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