Elastic buckling of web plates in I-girders under patch and wheel loading

Engineering Structures 27 (2005) 1528 156 www.elsevier.com/locate/engstruct Elastic buckling of web plates in I-girders under patch and wheel loading T. Ren, G.S. Tong Department of Civil Engineering, Zhejiang University, Hangzhou 10027, China Received 27 September 2004; received in revised form 11 May 2005; accepted 11 May 2005 Available online 1 June 2005 Abstract This paper makes a brief review of the earlier research on the elastic buckling of rectangular plates with simply supported and clamped boundary conditions, as well as the buckling of web plates in I-girders subjected to patch load. New investigation has been carried out to simulate the realistic load and the restraining conditions of the web plates in I-girders. The buckling of a large number of models under patch load was analyzed with ANSYS and formulae were proposed to predict the elastic buckling coefficients of webs in I-girders. The rotational restraints provided to the web plates by the flanges are considered accurately in the suggested formulae. The wheel load is another kind of patch load rarely touched in the literature. This paper suggests a simple model to determine the bearing stresses on the top edge of the web plates in I-girders where the effect of the crane rail rigidity is considered. Based on this model, the buckling of the web plate is analyzed and formulae with excellent accuracy are suggested to predict the buckling load. 2005 Elsevier Ltd. All rights reserved. Keywords: Patch loading; Wheel loading; Web plates in I-girders; Elastic buckling coefficients 1. Introduction Concentrated forces applied perpendicularly to the upper flanges of I-girders, a load case usually referred to as patch loading, are common in most steel structures, as shown in Fig. 1. Although a lot of concentration has been focused on this problem, the analytic solution has not been obtained due to the complexity of the problem. Therefore, most investigators used the finite element method to study the behavior of the webs under patch loading. The elastic buckling strength of a rectangular plate under patch loading can be expressed as follows: π 2 E tw F cr = k cr 12(1 µ 2 (1) ) h w in which k cr is the elastic buckling coefficient of the plate, h w and t w are height and thickness of the plate, and E and µ are modulus of elasticity and Poisson s ratio of the material, respectively. Corresponding author. Tel.: +86 571 87952259; fax: +86 571 87961697. E-mail address: tonggs@zju.edu.cn (G.S. Tong). Lagerqvist [1] made an excellent review on this topic. Only a brief review is given in the following. Girkmann in 196 was the first researcher to study the elastic stability of a rectangular, simply supported plate subjected to a single edge force. His results applies only to plates with aspect ratio, a/h w less than 1.1 and the solution was given in the form of a determinant which had to be evaluated for any particular case. Based on the energy method, Zetlin [2] investigated the elastic buckling of rectangular plates with different aspect ratios under patch loading. It was assumed that the applied load was equilibrated by parabolically distributed shear stresses along the two vertical edges of the plate. White and Cottingham [] studied this problem using the finite difference method. Rockey [4] presented an investigation of buckling of simply supported and clamped rectangular plates under partial edge loading using a finite element method. The study was expanded to webs in I-girders. For webs in I-girders, the stresses are assumed to be uniform along the vertical edges and the vertical edges were allowed to rotate as a rigid body about the neutral axis of the section so that the Bernoulli assumption is valid as in the web of an actual plate girder. After comparing with the 0141-0296/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2005.05.006

T. Ren, G.S. Tong / Engineering Structures 27 (2005) 1528 156 1529 (a) Patch loading. Fig. 2. The model of simply supported rectangular plate. (b) Wheel loading. Fig. 1. The model of the I-girder under patch loading and wheel loading. buckling of the rectangular plate with identical size, it was found that the buckling coefficient of the web increased significantly because of the flange, the buckling mode was also different from that of a rectangular plate. Based on the energy approach suggested by Alfutov and Balabukh, Khan [5 7] presented solutions for the buckling of webs in I-girders. The assumed stress distribution satisfied only the equilibrium conditions, and the corresponding strains may not satisfy the compatibility requirement. Using the same stress distribution as in Khan s theory and the assumed jπy sin h w, buckling displacement w = m nj=1 i=1 a ij sin iπ a x Robert [8] presented solutions for buckling coefficients of webs in I-girders under patch loading on the basis of the Galerkin method. Moriawaki and Takinoto [9] presented a formula for calculating the resistance for concentrated loads applied at one flange. The formula includes the critical buckling load and the author gave an expression for the buckling coefficients in which the flange thickness was included, therefore the influence of the flanges has been considered to some extent. Graves Smith [10,11] has also concentrated on this problem by using a finite stripe method. Kitipornchai [12] utilized a finite element method using thin-plate elements to investigate the problem studied by Rockey [4] andkhan[5]. The aim of doing this was to validate his finite element formulation, this validated also the results by Rockey and Khan. Based on a number of numerical analyses, Graciano and Lagerqvist [1] suggested the following equation for the buckling coefficients of webs in I-girders under patch loading. k cr = 5.82 + 2.1 ( hw a ) 2 + 0.46 4 β (2) in which β = b ftf is the factor considering the rotational h w tw restraint from flanges. The physical meaning of the factor β is the ratio of the flange free torsional stiffness to the bending stiffness of the web. We could imagine that when β is very small, the rotational restraint from flanges to webs is too slight to restrict the rotation of the web, the web, therefore, behaves like simply supported plates. When β becomes greater, approaching, the web behaves like a clamped plate. Eq. (2) gives an infinite buckling coefficient in this case which obviously violates the requirement of approaching a clamped plate. Therefore, it is worthwhile to further investigate the elastic buckling behavior of the webs in I-girders under patch loading and explain the restraining effects of the flanges properly. The webs of crane runway girders are subjected to wheel loadings, as shown in Fig. 1(b). The buckling of webs under wheel loading, although similar to the patch loading, is rarely reported in the literature. This paper will also address this problem. 2. Validation of the analysis method A finite element analysis with the general-purpose finite element package ANSYS was conducted to investigate the elastic buckling behavior of a simply supported and a clamped rectangular plate, as well as the webs in I-girders under patch loading. Uniform loads were applied on the top edge of the plate over a limited length or on the flanges of I-girders. For single rectangular plates, focus was put on the effects of the length height ratio, a/h w, on the elastic buckling coefficients, while the effects of the elastic restraints at the flange web juncture to the elastic buckling coefficients of webs in I-girders were well concerned. The four-node elastic shell element SHELL6 was adopted in ANSYS, it has six degrees of freedom at each node: translations in the nodal x, y, andz directions and rotations about the nodal x, y,andz-axes. Shown in Fig. 2 is the model adopted by Rockey [4], Shahabian and Robert [8] and Graciano and Lagerqvist [1] in their studies. The out-of plane displacements at the four edges are prevented and the vertical displacements at the left

150 T. Ren, G.S. Tong / Engineering Structures 27 (2005) 1528 156 and the right edges are restrained. Horizontal displacements of the nodes at the middle of the top and the bottom edges were restrained to prevent the rigid body motion of the plate. Table 1 shows the coefficients of the simply supported rectangular plates of a/h w = 1.0 with different c/h w, obtained from ANSYS as well as those cited from Rockey, Shahabian & Robert and Graciano & Lagerqvist. It is observed that the ANSYS results are in good agreement with others. Table 1 Elastic buckling coefficients of simply supported rectangular plates (a/h w = 1.0) c/h w k crs k crs k crs k crs (Rockey) (Shahabian) (Lagerqvist) (ANSYS) 0.0.25.2.22 0.1..27.26.26 0.2.45.4.6 0.25.51.41.4 0..60.49.51 0.4.70.68.67.71 0.5.95.90.92.97 Fig.. The elastic buckling coefficients of simply supported rectangular plates.. Elastic buckling of single rectangular plate.1. Simply supported rectangular plates In this paper, 1, 2, and 4, 5, 6 are used to denote the degrees of freedom of translations in x, y, andz directions and rotations about x, y,andz-axes, respectively. For simply supported rectangular plates, the first degree of freedom of nodes on the top and bottom edges and the first, second and sixth degrees of freedom of nodes on the left and the right edges were restrained. Horizontal displacements of the nodes at the middle of the top and the bottom edges were restrained to prevent the rigid body motion. The width over which the load is applied was expressed as s s = c/h w. Rectangular plates with a/h w = 1.0 4.0 ands s = 0 0.5 were analyzed and the coefficients calculated from ANSYS are shown in Fig.. Based on the analysis results and the formula suggested by Lagerqvist [1], the following equation is proposed to calculate the elastic buckling coefficients of simply supported plates under patch loading: k crs = 2.05 + 1.2 (a/h w ) 2 + s2 s [ 0.5 + 2.0 (a/h w ) 2 ]. () In Table 2, the coefficients computed from Eq. () are compared with those obtained from ANSYS and the difference between them is insignificant..2. Clamped rectangular plates If an I-girder has heavy flanges, the rotational restraint provided to the web by flanges is so significant that the Fig. 4. The elastic buckling coefficients of clamped rectangular plates. flange web junctures serve as clamped boundaries of the web. The left and the right edge of the webs are still in simply supported conditions due to the transverse stiffeners. In this paper, the plates with this boundary conditions are denoted as clamped plates. In order to simulate this kind of conditions, the first and sixth degrees of freedom of the nodes on the top and the bottom edges are restrained, the first, second and sixth degrees of freedom of the nodes on the left and the right edges are restrained. The longitudinal displacement of the plate was prevented by restricting the degree of freedom 1 of the node at middle of the top and the bottom edges. Clamped rectangular plates with a/h w = 1.0 4.0 and s s = 0 0.5 were analyzed and the coefficients calculated from ANSYS are shown in Fig. 4. Based on the analysis results, the following equation is proposed to determine the elastic buckling coefficients of clamped rectangular plates under patch loading: ( ) ] a 2 k crf = (1 + 0.65ss [6. 2 ) 0.05 + 0.6 (a/h w ) 2. (4) Eq. (4) is compared with ANSYS in Table, again the errors are very small. h w

T. Ren, G.S. Tong / Engineering Structures 27 (2005) 1528 156 151 Table 2 The elastic buckling coefficients of simply-supported rectangular plates a/h w ANSYS Eq. () Error (%) s s s s s s 0.0 0.1 0.2 0. 0.4 0.5 0.0 0.1 0.2 0. 0.4 0.5 0.0 0.1 0.2 0. 0.4 0.5 1.0.22.26.6.51.71.97.25.28.5.48.65.88 0.9 0.46 0.0 1.00 1.62 2.9 1.2 2.81 2.85 2.91.01.14.0 2.88 2.90 2.96.05.19.6 2.61 1.8 1.68 1.44 1.45 1.68 1.4 2.60 2.62 2.67 2.74 2.84 2.96 2.66 2.68 2.72 2.80 2.91.04 2.9 2.19 1.99 2.16 2.1 2.78 1.5 2.5 2.55 2.59 2.66 2.75 2.85 2.58 2.60 2.64 2.71 2.81 2.9 2.11 1.85 1.89 1.82 2.02 2.8 1.6 2.47 2.49 2.54 2.60 2.68 2.78 2.52 2.5 2.57 2.6 2.72 2.84 1.97 1.67 1.18 1.1 1.6 2.12 1.8 2.40 2.42 2.46 2.51 2.59 2.67 2.42 2.4 2.47 2.52 2.60 2.70 0.85 0.48 0.21 0.44 0.5 1.11 2.0 2.5 2.7 2.41 2.46 2.5 2.61 2.5 2.6 2.9 2.44 2.51 2.60 0.00 0.42 0.8 0.81 0.79 0.8 2.5 2.27 2.29 2. 2.8 2.44 2.52 2.24 2.25 2.27 2.2 2.7 2.45 1.2 1.74 2.7 2.70 2.74 2.90.0 2.20 2.22 2.26 2.1 2.7 2.45 2.18 2.19 2.21 2.25 2.0 2.6 0.76 1. 2.11 2.67.00.51.5 2.14 2.15 2.19 2.24 2.0 2.8 2.15 2.15 2.17 2.21 2.25 2.1 0.7 0.21 0.71 1.44 2.00 2.78 4.0 2.07 2.08 2.12 2.17 2.2 2.0 2.1 2.1 2.15 2.18 2.2 2.28 2.66 2.46 1.42 0.52 0.22 0.82 Table The elastic buckling coefficients of clamped rectangular plates a/h w ANSYS Eq. (4) Error (%) s s s s s s 0.0 0.1 0.2 0. 0.4 0.5 0.0 0.1 0.2 0. 0.4 0.5 0.0 0.1 0.2 0. 0.4 0.5 1.0 6.90 7.00 7.20 7.48 7.87 8.5 6.85 6.89 7.0 7.25 7.56 7.96 0.72 1.57 2.6.07.94 4.67 1.2 6.5 6.60 6.76 6.97 7.25 7.59 6.64 6.69 6.82 7.0 7.4 7.72 1.68 1.6 0.89 0.86 1.24 1.71 1.5 6.4 6.50 6.6 6.82 7.05 7. 6.45 6.50 6.62 6.8 7.1 7.50 0.1 0.00 0.15 0.15 1.1 2.2 2.0 6.2 6.8 6.52 6.71 6.94 7.22 6.25 6.29 6.41 6.62 6.90 7.27 1.11 1.41 1.69 1.4 0.58 0.69 2.5 6.1 6.20 6. 6.52 6.75 7.02 6.08 6.12 6.24 6.44 6.72 7.07 0.82 1.29 1.42 1.2 0.44 0.71.0 5.95 6.02 6.15 6. 6.56 6.8 5.92 5.96 6.07 6.26 6.5 6.88 0.50 1.00 1.0 1.11 0.46 0.7.5 5.77 5.8 5.96 6.1 6.6 6.62 5.74 5.77 5.89 6.07 6. 6.67 0.52 1.0 1.17 0.98 0.47 0.76 4.0 5.58 5.64 5.76 5.9 6.15 6.41 5.54 5.57 5.68 5.86 6.11 6.44 0.72 1.24 1.9 1.18 0.65 0.47 The buckling modes of simply supported and clamped rectangular plates of a/h w = 2.0 are illustrated in Fig. 5(a) and (b) and, obviously, they are in different forms... Elastic buckling of webs in I-girders The model as shown in Fig. 1 is used by Lagerqvist [1] to analyze buckling behavior of webs in I-girders under patch loading. The width of applied load is c, and all nodes in the area where the load is transmitted were controlled to displace only in the vertical direction, i.e. degrees of freedom 1,, 4, 5 and 6 were restrained and only displacement in direction 2 was allowed. The patch load is transmitted through the flanges to the edge of webs in I-girders. This model mixed three different contributions of the flange to the web buckling, one is that the flange helps to carry part of the load, the second is that the flange disperses the load to a greater length of the web edge, and the third is that the flange provides rotational restraint to the web against buckling. In order to highlight the rotational restraint provided by flanges, a different model is used in this paper. The flanges and the webs were established separately, only degrees of freedom 1 and 6, i.e. the rotation about the flange web juncture line and the out-of-plane displacement, were coupled between the flanges and the web. The patch load was applied on nodes at the top edge of the web. The degrees of freedom 1, 2, 5 of nodes on the left and the right edges were restrained and the same boundary conditions were used at the edges of the flange. The buckling of the webs is postponed because of existence of the rotational restraint of the flange web juncture. The ratio of rotational rigidity of the flange to bending rigidity of the web is used to consider the elastic restraints of flanges to webs. This ratio can be defined as GK Dh w,inwhichgk = 2(1+µ) E b f tf and Dh w = Et w h w 12(1 µ 2 ),after omitting the constant items we get: β = b ftf h w tw. (5) In order to validate whether the factor β can reflect the influence from the flanges to shear buckling coefficients accurately or not, a number of I-girders were analyzed with ANSYS. The web thicknesses were 4 and 8 mm, the flange widths were 150, 250 and 400 mm, respectively. The flange thickness was changed to obtain different β values. The curves of coefficients obtained from ANSYS versus the factor β are plotted in Fig. 6.

152 T. Ren, G.S. Tong / Engineering Structures 27 (2005) 1528 156 (a) Simply supported rectangular plate. (b) Clamped rectangular plate. Fig. 5. The first-order buckling wave modes of single rectangular plates (s s = 0.2). Fig. 6. The elastic buckling coefficients of different models versus the factor β(s s = 0.2). As can be seen from Fig. 6, for I-girders with the same aspect ratio, a/h w, the elastic buckling coefficients are almost identical in value as long as the factors β are equal. Therefore, the factor β can be well used to evaluate the elastic buckling coefficients of the webs in I-girders. As the value of β increases from zero, the elastic buckling coefficient of webs increases from the elastic buckling coefficient of a simply supported plate, and when β becomes greater, the elastic buckling coefficients approach the value of the elastic buckling coefficient of a clamped plate. The basic expression for elastic buckling coefficients of webs in I-girders can be obtained based on this phenomenon, i.e.: k cr = k crs + k crf β. (6) 1 + β The aspect ratio of webs and the width of applied load are other factors that must to be considered to evaluate the elastic buckling coefficients of the webs in I-girders. Because the width of applied load has been considered when calculating the elastic buckling coefficients of rectangular plates, namely k crs and k crf, we just need to introduce a factor λ to consider the influence of the aspect ratio. Based on the analysis results, the following equation is proposed to determine the elastic buckling coefficients of webs in I-girders under patch loading: k cr = k crs + k crf λβ 1 + λβ in which λ = 0.1 + 0.0a/h w + 1.6, β = b ft (a/h w ) 2 f. h w tw The elastic buckling coefficients for I-girders of different aspect ratios, different flange thicknesses and widths obtained from ANSYS and those computed from Eq. (7)are tabulated in Table 4. It may be seen that Eq. (7) has a good accuracy. Fig. 7(a) (c) shows the first-order buckling mode shapes of I-girders of a/h w = 2.0, s s = 0.2, t w = 4mm,β is 0.25, 6.75 and 128, respectively. It is worth noting that as the value of β increases, the buckling mode shapes are transformed from that of simply supported plate to clamped plate. 4. Elastic buckling of webs in I-girders under wheel load 4.1. The distribution of vertical compressive stresses on the edge of the web plates According to the authors previous work [14], the finite element analysis revealed that the girder length and the (7)

T. Ren, G.S. Tong / Engineering Structures 27 (2005) 1528 156 15 Table 4 The elastic buckling coefficients of webs in I-girders under patch loading h w = 1000, ANSYS Eq. (7) Error (%) h w = 1000, ANSYS Eq. (7) Error (%) t w = 4 t w = 4 b f = 250 b f = 250 s s β a/h w a/h w a/h w s s β a/h w a/h w a/h w 1.0 2.0 4.0 1.0 2.0 4.0 1.0 2.0 4.0 1.0 2.0 4.0 1.0 2.0 4.0 1.0 2.0 4.0 0.00.22 2.5 2.07.0 2.40 2.18 2.48 2.1 5.1 0.00.51 2.46 2.17.5 2.49 2.2 0.57 1.22 2.76 0.0.4 2.47 2.25.48 2.46 2.21 1.40 0.22 1.68 0.0.75 2.59 2.6.72 2.56 2.26 0.89 1.19 4.04 0.25 4.28 2.88 2.56 4.8 2.88 2.4 2.45 0.06 5.07 0.25 4.65.00 2.66 4.67.00 2.50 0.6 0.10 6.00 1.00 5.64.80.14 5.56.79.00 1.5 0.16 4.52 1.00 6.10.96.26 5.90.99.11.24 0.64 4.48 0.0 2.00 6.22 4.49.60 6.06 4.45.50 2.50 0.96 2.89 0. 2.00 6.71 4.70.75 6.4 4.69.65 4.22 0.1 2.62 6.75 6.70 5.57 4.5 6.57 5.45 4.48 1.87 2.10 1.08 6.75 7.25 5.87 4.75 6.96 5.77 4.72.98 1.79 0.72 16.00 6.82 5.98 5.0 6.7 5.87 4.99 1.5 1.87 0.72 16.00 7.8 6.2 5.1 7.12 6.21 5.27.49 1.74 0.76 54.00 6.88 6.21 5.40 6.81 6.1 5.6 0.97 1.2 0.79 54.00 7.45 6.59 5.72 7.21 6.49 5.66.20 1.5 1.00 128.00 6.89 6.27 5.50 6.8 6.20 5.46 0.81 1.15 0.72 128.00 7.47 6.66 5.84 7.2 6.56 5.77.17 1.44 1.1 0.00.26 2.7 2.08. 2.41 2.18 2.15 1.69 4.81 0.00.71 2.5 2.2.70 2.56 2.28 0.27 1.19 2.24 0.0.48 2.49 2.27.51 2.47 2.21 0.82 0.60 2.54 0.0.96 2.66 2.42.89 2.6 2.2 1.68 1.0 4.27 0.25 4. 2.90 2.57 4.42 2.89 2.4 2.0 0.27 5.5 0.25 4.91.08 2.72 4.88.10 2.57 0.62 0.62 5.69 1.00 5.71.82.16 5.60.81.01 1.92 0.14 4.89 1.00 6.41 4.07.4 6.16 4.1.21.88 1.51.81 0.1 2.00 6.0 4.52.62 6.10 4.47.51.14 1.05.10 0.4 2.00 7.06 4.84.84 6.71 4.87.78 5.01 0.56 1.56 6.75 6.79 5.62 4.56 6.61 5.49 4.50 2.60 2.7 1.28 6.75 7.62 6.06 4.89 7.26 6.00 4.90 4.72 0.96 0.26 16.00 6.91 6.0 5.08 6.77 5.91 5.02 2.06 2.07 1.21 16.00 7.76 6.5 5.48 7.4 6.47 5.49 4.28 0.9 0.1 54.00 6.97 6.28 5.45 6.85 6.17 5.9 1.68 1.79 1.18 54.00 7.8 6.82 5.92 7.52 6.76 5.90.96 0.84 0.1 128.00 6.99 6.4 5.56 6.87 6.24 5.49 1.66 1.62 1.26 128.00 7.85 6.89 6.05 7.54 6.84 6.02.91 0.71 0.51 0.00.6 2.41 2.12.40 2.44 2.20 1.19 1.24.77 0.00.97 2.61 2.0.9 2.65 2. 1.01 1.5 1.0 0.0.59 2.5 2.0.58 2.51 2.2 0.22 0.9 2.90 0.0 4.24 2.74 2.50 4.1 2.7 2.7 2.54 0.46 5.2 0.25 4.46 2.94 2.61 4.51 2.9 2.46 1.10 0.2 5.78 0.25 5.24.18 2.80 5.16.22 2.64 1.45 1.8 5.85 1.00 5.86.88.20 5.71.88.05 2.48 0.07 4.77 1.00 6.82 4.20.4 6.50 4.2. 4.62 2.92 2.89 0.2 2.00 6.46 4.59.67 6.2 4.55.56.61 0.86 2.92 0.5 2.00 7.50 5.00.95 7.07 5.11.94 5.67 2.12 0.26 6.75 6.97 5.72 4.6 6.75 5.59 4.58.18 2.0 1.01 6.75 8.09 6.28 5.06 7.65 6.1 5.14 5.8 0.54 1.67 16.00 7.10 6.15 5.17 6.91 6.02 5.11 2.74 2.18 1.08 16.00 8.24 6.79 5.70 7.8 6.81 5.77 4.98 0.2 1.26 54.00 7.17 6.41 5.56 6.99 6.28 5.49 2.48 1.96 1.25 54.00 8.2 7.09 6.17 7.9 7.12 6.22 4.74 0.48 0.75 128.00 7.18 6.47 5.67 7.01 6.6 5.60 2.1 1.76 1.28 128.00 8.4 7.16 6.0 7.95 7.21 6.4 4.67 0.66 0.68 web height have negligible effects on the distribution of vertical compressive stresses on the edge of the web plates of crane runway I-girders. The web thickness and the flexural rigidities of the crane rail and the top flange are the most important factors to influence the stress distribution. Eq. (8) was suggested to evaluate the distribution of the vertical compressive stresses on the edge of the web plates. P σ c = σ cmax f (γ, z) = 2.8 I x tw 2 e γ z (sin γ z + cos γ z). (8) Here I x is the sum of the flexural rigidities of crane rail and the top flange, z is the longitudinal coordinate of I-girders, whose origin is at the middle of I-girders and γ is given by 2 γ = 2.8. (9) I x /t w In Eq. (8), σ cmax is the maximum vertical compressive stress on the edge of the web plate and the function f (γ, z) = e γ z (sin γ z +cos γ z) determines the distribution of the stresses along longitudinal axis of I-girders. Fig. 8 shows the stress distributions of girders with different factor γ.eq.(9) indicates that as I x increases, the factor γ becomes smaller, the vertical compressive stresses disperse to a wider length along the edge of the web plate. This can be observed in Fig. 8. Fig. 9 shows a comparison of Eq. (8) and ANSYS, the cross section of the girder is H800 00 8 16, the member length is 2000 mm. The resultant forces P 1 can be obtained by integrating Eq. (8) over the whole girder length: a/2 P P 1 = a/2 2.8 e γ z (sin γ z + cos γ z) dz. (10) I x tw 2 In Fig. 10, the values of P 1 /P for 17 I-girders of different web thicknesses and aspect ratios investigated in [14] are shown, and they are all close to 1.0. Therefore it may be concluded that Eq. (8) can be used to simulate the variation of vertical compressive stresses along the top edge of web plates. 4.2. Elastic buckling of single rectangular plates The elastic buckling of single rectangular plates was first investigated. The discretization of finite elements and the

154 T. Ren, G.S. Tong / Engineering Structures 27 (2005) 1528 156 (a) β = 0.25. Fig. 8. Stress distribution along the top edge of the web plate. (b) β = 6.75. Fig. 9. Comparison of Eq. (8) and ANSYS. (c) β = 128. Fig. 7. The first-order buckling wave mode of I-girders with different value of β. loading arrangement are shown in Fig. 11. Theloadsare applied on nodes of the top edge of the plate, and their values are calculated from Eq. (8). The elastic buckling coefficients can be obtained by substituting resultant forces into Eq. (1). 4.2.1. Simply supported rectangular plates The boundary conditions are the same as in Section.1. The rectangular plates of a/h w = 1.0 4.0 and γ = Fig. 10. P 1 /P versus web thickness. 0.004 0.010 were analyzed with ANSYS. Based on the analysis results, the following equation is proposed to determine the elastic buckling coefficients of simply supported plates under wheel load: k sc = 2.0 + 1.2 (a/h w ) 2 + e 0.5 γ h w [ 1.5 + 6.5 (a/h w ) 2 ]. (11)

T. Ren, G.S. Tong / Engineering Structures 27 (2005) 1528 156 155 Table 5 Elastic buckling coefficients of simply supported rectangular plates under wheel load k sc (h w = 800 mm) a/h w ANSYS Eq. (11) Error (%) γ γ γ 0.004 0.005 0.006 0.007 0.008 0.010 0.004 0.005 0.006 0.007 0.008 0.010 0.004 0.005 0.006 0.007 0.008 0.010 1.0 4.88 4.48 4.14.86.66.41 4.82 4.28.9.69.5.5 1. 4.40 5.18 4.50.66 1.86 1.2 4.02.65.7.17.0 2.90 4.05.65.8.20.08 2.94 0.68 0.08 0.26 0.92 1.60 1.50 1.4.5.19 2.95 2.81 2.72 2.65.58.26.05 2.91 2.81 2.70 1.55 2.2.6.9.26 1.90 1.5.6.0 2.82 2.70 2.6 2.58.42.1 2.9 2.80 2.71 2.61 1.77.21.95.71.1 1.1 1.6.2 2.92 2.7 2.62 2.57 2.52.28.02 2.84 2.71 2.6 2.54 1.68.27.85.60 2.47 0.90 1.8.04 2.76 2.60 2.52 2.49 2.45.08 2.84 2.69 2.58 2.51 2.4 1.26.08.40 2.52 0.94 0.6 2.0 2.91 2.66 2.5 2.47 2.44 2.40 2.9 2.72 2.58 2.49 2.4 2.6 0.72 2.7 2.11 0.81 0.52 1.78 2.5 2.72 2.5 2.44 2.9 2.6 2.2 2.70 2.54 2.42 2.5 2.0 2.24 0.56 0.2 0.72 1.82 2.7.51.0 2.62 2.47 2.8 2. 2.29 2.25 2.58 2.4 2. 2.27 2.22 2.17 1.45 1.45 1.89 2.64 2.89.8.5 2.54 2.40 2.1 2.26 2.22 2.18 2.51 2.7 2.28 2.22 2.18 2.14 1.26 1.1 1.20 1.71 1.77 2.06 4.0 2.47 2. 2.24 2.19 2.16 2.12 2.46 2. 2.25 2.19 2.15 2.11 0.41 0.1 0.5 0.04 0.4 0.48 Table 6 Elastic buckling coefficients of clamped rectangular plates under wheel load k fc (h w = 800 mm) a/h w ANSYS Eq. (12) Error (%) γ γ γ 0.004 0.005 0.006 0.007 0.008 0.010 0.004 0.005 0.006 0.007 0.008 0.010 0.004 0.005 0.006 0.007 0.008 0.010 1.0 9.94 9.17 8.52 8.00 7.60 7.1 9.4 8.74 8.28 7.95 7.70 7.5 5.10 4.69 2.84 0.65 1.2.14 1.2 8.98 8.20 7.61 7.18 6.89 6.62 8.90 8.24 7.81 7.50 7.26 6.94 0.91 0.54 2.60 4.41 5.42 4.78 1.4 8.58 7.78 7.2 6.88 6.68 6.51 8.55 7.92 7.51 7.21 6.98 6.67 0.1 1.86.81 4.74 4.52 2.42 1.5 8.47 7.66 7.1 6.82 6.65 6.50 8.42 7.80 7.9 7.10 6.87 6.56 0.58 1.86.64 4.0.7 0.99 1.6 8.9 7.58 7.07 6.79 6.6 6.50 8.1 7.70 7.29 7.00 6.78 6.48 0.99 1.54.10.07 2.27 0.8 1.8 8.25 7.44 6.97 6.7 6.60 6.46 8.11 7.52 7.12 6.84 6.62 6. 1.64 1.05 2.16 1.59 0.6 2.08 2.0 8.08 7.28 6.87 6.65 6.52 6.8 7.95 7.7 6.98 6.70 6.49 6.20 1.56 1.22 1.59 0.77 0.42 2.82 2.5 7.58 6.92 6.59 6.40 6.27 6.11 7.61 7.05 6.68 6.41 6.21 5.9 0.4 1.9 1.7 0.22 0.89 2.87.0 7.15 6.61 6.1 6.12 5.98 5.8 7.0 6.76 6.40 6.15 5.96 5.69 2.04 2.27 1.46 0.44 0.41 2.45.5 6.7 6.28 5.99 5.81 5.68 5.5 6.97 6.46 6.11 5.87 5.69 5.4.5 2.79 2.07 1.04 0.1 1.78 4.0 6.9 5.91 5.65 5.48 5.7 5.2 6.61 6.1 5.80 5.57 5.40 5.15.48.67 2.70 1.67 0.52 1.44 coefficients of clamped plates under wheel load: ( k fc = 0.78 + 1.45 ) γ h w [ ( ) a 2 6.4 0.07 + 1.2 ] h w (a/h w ) 2. (12) In Table 6,Eq.(12) is compared with ANSYS and the errors are again very small. Fig. 11. The finite element model of rectangular plate under wheel load. In Table 5, elastic buckling coefficients calculated from Eq. (11) are compared with those from ANSYS and the errors are within 5%. 4.2.2. Clamped rectangular plates The boundary conditions are the same as in Section.2. The rectangular plates of a/h w = 1.0 4.0 and γ = 0.004 0.010 were analyzed with ANSYS. The following equation is proposed to determine the elastic buckling 4.. Elastic buckling coefficients of web plates in I-girders As discussed previously, the wheel load can be simulated by applying node loads in the form of Eq. (8) onthetop edge of web plates in I-girders. Therefore, considering the rotational restraint provided by the top flange and crane rail, the same method as that in Section. was used to analyze the buckling of web plates in I-girders under wheel load. The buckling analysis was carried out to investigate almost 800 I-girders with web thickness of t w = 8 mm, flange thicknesses of t f = 1 196 mm, web plate aspect ratio of a/h w = 1.0 4.0 and load factor of γ = 0.004 0.010.

156 T. Ren, G.S. Tong / Engineering Structures 27 (2005) 1528 156 5. Conclusion (a) γ = 0.006. New investigation has been carried out to simulate the realistic load and the restraining conditions of the webs in I-girders. The buckling of a large number of models under patch load was analyzed with ANSYS and formulae were proposed to predict the elastic buckling coefficients of web plates in I-girders. The rotational restraints provided to the web by the flanges are considered accurately in the suggested formulae. The wheel load is another kind of patch load rarely touched in the literature. According to the authors previous work, the bearing stresses on the top edge of the web plates in I-girders where the effect of the crane rail rigidity is considered can be determined by a simple model. Based on this model, the buckling of the web plate is analyzed and formulae with excellent accuracy were suggested to predict the buckling load. References (b) γ = 0.008. Fig. 12. Elastic buckling coefficients of web plates in I-girders under wheel load k crc. The resulting data of the representative series of γ = 0.00 and 0.008 are plotted in Fig. 12. Introducing a factor λ c to take the influence of the web plate aspect ratios into account, the following equation is proposed to determine the elastic buckling coefficients of web plates in I-girders under wheel loads: k crc = k sc + k fc λ c β (1) 1 + λ c β a/h where λ c = w 4.25a/h w.92, β = b ftf, k h w tw sc and k fc are the elastic buckling coefficients of simply supported and clamped plates given by Eqs. (11)and(12). The elastic buckling coefficients of the models with γ = 0.006 and 0.008 computed by Eq. (1)areshowninFig. 12. It can be seen that the Eq. (1) has good accuracy and it can be well used to evaluate the elastic buckling coefficients of web plates with aspect ratio a/h w < 2.0, which are frequently used in practice engineering. [1] Lagerqvist O. Patch loading-resistance of steel girders subjected to concentrated forces. Sweden: Lulea University of Technology; 1994. [2] Zetlin L. Elastic instability of flat plates subjected to partial edge loads. Proceedings, ASCE 1955;81(1): No. 795. [] White RN, Cottingham WS. Stability of plates under partial edge loadings. Proceedings, ASCE 1962;85(5):67 86. [4] Rockey KC, Bagchi DK. Buckling of plate girder webs under partial edge loadings. International Journal of Mechanics Science 1970;12: 61 76. [5] Khan MZ, Walker AC. Buckling of plates subjected to localized edge loadings. The Structural Engineer 1972;50(6):225 2. [6] Khan MZ, Johns KC. Buckling of web plates under combined loadings. Journal of Structural Division, ASCE 1975;101(10): 2079 92. [7] Khan MZ, Johns KC, Hayman B. Buckling of plates with partially loaded edges. Journal of Structural Division, ASCE 1977;10(): 547 58. [8] Shahabian F, Roberts TM. Buckling of slender web plates subjected to combinations of in-plane loading. Journal of Constructional Steel Research 1999;51:99 121. [9] Moriawaki Y, Takinoto T, Mimura Y. Ultimate strength of girders under patch loading. Transactions, JSCE 1985;15:187 90. [10] Graves Smith TR, Sridharan S. A finite strip method for the buckling of plate structures under arbitrary loading. International Journal of Mechanics Science 1978;20:685 9. [11] Graves Smith TR, Gierlinski JT. Buckling of stiffened webs by local edge loads. Proceedings, ASCE 1982;108(6):157 66. [12] Chin C-K, Al-Bermani FGA, Kitipornchai S. Finite element method of buckling analysis of plate structures. Journal of Structural Engineering 199;119(4):1048 68. [1] Graciano C, Lagerqvist O. Critical buckling of longitudinally stiffened webs subjected to compressive edge loads. Journal of Constructional Steel Research 200;59:1119 46. [14] Ren T, Tong G. Bearing stress of crane runway girders under wheel loads. Journal of Zhejiang University, Engineering Science 2005; 9(10) [in press] [in Chinese].