## 4 Approximate Solution and Galerkin Procedure  
Suppose that a spiral corrugated FG-CNTRC cylindrical shell is simply supported and subjected to radial loads \(q_{0}\) (N/m\({}^{2}\)). Thus the boundary conditions are of the form  
\[w=0,\;\;M_{x}=0,\;\;N_{x}=0,\;\;N_{xy}=0,\;\;at\;\;x=0;\;L. \tag{18}\]  
In this case, the deflection of the cylindrical shell can be chosen as a three-term solution form  
\[w=f_{0}+f_{1}\sin\frac{m\pi x}{L}\sin\frac{ny}{R}+f_{2}\sin^{2}\frac{m\pi x}{L}, \tag{19}\]  
in which \(f_{0}\) is the pre-buckling uniform deflection, \(f_{1}\) is the linear deflection amplitude, \(f_{2}\) is the nonlinear deflection amplitude, \(\sin\frac{m\pi x}{L}\sin\frac{ny}{R}\) is the linear buckling shape, \(\sin^{2}\frac{m\pi x}{L}\) is the nonlinear buckling shape in the axial direction, \(m\) is the number of half waves and \(n\) is the number of half waves in axial and circumferential directions, respectively. It can be seen that the simply supported boundary condition at \(x=0\) and \(x=L\) is satisfied in the approximate sense. Substituting Equation (19) into Equation (17) and solving the obtained equation for unknown \(\varphi\), leads to  
\[ \begin{array}{c}\varphi=\varphi_{1}\cos 2\overline{\alpha}x+\varphi_{2}\cos 2 \overline{\beta}x+\varphi_{3}\sin\overline{\alpha}x\sin\overline{\beta}y+ \varphi_{4}\sin 3\overline{\alpha}x\sin\overline{\beta}y\\ +\varphi_{5}\cos\overline{\alpha}x\cos\overline{\beta}y+\varphi_{6}\cos 3 \overline{\alpha}x\cos\overline{\beta}y-\frac{1}{2}\varphi_{0y}dx^{2},\end{array}  \tag{20}\]  
where  
\[ \begin{array}{c}\varphi_{1}=-\frac{1}{8}f_{2}\frac{1}{R\overline{\alpha}^{2} F_{22}}+\frac{1}{32}f_{1}^{2}\frac{\overline{\beta}^{2}}{\overline{\alpha}^{2} F_{22}},\;\;\varphi_{2}=\frac{1}{32}f_{1}^{2}\frac{\overline{\alpha}^{2}}{ \overline{\beta}^{2}F_{11}},\;\;\varphi_{3}=\frac{f_{1}M_{1}+f_{1}f_{2}M_{2}}{ M_{3}},\\ \varphi_{4}=f_{1}f_{2}\frac{M_{4}}{M_{5}},\;\;\varphi_{5}=\frac{f_{1}M_{6}+f_ {1}f_{2}M_{7}}{M_{3}},\;\;\varphi_{6}=f_{1}f_{2}\frac{M_{8}}{M_{5}},\\ \frac{m\pi}{L}=\overline{\alpha},\frac{\mu}{R}=\overline{\beta},\end{array}  \tag{21}\]  
\[ \begin{array}{c}M_{1}=\left[\overline{\alpha}^{4}F_{22}+\overline{\alpha}^{2 }\overline{\beta}^{2}(F_{12}+F_{21}+F_{66})+\overline{\beta}^{4}F_{11}\right] \frac{1}{R}\overline{\alpha}^{2},\\ M_{2}=-\overline{\alpha}^{2}\overline{\beta}^{2}\left[\overline{\alpha}^{4}F_{2 2}+\overline{\alpha}^{2}\overline{\beta}^{2}(F_{12}+F_{21}+F_{66})+\overline{ \beta}^{4}F_{11}\right]\\ M_{3}=\left[\overline{\alpha}^{4}F_{22}+\overline{\alpha}^{2}\overline{\beta}^{2 }(F_{12}+F_{21}+F_{66})+\overline{\beta}^{4}F_{11}\right]^{2}-\left[\overline{ \alpha}^{3}\overline{\beta}(F_{26}+F_{62})+\overline{\alpha}\overline{\beta}^ {3}(F_{16}+F_{61})\right]^{2},\\ M_{4}=\overline{\alpha}^{2}\overline{\beta}^{2}\left[81\overline{\alpha}^{4}F_{ 22}+9\overline{\alpha}^{2}\overline{\beta}^{2}(F_{12}+F_{21}+F_{66})+\overline{ \beta}^{4}F_{11}\right],\\ M_{5}=\left[81\overline{\alpha}^{4}F_{22}+9\overline{\alpha}^{2}\overline{\beta }^{2}(F_{12}+F_{21}+F_{66})+\overline{\beta}^{4}F_{11}\right]^{2}\\ -3\left[9\overline{\alpha}^{3}\overline{\beta}(F_{26}+F_{62})+\overline{\alpha }\overline{\beta}^{3}(F_{16}+F_{61})\right]^{2},\\ M_{6}=-\overline{\alpha}\overline{\beta}\left[\overline{\alpha}^{2}(F_{26}+F_{62 })+\overline{\beta}^{2}(F_{16}+F_{61})\right]\frac{1}{R}\overline{\alpha}^{2}, \\ M_{7}=\overline{\alpha}^{3}\overline{\beta}^{3}\left[\overline{\alpha}^{2}(F_{26 }+F_{62})+\overline{\beta}^{2}(F_{16}+F_{61})\right],\\ M_{8}=-3\overline{\alpha}^{3}\overline{\beta}^{3}\left[9\overline{\alpha}^{2}(F_{26 }+F_{62})+\overline{\beta}^{2}(F_{16}+F_{61})\right].\end{array}