# Buckling of Rectangular Composite Pipes under Torsion
## 3 Closed-Form Polynomial Equation  
To make quick and convenient estimates for torsional buckling of a composite tube, a simple closed form polynomial equation was derived by fitting the numerical solution of the orthotropic \([(0/90)_{r}(\pm 45)_{1,r}]_{8}\) layups. Here, \(r\) is the ratio of the 0/90 layer number to the total layer number. Because the \([(0/90)_{r}(\pm 45)_{1,r}]_{8}\) layup is often used, it is thus chosen. The numerical analysis described above is hard to use for design purposes and would entail a huge calculation cost if it were to be used for optimum design of the whole system. Thus, it is considered that a simple approximate design equation would still be useful. Table 2 shows the material properties of unidirectional pre-preg tape, and Table 3 shows the layup sequence used in the numerical analysis. The total ply number was 32, and the binding stiffness was systematically controlled by varying \(r\). Assuming that the number of layups was large enough, the bending stiffness was calculated as \(E_{ij}\)\(t^{3}/12=D_{ij}\) (\(i,j=x\), \(y\), \(s\)). Advanced stiffness properties like through thickness properties discussed in Refs. [19; 20; 21] are not used because of to make quick and convenient estimates. Figures 7-9 show the buckling load factor \(k_{s}\) for \(l/b\), \(h/b\), and \(r\). The ranges of \(l/b\), \(h/b\), and \(r\) are \(1\leq l/b\leq 6\), \(0.7\leq h/b\leq 1\), and \(0\leq r\leq 1\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Value** \\ \hline \(E_{L}\) [MPa] & 147,000 \\ \(E_{T}\) [MPa] & 9800 \\ \(G_{LT}\) [MPa] & 5096 \\ \(v_{L}\) [-] & 0.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material properties of unidirectional pre-preg tape. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Number of Layers** & **Ratio of 0/90 Layer** & \multicolumn{4}{c}{**Bending Stiffness [N-mm2]**} \\ \hline
0/90 & \(\pm 45\) & \(r\) & \(D_{xx}\) & \(D_{xy}\) & \(D_{yy}\) & \(D_{ss}\) \\
0 & 32 & 0 & 3845 & 2996 & 3845 & 3158 \\
4 & 28 & 0.125 & 4187 & 2654 & 4187 & 2816 \\
8 & 24 & 0.25 & 4529 & 2313 & 4529 & 2474 \\
12 & 20 & 0.375 & 4870 & 1971 & 4870 & 2133 \\
16 & 16 & 0.5 & 5212 & 1630 & 5212 & 1791 \\
20 & 12 & 0.625 & 5553 & 1288 & 5553 & 1450 \\
24 & 8 & 0.75 & 5895 & 946.4 & 5895 & 1108 \\
28 & 4 & 0.875 & 6237 & 604.7 & 6237 & 766.3 \\
32 & 0 & 1 & 6578 & 263.1 & 6578 & 424.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Layup sequence. Figure 6: FE analysis of torsional buckling. Figure 8: \(k_{\rm s}\) versus \(h/b\). Figure 7: \(k_{\rm s}\) versus \(l/b\). Figures 7 and 8 can be fitted by a second-order polynomial, while Figure 9 can be fitted by a first-order polynomial; thus, the following polynomial was employed. \[ \begin{split} k_{s}{}^{\prime}=\bigg{(}H_{0}\Big{(}\tfrac{h}{b} \Big{)}+H_{1}\Big{(}\tfrac{h}{b}\Big{)}\Big{(}\tfrac{l}{b}\Big{)}+H_{2}\Big{(} \tfrac{h}{b}\Big{)}\Big{(}\tfrac{l}{b}\Big{)}^{2}\Big{)}(R_{1}+R_{2}r)\\ H_{0}=H_{00}+H_{01}\Big{(}\tfrac{h}{b}\Big{)},\ H_{1}=H_{10}+H_{11} \Big{(}\tfrac{h}{b}\Big{)},\ H_{2}=H_{20}+H_{21}\Big{(}\tfrac{h}{b}\Big{)} \end{split}  \tag{27}\]  
The coefficients \(H_{00}\), \(H_{01}\),..., \(H_{21}\), \(R_{1}\) and \(R_{2}\) were determined by minimizing the following weighted residual and adjusting the weight \(w\) to make the non-safe side residual \(k_{s+\sigma}\) less than 5%. \[ \begin{split} k_{s+\sigma}=\tfrac{k_{s}{}^{\prime}-k_{s}}{k_{s}} \left(for\tfrac{k_{s}{}^{\prime}-k_{s}}{k_{s}}\geq 0\ \right),\\ k_{s-\sigma r}=\left|\tfrac{k_{s}{}^{\prime}-k_{s}}{k_{s}} \right|\left(for\tfrac{k_{s}{}^{\prime}-k_{s}}{k_{s}}<0\ \right)\\ \max\{k_{s+\sigma r},wk_{s-\sigma r}\}<0.05\end{split}  \tag{28}\]  
The minimization was conducted on a Microsoft Excel solver with the GRG Nonlinear option. The following coefficients shown in Table 4 were obtained with \(w=0.4\); \(k_{s+\sigma}\) was 4.77%, and \(k_{s-\sigma}\) was 11.9%. As shown in Figure 4, the buckling stress \(\tau_{\sigma r}\) does not change in the region \(l/b>6\).