# 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
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{_I_**(m)**}} & \multicolumn{4}{c}{**CL**} & \multicolumn{2}{c}{**CUDL**} \\  & \multicolumn{1}{c}{\(M_{\text{FEM (kN-m)}}\)} & \(M_{\text{th (kN-m)}}\) & \(M_{\text{th}}\)/\(M_{\text{FEM}}\) & \(M_{\text{FEM (kN-m)}}\) & \(M_{\text{th (kN-m)}}\) & \(M_{\text{th}}\)/\(M_{\text{FEM}}\) \\ \hline \multirow{2}{*}{1.5} & TF & 38.31 & 41.18 & 1.07 & 46.37 & 49.27 & 1.06 \\  & SC & 98.06 & 99.04 & 1.01 & 118.50 & 120.40 & 1.02 \\  & BF & 143.09 & 141.38 & 0.99 & 181.10 & 179.36 & 0.99 \\ \hline \multirow{2}{*}{2} & TF & 31.34 & 32.94 & 1.05 & 37.08 & 39.10 & 1.05 \\  & SC & 65.28 & 64.04 & 0.98 & 77.88 & 77.40 & 0.99 \\  & BF & 87.48 & 84.61 & 0.97 & 108.96 & 106.63 & 0.98 \\ \hline \multirow{3}{*}{3} & TF & 23.22 & 23.90 & 1.03 & 27.00 & 28.01 & 1.04 \\  & SC & 36.06 & 35.65 & 0.99 & 42.12 & 42.73 & 1.01 \\  & BF & 43.98 & 43.15 & 0.98 & 54.27 & 53.66 & 0.99 \\ \hline \multirow{3}{*}{4} & TF & 18.20 & 18.51 & 1.02 & 21.12 & 21.56 & 1.02 \\  & SC & 24.32 & 24.13 & 0.99 & 28.56 & 28.76 & 1.01 \\ \cline{1-1}  & BF & 28.28 & 27.88 & 0.99 & 34.56 & 34.27 & 0.99 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The critical moment predicted by finite element analysis (\(M_{\text{FEM}}\)) and theoretically predicted (\(M_{\text{th}}\)) critical moment for Section II—reinforced top flange. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Specimen** & \(M_{\text{ex (N-m)}}\) & \(M_{\text{th (N-m)}}\) & \(M_{\text{th}}\)/\(M_{\text{ex}}\) & **Specimen** & \(M_{\text{ex (N-m)}}\) & \(M_{\text{th (N-m)}}\) & \(M_{\text{th (N-m)}}\) & \(M_{\text{th (M_{ex}}\)}\)/\(M_{\text{ex (N-m)}}\) \\ \hline [PERSON] et al. 1972 [10] & & & & [PERSON] et al. 1990 [11] & & \\
1 Ae65 & 438.44 & 405.1 & 0.92 & 1 TA & 253. In Table 2, compared with the experimental results, the maximum value of the \(M_{\text{th}}/M_{\text{ex}}\) ratio is 1.04, and the minimum value is 0.92. In Tables 3-5, there are four sets of data with significant differences. These data came from SC under CL in Table 4. In reference [16], the distance between the SC and TF of the section II with a reinforced top flange is very small and the differences between their \(M_{\text{cr}}\) should not be significant. However, the differences in Table 4 are particularly significant and it can be concluded from reference [16] that \(M_{\text{th}}\) is more accurate than \(M_{\text{FEM}}\). Therefore, these four sets of data are not included in the analysis scope. In Tables 3-5, compared with the data of finite element analysis from reference [16], the maximum value of the \(M_{\text{th}}/M_{\text{FEM}}\) ratio is 1.07, and the minimum value is 0.97. The average of all the ratios is 0.98 with a standard deviation of 0.13. Therefore, critical moment can be accurately solved by the numerical method proposed in this paper. The predictions of the matrix equation and the data from existing literature are compared in Figure 5. The comparison method in Figure 5 uses the modified version of demerit point classification (DPC) [40]. DPC is used to evaluate the predictions of the matrix equation in terms of safety, accuracy and economic aspects. A specific explanation of the modified version of DPC can be found in Table 6. According to Table 6 and Figure 5, a penalty is assigned to each range of the \(M_{\text{th}}/M_{\text{ex}}\) or \(M_{\text{th}}/M_{\text{FEM}}\) ratio. The predictions based on Equation (34) are all safe in Figure 5. To sum up, the predictions of 100% of the specimens are within the appropriate safety range (total penalty = 0). Therefore, it can be concluded that Equation (34) resulted in a positive prediction in terms of safety, accuracy and economic aspects. Figure 5: Prediction capability of Equation (34) based on modified DPC [40]: **(a)** comparison with the experimental data from reference [10; 11; 18]; **(b)** comparison with the data of finite element analysis from reference [16]. (D: dangerous; AS: appropriate safety; C: conservative; EC: extra conservative).