# Lateral-Torsional Buckling of Cantilever Steel Beams under 2 Types of Complex Loads
## 3 Verification and Analysis of Matrix Equation  
### Verification of the Results and Discussion  
The proposed matrix equation is verified according to the experimental data [10; 11; 18] and finite element analysis data [16] of cantilever beams in existing literature. The dimensions of specimens are shown in Figure 4 and Table 1.  
Figure 3: Flowchart of the procedure. Because it is difficult to obtain the LTB load of a cantilever beam under the uniformly distributed load and other complex loads, most lateral-torsional buckling tests applied concentrated load at the free end of the cantilever beam. In order to verify the accuracy of the method in this paper and the applicability of the total potential energy equation based on the experimental data, it is necessary to make \(\alpha=0\) and \(\lambda=1\). At this time, the load case of the cantilever beam became a concentrated load at the free end. The experimental (\(M_{\text{ex}}\)) [10; 11; 18] and theoretically predicted (\(M_{\text{th}}\)) critical moment and the \(M_{\text{th}}/M_{\text{ex}}\) ratio are given in Table 2 for all the selected specimens. The specimens in reference [18] are not numbered, so the vertical position of the concentrated load is used instead of the specimen numbers. At the same time, as the experimental data can only verify the load case of CL, it is necessary to extract the data of finite element analysis from existing literature to verify the load case of CUDL. The critical moment calculated by finite element analysis (\(M_{\text{FEM}}\)) from reference [16] and theoretically predicted (\(M_{\text{th}}\)) critical moment and the \(M_{\text{th}}/M_{\text{FEM}}\) ratio are given in Tables 3-5 for all the selected specimens. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Reference** & **Section** & \(\mathbf{b_{1}}\) & \(\mathbf{t_{1}}\) & \(\mathbf{b_{2}}\) & \(\mathbf{t_{2}}\) & \(\mathbf{h}\) & \(\mathbf{f_{\text{w}}}\) & \(\mathbf{E}\) & \(\mathbf{G}\) \\  & & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(N/mm\({}^{2}\))** & **(N/mm\({}^{2}\))** \\ \hline
[10] & 1 & 31.52 & 3.13 & 31.52 & 3.13 & 72.44 & 2.19 & 69,400 & 26,000 \\  & 2 & 31.52 & 3.13 & 15.88 & 3.13 & 72.44 & 2.19 & 69,400 & 26,000 \\  & 3 & 31.47 & 3.14 & 31.45 & 1.15 & 71.44 & 2.19 & 69,400 & 26,000 \\ \hline  & 1 & 31.75 & 3.2 & 15.63 & 3.16 & 72.89 & 2.87 & 65,700 & 24,500 \\  & 2 & 31.67 & 2.18 & 15.63 & 3.118 & 72.711 & 2.25 & 64,800 & 24,200 \\  & 3 & 31.68 & 3.17 & 16.06 & 3.19 & 72.86 & 2.90 & 65,400 & 24,400 \\  & 4 & 31.75 & 3.21 & 15.75 & 3.19 & 72.75 & 2.89 & 65,400 & 24,400 \\ \hline
[18] & & 31.4 & 3.1 & 31.4 & 3.1 & 72.60 & 2.2 & 69,400 & 26,000 \\ \hline  & Section I & 82 & 7.4 & 82 & 7.4 & 152.6 & 5 & 200,000 & 76,923 \\  & Section II-reinforced & 82 & 7.4 & 41 & 7.4 & 152.6 & 5 & 200,000 & 76,923 \\  & top flange & & 41 & 7.4 & 82 & 7.4 & 152.6 & 5 & 200,000 & 76,923 \\  & Section II-reinforced & 41 & 7.4 & 82 & 7.4 & 152.6 & 5 & 200,000 & 76,923 \\  & bottom flange & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensions of specimens. Figure 4: Symbolic description of cross-sectional dimensions.