Microsoft Word - 2. K. Masuda and D. H. Chen--Prediction of Maximum Moment of Rectangular Tubes Subjected to Pure Bending.doc 214-225 Advances in Systems Science and Applications (2011), Vol.11, No.3-4 ISSN 1078-6236 International Institute for General Systems Studies, Inc. Prediction of Maximum Moment of Rectangular Tubes Subjected to Pure Bending K. Masuda and D. H. Chen Department of Mechanical Engineering, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan Abstract In this paper, the collapse behaviors of rectangular tube subjected to pure bending are studied by using the finite element method. Such bending collapse has been studied for a long time, including the landmark study by Kecman. According to these studies, there are two types of collapses. The first type is a collapse due to buckling at the compression flange. The second type is a collapse due to plastic yielding at the flanges. However, there may be another collapse. For a rectangular tube in which the web is wider than the flange, it is found that collapse due to buckling at the compression web may occur. Further, an approximation prediction method is proposed for estimating the maximum bending moment of rectangular tubes in which the web buckling is also taken into account. Its validity is verified by comparing with the numerical results by FEM under various conditions. Keywords FEM, Pure bending, Rectangular tube, Buckling, Effective width 1．Introduction Evaluation of a car’s crush behavior when the car is subjected to an oblique load, such as in an offset crash, is becoming increasingly important for general car design. It is thus vital to understand the crushing characteristic of rectangular tubes that are used as general components in car. Because the oblique load can be decomposed into axial load and pure bending, the pure bending of the rectangular tubes has been widely studied for a long time [1, 2, 3], including the landmark study by Kecman [1]. According to these studies, there are two types of collapses. The first type is a collapse due to buckling at the compression flange. The second type is a collapse due to plastic yielding at the flanges. However, when the web is wider than the flange, it is considered that a collapse due to buckling at the compression web may occur because it was already reported in bending of open section beams [4]. In the present study, the effects of the material and geometrical properties of rectangular tubes on their bending collapse are studied by using the finite element method. Further, based on the numerical results obtained, a method for estimating the maximum bending moment of rectangular tubes subjected to pure bending is proposed. In addition, a validity of FE analysis result under bending collapse has been already verified by comparing the experimental results by Kyriakides [5] with the previous numerical results by authors [6] under pure bending with cylindrical tubes. 2．Analytical method The commercial FEM analysis package MSC. Marc[7] is used in this study to analyze large elastoplastic bending of the rectangular tubes shown in Fig. 1. In the present calculation, one end of the rectangular tube was completely fixed to a rigid wall. Pure bending was applied from the other end by modeling a lid rotating about the axis of z under rotary control. The effects of various geometric parameters, such as tube thickness t, tube flange width c1, and tube web width c2, on the bending collapse were investigated. The value of lid thickness tf was set to five times t as referred to Guarracino [8] because the lid must be stiff enough to prevent distortion. Advances in Systems Science and Applications (2011), Vol.11, No.3-4 215 ISSN 1078-6236 International Institute for General Systems Studies, Inc. The tube material used in the analysis was assumed to be homogeneous and isotropic elastic perfectly plastic material that conforms to von Mises yield conditions. In this study, it was assumed that Young’s modulus E = 72.4 GPa, and Poisson’s ratio v = 0.3. The influence of the material properties on the bending collapse of the rectangular tube was investigated in terms of the yield stress yσ . In this study, the updated Lagrange method was used to formulate the geometric nonlinear behavior, and the algorithm based on the Newton-Raphson method and the return-mapping method were used to solve the nonlinear equation. The rectangular tubes were modeled using four-node quadrilateral thickness shell elements (Element type 75). The elements were divided the flange and wed width into 20 sub lengths, and divided the axial length in a way that the elements become almost square. In addition, the rectangular length used in the analysis was assumed to be long enough in order to exclude the influence of the boundary conditions. The ratio of the length and flange width L/c1 was set to 1/ 6L c > . L t t x y z2c 1c z y x M output zθ input tf =5t A AA-A Fig.1.Tube geometry and loading condition 3．Results and discussion 3.1 Comparison between proposal method by Kecman and results of present numerical analyses First, we show Kecman’s method for estimating the maximum bending moment of rectangular tubes subjected to pure bending. For a rectangular tube subjected to pure bending, the buckling stress bucσ of the compression flange was derived in the following equation. ( ) ( ) 22 2 5.23 0.16 1 12 1buc E a t b a πσ ν ⎛ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠ 　　 　　 b a 2 eaYσ 1y 1 1 Y b y y σ − (a) b Yσ (b) Yσ 2 b a Fig.2. Schematic representation of axial stress distribution proposed by Kecman: (a) buc yσ σ< ; (b) 2buc yσ σ≥ where E, v, a, b, and t are respectively, Young’s modulus, Poisson’s ratio, flange width, web width and tube thickness. In addition, a is 1c t+ and b is 2c t+ . Kecman presented a proposal method in which was decided by relations of the buckling stress bucσ and yield stress yσ . (1) In the case of buc yσ σ< If the buckling stress bucσ is less than the yield stress yσ , the compression flange buckles 216 Masuda: Prediction of Maximum Moment of Rectangular Tubes Subjected to Pure Bending and the edges stress come up to yield stress yσ . In order to consider this phenomenon, an effective width eσ is introduced in the following simplified equation. ( )0.7 0.3 2buc ea a σ σ ⎛ ⎞ = +⎜ ⎟⎜ ⎟ ⎝ ⎠y 　 　 　 As a result, stress distribution in the maximum moment is shown in Fig. 2(a). In the figure, y1 in which the distances from a compression flange to the neutral axis is derived from the condition of zero axial loads. Therefore, y1 is given by the following equation. ( )1 3 2e y a b b a a b + = + + 　 　 　 By summing moments through the cross-section, the maximum bending moment is derived in the following equation. ( ) ( )2 max 2 3 2 4 3 e aa b a bM t b a b σ ⎛ ⎞+ + ⋅ +⎜ ⎟ ⎝ ⎠⋅ ⋅ ⋅ +y= 　 (2) In the case of 2buc yσ σ≥ In this case, stress distribution in the maximum moment is shown in Fig. 2(b). Namely, it is assumed that the maximum moment is equal to a fully plastic moment Mp. The maximum bending moment is derived in the following equation. ( ) ( ) ( )2 max 0.5 2 5pM M t a b t b tσ ⎡ ⎤⋅ − + −⎣ ⎦y= = 　 (3) In the case of 2y buc yσ σ σ≤ < First, if the buckling stress bucσ is equal to the yield stress yσ , it is assumed that the maximum moment is equal to a elastic moment Me in which the stress of flanges are equal to the yield stress yσ . This elastic moment Me is derived in the following equation. ( )6 3e bM t b aσ ⎛ ⎞⋅ ⋅ ⋅ +⎜ ⎟ ⎝ ⎠ y= 　 And in the case of 2y buc yσ σ σ≤ < , the maximum bending moment is derived from linear interpolation: ( ) ( )max 7buc y e p eM M M M σ σ σ − − y = + 　 0 0.005 0.01 0.015 0.02 0.025 0 1 2 t / c1 M m ax / (σ y c 1 c 2 t ) σbuc<σy 2σy≤σbuc L = 300 mm c1 = 50 mm σy/ E = 1/1000 Kecman [1] c2/c1=1 c2/c1=2 c2/c1=1 c2/c1=2 2σy≤σbuc σbuc<σy Fig.3. Comparison between Kecman’s proposal and results of FEM with relation of t/c1 and ( )max 1 2/ yM c c tσ ⋅ ⋅ ⋅ Advances in Systems Science and Applications (2011), Vol.11, No.3-4 217 ISSN 1078-6236 International Institute for General Systems Studies, Inc. Figure 3 compares these proposal methods by Kecman and results of present numerical analyses for two levels of aspect ratio c2 /c1 with c1 =50mm, L=300mm, / 0.001y Eσ = . As can be seen from this figure, in the case of high-aspect ratio in which the web is wider than the flange, the results of maximum moment under various t/c1 between the Kecman’s proposal and results of FEM have a margin of error. In particular, the error increases with decreasing t/c1. Therefore, it is found that a region which does not apply to Kecman’s proposal exists. In order to estimate the maximum moment, it is vital to reveal the bending collapse mechanism of rectangular tubes. 3.2 Two types of collapse mechanism pointed out by Kecman An investigation of two types of collapse mechanism pointed out by Kecman was presented by using square tubes in which the aspect ratio c2 /c1 was set to 1. Figure 4 shows the relation of a tube curvature / Lκ θ= and moment M for a square tube with t=0.9mm, c1=50mm, c2=50mm, / 0.001y Eσ = ( 1.52buc yσ σ= ). And figure 4 also shows the relations of the tube curvature / Lκ θ= and axial stress /x yσ σ at point B and C (refer to schematic representation of cross-section in figure 4). As can be seen from the figure, the maximum moment is in good agreement with the value of Kecman’s proposal equation (7). The axial compression stress /x yσ σ at point B in the middle of compression flange increases until the moment becomes maximum moment, and the value /x yσ σ comes up to 1. In addition, the axial compression stress /x yσ σ at point C in the quarter of web width increases until the moment becomes maximum moment. Figure 5 shows the axial stress distribution of cross-section at phase ( )α and ( )β corresponding to 1/ 0.025L mθ −= and 10.065m− in Fig. 4. As can be seen from the figure, the absolute value of the axial stress when the maximum moment occurs is greater than the value at phase ( )α in all cross-section positions. In addition, the axial stress distribution when the maximum moment occurs is in good agreement with Kecman’s proposal. It is confirmed from the above investigation that in the case of 2 1/ 1c c = and y bucσ σ≤ , the collapse type is not due to buckling at the compression flange and web, but due to plastic yielding at the flanges. Therefore, to estimate the maximum moment by Kecman’s theory is possible in this case. 0 0.05 0.1 0 50 100 150 200 250 0 0.5 1 1.5 θ / L [m−1] M [ N m ] σ x /σ Y Moment σ x (Point B) σ x (Point C) eq.(7) L = 300 mm t = 0.9 mm c1 = 50 mm c2 = 50 mm extension side compression side B Cc2/4 σY / E = 1/1000 σx /σY =1(α ) (β ) Fig.4. Relations of / Lθ and , /x yM σ σ for square tube with t=0.9mm, c1=50mm, c2=50mm 218 Masuda: Prediction of Maximum Moment of Rectangular Tubes Subjected to Pure Bending 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5 σ x / σ Y s / (2c1+2c2 ) s extension side compression side a b a b L = 300 mm t = 0.9 mm c1 = 50 mm c2 = 50 mm σY / Ε =1/1000 Kecman (α ) (β ) Fig.5. Axial stress distribution of cross-section for the square tube shown in Fig.4 0 0.05 0.1 0 40 80 0 0.5 1 θ / L [m−1] M [ N m ] σ x /σ y Moment σ x (Point B) σ x (Point C) eq.(4) eq.(1) L = 300 mm t = 0.4 mm c1 = 50 mm c2 = 50 mm extension side compression side B Cc2/4 σy /E= 1/1000 (α ) (β ) Fig.6. Relations of / Lθ and , /x yM σ σ for square tube with t=0.4mm, c1=50mm, c2=50mm Figure 6 shows the relation of a tube curvature / Lκ θ= and moment M for a square tube with t=0.4mm, c1=50mm, c2=50mm, / 0.001y Eσ = ( 0.31buc yσ σ= ). And figure 6 also shows the relations of the tube curvature / Lκ θ= and axial stress /x yσ σ at point B and C (refer to schematic representation of cross-section in figure 6). As can be seen from the figure, the maximum moment is in good agreement with the value of Kecman’s proposal equation (4). The axial compression stress /x yσ σ at point B in the middle of compression flange decreases before the moment becomes maximum moment, and the maximum value /x yσ σ is in good agreement with the equation (1) of elastic buckling stress. In addition, the axial compression stress /x yσ σ at point C in the quarter of web width increases until the moment becomes maximum moment. Figure 7 shows the axial stress distribution of cross-section at phase ( )α and ( )β corresponding to 1/ 0.012L mθ −= and 10.038m− in Fig. 6. As can be seen from the figure, although the axial compression stress in the middle of compression flange decreases due to buckling in the middle of compression flange, the axial compression stress in both edges Advances in Systems Science and Applications (2011), Vol.11, No.3-4 219 ISSN 1078-6236 International Institute for General Systems Studies, Inc. of compression flange increase because buckling doesn’t occur in both edges. Immediately after buckling, stress increment in the both edges is greater than stress decrement in the middle of compression flange. Therefore, total force of compression side and moment increase. In addition, the stress distribution of the web changes linearly because buckling doesn’t occur in the web. Therefore, the axial stress distribution when the maximum moment occurs is in good agreement with Kecman’s proposal by using an effective width in the compression flange. It is confirmed from the above investigation that in the case of 2 1/ 1c c = and y bucσ σ> , the collapse type is due to buckling at the compression flange. Therefore, to estimate the maximum moment by Kecman’s theory is possible in this case. 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 Kecman σ x / σ y s / (2c1+2c2 ) s extension side compression side a b a b L = 300 mm t = 0.4 mm c1 = 50 mm c2 = 50 mm (α ) (β ) Fig.7. Axial stress distribution of cross-section for the square tube shown in Fig.6 3.3 Collapse mechanism which is different from Kecman’s indication In the case of high-aspect ratio in which the web is wider than the flange, it was confirmed that collapse due to buckling at the compression web occur. 0 0.005 0.01 0.015 0 100 200 300 0 1 2 θ / L [m−1] M [ N m ] σ x /σ y Moment σ x (Point B) σ x (Point C) eq.(5) L = 300 mm t = 0.5 mm c1 = 20 mm c2 = 100 mm extension side compression side B Cc2/4 σy / E = 1/1000 σx /σY =1 (α ) (β ) Fig.8. Relations of / Lθ and , /x yM σ σ for rectangular tube with t=0.5mm, c1=20mm, c2=100mm 220 Masuda: Prediction of Maximum Moment of Rectangular Tubes Subjected to Pure Bending Figure 8 shows the relation of a tube curvature / Lκ θ= and moment M for a rectangular tube with t=0.5mm, c1=20mm, c2=100mm, / 0.001y Eσ = ( 2 12.83 , / 5buc y c cσ σ= = ). And figure 8 also shows the relations of the tube curvature / Lκ θ= and axial stress /x yσ σ at point B and C (refer to schematic representation of cross-section in figure 8). As can be seen from the figure, the maximum moment is less than the value of Kecman’s proposal equation (5). In addition, the axial compression stress /x yσ σ at point B in the middle of compression flange increases until the moment becomes maximum moment, and the value /x yσ σ comes up to 1. And the axial compression stress /x yσ σ at point C in the quarter of web width decreases before the moment becomes maximum moment. Figure 9 shows the axial stress distribution of cross-section at phase ( )α and ( )β corresponding to 1/ 0.036L mθ −= and 10.048m− in Fig. 8. As can be seen from the figure, the axial stress distribution in the compression flange is constant value and the absolute value is almost 1 when the maximum moment occurs. And the axial stress distribution in the compression web doesn’t increase linearly. Therefore, the sum of axial stress when the maximum moment occurs is less than the Kecman’s proposal as much as it is shown by arrows of Figure 9. 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 1.5 σ x / σ Y s / (2c1+2c2 ) s extension side compression side a b a b L = 300 mm t = 0.5 mm c1 = 20 mm c2 = 100 mm σy / Ε =1/1000 Kecman (α ) (β ) Fig.9. Axial stress distribution of cross-section for the rectangular tube shown in Fig.8 It is found from the above investigation that in the case of high-aspect ratio and y bucσ σ< , the collapse type is not due to buckling at the compression flange but due to buckling at the compression web. Therefore, to estimate the maximum moment by Kecman’s theory is impossible in this case. Figure 10 shows the relation of a tube curvature / Lκ θ= and moment M for a rectangular tube with t=0.4mm, c1=50mm, c2=100mm, / 0.001y Eσ = ,( 0.30buc yσ σ= ), ( 2 1/ 2c c = ). And figure 10 also shows the relations of the tube curvature / Lκ θ= and axial stress /x yσ σ at point B and C (refer to schematic representation of cross-section in figure 10). As can be seen from the figure, the maximum moment is less than the value of Kecman’s proposal equation (4). The axial compression stress /x yσ σ at point B in the middle of compression flange decreases before the moment becomes maximum moment, and the maximum value /x yσ σ is in good agreement with the equation (1) of elastic buckling stress. In addition, the axial compression stress /x yσ σ at point C in the quarter of web width decreases before the moment becomes Advances in Systems Science and Applications (2011), Vol.11, No.3-4 221 ISSN 1078-6236 International Institute for General Systems Studies, Inc. maximum moment. Figure 11 shows the axial stress distribution of cross-section at phase ( )α and ( )β corresponding to 1/ 0.007L mθ −= and 10.016m− in Fig. 10. As can be seen from the figure, the axial stress in the compression flange is concentrated in the edges when the maximum moment occurs. And the axial stress distribution in the compression web doesn’t increase linearly. Therefore, the sum of axial stress in the maximum moment is less than the Kecman’s proposal as much as it is shown by arrows of Figure 11 because the equation (2) applies to the axial stress distribution of compression flange, and linearly approximation doesn’t apply to the axial stress distribution of compression web. It is found from the above investigation that in the case of high-aspect ratio and y bucσ σ≥ , the collapse type is not only due to buckling at the compression flange but also due to buckling at the compression web. Therefore, to estimate the maximum moment by Kecman’s theory is impossible in this case. 0 0.02 0.04 0.06 0 100 200 0 0.5 1 θ / L [m−1] M [ N m ] σ x /σ y Moment σ x (Point B) σ x (Point C) eq.(4) eq.(1) L = 300 mm t = 0.4 mm c1 = 50 mm c2 = 100 mm C B extension side compression side c2/4 σx /σy = 1/1000 (α ) (β ) Fig.10. Relations of / Lθ and , /x yM σ σ for rectangular tube with t=0.5mm, c1=20mm, c2=100mm 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 Kecman σ x / σ y s / (2c1+2c2 ) s extension side compression side L = 300 mm c1 = 50 mm c2 = 100 mm t = 0.4 mm a b a b (α ) (β ) Fig.11. Axial stress distribution of cross-section for the rectangular tube shown in Fig.10 222 Masuda: Prediction of Maximum Moment of Rectangular Tubes Subjected to Pure Bending 3.4 Proposal method of maximum moment considering the web buckling 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 FEM Kecman’s proposal σ x / σ y s / (2c1+2c2 ) s extension side compression side L = 300 mm c1 = 50 mm c2 = 30 mm t = 0.4 mm present proposal (b) 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 FEM Kecman’s proposal σ x / σ Y s / (2c1+2c2 ) s extension side compression side L = 300 mm c1 = 50 mm c2 = 100 mm t = 0.4 mm present proposal (a) Fig.12. Axial stress distribution in the range of buc yσ σ< : (a) by Kecman’s method; (b) by present method Figure 12 shows a schematic representation of axial stress distribution in the maximum moment after buckling of the compression flange ( buc yσ σ< ). Figure 12(a) shows Kecman’s proposal in which doesn’t consider the web buckling, and figure 12(b) shows present proposal in which considers the web buckling. As can be seen in the figure(b), an effective width ae applies to the compression web as well as the compression flange. It is assumed that the effective width ae is independent of initial web width as referring to Karman’s theory [9]. A coefficient α in which represents the axial tension stress is derived in the following equation. ( ) ( ) 1 2 8 2 ea t a b y t α − = + − − 　 　 　 Figures 13(a) and (b) show comparison between results of FEM and proposals with the axial stress distribution. As can be seen in the figures, in the case of high-aspect ratio 2 1/ 2c c = , present proposal in which considers the web buckling is in good agreement with the result of FEM. And in the case of low-aspect ratio 2 1/ 0.6c c = , Kecman’s proposal in which doesn’t consider the web buckling is in good agreement with the result of FEM. b a 2 eaYσ 1y 1 1 Y b y y σ − (a) b a 2 ea Yσ 1y Yασ b a 2 ea Yσ 1y Yασ (b) Fig.13. Axial stress distribution by results of FEM, Kecman’s proposal and present proposal: with (a) 2 1/ 2c c = ; (b) 2 1/ 0.6c c = Figure 14 shows a schematic representation of axial stress distribution in the maximum moment when the compression flange doesn’t buckle ( buc yσ σ≥ ). Figure 14(a) shows Kecman’s proposal in which doesn’t consider the web buckling, and figure 14(b) shows present proposal in which considers the web buckling. As can be seen in the figure, an effective width ae applies to only the compression web. A coefficient β in which represents the axial tension stress is derived in the following equation. Advances in Systems Science and Applications (2011), Vol.11, No.3-4 223 ISSN 1078-6236 International Institute for General Systems Studies, Inc. ( ) 1 2 9 2 ea a t a b y t β + = + − − - 　 　 　 Figures 15(a) and (b) show comparison between results of FEM and proposals with the axial stress distribution. As can be seen in the figures, in the case of high-aspect ratio 2 1/ 5c c = , present proposal in which considers the web buckling is in good agreement with the result of FEM. And in the case of low-aspect ratio 2 1/ 2c c = , Kecman’s proposal in which doesn’t consider the web buckling is in good agreement with the result of FEM. We show present method for estimating the maximum bending moment of rectangular tubes subjected to pure bending. In the case of buc yσ σ< , a position of the center of gravity in the tension web G is derived in the following equation. b aYσ / 2b Yσ (a) b / 2ea Yσ 1y Yβσ (b) a Fig.14. Axial stress distribution in the range of buc yσ σ≥ : (a) by Kecman’s method; (b) by present method 0 0.5 1 −1.5 −1 −0.5 0 0.5 1 FEM Kecman’s proposal σ x / σ y s / (2c1+2c2 ) s extension side compression side L = 300 mm c1 = 20 mm c2 = 100 mm t = 0.5 mm present proposal (a) 0 0.5 1 −2 −1 0 1 2 FEM Kecman’s proposal σ x / σ y s / (2c1+2c2 ) s extension side compression side L = 200 mm c1 = 20 mm c2 = 40 mm t = 0.5 mm present proposal (b) Fig.15. Axial stress distribution by results of FEM, Kecman’s proposal and present proposal: with (a) 2 1/ 5c c = ; (b) 2 1/ 2c c = ( )1 1 1 10 3 2 G b y⎛ ⎞= +⎜ ⎟ ⎝ ⎠ 　 　 Therefore, in the case of buc yσ σ< , the maximum moment in which the compression web buckles is derived in the following equation. ( )( ) ( ){ ( )( )max 1 1 2 2 2 2 (11) 2 2 4 e y e e abM t a t b t b y G a t b t aσ α α ⎫⎛ ⎞= − − + − + − − + − ⎬⎜ ⎟ ⎝ ⎠⎭ 　　　　　　　 where 1,y α and G are respectively, the value of equation (3), (8) and (10). It is found from the above investigation that in the case of buc yσ σ< , the maximum moment is derived in the following equation. ( ) ( )( ) ( )max . 4 , . 11 12M Min eq eq= 　 　 224 Masuda: Prediction of Maximum Moment of Rectangular Tubes Subjected to Pure Bending { }1 ( 2 )( ) 2 ( ) ( 2 )( ) 2 ( ) (13)max 12 2 4 ab eM t a t b t b y G a t b t ay eσ β β= − − + − + − − + − Moreover, in the case of buc yσ σ≥ , the maximum moment in which the compression web buckles is derived in the following equation. where 1,y β and G are respectively, the value of equation (3), (9) and (10). It is found from the above investigation that in the case of buc yσ σ≥ , the maximum moment is derived in the following equation. ( ) ( ) ( )( ) ( )max . 5 , . 7 , 13 14M Min eq eq eq= 　 Figure 16 shows comparison between results of FEM and Kecman’s proposal and present proposal with the maximum moment for two levels of /y Eσ . As can be seen in the figure, the lower values of Kecman’s proposal and present proposal is in good agreement with the results of FEM. 0 0.005 0.01 0.015 0.02 0.025 0 0.5 1 1.5 2 2.5 t / c1 M m ax / (σ y c 1 c 2 t ) L = 300 mm c1 = 50 mm Kecman’s proposal 2σy≤σbuc σbuc<σy σbuc<σy σy/ E = 1/1000 σy/ E = 1/500 present proposal c2 = 100 mm σy/ E = 1/1000 σy/ E = 1/500 Comparison between results of FEM and Kecman’s proposal and present proposal with relation of 1/t c and ( )max 1 2/ yM c c tσ ⋅ ⋅ ⋅ for two levels of /y Eσ . 4. Conclusion In this paper, the investigation of the bending collapse for rectangular tubes by using numerical analysis of the finite element method was presented. For a rectangular tube in which the web is wider than the flange, it is found that collapse due to buckling at the compression web may occur. It is possible to estimate the maximum moment under various the material and geometrical properties by using the present proposal in which applies an effective width to the web, and the Kecman’s proposal. References [1] Kecman D. Bending collapse of rectangular and square section tubes, International Journal of Mechanical Sciences, Vol.25 (1983) 623-636. [2] Kim T. H. and Reid S. R. Bending collapse of thin-walled rectangular section columns, Computers and Structures, Vol.79 (2001) 1897-1911. [3] Lu G. and Yu T. X. Energy absorption of structures and materials, (2003) Section5 Crc Pr I Llc. [4] Parh M. S. and Lee B. C. Prediction of bending collapse behaviours of thin-walled open section beams, Thin-Walled Structures, Vol.25(3) (1996) 185-206. [5] Kyriakides S. and Ju G. T. Bifurcation and localization instabilities in cylindrical shells Advances in Systems Science and Applications (2011), Vol.11, No.3-4 225 ISSN 1078-6236 International Institute for General Systems Studies, Inc. under bending, International Journal of Solids and Structures, Vol.29 (1992) 1117-1171. [6] Chen D. H. et al. Study on elastoplastic pure bending collapse of cylindrical tubes, Transactions of the Japan Society of Mechanical Engineers, Series A, Vol.74(740) (2008) 520-527. [7] MSC. Marc Manual, 2003. [8] Guarracino F. On the analysis of cylindrical tubes under flexure: theoretical formulations, experimental data and finite element analyses, Thin-Walled Structures, Vol.41 (2003) 127-147. [9] Karman V. T. et al. Strength of thin plates in compression, Trans ASME, Vol.54 (1932)