Elasto-plastic large deflection analysis of end support diaphragms in skew steel box girders

J. Construct.Steel Research32 (1995)173-190

© 1995ElsevierScienceLimited Printed in Malta. All rights reserved 0143-974X/95/$9.50 ELSEVIER

Elasto-plastic Large Deflection Analysis of End Support Diaphragms in Skew Steel Box Girders Shin-ichi Kimbara Res~.~arch Laboratory, Miyaji Iron Works Co., Ltd, Nihonbashi SK Building, 15-18 Nihonbashi, Kodenmacho, Chuo-ku, Tokyo 103, Japan

&

Shigeru Shimizu Department of Civil Engineering, Shinshu University, 500 Wakasato, Nagano 380, Japan (Received for publication 19 January 1994)

ABSTRACT Several studies on support diaphragms of steel box girders have been completed and some results of these studies have been incorporated into design rules. Almost all, however, assume support diaphragms placed orthooonally to the girder. Few studies have been done on skew bridge girder configurations, a number of which have recently been constructed. This paper reports on an elasto-plastic large deflection analysis of support diaphragms placed askew to the main girder axis and describes the behavior and strength of diaphragms under such conditions. The analysis was done on analytical models of steel box girders with two support diaphragms, with angle of diaphragm skew as a parameter. Results show that skew causes local out-of-plane deformation in a support diaphragm, which does not appear in an unskewed support diaphragm and w,~ich drastically reduces diaphragm strength.

NOTATION B H

Width of main girder (interval between webs) Height of main girder's web

0

Angle of skew (angle between plane of support diaphragm and girder axi,,;; see Fig. l(d)) 173

174

S. Kimbara, S. Shimizu

trvn Vertical in-plane stress in mid-plane of support diaphragm trhn Horizontal in-plane stress in mid-plane of support diaphragm trvb Vertical flexural stress in support diaphragm (difference between surface and mid-plane vertical in-plane stress intensity) trhb Horizontal flexural stress in support diaphragm (difference between surface and neutral-plane horizontal in-plane stress intensity)

1 INTRODUCTION As opposed to the 1-beam girder bridge with support under the main girder web, the box girder bridge uses a support diaphragm to transmit load to the web and flange of the main girder. These support diaphragms have been studied because they are directly subjected to large reactions and because failure of the support diaphragm causes failure of the entire bridge. There have been instances of box girder bridges under construction collapsing due to failure of their support diaphragms.1 Dowling et al. 2'3 conducted experiments focused on the collapse behavior of support diaphragms in box girders with rectangular and trapezoidal cross-sections and examined the effect of initial deflection on failure mode and strength. Analyzing the continuous straight box girder with intermediate support using the finite strip method, Shimizu et al. 4 studied support diaphragm stress distribution and the effect of loadbearing stiffeners. This work clarified the hyperstatic forces acting between the diaphragm and the web and flange of the main girder. Noting the importance of intermediate support for curved box girders, Shimizu and Yoshida 5 employed the method used in Ref. 4 to analyze the reaction allotment. They used the interval of intermediate diaphragms and the sectional form and curvature of the main girder as parameters. Yoda et al. 6 reported on the stiffness required of a support diaphragm and effective flange plate width, employing FEM analysis of a box girder subject to an eccentric load. Fukumoto et al., 7 intending to apply their results to actual bridges, conducted a series of experiments on the elastic behaviors and load-bearing capacities of support diaphragm specimens. The results of the experiments were compared with results of numerical analyses they performed using the method described in Ref. 4 and with results produced by the design formula given in Ref. 8. Dogaki et al. 9 performed elastic buckling analysis of support diaphragms with no load-bearing stiffeners. Kimbara et al. 1° performed a stress and elastic buckling analysis of support diaphragms for continuouscurve box girders. Shimizu, 1~ after determining the boundary conditions and load distribution of support diaphragms through the results presented

Elasto-plastic large deflection analysis

175

in Refs 4 and 7, analyzed the failure load of support diaphragms using elasto-plastic large deflection FEM. Kimbara and Simizu 12 used FEM to analyze stress distributions and reaction allotments of support diaphragms in skewed steel box girders. The literature on support diaphragms reviewed above provides little basis for practical applications. Some studies referred to above deal with curved box girders but assume that support diaphragms are placed orthogonally to the main girder, and almost no reports on support diaphralgms in skew box girders have been published except by the authors of this paper.12 Because of its excellent torsional stiffness, the box girder is appropriate for bridges with long spans and curves and has been adopted for many skew bridges. When a skew bridge is subjected to vertical loading, deformation such as twist and camber occurs in the main girder because of the skew---even if loading is symmetrical. The state of stress in a support diaphragm placed obliquely to the girder axis is thus different from that in a diaphragm orthogonal to the girder, and consequently girder strength is likely to be different as well. The authors of this paper performed elasto-plastic large deflection analysis on analytical models of support diaphragms placed obliquely in steel box girders for skew bridges, and examined the relationship between diaphragm strength and behavior. Komatsu et al. 13 proposed analysis by the load increment technique using a three-node planar shell. The analysis for this report also used a three-node planar shell, but improved the method used in Ref. 13 by using the arc-length technique rather than the load increment technique.

2 MODELS FOR NUMERICAL ANALYSIS Diagrams of the models used in this study are shown in Fig. 1. A simple single-box main girder with a rectangular section and supported at each end (a configuration frequently used in actual bridges) was assumed. The span of the main girder was set at 15 m, a size that can be applied to the design of actual bridges. Girder height was set at 1-2 m, less than one-tenth the span, to avoid creating a deep-beam structure. Investigation of the sectiona]L forms of bridges of this type built in the past showed that many had ratJios of girder height H to web interval B from 0.5 to 1.0 (0"5<~H/B<~I'O). For this reason, models reported on here have girder height to web interval ratios of 0.5 or 1.0 (H/B=0.5, 1.0). Angles of skew 0 of 60 ° and 75 ° (0 = 60 °, 75 °) were chosen for the models since they are often used for actual bridges. Models with no skew (0=90 °) were also used

S. Kimbara, S. Shimizu

176

i

"X

(a) Load

Bearing Stiffner

9

"1 2000 2400

~0

(b)

800 . 1200 (c) 24011Pa

A

o° 15,ooo

,i13oo ~

/~

strain

(d)

.....

24011Pa

(c) Fig. 1. Analytical model: (a) profile view; (b) section of H/B=0.5; (c) section of H/B= 1.0; (d) angle of skew; (e) assumed property of steel; (f) a typical mesh pattern.

for purposes of comparison. As shown in Fig. 1, there are load-bearing stiffeners 200 m m from the girder webs and the bearings of the girder are directly below them. The boundary conditions of the pivoting girder shoes allow rotation around the x, y and z axes but restrain movement along the x, y and z axes. The bottom flange elements within an area 200 x 200 m m around the bearing points were thought to be comparatively rigid, being

Elasto-plastic large deflection analysis

177

Fig. l.--Contd.

in contact with the diaphragm shoes, and the thickness of that part of the girder was increased to five times that of the flange. The load was distributed evenly over the webs of the main girder, as shown in Fig. l(a). Table 1 shows the names of the models, and the thicknesses of diaphragms and of load-bearing stiffeners, the angles of skew, and the initial deflections for the models used in this analysis. Deflection is defined as positive when convex toward the center of the span. In the analysis, initial deflection only in the central panel of a diaphragm 'was incorporated into the models because this study was intended to focus on support diaphragms and because the two panels between the load-bearing stiffeners and the girder webs are so narrow that almost no TABLE 1 Model Names and Parameters

Model name

H/B

Thickness of support diaphraam (cm)

Thickness of stiffeners (cm)

Anole of skew (degrees)

Initial deflection at center of diaphragm (cm)

P901 P751 P601 M601 P902 P752 P602

1"0 1"0 1-0 1-0 0"5 0"5 0"5

1>8 0-8 0-8 0"8 1-0 1"0 1"0

1-0 1-0 1"0 1"0 1-2 1"2 1"2

90 75 60 60 90 75 60

1"0 1"0 1"0 - 1-0 1-0 1"0 1-0

178

S. Kimbara, S. Shimizu

initial deflection of such panels has occurred in actual bridges. Initial deflection was zero at the edge of the central panel, rising to maximum at the panel's center such that a section through the center of the panel would be shaped like half a sinusoidal curve. The material assumed in the analysis was grade SS400 steel (tensile strength: 400 MPa, yield stress: 240 MPa) with a bilinear curve as shown in Fig. l(e). The mesh pattern of the analytical model is shown in Fig. l(f).

3 MAXIMUM LOAD AND DEFLECTION BEHAVIOR OF SUPPORT DIAPHRAGMS Figure 2 shows the deflection modes of support diaphragms after maximum load for analytical models P901, P601 and M601. Significant deflection of the center diaphragm panel between the two load-bearing stiffeners was generated. Non-skew model P901 shows deflection symmetrical around a peak at the center of the panel, while in skew models P601 and M601 a peak (or trough) of deformation runs from lower left to upper right. This resembles deformation after shear buckling. Comparing the results for P601 and those for M601, it appears that the direction of initial deflection determines the direction of deformation after maximum load is imposed. In the skew models, out-of-plane deformation also takes place in the panel between the load-bearing stiffener and the girder web on the

i III ~._ Mode s h s p e - ~

if----...

(b) Model F)601 (c) Model M601 Fig. 2. Deflectionmodes of support diaphragms after maximum loads: (a) model P901; (b) model P601; (c) model M601. (a)Model P901

Elasto-plastic large deflection analysis

179

obtuse angle side, but deformation of this panel does not appear in the non-skew model P901. The direction of the out-of-plane deformation of this panel is independent of the direction of the central panel's initial deflection; these panels always deform toward the center of the span. The maximum loads of the models are listed in Table 2. Figures 3 and 4 show ]oad~lisplacement curves for four models: P901, P751, P601 and M601. The vertical axes in these graphs represent the loads, and the horizontal axis in Fig. 3 represents the Euclidean norm (uTu) 1/2 of displacement vectors while that in Fig. 4 represents the horizontal x-axis displacement of a point (point A) on the free edge of a load-bearing stiffener 10 cm from the bottom of the stiffener. TABLE 2

Maximum Loads of Analyzed Models

Model name

Maximum load (kN)

Maximum load (skew) Maximum load (non-skew)

P9D1 P751 P601 M601

2 913"50 2 282"40 2 366-75 2 254-27

0'783 0"812 0"773

P902 P752 P6,02

3 759"65 3 150"85 3 139"86

0"838 0"835

kN

r,~ .~,pgOl

3, o o o '~ ,.I

P6Ol,,d

P751

A

2. ooo

. ~fM601

/

/

1. ooo

0

20 4'0 6'0 8'0 1 6 0 1 2 0 1'40 Displacement(era) Fig, 3, Load-displacement curves (Euclidean norm).

S. Kimbara, S. Shimizu

180

Load kN

3,, ooo FX301

d~.M601(.""~ \~ P601\,

-o~ os

-o.' 02

,

.....

2, 00

\ \ ~P751

-o~ ol

o

0.'01 Displacement(cm)

Fig. 4. Load--displacementcurves (x direction of point A).

As can be seen from Table 2 and Figs 3 and 4, the maximum loads for the skew models are about 80% those of otherwise-identical non-skew models, and no significant correlation between maximum load and angle of skew was observed. Comparison of model P601 with model M601 reveals that the former's maximum load is approximately 5% greater than the latter's. These models are identical except that P601 was given positive initial deflection and M601 negative initial deflection. It thus appears that the direction of the central panel's initial deflection has a certain correlation with the load capacity of a support diaphragm. Referring to Fig. 2, the deflection modes of these models differ in that M601 deflection is continuous through the load-bearing stiffeners while P601 deflection has a node at the stiffener. Thus, maximum load appears to have a correlation with deflection mode and hence with the direction of initial deflection. Moreover, the magnitude of x-axis displacement for point A in the non-skew models rises linearly in the positive direction as load increases, even after maximum load is imposed. In the skew models, however, point A is first displaced along the x-axis in the negative direction but displacement suddenly changes direction just before or just at maximum loading. Out-of-plane displacement at the edge of a load-bearing stiffener (such as displacement of point A along the x-axis) is thought to be governed by rotational displacement of the diaphragm panel around the z-axis. Therefore, from Figs 2 and 4, it is inferred that because in skew models

Elasto-plastic laroe deflection analysis

181

out-of-plane displacement occurs even in the panels between the stiffeners and the girder webs, the entire deformation mode of the diaphragm changes suddenly, resulting in failure of the diaphragm.

4 STRESS I N L O A D - B E A R I N G D I A P H R A G M S

4.1 Vertical in-plane stress (o,.: vertical stress intensity at mid-plane of support diaphragm) As an example, the vertical in-plane stress distribution of model P601 for maximum load is shown with 20-MPa contour lines in Fig. 5. Downward reaction is generated in the acute angle side and, thus, tensile stress occurs in the vicinity of the bearing on this side. In all model support diaphragms, a large amount of vertical in-plane stress appears near the bearing on the obtuse angle side where support reaction rises as the angle of skew grows. While this stress extends along the load-bearing stiffener on the obtuse angle side, little., stress is generated midway between the two bearings or at the P o s i t i o n of Load Bearing S t i f f e n e r

/

__ I

0 a) "c! ...

• ~

Ii

"1

CD

u

C', cu

;o .0

'20

0

-240

( U n i t :MPa) Fig. 5. Vertical in-plane stress distribution (model P601).

0 ~

S. Kimbara, S. Shimizu

182

center of the support diaphragm. Because this stress distribution pattern is common to all models, vertical stress in a skew support diaphragm, just as in the non-skew diaphragms which have been extensively studied, can satisfactorily be evaluated by dividing the panel in half and calculating the stress in one half. In the case of skew support diaphragms, the obtuse angle side with high stress levels should be singled out for this purpose. 4.2 Horizontal in-plane stress (ah: horizontal stress at mid-plane of diaphragm) As examples, the horizontal in-plane stress distributions of models P901 and P601 for maximum load are shown with 10-MPa contour lines in Fig. 6(a) and (b). Horizontal in-plane stress appears as tensile stress in the upper part of the diaphragms and as compressive stress in the lower part of the diaphragms. High compressive stress appears locally near the bearings, and its absolute value reaches about three times maximum stress in the upper part of each diaphragm. P o s i t i o n of Load Bearing S t i f f e n e r

I

J

10

10i

I I I

-60

60-10

0 t

~

-70 (a)

i

0

0 0

t

T

i

-70

(Un i t : M P a )

Fig. 6. H o r i z o n t a l in-plane stress distribution: (a) m o d e l P901; (b) m o d e l P601.

Elasto-plastic large deflection analysis

183

Position of Load Bearing S t i f f e n e r

0

0 o "0

o

qD

o

C

0 "-10

-10 0

) (b)

0 (Unit

:MPe)

-133

Fig. 6.--Contd. In the panel between the load-bearing stiffener and the girder web on the obtuse angle side of model P601, local tensile stress is greatest at one-fifth the height of the diaphragm. In the non-skew models the absolute value of this local stress is about the same as that of the local stress near the bearing, but in the skew models it is considerably greater. This indicates a need for caution when designing such configurations.

4.3 Vertical out-of-plane flexural stress (O,b) Figure 7 shows with stress contours the differences between the vertical stress intensity of the plate surface and that of the mid-plane, or vertical flexural stress components of the diaphragms, for maximum load. In non..skew model P901, significant flexural stress appears only in the central panel (which was given an initial deflection). In the central panels of the narrow-girder models (P901, P601 and M601), peaks of positive and negative flexural stress appear in turns from the upper flange down to the lower flange, while in model P602, which has a wider girder, positive stress is distributed over the center panel. In models P601 and M601, each of

,0/

P o s i t i o n of Load Bearing S t i f f e n e r

0

I

I

-5'

'5

5'

10

(a)

'-5

(Un i t : M P a ) P o s i t i o n of Load Bearing S t i f f e n e r

20

' (b)

'

-35 (Unit

0-5

0

20'

'0

:MPa)

Fig. 7. Vertical flexural stress distribution: (a) model P901; (b) model M60I; (c) model P601; (d) model P602.

Elasto-plastic large deflection analysis Poaition

of Load Dearino

185

Stiffener

0 Q

o

m

•

I

I

•

•

o

C

o

O u

(C)

-

~D

I

Position

u

t

30

1

(Un i t :MP8)

i

-10

of Load Bearing S t i f f e n e r

@ m w o 4-*--

t'i

(d) -15

,._l

/,

-5

(Unit :MPa) Fig. ?.--Contd.

I

I

.a

~"

0

m

~-25 '--15

-25

which have narrow girders and are skewed, a broad region of high stress in the central panel extends from lower left to upper right. This pattern resembles stress distribution after shear buckling. Moreover, while maximum stress

186

S. Kimbara, S. Shimizu

appears at about two-sevenths the height of the diaphragm in narrowgirder models P901, P601 and M601, in the wide-girder model P602 maximum stress appears at about four-fifths the height of the diaphragm. In the skew models P601, P602 and M601, comparatively high stress appears near the bottom of the panel between the load-bearing stiffener and the girder web on the obtuse angle side. In models P601 and P602, where initial deflection of central diaphragm panels is positive, this local flexural stress has two peaks--positive and negative. In contrast, model M601 with negative initial deflection of the central panel has only one positive peak of stress. From the fact that, even though models P601 and P901 have positive initial deflection, the positions of positive and negative peaks of local flexural stress in P601 are opposite those in P901, it is thought that local flexural stress between the stiffener and the girder web has some correlation with the direction of central panel deflection. 4.4 Horizontal out-of-plane flexural stress

(0"hb)

Figure 8 shows with stress contours the difference between horizontal stress intensity at the surface and that at the mid-plane, or the Position of Load 8earing S t i f f e n e r

0/

\0

0

(a) O '

'

(Unit

:MPa)

'

'0

Fig.& Horizontal flexural stress distribution: (a) model P901; (b) model P601; (c) model P602.

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188

s. Kimbara, S. Shimizu

typical of a continuous plate supported by stiffeners. In skew model P601, a large region of high negative flexural stress in the central panel runs from lower left to upper right. Two positive flexural stress peaks appear, one at the upper left and another at the lower right. Stress peaks also appear where load-bearing stiffeners intersect the support diap h r a g m - o n e in the upper portion of the diaphragm and another in the lower portion for each stiffener location, which have opposite signs. This distribution resembles a pattern found after shear buckling, as was the case with the distribution of vertical flexural stress. Furthermore, local flexural stress appears near the bottom of the girder web on the obtuse angle side. Though a similar distribution is observed in model P602, which has a wide girder, the difference between maximum and minimum stress is not as large as in model P601. Also, the location of peak local flexural stress near the web on the obtuse angle side is higher in model P901 than in model P601.

5 SUMMARY OF RESULTS The importance of end support diaphragms in skew steel box girders was noted and elasto-plastic large deflection analysis of these support diaphragms was conducted. In a support diaphragm, large amounts of vertical in-plane stress appear near the bearing on the obtuse angle side where reaction is larger at larger angles of skew. Figure 5, however, reveals that little stress is generated near the longitudinal midpoint of the main girder, even in models where negative reaction occurs near the bearing on the acute angle side. The authors think this suggests that vertical stress in a diaphragm can be evaluated by dividing a support diaphragm in half and performing calculations on one half (with one bearing point). Thus, in practice the design formulas presented in the existing literature (such as in Refs 4 and 5) may be applied to skew steel box girders without any modification if reaction can be calculated accurately. The most important factor for the load-carrying capacity of a support diaphragm seems to be vertical stress. In addition, the out-of-plane flexural stress (or out-of-plane deflection) described in Sections 4.3 and 4.4 of this paper may also be important. This report clarifies that, in a support diaphragm skewed to the girder axis, comparatively large local outof-plane flexural stress appears in the lower half of the diaphragm panel, between the load-bearing stiffener and the girder web on the obtuse angle side. As can be understood from the deformation modes shown in Fig. 2,

Elasto-plastic laroe deflection analysis

189

in skew models diaphragm failure involves out-of-plane deformation in the vicinity of the lower part of these panels between the stiffener and the girder web on the obtuse angle side, in addition to severe deformation of the diap]hragm's central panel. So far as the results of this analysis are concerned, the strength of a model non-skew support diaphragm depends on the strength of its central panel (which is imparted initial deflection), and, therefore, additional stiffeners, etc., at tile central panel would probably improve diaphragm strength. On the other hand, in a model support diaphragm skewed to the main girder, the post.-buckling strength imparted by a diagonal tension field using a load-bearing stiffener as anchorage should be effective, as is usually the case with shear buckling problems, because the central panel's behavior resembles ,;hear buckling. In the skew models used for this analysis, however, out-of-plane deformation in the lower half of the panels between the stiffeners and the girder webs, caused by local out-of-plane flexural stress generated near the girder web on the obtuse angle side, led to diaphragm failure. This indicates the need for additional stiffeners and the like---not only in the central panel but also in the panels between the load-bearing stiffeners and the girder webs--in order to prevent out-of-plane deflection.

6 CONCLUSION This paper reports the results of numerical analysis of analytical models of steel box girders with support diaphragms, focusing on the deflection behavior of support diaphragms placed obliquely to the girder axis. Because :the analysis focused on support diaphragm behavior in skew box girders, initial deflection was provided only to the central panel and was made relatively large. No models with stiffeners at the central panel were analyzed. Therefore, no simple and practical design formulas can be derived from the results of this study. In the future, studies on practical design methods, including methods of stiffening the panels between stiffeners and girder webs, should be conducted using the analytical techniques described here. ACKNOWLEDGEMENTS Numerical computations for this study were performed by the Computer Center, University of Tokyo, and the Shinshu University Computer Center.

190

S. Kimbara, S. Shimizu

REFERENCES 1. Massonnet, C., Tokyo seminar on some European contributions to the design of metal structures, with emphasis on plasticity and stability, ultimate strength and optimum design of steel buildings, and steel plate and box girders, Department of Civil Engineering, Nagoya University, Nagoya, 1974, pp. 313~8. 2. 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 Steel Box Girder Bridges. ICE, London, 1973, pp. 95-117. 3. Dowling, P. J., Strength of steel box girders, Proc. ASCE, 101 (ST9) (1975) 1929-44. 4. Shimizu, S., Kajita, T. & Naruoka, M., The stresses near intermediate support in continuous box girders. Proc. JSCE, No. 276, 1978 (in Japanese). 5. Shimizu, S. & Yoshida, S., Reaction allotment of continuous curved box girders. Thin-Walled Structures, 11 (1991) 319-41. 6. Yoda, T., Hirashima, M., Nakada, T. & Kanda, K., On the simplified design method of support diaphragms. Proc. Annual Meeting, JSCE, No. 42, 1987, pp. 440-1 (in Japanese). 7. Fukumoto, Y., Shimizu, S. & Furuta, H., Strength of steel box girder support diaphragms. Proc. JSCE, No. 318, 1982 (in Japanese). 8. The Committee of Inquiry into the Basis of Design and Method of Erection of Steel Box Girder Bridges: Interim Design and Workmanship Rules, Her Majesties Stationary Office, London, 1973. 9. Dogaki, M., Harada, M. & Yonezawa, H., Elastic buckling of support diaphragms with rectangular holes in steel box girders, Technical Report of Kansai University, No. 27, 1986, pp.173-85. 10. Kimbara, S., Shimizu, S. & Yoshida, S., A conclusion for buckling strength of intermediate support diaphragms in continuous box girders. Proc. Annual Meetino, JSCE, No. 35, 1980 (in Japanese). 11. Shimizu, S., An elasto-plastic analysis of steel box girder support diaphragms. Proc. Annual Meetin9, JSCE, No. 47, 1992, pp. 440-1 (in Japanese). 12. Kimbara, S. & Shimizu, S., Stresses of the support diaphragms in skew steel box girder. J. Structural Engineering, JSCE, 39A, 1993 (in Japanese). 13. Komatsu, S., Kitada, T. & Miyazaki, S., Elasto-plastic analysis of compressed plate with residual stress and initial deflection. Proc. JSCE, No. 244, 1975 (in Japanese). 14. British Standard Institution: BS 5400, Steel, Concrete and Composite Bridges, Part 3: Code of Practice for Design of Steel Bridges, 1982. 15. Japan Road Association, Japanese Specifications for Highway Bridges, Maruzen, Tokyo, 1990.