torsten

ELSEVIER

Thin-Walled Structures Vol. 29, Nos. 1-4, pp. 13 30, 1997 1998 Elsevier Science Ltd. All rights reserved

Printed in Great Britain 0263-8231/97 $17.00 + .00

P I I : S0263-8231 (97)0001 2 -8

Shear Buckling Resistance of Steel and Aluminium Plate Girders

Torsten H6glund

Department of Structural Engineering, The Royal Institute of Technology, Stockholm, Sweden

ABSTRACT

During the development of Eurocode 9 for aluminium alloy structures a number of design methods for the shear buckling resistance of plate girders were appraised, by comparison with experimental data. Among studied methods the so-called rotated stress fieM method [H6"glund, T., Design of thin plate I girders in shear and bending with special reference to web buckling. Royal Institute of Technology, Department of Building Statics & Structural Engineering, Stockholm, 1972], with some modifications, was found to give the best agreement with 366 tests on steel plate girders as well as 93 tests on aluminium alloy plate girders in shear. The method is simple to use and is applicable to unstiffened, transversally and longitudinally stiffened and trapezoidally corrugated webs. This paper presents the rotated stress field method and summarizes the result of the comparisons, including the design methods in Eurocode 3, Part 1.1, version April 1992, for steel plate girders. The rotated stress field method is also adopted in Eurocode 3, Part 1.5: plated structures, draft July 1996. 1998 Elsevier Science Ltd. All rights reserved

INTRODUCTION

The rotated stress field method is based on H6glund.l Originally it was developed for girders with web stiffeners at the supports only, a structure for which other tension field methods are very conservative.

In the design proposal I the allowable shear force was reduced for large slenderness ratios hw/tw because of limited number of tests, at that time only two for hw/tw > 210 for girders with no intermediate stiffeners. Since then a number of tests have been made, showing that this reduction is not needed.

13

14 T. HSglund

In this paper is presented:

the rotated stress field method; design formulae for girders with stiffeners at supports only; influence of transversal and longitudinal web stiffeners; influence of rigid flanges; influence of bending moment; comparison with other tension field methods and comparisons with tests.

Tests on aluminium alloy plate girders were found in Refs 2-10 and tests on steel plate girders in Refs 11-44.

WEB WITH STIFFENERS AT SUPPORTS ONLY

For webs in shear there is a substantial post-buckling strength provided that, after buckling, tension membrane stresses, anchored in surrounding flanges and transverse stiffeners, can develop. In a pure state of shear the absolute value of the principal membrane stresses al and a2 are the same as long as no buckling has occurred (z < zcr). After reaching the buckling load (Vcr = z~rhwtw) the web plate will buckle and redistribution of stresses starts. Increased load results in increased tensile stress al but only slightly, or not at all, increased compressive stress a2.

For a very thin web, after buckling, a2 is much less than trl and can be neglected. If the flanges are prevented from coming nearer each other (Fig. 1) then

z = trl sin ~b cos q5 = 0.5al sin 2tk. (1)

The direction of the tensile stresses is chosen to give z=maximum. Putting al equal to the yield strength of the web, fyw, then the result is

% ~l - - t * - t~- t~t - - t

V V

(a) Cry (b) Fig. 1. State of stress in a very thin web with transversely restrained edges (ideal tension

field).

o - 2 ~ o 1

Shear buckling resistance of steel and aluminium plate girders 15

,u_O.Sfyw_V~ - - fo r q~ = 450 (2) f. fv 2

where

_ fyw fv -- x/~" (3)

This theory, often called ideal tension f ie ld theory, is valid only if the flanges are prevented from moving towards each other by an external structure, for instance, an inner panel in a plate with rigid cross beams and stringers.

In a long beam, with transverse stiffeners at the ends only, nothing but the web prevents the flanges moving towards each other, this is why the membrane stresses in the transverse direction are zero. Equilibrium for a triangle according to Fig. 2(g) gives

17 0.1 -- (4)

tan ~b

02 = -~ tan q$ (5)

where ~b constitutes the direction of the principal stress. This state of stress has a stress component 0"h in the longitudinal direc-

tion

= z (ta~ ~ b ) 0"h tan q~ -~- 0"1 q'- 0"2" (6)

The total longitudinal force in the web is less than

/ ; ' , ' , \~ )/"d , X + h . _ _ . _ i "v i (a) (b~j / ~ (C) (d) (h)

Z" ,.- -,-- (

tl l i Gh t l O- h _

(e) shear (f) shear and (g) principal stresses only membrane stresses stresses

Fig. 2. State of stress in the web of a beam with transverse stiffeners at the ends only (rotated stress field).

16 T. H6glund

Nh = 0.hh+tw (7)

because close to the flanges there is more or less a pure shear state of stress.

This force has to be anchored at the ends of the beam by a transverse short beam called rigid endpost, in order to fully develop the rotated stress field (see Fig. 2(a)). This end post is supported by the flanges, which results in compressive forces in the flanges at the ends of the beam.

The ultimate shear strength of the beam can be derived using the yon Mises yield criterion

4 - + = (8)

and assuming that the compressive stress remains equal to the shear buckling stress after buckling, but acting in a smaller angle than 45

0"2 : -- '~cr" (9)

Furthermore, the slenderness parameter 2w is introduced

,F 2w = g~cr where ~cr = k+ 12(1 - v 2) (10) From eqns (3)(5), (8)(10) the ultimate strength % =~ can be derived as a function of 2w

f~zu--4~V~ ~ F 1 2 w 424 2X/32~1 for 2w _> 1.00. (11)

The large square-root in (11) is close to 1.00 if 2w>_2.5. Then

z. 1.32 -- ~ for 2w > 2.5 (12) f+ 2w

The inclination of the tension stress 0.1 defined by the angle ~b is decreased when the ratio "Cu/Zc+ is increased. That is why the theory is called the rotated stress field theory.

In the diagram in Fig. 3 z,,/f~ is given as a function of the slenderness parameter kw. The shear stress corresponding to the buckling load, 1/2w 2, is also given in the diagram, as well as a line corresponding to the ideal tension field and some test results on plate girders with stiffeners at the supports only.

The solid circles are tests on beams with rigid end posts. These tests agree very well with the curve for the rotated stress field theory.

The unfilled circles correspond to beams with only one stiffener at the girder end, non-rigid end post. Such end posts have only limited ability to serve as anchors for the longitudinal membrane stresses, and hence the

Shear buckling resistance of steel and aluminium plate girders

1

0,8 Vu

o,6

0,4

0,2

~ o /.~S'id.._ealten_.. siOn field' L

~; stress field"

o critical 1 ~,~ o stress "~w" ~

1 2 3 4 5 slenderness parameter Aw

rigid end post

vl ' 0 non-rigid end post

Fig. 3. Shear force resistance according to tension field theories and tests.

17

ultimate load is less than for beams with rigid end posts. But there is still a substantial post buckling strength.

In drafts of Eurocode 3, Part 1.5 (plated structures) and Eurocode 9 (aluminium structures) design formulas for the reduction factor

z,, (13) pv --L7

are given. The value of Pv (denoted Xv in Eurocode 3, Part 1.5) is somewhat reduced compared to the rotated stress field theory to allow for scatter in test results as a result of initial imperfections and plastic buckling.

The shear force capacity is

Vw = p~fywhwtw. (14)

The shear buckling reduction factor Pv is given in Table 1 and in the diagram in Fig. 4.

For small slenderness ratios, 2w < 0.48/7, strain hardening in shear can take place, giving larger strength than corresponding to initial yielding, r/= 1/v"3~0.58. Then

TABLE 1 Reduction Factor Pv for Shear Buckling

2., Rigid end post Non-rigid end post Steel Aluminium

2w < 0.48/~/ r! r 1 ~l 0-48/r/_

18 T. H6glund

Pv

r/ 0, / \ 0,6

Rigid end post 0,5 - - - a lumin ium . . . . . . . .

o,3 ~ ~ ' ~ "--

0,2 Non rigid end post I ~,~

0,1 . . . . l

0 0 1.0 2.0 ~ 3.0

F ig . 4. Reduct ion factor Pv for shear buckling.

~7 = 0.70 for $235, $275 and $355 z/= 0.60 for $420 and $460 ~7 = 0.4 + 0.2(f,,w)/(fyw) for aluminium wherefuw is the ultimate strength of the web material.

TRANSVERSELY STIFFENED WEB

Transverse stiffeners welded to the web have two main effects on the behaviour and strength of a girder in shear: first, they prevent the web from out- of-plane deflections, thus increasing the elastic buckling strength, and second, they prevent the flanges from coming closer to each other.

If the bending rigidity of the flanges about an axis in the centre of the flange is large, then the prevention of the flanges moving towards each other can be large (see later in this paper).

The shear buckling coefficient is, if simple supports are assumed all around the web panel of length a and width hw,

k~ = 5.34 + 4(hw/a) 2 when a/h~ >_ 1.00 (15a)

k~ = 4.00 + 5.34(hw/a) 2 when a/hw < 1.00. (15b)

LONGITUDINALLY STIFFENED WEB

Longitudinal stiffeners also protect the web from out-of-plane deflections. If they are rigid enough they divide the web panel between transverse


stiffeners into sub-panels which buckle individually. If not, web buckling includes lateral deflection of the longitudinal stiffeners.

In design the method for unstiffened webs is used but the buckling coefficient is increased.

In Crate and Lo 45 the buckling coefficient ks is given as a function of the stiffness parameter Yst = EI/hwD for one centric longitudinal stiffener in a long web panel. The theoretical curve can be approximated by

k, = 5.34 + 1.36 (16)

(see Fig. 5). This value of ks cannot directly be used in design because the post-buckling strength is much less in a stiffened web than in an unstiffened one. Therefore, the stiffness I of the stiffener is reduced with a factor of I/3. Inserting the reduced value I = Ist/3 and D = Et3/(12(1 - V2)) into (16) and using v=0.3 gives

k, = 5.34 + 2.1 V h- z w. (17)

In a web with closely spaced transverse stiffeners and a longitudinal stiffener with relative stiffness Vst, ks can be approximated with

k~ = 5.34-t 4 3.45),y 4 - - -+ (18) (a/hw) 2 (a/hw) 2 "

Comparisons between formula (18) and values according to K16ppel and Scheer 46 show good agreement (see Table 2).

Again, after reducing the stiffness of the longitudinal stiffener with a factor of 1/3 and after rearranging we have

40

kr 3O

20

10

I ~ 1 1 P

E/ ~l~hw I I 1/111 - - ~ ~ - - kr = 21. 4

//I ~! I1[I \ l" Crate & L (1948) I II

Eq (16) / / I I I I , , : . . . . . . ]]1 1 10 100 1000

F !/

10000 ~t

Fig. 5. Buckling coefficient k~ for a plate with one centric longitudinal stiffener.

20 T. HSglund

TABLE 2 Buckling Coefficient ks

~St 50 100 200 500 600

K16ppel and Scheer 46 75 120 190 375 430 eqn (18) 75 119 193 375 428

h 2 ( i s t~3/4 k~= 5.34+4(k~) +9.1 (~) 2 (19)

v - wj The largest ks according to eqn (17) and eqn (19) should be used, but Zcr should not be larger than that of the sub-panel giving the smallest value of ~C cr.

It is assumed that the increase in ~cr associated with a low value of the aspect ratio, a/hw, causes an increase in the load-carrying capacity, Vw, which corresponds to the increase in Zcr. Therefore, ks given by eqns (17) and (19) is substituted in eqn (10).

INFLUENCE OF RIGID FLANGES

As mentioned, transverse stiffeners prevent the web from deflecting and prevent the flanges from coming nearer to each other at the stiffeners.

In the stage of failure, four hinges, denoted E, H, G, and K, form in the top and bottom flanges (see Fig. 6(b)). A tension stress field, EHGK, develops in the web. This tension stress field differs from the tension field described, for instance, by Rockey and Skaloud 16, in that it is assumed to be developed between the flanges only. The other parts of the tension field dealt with by Rockey and Skaloud 16 may be imagined to be comprised in

a

I

(a) Vw I (b) Vf' K G

-'-n i ~ bf ~

(c)

Fig. 6. Model of web in the post-buckling range. (a) Shear force carried by the web by rotated stress field. (b) Shear force carried by truss action, the development of which is

dependent on the rigidity of the flanges. (c) Notations for the cross section.


the rotated stress model shown in Fig. 6(a). The shear force, Vf, which is transmitted by the tension stress field is obtained from the equation of equilibrium of the flange portion c. This equation gives

VU- 4Zffyf (20) c

The tensile stresses in the tension stress field produce a stiffening effect on the web (this effect is favourable to the load-carrying capacity) at the same time as the effective stress increases in the web (this effect is mostly unfavourable).

However, it is assumed that the shear resistance of the web, Vw, is not changed by the formation of the tension field between flanges. Then the shear resistance of the girder, Vu, is the sum of the shear resistance of the web, Vw, and the shear resistance contributed by the flanges, VT,

Vu = Vw + Vf. (21)

As has been shown by the test series carried out by Rockey and Skaloud 47 and by Skaloud 17 the distance, c, between the plastic hinges in the flanges varies from 0.16 to 0.55 times the spacing, a, of the stiffeners, depending on the rigidity of the flanges. The distance c is estimated by means of eqn (22) for steel plate girders

1.6bft~fyf~ c = 0.25 -+ twh~fy------7 )a. (22)

/

For aluminium plate girders a somewhat different formula gives better correlation with the test results. The coefficients 0.25 and 1.6 in eqn (22) are replaced by 0.08 and 4.4 for aluminium.

EFFECT OF BENDING MOMENT

If the girder is subjected to a shear force and at the same time to a bending moment of small magnitude (M < My, where Mf = AFlfy f is the moment capacity of the flanges, dis the distance between centroides of flanges), then it is assumed that the stresses in the web which are caused by the bending moment do not influence that portion, Vw, of the shear force which is resisted by the web. On the other hand, these stresses influence that portion, Vf, of the shear force which is resisted by the tension field between the flanges.

Vrea = V,, + Vf 1 - when M < Mr. (23)

If M > Mr, then the flanges cannot contribute to the shear capacity of the girder, and the capacity of the web to carry shear forces is reduced. The

22 T. H6glund

V w + Vf A 23)

0 0

/(24)

C I I I ! ! I

ID ! M Mpt

Mof Fig. 7. Interaction diagram for a girder subjected to a shear force and a bending moment.

interaction formula published by Basler 48 is applied (see the curve portion B- D in the interaction diagram, Fig. 7)

M=My+(Mp-My) 1- ~ . (24)

OTHER DESIGN METHODS

Many other design methods have been proposed for a plate girder in shear (see e.g. the survey in Galambos49). Most of these theories start with the elastic buckling load Vc,. and add a load corresponding to different types of diagonal tension fields. Many of these theories give good results for beams with small web panel aspect ratios, but conservative results when the distance between the transverse stiffener is large, because the contribution from the tension field is small. To meet this disadvantage, the tension field method in Eurocode 3, Part 1.1 (version April 1992) is supplemented with a so-called simple post-critical method, which is actu- ally very similar to the rotated stress field method (Vw) presented here.

As is shown in the comparisons with the tests the rotated stress field method (including Vf) gives good agreement with tests. The method is also much simpler to use.

COMPARISONS WITH TESTS

Comparisons are made for aluminium girders =-~ as well as steel girders. 11--44


The tests cover

girders with stiffeners at supports only; 9' 22, 23, 32 girders with transverse intermediate stiffeners; 2-1' 13-2~ girders with longitudinal and transverse stiffeners; s' 6, l l, 12, 31, 35-42 girders with trapezoidally corrugated webs. 9' 24, 30,43, 44

The total number of tests is 366, 93 in aluminium and 273 in steel. For some of these types of girder the number of tests is few. For instance, there are only three tests on aluminium girders with corrugated webs 9 and eight tests on longitudinally stiffened webs of aluminium. 5' 6

Especially the few aluminium girders with trapezoidally corrugated webs make the design method for aluminium girders uncertain. The method, proposed in the very first draft of Eurocode 9 for aluminium, is therefore much more conservative than the method used in this comparison.

A few of all tests in the reports are omitted in the diagrams and in the statistical evaluation. These tests are those where, according to the authors, failure was initiated by none of the modes mentioned in the following. Some tests were omitted because of failure due to insufficient lateral bracing of the flange or failure in a weld. Some of the tests 9 which failed in an overall web buckling mode failed at a load substantially larger than the theoretical ultimate load. The reason is probably initial web deflections in a different pattern than the overall buckling pattern which prevented long buckles forming. 9 Finally, some of the tests by Granholm 22 are omitted because of unclear support conditions.

FAILURE MODES

The resistance of plate girder webs depends on many parameters, for instance depth-to-thickness ratio hw/tw and web stiffener arrangements. In the test reports failure modes illustrated in Fig. 8 are observed.

This appraisal covers test beams with stiffened and unstiffened webs which failed due to shear buckling (failure modes b, e, f, h, i, j, k and 1) and overall web buckling caused by transverse forces and curvature-induced transverse stresses in the web (failure mode c). Girders failed due to patch loading (failure mode d or g) are, among others, appraised by Lagerqvist. 34

THEORIES

The tests are compared with two sets of design formulae

Eurocode 3: Design of steel structures--Part 1.1. General rule and rules for buildings. ENV 1993-1-1:1992 April 1992.

24 T. H6glund

~, ~ ~, no stiffener

]~ (e(~) ] (g) (f) (j) (i) (h) ''~' (o)

"~(k) (m) (I) (P) (q)

Fig. 8. (a)-(n) Web buckling modes and (o)-(q) flange buckling modes.

Eurocode 9: Design of aluminium structures. Draft June 1996 with slight modifications. In the diagrams it is called 'H6glund'. The same method is also proposed in Eurocode 3, Part 1.5: Plated structures, draft July 1996.

Reference is made to these documents and to the above presentation of the 'H6glund' formulation.

COMPARISON WITH DRAFTS OF EUROCODE 9 AND EUROCODE 3, PART 1.5: PLATED STRUCTURES (HOGLUND)

In the lower diagram in Fig. 9 test results on girders with flat webs are summarized.

A special buckling mode called overall web buckling has been found in aluminium girders without intermediate stiffeners 9 (see Fig. 8(c)). This buckling mode, not reported for steel girders, is a transverse column-type buckling mode with no post-buckling strength. Theoretically, 33 it has been found that this buckling mode can occur in long girders without intermediate stiffeners loaded with distributed load or closely spaced concentrated loads.

For none of the tests is Vexp/VR less than 0.98 and the scatter is small, the coefficient of variation being 0.10.

In Fig. l0 test results on girders with trapezoidaUy corrugated webs are summarized. In Fig. 10(a) the value on the x-axis is a sort of measure of the overall slenderness of the web. Iz is the second moment of area for the corrugated web per unit width. VR is the shear force resistance provided that only local buckling of flat parts, width bc of the corrugations, was considered, using the buckling reduction factor Pv according to Fig. 4, non-rigid end post. If global buckling is the reason why Vexp is much less than VR, then tests with larger bw/Iz 1/3 should have less shear force capacity. But there is no


1.5

1.0

0.5

0 6

Q 0

. ;e ~d~. J * ~

~ t, ~oO.+ ~ . " ~, ~ o v

+~ ~"

~0 ~/x ~ - Q

%

o Rockey/Ev. T,A zx Hamoodi T,A o Burt 7019 T,A ~, Basler T,S

I Eurocode 3 I Cooper T,S I I Benson N,A " Carskaddan T,S Combined method Benson O,A Rockey/S I1 T,S

Benson Y,A m Skaloud T,S Number o f tests 187 * HOglu/Frey N,S Average va lue m 1.394 ,, Seah T,A * Okumura & T,S Coef f . o f var iat ion s 0.191 Brown 7019 T,A : Oral~olmFujii & at T,sT'S m - 2s 0 .863 + Evans/Lee T,A o Roekey/S I T,S

,~ Evans/Lee 7 T,A

I I f I i .... i J 1 2 3 4 5 6 7

Xw

1"5 / , ,, g *

I- "_W,-'. ; , *+ ,e , J . .,

26 T. H6glund

1.5

a)

1.0 V~p

local buckling not reduce

b)

II dOCualu buckling) eea)

c)

~ - - . . . . . .. . no

.:" 7." .,:.~ -" : " ' - am

00 I t I I I I I I t I I I ~ 2~

1.5

1.0 f

00t

I I I [ I 30O

!

s* " * |

I I I

1.5

10 - 0.5

oca l buck l ing)

0.5

I tw I

Number of tests 63 Average value m 1.1"/9 Coeff. of variation s 0.104 m - 2s 0.935

I I I 400

I I

hc .0 Fc

# Benson C ,A Le iva& al C,S

1 2 3 4 5 6 7 8 h w ( loca l buck l ing)

Fig. 10. Girders with trapezoidally corrugated webs; s,24-30,43,44 (a) not reduced, (b) and (c) reduced local buckling resistance.

tendency in that direction. The reduced strength is more likely to be local buckling including adjacent flat panels thus initiating global or zonal buckling. If the local buckling load is reduced with a factor of 0.72, then the result is much more coherent (see Fig. 10(b) and (c)). The slenderness parameter 2w for local buckling is in the transition slenderness region 0.57 < 2,:< 0.95 which means that there is no post-buckling strength for local buckling.

The influence of different parameters is good and the scatter is reason- ably small also for girders with trapezoidal webs. The coefficient of variation for Vexp/VR,red is O. 1 1.

Shear buckling resistance of steel and alurninium plate girders

TABLE 3 Statistical Results and yM*-values for Tests on Steel Plate Girders 5~

27

Stiffeners Reference No. of tests Std dev. Safety factor, 7M*

Non-rigid end post Stiffeners at support 23, 22, 35 5 0'063 0"88 only Transverse stiffeners J3.16 26 0.073 0.99 Rigid end post Stiffeners at support 23, 35 8 0-060 1.04 only Transverse stiffeners ll, 13, 16-21,35, 39-41 52 0-084 1.01 Longitudinal stiffeners l l , 12,31,35--42 42 67 0.030 1.08 Interaction between shear and bending moment Transverse stiffeners 18, 21 3 0-01 1' 10

COMPARISON WITH EUROCODE 3, PART 1.1

In the upper diagram of Fig. 9, the tests are compared with Eurocode 3, Part 1.1. The larger of the resistance according to the simple post-critical method and the tension field method is used, in the figure called 'combined method'. In the tension field method, iteration is used to find the optimum value of the inclination of the tension field. As longitudinally stiffened webs and trapezoidally corrugated webs are not covered in Eurocode 3, Part 1.1, tests on girders with such webs are omitted.

For a single test series the scatter is small, but the average values of V~xp/VR for the different series is different, resulting in larger total scatter, the coefficient of variation being 0.19.

The simple post-critical method gives in itself conservative results if aspect ratio web panel a/bw is small and the flanges are large. On the contrary the tension field method is conservative for girders with large a/bw.

Details of the appraisal is given in H6glund. 5 The tests on steel girders are statistically evaluated in a background document to Eurocode 3, 51 showing that a safety factor 7~t-- 1.1 can be used (see Table 3).

REFERENCES

l. H6glund, T., Design of thin plate I girders in shear and bending with special reference to web buckling. Royal Institute of Technology, Department of Building Statics & Structural Engineering, Stockholm, 1972.

2. Evans, H. R. and Hamoodi, M. J. The collapse of welded aluminium plate girders--an experimental study. Thin-Walled Structures, 5 (1987) 247-275.

28 T. H6glund

3. Evans, H. R. and Burt, C., Ultimate load determination for welded aluminium plate girders. In Aluminium Structures: Advances, Design and Construction (Edited by R. Narayanan), pp. 70-80. Elsevier Applied Science, London, 1990.

4. Rockey, K. C. and Evans, H. R., An experimental study of the ultimate load capacity of welded aluminium plate girders loaded in shear. Research Report, University of Wales, College of Cardiff, 1970.

5. Hamoodi, M. J., The behaviour of reinforced aluminium web plates in a shear loading. MSc thesis, University of Wales College of Cardiff, 1983.

6. Seah, M. H., The behaviour of welded aluminium alloy plate girders reinforced with carbon fibre reinforced plastic. MSc thesis, University of Wales, College of Cardiff, 1984.

7. Burt, C. A., The ultimate strength of aluminium plate girders. Ph.D thesis, University of Wales, College of Cardiff, 1987.

8. Brown, K. E. P., The post-buckling and collapse behaviour of aluminium girders. Ph.D thesis, University of Wales, College of Cardiff, 1990.

9. Benson, P. G., Shear buckling and overall web buckling of welded aluminium girders. Ph.D thesis, Royal Institute of Technology, Division of Steel Struc- tures, Stockholm, 1992.

10. Evans, H. R. and Lee, A. Y. N., An appraisal, by comparison with experimental data, of new design procedures for aluminium plate girders. Proceedings of the Institute of Civil Engineers, Structures & Buildings, February 1995.

11. D'Apice, M. A., Fielding, D. J. and Cooper, P. B., Static tests on longitudinally stiffened plate girders. Welding Research Council, New York, Bulletin No. 117, October 1966.

12. Komatsu, S., Ultimate strength of stiffened plate girders subjected to shear. In IABSE Colloquium, Vol. 11, pp. 49-65. IABSE, London, 1971.

13. Basler, K., Yen, B. T., Mueller, J. A. and Thiirlimann, B. Web buckling tests on welded plate girders. Welding Research Council, New York, Bulletin No. 64, September 1960.

14. Cooper, P. B., Lew, H. S. and Yen, B. T. Welded constructural alloy steel plate girders. ASCE Journal, 1 (1964).

15. Carskaddan, P. S., Shear buckling of unstiffened hybrid beams. ASCE Jour- nal, 8 (1968) 1965-1992.

16. Rockey, K. and Skaloud, M., Influence of the flexural rigidity of flanges upon the load-carrying capacity and failure mechanism in shear. Acta Tech- nica CSA V, 1969, 3.

17. Skaloud, M., Ultimate load and failure mechanism of webs in shear. In IABSE Colloquium, Vol. 11, pp. 115-130. IABSE, London, 1971.

18. Okumura, T. and Nishino, F., Failure tests of plate girders using large-sized models. Structural Engineering Laboratory Report, Department of Civil Engineering, University of Tokyo, 1966.

19. Okumura, T., Fujii, T., Fukumoto, Y., Nishino, F. and Okumura, T. Failure tests on plate girders. Structural Engineering Laboratory Report, Depart- ment of Civil Engineering, University of Tokyo, 1967.

20. Nishino, F. and Okumura, T., Failure tests of plate girders using large-sized models. IABSE Eight Congress, Final report, New York, 1968.

21. Fujii, T., Comparison between the theoretical shear strength of plate girders and the experimental results. Contribution to the prepared discussion. In IABSE Colloquium, Vol. 11, pp. 161-172. IABSE, London, 1971.


22. Granholm, C.-A., L/ittbalkar (Light girders). Teknisk Tidskrift, Stockholm (in Swedish), 1960-63.

23. Hrglund, T., Livets verkningss~it och b~irfrrm~ga hos tunnv~iggig I-balk. (Behaviour and strength of the web of thin plate I-girders.). Royal Institute of Technology, Department of Building Statics & Structural Engineering Report No. 93, Stockholm (in Swedish), 1971.

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