1 122 CHAPTER 5 PROPOSED WARPING CONSTANT 5.1 INTRODUCTION Generally, lateral torsional buckling is a major design aspect of flexure members composed of thin-walled sections. When a thin walled section is subjected to flexure about its strong axis with insufficient lateral bracing, out- of plane bending and twisting can occur as the applied load approaches its critical value. At this critical value, lateral torsional buckling occurs. The equations used to calculate the critical lateral-torsional buckling strength of the I-girder with flat webs would underestimate the capacity of the I-girder with corrugated web Linder (1990) proposed an empirical formula for the warping constant of I-girder with corrugated web on the basis of test results. The warping constant of I-girder with corrugated web is larger than that of I-girder with flat web. Sectional warping constant C w is determined either by mathematical integration proposed by Galambos (1968) or by complex formulas. Computation of warping constant for open thin walled section is greatly simplified by recognizing the linear variation of unit warping constant properties (w, w 0, W n, Figure 5.4) between the two consecutive intersection points of plate elements. As a result, the sophisticated integral form for C w is represented by numerical expression suitable for computer coding.
2 123 The cross-section of the beam varies along the span of the beam due to the corrugation profile. It is difficult to find the warping constant for varying depths of corrugation. No design rules are available in Australian/ New Zealand Standards (AS/NZS-4600:2005) and North American Specifications (AISI-S100:2007) for calculating moment carrying capacity of I-section with trapezoidal corrugated web. In order to find lateral buckling moment capacity, elastic buckling stress for flexural buckling about the y-axis f oy and elastic buckling stress for torsional buckling f oz have to be calculated. It is found that properties such as polar radius of gyration of the cross section about the centre, radius of gyration, Section modulus, Warping constant etc varies along the longitudinal direction due to a change in depth of in change of properties. Due to change in depth y-axis also changes along the length and the geometric properties are not constant at all sections. In this chapter, the procedure to find the warping constant for lipped I-beam with trapezoidal corrugated web and to locate the shear centre have been proposed. The proposed warping constant is also validated by using finite element analysis. 5.2 SHEAR MODULUS & TORSIONAL RIGIDITIES OF LIPPED I-BEAM WITH CORRUGATED WEB Generally, the shear modulus of the corrugated plates is smaller than that of the flat plates. The shear modulus of the corrugated plates used in this study was proposed by Samanta &Mukhopadhyay (1999). The shear modulus G cog of the corrugated plate is defined as
3 124 a b Gcog G G a c (5.1) where G is the shear modulus of the flat plates and is the ratio of the projected length (a+b) to the actual length of the corrugated plates (a+c). Pure torsional constant J cog of the lipped I-beam with corrugated web is the same as that of the lipped I-beam with flat webs. Because the pure torsional constant of a section is equal to the sum of the pure torsional constants of each individual element, in the case of lipped I-beam with corrugated web, J cog can be expressed as the sum of the pure torsional constants of the two flanges, four lips and corrugated web. Therefore, (5.2) 5.3 SHEAR CENTER OF LIPPED I-BEAM WITH CORRUGATED WEB It is presumed that the shear flow is evenly distributed over the total depth of the web as shown in Figure 5.1 and it is given by q w =V/h w, where q w V shear force acting on the cross-section.
4 125 Figure 5.1 Shear flow distribution Figure 5.2 Shear flow distribution and location of shear center The shear flow can be determined by using the relationship. due to the change of the bending normal stress q = - (V*Q x )/(I xx ) (5.3) where x is the first moment of the area about the x-axis. From Figure 5.2, it is found that the unbalanced shear force on the flange V f is generated due to the corrugation depth. The magnitude of f can be determined by the sum of the shear flows acting on the flanges and expressed as V f = (V/h w )*d = q w *d (5.4) f is proportional to the corrugated depth d. For lipped I-beam with flat web f is equal to zero. Figure 5.2 shows the shear force acting on the cross-section of lipped I-beam with corrugated web. The location of the shear center of this cross-section is determined by the moment equilibrium. There is no twisting of the cross-section, when the applied load passes through the
5 126 shear center. Therefore, the suation of the moment about shear centre is equal to zero. The location of shear centre is obtained as X o = - d. It is found that this shear center is located at a distance of 2d from, the center of upper and lower flange (Figure 5.2). 5.4 WARPING CONSTANT OF LIPPED I-BEAM WITH CORRUGATED WEB The warping constant is determined either by integration forms or by numerical forms. The open section is made up of thin plate elements. Warping constant is determined by numerical forms, considering the section is composed of a series of inter-connected plate elements. If a plate element of length L ij and thickness t ij is considered, then the normalized unit warping at points i and j of any element (i-j) is given by W ni =[( ) (w oi +w oj )t ij L ij ]-w oi (5.6a) W nj =[( ) (w oi +w oj )t ij L ij ]-w oj (5.6b) w oj =w oi oij * L ij (5.6c) where, oij is the distance between the shear center to the tangent of element ij (Figure 5.3), w o is the unit warping with respect to shear center and w oi andw oj are thecorresponding values of w o at the ends of element i and j (Figure 5.4)
6 127 Figure 5.3 Coordinates and tangential Figure 5.4 Distribution of Won distances element in an a plate open thin-walled section Figure 5.5 Direction for path for calculating warping constant of lipped I-beam with corrugated web Figure 5.5 shows the direction of the path for calculating the warping constant of the lipped I-beam with corrugated web. Using the
7 128 equation (5.6) and calculating the path as shown in Figure 5.5, the simplified form W ni of the lipped I-beam with corrugatedweb can be expressed as (5.7a) (5.7b) (5.7c) (5.7d) (5.7e) (5.7f) (5.7g) (5.7h) (5.7i) (5.7j) The general formula of C w of any arbitrary section composed of thin plate is given by (5.8) C w,cog is obtained by using equation (5.8) and W ni as described in equation (5.7). It is found that W ni varies along the longitudinal direction due to a change in d, which results in a change in C w,cog.the average corrugation depth d avg suggested in this study for considering the change in d. d avg is given by
8 129 d avg =[(2a+b)*d max ] / [2*(a+b)] (5.9) Procedure for calculating C w,cog is as follows a) Calculation of average corrugation depth d avg byusing equation (5.9) b) Evaluation of the normalized unit warping at point i and W ni of the lipped I-beam with corrugated webs by using equation (5.7 ) with d avg in Step (a) c) Determination of the warping constant of the lipped I-beam with corrugated webs C w, cog by using equation (5.8) with W ni obtained in Step (b) 5.5 LATERAL-TORSIONAL BUCKLING STRENGTH OF LIPPED I-BEAM WITH CORRUGATED WEB If a uniformly distributed load or any other transverse load acts on the I-beam, shear force induced is taken up by web. In the case of the lipped I- beam with corrugated web, the attachment of corrugated web to the flanges is not along the center line of the beam. It is attached eccentrically, as the profile of web varies along the span. Due to this, force derived from shear in the corrugated web causes out-of-plane transverse bending of the upper and lower flanges (Figure 5.6). In this study, uniform bending is adopted to investigate the lateraltorsional buckling strength. The boundary condition used in this study is simple support in flexure and torsion. It is assumed that the formula of the lateral-torsional buckling strength of the lipped I-beam with flat web can be applied to lipped I-beam with corrugated webs under uniform bending. The beam under uniform
9 130 bending deflects in-plane without any torsional behavior, even if the shear center and the center of beam do not coincide. Figure 5.6 Deformed shape of I- beam with corrugated webs under uniformly distributed loadmoon J et al. (2009) Figure 5.7 Deformed shape of I- beam with corrugated webs under uniform bendingmoon J et al. (2009) The elastic lateral torsional buckling strength of the beam is expressed as 2 Mcr ( EI ygcog Jcog )( I W T ) (5.10a) 4 WT ( ECw, cog / Gcog Jcog ) L (5.10b) where L is the length of the beam and W T represents the effect of the warping torsional stiffness. 5.6 VERIFICATION OF THE PROPOSED EQUATIONS
10 131 In this section, the proposed warping constant of the lipped I-beam with corrugated webs is verified with finite element analysis. The warping constant from the FEA C w,fem for the lipped I-beam with corrugated web is calculated by C M * L ( Gcog * Jcog L E I E 2 2 CrFEM w, FEM 2 2 ( * yy * 2 (5.11) where M cr,fem is the elastic lateral torsional buckling strength from FEA. Linder(1990) suggested the following empirical formula for the warping constant of I-section with corrugated web C wlinder = C w, flat + (C w *L 2 2 ) (5.12a) C w = [(2*d max )*h w ]/[8*u x *(a+b)] (5.12b) U x h h *( a b) *( I I ) w 2* G * a* t 600* a * E * I * I 2 3 w xx yy 2 w xx yy (5.12c) where C w,flat is the warping constant of the I-section with flat web. Equation (5.12) suggested by Linder(1990) without lip in flanges is compared with the proposed warping constant in line with Moon (2009). An Eigen-value analysis is performed by using ANSYS.12 to evaluate the lateral-torsional buckling strength of the lipped I-beam with corrugated web. Four node Shell-181 element is used. Figures 5.8, 5.9 and 5.10 show a typical loading and boundary condition of the FE model. The end moments are applied in the form of compression and tension on the top and bottom flanges, respectively. The beams are considered to be simply
11 132 supported in the flexure and torsion and the following boundary conditions are implemented. Figure 5.8 Loading and Boundary Condition of the FE Model Figure 5.9 Loading and Boundary Condition of the FE Model at the left end
12 133 Figure 5.10 Loading and Boundary Condition of the FE Model at the right end Displacements about directions x, y and z(u x, u y and u z ) and the rotation about direction Z ( z ) at point A are restrained. Displacement about directions x and y( u x, u y ) and the rotation about direction Z ( z) at point B are restrained. u x at the line a and b are restrained and u y at the line c and d are also restrained. To verify the finite element model used in this study, the lipped I-beam with flat web is modeled and the result is compared with the theoretical lateral-torsional buckling strength. The dimensions and result of the analysis are shown in Table 5.1.
13 134 Table 5.1 Dimensions and Result of FE and Theory Moment for flat web t w h w b f t f L b l t l FE Moment N 10 6 Theory Moment N 10 6 Error % An appropriate mesh size of is chosen after a mesh sensitivity analysis in order to get accurate results. Table 5.2 Dimensions of models with corrugated web Model No. a b c d max t w h w b f t f b l t l L Aspect Ratio TCIAE TCIAE TCIAE TCIAE TCIAR TCIAR TCIAR TCIAR Table 5.2 shows the dimensions of the models with corrugated web. In the TCIAE series models, the corrugation angle is varied from 15º to 60º with increase in d max keeping aspect ratio (a:c) as one. In the TCIAR series models, the aspect ratio is varied by keeping a corrugation angle of 45º with an increase of d max. Figures 5.11 and 5.12 show the lateral torsional buckling shape obtained from FEA for TCIAE4 and TCIAR2 models respectively.
14 135 Table.5.3 shows the comparison of warping constant of C w,cog, C w,linder and C w,fem. Figures 5.13 and 5.14 show the comparison of results for varying aspect ratio and corrugation angle respectively. It is found that C w,cog is in good agreement with C w FEM, while C w,linder generally overestimates the warping constant of the lipped I-beam with corrugated webs. The difference between C w FEM and C w,linder (Figure 5.14) and also the difference between C w FEM and C w,linder increases with increasing aspect ratio(figure 5.13). Table 5.3 Comparison of warping constant of proposed method, FEA and Linder Model No. C w,cog C w,linder C w,fem C w,fem /C w,cog C w,fem /C w,linder TCIAE TCIAE TCIAE TCIAE TCIAR TCIAR TCIAR TCIAR Mean Standard Deviation
15 136 Figure 5.11 Lateral-torsional buckling shape from finite element analyses for TCIAE4 Figure 5.12 Lateral-torsional buckling shape from finite element analyses for TCIAR2
16 C Aspect Ratio (a:c) Cw,FEM/Cw,cog Cw,FEM/Cw,linder Figure 5.13 Comparison of the warping constant of corrugated web with respect to aspect ratio C w,fem /C w,cog or C w,fem /C w,linder Corrugation Angle Cw,FEM/Cw,cog Cw,FEM/Cw,linder Figure 5 14 Comparison of the warping constant of corrugated web with respect to corrugation angle
17 138 Linder(1990) proposed an equation for warping constant under a transverse loading. Lateral displacement occurs along the flange locations of the corrugated web lipped I-beam under such transverse loading, because the shear forces in the corrugated web in inclined panel should be divided into a lateral component in the out-of-plane direction due to the corrugated shape. (1990) work includes the effect of lateral displacement component. Therefore, Linder(1990) results do not match with proposed warping constant result under uniform bending condition. 5.7 CONCLUSION A simplified method for estimating the warping constant of the lipped I-beam with corrugated web is evaluated. In this study, the depth of corrugation varies along the span of the beam for which the average depth is taken for calculating the warping constant. upper and lower flanges. from the middle of the Numerical validation has been carried out to verify the appropriateness of the warping constant proposed in this study by using the FEA software ANSYS. 12 and it is found that the numerical results are quite closer to the proposed method.