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Web crippling with bending and shear of thin-walled plate girders
J. Construct. Steel Research 22 0992) 87-97
Web Crippling with Bending and Shear of Thin-Walled Plate Girders Max A. M. Herzog Rohrerstrasse 3, 5000 Aarau, Switzerland (Received 16 December 1991; revised version received 4 February 1992; accepted 24 February 1992)
ABSTRACT Even the newest versions of different standards do not yet solve the title problem of this paper adequately for practical requirements. Based on the posterior analysis of 340 tests reported in the literature some empirical formulas are presented which considerably simplify the designer's routine work. They have been obtained by reoression analyses and allow all influences on the ultimate strenoth of thin-walled plate oirders subjected to concentrated loads to be considered explicitly.
NOTATION a
A AL br c Co
C,. CL d
f. f,
M Mf
Distance of transverse stiffeners Arithmetic mean Sectional area of longitudinal stiffener Flange width Distribution length of concentrated load Effective contact length of concentrated load Coefficient accounting for the influence of distribution length Coefficient accounting for the influence of longitudinal stiffeners Web depth Ultimate tensile strength Yield strength Bending moment Bending moment for fully plastic flanges
87 J. Construct. Steel Research 0143-974X/92/$05.00 © 1992 Elsevier Science Publishers Ltd, England. Printed in Malta
88
Max A. M. Herzog
P eb Pc P~ Q Q~ r R
1/R S tf tr tw V
Concentrated load Buckling load Crippling load Squash load Shear force . Ultimate shear capacity of a transversely stiffened plate girder Fillet radius Resistance factor (larger than unity) Calibration factor (smaller than unity) Standard deviation Flange thickness Depth of crane rail Web thickness Coefficient of variation Efficiency of stiffener
INTRODUCTION Experienced designers will agree that in general new standards render their task more difficult instead of simplifying it. In many cases, simpler solutions are known, with an accuracy only a few percent lower than that of more demanding scientific methods. There is no advantage to the owner if the expenditure for the more demanding analysis exceeds the savings realizable in steel tonnage. This point is sometimes overlooked by pure academics.
THE PROBLEM Thin-walled plate girders subjected to a concentrated l o a d - - a s in the case of crane girders or of bridge girders during erection by longitudinal launching--can fail locally by either web squashing, web crippling or web buckling.
Web squashing If a uniform distribution of the concentrated load is assumed over the idealized length (Fig. 1), c = Co+ 6(t~ + tf -I- r)
(1)
Web crippling with bending and shear of thin-walled plate girders
I("
.....
Fig.
1. S q u a s h i n g
o
of a web
89
_i plate.
the squash load of the web plate (yield strength fy) amounts to
P, =fyctw
(2)
Equation (2) is relevant to rolled sections and thick-walled plate girders only. A posterior analysis of 28 tests with rolled sections in Zurich (2 by Basler I and 26 by Duba's & Gehri 2) whose relevant parameters lay in the following ranges: - - web slenderness d/tw= 13.3-45.8; -web thickness tw =4.1-18.0 mm; -ratio of flange to web thickness tf/tw= 1.34-3.75; -yield strength of steel fy = 242-856 N/mm 2, calls for the factor 6 in eqn (1). The ratio of measured to predicted squash load (Fig. 2) is characterized by the following: arithmetic mean A = 1.119; - - standard deviation S = 0.237; - - coefficient of variation V=0-212.
--
The calibration factor 1/R--as the inverse value of the resistance factor R used in modern L R F D - - i s defined as the 5% fractile of the test results 10000
2 Zurich1973 Zurich~
kN
.~
26
1000 o
,
o,
-
A =1.119 V = 0,212
100 100
1000
, , I .... k N 10000
Prediction
Fig. 2. Squash load according to 28 tests with rolled sections.
90
Max A. M. Herzo O
(only 5% of all tests lie below the prediction). For the squash load it amounts to 1/R=0.84 of eqn (2).
Pure web crippfing Pure web crippling without interaction of bending and shear can be observed only in short girders without intermediate transverse stiffeners. Five years ago the author presented 3 the simple empirical formula Pc= 25 fyt 2 ~
~/a
(3)
for the 50% fractile of the web crippling load of short, thin-walled plate girders. The value of the square root must not be smaller than unity. It may seem surprising at first glance that the web slenderness does not appear in eqn (3). However, already 20 years ago it was shown in G6teborg 4 that the influence of this parameter is of little importance, because web crippling is a local phenomenon that is restricted to a portion of the web depth only. The calibration factor of eqn (3) amounts to 1/R=0.87. Equation (3) was deduced from a regression analysis of 136 tests with welded plate girders (Fig. 3) whose relevant parameters lay in the following ranges: - - web slenderness d/tw= 50.3-505; - - aspect ratio a/d=0.75-14.0; - - ratio of flange to web thickness tf/tw=1.0-12.4; 10000 kN :: 136 old t~sts
tO00
,o
/
E E
.~ 100 ul
10
./
-
p,
e.
A= 1.120 S = 0'266 v_-o.2
. . . . . . . . . . . .
10
100
1000
kN 10000
Prediction
Fig. 3. Pure web crippling load according to 136 old tests with welded plate girders.
Web cripplin0 with bending and shear of thin-walled plate girders
91
lOOO = 15 L e i p Z i g ° 12 B r u n s w i c k a 1 Pari~
"E5oo
•
63 t e s ,
/ *
e0
• ~*OZo* / . 0 ~
200 " :
%o
d,
A=0890 S=0"139 V =0"156
~,
....
200
I
,
f ,
ooo
Prediction
Fig. 4. Pure web crippling load according to 63 new tests with welded plate girders.
web thickness tw =0.99-12.4 mm; - - web depth d = 250-1300 mm; yield strength of steel fy = 192-360 N/mm 2. -
-
-
-
In the meantime, eqn (3) is supported convincingly (Fig. 4) by data from 63 new tests (1 by Galea et al., 5 12 by Scheer et al., 6 15 by Glas & Johne 7 and 35 by Tschamper s) within the following parameter ranges: d/tw = 64.2400, a/d=0.75-3.47, tf/tw= 1.60-7.5, tw=2.0-7.41 mm, d=282-1274 mm and fy = 238-378 N/mm 2. The two sets of tests are characterized by: A
S
V
136 older tests 63 new tests
1.120 0-890
0.266 ff139
0.238 0.156
199 tests
1.047
0.233
0.223
The reason for the observed difference of accuracy between old tests (before 1985) and new tests (after 1986) cannot be offered. Due to the lack of symmetry the web crippling load of unstiffened girder ends amounts to only one half of the prediction by eqn (3). The use of longitudinal stiffeners makes sense only in the case of closely spaced transverse stiffeners (a/d<2). To achieve full efficiency, they must be located near the compression flange (distance less than d/5), where they hinder the formation of a web buckle most effectively. It can be deduced from 43 tests (13 by Bergfelt, 9 2 by Shimizu et al., 1° 12 by Karnikovfi et al.11 and 16 by Tschamper 8) that the increase of the web crippling load is proportional to the ratio of cross sections (sectional area of the longitudinal stiffener AL) c L = (dr. +
(4)
Max A. M. Herzoo
92 1000
o
kN
I 13 G6iebor, g
I
I
* 16 Zurich A
E E
I
2 Nagano
¢
v 12 P r a c j u e l
_h IO0
43 t e s t s
~"
ul
® ~E
A = 1.060 S = 0,159
v = o.15o ,
k , I,,,,
1010
i
100
, , I,,
kN 1000
Prediction
Fig. 5. Increase of web crippling load by the use of longitudinal stiffeners according to 43 tests.
The efficiency of open section stiffeners can be given the value r/= 1 and that of closed ones at least r/= 2, due to their torsional stiffness. The ratio of measured to predicted web crippling loads (Fig. 5) is then characterized by the arithmetic mean A = 1.060, the standard deviation S = 0.159 and the coefficient of variation V=0.150. In practice, it will always be more economical to increase the web thickness instead of using longitudinal stiffeners.
Web crippling with bending and shear This design problem, which looks rather complicated at first glance, can be avoided elegantly by using the well-known design rule which assigns the bending moment to the flanges and the shear force to the web. No interaction of bending with shear takes place 12 if this design rule is followed. If the acting concentrated load P is related to the pure web crippling load according to eqn (3) and the acting bending moment M to the one for fully plastic flanges without web contribution M f =fybftf(d + tf)
(5)
then the interaction formula
(p/p=)2 + M / M f = 1
(6)
holds. As can be seen in Fig. 6, the 5% fractile of test values is reliably given by the modified interaction formula
(p/p~)2 + M / M f = 0"9
(7)
Web crippling with bending and shear of thin-walled plate girders
93
i
. •
1.5
4 Gbteborg 3 Paris
/
" 10 Nagan¢~ * 17 Brunswick _ • 8 Zurich 42 tests
1.0
v v
\-, 0.5
.E~6)
i
0.5
1.0
i
1.5
M/MF Fig. 6. Reduction of web crippling load by interaction with bending according to 42 tests.
This statement is based on 42 tests (4 by Bergfelt, 4 3 by Galea et al., 5 10 by Shimizu et al., ~° 17 by Scheer et al. 6 and 8 by TschamperS). If the concentrated load P near the end bearing of a transversely stiffened plate girder is related to the pure web crippling load Pc by eqn (3) and the average shear force Q of the end panel to the ultimate shear capacity ~2
(8)
q dt-;7;
then it is demonstrated by three relevant tests (by Oxfort & Gauger 13) in Stuttgart (Fig. 7) that the interaction of web crippling and shear without 1'0 . . . _ _ . . _ = ~ = ~
v4 VSO 0V3
0-
o.s
0
.
~ 0
.
.
.
.
.
.
.
.
.
.~L--L----t~L-~ 0"5 1-0
O/Qu Fig. 7. Reduction of web crippling load by interaction with shear according to 3 tests.
94
Max A. M. Herzoo
bending is given with sufficient accuracy by
(p/p¢)3 +(Q/Q~)3 = 1
(9)
Web buckfing
If the distribution length of the concentrated load is great (c/d > 0.5), web buckling rather than web crippling is to be expected, as already observed in the bearing diaphragms of box girder bridges (Fig. 8). The greater distribution length increases the domain concerned with web instability and finally involves the whole web depth; therefore the wording 'web buckling' is used by comparison with that of 'web crippling'. If the transverse stiffeners of a plate girder are closely spaced (aid < 2), the web buckling load is either a function of the horizontal ultimate tensile (not yield) strength of the web plate 15
Pb =fu(twd2)/a
(10)
or it is equal to the squash load according to eqn (2). The posterior analysis of 13 tests in London (12 by Dowling et al. 16 and 1 by Einarsson & Dowling 17) whose relevant parameters lay within the following ranges: - - panel aspect ratio a/d=0.71-2.00; - - web slenderness d/tw = 31.2-381; - - contact length ratios co/a=0"104-0"603 and co/d=0.139-0.883;
Fig. 8. Web buckling of the bearing diaphragm of a box girder. (From Ref. 14, pl. 223, Fig. C27.)
Web crippling with bending and shear of thin-walled plate girders 10
/ /
o 9 LonJon 1973 a 3 Lon:lon 1973 " + 1 Lon:lon 1979 5~13 tests-_
MN
i
95
/
J/
° • / ( 2 ; '°-
1
~
/' ....
A = 1'007 S = 0"154 V = 0"153 I
,
2
I ,
,l,l,l,I, 5
MN
10
Prediction
Fig. 9. Web buckling load of unstiffenedand stiffeneddiaphragmsaccording to 13 tests. 1000 kN:--
i
0 2 GOtebor~ • 3 Zurich I ~// * 3 Paris / //7 [] 4 NoClctno~ / /
-// lO 10
I
s = ot46
100
kN
1000
Prediction
Fig. 10. Web buckling load of unstiffened plate girders with significant distribution length of the concentrated load, according to 12 tests.
- - yield strength of steel fy = 257--458 N/ram2; - - ultimate tensile strength of steel f , = 387-583 N/ram2; shows (Fig. 9) that the calibration factor (or 5% fractile of measured web buckling loads) amounts to 1/R =0-85 of the predictions by eqns (10) or (2). If no transverse stiffeners are used, the tensile action according to eqn (10) cannot occur. Therefore the web buckling load cannot exceed the web crippling load by eqn (3). The posterior analysis of 12 tests (2 by Bergfelt & H6vik, Is 3 by Galea et al., s 4 by Shimizu et al. 1° and 3 by Tschamper s) shows that the measured web buckling load is increased due to a significant contact length by the factor (Fig. 10)
1
(11)
96
Max A. M. Herzog
COMMENTARY In designing plate girders for a concentrated load we have to distinguish between web squashing of thick webs in rolled sections and thick-walled plate girders, and web crippling or web buckling of thin-walled plate girders. For small contact lengths, as in crane and bridge girders, web crippling is a function of the following parameters: yield strength of steel, - - web thickness, ratio of flange to web thickness, panel aspect ratio, -
-
-
-
-
-
but not of the web slenderness. If the distance between transverse stiffeners is small (aid < 2), the web crippling load can be increased by using longitudinal stiffeners of open or closed section near the compression flange. Nevertheless, it will always be more economical to increase the web thickness. For great contact lengths (c/d>0.5) of the concentrated load, as in bearing diaphragms of box girder bridges, the actual failure mechanism is better described by the word 'web buckling' because, generally, a single large buckling deformation can be observed. The web buckling load is limited by a horizontal tensile action of the web plate which can be weakened by a cut-out in the web. The web buckling load is a function of the ultimate tensile strength of steel and not of the yield strength, which is relevant only in the cases of squash and of web crippling loads. In the absence of transverse stiffeners the web buckling load for great contact lengths exceeds the web crippling load according to eqn (3). Contrary to the case of short girders, the web crippling load of long girders is reduced by the interaction with bending or shear. Dealing with the interaction of web crippling, bending and shear can be avoided by using the well-known design rule which assigns the bending moment to the flanges and the shear force to the web plate. The calibration factor is put equal tO the 5% fractile of test values, which means that only 5% of the test values lie below the prediction by the formulas presented in this paper.
CONCLUSION A few very simple formulas are presented for the analysis of thin-walled plate girders subjected to a large concentrated load. As they were obtained from 340 tests by classical regression analyses they can be used with confidence.
Web crippling with bending and shear of thin-walled plate girders
97
REFERENCES 1. Basler, K., Rippenlose Verbindunoen im Stahlhochbau. Schweiz. ZentralsteUe fiir Stahlbau, Zurich, 1973. 2. Dubas, P. & Gehri, E., Behaviour of webs under concentrated loads acting between widely spaced vertical stiffeners. ECCS Commission 8.3, Zurich, 1978. 3. Herzog, M., Die Kriippellast yon Blechtriiger- und Walzprofilstegen. Stahlbau, 55(3) (1986) 87-8. 4. Bergfelt, A., Studies and tests on slender plate girders without stiffeners-shear strength and local web crippling. IABSE Colloquium on Design of Plate and Box Girders for Ultimate Strength, London, Report, 1971, pp. 67-83. 5. Galea, Y., Godart, B., Radouant, I. & Raoul, J., Tests of buckling of panels subjected to in-plane patch loading. In Proceedings of International Colloquium on Stability of Plate and Shell Structures, Ghent, 1987, pp. 65-71. 6. Scheer, J., Liu, X. L., Falke, J. & Peil, U., Tragiastversuche zur Lasteinleitung an I-fSrmigen geschweissten Biegetr/igern ohne Steifen. Stahibau, 57(4) (1988) 115-21. 7. Glas, H. D. & Johne, H., Noch einmal: Beultraglast yon Vollwandtr/igern unter Einzellast. Stahlbau, 60(4) (1991) 111-20. 8. Tschamper, H., Konzentrierte Lasteinleitung und Biegung an unversteiften schlanken Tr/igern. Stahlbau, 60(1) (1991) 5-14. 9. Bergfelt, A., Patch loading on a slender web--influence of horizontal and vertical stiffeners on the load carrying capacity. Publication $79:1, Inst. f. Konstruktionstekn., Chalmers Tekn. H6gskola, G6teborg, 1979. 10. Shimizu, S., Yoshida, S. & Okuhara, H., An experimental study on patchloaded web plates. In Proceedings of International Colloquium on Stability of Plate and Shell Structures, Ghent, 1987, pp. 85-94. 11. Karnikovfi, I., Novfik, P. & ~kaloud, M., Ultimate load behaviour of longitudinally stiffened steel webs subjected to partial edge loading. Stavebnickf~ (~asopis, 27(10) (1979) 752-5. 12. Herzog, M., Tragfiihigkeit und Bemessung unversteifter und versteifter Blechtr~iger auf Schub in einfachster N/iherung. Bauingenieur, 63(3) (1988) 133-7. 13. Oxfort, J. & Gauger, H. U., Beultraglast von Vollwandtr/igern unter Einzellasten. Stahlbau, 58(11) (1989) 331-9. 14. Proceedings of ICE Conference on Steel Box Girder Bridges, London, 1973, pp. 95-117, 173-91 and 222-4. 15. Herzog, M., Die Traglast der Lagerquerscheiben st/ihlerner Kastentr/iger nach Versuchen. Bauingenieur, 52(7) (1977) 263-5. 16. Dowling, P. J., Loe, J. A. & Dean, J. A. The behaviour up to collapse of load bearing diaphragms in rectangular and trapezoidal stiffened steel box girders. In Proc. lnternat. Conf. on Steel Box Girder Bridges, at the ICE London, 1973, pp. 95-117. 17. Einarsson, B. & Dowling, P. J., Tests on simply stiffened rectangular diap h r a g m s - M o d e l 1. CESLIC Report BG54, Imperial College of Science & Technology, London, 1979. 18. Bergfelt, A. & H6vik, J., Thin-walled deep plate girders under static loads. 8th IABSE Congress, New York, Final Report, 1968, pp. 465-78.
Web Crippling with Bending and Shear of Thin-Walled Plate Girders Max A. M. Herzog Rohrerstrasse 3, 5000 Aarau, Switzerland (Received 16 December 1991; revised version received 4 February 1992; accepted 24 February 1992)
ABSTRACT Even the newest versions of different standards do not yet solve the title problem of this paper adequately for practical requirements. Based on the posterior analysis of 340 tests reported in the literature some empirical formulas are presented which considerably simplify the designer's routine work. They have been obtained by reoression analyses and allow all influences on the ultimate strenoth of thin-walled plate oirders subjected to concentrated loads to be considered explicitly.
NOTATION a
A AL br c Co
C,. CL d
f. f,
M Mf
Distance of transverse stiffeners Arithmetic mean Sectional area of longitudinal stiffener Flange width Distribution length of concentrated load Effective contact length of concentrated load Coefficient accounting for the influence of distribution length Coefficient accounting for the influence of longitudinal stiffeners Web depth Ultimate tensile strength Yield strength Bending moment Bending moment for fully plastic flanges
87 J. Construct. Steel Research 0143-974X/92/$05.00 © 1992 Elsevier Science Publishers Ltd, England. Printed in Malta
88
Max A. M. Herzog
P eb Pc P~ Q Q~ r R
1/R S tf tr tw V
Concentrated load Buckling load Crippling load Squash load Shear force . Ultimate shear capacity of a transversely stiffened plate girder Fillet radius Resistance factor (larger than unity) Calibration factor (smaller than unity) Standard deviation Flange thickness Depth of crane rail Web thickness Coefficient of variation Efficiency of stiffener
INTRODUCTION Experienced designers will agree that in general new standards render their task more difficult instead of simplifying it. In many cases, simpler solutions are known, with an accuracy only a few percent lower than that of more demanding scientific methods. There is no advantage to the owner if the expenditure for the more demanding analysis exceeds the savings realizable in steel tonnage. This point is sometimes overlooked by pure academics.
THE PROBLEM Thin-walled plate girders subjected to a concentrated l o a d - - a s in the case of crane girders or of bridge girders during erection by longitudinal launching--can fail locally by either web squashing, web crippling or web buckling.
Web squashing If a uniform distribution of the concentrated load is assumed over the idealized length (Fig. 1), c = Co+ 6(t~ + tf -I- r)
(1)
Web crippling with bending and shear of thin-walled plate girders
I("
.....
Fig.
1. S q u a s h i n g
o
of a web
89
_i plate.
the squash load of the web plate (yield strength fy) amounts to
P, =fyctw
(2)
Equation (2) is relevant to rolled sections and thick-walled plate girders only. A posterior analysis of 28 tests with rolled sections in Zurich (2 by Basler I and 26 by Duba's & Gehri 2) whose relevant parameters lay in the following ranges: - - web slenderness d/tw= 13.3-45.8; -web thickness tw =4.1-18.0 mm; -ratio of flange to web thickness tf/tw= 1.34-3.75; -yield strength of steel fy = 242-856 N/mm 2, calls for the factor 6 in eqn (1). The ratio of measured to predicted squash load (Fig. 2) is characterized by the following: arithmetic mean A = 1.119; - - standard deviation S = 0.237; - - coefficient of variation V=0-212.
--
The calibration factor 1/R--as the inverse value of the resistance factor R used in modern L R F D - - i s defined as the 5% fractile of the test results 10000
2 Zurich1973 Zurich~
kN
.~
26
1000 o
,
o,
-
A =1.119 V = 0,212
100 100
1000
, , I .... k N 10000
Prediction
Fig. 2. Squash load according to 28 tests with rolled sections.
90
Max A. M. Herzo O
(only 5% of all tests lie below the prediction). For the squash load it amounts to 1/R=0.84 of eqn (2).
Pure web crippfing Pure web crippling without interaction of bending and shear can be observed only in short girders without intermediate transverse stiffeners. Five years ago the author presented 3 the simple empirical formula Pc= 25 fyt 2 ~
~/a
(3)
for the 50% fractile of the web crippling load of short, thin-walled plate girders. The value of the square root must not be smaller than unity. It may seem surprising at first glance that the web slenderness does not appear in eqn (3). However, already 20 years ago it was shown in G6teborg 4 that the influence of this parameter is of little importance, because web crippling is a local phenomenon that is restricted to a portion of the web depth only. The calibration factor of eqn (3) amounts to 1/R=0.87. Equation (3) was deduced from a regression analysis of 136 tests with welded plate girders (Fig. 3) whose relevant parameters lay in the following ranges: - - web slenderness d/tw= 50.3-505; - - aspect ratio a/d=0.75-14.0; - - ratio of flange to web thickness tf/tw=1.0-12.4; 10000 kN :: 136 old t~sts
tO00
,o
/
E E
.~ 100 ul
10
./
-
p,
e.
A= 1.120 S = 0'266 v_-o.2
. . . . . . . . . . . .
10
100
1000
kN 10000
Prediction
Fig. 3. Pure web crippling load according to 136 old tests with welded plate girders.
Web cripplin0 with bending and shear of thin-walled plate girders
91
lOOO = 15 L e i p Z i g ° 12 B r u n s w i c k a 1 Pari~
"E5oo
•
63 t e s ,
/ *
e0
• ~*OZo* / . 0 ~
200 " :
%o
d,
A=0890 S=0"139 V =0"156
~,
....
200
I
,
f ,
ooo
Prediction
Fig. 4. Pure web crippling load according to 63 new tests with welded plate girders.
web thickness tw =0.99-12.4 mm; - - web depth d = 250-1300 mm; yield strength of steel fy = 192-360 N/mm 2. -
-
-
-
In the meantime, eqn (3) is supported convincingly (Fig. 4) by data from 63 new tests (1 by Galea et al., 5 12 by Scheer et al., 6 15 by Glas & Johne 7 and 35 by Tschamper s) within the following parameter ranges: d/tw = 64.2400, a/d=0.75-3.47, tf/tw= 1.60-7.5, tw=2.0-7.41 mm, d=282-1274 mm and fy = 238-378 N/mm 2. The two sets of tests are characterized by: A
S
V
136 older tests 63 new tests
1.120 0-890
0.266 ff139
0.238 0.156
199 tests
1.047
0.233
0.223
The reason for the observed difference of accuracy between old tests (before 1985) and new tests (after 1986) cannot be offered. Due to the lack of symmetry the web crippling load of unstiffened girder ends amounts to only one half of the prediction by eqn (3). The use of longitudinal stiffeners makes sense only in the case of closely spaced transverse stiffeners (a/d<2). To achieve full efficiency, they must be located near the compression flange (distance less than d/5), where they hinder the formation of a web buckle most effectively. It can be deduced from 43 tests (13 by Bergfelt, 9 2 by Shimizu et al., 1° 12 by Karnikovfi et al.11 and 16 by Tschamper 8) that the increase of the web crippling load is proportional to the ratio of cross sections (sectional area of the longitudinal stiffener AL) c L = (dr. +
(4)
Max A. M. Herzoo
92 1000
o
kN
I 13 G6iebor, g
I
I
* 16 Zurich A
E E
I
2 Nagano
¢
v 12 P r a c j u e l
_h IO0
43 t e s t s
~"
ul
® ~E
A = 1.060 S = 0,159
v = o.15o ,
k , I,,,,
1010
i
100
, , I,,
kN 1000
Prediction
Fig. 5. Increase of web crippling load by the use of longitudinal stiffeners according to 43 tests.
The efficiency of open section stiffeners can be given the value r/= 1 and that of closed ones at least r/= 2, due to their torsional stiffness. The ratio of measured to predicted web crippling loads (Fig. 5) is then characterized by the arithmetic mean A = 1.060, the standard deviation S = 0.159 and the coefficient of variation V=0.150. In practice, it will always be more economical to increase the web thickness instead of using longitudinal stiffeners.
Web crippling with bending and shear This design problem, which looks rather complicated at first glance, can be avoided elegantly by using the well-known design rule which assigns the bending moment to the flanges and the shear force to the web. No interaction of bending with shear takes place 12 if this design rule is followed. If the acting concentrated load P is related to the pure web crippling load according to eqn (3) and the acting bending moment M to the one for fully plastic flanges without web contribution M f =fybftf(d + tf)
(5)
then the interaction formula
(p/p=)2 + M / M f = 1
(6)
holds. As can be seen in Fig. 6, the 5% fractile of test values is reliably given by the modified interaction formula
(p/p~)2 + M / M f = 0"9
(7)
Web crippling with bending and shear of thin-walled plate girders
93
i
. •
1.5
4 Gbteborg 3 Paris
/
" 10 Nagan¢~ * 17 Brunswick _ • 8 Zurich 42 tests
1.0
v v
\-, 0.5
.E~6)
i
0.5
1.0
i
1.5
M/MF Fig. 6. Reduction of web crippling load by interaction with bending according to 42 tests.
This statement is based on 42 tests (4 by Bergfelt, 4 3 by Galea et al., 5 10 by Shimizu et al., ~° 17 by Scheer et al. 6 and 8 by TschamperS). If the concentrated load P near the end bearing of a transversely stiffened plate girder is related to the pure web crippling load Pc by eqn (3) and the average shear force Q of the end panel to the ultimate shear capacity ~2
(8)
q dt-;7;
then it is demonstrated by three relevant tests (by Oxfort & Gauger 13) in Stuttgart (Fig. 7) that the interaction of web crippling and shear without 1'0 . . . _ _ . . _ = ~ = ~
v4 VSO 0V3
0-
o.s
0
.
~ 0
.
.
.
.
.
.
.
.
.
.~L--L----t~L-~ 0"5 1-0
O/Qu Fig. 7. Reduction of web crippling load by interaction with shear according to 3 tests.
94
Max A. M. Herzoo
bending is given with sufficient accuracy by
(p/p¢)3 +(Q/Q~)3 = 1
(9)
Web buckfing
If the distribution length of the concentrated load is great (c/d > 0.5), web buckling rather than web crippling is to be expected, as already observed in the bearing diaphragms of box girder bridges (Fig. 8). The greater distribution length increases the domain concerned with web instability and finally involves the whole web depth; therefore the wording 'web buckling' is used by comparison with that of 'web crippling'. If the transverse stiffeners of a plate girder are closely spaced (aid < 2), the web buckling load is either a function of the horizontal ultimate tensile (not yield) strength of the web plate 15
Pb =fu(twd2)/a
(10)
or it is equal to the squash load according to eqn (2). The posterior analysis of 13 tests in London (12 by Dowling et al. 16 and 1 by Einarsson & Dowling 17) whose relevant parameters lay within the following ranges: - - panel aspect ratio a/d=0.71-2.00; - - web slenderness d/tw = 31.2-381; - - contact length ratios co/a=0"104-0"603 and co/d=0.139-0.883;
Fig. 8. Web buckling of the bearing diaphragm of a box girder. (From Ref. 14, pl. 223, Fig. C27.)
Web crippling with bending and shear of thin-walled plate girders 10
/ /
o 9 LonJon 1973 a 3 Lon:lon 1973 " + 1 Lon:lon 1979 5~13 tests-_
MN
i
95
/
J/
° • / ( 2 ; '°-
1
~
/' ....
A = 1'007 S = 0"154 V = 0"153 I
,
2
I ,
,l,l,l,I, 5
MN
10
Prediction
Fig. 9. Web buckling load of unstiffenedand stiffeneddiaphragmsaccording to 13 tests. 1000 kN:--
i
0 2 GOtebor~ • 3 Zurich I ~// * 3 Paris / //7 [] 4 NoClctno~ / /
-// lO 10
I
s = ot46
100
kN
1000
Prediction
Fig. 10. Web buckling load of unstiffened plate girders with significant distribution length of the concentrated load, according to 12 tests.
- - yield strength of steel fy = 257--458 N/ram2; - - ultimate tensile strength of steel f , = 387-583 N/ram2; shows (Fig. 9) that the calibration factor (or 5% fractile of measured web buckling loads) amounts to 1/R =0-85 of the predictions by eqns (10) or (2). If no transverse stiffeners are used, the tensile action according to eqn (10) cannot occur. Therefore the web buckling load cannot exceed the web crippling load by eqn (3). The posterior analysis of 12 tests (2 by Bergfelt & H6vik, Is 3 by Galea et al., s 4 by Shimizu et al. 1° and 3 by Tschamper s) shows that the measured web buckling load is increased due to a significant contact length by the factor (Fig. 10)
1
(11)
96
Max A. M. Herzog
COMMENTARY In designing plate girders for a concentrated load we have to distinguish between web squashing of thick webs in rolled sections and thick-walled plate girders, and web crippling or web buckling of thin-walled plate girders. For small contact lengths, as in crane and bridge girders, web crippling is a function of the following parameters: yield strength of steel, - - web thickness, ratio of flange to web thickness, panel aspect ratio, -
-
-
-
-
-
but not of the web slenderness. If the distance between transverse stiffeners is small (aid < 2), the web crippling load can be increased by using longitudinal stiffeners of open or closed section near the compression flange. Nevertheless, it will always be more economical to increase the web thickness. For great contact lengths (c/d>0.5) of the concentrated load, as in bearing diaphragms of box girder bridges, the actual failure mechanism is better described by the word 'web buckling' because, generally, a single large buckling deformation can be observed. The web buckling load is limited by a horizontal tensile action of the web plate which can be weakened by a cut-out in the web. The web buckling load is a function of the ultimate tensile strength of steel and not of the yield strength, which is relevant only in the cases of squash and of web crippling loads. In the absence of transverse stiffeners the web buckling load for great contact lengths exceeds the web crippling load according to eqn (3). Contrary to the case of short girders, the web crippling load of long girders is reduced by the interaction with bending or shear. Dealing with the interaction of web crippling, bending and shear can be avoided by using the well-known design rule which assigns the bending moment to the flanges and the shear force to the web plate. The calibration factor is put equal tO the 5% fractile of test values, which means that only 5% of the test values lie below the prediction by the formulas presented in this paper.
CONCLUSION A few very simple formulas are presented for the analysis of thin-walled plate girders subjected to a large concentrated load. As they were obtained from 340 tests by classical regression analyses they can be used with confidence.
Web crippling with bending and shear of thin-walled plate girders
97
REFERENCES 1. Basler, K., Rippenlose Verbindunoen im Stahlhochbau. Schweiz. ZentralsteUe fiir Stahlbau, Zurich, 1973. 2. Dubas, P. & Gehri, E., Behaviour of webs under concentrated loads acting between widely spaced vertical stiffeners. ECCS Commission 8.3, Zurich, 1978. 3. Herzog, M., Die Kriippellast yon Blechtriiger- und Walzprofilstegen. Stahlbau, 55(3) (1986) 87-8. 4. Bergfelt, A., Studies and tests on slender plate girders without stiffeners-shear strength and local web crippling. IABSE Colloquium on Design of Plate and Box Girders for Ultimate Strength, London, Report, 1971, pp. 67-83. 5. Galea, Y., Godart, B., Radouant, I. & Raoul, J., Tests of buckling of panels subjected to in-plane patch loading. In Proceedings of International Colloquium on Stability of Plate and Shell Structures, Ghent, 1987, pp. 65-71. 6. Scheer, J., Liu, X. L., Falke, J. & Peil, U., Tragiastversuche zur Lasteinleitung an I-fSrmigen geschweissten Biegetr/igern ohne Steifen. Stahibau, 57(4) (1988) 115-21. 7. Glas, H. D. & Johne, H., Noch einmal: Beultraglast yon Vollwandtr/igern unter Einzellast. Stahlbau, 60(4) (1991) 111-20. 8. Tschamper, H., Konzentrierte Lasteinleitung und Biegung an unversteiften schlanken Tr/igern. Stahlbau, 60(1) (1991) 5-14. 9. Bergfelt, A., Patch loading on a slender web--influence of horizontal and vertical stiffeners on the load carrying capacity. Publication $79:1, Inst. f. Konstruktionstekn., Chalmers Tekn. H6gskola, G6teborg, 1979. 10. Shimizu, S., Yoshida, S. & Okuhara, H., An experimental study on patchloaded web plates. In Proceedings of International Colloquium on Stability of Plate and Shell Structures, Ghent, 1987, pp. 85-94. 11. Karnikovfi, I., Novfik, P. & ~kaloud, M., Ultimate load behaviour of longitudinally stiffened steel webs subjected to partial edge loading. Stavebnickf~ (~asopis, 27(10) (1979) 752-5. 12. Herzog, M., Tragfiihigkeit und Bemessung unversteifter und versteifter Blechtr~iger auf Schub in einfachster N/iherung. Bauingenieur, 63(3) (1988) 133-7. 13. Oxfort, J. & Gauger, H. U., Beultraglast von Vollwandtr/igern unter Einzellasten. Stahlbau, 58(11) (1989) 331-9. 14. Proceedings of ICE Conference on Steel Box Girder Bridges, London, 1973, pp. 95-117, 173-91 and 222-4. 15. Herzog, M., Die Traglast der Lagerquerscheiben st/ihlerner Kastentr/iger nach Versuchen. Bauingenieur, 52(7) (1977) 263-5. 16. Dowling, P. J., Loe, J. A. & Dean, J. A. The behaviour up to collapse of load bearing diaphragms in rectangular and trapezoidal stiffened steel box girders. In Proc. lnternat. Conf. on Steel Box Girder Bridges, at the ICE London, 1973, pp. 95-117. 17. Einarsson, B. & Dowling, P. J., Tests on simply stiffened rectangular diap h r a g m s - M o d e l 1. CESLIC Report BG54, Imperial College of Science & Technology, London, 1979. 18. Bergfelt, A. & H6vik, J., Thin-walled deep plate girders under static loads. 8th IABSE Congress, New York, Final Report, 1968, pp. 465-78.