# Lateral-Torsional Buckling of Cantilever Steel Beams under 2 Types of Complex Loads
## 2 Calculation of Total Potential Energy Equation under CL and CUDL
### The Buckling Deformation Functions  
When using the Rayleigh-Ritz method for stability analysis, the assumption of the buckling deformation functions is very critical, which directly affects the calculation accuracy. First, the buckling deformation functions should match the actual deformation form as much as possible and can be expressed by infinite series. Secondly, the buckling deformation functions must meet the displacement boundary conditions. If it can also meet the mechanical boundary conditions, the accuracy will be higher. Based on the above situation, the following trigonometric series are used to describe the lateral deflection \(u(z)\) and twist angle \(\varphi(z)\) of cantilever steel beams [4, 12]. \[u(z)=h\sum_{n=1}^{\infty}A_{n}\bigg{[}1-\cos\frac{(2n-1)\pi\mathrm{z}}{2l} \bigg{]} \tag{10}\]  
\[\varphi(z)=\sum_{n=1}^{\infty}B_{n}\bigg{[}1-\cos\frac{(2n-1)\pi\mathrm{z}}{2l }\bigg{]} \tag{11}\]  
where \(h\) is the distance between the centroids of the top and bottom flange, and \(A_{n}\) and \(B_{n}\) are mutually independent generalized coordinates. From the perspective of structural dynamics, it can be seen that the dimensions of \(A_{n}\) and \(B_{n}\) are different. Therefore, \(h\) is introduced into the expression of \(u(z)\) to eliminate the dimension of \(A_{n}\), so that \(A_{n}\) and  
\[ \begin{array}{l}\Pi_{3}=\frac{1}{2}\int_{0}^{l}2M_{\rm x}u^{\prime\prime}\varphi{ \rm d}z\cdot\frac{l^{3}}{h^{2}E\Gamma_{y}}\\ =\sum\limits_{n=1}^{\infty}-\frac{\pi\widetilde{M}_{0}A_{n}B_{n}}{2(-1+2n)(a+2 \lambda)}\{96\alpha\cos(n\pi)-(-1+2n)\pi\Big{[}-42(1+\alpha)+(1-2n)^{2}\pi^{2} (\alpha+3\lambda^{2})+\\ 48\cos(\frac{-1+2n}{2}\pi\lambda)-6\cos((-1+2n)\pi\lambda)\Big{]}\}+\sum\limits _{n=1}^{\infty}\ \ \ \sum\limits_{\begin{array}{c}r=1\\ r\
eq n\end{array}}^{\infty}\ \ \frac{\pi\widetilde{M}_{0}A_{n}B_{r}}{4(-r+n)^{2}(-1+r+n)^{2}(-1+2n)(a+2 \lambda)}.\\ \{\{-16\alpha(r-n)^{2}(-1+r+n)^{2}\cos(\pi\lambda)+\pi(-1+2n)[-(1-2r)^{2}(-1-2r +2r^{2}+6n-6n^{2})(1+\alpha)+\}\\ 8(r-n)^{2}(-1+r+n)^{2}\cos(\frac{1}{2}\pi(1-2n)\lambda)+(1-2n)^{2}(-(-1+r+n)^{ 2}\cos(\pi(r-n)\lambda)-\\ (r-n)^{2}\cos(\pi(-1+r+n)\lambda))]\}\\ \end{array}  \tag{18}\]  
\[ \begin{array}{l}\Pi_{4}=\frac{1}{2}a_{\rm P_{y}}P_{y}\varphi(l)^{2}\cdot \frac{l^{3}}{h^{2}E\Gamma_{y}}\\ =-\sum\limits_{n=1}^{\infty}\frac{\widetilde{a_{\rm qy}}\widetilde{M}_{0}\pi a \Gamma_{n}^{2}}{2(-1+2n)(a+2\lambda)}(3\pi-6n\pi-8\cos(n\pi))+\sum\limits_{n=1} ^{\infty}\ \ \sum\limits_{\begin{array}{c}r=1\\ r\
eq n\end{array}}^{\infty}\frac{\widetilde{a_{\rm qy}}\widetilde{M}_{0}\pi^{2 }aB_{n}B_{r}}{2(\alpha+2\lambda)}(2-\frac{4\cos(\pi n)}{\pi(1-2r)}-\frac{4 \cos(\pi n)}{\pi(1-2n)})+\\ \sum\limits_{n=1}^{\infty}\frac{\widetilde{a_{\rm py}}\widetilde{M}_{0}\pi^{2 }b_{n}^{2}}{a+2\lambda}(1-\cos(\frac{2n-1}{2}\pi\lambda))^{2}+\sum\limits_{n= 1}^{\infty}\ \ \ \sum\limits_{\begin{array}{c}r=1\\ r\
eq n\end{array}}^{\infty}\frac{\widetilde{a_{\rm py}}\widetilde{M}_{0}\pi^{ 2}b_{n}B_{r}}{a+2\lambda}(1-\cos(\frac{2n-1}{2}\pi\lambda))(1-\cos(\frac{2r-1 }{2}\pi\lambda))\\ \end{array}  \tag{19}\]  
The total potential energy equation can be written as the sum of Equations (16)-(19):  
\[\Pi=\Pi_{1}+\Pi_{2}+\Pi_{3}+\Pi_{4}=\sum\limits_{i=1}^{4}\Pi_{i} \tag{20}\]  
According to the principle of stationary total potential energy, the stability conditions of the system can be obtained as follows. \[\left\{ \begin{array}{l}\frac{\partial\Pi}{\partial A_{n}}=0,\ n{=}1,2,3\dots.\dots,\infty;\\ \frac{\partial\Pi}{\partial B_{n}}=0,\ n{=}1,2,3\dots.\dots,\infty;\end{array} \right. \tag{21}\]  
The following expressions are made available by Equation (21). \[