## 3 Numerical Analysis
The Assign Element Type Command was used for assigning the S8R5 mesh element to flanges and the web. With the Seed Edges command, the flanges and web were divided into finite elements. Using the Create Mesh Part command, an independent part was created based on the meshed geometry. The original CAD geometry was discarded, and further geometric operations such as partitioning, sketching, and extrusion could not be performed. However, nodes, shell elements, and sets remained in the new part. The new part was imported into the Assembly module. The Linear Perturbation procedure and Buckle were chosen from the Step menu for lateral-torsional buckling analysis. Loads were applied to the nodes. Boundary conditions were applied to nodes located in the two end cross-sections. Vertical and lateral translations that were perpendicular to the beam's longitudinal axis were restrained. Finally, the analysis was executed, and the buckling load results were visualized. Tables 3-6 show the elastic LTB loads of web-tapered I-section beams subjected to six different types of loads and three load height levels, including the top (TF) and bottom flanges (BF) and shear center (SC) loadings, calculated using LTBWTB and FEA. The R in Tables 3-6 refers to the ratio of LTBWTB to FEA. Figure 7 illustrates the LTB shapes of web-tapered beams with I-sections under the point load at their mid-span. Figure 6: The shell finite element model of web-tapered beam 2 (WTB2). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Section**} & \multicolumn{8}{c}{**WTB2**} \\ \hline \multirow{2}{*}{**Load Types**} & \multicolumn{3}{c}{**TF**} & \multicolumn{3}{c}{**SC**} & \multicolumn{3}{c}{**BF**} \\ \cline{2-9}  & **LTBWTB** & **FEA** & **R** & **LTBWTB** & **FEA** & **R** & **LTBWTB** & **FEA** & **R** \\ \hline P (kN) & 44.7 & 42.7 & 1.05 & 48.7 & 46.5 & 1.05 & 82.2 & 79.0 & 1.04 \\ \hline q (kN/m) & 13.0 & 12.8 & 1.02 & 14.0 & 13.7 & 1.02 & 21.8 & 21.5 & 1.01 \\ \hline q + 0.5 qL (kN/m) & 7.0 & 6.8 & 1.03 & 7.5 & 7.3 & 1.03 & 12.2 & 12.0 & 1.02 \\ \hline q + qL (kN/m) & 4.8 & 4.6 & 1.03 & 5.1 & 5.0 & 1.03 & 8.5 & 8.2 & 1.03 \\ \hline P + P (kN) & 27.8 & 26.9 & 1.03 & 29.9 & 29.0 & 1.03 & 47.8 & 46.4 & 1.03 \\ \hline M (kNm) & - & - & - & 65.7 & 65.6 & 1.00 & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The LTB loads of WTB2 obtained by LTBWTB and FEA. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Section**} & \multicolumn{8}{c}{**WTB1**} \\ \hline \multirow{2}{*}{**Load Types**} & \multicolumn{3}{c}{**TF**} & \multicolumn{3}{c}{**SC**} & \multicolumn{3}{c}{**BF**} \\ \cline{2-9}  & **LTBWTB** & **FEA** & **R** & **LTBWTB** & **FEA** & **R** & **LTBWTB** & **FEA** & **R** \\ \hline P (kN) & 48.3 & 48.3 & 1.00 & 69.9 & 68.5 & 1.02 & 100.4 & 99.6 & 1.01 \\ \hline q (kN/m) & 14.4 & 14.7 & 0.98 & 19.3 & 19.6 & 0.98 & 25.8 & 26.1 & 0.99 \\ \hline q + 0.5 qL (kN/m) & 7.6 & 7.7 & 0.99 & 10.6 & 10.6 & 1.00 & 14.6 & 14.7 & 1.00 \\ \hline q + qL (kN/m) & 5.2 & 5.2 & 1.00 & 7.3 & 7.3 & 1.00 & 10.2 & 10.2 & 1.00 \\ \hline P + P (kN) & 30.7 & 30.7 & 1.00 & 42.0 & 41.5 & 1.01 & 57.3 & 57.0 & 1.00 \\ \hline M (kNm) & - & - & - & 76.2 & 76.1 & 1.00 & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The LTB loads of WTB1 obtained by LTBWTB and FEA. Figure 7: The buckled shape of web-tapered I-beams under concentrated load (Units are in N). An examination of Tables 3-6 shows that the maximum difference between the LTBWTB and shell-based FEA results reached 5%, with an average difference of only 1%. It is considered that there is a strong correlation between analytical and numerical results for LTB problems [13; 14; 32; 38; 42; 44]. The comparative LTB loads presented in Tables 3-6 indicate that LTBWTB can calculate the elastic LTB loads of web-tapered beams with both doubly symmetric and monosymmetric I-sections entirely in accord with the FEA across all six transverse load types and three load height levels (TF, SC, BF).