## 3 Fundamental Equations
Based on our formulation, the corresponding strain energy is expressed as [52]  
\[\delta U_{s} =\frac{1}{S_{s}}\int_{0}^{L}\int_{0}^{\alpha}\left(\int_{A_{s}} \sigma_{s}\delta\varepsilon_{s}dA\ A\ +\ G_{s}J_{s}\frac{\partial^{2}w_{l}}{\partial x\partial y}\frac{\partial^{2} \delta w_{l}}{\partial x\partial y}\right)dy\,dx \tag{19}\] \[\delta U_{r} =\frac{1}{S_{r}}\int_{0}^{L}\int_{0}^{\alpha}\left(\int_{A_{r}} \sigma_{r}\delta\varepsilon_{r}dA\ A\ +\ G_{r}J_{r}\frac{\partial^{2}w_{l}}{\partial x\partial y}\frac{\partial^{2} \delta w_{l}}{\partial x\partial y}\right)dy\,dx\]  
Substituting Equations (17) and (18) into Equation (19), it is  
\[\delta U_{s}=\int_{\Omega}\frac{1}{S_{s}}\left\{\left( \begin{array}{cc}E_{s }A_{r}\left(\frac{\partial u_{l}}{\partial x}\ +\ \frac{1}{2}\left(\frac{\partial w_{l}}{\partial x}\right)^{2}\ -\ \frac{n_{s}\Delta T}{1-2\varepsilon_{s}}\right)\left(\frac{ \partial u_{l}}{\partial x}\ +\ \frac{\partial w_{l}}{\partial x}\frac{ \partial u_{l}}{\partial x}\right)+\\ -E_{s}A_{s}\frac{\partial^{2}\delta w_{l}}{\partial x^{2}}\left(\frac{ \partial u_{l}}{\partial x}\ +\ \frac{1}{2}\left(\frac{\partial w_{l}}{\partial x}\right)^{2}\ -\ \frac{n_{s}\Delta T}{1-2 \varepsilon_{s}}\right)+\\ -E_{s}A_{s}\frac{\partial^{2}w_{l}}{\partial x^{2}}\left(\frac{\partial u_{ l}}{\partial x}\ +\ \frac{\partial w_{l}}{\partial x}\frac{ \partial u_{l}}{\partial x}\right)\ +\ E_{s}I_{r}\frac{\partial^{2}w_{l}}{\partial x^{2}}\frac{ \partial^{2}\delta w_{l}}{\partial x^{2}}\end{array} \right\}dxd\theta \tag{20}\]  
\[\delta U_{r}=\int_{\Omega}\frac{1}{S_{r}}\left\{\left( \begin{array}{cc}E_{r }A_{r}\left(\frac{\partial w_{l}}{\partial y}\ +\ \frac{1}{2}\left(\frac{\partial w_{l}}{\partial y}\right)^{2}\ +\ \frac{n_{l}}{\Gamma}\ -\frac{n_{s}\Delta T}{1-2 \varepsilon_{s}}\right)\left(\frac{\partial u_{l}}{\partial y}\ +\ \frac{\partial w_{l}}{\partial y}\frac{ \partial w_{l}}{\partial y}\ +\ \frac{\partial w_{l}}{\partial y}\right)+\\ -E_{r}A_{r}\frac{\partial^{2}\delta w_{l}}{\partial y^{2}}\left(\frac{ \partial w_{l}}{\partial y}\ +\ \frac{1}{2}\left(\frac{\partial w_{l}}{\partial y}\right)^{2}\ +\ \frac{n_{l}}{\Gamma}\ -\frac{n_{s}\Delta T}{1-2 \varepsilon_{s}}\right)+\\ -E_{r}A_{r}\frac{\partial^{2}w_{l}}{\partial y^{2}}\left(\frac{\partial w_{l} }{\partial y}\ +\ \frac{\partial w_{l}}{\partial y}\frac{ \partial w_{l}}{\partial y}\ +\ \frac{\partial w_{l}}{\Gamma}\ \right)+E_{r}I_{r}\frac{\partial^{2}w_{l}}{\partial y^{2}}\frac{ \partial^{2}\delta w_{l}}{\partial y^{2}}\end{array} \right\}dxd\theta \tag{21}\]  
\(I_{r}=b_{r}h_{r}{}^{3}/12+\frac{2}{2}_{r}^{2}A_{r}\) and \(I_{s}=b_{s}h_{s}{}^{3}/12+\frac{2}{2}_{s}^{2}A_{s}\) being the moments of inertia for the rings and stringers at a distance \(\Xi_{r},\Xi_{s}\) from the reference surface (\(z=0\)), with cross sections \(A_{r},A_{s}\), heights \(h_{r},h_{s}\), and widths \(b_{r},b_{s}\). Besides, \(J_{s}=h_{s}b_{s}{}^{3}/3\) and \(J_{r}=h_{r}b_{r}{}^{3}/3\) are the polar moments of inertia of the rings and stringers, while \(S_{s}=2\pi R/N_{s}\), \(S_{r}=L/N_{r}\) refer to the distances between two consecutive stringers and rings, whose total number is \(N_{s}\) and \(N_{r}\), respectively.