## 3 Numerical Analysis  
In numerical analysis, four different simply supported web-tapered beams with I-sections were designed to validate the LTB loads obtained from LTBWTB. Two had doubly symmetric I-sections, and the other had mono-symmetric I-sections. The lengths of the  
Figure 4: The inputs and outputs of the LTBWTB software. Figure 5: The flow chart of the LTBWTB software. beams were 6.0 m and 8.0 m, resulting in beams with different slenderness ratios. The geometric dimensions of the four simply supported web-tapered beams with I-sections are presented in Table 1. In the numerical examples, the elastic modulus was assumed to be 210,000 MPa. Poisson's ratio was 0.3. The LTB loads for all beams were computed under six different transverse load types, which are applied to the top and bottom flanges and the shear center. Shell-based finite element analysis (FEA) using the S8R5 element, which accounts for warping effects, was performed in ABAQUS/CAE (version 2021) software to assess the LTB of simply supported web-tapered beams with I-sections. The S8R5 shell element is an 8-node doubly curved thin shell with reduced integration using five degrees of freedom per node. Element type S8R5 has three displacements and two rotation variables at an internally generated midbody node [48]. Several studies have demonstrated that the S8R5 shell element yields accurate results for LTB evaluation [11; 13; 14; 22; 30; 32; 44]. The formulation of such reduced-integration quadrilateral shell elements and their capability to capture instability phenomena, including local-global mode interaction, is extensively detailed in the study of [PERSON] et al. [49]. The beams were discretized into 60, 8, and 4 mesh elements along the length, web, and flange, respectively. Additionally, further mesh refinement did not significantly affect LTB loads, although it increased computation time. While eight and four meshes along the web and flange were kept constant, the mesh through the beam's length was changed for the WTB1, subjected to a point load on its top flange, for the mesh convergence study. The mesh convergence results are summarized in Table 2.  
For instance, the finite element model of the web-tapered beam 2 (WTB2) is presented in Figure 6. In the modeling phase, first, the 2D geometry of the tapered web was generated. Flange widths were arranged with the Shell Extrude command by selecting the top and \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Properties** & **WTB1** & **WTB2** & **WTB3** & **WTB4** \\ \hline \multirow{2}{*}{I-section} & doubly & mono- & doubly & mono- \\  & symmetric & symmetric & symmetric & symmetric \\ \hline b\({}_{\text{f1}}\) (mm) & 150 & 150 & 150 & 150 \\ \hline b\({}_{\text{f2}}\) (mm) & 150 & 75 & 150 & 75 \\ \hline t\({}_{\text{f1}}\) (mm) & 10.7 & 10.7 & 10.7 & 10.7 \\ \hline t\({}_{\text{f2}}\) (mm) & 10.7 & 10.7 & 10.7 & 10.7 \\ \hline t\({}_{\text{w}}\) (mm) & 7.1 & 7.1 & 7.1 & 7.1 \\ \hline d\({}_{\text{i}}\) (mm) & 240 & 240 & 240 & 240 \\ \hline d\({}_{\text{m}}\) (mm) & 300 & 300 & 300 & 300 \\ \hline L (mm) & 6000 & 6000 & 8000 & 8000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The geometric dimensions of web-tapered I-beams. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mesh Elements Along the Length** & **120** & **60** & **30** \\ \hline Number of nodes & 6033 & 3033 & 1533 \\ \hline Number of elements & 1920 & 960 & 480 \\ \hline P\({}_{\text{cr}}\) (N) & 48,326 & 48,343 & 48,503 \\ \hline Time (s) & 4.7 & 2.1 & 1.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The results of the mesh convergence study. bottom lines of the web in two directions. After the Shell Extrude operation, the model consisted of three shell parts. This process allowed for three sections to be connected from their intersections without any extra constraint properties, such as tie contact. Homogeneous shell sections were modeled for flanges and web by defining elastic material properties and shell thickness. Shell sections generated were assigned to the flanges and the web.