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Short Paper Proc. of Int. Conf. on Structural and Civil Engineering 2011

Shear Behaviour of Stiffened Plate Girders Ajeesh S S1 and Sreekumar S2 1

College of Engineering, Trivandrum Email: ss_ajeesh@yahoo.co.in 2 College of Engineering, Trivandrum Email: sks101y@yahoo.com usually slender in cross section and are susceptible to buckling phenomenon. The elastic buckling stress (ôcr) of a rectangular web plate (width c, depth d and thickness tw) was given [5] as

Abstract—This paper deals with the shear behaviour of steel plate girders under varying parameters such as aspect ratio, web slenderness ratio, ratio of flange to web thickness and the position of longitudinal stiffener. The effect of these parameters on critical shear strength was compared analytically using finite element analysis. It was numerically demonstrated that the shear resistance of plate girder decreases with increase in aspect ratio and web slenderness ratio. The presence of thick flange improves the shear buckling strength of plate girder. The effective position of longitudinal stiffener is at mid height of the web panel so that maximum shear resistance can be achieved.

Where E=Young’s modulus of elasticity, µ=Poisson’s ratio and k=web shear buckling coefficient.

The web shear buckling coefficient ‘k’ is a key factor influencing the shear strength of plate girder. The boundary of web panel was assumed to be simply supported in order to find the value of shear buckling coefficient in theoretical formulations. The shear buckling coefficient for simply supported edge condition is given by

Index Terms—Girder; Shear resistance; Steel

I. INTRODUCTION Plate girder is a deep flexural member fabricated using steel plates by riveting, bolting or welding. The primary functions of the web plate in a plate girder are to maintain the relative distance between the top and bottom flanges and to resist the introduced shear forces. In most practical ranges of span lengths for which a plate girder is designed, the induced shear force is relatively low as compared with the axial forces in the flanges resulting from flexure. As a result, the thickness of the web plate is much smaller than that of the flanges. Consequently, the web panel buckles at a relatively low value of the applied shear loading. The web of plate girder is often reinforced with transverse and longitudinal stiffeners to increase their buckling strength, and the web design involves finding a combination of optimum plate thickness and stiffener spacing that renders economy in terms of material and fabrication cost. Finite element analysis of longitudinally stiffened plates subjected to shear loading was discussed in the literature [1 and 4].

Experimental and numerical investigation was conducted to study the shear response of stainless steel plate girders [2].

Nonlinear finite element analysis on three dimensional models of transversely stiffened plate girder models subjected to pure shear was reported [3]. The present analytical study is mainly concentrated on the shear resisting parameters of the plate girder namely aspect ratio of the web panel, web slenderness ratio, ratio of flange to web thickness and the position of longitudinal stiffener.

The shear stress (τ) acting on the web panel of plate girder is resisted by the principal compressive stress and tensile stress (σ) in the elastic range. This mechanism continues till shear buckling stress (τcr) of material is reached. Once buckling occurs, the web panel has no more compression capacity and it remains equal to critical shear stress. Therefore further increase in load is resisted by an increase in the principal tensile stress and the web panel behaviour is governed by a tension field due to the formation of plastic hinges (Fig. 1).

The nonlinear shear stress and normal stress interaction that takes place from the onset of elastic shear buckling to the ultimate strength state (Vu) is known as postbuckling shear strength. The ultimate failure of the girder is due to yielding of the web panel and it is governed by von Mises yield criterion. The von Mises yield stress (fvn) for pure shear condition depends on the yield stress of web panel (fyw) and it is given by

II. SHEAR RESISTANCE OF PLATE GIRDER The shear design methods of plate girder web panel was divided into two categories: (1) allowable stress design based on elastic buckling as a limiting criterion; and (2) strength design based on ultimate strength, including postbuckling shear strength as a limit state. The web panel of plate girder is © 2011 ACEE DOI: 02.SCE.2011.01. 3

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Short Paper Proc. of Int. Conf. on Structural and Civil Engineering 2011 the analysis was restricted in the elastic range. The material was assumed to be linearly elastic and isotropic. The material constants were conforming to the provisions of IS 800:2007 and is listed in Table I. TABLE I. MATERIAL PROPERTIES OF STRUCTURAL STEEL

Figure 1. Shear mechanism of web panel: (a) Unbuckled behaviour (b)Postbuckled behaviour (c) Collapse behaviour

Loading on the plate girder model was done on the web panel to compute the shear buckling resistance. Eigen buckling analysis was performed to compute the shear buckling load by applying a unit concentrated load at mid span. Then the loading was increased in several sub steps so that buckling of the plate occurs. The load required to obtain the first buckling mode was the critical shear buckling load. The buckled mode of the web panel of the girder is shown in Fig.3.

III. FINITE ELEMENT ANALYSIS Finite element modelling of transversely as well as longitudinally stiffened plate girder was done using the package ANSYS-10. For the purpose of this study, three dimensional finite element models were developed. The finite element model adopted for the present study is shown in Fig. 2. The web panels, flanges and stiffeners were modelled in the global coordinate system and these components were glued together. Separate element sizes were adopted for web, flanges and stiffeners depending upon the size of model.

The model was susceptible to out of plane deflections. So the most ideal element for modelling the girder is shell element.

The shell element in ANSYS element library used for the present study was SHELL-181. The element is suitable for analyzing thin to moderately thick shell structures. It consists of four nodes with six degrees of freedom at each node: translations along x, y, and z directions and rotations about the x, y, and z axes. The shell thickness can be specified at each of the 4 nodes. The element is well suited for linear, large rotation, and/or large strain nonlinear applications.

Figure 3 . Buckled mode of the web panel of girder

In the present study, finite element analysis of plate girder was done by varying the aspect ratio (c/d) of web panel, web slenderness ratio (d/tw), ratio of flange to web thickness (tf/ tw) and position of longitudinal stiffener (h lst/d). Here hlst is the distance of longitudinal stiffener from top flange. The critical shear buckling strength (Vcr,FEM) computed using finite element analysis was compared with theoretical prediction [5].

IV. RESULTS AND DISCUSSIONS A. Variation in shear strength with aspect ratio The plate girders were modelled by varying the aspect ratio (c/d) of web panel from 0.3 to 3. Variation in aspect ratio was obtained by varying the width (c) of the web panel.

The critical shear buckling load (Vcr,FEM) was obtained using finite element analysis and compared with theoretical values.

Beyond aspect ratio 3, the web panel is considered as unstiffened [IS 800:2007, Clause 8.6.1.1.]. So the analysis was restricted to an aspect ratio of 3.

Figure 2. Finite element model of plate girder

Simply supported end condition was assumed for the present finite element model. The bottom flange was restrained against translation along vertical direction at both ends. The edges of end stiffeners were restrained against motion along lateral direction. Also the movement along longitudinal direction was arrested for one of the support. Structural steel was used as the basic material for the analysis of plate girder and Š 2011 ACEE DOI: 02.SCE.2011.01. 3

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Short Paper Proc. of Int. Conf. on Structural and Civil Engineering 2011 buckling for web slenderness ratio greater than or equal to 1.

The percentage variation of FEM results from theoretical values was less than 5%, for slenderness ratio closer to unity.

TABLE II. COMPARISON OF SHEAR STRENGTH WITH WEB SLENDERNESS RATIO

Figure 4. Critical shear force versus aspect ratio

The comparison of critical shear buckling strength with aspect ratio is shown in Fig.4. It was observed that the shear strength of plate girder decreases with increase in aspect ratio. For aspect ratios 0.3 and 0.5, the von Mises yield strength was lower than the shear buckling load which governs the failure of girder. When the aspect ratio of web panel increased from 0.3 to 0.5, the ultimate shear force decreased about 53%. This percentage variation gradually reduced and reached about 23%, when the aspect ratio is 1.5. There was not much reduction in shear buckling load (< 5%) when the aspect ratio is greater than 1.5.

Figure 5. Maximum lateral deflection versus aspect ratio

The maximum lateral deflection of the web panel during critical buckling stage was compared based on the variation in aspect ratio as shown in Fig. 5. As the aspect ratio increases, the maximum lateral deflection also increases linearly upto an aspect ratio 1. The maximum lateral deflection remains almost constant for c/d greater than 1.5.

B. Effect of web slenderness ratio on shear strength The variation in web slenderness ratio (ëw= d/tw) was obtained by varying the thickness of the web panel (tw) from 3mm upto 12mm. Web slenderness was varied for aspect ratios ranging from 0.3 upto 3. The ultimate shear buckling load obtained during finite element analysis was compared with theoretical shear buckling load. Also a comparison of von Mises yield strength (Vvn) was made. The variation in shear buckling load with web slenderness ratio is presented in Table II. The shear buckling load decreases with increase in web slenderness ratio in both finite element analysis and theoretical computation. The failure of the web panel was due to shear © 2011 ACEE DOI: 02.SCE.2011.01. 3

The maximum vertical deflection of girder at critical buckling stage was compared for various values of aspect ratio by maintaining the thickness of the web panel constant (tw=12 mm) as shown in Fig.6. The maximum vertical deflection of the girder increases with increase in aspect ratio of web panel.

This phenomenon was observed for lower values of web slenderness ratio (denoted by ‘*’ in Table II). Also it was observed that the maximum vertical deflection was on the flanges. Hence it is clear that flanges were subjected to instabilities such as local buckling and torsional buckling for lower values of web slenderness and for high aspect ratio.

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Short Paper Proc. of Int. Conf. on Structural and Civil Engineering 2011

Figure 8. Critical shear strength versus flange to web thickness

The comparison of maximum lateral deflection of girder with tf/tw is shown in Fig. 9. As the ratio of flange to web thickness increases, the maximum lateral deflection of web panel increases. When tf/tw increased from 1 to 2, the increase in maximum lateral deflection of the web panel was about 16%.

When the ratio of flange to web thickness is greater than 8, the variation in maximum lateral deflection was less than 1%.

Figure 6. Maximum vertical deflection versus aspect ratio

Figure 7. Maximum lateral deflection versus slenderness ratio

The comparison of maximum lateral deflection of the girder for variation in slenderness ratio is shown in Fig.7. For lower values of aspect ratios (c/d<1.5), the maximum lateral deflection remains almost constant with increase in slenderness. For c/deâ&#x20AC;?1.5, the maximum lateral deflection decreases and then remains almost constant with increase in web slenderness ratio. This lower bound value of maximum lateral deflection for lower values of slenderness ratio was due to the torsional buckling of web panel in addition to flange buckling.

Figure 9. Maximum lateral deflection versus flange/web thickness

D. Effect of position of longitudinal stiffener on shear strength By maintaining the aspect ratio 1 and by maintaining a constant web thickness (4mm), a longitudinal stiffener was introduced in the web panel of the plate girder in addition to transverse stiffener. The thickness of the longitudinal stiffener was maintained as 20 mm. The ratio of position of longitudinal stiffener to the depth of the web panel (hlst/d) was varied from 0.2 to 0.6 in order to study the variation in shear strength.

The effect of the position of longitudinal stiffener on critical shear buckling strength is shown in Fig. 10.

C. Variation in shear strength with ratio of flange to web thickness By maintaining the aspect ratio 1 and by maintaining the thickness of web panel as 4 mm, the ratio of flange to web thickness (tf/ tw ) was varied from 1 to 10 by varying the thickness of flange (tf). The comparison of critical shear strength with variation in ratio of flange to web thickness is shown in Fig. 8. The shear strength of the plate girder increases with increase in ratio of flange to web thickness.

The critical shear strength of girder increases by 55%, when the ratio of flange to web thickness was varied from 1 to 2.

The percentage variation in shear strength was less than 5% for tf/ tw greater than 3. The increase in shear strength was due to the stiffening of the boundary of web and flange juncture towards fixed end condition from simply supported edge condition.

Figure 10. Plot of critical shear force versus position of longitudinal stiffener

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Short Paper Proc. of Int. Conf. on Structural and Civil Engineering 2011 CONCLUSIONS The various conclusions obtained from the present analysis of plate girder model are summarised below  Shear resistance of the plate girder decreases with increase in aspect ratio of the web panel. The aspect ratio should be greater than 0.75 in order to ensure shear buckling of the web panel. When c/de”1.5, the plate girder is susceptible to torsional buckling.  Shear strength of the plate girder decreases with increase in web slenderness ratio. It is desirable to maintain the web slenderness ratio greater than 1 in order to ensure shear buckling and to avoid flange buckling and torsional buckling.  The presence of thicker flanges and longitudinal stiffener improve the shear buckling strength of plate girder.

The effective position of longitudinal stiffener is at mid height of web panel (hlst/d=0.5) for maximum shear strength and to avoid instability.

Figure 11. Maximum lateral deflection versus position of longitudinal stiffener

The presence of longitudinal stiffener in addition to transverse stiffener enhanced the critical shear strength of plate girder.

As hlst/d increases, the shear strength of the plate girder increases. When hlst/d is greater than 0.2, the buckling of the web panel was observed in the end panel rather than the loaded panel. The graph of maximum lateral deflection of model for various position of longitudinal stiffener is shown in Fig. 11. There was an increase of about 400% for maximum lateral deflection, when hlst/d is varied from 0.6 to 0.8 In this case, maximum lateral deflection was observed in the flanges and which will affect the stability of the model. Hence the effective position of longitudinal stiffener is at the mid height of web panel (hlst/d=0.5), considering the stability aspects.

© 2011 ACEE DOI: 02.SCE.2011.01. 3

REFERENCES [1] M. M. Alinia and S. H. Moosavi, “Stability of Longitudinally Stiffened Web Plates under Interactive Shear and Bending Forces,” J. Thin Walled Structures, Vol. 47, pp. 53-60, 2009.

[2] I. Estrada, E. Real, and E. Mirambell, “General Behavior and Effect of Rigid and Non-rigid End Post in Stainless Steel Plate Girders Loaded in Shear. Part I: Experimental Study,” J. Constructional Steel Research, Vol. 63, pp. 970–984, 2007.

[3] S. C. Lee and C. H. Yoo, “Strength of Plate Girder Web Panels under Pure Shear,” J. Structural Engineering, Vol. 124, pp. 184– 194, 1998.

[4] L. Pavlovcic, D. Beg, and U. Kuhlmann, “Shear Resistance of Longitudinally Stiffened Panels Part 2: Numerical Parametric Study,” J. Constructional Steel Research, Vol. 63, pp. 351-364, 2007.

[5] S. P. Timoshenko and J. M. Gere, Theory of elastic stability, McGraw-Hill Book Co, Newyork, 1961.

[6] IS: 800-2007, Code of practice for general construction in steel, BIS, New Delhi-110002.

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