## 3 Fundamental Equations
To proceed with the stability equations of the CNTRC stiffened cylindrical shell, the displacement components can be written as  
\[u_{0}=u_{0}^{0}+u_{0}^{1},\quad v_{0}=v_{0}^{0}+v_{0}^{1},\quad w_{0}=w_{0}^{0}+ w_{0}^{1},\quad\psi=\psi^{0}+\psi^{1},\quad\phi=\phi^{0}+\phi^{1} \tag{33}\]  
where \(u_{0}^{0}\), \(v_{0}^{0}\), \(w_{0}^{0}\), \(\psi^{0}\), \(\phi^{0}\) represent the displacement components at the equilibrium configuration. The displacement fields near the stable equilibrium are expressed by \(u_{0}^{1}\), \(v_{0}^{1}\), \(w_{0}^{1}\), \(\psi^{1}\), \(\phi^{1}\). Similarly, the stress resultants associated with the equilibrium condition read  
\[ \begin{array}{l}N^{1}{}_{xx}\to N^{0}{}_{xx}+\Delta N_{xx}\quad M^{1}{}_{xx} \to M^{0}{}_{xx}+\Delta M_{xx}\quad P^{1}{}_{xx}\to P^{0}{}_{xx}+\Delta P_{xx} \\ N^{1}{}_{yy}\to N^{0}{}_{yy}+\Delta N_{yy}\quad M^{1}{}_{yy}\to M^{0}{}_{yy}+ \Delta M_{yy}\quad P^{1}{}_{yy}\to P^{0}{}_{yy}+\Delta P_{yy}\\ N^{1}{}_{xy}\to N^{0}{}_{xy}+\Delta N_{xy}\quad M^{1}{}_{xy}\to M^{0}{}_{xy} +\Delta M_{xy}\quad P^{1}{}_{xy}\to P^{0}{}_{xy}+\Delta P_{xy}\\ Q^{1}{}_{xz}\to Q^{0}{}_{xz}+\Delta Qz\quad R^{1}{}_{xz}\to R^{0}{}_{xz}+ \Delta R_{xz}\quad\overline{N}^{1}{}_{xx}\to\overline{N}^{0}{}_{xx}+\Delta \overline{N}{}_{xx}\\ Q^{1}{}_{yz}\to Q^{0}{}_{yz}+\Delta Q_{yz}\quad R^{1}{}_{yz}\to R^{0}{}_{yz} +\Delta R_{yz}\quad\overline{N}^{1}{}_{yy}\to\overline{N}^{0}{}_{yy}+\Delta \overline{N}{}_{yy}\end{array}  \tag{34}\]  
The superscripts \(0\) and \(1\) refer to the equilibrium and the stability states, respectively. It is also worth noticing that the applied load enforced at a certain configuration corresponds to the critical buckling load if the relation \(\delta^{2}U=0\) is satisfied. By substitution of Equations (11) and (12) into Equation (32) and by combination with Equation (14), we obtain the second variation of the total potential energy, which leads to the stability equations after some mathematical manipulations, i.e.,  
\[N^{1}_{xx,x}+N^{1}_{xy,y}+\frac{\partial}{\partial x}\Bigg{(}\frac{1}{S_{s}} \Bigg{(}E_{r}A_{r}\Bigg{(}\frac{\partial u_{I}^{1}}{\partial y}+\frac{w_{I}^{ 1}}{R}\Bigg{)}-\overline{z}_{r}A_{r}E_{r}\frac{\partial^{2}w_{I}^{1}}{\partial y ^{2}}\Bigg{)}\Bigg{)}=0 \tag{35}\]  
\[N^{1}_{xy,x}+N^{1}_{yy,y}\,+\frac{\partial}{\partial y}\Bigg{(}\frac{1}{S_{r}} \Bigg{(}E_{r}A_{r}\Bigg{(}\frac{\partial v_{I}^{1}}{\partial y}+\frac{w_{I}^{ 1}}{R}\Bigg{)}-\overline{z}_{r}A_{r}E_{r}\frac{\partial^{2}w_{I}^{1}}{\partial y ^{2}}\Bigg{)}\Bigg{)}=0 \tag{36}\]  
\[ \begin{array}{l}\frac{1}{R}N^{1}_{yy}-\frac{1}{RS_{r}}\Bigg{(}E_{r}A_{r} \Bigg{(}\frac{\partial v_{I}^{1}}{\partial y}+\frac{w_{I}^{1}}{R}\Bigg{)}- \overline{z}_{r}A_{r}E_{r}\frac{\partial^{2}w_{I}^{1}}{\partial y^{2}}\Bigg{)} \\ +N^{0}_{xx}w_{1,xx}+2N^{0}_{xy}w_{1,xy}+\frac{1}{S_{s}}\overline{N}^{0}_{xx} \frac{\partial^{2}w_{I}^{1}}{\partial x^{2}}+\frac{1}{S_{r}}\overline{N}^{0}_{ yy}\frac{\partial^{2}w_{I}^{1}}{\partial y^{2}}-N^{0}_{S}w_{1,xx}-N^{0}_{r}w_{1,yy} \\ +N^{0}_{yy}w_{1,yy}+Q^{1}_{xx,x}+Q^{1}_{yz,y}-3c_{1}\Big{(}R^{1}_{xx,x}+R^{1}_{ yz,y}\Big{)}+c_{1}\Big{(}P^{1}_{xx,xx}+2P^{1}_{xy,xy}+P^{1}_{yy,yy}\Big{)}\\ -\frac{\partial^{2}}{\partial x^{2}}\Bigg{(}\frac{1}{S_{s}}\Big{(}E_{s}I_{s} \frac{\partial^{2}w_{I}^{1}}{\partial x^{2}}-\overline{z}_{s}A_{s}E_{s}\frac{ \partial u_{I}^{1}}{\partial x}\Big{)}\Bigg{)}\\ -\frac{\partial^{2}}{\partial y^{2}}\Bigg{(}\frac{1}{S_{r}}\Bigg{(}E_{r}I_{r} \frac{\partial^{2}w_{I}^{1}}{\partial y^{2}}-\overline{z}_{r}A_{r}E_{r}\Bigg{(} \frac{\partial v_{I}^{1}}{\partial y}+\frac{w_{I}^{1}}{R}\Bigg{)}\Bigg{)}\Bigg{)} \\ -\frac{\partial^{2}}{\partial xdy}\Bigg{(}\frac{1}{S_{s}}G_{s}I_{s}\frac{ \partial^{2}w_{I}^{1}}{\partial xdy}+\frac{1}{S_{r}}G_{r}I_{r}\frac{\partial^{ 2}w_{I}^{1}}{\partial xdy}\Bigg{)}=0\\ \end{array}