## 2 Equivalent Model of Spiral Corrugated FG-CNTRC Cylindrical Shells
Stress-strain relations for a non-corrugated CNTRC shell are expressed as (in the local coordinate system)  
\[\left( \begin{array}{c}\sigma_{n}\\ \sigma_{s}\\ \sigma_{ns}\end{array} \right)=\left( \begin{array}{cccc}Q_{11}&Q_{12}&0\\ Q_{12}&Q_{22}&0\\ 0&0&Q_{66}\end{array} \right)\left( \begin{array}{c}\varepsilon_{n}\\ \varepsilon_{s}\\ \gamma_{ns}\end{array} \right), \tag{6}\]  
where  
\[Q_{11}=\frac{E_{11}}{1-\
u_{12}\
u_{21}},\ Q_{22}=\frac{E_{22}}{1-\
u_{12}\
u_ {21}},\ Q_{12}=\frac{\
u_{21}E_{11}}{1-\
u_{12}\
u_{21}},\ Q_{66}=G_{12}.\]  
In the local coordinate system, the internal force expressions of a CNTRC shell are presented by using the homogenization technique for corrugated panels reported by [PERSON] et al. 's model [48] with the kinematic assumptions of the Kirchhoff theory, leading to  
\[\left\{ \begin{array}{l}N_{n}\\ N_{s}\\ N_{ns}\\ M_{n}\\ M_{s}\\ M_{ns}\end{array} \right\}=\left\{ \begin{array}{llll}\overline{A}_{11}& \overline{A}_{12}&0&0&0&0\\ \overline{A}_{12}&\overline{A}_{22}&0&0&0&0\\ 0&0&\overline{A}_{66}&0&0&0\\ 0&0&0&\overline{D}_{11}&\overline{D}_{12}&0\\ 0&0&0&\overline{D}_{12}&\overline{D}_{22}&0\\ 0&0&0&0&\overline{D}_{66}\end{array} \right\}\left\{ \begin{array}{l}\varepsilon _{n}^{0}\\ \varepsilon_{s}^{0}\\ \gamma_{ns}^{0}\\ \chi_{n}\\ \chi_{s}\\ 2\chi_{ns}\end{array} \right\}, \tag{7}\]where the stiffness matrix constituents of ring corrugated shells are determined as  
\[\overline{A}_{11}=\frac{2c}{\frac{I_{1}}{\hat{A}_{11}}+\frac{I_{2}}{\hat{D}_{11}}},\ \overline{A}_{12}=\frac{\hat{A}_{12}}{\hat{A}_{11}}\overline{A}_{11}\ \ \overline{A}_{22}=\frac{\overline{A}_{12}\hat{A}_{12}}{\hat{A}_{11}}+\frac{l}{c} \frac{\hat{A}_{11}\hat{A}_{22}-\hat{A}_{12}^{2}}{\hat{A}_{11}}\ \ \overline{A}_{66}=\frac{c}{l}\hat{A}_{66}\]  
\[\overline{D}_{11}=\frac{c}{l}\mathcal{D}_{11},\ \ \overline{D}_{12}=\frac{\hat{D}_{12}}{ \hat{D}_{11}}D_{11}\ \ \overline{D}_{22}=\frac{1}{2c}\big{(}I_{2}\hat{A}_{22}+I_{1}\hat{D}_{22}\big{)} \ \ \overline{D}_{66}=\frac{l}{c}\hat{D}_{66}\]  
with \(l=\frac{2f}{\sin\alpha}+c-\frac{2f}{\tan\alpha}\), \(I_{1}=\frac{4f\cos\alpha}{3\sin\alpha}+2c-\frac{4f}{\tan\alpha}\), \(I_{2}=\frac{4f^{3}}{3\sin\alpha}+2f^{2}\bigg{(}c-\frac{2f}{\tan\alpha}\bigg{)}\), for trapezoidal corrugation and \(l=\pi r+2d\), \(c=2r\), \(I_{1}=\pi r\), \(I_{2}=\frac{4d^{3}}{3}+2\pi d^{2}r+8 dr^{2}+\pi r^{3}\), for round corrugation; and the stiffness matrix constituents of pre-corrugated shell panels are calculated by  
\[\hat{A}_{ij}=\int\limits_{-\frac{h}{2}}^{\frac{h}{2}}Q_{ij}dz,\ \ \ \ \hat{D}_{ij}=\int\limits_{-\frac{h}{2}}^{\frac{h}{2}}Q_{ij}z^{2}dz. \tag{8}\]  
By using the coordinate transformation procedure, the internal force terms of a CNTRC cylindrical shell in global coordinate system (\(Oxy\)) are expressed by  
\[\left\{ \begin{array}{c}N_{x}\\ N_{y}\\ N_{xy}\\ M_{x}\\ M_{y}\\ M_{xy}\end{array} \right\}=\left\{ \begin{array}{cccccc}A_{11}&A_{12}&A_{16}& 0&0&0\\ A_{21}&A_{22}&A_{26}&0&0&0\\ A_{61}&A_{62}&A_{66}&0&0&0\\ 0&0&0&D_{11}&D_{12}&D_{16}\\ 0&0&0&D_{21}&D_{22}&D_{26}\\ 0&0&0&D_{61}&D_{62}&D_{66}\end{array} \right\}\left( \begin{array}{c}\varepsilon _{x}^{0}\\ \varepsilon_{y}^{0}\\ \gamma_{xy}^{0}\\ -w_{,xx}\\ -2w_{,xy}\end{array} \right), \tag{9}\]  
where  
\[