## 4 Approximate Solution and Galerkin Procedure
\]  
Substituting the deflection form (19) and stress function (20) into Equation (15) and then applying the [PERSON] method in the ranges \(0\leq x\leq L\) and \(0\leq y\leq 2\pi R\) leads to  
\[-2\sigma_{0y}\frac{h}{R}+2q_{0}=0, \tag{22}\]\[f_{1}\bigg{(}N_{1}+\sigma_{0y}h\frac{n^{2}}{R^{2}}\bigg{)}+f_{1}f_{2}N_{2}+f_{1}f_ {2}^{2}N_{3}+f_{1}^{3}N_{4}=0, \tag{22}\]  
\[f_{1}^{2}N_{5}+f_{1}^{2}f_{2}N_{6}+f_{2}N_{7}-\frac{\sigma_{0y}h}{R}+q_{0}=0, \tag{23}\]  
where  
\[ \begin{array}{l}N_{1}=-\frac{M_{1}}{M_{3}}\frac{\overline{\alpha}^{2}}{R}- \bigg{[}\overline{\alpha}^{4}D_{11}+\overline{\alpha}^{2}\overline{\beta}^{2} (D_{12}+4D_{66}+D_{21})+\overline{\beta}^{4}D_{22}\bigg{]},\\ N_{2}=-\frac{M_{2}}{M_{3}}\frac{\overline{\alpha}^{2}}{R}+\frac{M_{1}}{M_{3}} \overline{\alpha}^{2}\overline{\beta}^{2}+\frac{1}{4F_{22}}\frac{\overline{ \beta}^{2}}{R}N_{3}=\frac{M_{3}}{M_{3}}\overline{\alpha}^{2}\overline{\beta}^ {2}-\frac{M_{4}}{M_{5}}\overline{\alpha}^{2}\overline{\beta}^{2},\\ N_{4}=-\frac{1}{16F_{11}}\overline{\alpha}^{4}-\frac{1}{16F_{22}}\overline{ \beta}^{4},\ N_{5}=\frac{1}{16F_{22}}\frac{\overline{\beta}^{2}}{R}+\frac{1}{2 }\frac{M_{1}}{M_{3}}\overline{\alpha}^{2}\overline{\beta}^{2},\\ N_{6}=\frac{1}{2}\Big{(}\frac{M_{2}}{M_{3}}-\frac{M_{4}}{M_{5}}\Big{)}\overline{ \alpha}^{2}\overline{\beta}^{2},\ N_{7}=\frac{F_{21}}{F_{22}}\frac{\overline{ \alpha}^{2}}{R}-\frac{1}{4F_{22}R^{2}}-4\overline{\alpha}^{4}D_{11}.\end{array} \]  
For the revolution shell characteristics, the corrugated FG-CNTRC cylindrical shell should satisfy the circumferential closed condition, expressed as [5]  
\[\int\limits_{0}^{L}\int\limits_{0}^{2\pi R}\Big{(}\epsilon_{y}^{0}+\frac{w}{R} -\frac{1}{2}w_{y}^{2}\Big{)}dxdy=0. \tag{24}\]  
Using Equations (16), (19), (23) and (24), this integral becomes  
\[-2F_{22}\sigma_{0y}h+2\frac{f_{0}}{R}+\frac{f_{2}}{R}-\frac{1}{4}f_{1}^{2} \frac{n^{2}}{R^{2}}=0. \tag{25}\]  
Eliminating \(\sigma_{0y}\) from Equations (21)-(23) and the condition of closed form (25), leads to  
\[q_{0}=-f_{2}^{2}\frac{N_{3}R}{n^{2}}-f_{2}\frac{N_{2}R}{n^{2}}+\frac{f_{2}}{f_ {2}N_{6}+N_{5}}\frac{N_{4}N_{7}R}{n^{2}}-\frac{N_{1}R}{n^{2}}. \tag{26}\]  
Ignoring the nonlinear term \(f_{2}\) in Equation (26), leads to  
\[q_{0}^{upper}=-\frac{N_{1}}{N_{4}N_{7}}, \tag{27}\]  
where \(q_{0}^{upper}\) is the linear upper static buckling load of corrugated and non-corrugated FG-CNTRC cylindrical shells. The linear static critical buckling loads of spiral corrugated FG-CNTRC cylindrical shells are determined by conditions \(q_{\sigma}=\min q_{0}^{upper}\) vs. \((m,n)\). The maximal deflection of the shell can be calculated by taking the sum of three amplitude terms, as  
\[W_{\max}=f_{0}+f_{1}+f_{2}. \tag{28}\]  
By using Equations (21)-(23), Equation (28) is rewritten by respecting to nonlinear deflection amplitude \(f_{2}\), as  
\[W_{\max}=-\frac{N_{7}n^{2}f_{2}}{8R(f_{2}N_{6}+N_{5})}+\frac{f_{2}}{2}+\left( -\frac{f_{2}N_{7}}{f_{2}N_{6}+N_{5}}\right)^{\frac{1}{2}}+F_{22}R^{2}q_{0}. \tag{29}\]  
The postbuckling curves of spiral corrugated FG-CNTRC cylindrical shells may be obtained by combining Equation (26) and Equation (29).