## 3 Fundamental Equations
In addition, the elastic coefficients \(Q_{ij}\) (for \(i,j=1,\ldots,6\)) are the reduced material stiffness coefficients compatible with the plane-stress conditions. They are obtained as follow [52; 53; 54]  
\[ \begin{split} Q_{11}&=\frac{E_{11}(z,T)}{1-v_{12}(z,T )v_{21}(z,T)}\,,\\ Q_{22}&=\frac{E_{22}(z,T)}{1-v_{12}(z,T)v_{21}(z,T )}\,,\\ Q_{12}&=\frac{v_{21}(z,T)E_{11}(z,T)}{1-v_{12}(z,T )v_{21}(z,T)},\\ Q_{44}&=G_{23}(z,T)\\ Q_{55}&=G_{13}(z,T)\\ Q_{66}&=G_{12}(z,T)\end{split}  \tag{13}\]  
The equilibrium equations for a FG wavy CNTRC stiffened cylindrical shell under thermal loading will be derived from a stationary potential energy criterion. Thus, we define the total potential energy for a cylindrical shell stiffened by ring and stiffeners as  
\[U=U_{Sh}\ +\ U_{r}\ +\ U_{s} \tag{14}\]  
where \(U_{Sh},U_{r},U_{s}\) are the single components of the strain energy for the shell, the ring, and the stiffeners, respectively. More specifically, according to the TSDT, the strain energy component \(U_{Sh}\) for the cylindrical shell can be defined as  
\[ \begin{split} U_{Sh}=\frac{1}{2}\int\Bigl{\{}& \sigma_{xx}\Bigl{(}\epsilon_{xx}^{(0)}\ +\ z\epsilon_{xx}^{(1)}\ +\ z^{3}\epsilon_{xx}^{(3)}\Bigr{)}\ +\ \sigma_{yy}\Bigl{(}\epsilon_{yy}^{(0)}\ +\ z \epsilon_{yy}^{(1)}\ +\ Z^{3}\epsilon_{yy}^{(3)}\Bigr{)}+\\ &\tau_{xy}\Bigl{(}\gamma_{xy}^{(0)}\ +\ z\gamma_{xy}^{(1)}\ +\ z^{3}\gamma_{xy}^{(3)}\Bigr{)}\ +\ \tau_{xz}\Bigl{(}\gamma_{xz}^{(0)}\ +\ z^{2}\gamma_{xz}^{(2)}\Bigr{)}\ +\ \tau_{yz}\Bigl{(}\gamma_{yz}^{(0)}\ +\ z^{2}\gamma_{yz}^{(2)} \Bigr{)}\Bigr{\}}dV\end{split}  \tag{15}\]  
As is different from the available works [55; 56], where the discrete stiffener are smeared out over the respective spacing, in the present paper, we model the stiffeners as Euler-Bernoulli beams under the hypothesis that normals to the cylindrical shell and stiffeners remain normal after the deflection. Thus, following the Kirchhoff-Love classical assumptions [52] the displacement field can be expressed as  
\[ \begin{split}\overline{u}(x,y)&=u_{l}(x,y)\ -\ z\frac{\partial w_{l}(x,y)}{\partial x}\\ \overline{v}(x,y)&=v_{l}(x,y)\ -\ z\frac{\partial w_{l}(x,y) }{\partial y}\\ \overline{w}(x,y)&=w_{l}(x,y)\end{split}  \tag{16}\]  
where \(u_{l},v_{l}\), and \(w_{l}\) denote the displacement components of points located through the cylindrical shell thickness. The following expressions for the strain field \(\epsilon_{s},\epsilon_{r}\) are derived for stringer and ring stiffeners, respectively:  
\[ \begin{split}\epsilon_{s}=\frac{\partial u_{l}}{\partial x}\ +\ \frac{1}{2}\Bigl{(}\frac{\partial w_{l}}{\partial x}\Bigr{)}^{2}\ -\ z\frac{\partial^{2}w_{l}}{\partial x^{2}}\\ \epsilon_{r}=\frac{\partial v_{l}}{\partial y}\ +\ \frac{1}{2}\Bigl{(} \frac{\partial w_{l}}{\partial y}\Bigr{)}^{2}\ +\ \frac{w_{l}}{R}\ -\ z\Bigl{(}\frac{ \partial^{2}w_{l}}{\partial y^{2}}\Bigr{)}\end{split}  \tag{17}\]  
Moreover, we propose the following form to define the stress field of stringer and ring stiffeners in thermal environments (see [57; 58]). \[ \begin{split}\overline{\sigma}_{xx}&=E_{s}\epsilon _{s}-\frac{E_{s}}{\Gamma-2v_{s}}\alpha_{s}(z,T)\Delta T\\ \overline{\sigma}_{yy}&=E_{r}\epsilon_{r}-\frac{E_{ r}}{\Gamma-2v_{r}}\alpha_{r}(z,T)\Delta T\end{split}  \tag{18}\]  
\(E_{s},E_{r}\) being the elasticity modulus for stringer and ring stiffeners, respectively. More complex models could be considered to capture the actual orthotropic behavior of stiffeners or ribs, as proposed in [59]. This will be considered as a further development of the proposed formulation, here explored in a simplified manner.