Finite element modelling of box girder diaphragms at supports

Thin-Walh, d Structures 22 (1995) 25 37 ~'~ 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 0263-8231/95/$9.50

ELSEVIER

Finite Element Modelling of Box Girder Diaphragms at Supports T. H. G. Megson Department of Civil Engineering, University of Leeds, Leeds, UK, LS2 9JT

& G. Hallak Department of Civil Engineering, Teshreen University, Lattakia, Syria (Received 15 June 1993: accepted 15 December 1993)

A BS TRA C T A finite element modelJor a rectangular, st(ffened, load-bearing, box girder diaphragm is proposed and used for the prediction of" deflections aJ,d stress distributions in the elastic range andJor the prediction qf failure modes and collapse loads'. The method, using the A BA Q US package, is shown to give good agreement with the results obtained from two experimental investigations

1 INTRODUCTION Until the comparatively recent advances in the power and capacity of computers it has been difficult to model the non-linear behaviour associated with the collapse analysis of structures having complex geometries and b o u n d a r y conditions. In 1973 Crisfield 1 carried out an elasto-plastic buckling analysis of flat plates using the finite element method. This was followed in 1977 by a two-dimensional idealisation by Crisfield and Puthli 2 of a load-bearing diaphragm with parts of the adjacent webs and flanges incorporated in the model. Different assumptions were made concerning the interaction between the different components and the boundaries of the diaphragm were assumed to be either fully restrained or simply supported, neither 25

26

T. H. G. Megson, G. Hallak

assumption representing the actual conditions accurately. Also, in 1977, Puthli 3 used a three-dimensional model of an unstiffened diaphragm in a length of unstiffened box girder; in order to reduce computer running time and storage a coarse mesh was used. This paper describes part of a research programme4 conducted to investigate the optimum size and arrangement of stiffeners in rectangular box girder diaphragms. A finite element model, based on the ABAQUS package, 5 is used to predict stress distributions in the elastic range and to predict collapse modes and loads for a rectangular, stiffened, load-bearing diaphragm. The results from the finite element analysis are compared with the results from two experimental investigations and shown to give good agreement. 2 E X P E R I M E N T A L INVESTIGATION

2.1 Experimental model A general view of the experimental model is shown in Fig. 1. The model was designed so that normal fabrication procedures could be used in its construction and also so that it would fit into the largest available loading rig in the laboratory. The diaphragm was positioned centrally in a length of box girder such that a length of box girder equal to two-thirds of the depth of the diaphragm was provided on either side. The diaphragm comprised a 6 mm thick steel plate which was stiffened symmetrically by two pairs of stiffeners, each 65mm x 10mm in cross-section placed on each side of the diaphragm and directly under the bearing plates; these were of high yield steel 60 mm thick. The flanges of the length of box girder were steel plates 12-75 mm thick, stiffened longitudinally and transversely by 100mm x 12.75mm flat stiffeners; the webs were 10mm thick and stiffened horizontally by 65mm x 10mm flat stiffeners. Transverse steel frames, 100mm x 12.75mm flat plate, were provided at each end of the box girder to minimise distortion.

2.2 Instrumentation The initial geometric imperfections of the diaphragm were measured using the method described in Ref. 6. Strain measurements were made using 144 electrical resistance strain gauges and the out-of-plane deflections of the diaphragm were obtained from five linear variable differential transformer transducers of +25 mm travel which were mounted on a separate frame at the west end of the box girder.

27

Finite element modelling of box girder diaphragms cast end

southweb

• ¢a.ri ng

"~-"

J north web

I

~"P'~'~ lq

[3= 1200

=~ldil~ensl()lls

in ram.

Fig. 1. General view of the length of box girder in the testing position.

2.3 Test procedure The box girder was inverted in the loading rig and supported as shown in Figs 1 and 2 by two steel line supports, a roller and a rocker, which in turn rested on two heavy steel beams. The load was applied by a hydraulic jack via a spreader beam to the two bearings. Initially a small load was applied to ensure that the box girder was resting properly on the supports and to check the functioning of the instrumentation. This load was then removed and a zero reading of the instrumentation taken with the box girder supporting the spreader beam only. Subsequently the strain gauges were scanned by data loggers at each increment of load and the readings output on a line printer.

2.4 Results Stress distributions in the elastic range are compared with those determined by the finite element analysis in Section 4.1 (see also figures within Section 4.1). The shear stresses were generally uniform in the outer panels ( o u t b o a r d

28

T. H. G. Megson, G. ttallak

~ :~ ~i~ ii!iiii~ ~

Fig. 2. Box girder in the testing rig. of the stiffeners)~ decreased rapidly above the bearings and were low throughout the centre panel (between the stiffeners). The horizontal direct stresses were generally low except in the vicinity of the bearings where a local increase in compressive stress occurred. This was caused by the Poisson effect of the large vertical direct stresses in this region. In other regions of the diaphragm the vertical direct stresses were low. The diaphragm failed at a load of 1290 kN per bearing although yielding commenced at a load of 760 kN per bearing in the south load-bearing

Finite element modelling of box girder diaphragms

29

stiffener. At a load of 890 kN per bearing yielding began in the north load bearing stiffener. This yielding spread along the stiffeners and into the diaphragm plate. The mode of failure was a combination of the shear buckling of the outer panels of the diaphragm and the compression and bending of the load-bearing stiffeners. The diaphragm buckled symmetrically about the centre line with both outer panels buckling to the west in a typical shear failure that forced the central panel to buckle to the east in the lower third of the diaphragm. The buckling of the diaphragm extended diagonally from the lower corners of the outer panels to a point 130mm above the bearings in the stiffeners. At this point the stiffeners buckled locally between the intermittent welds. This mode of failure is associated with the bowing across the diaphragm and the small a m o u n t of twisting in the upper parts of the stiffeners. The deflection gauges showed that the bowing began at the start of the loading. Figure 3 shows the diaphragm in its final buckled state after part of the box girder had been removed.

2.5 Imperial College diaphragm Additional experimental data were obtained from the published results of tests carried out on a full length box girder at Imperial College by Einarsson, Crisfield and Dowling. 7"8, 9

3 THEORY 3.1 ABAQUS After a preliminary investigation it was decided that the A B A Q U S finite element package 5 would be used to predict the behaviour of the stiffened diaphragms. This package is available on the VP1100 and VP1200 at the University of Manchester Computing Centre and is designed to analyse advanced structural and heat transfer problems. The ABAQUS system possesses a large element library, covers a wide range of linear, non-linear elastic and elasto-plastic materials and offers extensive and flexible loading options.

3.2 Finite element idealisation Non-conforming, eight-noded, quadrilateral shell elements (Type $8R5 in the A B A Q U S system) were used to model the diaphragm plate. This

30

T. H. G. Megson, G. Hallak

Fig. 3. Buckled diaphragm and stiffeners.

element is designed so that the shear lock problem usually encountered in thin shell elements is circumvented by employing, firstly, a reduced integration along the surface but an exact integration through the thickness and, secondly, the discrete Kirchoff assumption using the penalty method (see Ref. 5). Based on a kinematic formulation this element allows variable displacements within its boundary; these are quadratic over the surface and linear across the thickness. In addition the element has five degrees of freedom, three of translation and two in-plane rotations at most nodes. It has six degrees of freedom at any node if the node has a rotational constraint in a direction perpendicular to the plane of the element or if it is attached to a beam or shell element which has six degrees of freedom. The flange and web plates of the box girder were modelled using fournoded quadrilateral shell elements (Type $4R5 in the A B A Q U S system). To ensure continuity of displacement along the interface between the eight- and four-noded quadrilateral shell elements (Type STRI35 in the A B A Q U S system) were used along the interaction lines. The $4R5 and STRI35 shell elements have the same degrees of freedom as the $8R5 shell elements. Three-noded curved beam elements (Type B32 in the A B A Q U S system) were used to model the stiffeners on the diaphragm. This element allows a quadratic variation of displacement along its length and its formulation is

Finite element modelling of box girder diaphragms

31

based on Timoshenko beam theory where the shear deformation effect is included. The element has six degrees of freedom at each node so that continuity of displacement is satisfied at the junction nodes with the $8R5 shell element at the mid-plane of the diaphragm. The box girder flange and web stiffeners and also the end frames were modelled using two-noded straight beam elements (Type B31 in the ABAQUS system). This element has six degrees of freedom at each node and rigid links were used to attach these beam elements to the shell elements by using Code 7 of the *MPC option in the ABAQUS system (see Ref. 5). The finite element mesh is shown in Fig. 4 and was selected so that nodes coincided with the strain gauge positions on the experimental box girder. Also a finer mesh was used in the vicinity of the bearings, around the cut-outs and along the diaphragm/web and diaphragm/flange junctions where higher stress gradients were expected. The bearing plates were not modelled as thick plates but were assumed to be rigid so that their effect was imposed by prescribed displacements. The finite element model was supported in an identical manner to the actual box girder and the loads applied to the centre of each bearing (see Fig. 4). Due to the asymmetrical nature of the measured initial deflections of the actual diaphragm the complete finite element model was analysed.

+Z

_~Uz=Uy=Ux=O

Fig. 4. Finite element mesh.

32

T. H. G. Megson, G. Hallak

3.3 Load application In the actual box girder the load was applied to the bearing plates without eccentricity. Therefore, since the bearings are assumed to be rigid, all nodes within and on the boundary of the bearings must be subjected to equal vertical displacements. This condition was introduced into the finite element model using the * E Q U A T I O N option in the A B A Q U S system. Then all nodes in each bearing were subjected to the same vertical displacement as the central node, which meant that only a single concentrated load need be applied at this central node rather than a uniformly distributed load over the surface of the bearing.

3.4 Material modelling The steel plates and stiffeners of the actual box girder were modelled as an elasto-plastic material; the material properties were obtained from tests. Plasticity in the plates and stiffeners was treated using the von Mises yield criterion, since using this criterion in combination with an integration through the thickness of a shell element or through the cross-section of a beam element gives an accurate method of tracing the spread of plasticity in stiffened plate structures.

3.5 Initial deflections and type of analysis The measured initial deflections of the actual diaphragm were inserted directly into the program by defining the spatial coordinates of each node for which the initial deflection had been measured. The initial deflection of other nodes was found by linear interpolation. A non-linear static analysis was carried out in which a large deflection analysis and a Riks analysis were used in combination. The loads were applied in a series of small increments for each of which the program sought to give a convergent solution with minimum computer time. A detailed description of this procedure is given in Ref. 4.

3.6 Imperial College diaphragm The idealisation of the Imperial College diaphragm and its associated length of box girder was carried out in an identical m a n n e r to that described above except that additional stub stiffeners were attached to the diaphragm and there was a known eccentricity of loading of the bearing plates.

Finite element modelling of box girder diaphragms

33

The stub stiffeners were idealised using the three-noded curved beam elements (Type B32) which were used to idealise the full depth stiffeners. The eccentricity of loading of the bearing plates was allowed for by replacing the eccentric load by a concentrated load applied at the central node together with a moment. Again the bearing plates were assumed to be rigid so that a linear variation of vertical displacement was prescribed across the width of each plate using the *EQUATION option in the ABAQUS system (see Ref. 4 for details). Initial deflections were input in the mathematical form suggested in Ref 8, i.e. w

=0.8 sin 7t(X + 0.5 B) roY 3rc(X + 0.5 B) s i n - - sin × B D B

[

1-2 sin ~-- -t- 0.2 sin D

in which w is the initial deflection, X and Y are nodal coordinates and B and D are the breadth and depth of the diaphragm, respectively. 4 COMPARISON OF RESULTS 4.1 Experimental method

Figures 5, 6 and 7 show experimental and theoretical stress distributions in the elastic range for a section of the diaphragm immediately inboard of a bearing plate; the applied load was 300kN per bearing. It can be seen that there is good agreement between experimental and theoretical values. Stress distributions were also obtained at other sections of the diaphragm at the applied load of 300 kN per bearing; again there was good agreement between the experimental and theoretical results (see Ref. 4). Experimental and theoretical values of vertical direct stress are compared in Fig. 8 for the south load-bearing stiffener at an applied load of 300 kN per bearing; again it can be seen that the theoretical results are in reasonable agreement with the experimental values. The theoretical failure load was 1253 kN per bearing compared with the actual failure load of 1290kN per bearing, a very close agreement. The theoretical collapse mode is shown in Fig. 9 and is similar to that of the actual diaphragm except that local buckling of the load-bearing stiffeners did not occur. As a consequence the shear buckling of the outer panels of the diaphragm did not develop as fully as it did in the experimental model. This discrepancy could be due to the fact that the actual stiffener/ diaphragm welds were intermittent, whereas in the finite element model

T. H. G. Megson, G. Hallak

34

I Finite element I •

Experimental.

/

Boo

/

400 ~

200 o

/

J

Y,

-80 -60 -~0 -2.,O Vertical stress (N/mm")

-100

E

600

0

0

Fig. 5. Vertical direct stress distribution in d i a p h r a g m at a load of 300 kN per bearing.

I FiniteI etement I

--

BOO

f/ ~

• Experimental.

E

600

•

._m J~

200

"6 ll

c~

_ ____._ ___..__~ - - . a t "

-50

-40

-30 -20

-10

0 1(1

0

Horizontal stress (N/mm 21 Fig. 6. H o r i z o n t a l direct stress distribution in d i a p h r a g m at a load o f 300 kN per bearing.

I - A

I

A

Finite e l e m e n t ' Experimental

8oo E GO0 ~_ .~_ -o /.,UO t--

J ~

A

J

-60

"6 200 _c

-40 -20 0 Shear stress (N/mm 2)

0 2O

Fig. 7. Shear stress distribution in d i a p h r a g m at a load of 300 kN per bearing.

they were assumed to be continuous. Thus no gaps were present to allow such buckling to develop. An attempt was made to model the intermittent welding by allowing relative movement between the stiffener nodes and the corresponding diaphragm nodes in the regions between the welds.

Finite element modelling of box girder diaphragms

]

I11 A

/. /

/. /

800

/

Finite eternent Exper mental

/

35

/.

7o0 -~

/

6OO

/

t-.

soo

/

400 o

J

J

._1

117o

J -111

-100

E

-90 -80 -70 -60 -50 -40 -30 -20 Vertical stresses in eastern outstand (N/ram 2)

0 -10

Fig. 8. Direct stress distribution in a stiffener.

However, the theoretical results were unchanged. Possibly full modelling of the welds is required using plane strain, triangular shell or brick elements. In the theoretical analysis plasticity commenced in the diaphragm above the outer corners of the bearings at a load of 853 kN per bearing. In the experimental model the same yielding occurred at a load of 880 kN per bearing. At 900 kN per bearing compressive yielding occurred near the base of each load-bearing stiffener; experimentally this occurred at 760 kN per bearing. As the load increased plasticity spread to cover most of the bottom part of the outer panels and the areas above the bearings including the load bearing stiffeners; experimentally a similar pattern of yielding occurred.

4.2 Imperial College diaphragm A full account of the theoretical analysis of the Imperial College diaphragm is given in Ref. 4. Results were obtained in the elastic range and also at failure; these were compared with the experimental values for both elastic and plastic conditions; again good agreement was obtained between experimental and theoretical results. 5 CONCLUSIONS It has been shown that the ABAQUS finite element package is capable of accurately predicting the behaviour of a stiffened diaphragm in a length of

36

T. H. G. Megson, G. Hallak

W~

phragrn

Flange

Fig. 9. Predicted failure mode of diaphragm. box girder, both in the elastic range and in the plastic range, up to failure. This package will therefore be subsequently used in a parameter study to determine the o p t i m u m size and arrangement of stiffeners in load-bearing diaphragms.

REFERENCES 1. 2.

3. 4.

Crisfield, M. A., Large-deflection, elasto-plastic buckling analysis of plates using finite elements. Department of the Environment, TRRL Report LR593, Crowthorne, 1973. Crisfield, M. A. & Puthli, R. S., A finite element method applied to the collapse of stiffened box girder diaphragms. In Steel Plated Structures. An International Symposium, ed. P. Dowling, J. E. Harding & P. A. Frieze. Crosby Lockwood Staples, London, 1977. Puthli, R. S., Collapse analysis of thin-walled structures. PhD Thesis, University of Surrey, 1977. Hallak, G., Optimum design of box girder diaphragms. PhD Thesis, University of Leeds, 1991.

Finite element modelling of box girder diaphragms

5. 6. 7. 8. 9.

37

Hibbitt, Karlsson and Sorenson Inc., A B A Q U S Theory Manual and User's Manual. Manchester Computing Centre, September, 1989. Megson, T. H. G. & Hallak, G., Measurement of the geometric initial imperfections in diaphragms. Thin- Walled Structures, 14 (1992) 381-94. Einarsson, B. & Dowling, P. J., Steel box girders: Tests on simply stiffened rectangular diaphragms - - Model 1. Imperial College, London. CESLIC Report BG54, February, 1979. Einarsson, B., Ultimate load behaviour of rectangular steel box girder diaphragms. PhD Thesis, Imperial College, London, 1983. Einarsson, B., Crisfield, M. A. & Dowling, P. J., Collapse of stiffened box girder diaphragms. Journal of Constructional Steel Research, 2(3) (1982).