August 1969

Alexis Ostapenko

byChingmiin Chernand

- ------~~---~

Unsymmetrical Plate Girders

ULTIMATE STRENGTHOF PLATE GIRDERS

UNDER SHEAR

Fritz Engineering Laboratory Report No. 328~7

328 September 19, 1969

For the ease of calculation,

PAGE

Abstract

1

2

3

4

6

9

10

.12

11+

15

16

'LINE

8

1

,14

8

11

19

14

·14

23

5

Eq •. 5d

11

5

16

1

7

8

11

. IS

filed

consist

~ had

the methods'

was first

1961

stresses were

them.

like many others

shear.

flanges as shown in thet:ig'ure.

can increase

of the inclination

there is no

taken to be

~ig. 9b may

in strain

.. SHOULD 'BE

field

consists

,has

methods

was the first

1963

stress was

Rockey and Skaloud.

. like others

shear which is to be added to theweb stre.ngth.

\/ '

'" V ~..z~-? \tt\: 12-._

, fla:nges •

For ease in calculation,

can then increase

of inclination

there are no

taken for. this case to be

F.ig • 9b, may

iIi the strain

PAGETJ7

18

P,age 2 of 2

LINE 'IS . 'SHOULD BE-1- of bendi,ng of the bendi.ng

2 thus strictly thus s' strictly

9 F.ig. 3a and 3b Tables '3a and 3b

12 occuri.ng occurri.ng

6 lined line

12 of experimental of the experimental

Add at the bottom of page 18: This 'supports the ~e­

commendation to ne'glect the post-buckli;ng stre.ngthfor

19 7 applicable to symmetrical applic~ble to homogeneous,hybrid) symmetrical

Item 4. Delete (4.) and make a pa~agraph out of it.

20

21 '

39,

,42.

43 s44,

45

46

1

Footnote

F,ig. ·14

. 8

20

,17

(a combination.

v

Consecutive

w. B. McLean

(this will lead to a combination ofpanel and beam.mechanisms in the

. fla.n~es ) •

~{The 45° inclination co~responds toa full tension field.

Make the t~i~ngle for Gl-l solid.

Ma~y editorial style corrections.

consecutive

J.F.Oyler (for W. B. McLean)

328.7

Unsymmetrical Plate Girders

. ULTIMATE STRENGTH OF PLATE GIRDERS UNDER SHEAR

by

Chingmiin Chern

and

Alexis Ostapenko

This work was conducted as part of the projectUnsymmetrical Plate Girders, sponsored by the AmericanIron and Steel Institute, the Pennsylvania Departmentof Highways, Bureau of Public Roads (Department ofTransportation), and the Welding Research Council. Thefindings and conclusions expressed in this report arethose of the authors, and not necessarily those of thesponsors.

Fritz Engineering LaboratoryDepartment of Civil Engineering

Lehigh UniversityBethlehem, Pennsylvania

AugU$t 1969

Fritz Engineering Laboratory Report' No. 328.7

328.7

TABLE. OF CONTENTS

Page

ABSTRACT

1. INTRODUCTION 1

2. ANALYSIS 7

. 3, COMPARISON WITH TEST RESULTS 16

4. CONCLUSIONS AND COMME,NTS 19

5. APPENDIX 21

6. TABLE·S AND FIGURE·S 22

7. NOMENCLATURE 41c-

8. REFERENCES 43

9. ACKNOWLE·DGME·NTS 45

328.7

ABSTRACT

A new approach is presented for the ultimate ~trength

analysis of plate girder panels subjected to shear. The ultimate

shear strength is assumed to be equal to the sum of the strengths

of the beam action, the tension field action, and the frame action

In the web plate buckling computation for the beam action, the

web plate is assumed to be fixed at the flanges and simply sup-

ported at the stiffeners. After buckling, the web develops the

tension f;Y~d of a pattern different than those proposed by others.f!e(~

SimUltaneously, the flanges develop the frame action--a panel mech-

aniSffi.with plastic hinges forming in the flanges at each corner of

the girder panel. The method is applicable to symmetrical, un-

symmetrical, homogeneous, and hybrid plate girders.

A comparison with the available 33 test results indicates

a better and more consistent accuracy of the proposed method than

of the methods developed by other researchers.

(.)

328.7 -1

1. INTRODUCTION

A typical plate girder panel consis~of the top and bottom

flanges, two transverse stiffeners and the web plate. Thus,

the stiffeners serve as boundary members of the rectangular web

plate~ When both flanges have equal areas, the location of the

centroidal axis is at the middepth of the web and the cross

section is considered as symmetrical. If the flanges have

different areas, the centroidal axis is located closer to the

larger flange and the cross section is unsymmetrical with respect

to the centro idal axis. A girder in this case is de,fined as

an unsymmetrical girder.

Design of plate girders has been based on the assumption

that the load carrying capacity of the web plate is limited by

buckling. However, experiments indicated that the web plate

ha« considerable post-buckling strength, and this fact was recog-'5 .

nized by using a lower factor qf safety in establishing the

allowable stress for the web than for other structural components.

In aircraft design the post-buckling strength of the web has been

utilized directly.( 15 )

It is oniy in the past dozen years that a direct utilization

of the post-buckling web strength has been studied for plate '

gir~ers of the proportions encountered in civil engineering

328.7 -2

practice where unlike the aircraft struct.ures the girder flanges

are very flexible. An exact analysis of the extremely complex

behavior of the web in the post-buckling inelastic range has not

yet been possible. And all attempts have been limited to

developing simplified physical models which would facilitate

the evaluation of the ultimate .strength of a girder panel. Although

the primary interest has been in girder panels subjected to shear,

~'methods were also developed for panels under bending or a

combination of bending and shear. This paper deals with the

case of shear alone.

-JBa~ler wa~\first to present in 1959 a theory for the ultimate

shear strength of welded plate girders.(2,4) Ultimate load tests

assisted in developing the theory.(4) It was assumed that the

panel acts according to the beam theory up to the web buckling

load and thereafter in a tension field manner up to the point

of initiating yielding. in the tension diagonal. The strength of

the flanges was not'taken into account. The method gave adequate

correlation with test results to be used as the basis for the

design procedure incorporated in the 1967 AISC specification.(l)

oIn 1964, Cooper, Lew and Yen applied Basler's approach to

high strength alloy steel plate girders and conducted some tests.(7)

It was found by observing the girder behavior that in evaluating

the buckling strength ,it may be more reasonable to assume that the

328.7 -3

web plate is fixed at the flanges and simply supported at the

stiffeners instead of being simply supported at all edges as

assumed "by Basler. However, an inclusion of this effect ,into

BaslerTs formulation led to a poorer correlation with test results.

The same year, Takeuchi suggested a modification of Basler's

approach by proposing the tension field model shown in Fig. 1.(14)

The tension field was assumed here to extend distances a1 and a2

into the top and bottom flanges, respectively. Distances a1 and

a2

were suggested to be proportional to the moments of inertia

of the flanges about their own axes. This was an attempt to

include the effect of the flanges on the ultimate shear strength.

Howeve~, the research based on this model was not continued.

In 1967, Fujii presented a tension field model in which the

tension field stress~8/we,£e assumed to be uniform in the panel,,// "~ (,,":'J (','/ ~,

and controlled by the vertical web stresses needed to develop

three-hinge plastic beam mechanisms in the flanges.(9,lO,11)

This theory gave better agreement with the results of the tests

conducted in the United States and Japan than Basler's theory.(ll)

However, in its present form, the theory cannot be applied to

unsymmetrical gird~rs.

In 1968, Lew simplified Takeuchi's tensjon field model into

the special case of symmetrical girders. (16 ) Parameter al

= a2

was determined from test results as a function ,of the-aspect ratio.

328.7 -4

In 1968, Nishino and Okumura reported several tests to

f · h h did b F··' (18)con lrm t e approac eve ope y UJ11. They suggested

that the buckling load (beam action contribution) may be more

reasonably calculated by considering the web to be under the

presence of shearing and bending stresses, even if bending

stresses are very small.

The same year, Rockey and Skaloud(2l) presented experimental

results which indicated that the actual mode of failure of a

girder panel under shear may be dissimilar from that assumed by

Basler or Fujii (they were not aware of Fujiits work, however).

Th~y showed th~t the flange rigidity affects the ultimate shear

strength. This effect was not included in BaslerTs approach; it

was considered by Fujii, but in a different failure mode than

suggested by t~ '" The results of their theoretical investigation/cock.e"v SkeAt c>!"cd

have not yet been published (August 1969).

Except for TakeuchiTs suggestion which implied that girders

may be unsymmetrical, all of the above described research has

been directed to symmetrical plate girdersG

This paper describes a new formulation of the ultimate

shear strength developed in the course of a theoretical and

experimental research on unsymmetrical plate girders conducted

at Lehigh University since 1966(8,22) The method is based,

like mJ~ others, on the model which gives the ultimate strength,I

,/

328.7 -5

of the web as a sum of the buckling stre~gth and the post-

buckling tension field strength. These contributions are,

however, evaluated using somewhat different assumptions t0an

proposed by others. Furthermore, the strength of the flanges

is added as a separate contribution. It is visualized that

the behavior of a plate girder panel subjected to a gradually

increasing shear force is as described in the following.

When the panel is loaded by shear alone, the shear is

assumed to be carried by the web plate up to the point of web

buckling. As the shear force is increased beyond the buckling

load, the web plate starts forming a wave-like wrinkle along

the direction of the principal tension stress. The web plate

then behaves like a diagonal tension member in a truss panel,

with the flanges and stiffeners being the other members. This

type of behavior is' called tension field action, and the shear

capacity in excess of the buckling load contributed by the web.

- is called the tension field action shear.

The tension field action cannot be achieved without the

stretching of the tension diagonal. However, the stretching of

the tension diagonal will ,be accompanied by the distortion of the

pan~l shape from a rectangle to a parallelogram. The continuity

of the flanges and of the web plate into the neighboring pane~s

provides sufficient restraint to prevent any significant distortion

328.7 -6

of the transverse stiffeners. Thus, when the distortion of

the panel takes place, the:tJ-anges behave like beams with both~_._._,.. _.. --_... - ....._'... , ... - ---- -_.... _.. ,y ..•....... _ .•.........•..•

8!l9s__ fixed ~~/- The shear capacity contributed by the formation of

plastic hinges at the ends of the flanges is called the frame

action sheaIj\d

328.7 -7

2. ANALYSIS

In the approach presented here, the ultimate shear strength

of a transversely stiffened plate girder is given by the following

three contributions: (a) the beam action shear V , (b) the'T

tension field action shear Va' and (c) the frame action shear Vfo

v = V + V + VfU 'T IT( 1)

I.

The basic assumptions are analogous to those made, say, by

Basler(4), but the individual contributions are computed differently

and the frame action contribution has not been considered- previously.

Beam action shear - A web plate element subjected to shearing

stress T is shown at the left in Fig. 2a. These stresses

correspond to the principal stresses shown at the right in the

figure, where the tensile principal stress 01 is numerically equal to

both the.compressive prin~ipal stress d 2 and th~ shearing stress T.

The maximum beam action contribution is defined as the shear

force carried by the web at the web buckling load. It is given by

where the shear buckling stress is

(2)

'fer = kv

2'IT E

212 (l-'J )

(3)

328.7 -8

The shear buckling coefficient k is a function of thev

aspect ratio a and the boundary conditions.

According to Ref. 19, for a rectangular plate fixed at its

longitudinal edges and pinned at its transverse edges and

subjected to pure shear, the k -values can be computed fromv

the following equations:

2.31 3,L14 8.3gk

5.34 ~~+ ~O!= -2- +--v (i

Q'

for Q' < 1.0

and 5.6\ 1.99·k = 8.98 +~-~v 2 3

Q' Q'

(4a)

(4b)

for ¥ >' 1.0eX:

The graphical presentation of Eq. 4 is shown by curve (1) in

Fig. 3. Curve (2) in the figure represents a rectangular plate

simply supported at its four edges. It is seen that the buckling

stress is greater when the plate is assumed fixed at the horizontal

edges.

The theoretical buckling stress Tcr computed from Eq. 3 is?I

considered to be valid only when it does not exceed 0.5 'T'\", wherey,'

When the calculated T cr isT is the web shear yield stress.y

greater than 0.5 T , a reasonably smooth transition c~rve Eq. 5b and c,y

which is analogous to the plate buckling curve in Ref." 20, is

328.7 -9

employed to take into account the effect of residual stresses,

initial imperfections, and strain hardening. The buckling curves

in the elastic, inelastic, and strain-hardening ranges are

expressed as follows:

for A > /2.

for 0.58 < A < /2-

1'"_ 0.58)1.18cr 1 - Oc615 (A=

1'"Y

for A. < 0.58

( Sa)

(Sb)

TY

where A = 8

1 + 4.30 (0.58 - A)1.58 ( 5c)

(5d)

The graphical presentation of Eq. 5 is in Fig. 4. The figure

also shows the following curves: (a) the elastic shear buckling

curve, (b) the inelastic shear buckling curve suggested by

Basler and (c) Johnson's column curve employed by Fujii.

Tension field action shear - After the applied shear force

reaches the plate buckling load, stress cr2 cannot be expected to

328.7 ~lO

increase to any significant degree. However, the stress in the

direction of the tension diagonal continues to· grow beyond the

buckling value. A field of tensile stresses of the type shown

in Fig. 2b develops. This is the source of the post-buckling

strength of the girder web.

According to the evaluation of the tests conducted at Lehigh

University(3,7,8,22), a probable tension field stress distribution

over the web depth may be assumed to be the one shown in .Fig. Sa.

This tension field model consists of two parts: a fully plas-

tified zone along the tension diagonal and two elastic triangular

The tension field stress in the plastified zone is crt' the stresses

in the elastic triangular portions vary depending on the rigidity

of the longitudinal boundaries. For /~e ease ~ calculation,In

the unevenly distributed stresses in the triangular ~ortions

can be replaced by a uniform stress Pat' constant in both

triangles as shown,in Fig. Sb. p is ~hus a parameter dependiDg

on the rigidity of the flanges.

The tension field action shear is given by ~~~0_e__v_e_r~t~1

~mpo~~~_~_~~~_!?~ce and it i fou d from Fig. Sb

to be

Vcr = ~ Aw crt [sin 2C+? - (1 - P)O'+ (1 - p)Q' cos 2cp J (6)

328.7

The

-11

imum tension field stress corresponding to -the

initiation of yielding in the web is a function of T cr ' ayw '

and ~, where a is the yield stress of the web.(4)yw

3~ '2 'fer sin 2C',O (7)

When the tension field force of the type shown in Fig. 6a

is set equal to that in Fig. 6b, the parameter p is found to be

p =ads S ads+ 8 3 (Sa)

The param~ter p can be conservatively approximated by

assuming that the stress in the triangular portions in Fig. 6a

varies linearly,

p = 0.5 + 0.5crt S + cr Tt StIt 3at (sl +: CBb)

The boundary stresses cr~ and cr~ shown in Fig. 6a are the

resultants of the vertical arid horizontal stresses existing

between the flange and the web. The tensile web stresses and

their vertical components acting on the horizontal boundaries,

q, are shown in Fig. 7. The variation of the stress q is plotted

as a function of the applied shear in Fig. 8. The stress q is

ze~o prior,to the shear force reaching the theoretical buckling

load Vcr After the web plate buckles, the tension field forms

alortg the tension d~agonal, and then spreads out as the load

328.7 -12

continues to increase. Thus, the stress q is developed. If the

flanges are perfectly rigid, as shown by curve (1) in Fig. 8,

the stress q will be able to increase until the web develops its

full tension field (Fig. 7b). In other words, the boundary

stresses a~ and (J~ can~increase until they reach at' Thus,

Eq. 8b be come s ~f lto'./V\.".,/i

p = 1.0 ( Be)

On the other hand, if the flanges are perfectly flexible,

only partical tension field will develop (Fig. 7d) and the

intensity of q will remain zero as indicated by curve (2). Thus,

the boundary stresses cr~ and cr~ will be zero, and Eq. 8b gives

p = 0.5 (8d)

The flanges used in the conventional welded plate girders

are not rigid, nor are they perfectly flexible. Therefore, it

- is conservative to use Eq. 8d.

The variable ~ in Eq .. 6 is the angle of t~ inclination of/' "."'"

the tension field, ~t__ "Qg:rameterp. assumed. c()ns~arlt, at is a

function of ~ for a panel.with given geometrical and material

properties. Hence, Eq. 6 can be rewritten as

v = V (cp)IT 0"

in which ~ is the only variable. The maximum value of V is0"

obtained by setting its derivative with respect ~o ~ equal to zero,

328.7

dV Ide+> = 0cr

-13

This gives the following expression for ~ , the value of ~o

for which V is a maximum:cr

2(~)

2sin 4ep

o - 3 (1"cr ) cos 2'+'0} ~ f sin(Jyw

2cpo

- .~ (:~:) sin 2cpo 1· {2 cos 2'+'0 - 2 (1 - p) Q' sin 2'+'01= 0 (9)

Equation 9 can be solved for ~ by iteration. (A method foro

performing such a solution is explained in the Appendix.)

Substituting ~ back into Eqs. 7 and then into Eq. 6, the maximumo

tension field action contribution is obtained to be

where

(10)

0'yw 1 + (1"cr)2{ [-23 sin 2CP12

- 3} - ~ T sin 2cpcryw 0 2 cr 0

(11)

Equation 10 is considered to be applicable only to panels

with aspect ratios in the range of 0.5 to 3.0. When the aspect

ratio ~ becomes larger than about three, two shear buckles tend

328.7 -14

to form in the web leading to the tendency for developing two

tension field bands. However, since there}~ no vertical stif-,~/C/r«?,_~ .

feners between the buckles, the vertical component of the tension

fields must be carried by the flanges in bending and by a verticgl

strip of the web. The flanges are relatively flexible and are

supported by stiffeners at the far ends only. This mode of the

post-buckling behavior has not yet been investigated. Pending

further research on such panels, it is proposed here to conser-

vatively neglect the post-buckling strength of the web when

ex > 3.0.

Frame action strength - As it was described in the introduction,

part of the strength of a plate girder panel is contributed by

the panel mechanism formed by the flanges. The assumed mechanism

is shown in Fig. 9. The cross sections of the frame members, as

shown in Fig. 9b, consist of the flanges and effectiye strips

of the web plate. Each flange behaves like a beam with both

ends fixed. The maximum frame action contribution is

(12) ,

where mpt and mpb are the plastic moments of the top and bottom

frame me'mbers.

There is little information available about the 'effective

strtp of a rectangular plate subjected to shear alone; However,

when, due to a low web slenderness ratio B~ the plate buckling

328.7 -15

stress T exceeds the proportional limit (which is taken to becr '~

0.8 ~y)' the web is in the inelastic or strain hardening range,

and only a very small part of the web area can contribute to

act with the flange. In another extreme case, when the web

slenderness ratio '8 is very high, and T «If, a portion of. ~ . cr y

.the web equal to ten times the, plate thickness is assumed to be

effective in this study. Thus, an effective web depth C tc:aS2~­V)

shown in Fig 9b/may be taken without serious error to be/~~I

when

'T= 12.5 (0.8 _ cr)

Ty

i cr < 0.8T y

( 13)

Ultimate shear strength - The final form of the static ultimate

. shear strength formula is obtained when the contribu~ions of Eqs,

2, 10, and 12 are added together according to Eq. 1

(14)

328.7 -16

3. COMPARISON WITH TEST RESULTS

The proposed ultimate shear strength theory is checked here

against all experimental, shear test results that could be found

- 1- (3,5,7,8,9,18,22) h h - also d wl-thIn lterature. . T e t eory 1S compare

the theories suggested by Basler and Fujii by applying them to

the test specimens listed in Tables 1, 2, 3a and 3b. Lew's

approach is not shown because parameter a1

= a2 in his approach

was obtained by matching test results for some symmetrical

girders only and thus may not be applicable to the tests shown

in the Tables, especially to unsymmetrical ,girders.

Table 1 summarizes the shear tests in strain hardening range.1\

It is seen that the proposed and Basler's approach are in good

agreement with the test results. Fujii's approach is not shown

because the strain hardening is not taken into account in his

approach.

Ten tests in the inelastic range of shear buckling are

shown in Table 2. The proposed approach provides an average

deviation of 8% with the maximum deviation of 14%, It should

be pointed out that the proposed approach consistently over-

estimates the ultimate shear strength in this range. This

indicates that the actual panel capacity must be reduced by the

Selected are the end panels or panels with zero moment at the, mid-span I'

-1.--

328.7

presence of(\ bending effect.

-17

All of these panels were subjected

to a small bending moment in addition to shear and thus strictly;J

speaking, should be classified as combined loading caseSn Basler's

and Fujiits approaches both give 7% average deviations with the

maximum of 20% and 22%, respectively. It is interesting to note

that most of the Fujiits predictions under-estimate the panel

strength.

Seventeen shear test results in the elastic range of shear

buckling are shown in ~~. 3a and 3b. The proposed approach~""-T--th 2') I

gives consistent predictions with an average of 3% deviation.

(B-panel indicates a deviation of 21%, but this deviation is

justified as occurjng under special conditions ). Several('r~~-·

large deviations of the test results (all of them are under-

estimates) indicate that Basler's approach may lead to a substan-

tially lower estimate of the panel strength. Fujii t s, approach

gives good agreement with tests for symmetrical girders, but it

is not applicable to unsymmetrical sections (at least, in the

formulation of Reference 11) 0

Girder panels with aspect ratios larger than 3.0 are shown

in Table 4. The proposed approach ~ives an average deviation of

of 4% with the maximum deviation of 7%

approaches are not compared since they, are· not applicable to this case.

The test results shown in Table 2 are considered as panels undercombined shear and bending in Refe 6 and give considerably bettercorrelation than here.

See Reference 15.

328.7

- --- -- -- - - -- - --- -----------------o--~"~

-18

The shear test series of symmetrical girders G6 and G7

of Ref. 3 and unsymmetrical" girder UGl, UG2 and UG3 of Ref. 8,

respectively, h~ve the same cross section and material properties.

The only variable parameter in each series is the aspect ratio a.

The ultimate shear strength is plotted against the aspect ratio

in Figs. 10 and 11. The long dashed line~~in the figures

represents the beam action strength, the short dashed line

represents the sum of the beam action and the tension fie~

action strengths and the solid line represents the ultimate

shear strength. A good agreement with the theoretically

predicted curves can be observed.

A comparison ofAexperimental web shear strength (experimental

ultimate shear reduced by the frame action shear) with theoretical

web shear strength for each tested panel is shown to proper scale

in Figs. 12 to 15. Figure 12 (Table 1, A < 0.58) sho~s the effect

of strain hardening on the strength of panels with stocky web.

Figure 13 (Table 2, O~58 ~ A ~ 12) indicates that most of the

panels with thicker webs failed primarily due to inelastic

shear buckling and there was little post-~uckling strength developed.

Figure 14 (Tables 3a and 3b, A > /2) shows that in panels with

thin web~ the web buckles elastically and then develops a significant

amount of post-buckling tension field strength .

. Figure 15 shows that for panels with aspect ratios larger

than 3.0 the post-buckling strength is very, small if not zero.,

f

T~\15 SV"{)fJ o d'5" cl

c" bucld ' 'S··hcvt,~jl! fol-·

---------------------------=.

328.7 -19

4. CONCLUSIONS AND COMMENTS

The conclusions drawn as a result of this investigation

are as follows:

1. The ultimate shear strength may be accurately obtained

by the summation of the strengths of beam actiori,

tension field action and frame action.!tD VkL0j'e-V) ccneS j !~)yb vi

2. The proposed approach is applicable to symmetrical andr\.

unsymmetrical plate girders. In fact, symmetrical

girders are just a particular case of unsymmetrical

girders.

3. The proposed approach is applicable to girder panels

with aspect ratios less than 3.0. For girder panels

with aspect ratios greater than 3.0, it is suggested

to neglect the tension field contribution.

'~ There are some modes of failure which are not covered

by the proposed theory and they should be guarded

against by making the girder end panels of such aspect

ratios that the panel shear strength given by the beam

and frame actions (tension field action is excluded)

. is gr~ater than the r~quired sheare

It appears that the theory can be further improved and

gen~ralized through a model combining the principal features of

the model proposed in this paper and of the, model used by

328.7

J/

/

!

c:::::"-"l""" C::"t5'>--/V] Jb /'

-20

Fujii (Ref. 9) (a ?3~~~1tion(yfPanelanf~/(J?,~~~>'~~:~j~~~S'/~);/II ~r(OVll') es

Observations of Rockey and Skaloud (Ref.2l) on the location of

the intra-panel plastic hinges in the flanges can be extended

and incorporated in this model also.

328.7 -21

5. APPENDIX

Optimum Inclination of the Tension Field

For a panel with given material and geometrical properties,

Eq. 9 can be rewritten in the form F(0) = 0 where 0 is the only

variable. A way to solve for ·the variable 0 in F(¢) = o .is to

select a trial angle 01 somewhereo 0 ~t:

between 0 and 45 . By

substituting 01 into the equation a corresponding value FI is

found. FI is then checked whether it is within the required

accuracy.

If the trial is not satisfactory, a suitable increment is

added to ¢l to obtain the second trial angle O2 and the c~rres­

ponding value F2 . If F2 is still not within the required

accuracy, then the following recursion equation can be used to

obtain a better value of 0.

The computation of successive values of ¢ and F is repeated till

the required accuracy is reached.

~t::

The 45° inclination of the tension field corresponds to fulltension field.

328.7 -22

6. TABLES AND FIGURES

Source, Test Q' 8 Web Compression Tension V V V VNumber Flange Flange ex u ex ex

V VUBb x t 2c x d 2ct x dt

uIT aye CJytyw c - c

in .. x in. ksi in. x in .. ksi" in .. x in .. ksi -kips kips

WB-l 2 .. 64 56 14.00x.25 43.3 10. DOxl. 55 33.0 lO .. OOxl.55 33.0 109. 109.5 1 .. 00 1.01

WB-2 2 .. 64 55 " 14 .. 00x.25 47.8 10 .DOxl. 56 33.0 lO.OOxl.56 33.0 128 120.8 1.06 1.07

Ref. WB-3 2.56 59 16.03x.27 49.6 lO.06xl.50 33.0 lO.06xl.50 33.0 139 142.7 .98 1.00

17WB-6 2.45 70 17.56x.25 33.1 lO.D2xl.51 33.0 lO.02xl.51 33.0 96 101.6 .95 .99

~WB-7 2.51 61 15 .34x .25 33.7 10 . 07x1. 50 33.0 10 .07xl. 50 33.0 95 98.6 .96 .95

WB-8 2 .. 46 60 15 .. 65x~26 29 .. 7 lO .. 07xl.51 33.0 10. D7xl. 51 33.0 100 97.9 1.02 .99

WB-9, 2.68 50 12.50x.25 30.3 -10 . 04xl. 50 33.0 10.04xl.50 33.0 92 91.7 1.00 1.00

WB-IO 2.68 50 12.50x.25 30.3 lO.Olxl.51 33.0 10 .Dlxl. 51 33.0 94 92.9 1.01 1.00

Note: V = ultimate shear evaluated by Proposed Approachu -VUB = ultimate shear evaluated by Baslerfs Approach

Table Ie Shear Tests in Strain-Hardening Range

lJ-JrvOJ.-..J

If'0v.J

I

WebCompression Tension

Flange Flange V V VTest Q' S V V ex ex ex

N.umber b x t 2c x d 2c x d ex u V- VuE VuFSource CTyw c c (}yc c c aye u

K K Kmm x mm --.L mm x mm --.L mm x mm -L Tons Tons

2 2 2mm mm mm

Gl 2 . .61 55 440x8.0 44 .. 0 160x30 42.0 160x30 42".0 82 94~1 .87 .85 1.06

G2 2.61 55 440x8.0 44.1 200x30 42.0 200x30 42.0 84 95.4 .88 .87 1.08

G3 2.63 70 560x8.0 44.0 160x30 42.0 160x30 42.0 99 112.2 .88 .97 1.01

Ref. 9 G5 2.68 70 560x8.0 44 .. 0 250x30 42.0 250x30 42.0 107 114.3 .94 1 .. 05 1.09

G6 1.25 70 560x8.0 ·44.0 250x30 42.0 250x30 42.0 120 124.9 .96 1.06 1.22

G7 2.68 70 560x8.0 44.0 250x30 42.0 250x30 42.0 107 114.3 .94 1.07 1.09

G9 2.78 90 720x8.0 44.0 250x30 42.0 250x30 42.0 118 130.5 .90 1.20 .95

Gl 2.67 60 543x9.1 38.0 301x22.4 44.0 301x22.4 44.0 110.5 112.1 ~98 1.04 .94

Ref. '18 G2 2.67 60 543x9.1 38.0 220x22 .. 4 44.0 220x22.4 44.0 104 111.5 .93 .98 1.11

G3 2.63 77 722x9.4 38.0 302x22.2 44.0 302x22.2 44.0 124.5 144.7 .86 1.00 .98

v = Ultimate shear evaluated by Proposed ApproachU

VUE = Ultimate shear evaluated by Basler's Approach

VUF

= Ultimate shear evaluated by Fujii's Approach

Table 2 Shear Test Results in the Inelastic Range of Shear Buckling

LN1"0co

'-J

Irv~

Web Compression TensionV V V

Source Test 8Flange Flange

V V ex ex exQ' V- VuE VuFNumber

ex ub x t 2C x d 2C

tx d

tu

csyw c C (Jyc rJyt

in. x in. ksi in. x in. ksi in. x in. ksi kips kips

G6-Tl 1.5 259 50.0x.193 36.7 12.13x.778 37.9 12.13x.778 37.9 116 '121.3 .96 1.04 1.08

G6-T2 .75 259 50.0x.193 36.7 12.13x.778 37.9 12;13x.778 37.9 150 159.5 .94 .95 .97

Ref. 3 G6-T3 .50 25S 50.0x.193 36.7 12.13x.778 37.9 12 .13-x .77E 37.9 177 190 .6 .93 .98 1.00

G7-Tl 1.0 25: 50.0x.196 36.7 12.19x.769 37.6 12.19x.76S 37.6 140 145.3 .96 .98 1.05

G7-T2 1.0 255 50.0x.196 36.7 12.19x.769 37.6 12.19x.76~ 37.6 145 145.3 1.00 1.02 1.09I

HI-Tl 3.0 127 SO.Ox.393 108.117.03x.982

lD2.017.03x.98L

lO2.0 630 . 630 1.00 1.33 .96Ref. 7 18.06x.977 18.06x.983

HI-T2 1.5 12/ 50.0x.393 108.1 .18 .06x. 977 l02D 18 .06x. 983 102.0 769 793 .97 1.08 .92

UGl.l .80 30C 36.0x.120 44.4 8.0 x.625 34.28.0 x.52:

34.2 88.8 86.9 1.02 1.1210.5 x.75C

Ref.' 8 i]G2.1 1.20 29: 36.0x.122 43.2 8.0 x.625 36.7 8.0 x.52: 36.7 76 72.6 1.04 1.1710.5 x.75C

TJG3.1 1.60 29: 36.0x.122 43.5 8.0 x.625 33.~8.0 x.52: 33 .. 3 65.5 62 .. 3 1.05 1.19

10.5 x.75C

JG4.1 1.77 41Ll 48.0x.116 56.1 10.0 x.750 34.1 13 .Oxl. 38LJ 34.1 81.6 82 .. 5 .99 1.02Ref .22

UG4.6 1.77 263 48.0x.183 35-.5 13.0x1.384 34.1 10.0x.750 34.. 1 98.8 102 1.00 1.00

Note: V = Ultimate shear evaluated by Proposed Approachu

VUB = Ultimate shear evaluated by BaslerTs Approach

VuF = Ultimate shear evaluated by Fujii's Approach

Table 3a Shear Test Results in Elastic Shear Buckling Range

_"'~=, __"""m ,;=, ','>" .._._.=_._=.. =...0,·.·=......='==~ ......... _

eNl'VOJ

.......,J

I1'0lfl

Web Compression TensionFlange Flange V V V

Source Test S V Vex ex ex

Ci V VuB

VuFNumber b x t 2C x d 2C

tx d

tex u

Ciyw rT crytU

C C yc

K K Kmm x mm g mm x mm -3. mm x mm ~ Tons Tons-2 2 2

mm mm mm

B 1.0 26; 1200x4.5 50.0 240x12 50.0 240x12 50.0 76 . 96 .79 .81 1.02

Gl-l 3.0 18~ 1200x6.6 49.6 250x23 51.0 250x23 51.0 99 99.8 .99 1.21 1.07

Re"f .18 Gl-2 1.50 18~ 1200x6.6 49.6250x23 51.0 250x23 51.0

129 142.7 .91 1.03 1.04250x13 46.0 250x13 46.0

,G2-1 3.0 14L 950x6.6 49.6 250x19 53.0 250x19 53.0 98 96.4 1.02 1.34 .96

G2~2 1.5 14L 950x6.6 49.6 250x19 53.0 250x19 53.0125 130.2 .96 1.17 1.00

250x13. "-

46.0 250x13 46.0

LNtvCD.........j

Note: 'V = Ultimate Shear evaluatedu

VUB = Ultimate Shear evaluated

VuF

= Ultimate Shear evaluated

by Proposed Approach

by Basler's Approach

by FujiiTs Approach

11'0())

Table 3b Shear Test Results in Elastic Shear Buckling Range

Web Compo Flange Tens. Flange

Girder tEnd Port. Center End Port. Center Vex

Source Number f3 b x t 2c x d 2c x d 2ct

x dt

2ct

x dt

V V Va fJyw (J O"ytc c c c ~yc ex u u

in x in ksi in x in in x in ksi in x in in x in ksi kips kips

C-AC2 . 5115 143 17.88x.12 30.6 3.67x.38 5.52x.38 109.3 3.69x.38 5.52x.38 109.3 26.7 26.2 1.02

C-AC3 5.5 71 17.93x.25 36.5 5.51x.51 8.49x.51 108.0 5.52x.51 8.49x.51 108.0 89 .. 2 95.2 0.94

Ref. 5 C-AC4 5.5 102 17.93x.17 33.6 5.27x.64 7.92x.64 113.2 5.27x.64 7.92x.64 113.2 55 53.8 , 1.02

C-AC5 5.5 103 17x96x~17 33.6 5.18x.75 7.79x.75 113.6 5.18x.75 7.79x,,75 113.6 52.4 54.4 0.96

C-AHI 5.5 69 17.96xp26 48.4 5.57xl.O 8.33xl.O 105.9 5.57x1.0 8. 34xl. O' 105.9 130 130 1.00

k'g/m~ J<g/mm2 2 Tonsrom.x mm mm x mm mm x mm mm x mm mm x rom kg/mm Tons

Ref. 9 G4 3.7 7 560x80 44.0 250x30 250x30 42.0 250x30 250x30 42.0 97 104.7 0 .. 93

Table 4 Test Results of Panels with a > 3.0

CJJrvOJ.-...,J

ff\..)

-.....J

328.7

Fig. 1 Tension Field Model Proposed by Takeuchi

-28

328.7 -29

(0) Beam Action Shear Stress

(b) Tension Field Stress

2 .States of Stress in Plate Girder Web

328.7 -30

30

25

20

k v• •

15

~~tI....-

10,

.. 5• •

J.o 2.0

Fig. 3 Shear Buckling Coefficient vs. A5pect Ratio

WNco-J

3.02.52.0- 1.51.00.5

\~:ai T:\ ~... l- Tcr =1+4.3 ( 0.58-A )1.56 .

:;;{' \ y

~ \\~ ~\ ~. -\~" ~.

\<p ~

~ \<po \% <;c~ ~.:L~ ~- -~.

(> \6) ~~......--...; .::::._ .:- _ 6l.1"4':-:- \.... .-P

~ --........ qll" \ T:OhnSons -... ......... p C'l.Ir.~\ cr=I-O.615(A-0.58)1.I8

Colli", '" "'6 T YI 'f1 Cllr.~ \.I vi ~ ~'\.

I I---1·----1 --I I I

Strain ! .I . ~ ElasticHardening i,ne,ast'j BUCkltngiBuckling

0.58

o

1.0

1.5

0.5

TcrT y

A.= Ty =~

Tcri

rEy ) 12( l-v2 )

\./3 .".2 k"I

Wf-I

Fig. 4 Proposed Shear Buckling Curve

328.7 -32

(a) Probable Tension Field (b) Equivalent Tension Field

Fig. 5

I

Of

Proposed Tension Field Model

cr."t

(0) Probable Tension FieldStress Distribution

(b) Equivalent Tension FieldStress Distribu1ion

Fi<,·. 6 Tension Field stress Distribution

328.7

q

(a)

q

(e)

-33

(b) (d)

Perfectly Flexible Flange

Fig. 7 Tension Field Stress Distribution Influenced byFlange Rigidity

q

Perfectly Rigid~. )~ .........Flange ~

//

// (2)

O-------~........-_=:lllaIIIliIIII---------VVcr

BeamAction Tension Field Action

Fig. 8 Schematic Curve of Load vs. Vertical Com~onent ofTension Field StresS Along Web-Flange Junction

328.7 -34

mptmpt

Afe J~

~Vf~vt

b

--------,-tvt

~t

I~Pbmpb Aft

a .~(0) Fixed End8~am Behavior (b) Cross Sections of

Effective Flanges

Fig. 9 Frame Action Model

(;.)

NCD.-.J

1.0

3.02.5

Flange Strength

2.0

Tension Field Strength

1.51.00.5

G6-T3\,,G6-T2 -- ~G7-T2

G7::r, t:"......... -

\ G6-TI----

\\

""o

0.5vVp

a IW(J1

Fig. 10 Shear Strength vs. Aspect Ratio - Symmetrical Plate Girders

wI\..)

CD

-J

\

""'---Tension Field Strength

3.0

--

2.5

Strength

Flange Strength

---

2.01.5

-........

1.00.5o

1.0

0.5vVp

aI

Wm

Fig. 11 Shear Strength vs. Aspect Ratio - Unsymmetrical Plate Girders

Iw.....,J

Wtvm.-J

a=2.03.0

1.00.9

{& Theoretical - Proposed

a Experimental- Ref. 17

0.80.7

:'1=1-0.615(A.-O.8)1.38

0.6

T Y =f3 A- Ey )_12_(_1-_112......>

Tcr V~.J3 1f2 kv

IIIIII

0.58

A=

TT'Y =I + 4.36 ( 0.58-A)'-58

0.50.4

I. I

1.2

1.3

1.0

0.9

0.8' , ' , , ' , ,,,..., ,0.3

T

. 'y

Figw 12 Comparison of Theorv With Tests in Strain-Hardening Range_1

T

Ty

0.9

0.8

0.7

0.6

r·

IIIIIIIIIIIIIII

0.58

{e Theoretical - ProposedoExperimenfal- Ref. 9

{ATheoretical - ProposedAExperim.ental- Ref. 18

TT =1-O.615(X-O.58t18

y

a=I.O

2.0

3.0

(;.)

tv00

.....,J

ZE y )12 (l-v 2 )

\./3 .".2 kv

0.5 0.6 007

X=

0.8 0.9 1.0 1.1 1.2 IW00

Fig~ 13 Comparison of Theory With Tests in Inelastic Buckling Range

T-"'~rT lLl ,~ RC"''; t':" '2""" .-, -t: Tr-: ~~ 'V">"\'" r,T ; + 'h, r-r ;::? <"':':' +- D ~ .-- ~ ~ 1 --I- ~ i ~ F ~._ -3 ~ t: i= E' ~ 2J'= ~.. ! "':"' ~-; D ? ~ 2'::::

1.'0r---~ wtvro

I ~.-J

I ! -{ A Theoretical - Proposed

0.91- I C-AC3

I6. Experimental - Ref. 5

I0.8~

[I .

T IT yI

0.71- I 2.. =I-O.615( A-a.S8)us

ITy

I0.61- I

II

0.58I I I

0.5 0.6 0.7 0.8 0.9 I ~O 1.1 1.2I

Ey )12~I-V2)+0

A= j -L"-=B jf-e/3 1T k"

Figo 15 Comparison of Theory With Tests for Unstiffened Hybrid Beams

328.7 -41

7. NOMENCLATURE

1. Lower Case Letters

a panel width or distance between transversestiffeners

b panel depth or distance between flanges

k plate buckling coefficient under pure shearv

mpt ' mpb plastic moments of top and bottom boundarymembers

t web thickness

2. Capital Letters

A area of the cross section

Afc

area of Lhe compression flange

Aft area of [he tension flange

A web areaw

E Young modulus, 29600 ksi

V experimental ultimate shearexV beam action shear

T

V tension field action shearcr

Vf frame action shear

V theoretical ultimate shear strength under pure shearu

3. Greek Letters

panel aspect ratio

B web slenderness ratio

poisson's ratio, 0.3

328.7

p

T

(=\.j v

-42

coefficient of equivelent tension field stress

beam action shear stress; with subscript "cr" ,shear buckling stress; with "y", web shear yield stress

yield. stress fa web

maximum tension field stress

inclination of tension field; with subscript Hott,the optimum inclination of tension field

coefficient of the eff,ective web depth for aplate under pure shear

parameter used in shear buckling curve

328.7

8. REFERENCES

1. American Institute of Steel Construction, Inc.SPECIFICATION FOR THE DESIGN, FABRICATION ANDERECTION OF STRUCTURAL STEEL FOR BUILDING, AI~C,

New York, 1963.

-43

2. Basler, K.STRENGTH OF PLATE GIRDERS, Ph.D Dissertation, LehighUniversity, Univers~ty Microfilms, Ann Arbor,Michigan, 1959.

3. Basler, K., Yen, B. T., Mueller, J. A. and Thur1iman, B$WEB BUCKLING TESTS ON WELDED PLATE GIRDERS, Bulletin NOe63, Welding Research Council, Sept., 1963.

4. Basler, K.STRENGTH OF PLATE GIRDERS IN SHEAR, Trans. ASeE,Vol. 128, Part II, 1963, p. 683

5. Carskaddan, P. S.SHEAR BUCKLING OF UNSTIFFENED HYBRID BEAMS, J.ASCE,Vol. 94, No. 8T8, Aug., 1968

6. Chern, C. and Ostapenko, A.ULTIMATE STRENGTH OF PLATE GIRDERS UNDER SHEAR ANDBENDING, Fritz Engineering Laboratory Report No.328.9, Sept., 1969.

7. Cooper, P. B., Lew, H S and Yen, B. T.WELDED CONSTRUCTIONAL ALLOY STEEL PLATE GIRDERS, J.ASeE, No. STl, February, 1964.

8. Dimitri, J. R. and Ostapenko, A.PILOT TESTS ON THE ULTIMATE STATIC STRENGTH OF

'UNSYMMETRICAL PLATE GIRDERS, fritz Engineering­Laboratory Report No. 328.5, June 1968.

9. Fujii, T.MINIMUM WEIGHT DESIGN OF STRUCTURES BASED ON BUCKLINGSTRENGTH AND PLASTIC COLLAPSE, Institute of Shipbuilding,No. 122, Japan, Nov., 1967

10. Fujii, T. and Akita, Y.ON ULTI}ffiTE STRENGTH OF PLATE GIRDERS, Jap~n Ship­building and Marine Engineering, May 1968

11. Fujii, T.ON AN IMPROVED THEORY FOR DR. BASLERTS THEORY, 8thCongress of the International 'Association for Bridgeand Structural Engineering,. Theme IIc, New York,Sept., 1968, p. 479.

328.7

12.

13.

14.

15.

16.

17 It

18.

19.

20.

21.

22.

-44

Gaylord, E. H.Discussion of STRENGTH OF PLATE GIRDERS IN SHEAR,Prac. ASCE, Vol. 88 (8T2), April, 1962.

Kollbrunner, C. F. and Meister, M.AUSBEULEN, Springer-Verlag, Berlin, 1958

Konishi, I. et alTHEORIES AND EXPERIMENTS ON THE LOAD CARRYINGCAPACITY OF PLATE GIRDERS, Report of Western JapanResearch Society for Bridges, Steel Frames andWelding, July, 1965. (in Ja panes e)

Kuhn, P.STRESSES IN AIRCRAFT AND SHELL STRUCTURES, New York,McGraw-Hill, 1956.

Lew, H. S. and Toprac, A. A.RESEARCH ON THE STATIC STRENGTH OF HYBRID PLATEGIRDERS, Structures and Fatigue Research Laboratory,Tech. Rept. p. 550-11, University of Texas, Austin,Jan~ary, 1968

Lyse, I. and Godfrey, H. J.INVESTIGATION OF WEB BUCKLING IN STEEL BEAMS, Trans.ASeE, Vol. 100, 1935, p. 675

Nishino, F. and Okumura, T.EXPERIMENTAL INVESTIGATION OF STRENGTH OF PLATEGIRDERS IN SHEAR, 8th Congress of the InternationalAssociation for Bridges and Structural Engineering,Theme IIc, New York, Sept., 1968, p. 451.

Ostapenko, A. and Dimitri, J. R.BUCKLING OF PLATE GIRDER WEBS, Fritz EngineeringLaboratory Report No.328.3 (in preparation)

Ostapenko, A.LOCAL BUCKLING, Chapter 17 in STRUCTURAL STEELDESIGN, Ronald Press, New York, 1964.

Rockey, K. C. and Skaloud, M.INFLUENCE OF FLANGE STIFFNESS UPON THE LOAD CARRYINGCAPACITY OF WEBS IN SHEAR, 8th Congress, lABBE,Theme IIc, New York, Sept., 1968, p. 429

Schueller, Wlt and Ostapenko, A.. STATIC TESTS ON UNSYMMETRICAL PLATE. GIRDERS MAIN

TEST SERIES, Fritz Engineering Laboratory ReportNo. 328.6, September, 1968

328.7 -45

9. ACKNOWLEDGEMENTS

The work described here covers part of the research project

on unsymmetrical plate girders carried out at Fritz Engineering

Laboratory, Civil Engineering Department, Lehigh University,

Bethlehem, Pennsylvania. Dr. David A. VanHorn is Chairman of

the Department and Dr. Lynn S. Beedle is Director of the

Laboratory.

Thanks are due to Robert P. Kerfoot who read the manuscript

and offered many helpful suggestions.

The sponsors of this research project are the American Iron

and Steel Institute, the Pennsylvania Department of Highways,

Bureau of Public Roads (Department of Transportation) and the

Welding Research Council. Their interest in and support of this

project are gratefully acknowledged.

This research has been conducted under the general guidance

of the Welding Research Council Subcommittee for Welded Plate

Girders (M. Deutermann, Chairman) and a close technical monitoring

was exercised by the Task Group of this Subcommittee appointed

specifically for this project. (C. Z. Zwissler and L. H. Daniels,

)(onsecutive Chairmen). The authors gratefully ~cknowledge th,e(j

guidance of these two g~oups.

328.7

Members of the Subcommittee are:

-46

M. Deuterman, Chairman

J. H. Adams

A. Amirikian

L. S. Beedle

J. L. Durkee

L. H. Daniels(for C A, Zwissler)

E·. R. Estes, Jr.

G. F. Fox

J. A. Gilligan

T. R. Higgins

B. G. Johnston

H. G. Juhl

M. L. Koehler

K. H. Koopman

W. B. McLean

W. H. Munse

E·. G. Paulet

E·, Pisetzner

F. C. Sankey

C. F. S,cheffey

J. Vasta

I. M. Viest

C. ·F. Larson, Secretary

Bureau of Public Roads

Pittsburgh-Des Moines Steel. Co.

Bureau of Yards and Docks, U. S. Navy

Lehigh University

Bethlehem Steel Corporation

Kaiser Steel Corporation

American Iron and Steel Institute

Howard, Needles, Tammen & Bergendoff

United States Steel Corporation

American Institute of Steel Construction

University of Michigan

Pennsylvania Department of Highways

The Pennsylvania Railroad Company

Welding Research Council

Dravo Corporation

University of Illinois.

Bureau of Public Roads

Weiskopf a~d Pickworth

Pennsylvania Department of Highways

BUFeau of Public Roads

Litton Industries-AMTD

Bethlehem Steel Corporation

Welding Research Council

328.7

Members of the Task Group are:

-47

L. S. Daniels(for C. A. Zwissler)

G. F. Fox

T. R. Higgins

B. G. Johnston

H. G. Juhl

K. H. Koopman

F. C. Sankey

C. F. Scheffey