combinfd torsion - lehigh university
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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
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"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
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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
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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
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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.)