# Lateral-Torsional Buckling of Cantilever Steel Beams under 2 Types of Complex Loads
## 2 Calculation of Total Potential Energy Equation under CL and CUDL  
### Introduction and Comparison of Total Potential Energy Equation  
Figures 1 and 2 illustrate the LTB of a singly symmetric cantilever I-beam. As shown in Figures 1 and 2, \(L\) is cantilever length. \(P_{y}\) is concentrated load and \(q_{y}\) is uniformly distributed load, their coordinates are respectively \((0,y_{py})\) and \((0,y_{qy})\), the shear center's coordinates are \((0,y_{s})\), where \(a_{Py}=y_{Py}-y_{s}\), \(a_{qy}=y_{qy}-y_{s}\). \(\lambda\) is the ratio of distance between the action point of \(P_{y}\) and the fixed end to the cantilever's length. [PERSON] and [PERSON] [14; 46] thought that the LTB theory of steel beams should consider linear strain energy, nonlinear longitudinal strain energy, nonlinear shear strain energy and nonlinear transverse strain energy at the same time. Compared with the experimental results in reference [10] and reference [11], the total potential energy proposed by [PERSON] is accurate in calculating the \(M_{\text{cr}}\) of the cantilever beam. At the same time, it is also confirmed that the total potential energy proposed by [PERSON] and [PERSON] is applicable to the boundary conditions of cantilever beam through the discussion in reference [14]. For the load case shown in Figures 1 and 2, the total potential energy of [PERSON] and [PERSON] can be expressed as Equation (1). \[\Pi=\frac{1}{2}\int_{0}^{l}\left(EI_{y}u^{\prime\,2}+EI_{w}\varphi^{\prime\,2}+ GI_{y}\varphi^{2}+GI_{y}\varphi^{2}-2M_{x}u^{\prime}\varphi^{\prime}+2\beta_{x}M_{x} \varphi^{\prime\,2}+2\beta_{x}M_{x}\varphi\varphi^{\prime}-2M_{x}u^{\prime} \varphi-q_{y}(\beta_{x}-a_{y_{y}})|\varphi^{2}\right)\mathrm{d}z-\frac{1}{2} \left(\beta_{x}-a_{y_{y}}\right)P_{y}\varphi^{2}(\lambda l) \tag{1}\]  
where \(E\) is [PERSON]'s modulus, \(G\) is the shear modulus, \(I_{w}\) is the warping constant, \(I_{\mathrm{t}}\) is the torsional constant, \(I_{y}\) is the moment of inertia about the \(y\)-axis, \(\beta_{x}\) is [PERSON]'s coefficient, \(u\) and \(\varphi\) are the lateral deflection and twist angle of section, respectively, \(\varphi(\lambda l)\) is the twist angle at the point where the transverse concentrated load acts during the lateral-torsional buckling of the beam, and \(M_{x}\) is the moment around the \(x\)-axis. The ratio of the uniformly distributed load to concentrated load in Figure 1 is assumed to be as follows:  
\[\alpha=\frac{q_{y}l}{P_{y}} \tag{2}\]  
The magnitude of the moment at any position of cantilever (\(M_{x}\)) is expressed by  
According to Equation (3), boundary conditions and the properties of the integral, the following equations can be obtained. \[\int_{0}^{l}2\beta_{\mathrm{x}}M_{\mathrm{x}}{}^{\prime}\varphi\varphi^{\prime} \mathrm{d}z=-2M_{\mathrm{x}}u^{\prime}\varphi\Big{|}_{0}^{l}+\int_{0}^{l}2(M_{ \mathrm{x}}u^{\prime})^{\prime}\varphi\mathrm{d}z=\int_{0}^{l}2(M_{\mathrm{x}} u^{\prime\prime}+M_{\mathrm{x}}{}^{\prime}u^{\prime})\varphi\mathrm{d}z \tag{4}\]  
\[\int_{0}^{l}2\beta_{\mathrm{x}}M_{\mathrm{x}}{}^{\prime}\varphi\varphi^{\prime} \mathrm{d}z=M_{\mathrm{x}}{}^{\prime}\beta_{\mathrm{x}}\varphi^{2}\Big{|}_{0}^ {l}-\int_{0}^{l}(M_{\mathrm{x}}{}^{\prime}\beta_{\mathrm{x}})^{\prime}\varphi^{ 2}\mathrm{d}z \tag{5}\]  
\[M_{\mathrm{x}}{}^{\prime\prime}=-q_{y} \tag{6}\]  
If \(\lambda=1\), then \(M_{\mathrm{x}}^{\prime}(l)=P_{y}\) and \(\varphi(0)=0\). The following equation can be obtained by Equations (5) and (6).