## 3 Finite Element Modeling  
The elastic critical LTB of steel beams subjected to transverse loading with negative end moments was achieved using the finite element analysis (FEA) program ABAQUS [31], applying the four-node shell element (S4R), which was utilized to model the investigated steel beam. It should be mentioned that the S4R element, with six degrees of freedom at each node, which was also used in previous studies [32; 33; 34; 35; 36; 37], can accurately model the elastic buckling behavior of steel beams and provide quick results with high accuracy. In addition, the steel beam with an H700x300 cross-section was chosen in this study according to [38], with detailed properties listed in Table 1. Regarding the cross-section size of the beams, the beam height (\(h\)), flange width (\(b_{f}\)), flange thickness (\(t_{f}\)) and web thickness (\(t_{w}\)) were 700, 300, 24 and 13 mm, respectively, as shown in Figure 2. In addition, the mechanical properties of the beams, i.e., elastic modulus (\(E\)) and yield stress (\(f_{y}\)), are 210 and 275 Mpa, respectively. Meanwhile, the Poisson's ratio (\(m\)) was 0.3, and all the geometric and mechanical properties of the beams were taken from [38]. In order to consider the various beam lengths, a range of length-to-height ratios (_L/h_) from 10 to 40 with an interval of 5 was employed. Moreover, a convergence analysis was conducted to determine the proper mesh size of the steel beam models. Furthermore, based on the analysis results, the meshing of beam models was achieved with 30 mm \(\times\) 30 mm elements. The elastic critical LTB of steel beams was investigated by using eigenvalue analyses. This study aimed to investigate the LTB strength of simply supported beams subjected to transverse loadings on top flanges with negative end moments. In addition, the beam model and boundary condition of the simply supported beam are given in Figure 3. Here, both tips of the bottom flanges and both centroids at the ends of the web were fixed to ensure the prevention of vertical displacements. Likewise, lateral displacements for both tips of the web were also prevented. Moreover, the displacement along the axial direction of one centroid at the ends of the web was prevented for the hinge support and another one was free at the position of the roller support. Figure 4 shows the loading conditions used in this study. It can be seen that the steel beams were subjected to a uniformly distributed load or concentrated load on the top flange with negative end moments, which are denoted LC\({}_{1}\) and LC\({}_{2}\), respectively. In Figure 4, \(M_{1}\) and \(M_{2}\) are the greater and smaller moments at the ends of the beam, and \(\alpha\) is the ratio of \(M_{1}\) taking into account the end moment variation with the values of \(\alpha\) as 0, 0.5, 0.75 and 1. In addition, \(M_{q}\) and \(M_{p}\) are, respectively, the moment at the midspan of the simple beam triggered by the uniformly distributed load and the concentrated load with respect to unbraced length \(L\). To take into account the effect of the variation in the negative end moments and transverse loadings, the \(M_{q}\)/\(M_{1}\) and \(M_{p}\)/\(M_{1}\) ratios were investigated ranging from 0.5 to 3 with an interval of 0.5 in this study. The negative end moments were simulated by using tension forces at the top flange and compression forces at the bottom flange, as shown in Figure 5. In addition, the typical buckling mode shapes are illustrated in Figure 6. Additionally, for the case of the \(M_{q}\)/\(M_{1}\) and \(M_{p}\)/\(M_{1}\) ratios greater than 1, the buckling of the beam occurs in the top flange, as shown in Figure 6a. In the case of \(M_{q}\)/\(M_{1}\) and \(M_{p}\)/\(M_{1}\) ratios less than 1, the buckling of the beam occurs in the bottom flange (i.e., compression flange), as shown in Figure 6b.