International Journal of Civil Structural 6 Environmental And Infrastructure Engineering Research Vol.1, Issue.1 (2011) 1-15 TJPRC Pvt. Ltd.,. INFLUENCE OF FLANGE STIFFNESS ON DUCTILITY BEHAVIOUR OF PLATE GIRDER K. Baskar Associate Professor Department of Civil Engineering, National Institute of Technology, Tiruchirappalli 620 015, Tamilnadu, India ABSTRACT The ultimate strength of plate girders designed using tensional field theory is assumed to depend upon critical buckling strength, post buckling strength of web panel and yield strength of flanges. Though the load carrying mechanism depends on the above three contributions, the post yield failure of the girder is primarily governed by the flange stiffness. Experimental and analytical studies on plate girders by various researchers show that the flange parameter influences the post yield behaviour of girders significantly. In the present numerical study, a 3-D finite element model developed using ANSYS was employed to analyze plate girders in order to investigate further the influence of flange stiffness on the behaviour of plate girders. It was observed from the results that the girders with larger M p /M ratio provide more ductile compared to the girders having lesser M p /M thus confirming the influence of flange parameter on ductility behaviour of plate girders. Also, it was noted that the girders with larger d/t ratio provide more ductility compared to the girders with smaller d/t ratios. The paper presents the results obtained from the finite element analyses on girders having different values of flange stiffness. 1
K. Baskar and Chitra Suresh Keywords: Plate girder, Flange, Buckling, Tension field, Stiffness, Post buckling. 1. INTRODUCTION Rockey and Skaloud (1968, 1972) showed that for plate girders having proportions similar to those employed in civil engineering, the ultimate load capacity is greatly influenced by the flexural rigidity of the flanges. They showed that the collapse mode of the plate girders involve development of plastic hinges in tension and compression flanges. Rockey and Skaloud conducted ultimate load tests on three series of plate girders in each of which only the size of the flanges, and therefore their flexural rigidity, was varied. The position of the internal hinges was found to vary with flange stiffness, the value c, which defines the position of plastic hinge, increasing from near zero in the case of flexible flanges to approximately 0.5b when the flanges are strong. It is also proved experimentally that it is possible to increase the ultimate shear strength of the web to the extent of 60% by simply increasing the flexural rigidity of the flange. Porter et al (1975) presented an equilibrium solution to calculate the ultimate strength of plate girders. It is assumed that the web panels are simply supported along its boundaries; this assumption obviously leads to lower limit. Fujii (1968), Chern and Ostapenko (1969) and Komatsu (1971), on the other hand assumed that the flanges provide a fully clamped condition and the vertical stiffeners providing a simple edge support. The correct buckling solution lies somewhere between these two extreme solutions. Porter et al, also assumed that the effect of bending stresses on the shear buckling stress of the web and the variation of over the web panel could be ignored. Based on their experimental observation and equilibrium approach, Porter et al (Ref) proposed the expression shown in Eqn. 1 to calculate the ultimate shear carrying capacity of steel plate girders. 2
Influence Of Flange Stiffness On Ductility Behaviour Of Plate Girder V s 4M y pf cr dt 2 = τ + σ t t sin θ ( d cot θ b + c) + (1) c This expression has been shown as more effective and further reduced to extreme cases such as girders with weak flanges, strong flanges and thick web. In the case of weak flanges, the value of flange strength becomes very small and it is neglected from the equation. When the flanges are very strong, it was assumed that the distance of the plastic hinge, c, away from the end of the panel increases and becomes equal to the panel width, b, the hinges formed at four corners of the panel to form a picture frame mechanism. The tension field was assumed to act at an angle of 45 o and the value of c was considered as equal to b to obtain the limiting value of shear capacity. In the case of a girder with thick web, they assumed that the web may yield before it buckles so that no tension field action would develop. The theory proposed by Porter et al has been validated by checking with experimental results reported by a number of researchers. The development of tension field and collapse mechanism of shear panels isolated from plate girders were studied using the finite element method by Kuranishi et al (1988). Special attention was paid to the influence of the rigidity of flanges and the boundary conditions of web panel. It was found that no plastic hinge appeared in flanges ever in the ultimate state, and that a collapse mechanism was formed when the yielded zones propagate completely in the diagonal direction of the panel. Kuranishi et al (1988) considered four different panel aspect ratios viz. 0.5, 0.75, 1.0 and 1.5 and three different web slenderness ratios such as 152, 180 and 250 in their numerical study. Shanmugam and Baskar (2003), vice-verse, have carried out experimental and numerical investigation on steel-concrete composite plate girders. Two bare steel plate girders and ten steel-concrete composite plate girders with two different d/t ratios of web plate and varying Mp/M ratio of flanges were considered in the investigation. It was noticed from these experimental and 3
K. Baskar and Chitra Suresh numerical studies that the stiffness of the flanges greatly influences the post yield behaviour of the plate girders. Further numerical investigation is made in this present study using four different bare steel plate girders with d/t ratios of 250, 150, 125 and 94 which are named as SPG1, SPG2, SPG3 and SPG4, respectively. A constant panel aspect ratio 1.5 is considered for all the girders. The previously tested girders (Baskar & Shanmugam, 2003), SPG1 and SPG2 are considered as reference girders and the numerical investigation is made by varying the flange stiffness over the girders. SPG3 and SPG4 are subjected to only numerical study and the results obtained from the studies are presented herein. 2. DETAILS OF THE PLATE GIRDER The girders were designed using tension field theory in accordance with BS5950: Part 1: 1990. When d/t ratio is less than 63ε (ε = (275/ρ y ) 0.5 ), the girder has to be considered as a beam in which no tension field effect can be considered; when the ratio exceeds the above value the effect of tension field action can be included and designed as plate girder. In normal practice plate girders are designed with a d/t ratio ranging from 120 to 160 and BS5950: Part1 allows up to a maximum value of 250. It is also noted (Evans and Moussef, 1988) that the contribution from the post buckling reserve strength of the web plate increases with increasing d/t ratio. In view of the above factors, Shanmugam and Baskar (2003) considered two different d/t ratios viz. 250 and 150 in order to study the behaviour of plate girders. From a practical consideration of welding and availability of minimum thickness of plates, a 3mm thick plate was chosen for the web for girders with a d/t ratio of 250 (SPG1), and a 5mm thick plate for girders with a d/t ratio 150 (SPG2). The panel aspect ratio of the web was restricted to 1.5 in all girders. SPG3 and SPG4 are provided with a web thickness of 6mm and 8mm, respectively, such that their d/t ratios are 125 and 94. The d/t ratios of all the four girders are more than 63ε and therefore, the tension field theory can be applied. 4
Influence Of Flange Stiffness On Ductility Behaviour Of Plate Girder Proportioning the flange dimension is critical in the case of girders subjected to tension field action. The shear carrying capacity is calculated from the three different contributions such as critical buckling strength of web, post buckling reserve strength of web panel and the yielding of flanges. At the beginning, the flange sections are designed by beam theory; but such a calculation provides a minimum flange dimension which leads to the ultimate load in perfect cases and lateral torsional mode of failure in some other cases. Therefore, the SPG1 and SPG2 were designed with larger M p /M ratio where M p is the plastic moment capacity of section provided and M is the bending moment due to the load which is calculated from the ultimate shear capacity of the girder. The minimum flange thickness required from the beam theory is shown in Table 1. In view of controlling the experimental behaviour of SPG1 and SPG2, the flange dimensions were taken as 200mm x 20mm and 260mmx20mm, respectively. These dimensions were decided based on a parametric study through FE analyses. The details of the considered girders are shown in Fig.1 and Table-1. 3. FINITE ELEMENT ANALYSIS In view of carrying out a parametric study with various flange stiffness, a three dimensional finite element (FE) model of a plate girder was developed using the multi-physics finite element software ANSYS (Version-11). Girder SPG1 was employed in developing the finite element model. The web, flanges and stiffener plates were modelled with 8-noded shell element with six degrees of freedom at each node which is referred as SHELL281 in ANSYS element library. SHELL281 is identified as well-suited for linear, large rotation, and/or large strain nonlinear applications and for analyzing thin to moderately-thick shell structures. The element also provides special features such as stress stiffening, large deflection and large strain capabilities. The elastic and inelastic properties of the girder materials were provided to represent the entire behaviour. In the elastic region the material was 5
K. Baskar and Chitra Suresh treated as linear isotropic and in the plastic region the von-mises yield criterion invoked with Multi-linear Isotropic Hardening. The first mode shape from buckling analysis was considered as the initial imperfection in the web panel. Non-linear behaviour of material and geometric non-linearity due to large deformations were considered in the analyses. The load was applied in steps with smaller increment to simulate the monotonic ramp loading of the experiment. The load vs deformation pattern was analyzed under all the critical load steps. 4. RESULTS AND DISCUSSION The girder was initially analyzed without considering the web imperfection and noticed that the webs reached to a yielding mode of failure rather than reaching the buckling mode as expected in tension field theory. Further analysis was made with assumed initial imperfection in the web panels and noticed the expected tension field action in the web panels. Modelling the initial imperfection plays a major role in achieving the post yield behaviour of the girder. Out of various available methods, the mode shape superposition was employed to simulate the web imperfection. A buckling analysis was carried out at the first stage and the deformed shape from the first mode of buckling analysis was imported as the imperfection of web panels. In the second stage, the nonlinear analysis was carried out. The mid-span deflections under various load steps were monitored and retrieved from the FE analysis. A graph was plotted between load and the midspan deflection and was compared with the experimental prediction. The FE model was able to predict the ultimate failure load and the behaviour of girder to an acceptable accuracy. The same modelling technique was adapted to model the previously tested girder SPG2 and the numerically predicted results were compared with experimental results as shown in Fig.2. From these comparisons of experimental vs numerical results of previously tested girders, SPG1 and 6
Influence Of Flange Stiffness On Ductility Behaviour Of Plate Girder SPG2, it is concluded that the FE model can predict the ultimate load and its behaviour to an acceptable accuracy and thus the FE model was validated. Further to the validation of model, the parametric study was carried out. The flange thickness alone varied keeping the other parameters constant. The girders were investigated from the minimum flange thickness which is required from the beam theory to a maximum limit of 30mm. The minimum thickness shown in Table 1 may be sufficient if the girder webs also designed by beam theory. Since, the present girder webs are designed using tension field theory, the web start buckles and further leads to lateral torsional buckling mode of failure under lower flange thicknesses. The FE model predicted a premature failure of girders without reaching the ultimate load. Therefore, further increase in flange thickness was considered. The girder SPG1 with a flange thickness of 10mm reached the desired load of 420kN and showed reduction in load carrying capacity after reaching the ultimate load. It is proved that the minimum thickness of flanges could reach the ultimate load but no ductility would be obtained. Further increase in flange thickness increased the ultimate to certain extent and the ductility to a greater extent. The uniform increase in the loaddeflection profiles can be clearly seen from Fig-3. It is observed that the yield load as well as the ultimate load increases with the flange stiffness. It can also be noted that the increase in flange stiffness widens the curve and becomes more flatten with considerable increase in the deflection range. It reveals that the ductility factor increases i.e. it undergoes large deformations without decrease in the load. The ductility factor and ultimate load of the plate girders with various flange thicknesses were calculated and listed below in Table 2. Similar behavior was observed in Girder SPG2. The minimum flange thickness required for SPG2 is 10mm as shown in Table-1. Referring to the Fig- 7
K. Baskar and Chitra Suresh 4, though the 10mm thickness flange reached the required ultimate load, it loses its load carrying capacity immediately after the yield load which indicate the lower thickness flange is not able to provide any ductile behavior. The same behavior is observed in SPG2 up to a flange thickness of 18mm. A lateral torsional buckling mode of failure was observed for the flange thicknesses varies from 10mm to 18mm. On the other hand the ductility behavior is observed only when the flanges are provided such that the M p /M ratio more than 2.5. The ductility factor increases with increase in M p /M ratio and shows a direct proportion. This can be noticed from Table-3 Results obtained from SPG3 and SPG4 indicated a different behavior from SPG1 and SPG2. The girders SPG3 and SPG4 are not reached the expected ultimate loads which were calculated through the tension field theory. A sudden decrease in load capacity is noticed after the yield load. No ductile behavior was observed for SPG3 and SPG4. These can be revealed from Fig-5 and Fig-6. No increase in ultimate load also noticed for smaller d/t girders. From the Table-2 it can be seen that the ultimate load capacity increases even up to 41.5% with the increase in flange stiffness for the girder SPG1 with a d/t ratio of 250. But Table -3 to Table -5 indicate that the increase in ultimate load w.r.t. flange thickness is decreasing with decreasing d/t ratios. 5. CONCLUSION From the present study the following conclusions are drawn. The ductility factor is less for the girders with M p /M ratio less than 2.5, and it increases drastically with increase in M p /M ratio more than 2.5. This observation is made only for girders with d/t ratios equal to or greater than 150. For girders with d/t ratio 125 and 94, no ductility behaviour is observed. Girders with larger d/t ratio (i.e for 250 and 150) are sensitive to the flange stiffness. These types of girders provide more ductile behaviour for the 8
Influence Of Flange Stiffness On Ductility Behaviour Of Plate Girder increasing flange thickness. Also, showed up to 41% increase in ultimate load for the increased flange thickness. For girders with smaller d/t ratios, not much difference is observed between M p and M. Consideration of small increase in flange thickness provides better ductility behaviour and therefore well suited for seismic regions. REFERENCES 1. Baskar. K and Shanmugam. N. E.(2003). Steel concrete composite plate girders subject to combined shear and bending. Journal of Constructional Steel Research, Volume 59, Issue 4, pp 531-557. 2. Chern, C. and Ostapenko, A. (1969). Ultimate Strength of Plate Girders under Shear. Lehigh Univ, Dept Civ. Eng. Fritz Eng Laboratory Report 328.7. 3. Evans, H.R. and Moussef, S. (1988) Design Aid for Plate Girders. Proceedings of the Institution of Civil Engineers, V.85, Pt. 2, pp.89 104. 1988. 4. Fujii, T. (1968), On an Improved theory for Dr. Basler s theory. Proc. 8th Congress, IABSE, New York, Sep.pp.477-487. 5. Komatsu, S. (1971). Ultimate Strength of Stiffened Plate Girders Subjected to Shear. Proc., Colloquium on Design of Plate and Box Girders for Ultimate Strength. IABSE, London, pp.49-65. 6. Kuranishi, S. et al. (1988). On the Tension Field Action and Collapse Mechanism of a Panel under Shear. Structural Eng. /Earthquake Eng., Japan Society of Civil Engineers, V.5, N.1, pp.183-193. 7. Porter, D.M., Rockey, K.C. and Evans, H.R. (1975). Collapse Behaviour of Plate Girders Loaded in Shear. Struct Eng, Volume 53, Issue 8, pp.313-325. 9
K. Baskar and Chitra Suresh 8. Rockey, K.C and Skaloud, M. (1972). The Ultimate Load Behaviour of Plate Girders Loaded in Shear. The Structural Engineer, 50(1), pp.29-48. 9. Rockey, K.C. and Skaloud, M. (1968). Influence of Flange Stiffness upon the Load Carrying Capacity of Webs in Shear. Final Report, Proc. 8th Congress, IABSE, New York, pp.429-439. 10. Shanmugam, N E and Baskar, K. (2003) Steel-concrete Composite Plate Girders Subject to Shear Loading. Journal of Structural Engineering, ASCE, Volume 129, Issue 9, pp. 1230-1242. Table 1: Dimensions of the Plate Girders Considered l. No Girder d/t ratio Thickness of web provided, mm Min. Flange size required, mm Actual flange size of tested girders, mm b f t f b f t f 1 SPG1 250 3 200 6.71 200 20 2 SPG2 150 5 260 9.99 260 20 3 SPG3 125 6 260 13.35 - - 4 SPG4 94 8 260 21.72 - - Table 2: Ductility Factor and Increase in Ultimate Load of SPG1 Sl.No Flange thickness, mm Mp/M ratio Deflection at My 1 10.57 5 - - 422.77 0.00 2 14 2.20 5 - - 440.47 4.20 3 16 2.51 5 - - 453.31 7.20 4 18 2.83 5 45 9.0 467.57 10.6 5 20 3.14 5 81 16.2 483.68 14.4 6 22 3.46 5 11 22.2 502.28 18.8 7 24 3.77 5 >126 >25.2 523.36 23.8 8 26 4.08 5 >126 >25.2 546.86 29.4 Deflction at Mult Ductility factor Ultimate load, kn % increase in ultimate load w.r.t first girder 10
Influence Of Flange Stiffness On Ductility Behaviour Of Plate Girder 9 28 4.40 5 >126 >25.2 572.28 35.4 10 30 4.71 5 >126 >25.2 598.40 41.5 Table 3: Ductility Factor and Increase in Ultimate Load of SPG2 Sl.No Flange thickness, mm Mp/M ratio Deflection at My Deflectionat Mult Ductility factor Ultimate load,kn % increase in ultimate load w.r.t first girder 1 10 1.13 5 - - 754.66 0.00 2 14 1.58 5 - - 766.09 1.51 3 16 1.80 5 - - 767.47 1.69 4 18 2.03 5 - - 780.68 3.41 5 20 2.25 5 34 6.8 788.79 4.45 6 22 2.48 5 59 11.8 806.31 6.67 7 24 2.70 5 88 17.6 825.04 8.99 8 26 2.93 5 >88 >17.6 846.39 11.58 9 28 3.15 5 >88 >17.6 871.38 14.53 10 30 3.38 5 >88 >17.6 900.32 17.86 Table 4: Ductility Factor and Increase in Ultimate Load of SPG3 Sl.No Flangethickness, mm Mp/M ratio Deflection at My Deflection at Mult Ductilityfactor Ultimate load, kn % increase in ultimate load w.r.t first girder 1 4 1.13 4 - - 1006.79 0.00 2 6 1.29 4 - - 1009.35 0.25 3 8 1.45 4 - - 1023.26 1.63 4 0 1.61 4 - - 1027.43 2.04 5 2 1.77 4 - - 1034.76 2.75 6 4 1.93 4 - - 1041.09 3.37 11
K. Baskar and Chitra Suresh 7 6 2.10 4 - - 1046.72 3.91 8 28 2.26 4 - - 1052.79 4.49 9 30 2.42 4 - - 1070.40 6.16 Table 5: Ductility Factor and Increase in Ultimate Load of SPG4 Sl.No Flange thickness, mm Mp/M ratio Deflection at My Deflection at Mult Ductility factor Ultimate load, kn 1 24 1.10 5 - - 1591.42 0.00 2 26 1.19 5 - - 1603.20 0.74 3 28 1.28 5 - - 1614.16 1.42 4 30 1.37 5 - - 1622.90 1.97 % increase in ultimate load w.r.t first girder Fig. 1 Variation of Shear Strength with Web Slenderness 12
Influence Of Flange Stiffness On Ductility Behaviour Of Plate Girder Fig. 2: Details of Plate Girder 450 400 350 Stress [N/mm2] 300 250 200 150 100 50 0 0 0.05 0.1 0.15 0.2 0.25 Strain Fig. 3 Typical Stress-Strain Curve for Steel SPG 1 (Exp t) SPG 1 (An sys) SPG 2 (Exp t) SPG2 (An sys) SPG3 (Ansys) SPG4 (Ansys) 1 8 0 0 1 6 0 0 1 4 0 0 1 2 0 0 N k 1 0 0 0, o a d 8 0 0 L 6 0 0 4 0 0 2 0 0 0 0 2 0 4 0 6 0 8 0 1 0 0 D eflection, m m Fig. 4: Load vs Deflection Behaviour of Girders SPG1 to SPG4 (Ansys vs Experimental results) 13
K. Baskar and Chitra Suresh Fig.5. Typical View of the Girder SPG1 at Ultimate Load [Shanmugam, N.E. and Baskar, K., 2003] Fig.6. Deformed Shape of the Girder SPG1 Predicted through FE Model 600 Load, kn 500 400 300 200 100 0 0 20 40 60 80 100 Deflection, mm 30mm 28mm 26mm 24mm 22mm 20mm 18mm 16mm 14mm 10mm Figure 7: Load-deflection Behaviour of SPG1 with Various Flange Thickness Load, kn 1000 900 800 700 600 500 400 300 200 100 0 0 20 40 60 80 100 Deflection, mm 30mm 28mm 26mm 24mm 22mm 20mm 18mm 16mm 14mm 10mm 14
Influence Of Flange Stiffness On Ductility Behaviour Of Plate Girder Figure 8: Load-deflection Behaviour of SPG2 with Various Flange Thickness 1200 Load, kn 1000 800 600 400 200 0 0 20 40 60 80 100 Deflection, mm 30mm 28mm 26mm 24mm 22mm 20mm 18mm 16mm 14mm 10mm Figure 9: Load-deflection Behaviour of SPG3 with Various Flange 2000 Thickness 1750 1500 30mm 28mm Load, kn 1250 1000 750 500 26mm 24mm 22mm 20mm 18mm 16mm 250 0 0 20 40 60 80 100 Deflection, mm 14mm 10mm 15
K. Baskar and Chitra Suresh Figure 10: Load-deflection Behaviour of SPG4 with Various Flange Thickness 16