## 3 Fundamental Equations
Furthermore, the internal actions of the cylindrical shell can be expressed in terms of forces and moments as follows:  
\[\left(N_{ij},M_{ij},P_{ij}\right) =\int_{-h/2}^{h/2}\sigma_{ij}(1\,,\,z,\,z^{3})dz ij =xx\,,\,yy \tag{22}\] \[\left(N_{ij},M_{ij},P_{ij}\right) =\int_{-h/2}^{h/2}\sigma_{ij}(1\,,\,z,\,z^{3})dz ij =xy\] \[\left(Q_{ij},R_{ij}\right) =\int_{-h/2}^{h/2}\sigma_{ij}(1,\,z^{2})dz ij =xz\,,\,yz\]  
Using the Equations (11), (12) and (22), the internal actions for the FG-CNTRC stiffened cylindrical shell become  
\[N_{xx} =A_{11}\varepsilon_{xx}^{(0)}+A_{22}\varepsilon_{xx}^{(1)}+A_{44} \varepsilon_{xx}^{(3)}+B_{11}\varepsilon_{yy}^{(0)}+B_{22}\varepsilon_{yy}^{(1) }+B_{44}\varepsilon_{yy}^{(3)} \tag{23}\] \[N_{yy} =B_{11}\varepsilon_{xx}^{(0)}+B_{22}\varepsilon_{xx}^{(1)}+B_{44} \varepsilon_{xx}^{(3)}+D_{11}\varepsilon_{yy}^{(0)}+D_{22}\varepsilon_{yy}^{(1) }+D_{44}\varepsilon_{yy}^{(3)}\] \[N_{xy} =E_{11}\gamma_{xy}^{(0)}+E_{22}\gamma_{xy}^{(1)}+E_{44}\gamma_{xy} ^{(3)}\] \[P_{xx} =A_{44}\varepsilon_{xx}^{(0)}+A_{55}\varepsilon_{xx}^{(1)}+A_{77} \varepsilon_{xx}^{(3)}+B_{44}\varepsilon_{yy}^{(0)}+B_{55}\varepsilon_{yy}^{(1) }+B_{77}\varepsilon_{yy}^{(3)}\] \[P_{yy} =B_{44}\varepsilon_{xx}^{(0)}+B_{55}\varepsilon_{xx}^{(1)}+B_{77} \varepsilon_{xx}^{(3)}+D_{44}\varepsilon_{yy}^{(0)}+D_{55}\varepsilon_{yy}^{(1) }+D_{77}\varepsilon_{yy}^{(3)}\] \[P_{xy} =E_{44}\gamma_{xy}^{(0)}+E_{55}\gamma_{xy}^{(1)}+E_{77}\gamma_{xy} ^{(3)}\] \[M_{xx} =A_{22}\varepsilon_{xx}^{(1)}+A_{33}\varepsilon_{xx}^{(1)}+A_{55} \varepsilon_{xx}^{(3)}+B_{22}\varepsilon_{yy}^{(0)}+B_{33}\varepsilon_{yy}^{(1) }+B_{55}\varepsilon_{yy}^{(3)}\] \[\left(N_{ij},M_{ij},P_{ij}\right) =\int_{-h/2}^{h/2}\sigma_{ij}(1\,,\,z,\,z^{3})dz ij =xy\]0 \[\left(N_{ij},M_{ij},P_{ij}\right) =\int_{-h/2}^{h/2}\sigma_{ij}(1\,,\,z,\,z^{3})dz ij =xy\]1 \[\left(N_{ij},M_{ij},P_{ij}\right) =\int_{-h/2}^{h/2}\sigma_{ij}(1\,,\,z,\,z^{3})dz ij =xy\]2 \[\left(N_{ij},M_{ij},P_{ij}\right) =\int_{-h/2}^{h/2}\sigma_{ij}(1\,,\,z,\,z^{3})dz ij =xy\]3 \[\left(N_{ij},M_{ij},P_{ij}\right) =\int_{-h/2}^{h/2}\sigma_{ij}(1\,,\,z,\,z^{3})dz ij =xy\]4 \[\left(N_{ij},M_{ij},P_{ij}\right) =\int_{-h/2}^{h/2}\sigma_{ij}(1\,,\,z,\,z^{3})dz ij =xy\]5where the constants are determined as  
\[ \begin{array}{l}(A_{11},A_{22},A_{33},A_{44},A_{55},A_{77})=\int_{-h/2}^{h/2}Q_{ 11}(z,T)\big{(}1,z,z^{2},z^{3},z^{4},z^{6}\big{)}dz\\ (D_{11},D_{22},D_{33},D_{44},D_{55},D_{77})=\int_{-h/2}^{h/2}Q_{22}(z,T)\big{(}1,z,z^{2},z^{3},z^{4},z^{6}\big{)}dz\\ (B_{11},B_{22},B_{33},B_{44},B_{55},B_{77})=\int_{-h/2}^{h/2}Q_{12}(z,T)\big{(}1,z,z^{2},z^{3},z^{4},z^{6}\big{)}dz\\ (H_{11},H_{22},H_{33},H_{44},H_{55},H_{66})=\int_{-h/2}^{h/2}Q_{44}(z,T)\big{(}1,z,z^{2},z^{3},z^{4},z^{5}\big{)}dz\\ (E_{11},E_{22},E_{33},E_{44},E_{55},E_{66},E_{77})=\int_{-h/2}^{h/2}Q_{66}(z,T )\big{(}1,z,z^{2},z^{3},z^{4},z^{5},z^{6}\big{)}dz\\ (F_{11},F_{33},F_{55})=\int_{-h/2}^{h/2}Q_{55}(z,T)\big{(}1,z^{2},z^{4}\big{)} dz\end{array}  \tag{24}\]  
By substitution of Equations (15), (20) and (21) into Equation (14), the equilibrium equation for the FG wavy CNT stiffened cylindrical shell can be obtained with a variational approach.