# 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
\[\{A\}=\left[\begin{array}{cccc}A_{1}&A_{2}&A_{3}&\dots&\right]^{T} \tag{24}\] \[\{B\}=\left[\begin{array}{cccc}B_{1}&B_{2}&B_{3}&\dots&\right]^{T}\]  
\[R^{0}_{n,n}=\frac{(-1+2n)^{4}\pi^{4}}{32};R^{0}_{n,r}=0,n\
eq r \tag{25}\]  
\[S^{0}_{n,r}=T^{0}_{n,r}=0 \tag{26}\]  
\[Q^{0}_{n,n}=\frac{(-1+2n)^{4}\pi^{4}\eta}{32(1+\eta)^{2}}+\frac{(-1+2n)^{2} \pi^{4}\eta}{8K^{2}(1+\eta)^{2}};Q^{0}_{n,r}=0,n\
eq r \tag{27}\]  
\[R^{1}_{n,r}=0 \tag{28}\]  
\[S^{1}_{n,n}= -\frac{\pi}{24(-1+2n)(\alpha+2\lambda)}\{-96\alpha\cos(n\pi)+(-1+2n) \pi\Big{[}-42(1+\alpha)+(1-2n)^{2}\pi^{2}(\alpha+3\lambda^{2})+ \tag{29}\] \[48\cos(\frac{-1+2n}{2}\pi\lambda)-6\cos((-1+2n)\pi\lambda)\Big{]}\}\] \[S^{1}_{n,r}= \frac{-\pi}{4(-r+n)^{2}(-1+r+n)^{2}(-1+2n)(\alpha+2\lambda)}\{-16 \alpha(r-n)^{2}(-1+r+n)^{2}\cos(n\pi)+\pi(-1+2n)\cdot\] (30) \[\left[-(1-2r)^{2}(-1-2r+2r^{2}+6n-6n^{2})(1+\alpha)+8(r-n)^{2}(-1+ r+n)^{2}\cos(\frac{1}{2}\pi(1-2n)\lambda)+\right.\] \[\left.\left. \begin{array}{rl}(1-2n)^{2}(-(-1+r+n)^{2}\cos(\pi(r-n) \lambda)-(r-n)^{2}\cos(\pi(-1+r+n)\lambda))]\right\},n\
eq r\end{array} \right.\] \[\left. \begin{array}{rl}T^{1}_{n,r}=S^{1}_{r,n}\end{array} \right. \tag{31}\]  
\[Q^{1}_{n,n}= \frac{\widetilde{\overline{\kappa}_{n}}\pi^{2}}{\overline{12(\alpha+2 \lambda)}}\left\{(\alpha+3\lambda^{2})(1-2n)^{2}\pi^{2}-6(1+\alpha)+6\cos[(-1+2 n)\pi\lambda]\right\}+ \tag{32}\] \[\frac{\widetilde{\overline{\kappa}_{n}}\pi\alpha}{(-1+2n)(\alpha+ 2\lambda)}\left(3\pi-6n\pi-8\cos(n\pi)\right)-\frac{2\widetilde{\overline{\kappa} _{n}}\pi^{2}}{\overline{\kappa}+2\lambda}(1-\cos(\frac{2n-1}{2}\pi\lambda))^{2}\]\[ \begin{array}{ll}Q_{n,r}^{1}=&\frac{(-1+2r)(-1+2n)\widetilde{\beta_{\rm F}}\pi^{2 }}{2(r-n)^{2}(-1+r+n)^{2}(a+2\lambda)}\{(-1+2r)(-1+2n)(1+\alpha)-(-1+r+n)^{2} \cos(\pi(r-n)\lambda)+\\ &(-r+n)^{2}\cos[(-1+r+n)\pi\lambda]\}-\\ &\frac{\widetilde{a_{\rm M}}\pi^{2}\pi^{2}}{(a+2\lambda)}(2-\frac{4\cos(\pi r )}{\pi(1-2r)}-\frac{4\cos(\pi n)}{\pi(1-2n)})-\frac{2\widetilde{a_{\rm F}}\pi^ {2}}{a+2\lambda}(1-\cos(\frac{2n-1}{2}\pi\lambda))(1-\cos(\frac{2r-1}{2}\pi \lambda)),n\
eq r\end{array}  \tag{33}\]  
Equation (21) can be written as the following matrix equation. \[ \begin{bmatrix}R^{0}-R^{1}\widetilde{M_{0}}&S^{0}-S^{1}\widetilde{M_{0}}\\ T^{0}-T^{1}\widetilde{M_{0}}&Q^{0}-Q^{1}\widetilde{M_{0}}\end{bmatrix}\begin{cases} A\\ B\end{cases} = \begin{cases}0\\ 0\end{cases}  \tag{34}\]  
Equation (34) is derived from Equation (1) through integration. In the integration process, the buckling deformation functions Equations (10) and (11) are substituted into Equation (1). Since Equations (1), (10) and (11) meet the boundary conditions of cantilever beams, Equation (34) is applicable to the calculation of the \(M_{\rm cr}\) of cantilever beams. For convenience of writing, the following matrices can be defined. \[D_{0}= \begin{bmatrix}R^{0}&S^{0}\\ T^{0}&Q^{0}\end{bmatrix}  \tag{35}\]  
\[D_{1}= \begin{bmatrix}R^{1}&S^{1}\\ T^{1}&Q^{1}\end{bmatrix}  \tag{36}\]  
It can be seen from Equation (34) that the solution of the dimensionless critical moment (\(\widetilde{M_{0}}\)) is the smallest positive real number of the generalized eigenvalues of the matrices \(D_{0}\) and \(D_{1}\). The matrix \(\left\{ \begin{array}{c}A\\ B\end{array} \right\}\) formed by the generalized coordinates is the generalized eigenvector corresponding to the smallest positive real number of the generalized eigenvalues of the matrices \(D_{0}\) and \(D_{1}\). Our MATLAB program can be compiled to solve \(M_{\rm cr}\) and \(\left\{ \begin{array}{c}A\\ B\end{array} \right\}\). The flowchart given in Figure 3 elaborates on the idea of the procedure.