# Lateral-Torsional Buckling of Cantilever Steel Beams under 2 Types of Complex Loads
## 6 Conclusions  
A novel numerical method and a new closed-form solution for the elastic LTB analysis of cantilever steel beams have been successfully developed and validated in this paper. The numerical method was developed based on postulated buckling deformation functions and the Rayleigh-Ritz method. The closed-form solution was developed based on the assumption that the torsion angle is small and based on the numerical method. The values of \(M_{\rm cr}\) predicted by the proposed numerical method are sufficiently comparable to the data from existing literature and the following findings can be drawn:  
1. \(M_{\rm cr}\) and the generalized coordinates \(A_{n}\) and \(B_{n}\) in the buckling deformation functions of cantilever I-beams under CL and CUDL can be accurately calculated by the numerical method proposed in this paper. Compared with the experimental results and the finite element results of relevant reference, the maximum of the \(M_{\rm th}/M_{\rm ex}\) and \(M_{\rm th}/M_{\rm FEM}\) ratio is 1.07, and the minimum error is 0.92. The average of all the ratios is 0.98 with a standard deviation of 0.13. 2. The safety, accuracy and economic aspects of the prediction of the numerical method are assessed using DPC. The results show that the numerical method can effectively predict \(M_{\rm cr}\) of cantilever steel beams under CL and CUDL. 3. The number of generalized coordinates determines the accuracy of \(M_{\rm cr}\). The number of generalized coordinates is defined as \(n\). The obtained errors of \(M_{\rm cr}\) do not exceed 3.96% when \(n\) is greater than or equal to 10. The errors of \(M_{\rm cr}\) exceed 17% when \(n\) is equal to 1 or 2. 4. The degree of \(M_{\rm cr}\) in Equation (34) varies with the number of generalized coordinates of twist angle function. When the number of generalized coordinates of twist angle function is \(n\), the degree of \(M_{\rm cr}\) in Equation (34) is \(2n\). 5. The torsional buckling mode of the cantilever steel beam is mainly determined by \(K\), which is the torsional stiffness coefficient. However, in order to provide reference for the calculation of the \(M_{\rm cr}\) of cantilever steel beams matching the specification of ANSI/AISC 360-16, a large number of numerical calculations and statistical evaluations should be performed to determine the expression of \(C_{\rm b}\), which is the lateral-torsional buckling modification factor. In addition, further investigations are needed on the LTB performance of beams composed of other materials that are not the isotropic material, such as Glass Fiber Reinforced Polymer (GFRP), laminate composite material and pultruded fiber reinforced polymer (PFRP) material, which are increasingly used in beams. Conceptualization, software, validation, writing--original draft preparation, A.L. ; writing--review and editing, supervision, Y.C. and X.L. All authors have read and agreed to the published version of the manuscript. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(\Lambda\)} & \multicolumn{4}{c}{**Section I**} & \multicolumn{4}{c}{**Section II-Reinforced**} \\  & & \(M_{\rm cr1}\) & \(M_{\rm cr2}\) & \(M_{\rm cr1}/M_{\rm cr2}\) & \(M_{\rm cr1}\) & \(M_{\rm cr2}\) & \(M_{\rm cr1}/M_{\rm cr2}\) \\ \hline \multirow{3}{*}{1/2} & TF & 34.75 & 30.86 & 1.13 & 21.14 & 18.09 & 1.17 \\  & SC & 54.70 & 54.56 & 1.00 & 43.21 & 43.03 & 1.00 \\  & BF & 72.37 & 64.79 & 1.12 & 45.67 & 43.79 & 1.04 \\ \hline \multirow{3}{*}{2/3} & TF & 29.22 & 27.12 & 1.08 & 18.09 & 15.93 & 1.14 \\  & SC & 43.30 & 43.1 & 1.00 & 33.53 & 33.36 & 1.01 \\  & BF & 54.58 & 50.12 & 1.09 & 35.13 & 34.33 & 1.02 \\ \hline \multirow{3}{*}{1} & TF & 21.53 & 20.11 & 1.07 & 13.93 & 13.78 & 1.01 \\  & SC & 28.74 & 28.56 & 1.01 & 21.68 & 21.5 & 1.01 \\ \cline{1-1}  & BF & 34.26 & 32.56 & 1.05 & 22.48 & 22.12 & 1.02 \\ \hline \hline \end{tabular} Note: \(M_{\rm cr1}\) is the calculated value by Equation (59). \(M_{\rm cr2}\) is the calculated value by ABAQUS.  
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