Congrès annuel de la Société canadienne de génie civil Annual Conference of the Canadian Societ for Civil Engineering oncton, Nouveau-Brunswick, Canada 4-7 juin 2003 / June 4-7, 2003 LATERAL STABILITY OF LATE GIRDERS WITH CORRUGATED STEEL WEBS Ezzeldin Yazeed Saed-Ahmed Associate rofessor, Ain Shams Universit, Facult of Engineering, Structural Engineering Department, Cairo, Egpt (On leave to Universit of Qatar, Civil Engineering Department, Doha, Qatar) ABSTRACT: late girders with corrugated steel webs have recentl been used in different structural applications. The girder s flanges provide the girder s flexural capacit with no contribution from its corrugated web which provides the girder s shear capacit. Lateral torsion-flexure buckling of steel girders with plane webs subject to flexure is a ver important design aspect. Codes of practice allow designers to use the critical moment of a simpl supported beam subject to a constant moment and relate it to the critical moment of an other loading case using an equivalent moment factor. Lateral buckling of plate girders with corrugated steel webs still needs to be investigated. In this paper, a numerical analsis based on the finite element method is performed on these girders. The critical moment causing lateral buckling is determined numericall for corrugated web girders and compared to the critical moment of plate girders with plane webs. The model is also used to investigate the applicabilit of the equivalent moment factor concept to plate girders with corrugated webs and to determine the values of this factor for different loading configuration. 1. INTRODUCTION Corrugated steel webs were recentl proposed to replace the stiffened steel plates of plate girders to improve both the aesthetics and the econom of the structure (Elgaal et al. 1996; Cherez and Combault 1990; Lebon 1998; Saed-Ahmed 2001). The practical web height to thickness ratio of a corrugated steel web ranges between 150 and 260 for beams in buildings and reaches 450 for bridge girders. The most commonl used corrugation profile for corrugated web plates is the trapezoidal one for which the geometric characteristics are shown in Figure 1. Flexural strength of a steel plate girder with a corrugated web plate is provided b the flanges with almost no contribution from the web and with no interaction between flexure and shear behaviour (Elgaal et al. 1997; Johnson and Cafolla 1997; Leiva-Aravena 1987; Luo and Edlund 1994). The corrugated steel web solel provides the shear capacit of the girders where the shear strength is controlled b buckling and/or steel ielding of the web (Elgaal et al. 1996; Saed-Ahmed 2001; Elgaal et al. 1997; Johnson and Cafolla 1997; Leiva-Aravena 1987; Luo and Edlund 1994). The flanges provide boundar supports for the web which lie somewhere between a simpl supported boundar and a clamped one. Generall, lateral torsion-flexure buckling is a major design aspect for flexure members composed of thinwalled I-sections subject to flexure. When a slender steel I-beam is subjected to flexure about its axis of greatest flexural rigidit with insufficient lateral bracing, out-of-plane bending and twisting ma occur when the applied load reaches its critical value (Figure 2). At this critical value of the load, in-plane bending GCF-231-1
deformation ceases to be a stable configuration for the beam and lateral buckling takes place (Chen and Lui 1987; Galambos 1998). The critical load at which lateral buckling initiated depends on the crosssection mechanical properties, the laterall unbraced length of the beam, the support conditions, the tpe of loading, and the location of the load with respect to the shear centre of the cross-section (Figure 3). c s h w a b d α h c 2 (b + d) s 2 (b + d / cos α) h d tan α β a/b Figure 1. Geometric characteristics of the trapezoidal profile of corrugated web plates. Lateral torsion-flexure buckling has been explicitl investigated for steel I-girders with plane webs. For design purposes of such girders, codes of practice (e.g. AISC 2000; CISC 2000) allow designers to evaluate the critical moment causing lateral buckling for a simpl supported beam subjected to uniform bending and modif it using an equivalent moment factor which depends on the tpe of loading. x o C.L. o z C.L. x w z γ Figure 2. Lateral buckling of a beam subject to constant bending about the z-z axis. GCF-231-2
w Shear centre w Shear centre h h Load acts above the shear centre Load acts below the shear centre γ γ Figure 3. Location of the load with respect to the shear centre of the section On the contrar, lateral buckling of steel girders with corrugated webs still needs to be investigated. Will the same concept used for steel girders with plane webs still be applicable to steel girders with corrugated webs? A question which et to be answered. In this paper, a numerical analsis based on the finite element method is adopted to determine the critical moment which causes lateral buckling for steel girders with corrugated webs and compare it to the critical moment of plate girders with plane webs. Girders subjected to constant moments are first scrutinized using this model. Then, the analsis is extended to investigate the values of the critical moments for steel girders with corrugated webs which are subjected to end moments causing moment gradient through the girder s span. The validit of the equivalent load factor concept established earlier for girders with traditional plane webs to handle the lateral buckling is verified for girders with corrugated webs using the numerical analsis results. 2. LATERAL BUCKLING OF TRADITION LATE GIRDER WITH LANE WEBS The critical moment causing lateral torsion-flexure buckling to initiate for a simpl supported I-beam subjected to two equal and opposite end moments is given b (Chen and Lui 1987; Galambos 1998): [1] ocr π L EI GJ (1 + W 2 R ) W R π L EC GJ w where E is the Young s modulus, G is the elastic shear modulus, and L is the beam span. I, J and C w are the second moment of area about the weak axis of inertia, the torsional constant, and the warping constant of the beam s cross-section respectivel. In Equation 1, W R represents the warping restraint contribution to the girder s resistance. Equation 1 neglects the effect of in-plane bending deformation on the lateral stabilit of the girder which increase the critical moment to (Allen and Bulson 1989): [2] ocr L π I r EI GJ (1 + W 2 R ) I r I 1 I z However, for a steel girder composed of thin-walled I-section, the ratio between I and I x is small and thus, the amount of increase in ocr due to in-plane bending deformation is commonl ignored. For beams subjected to other tpe of loading, the effect of the moment gradient on the critical moment is accounted for b using of an equivalent moment factor C b (Salvadori 1955). This concept has been adopted in design b most codes of practice (e.g. AISC 2000, CISC 2000). Thus, the critical moment equation is generalized to be: GCF-231-3
[3] ocr Cbπ L EI GJ ( 1 + W 2 R ) W R π L EC GJ w For girders subjected to unequal end moments ( A and B ), the equivalent moment factor C b ma be given b (Chen and Lui 1987; Galambos 1989): A A 2 [4] Cb 1.75 + 1.05 + 0.3( ) 2. 3 B B where A is the smaller end moment and the ratio A / B is positive for beams bent in double curvature and negative for beams bent in single curvature. CAN/CSA-S16.1-94 adopted Equation 4 in design with a change in its limiting value from 2.3 to 2.5 (CISC 2000). The same specifications also use C b of 1.0 when the bending moment between the end supports becomes greater than the end moment. On the other hand, the AISC (2000) defines the following equation for the equivalent moment factor: [5] C b 3 1 12.5 + 4 + 3 2 max 3 + 2.5 max where 1, 2, 3 are the absolute values of the moments at the quarter point, midpoint and three-quarter point of the beam, respectivel and max is the maximum moment acting on the beam. Table 1 summarizes the values of the equivalent moment factor for simpl supported beams subject to different tpes of loads (Chen and Lui 1987). In all the previousl mentioned values of the equivalent moment factor, the load was assumed to act along the shear centre of the cross-section. The effect of the applied load location with respect to the shear centre of the section is ignored despite its importance. Table 1. Values of C b for different load configuration (Chen and Lui 1987). Loading Bending oment Diagram C b 1.0 1.75 2.3 L/4 1.35 L/4 L/2 L/4 wl 2 /8 L/4 1.13 1.04 GCF-231-4
3. LATERAL BUCKLING OF LATE GIRDERS WITH CORRUGATED WEBS Lateral torsion-flexure buckling of steel plate girders with corrugated steel webs are et to be investigated. In the following sections the critical moment causing lateral buckling is numericall determined using a numerical model based on the finite element method. The applicabilit of the equivalent load factor established earlier for traditional I-girders to such girders is also investigated using the numerical model. 3.1 Finite Element odel: Verification Shell elements with 8 nodes and 6 degrees of freedom per node are used to model all the corrugated web plate girders. The analzed girders are considered to be simpl supported in flexure and in torsion: at the beam s ends rotation and warping about the weak axis are free while rotation about the centroidal axis is restrained. Eigen buckling analsis using the finite element package ANSYS 5.4 is performed to evaluate the critical moment which initiates lateral torsion-flexure buckling. The model is first verified b analzing plate girders with traditional plane webs which is subjected to end moments and compare the results to the theoreticall predicted moments (Equations 1 and 3). The girders span is 12.0 m and their cross-section is composed of two flange plates 229x19.6 and a web plate 573x11.9 (Figure 4): this cross-section approximatel matches the dimensions of a W610x125 hot rolled section (CISC 2000). The mechanical properties of the cross-section considered in the analsis are listed in Table 2. The dimensions and the span of the analzed girders are chosen such that the critical moment due to lateral torsion-flexure buckling will be reached before ielding initiates in an part of the girder s cross-section: based on F 300 a, the ield moment of the considered section is 1006.5 kn.m and the plastic moment p is 1110.6 kn.m (Figure 4). F 300 a - F 300 a - 229x19.6 612 572.8x11.9 + F 300 a p 1110.6 kn.m + F 300 a 1006.5 kn.m 229x19.6 Figure 4. Cross-section of the analzed girders (right), and stress distribution through the section at the ield moment (centre) and at the plastic moment (left). Table 2. echanical and aterial properties of the I-section adopted in the verification of the numerical model. A I x I S ropert (10 3 ) (10 6 ) (10 6 r ) x r x S Z x Z C w J (10 3 ) (10 3 ) (10 6 ) (10 6 ) (10 9 ) (10 3 ) mm 2 mm 4 mm 4 mm mm mm 3 mm 3 mm 3 mm 3 mm 6 mm 3 Value 16.03 993.7 39.2 249 49.5 3355 343 3703 513.9 3448 1482 echanical Elastic modulus E Shear modulus G oisson s ratio Uniaxial ield strength ropert (Ga) (Ga) µ (a) Value 200 76.9 0.3 300 GCF-231-5
A tpical finite element mesh adopted in the analsis and a lateral buckling mode are shown in Figure 5. Results of the verification analsis are summarized in Table 3 and plotted in Figure 6. These results reveal a good match between the theoreticall predicted behaviour and the numericall obtained one. Girder G1 Figure 5. A tpical mesh and lateral buckling mode for Girder G1 (Table 3). Table 3. Results of the verification analsis for the numerical model Loading δ max-fe (mm) δ theo (mm) δ max-fe / δ theo cr-fe kn.m cr-eq. 3 kn.m cr-fe/ cr-eq. 3 C b-theoretical C b-fe Eq. 4 Eq. 5 Girder G1 12.303 12.283 1.002 287.7 294.5 0.98 0.98 1.0 1.0 /2 Girder G2 9.255 9.212 1.004 378.7 382.9 * 0.99 1.286 1.3 1.25 Girder G3 6.312 6.305 1.001 520.5 515.4 * 1.01 1.767 1.75 1.67 /2 Girder G4 3.639 3.617 1.006 707.6 736.3 * 0.96 2.403 2.3 2.17 Girder G5 1.565 1.573 0.995 770.4 736.3 * 1.05 2.616 2.3 2.27 * The equivalent moment factor is calculated using Equation 4. GCF-231-6
2.50 2.25 A B Eq. 4 - CISC Eq. 4 F.E. Cb 2.00 1.75 Eq. 5 - AISC 1.50 1.25 1.00-1.00-0.80-0.60-0.40-0.20 0.00 0.20 0.40 0.60 0.80 1.00 A / B Figure 6. The equivalent moment factor for girders with plane webs subjected to end moments. 3.2 Numerical odeling of Corrugated Web Girders The same previous finite element model is used to analze plate girders with corrugated steel webs. The analzed girders are considered to be simpl supported in flexure and in torsion and the critical moments initiating lateral torsion-flexure buckling are numericall evaluated. A linear elastic buckling analses is performed on girders having the cross-sections shown in Figure 7. Two panel widths for the corrugated web (200 mm and 400 mm) are adopted in the analsis. The analzed panel widths correspond to panel width to web height ratios of 0.42 and 0.83. Web thicknesses of 2 mm and 4 mm are considered for the 200 mm and 400 mm panel widths respectivel: complete geometric properties of the corrugated web are shown in Figure 7. Stiffener plates 20 mm thick were added at the support locations for all the analzed girders. The mechanical properties of the girders cross-section are listed in Table 3. Using data from Table 3, the theoretical critical moments (Equation 1) of such girders when the are subjected to constant bending moments are 526.1 kn.m and 527.4 kn.m for web thickness of 2 mm and 4 mm respectivel. It is clear that, theoreticall, neither the web thickness nor the corrugation panel width of the web would have significant effect on the value of the critical moment. 300 Stiff l. 1 20 b x t w 200x2 400x4 t w Sec. 1-1 520 a L 11.52 m b d 1 α h a b β a/b 1.0 h d tan α c b + d s b + d/cosα t eq (s/c) t w Figure 7. Geometric properties of the analzed corrugated web girders GCF-231-7
Table 3. echanical properties of the analzed corrugated web girders cross-section is considered as a double smmetric I-section with an equivalent thickness t eq s/c t w (Figure 7). ropert A (10 3 ) I x (10 6 ) I (10 6 ) r x r S x (10 3 ) S (10 3 ) Z x (10 6 ) Z (10 6 ) C w (10 9 ) J (10 3 ) mm 2 mm 4 mm 4 mm mm mm 3 mm 3 mm 3 mm 3 mm 6 mm 3 t w 2 mm 13.07 770.5 90.0 243 83.0 2963 600 3127 900 5625 1602 t w 4 mm 14.13 791.0 90.0 236 79.8 3042 600 3255 900 5625 1614 3.3 Results of the Numerical Analsis Corrugated web girders subjected to the end moments shown in Table 4 are numericall analzed. The critical moments which initiate lateral torsion-flexure buckling of the flange in each case are listed in Table 4. Results of the analsis are also plotted in Figure 8. A tpical finite element mesh and lateral torsionflexure buckling mode encountered in the analsis are shown in Figure 9. Table 4. Results of the finite element buckling analsis for girders with corrugated steel webs. b x t w 200 mm x 2 mm b x t w 400 mm x 4 mm Loading C cr-fe b-eq.4 C cr-fe / cr-fe kn.m b-fe C cr-fe / cr-eq 3 kn.m b-fe cr-eq 3 1.00 606.0 1.00 1.152 635.1 1.00 1.204 Girder G6 /2 Girder G7 1.30 797.9 1.32 1.167 835.6 1.31 1.219 Girder G8 1.75 1097.9 1.81 1.192 1144.5 1.80 1.240 /2 Girder G9 2.30 1522.5 * 2.51 1.258 1558.8 * 2.45 1.285 Girder G10 2.30 1661.1 * 2.74 1.373 1663.8 * 2.62 1.372 * Lateral buckling is not the first buckling mode. It is preceded b flange and/or web local buckling mode. For each of the analzed girders, the theoretical values for the critical moments ( cr-eq.3 ) are calculated using Equation 3 with the equivalent moment factor C b evaluated using Equation 4. In these calculations, the girders were assumed to have plane webs with a thickness equivalent to the corrugated web thickness (t eq t w s/c). A comparison between the critical moment resulting from the numerical analsis ( cr-fe ) and the theoreticall calculated critical moment ( cr-eq.3 ) is presented in Table 4 and Figure 8. The numericall obtained critical moments ( cr-fe ) are related to the critical moment of Girder G6 (Table 4). Thus, equivalent moment factors for the analzed corrugated web girders are evaluated: those are listed in Table 4 and plotted in Figure 8 versus the equivalent moment factors calculated using Equation 4. It is evident from both Table 4 and Figure 8 that resistance to lateral torsion-flexure buckling of steel plate girders with corrugated webs is different from that of plate girders with traditional plane webs. The critical moment causing lateral buckling to initiate for corrugated web girders is larger than the one of traditional girders. The numerical analsis reveals that cr for corrugated web girders is 15% to 37% higher than cr for girders with plane webs. The main reason for this increase is the stiffening effect provided to the flange GCF-231-8
Cb 2.75 2.50 2.25 2.00 1.75 1.50 A bxt w 400mmx4mm B C b -Eq. 4 - CISC limit C b -F.E. cr-fe / cr-eq. 3 C b -Eq. 4 1.70 1.60 1.50 1.40 1.30 1.20 cr-fe / cr-eq. 3 1.25 1.10 1.00-1.00-0.80-0.60-0.40-0.20 0.00 0.20 0.40 0.60 0.80 1.00 A / B Figure 8. The equivalent moment factor and the critical moment for girders with corrugated webs. b the geometricall corrugated web. In the matter of fact this stiffening effect is mutual between the web and the flange: the flanges provide a boundar condition to the web to resist local/overall buckling (Saed- Ahmed 2001) and the corrugated web laterall stiffens the flange to resist lateral buckling. The numerical analsis results (Figure 8 and Table 4) also reveals that the concept of the equivalent moment factor which is used for traditional I-girders with plane web to account for the moment gradient is applicable to girders with corrugated webs. Table 4 shows that Equation 4 which is used to calculate the 1.00 Girder G6 Figure 9. A tpical mesh and lateral buckling mode for Girder G6 (bxt w 400 mm x 4 mm -Table 4). GCF-231-9
equivalent moment factor is also applicable to steel plate girders with corrugated webs. Furthermore, Figure 8 shows that Equation 4 with the CAN/CSA-S16.1-94 (CISC 2000) provides the best match to the results of the numerical analsis. 5. CONCLUSIONS Lateral stabilit of plate girders with corrugated steel webs have been numericall investigated. It is concluded that resistance to lateral torsion-flexure buckling of such girders is 15% to 37% higher than the resistance of plate girders with traditional plane webs to lateral buckling. Thus, the equation used to calculate the critical moment of girders with plane webs would underestimate the capacit of plate girders with corrugated web to resist lateral buckling but would be conservative for design purposes. It is also concluded that the equivalent moment factor concept which is used for the traditional plate girders with plane web is equall applicable to plate girders with corrugated webs. Thus, all the equations and tables which are currentl used to determine the equivalent moment factor for girders with plane web ma also be used for plate girders with corrugated steel webs. 6. REFERENCE American Institute of Steel Construction (AISC). 2000. Load and Resistance Factor Design- Vol. 1: Structural members, Specifications and Codes. Illinois, USA. Canadian Institute of Steel Construction (CISC). 2000. Handbook of Steel Construction 350W. 7th edition, Ontario, Canada. Chen, W.F., and Lui, E.. (1987) Structural Stabilit: theor and Implementation. Elsevier Science ublishing Co., Inc. N.Y., USA. Cherez,. and Combault, J. (1990) Composite Bridges with Corrugated Steel Webs - Achievement and rospects. IABSE Smposium, ixed Structures: Including New aterials, IABSE Reports, Brussels, pp.479-484. Galambos, T. V. (1998) A guide to Stabilit design criteria for metal structures. 5th edition. John Wile & Sons, Inc., N.Y. USA. Elgaal,., Hamilton, R.W. and Seshadri, A. (1996) Shear Strength of Beams with Corrugated Webs. Journal of Structural Engineering, ASCE, 122(4): 390-398. Elgaal,., Seshadri, A. and Hamilton, R.W. (1997) Bending Strength of Steel Beams with Corrugated Webs. Journal of Structural Engineering, ASCE, 123(6): 772-782. Johnson, R.. and Cafolla, J. (1997) Corrugated Webs in late Girders for Bridges. roceedings of the Institution of Civil Engineering, Structures and Buildings, 123: 157-164. Lebon, J. (1998) Steel Corrugated Web Bridges - First Achievements. 5 th International Conference on Short and edium Span Bridges (SSB V), CSCE, Calgar, Canada, CD-roceedings. Leiva-Aravena, L. (1987) Trapezoidall Corrugated anels - Buckling Behaviour under Axial Compression and Shear. Division of steel and Timber Structures, Chalmers Universit of Technolog, Gothenburg, Report S84:2, Sweden, ublication S87:1. Luo, R. and Edlund, B. (1994) Buckling of Trapezoidall Corrugated anels using Spline Finite Strip ethod. Thin Walled Structures, Elsevier Science Limited, 18: pp. 209-240. Salvadori,. G. (1955) Lateral buckling of I-beams. ASCE Transaction. Vol. 120, pp. 1165-1177. Allen, H.G. and Bulson,.S. (1989) Background to buckling. cgraw Hill Book Compan, London, UK. Saed-Ahmed, E.Y. (2001) Behaviour of Steel and/or Composite Girders with Corrugated Steel Webs. Canadian Journal of Civil Engineering, 28(4): 656-672. GCF-231-10