# Buckling of Rectangular Composite Pipes under Torsion
## 2 Numerical Calculation
### Case of a Variable Edge Angle  
Supposing that the edge angle is variable, substituting Equations (1) and (2) into Equation (14) yields the total strain energy \(U\) and the work \(W\) of the external torsional stress \(\tau_{cr}\) as follows:  
\[ \begin{array}{l}U=\frac{\pi^{4}\mu^{2}}{4\hbar^{3}}\sum\limits_{m=1}^{\infty} \sum\limits_{n=1}^{\infty}B_{mn}{}^{2}\bigg{\{}D_{xx}m^{4}+2\Big{(}\frac{l}{b} \Big{)}^{2}(2D_{ss}+D_{xy})m^{2}n^{2}+\Big{(}\frac{l}{b}\Big{)}^{4}D_{yy}n^{4} \bigg{\}}\\ +\frac{\pi^{4}\mu^{2}}{4\hbar^{3}}\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^ {\infty}H_{mn}{}^{2}\bigg{\{}h^{4}D_{xx}m^{4}+2\Big{(}\frac{l}{\hbar}\Big{)}^{2 }(2D_{ss}+D_{xz})m^{2}n^{2}+\Big{(}\frac{l}{\hbar}\Big{)}^{4}D_{zz}n^{4}\bigg{\}} \end{array}  \tag{22}\]  
\[ \begin{array}{l}W=\quad\quad 8t\tau_{cr}\Big{(}\frac{b}{b}\Big{)}^{2}\sum \limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\sum\limits_{\sum\limits_{j}} \sum\limits_{\begin{subarray}{c}\infty\\ (\overline{i}^{2}-m^{2})(\overline{j}^{2}-n^{2})\end{subarray}}\\ +8t\tau_{cr}\Big{(}\frac{b}{\overline{i}}\Big{)}^{2}\sum\limits_{m=1}^{\infty} \sum\limits_{n=1}^{\infty}\sum\limits_{\sum\limits_{j=1}}^{\infty}\sum\limits_ {\begin{subarray}{c}\infty\\ (\overline{i}^{2}-m^{2})(\overline{j}^{2}-n^{2})\end{subarray}}^{\infty}\end{array}  \tag{23}\]  
The buckling torsional stress is obtained by minimizing the total potential energy. \[\frac{\partial}{\partial B_{mn}}(U-W)=0,\quad\frac{\partial}{\partial H_{mn}} (U-W)=0 \tag{24}\]  
The following equation is obtained from Equation (17). \[ \begin{array}{l}\frac{\pi^{4}}{32\tau_{cr}\hbar^{2}}\bigg{\{}\Big{(}\frac{b }{\overline{l}}\Big{)}^{3}\bigg{[}D_{xx}m^{4}+2(2D_{ss}+D_{xy})\Big{(}\frac{l} {b}\Big{)}^{2}m^{2}n^{2}+D_{yy}\Big{(}\frac{l}{b}\Big{)}^{4}n^{4}\bigg{]} \bigg{\}}H_{mn}\\ -mn\Big{(}\frac{b}{\overline{h}}\Big{)}\sum\limits_{i=1}^{2}\sum\limits_{j=1} ^{\infty}\frac{ijH_{ij}}{(\overline{i}^{2}-m^{2})(\overline{j}^{2}-n^{2})}=0 \end{array}  \tag{25}\]  
\[ \begin{array}{l}\frac{\pi^{4}}{32\tau_{cr}\hbar^{2}}\bigg{\{}\Big{(}\frac{h }{\overline{l}}\Big{)}^{3}\bigg{[}D_{xx}m^{4}+2(2D_{ss}+D_{xx})\Big{(}\frac{l} {\overline{h}}\Big{)}^{2}m^{2}n^{2}+D_{zz}\Big{(}\frac{l}{\overline{h}}\Big{)} ^{4}n^{4}\bigg{]}\bigg{\}}B_{mn}\\ -mn\Big{(}\frac{h}{\overline{h}}\Big{)}\sum\limits_{i=1}^{2}\sum\limits_{j=1} ^{\infty}\frac{ijB_{ij}}{(\overline{i}^{2}-m^{2})(\overline{j}^{2}-n^{2})}=0 \end{array}  \tag{26}\]  
The normalized torsional buckling load factor \(k_{s}\) was obtained by setting the determinant of the coefficient matrix for \(B_{mn}\) and \(H_{mn}\) to zero. To check the accuracy relative to the number of terms of \(n\) and \(m\), a similar numerical calculation as shown in Figure 4 was conducted, and the same conclusion was obtained for the terms of \(n\) and \(m\). The numerical results for Equations (18) and (26), and FEM for \(h/b=0.7\) are shown in Figure 6. The results for Equations (18) and (26) for \(h/b=1.0\) are also plotted for comparison. In the case of \(l/b=1\), the FEM result of \(h/b=0.7\) is close to the constant-angle case; however, in the region \(l/b\geq 2\), the FEM result of \(h/b=0.7\) is close to the variable-angle case. There are no apparent differences between the results of Equation (18) for \(h/b=1.0\) and those of Equation (26) for \(h/b=1.0\) because of the section is rectangular and thus the edge angle is constant. These results show that the assumption of a constant edge angle is a stiff constraint for the region \(h/b<1.0\). Thus, the assumption of a variable edge angle, Equation (26), was employed in the subsequent analysis because of its more accurate and safe-side estimation. Note that both FE analysis and the method of minimizing the total potential energy are essentially the same because they are based on energy principle. If the difference appears, the reason is convergence due to mesh size of FE model or number of terms of displacement function of Equations (1) and (2).