## 4 Conclusions  
Structural designers often resort to finite element analyses (FEA) to determine the lateral-torsional buckling (LTB) loads of web-tapered simply supported doubly symmetric and mono-symmetric I-section beams under various transverse loads, as there is no detailed and easily applicable analytical method or closed-form formula available in the literature. However, conducting an FEA requires specialized expertise due to the complexity of several steps, including geometry generation, material property definition, section assignment, finite element meshing, the application of loads and boundary conditions, analysis execution, and result interpretation. Moreover, the increased modeling and computation time poses a significant challenge for practical applications. This study proposes an analytical method based on an energy approach to compute the LTB load of web-tapered simply supported I-section beams. The analytical method accounts for the monosymmetry of the I-section, the moment distribution through the beam due to transverse loads, load height level along the cross-section, web tapering levels, and slenderness effects. The analytical method was programmed with Mathematica software, and a package named LTBWTB was developed. Through the LTBWTB, only the geometric dimensions of the web-tapered beam and elastic material properties are required to determine the LTB load under the given loading type and position. In comparison to shell-based FEA, LTBWTB offers a faster and more practical means of evaluating the LTB load of web-tapered simply supported I-section beams. A numerical study was conducted in which the LTB loads were computed using both LTBWTB and FEA. The maximum and average differences between the two methods were 5% and 1%, respectively. The results demonstrate that LTBWTB accurately accounts for I-section mono-symmetry, loading type, loading position, web tapering level, and slenderness in LTB load calculation. The strong agreement between the LTBWTB  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Q Load on Shear Center (kN)** & \\ \hline
**Section** & **PS** & **AS** & **R** \\ \hline WTB1 & 19.3 & 18.9 & 1.02 \\ \hline WTB2 & 14.0 & 14.1 & 0.99 \\ \hline WTB3 & 7.6 & 7.5 & 1.01 \\ \hline WTB4 & 5.5 & 5.6 & 0.98 \\ \hline \hline \end{tabular}
\end{table}
Table 7: The LTB loads of tapered beams subjected to q loads on their shear center obtained from LTBWTB and [PERSON] et al. [38].  
results and the shell-based FEA results confirms that the developed software can be reliably used for determining the LTB load of web-tapered I-section beams.  
The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app15137572/s1](https://www.mdpi.com/article/10.3390/app15137572/s1)]([https://www.mdpi.com/article/10.3390/app15137572/s1](https://www.mdpi.com/article/10.3390/app15137572/s1)), File LTBWTB: A Mathematica Software to Evaluate the Lateral-Torsional Buckling Load of Web-Tapered Mono-Symmetric I-Section Beams.  
This research received no external funding.  
Not applicable.  
The original data presented in the study are openly available in article/Supplementary Material. Further inquiries can be directed to the author.  
The author declares no conflict of interest.