# Lateral-Torsional Buckling of Cantilever Steel Beams under 2 Types of Complex Loads
## 3 Verification and Analysis of Matrix Equation
### Comparative Analysis between Matrix Equation and Equations Proposed in Other References  
In past studies, many scholars have also proposed the equation of the \(M_{\text{cr}}\) of cantilever beam, such as references [12; 17]. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(M_{\text{th}}/M_{\text{ex}}\) or \(M_{\text{th}}/M_{\text{FEM}}\) & **Classification** & **Penalty (PEN)** \\ \hline \(>\)2 & Extra dangerous & 10 \\ \([1.176\text{--}2]\) & Dangerous & 5 \\ \([0.869\text{--}1.176]\) & Appropriate safety & 0 \\ \([0.5\text{--}0.869]\) & Conservative & 1 \\ \(\leq\)0.5 & Extra conservative & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Modified version of the Demerit Points Classification (DPC) criteria. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(n\)} & \multicolumn{5}{c}{**Load Cases: CUDL, Section: II—Reinforced Top Flange**} \\  & \multicolumn{5}{c}{\(M_{\text{th}}/\text{kN\text{\textminus}}\)} & \multicolumn{5}{c}{**Error**} \\  & **TF** & **SC** & **BF** & **TF** & **SC** & **BF** \\ \hline \(1\) & 30.47 & 35.29 & 116.13 & 164.52\% & 188.35\% & 680.47\% \\ \(2\) & 14.06 & 14.81 & 20.18 & 22.03\% & 20.97\% & 35.62\% \\ \(5\) & 11.89 & 12.35 & 15.20 & 3.22\% & 0.92\% & 2.16\% \\ \(10\) & 11.78 & 12.24 & 15.05 & 2.28\% & 0.02\% & 1.13\% \\ Convergence term & 11.77 & 12.22 & 15.03 & 2.16\% & 0.14\% & 0.12\% \\ \hline \(M_{\text{FEM}}/\text{kN\textminus}\)m & 11.52 & 12.24 & 14.88 & ———————————————————————————————————  
The method used in reference [17] is the finite difference approach. It established the governing equation of cantilever beams using the [PERSON] method. The finite difference method was used to discretize the beam model and solve \(M_{\text{cr}}\). In order to compare the advantages and disadvantages of the equations in reference [12; 17] and this paper, section I in Table 1 is selected for calculating \(M_{\text{cr}}\). The lengths of the cantilever is 1.5, 2, 3 and 4 m, respectively, and the other properties of the beam are shown in Table 1. The concentrated load or uniformly distributed load is applied at the TF, SC and BF of the beam. The specific calculation results are shown in Figure 7. In Figure 7, the horizontal coordinates represent different load cases. When the numbers are 1, 2, 3 and 4, the position of the load is TF and the lengths of the beams are 1.5 m, 2 m, 3 m, and 4 m, respectively. When the numbers are 5, 6, 7 and 8, the position of the load is SC and the lengths of the beams are 1.5 m, 2 m, 3 m, and 4 m, respectively. When the numbers are 9, 10, 11, and 12, the position of the load is BF and the lengths of the beams are 1.5 m, 2 m, 3 m, and 4 m, respectively. It can be seen from Figure 7 that the calculation results of Equation (34) are similar to those of reference [12], but are slightly different from those of reference [17]. Through comparison and analysis, it can be concluded that Equation (34) in this paper has the following advantages: (i) Equation (34) has higher accuracy than reference [17]; (ii) Equation (34) can calculate \(M_{\text{cr}}\) under CUDL, CL or uniformly distributed load, while the equation proposed in reference [12] can only calculate \(M_{\text{cr}}\) under a single load and the equation proposed in reference [17] can only calculate \(M_{\text{cr}}\) under CL and CUDL when the concentrated load is limited to the free end; (iii) Equation (34) considers the ratio of uniformly distributed load to concentrated load, the symmetry of the section, the action position of the concentrated load in the direction of the beam length and the vertical position of two loads. It considers very comprehensive factors and can calculate \(M_{\text{cr}}\) in all cases. The specific comparison results can be seen in Tables 8-11. In Tables 8-11, \"\(\surd\)\" indicates that it can be calculated and \"\(\times\)\" indicates that it cannot be calculated.