# Lateral-Torsional Buckling of Cantilever Steel Beams under 2 Types of Complex Loads
## 5 Practical Expressions for Design under CL and CUDL
### Example Verification  
Section I, Section II that reinforced bottom flange and their properties in Table 1 are selected for calculating \(M_{\rm cr}\). The length of cantilever is 4 m. It is assumed that the material is elastic and the component is free from defects. \(M_{\rm cr}\) is obtained according to Equation (59) and the finite element analysis respectively when \(\lambda\) takes different values.  
Abaqus software is used to generate finite element models of the cantilever beams. The Finite Element Method (FEM) is applied to this software. The models of cantilever beams were generated in the CAE module and calculations were carried out in the STANDARD module [50]. Cantilever beams are divided to 13,000 finite elements. S8R5 shell elements are used in this model and results are obtained by eigenvalue. In order to keep the original shape of the section and better conform to the rigid perimeter assumption, one stiffener is set for each \(l/6\) in the length direction and a total of 5 stiffeners are set. In order not to create unnecessary constraints on other deformations of the cantilever beams, the stiffener is only rigidly connected with the web, and only couples with the in-plane displacement of top and bottom flanges. Tables 14 and 15 show that the results obtained by Equation (59) and Abaqus are nearly identical at SC. For TF and BF, difference between Abaqus results and Equation (59) results increases as \(\lambda\) decreases. [PERSON] and [PERSON] [13] explained this phenomenon. They verified that 1D frame element assumption loses its validity as the beam length decreases. And the situation of \(\lambda\) decreases is similar to this situation.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline  & & \multicolumn{3}{c}{\(\lambda\) = 1/3} & & & \multicolumn{3}{c}{\(\lambda\) = 1/2} & \\ \(K\) & \(R_{1}\) & \(R_{2}\) & \(R_{3}\) & \(R_{4}\) & \(R_{5}\) & \(R_{1}\) & \(R_{2}\) & \(R_{3}\) & \(R_{4}\) & \(R_{5}\) \\ \hline
2.5 & 0.0168 & \(-\)0.2722 & 0.2426 & 0.6697 & 0.6114 & 0.0244 & \(-\)0.3336 & 0.3024 & 0.7721 & 0.6263 \\
2.22 & 0.0208 & \(-\)0.2570 & 0.2333 & 0.5279 & 0.5562 & 0.0305 & \(-\)0.3168 & 0.2929 & 0.6132 & 0.5718 \\
1.81 & 0.0282 & \(-\)0.2148 & 0.2028 & 0.3323 & 0.4314 & 0.0384 & \(-\)0.2453 & 0.2363 & 0.3581 & 0.4096 \\
1.57 & 0.0298 & \(-\)0.1611 & 0.1573 & 0.2085 & 0.3053 & 0.0411 & \(-\)0.1852 & 0.1849 & 0.2263 & 0.2919 \\
1.4 & 0.0310 & \(-\)0.1268 & 0.1275 & 0.1434 & 0.2289 & 0.0432 & \(-\)0.1465 & 0.1509 & 0.1564 & 0.2203 \\
1.19 & 0.0326 & \(-\)0.0893 & 0.0941 & 0.0844 & 0.1509 & 0.0460 & \(-\)0.1038 & 0.1125 & 0.0926 & 0.1470 \\
1 & 0.0340 & \(-\)0.0607 & 0.0676 & 0.0476 & 0.0965 & 0.0487 & \(-\)0.0709 & 0.0814 & 0.0525 & 0.0954 \\
0.84 & 0.0352 & \(-\)0.0409 & 0.0481 & 0.0267 & 0.0619 & 0.0511 & \(-\)0.0478 & 0.0584 & 0.0295 & 0.0622 \\ \hline \end{tabular}
\end{table}
Table 13: Specific values of \(R_{1}\), \(R_{2}\), \(R_{3}\), \(R_{4}\) and \(R_{5}\) under CUDL.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline  & & \multicolumn{3}{c}{\(\lambda\) = 1/3} & & & \multicolumn{3}{c}{\(\lambda\) = 1/2} & \\ \(K\) & \(R_{1}\) & \(R_{2}\) & \(R_{3}\) & \(R_{4}\) & \(R_{5}\) & \(R_{1}\) & \(R_{2}\) & \(R_{3}\) & \(R_{4}\) & \(R_{5}\) \\ \hline
2.5 & 0.0062 & \(-\)0.2548 & 0.2257 & 1.7400 & 0.7071 & 0.0191 & \(