## 3 Fundamental Equations
Thus, the adoption of the Euler equation [60] yields to the following set of equilibrium equations  
\[N_{xx,x}+N_{xy,y}+\frac{\partial}{\partial x}\Bigg{(}\frac{1}{S_{s}}\Bigg{(}E _{s}A_{s}\Bigg{(}\frac{\partial u_{I}}{\partial x}+\frac{1}{2}\Bigg{(}\frac{ \partial u_{I}}{\partial x}\Bigg{)}^{2}\Bigg{)}-\overline{z}_{s}A_{S}E_{S} \Bigg{(}\frac{\partial w_{I}}{\partial x}\Bigg{)}^{2}\Bigg{)}\Bigg{)}=0 \tag{25}\]  
\[N_{xy,x}+N_{yy,y}+\frac{\partial}{\partial y}\Bigg{(}\frac{1}{S_{r}}\Bigg{(}E _{r}A_{r}\Bigg{(}\frac{\partial v_{I}}{\partial y}+\frac{w_{I}}{R}+\frac{1}{ 2}\Bigg{(}\frac{\partial w_{I}}{\partial y}\Bigg{)}^{2}\Bigg{)}-\overline{z} _{r}A_{r}E_{r}\Bigg{(}\frac{\partial^{2}W_{I}}{\partial y^{2}}\Bigg{)}\Bigg{)} \Bigg{)}=0 \tag{26}\]  
\[\begin{array}{l}-\frac{1}{R}N_{yy}-\frac{1}{R_{Sr}}\Bigg{(}E_{r}A_{r} \Bigg{(}\frac{\partial v_{I}}{\partial y}+\frac{w_{I}}{R}+\frac{1}{2}\Big{(} \frac{\partial w_{I}}{\partial y}\Big{)}^{2}\Bigg{)}-\overline{z}_{r}A_{r}E_{ r}\Bigg{(}\frac{\partial^{2}w_{I}}{\partial y^{2}}\Bigg{)}\Bigg{)}+N_{xx}w_{0,xx} \\ +N_{yy}w_{0,yy}+2N_{xy}w_{0,xy}-N_{0}\frac{\partial^{2}w}{\partial y^{2}}+ \frac{1}{s_{s}}\overline{N}_{xx}\frac{\partial^{2}w_{I}}{\partial x^{2}}\\ -N_{0x}\frac{\partial^{2}w}{\partial x^{2}}+\frac{1}{s_{r}}\overline{N}_{yy} \frac{\partial^{2}w_{I}}{\partial y^{2}}+Q_{xz,x}+Q_{yz,y}-3c_{1}\big{(}R_{xz,x}+R_{yz,y}\big{)}\\ +c_{1}\big{(}P_{xx,xx}+2P_{xy,xy}+P_{yy,yy}\big{)}-\frac{\partial^{2}}{\partial x ^{2}}\Bigg{(}\frac{1}{S_{s}}\bigg{(}\bigg{(}E_{s}I_{s}+\overline{z}_{s}E_{s}A_ {s}c_{1}\Big{(}\frac{h}{2}\Bigg{)}^{3}\bigg{)}\frac{\partial^{2}w_{I}}{ \partial x^{2}}\\ -\bigg{(}\overline{z}_{s}A_{s}E_{s}+E_{s}A_{s}c_{1}\Big{(}\frac{h}{2}\Big{)}^{3} \bigg{)}\Bigg{(}\frac{\partial u_{I}}{\partial x}+\frac{1}{2}\Big{(}\frac{ \partial w_{I}}{\partial x}\Big{)}^{2}\bigg{)}\Bigg{)}\Bigg{)}-\frac{\partial^{2} }{\partial y^{2}}\Bigg{(}\frac{1}{S_{r}}\bigg{(}\bigg{(}E_{r}I_{r}+\overline{z }_{r}E_{r}A_{r}c_{1}\Big{(}\frac{h}{2}\Big{)}^{3}\bigg{)}\frac{\partial^{2}w_{I }}{\partial y^{2}}\\ -\bigg{(}\overline{z}_{r}A_{r}E_{r}+E_{r}A_{r}c_{1}\Big{(}\frac{h}{2}\Big{)}^{3} \bigg{)}\Bigg{(}\frac{\partial v_{I}}{\partial y}+\frac{1}{2}\Big{(}\frac{ \partial w_{I}}{\partial y}\Big{)}^{2}+\frac{w_{I}}{R}\bigg{)}\bigg{)}\bigg{)} \Bigg{)}-\frac{\partial^{2}}{\partial x\partial y}\Big{(}\frac{1}{S_{s}}G_{s}J_{s }\frac{\partial^{2}w_{I}}{\partial x\partial y}+\frac{1}{S_{r}}G_{r}J_{r} \frac{\partial^{2}w_{I}}{\partial x\partial y}\Big{)}=0 \tag{28}\]  
\[-Q_{xz}^{1}-3c_{1}R_{xz}^{1}+M_{xx,x}^{1}+M_{xy,y}^{1}-c_{1}\Big{(}P_{xx,x}^{1} +P_{xy,y}^{1}\Big{)}=0 \tag{29}\]  
\[-Q_{yz}^{1}-3c_{1}R_{yz}^{1}+M_{xy,x}^{1}+M_{yy,y}^{1}-c_{1}\Big{(}P_{xy,x}^{1} +P_{yy,y}^{1}\Big{)}=0 \tag{30}\]  
where \(\overline{N}_{xx}\) and \(\overline{N}_{yy}\) are the axial forces applied to the ring and stringer stiffeners, defined as  
\[\overline{N}_{xx}=\int_{A_{S}}\sigma_{xx}dA=\int\limits_{h/2}^{h/2+h_{s}}b_{s} \sigma_{s}dA= E_{s}A_{s}\Bigg{(}\frac{\partial u_{I}}{\partial x}+\frac{1}{2} \Bigg{(}\frac{\partial w_{I}}{\partial x}\Bigg{)}^{2}\Bigg{)}-\overline{z}_{s}A_{ s}E_{s}\frac{\partial^{2}w_{I}}{\partial x^{2}} \tag{31}\]  
\[\overline{N}_{yy}=\int_{A_{S}}\sigma_{yy}dA=\int\limits_{h/2}^{h/2+h_{r}}b_{r} \sigma_{r}dA= E_{r}A_{r}\Bigg{(}\frac{\partial v_{I}}{\partial y}+\frac{1}{2} \Bigg{(}\frac{\partial w_{I}}{\partial y}\Bigg{)}^{2}+\frac{w_{I}}{R}\Bigg{)}- \overline{z}_{r}A_{r}E_{r}\frac{\partial^{2}w_{I}}{\partial y^{2}} \tag{32}\]  
By using the Taylor series, the total strain energy of the shell can be expanded about the equilibrium state as  
\[\Delta U=\delta U+\frac{1}{2}\delta^{2}U+\frac{1}{6}\delta^{3}U+\ldots \tag{33}\]  
where the first variation, \(\delta U\), is related to the equilibrium configuration. The stability condition of the stiffened cylindrical shell near the equilibrium state is known to be related to the sign of the second variation \(\delta^{2}U\). The condition \(\delta^{2}U=0\) is enforced to determine the stability equations of the buckling problem [60].