Comb.of

Constants

.e =10"

a =27.5"

Normal Stress (k.s.i.)

B 1

T 1I

S 1

P 1

B 2

T 2II

S 2

P 2

B

B

B

B

A

A

A

3.38

3.39

3.39

3.39

3.83

3.82

3.83

3.83

1.53

1.75

1.82

2.01

2.26

2.26

11.60

11.60

12.90

13•.30

12.97

12.96

14.50

14.40

14.98

16.52

18.04

18.51

16.80

18.79

20.59

20.49

Method of Comwtation

B - B. So Co. Formula

T - Timoshenko

S - Sourochnikoff "

Combination of Constants

A - Test K, Test Ix, Test Iy

B - Design K, Gross Ix, Gross I y

Ciix - VerticC1.l bending stress

P - Pettersson O'(;y - Lateral It

0; - Torsional "

Combined Bending and Torsion of Plate GirdersTesting Program conducted at Swarthmore CollegeLehigh un!varsity - Fritz IDngineering Laboratory Proj ect 215B

68

j \____._.. . M;....;.q1: u <l.=-....·--

Q

pe

'2.0

C. B. & T.

e = 511 '

P = 79 kipa. = 27.5"Bolted

Ie.

.- ...·~-----i

Notes: .", Computed' Value

8 Pettersson Formula PI

I•• -. -~._-.-.--.- ~r_ •.--.-.•. - .-.

J

I

: l

0 --L

0 4- '(3

5

30

r---Lf)

~!'

~~.

Cl.

f'ig. ::): 11 CGI{PONEl(TS 'OF FLANGE STRESS AT CENTERLINE VS. LV,W C0i·1FUTED

69

II_.L , ,_ ,I

e = '5"P = 79 kip.­a. =27.5"Bolted

T10-B-102 -

C. B.·&T.

'j .\;,

, '

"• I

i______ ..1 _

o

50

GO

40

-----, "I

I, 'iI ': I iI \ ' , i ' ,!

80 :=-:-" .::~ ~::t:-.::--=:-::=':=":=:-=::"~==--==t~"=~--=!/"==F c:

, 'I I '/:i :.;; ,; ,

70 - - -- ----.--- +- -.------------------ ------- ..---r/, // _. .L~" .._-.--__:_

, :,' . /./;. ,

/ //; ;'/- I I

/" /I---!-------j----- ------ "( • ! '

';; ;! / "----r----'-'i I-~-----·---------·-·----~------- -;,".f i' ,

~ 'Notes:

-- /1/ , '. ,,~~mr.t~.v~~aBl-!jJ -0- Timoshenko,n Tl

A'V EJ Soul'Oopnlkoff It '51$8Pettersson n - Pl

20 ., ,., ---. -.- /f--,

I, Ii, II

10 ~-

Q

,0 4 12 1,6 20

Fi g " 5: 12 MAXIMUM COMBINED F11>NGE STRESS ·VS. LOAD __ COMPUTED

70

0·03

c. B'. & T.

TlO-B-102

e'= 5"p= 79 kip'i' = 27.5",Bolted

0,02.

. i _ _ .!i, , _.._,,__.,_~j-

I 1i,1" "

_... _._ .._' _. . .' _.h_.. .. _.__

,

o

I "I

~,-:.- i _. ....., __-"-- -,---.:-'-'---Lr--------'--:--,-----l

0.0/

o

30

/0

50

70

60

, 80

'"g:-... .:.c:'V

40" (L

{/j (ROd).·r1. '

Fig. 5:13 LOAD' VS. MAXIMUM ANGLE OF TWIST AT CENTERLINE -- COMPUTED---,.---: -------'-~. '

"

---~=---==---,------,-----._------_.

20

/6

------~-------- L/2 =: 220';

T10-B-10~ ~-~-J P \', Not·p:=.~··· .'-' - - .- -- -I~I l-r; - Computed V.s.lues

e :: 5n . W1i - _ _. Tim;shenko Formula TlP _. 70 .. 5 IrNa Top South Corner T8a ,27. ~~,P-jt·_--_L Top North Corne~ TN1'(.1ted :l h Bot tor..: North Corner BN

r--"IIL-J Bottom South Corner BS,I II

C. n. ,:j T. (+) Tel1sionC-) Compression

I .

.;I

i

. _. - -- -- .. T

i.

iI

o~

12-

~II)\(

"'--'

I

D8 l~'

I

II

4--!

o

Fig. 5;14

I

t-i

CONPUTED CGiv'l1jINElJ FLf>NGE STRESS AT Ei"CH COE1\jT:iR LLOf,:G SPAN

FIG. 5:15

\

EFFECT OF VARIATION OF ECCENTRICITY

ON CONPUTED NiiIDIUM FLi..NGE .sTRESS--

I I l'lAXIHUN C01'~BINED FLnNGE .sT1ili33 itT <t,i:. rksi)! !

"! HErROD & COHPUTil.TIONECCEN-- VERTICiJ,

TRICITY LOADBEl'HFl..EHEM TIJ.'10:-.JHENKO PErTERSSON SoUEO:::HNIKOFF

e (in. ) p (t"l n ) STEEL CO...__~\l

~ ? h T ? h P ? h C " h

5 79 16.29 18.76 21.55 21.61

10 48.5 16.29 18•.'20 19.82 19.89

15 35 16.29 '17.73 18.82 18.92

-20 27.3 16.2.9 17·.39 18.24 18.37

I • e.J PNotes : I'l_

'-t j' .Specimen T 10 - B .' 1q.

~.

---'-,a =27.5" ' ~ -

Compression stress 3.t top J h.Fe::::.: -"-=-i

south comer,

..

.!

,E ~~ .,'_.__~ _....,.._--~e I-~--,.

~-=-----------'~------- ------- -~ -'~----------.----------~~------:---' -------1~ - .

, .,l

Z.c.J--i--~-1-2---R---~~P -'---,.- Y/C"CT" . I" 1 - /

e =10· T'. a. --.I--.-----.~.-..- .. j ..' t +./~<P = 50 kip -eIiL I - .It! r //1

L== J Ii i'

" /' I

C. B..:T~ .. _.... L.,-".:t._._.t. . ..J ... ? f'/ .:.1. .: ! j' I' !,// I

: I I : - 1 L,./ I',I I I ' ,I I ': '/\ .; ,t " i..:- 7 5 ~/ 1 ' I"

i I I i a = 2 •• /' ~.__ ..~.,__ .~+__ .~_. .,I

~ " '! " ,/ 'II ! . IiI

I/., I

I I~' ! t - :

4 -"1'" "" ·t~~~T --- -'j··_··~·-t-·'-'-·:·'---~-r·-·'-i··~-·j ..&.~ I r I II

! I ; , , !' "to _,~__--'- ...l----'r--"""__..J.-.. J-- --; -":' ---4._~--r-__' .;...1------1

/6

/2

---....:Vj~\::.

b e

STATION6

FIG. 5:16.

EFFECT OF, CHANGE O~ HEIGHT Or LOAD APPLiCATION

Comparison of Computed Maximum Flange Stress Along Span -Petters,sonTheo1"1.' _ t.

";'

'. ' .. ..

t.<>.

e- - - ~ - -

)- v. '

/<:)

/

•

---'---_._---'-'--------,-'----- -_.._-----

T252P2"Pettersson

Compo "H"

Exp. Value -- Top Flg.-50.Computed Values

B. S. Co. Formula B2Timoshe.'1ko "50urochnikoff "

+C. B. & T.

-+-~---------r--------- L/2 =220"----;-------:------;-------.:~

.-...... /2--....:Lj

~~

b 8

.j II i-,- -------.,.--- -_._----; i.

!

.--+

i!

II

II

/0, ~ I____ 1- ' ~ _. +____

---'J ---------l----//--~ !J ~ @ i

IO-""'''-----t-------'----r-----'''------r----'---------.-..y.------'--,-----'------,----"----r-----1

4

t' •

•

,,'1

.. , '----...,.----_._-- ---;--;.-

-- ------ -L -- 1!II

~ .-I,

Exp. Value -- Top Flg.-So.

Computed Valuesx B. S. Co. Formula B2

_._Timoshenko " T2El Sourochnikoff" S28, Pettersson " P2+ Compo "HI! --- .. --

Notes:

(oJ

C. b. &T.

T12-R-401

e = 10"P = 50 kipa:: 0Riveted

tf.7\

e-. -- - ------ - - - >- -

(

u L/2::220", , I

/+/ (,)

./ d':.../

././

/8- -- ---~./~ - - - - -- .. ~

0/ I

1'/ !: // .

1/ i

/f i--- --- ._------£_-,./ ! - -- ---- -- - --j

./

.4'i......, /. Il . I ./ "

• 'I /'

__________________1 l_. ~-------J/-~ : i __ . .. _~---------------- ~-_-------------. : : ~} I /~ j ! . i

I I H,// I I! I / I II I/'"

I ,,0~1" I Ii I ~ ! r i

--------~--- -----t~4-'-//l~ ~-"----i--- -------Li--------1-·-- --~---I--~- ~-- t--~

~~~~1 I II I I

O.......u~__~------'---r__----L---__._------____,--------!.--_.__---J....---_._-~--__,_--__I

4

/6

/2

~ 3 4.L '6 7j 9 @ ctzST4T/O/V

;=/r-; FA'PERJMENTA L NaRMAL FLANc)£-.J

5:/8 v~s: COMPUTED V' - MAX. ST/fESS IN _..J. ._----- ...._---,_._-......-.-......--................_---_.

.. • •

-Mb~--------- ---xx

-- x----------x

e--x --,------1

; X

I/Z:: 220 1/ -----~------_'>_l

Computed Value- . - Timoshenko Formula Tl

Ie = 10"I,p = 50 kipI a = 27.5". Bolted

. iIiii

I<5 ----------.--.---------j

i

Notes~

Q-0

. TIO-B-I03Exp. Value -- Flange centerline

" "-- Outside gage line

·c. B. & T.

-I

_ -0--I-

I .j

---

. I - 'I' !? ' !

----~-- ---- -- ------------ -- -----l----~- --- ---j . II II I

-0-

-_..:._-------------+------------------ - ----- ---------~~-- - - --_..--.--------- i I .

-1--------- --- ---------!--------iI

I-{)-

01- _

~

2 =~~-=-~--=--t--------------! --. - - - ---:-- ----- --

<0

..-...--.::Vi~-

"-

4(-

-.fTAT/ON

Fig. 5; 19 EXPERIMENTAL VS. C0I11'UTED COHBINED FLANGE SHEAR

.'• '. • .•

--+-==-=.--=.-=-------- -'-j( -------~_._~-

x*- :-.----:-*------ --)< ---~:-----'

X X

1/2-:: 2 20/1

Computed Value- . -Timoshenko Fonnula Tl

8

. N0.1.es;.. ~ .Exp.

-0- VI.

Value -- Flange centerline

II Outside gage line

T12-R-40l

e = 10"P = 50 kipa = 0Riveted

C. B. & T.

I.

. I._. -- -_.-'..!_.. - .... -'.i

. !1 -.,

:~, .Q

--'.

--- --... ·'···---r--·-' .-.IIi.

I!'.'

., f

I

¢- -;- .....•..... Q

-. _._. - -.... -_ ..~ !

-0-

, ­.

---_._-_ .._---.:iII,j

£.

0-t----------,----.-----r----~---'--'__r-~-~__:_----'---,__---'--~r__---'-'__r--.

2

6

4

; .~:X

Fi-g. 5;20

STATION

EXPERIMENT"AI.. VS. COMPUTED CmmINED FLANGE SHEAR .,

• (

p

pe-

'j'

C.B.&T.

e = 10"p = 50 ki.a = 27.5"Bolt('d

T10- -10;

~, ,

;,,'

Exp. Value -- Avg. Yeb ShearNorth Side

_ Liz - 220 "

Con puted ValueTimoshenko Forrr~la T1

Note::::o

~-----_ ..

, /)(, ,

-

o

tY/---f--

;3 -

f

oi---L------C1

._- ------- - -~I

~"TATION

Fig. 5:21 EXPrRIHl' TAL V2. CQ·1PUTED CO .BlNED i\"EB ..,nEAR (MAX.)

•

x ......:. x

p

ae = 10"P:= 50 kipa= 27.5"Bolted

TIO-B-IO~

L/2 =Z20u------------,--------~__i

2.0

C. B.·& T. E~:3

iIi

i- --'--. II

,1 ,-- .c •• ; 1

o.:"... . .~, .

iii

.!

i . . . .- . _.- ..- ---.. - '---r' .--.-.-.-------.-..-.-..! •. --.~---.--.-.:- --.IIII

iI

iI

I!

.12; I •-' -·-i-·

i.L..;IJ

'" - .._.. --_ ... --··-r-:---·_·j

!

,J

o

----- ---.....,~--_...-.- .• - ~ •. _. - 1"

II

I.1i­Ii

iIj

II,i-/. 0 +-------....L-~-_r_--------.:...--"---_,__--;--...;......'--------:-~----_r-l.------'---"--4

-.~

'-:.8

tr;)W(7::( I !

o - - - - k - ..=-=--.::.±-l=-=:..-:::=--r-:=---2. l-.4

"

STATION

Fig.. 5a?2. EXPERIMENTAL VS. COMPUTED COMBINED WEB SHEAR (l'IJ:N.)