## 3 Numerical Analysis
Additionally, it can be concluded that LTBWTB can capture the slenderness effect, as the LTB loads of WTB3 and WTB4, which were relatively more slender than WTB1 and WTB2, respectively, also aligned with those obtained by FEA. The average and maximum differences between the LTB loads calculated by LTBWTB and FEA for WTB1 and WTB2 were 1% and 3%, respectively. These differences were 1% and 2% for the more slender beams, WTB3 and WTB4, respectively. Despite there being negligible differences between the elastic LTB loads obtained from LTBWTB and FEA, the LTB loads determined with LTBWTB, based on the 1D frame element formulation using the Ritz method, show greater agreement with shell-based FEA as slenderness increased, and the differences tended to diminish. This tendency occurred because the one-dimensional frame element assumption became more valid as the beams' length increased, thereby increasing slenderness. In the analysis of LTB behavior for relatively short beams, the one-dimensional frame element model underlying LTBWTB may not adequately capture localized web buckling or significant web/flange distortions near the load application region, which are more accurately represented in the FEA [30; 32; 42; 44]. For further discussion on the interaction between local plate instability and global buckling, refer to the classical works of [PERSON] and [PERSON] [50; 51], which are available for consultation. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Section**} & \multicolumn{8}{c}{**WTB4**} \\ \hline \multirow{2}{*}{**Load Types**} & \multicolumn{2}{c}{**TF**} & \multicolumn{2}{c}{**SC**} & \multicolumn{2}{c}{**BF**} \\ \cline{2-10}  & **LTBWTB** & **FEA** & **R** & **LTBWTB** & **FEA** & **R** & **LTBWTB** & **FEA** & **R** \\ \hline P (kN) & 24.3 & 23.4 & 1.04 & 25.9 & 25.0 & 1.04 & 39.6 & 38.4 & 1.03 \\ \hline q (kN/m) & 5.3 & 5.2 & 1.01 & 5.5 & 5.5 & 1.01 & 7.9 & 7.8 & 1.01 \\ \hline q + 0.5 qL (kN/m) & 2.8 & 2.8 & 1.01 & 3.0 & 2.9 & 1.03 & 4.4 & 4.3 & 1.03 \\ \hline q + qL (kN/m) & 1.9 & 1.9 & 1.01 & 2.1 & 2.0 & 1.03 & 3.1 & 3.0 & 1.02 \\ \hline P + P (kN) & 15.0 & 14.6 & 1.03 & 15.9 & 15.5 & 1.02 & 23.0 & 22.5 & 1.02 \\ \hline M (kNm) & - & - & - & 44.5 & 43.9 & 1.01 & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The LTB loads of WTB4 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}{**WTB3**} \\ \hline \multirow{2}{*}{**Load Types**} & \multicolumn{2}{c}{**TF**} & \multicolumn{2}{c}{**SC**} & \multicolumn{2}{c}{**BF**} \\ \cline{2-10}  & **LTBWTB** & **FEA** & **R** & **LTBWTB** & **FEA** & **R** & **LTBWTB** & **FEA** & **R** \\ \hline P (kN) & 27.3 & 27.3 & 1.00 & 36.9 & 36.5 & 1.01 & 49.5 & 49.3 & 1.00 \\ \hline q (kN/m) & 6.0 & 6.1 & 0.99 & 7.6 & 7.8 & 0.98 & 9.6 & 9.8 & 0.98 \\ \hline q + 0.5 qL (kN/m) & 3.2 & 3.2 & 1.01 & 4.2 & 4.2 & 1.00 & 5.4 & 5.5 & 0.99 \\ \hline q + qL (kN/m) & 2.2 & 2.2 & 1.00 & 2.9 & 2.9 & 0.99 & 3.8 & 3.8 & 1.00 \\ \hline P + P (kN) & 17.2 & 17.2 & 1.00 & 22.2 & 22.0 & 1.01 & 28.4 & 28.4 & 1.00 \\ \hline M (kNm) & - & - & - & 53.7 & 53.7 & 1.00 & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The LTB loads of WTB3 obtained by LTBWTB and FEA. Table 7 presents a comparison of the LTB loads of web-tapered beams subjected to distributed load on their shear center, which was obtained from LTBWTB and [PERSON] et al. [38]. The present analytical model is based on the Ritz method, while [PERSON] et al. [38] have utilized [PERSON]'s method. The maximum and average discrepancies between the LTB loads obtained by these two energy-based methods were 2% and 1.5%, respectively. This is another demonstration that LTBWTB accurately predicted the LTB loads of web-tapered beams with I-section. Consequently, the proposed analytical method, along with the developed Mathematica-based software package LTBWTB, can accurately predict the lateral-torsional buckling (LTB) loads of web-tapered I-section beams.