## 2 3D Exact Static Analysis of FGM Structures
With these parametric coefficients, the geometrical relations for each k layer of the multilayered FGM shell may be written as:  
\[\epsilon^{k}_{\alpha\alpha} =\frac{1}{H_{\alpha}(z)}\frac{\partial u^{k}}{\partial\alpha}+ \frac{w^{k}}{H_{\alpha}(z)R_{\alpha}}\,\] (4) \[\epsilon^{k}_{\beta\beta} =\frac{1}{H_{\beta}(z)}\frac{\partial v^{k}}{\partial\beta}+ \frac{w^{k}}{H_{\beta}(z)R_{\beta}}\,\] (5) \[\epsilon^{k}_{zz} =\frac{\partial w^{k}}{\partial z}\,\] (6) \[\gamma^{k}_{\beta z} =\frac{1}{H_{\beta}(z)}\frac{\partial w^{k}}{\partial\beta}+ \frac{\partial v^{k}}{\partial z}-\frac{v^{k}}{H_{\beta}(z)R_{\beta}}\,\] (7) \[\gamma^{k}_{\alpha z} =\frac{1}{H_{\alpha}(z)}\frac{\partial w^{k}}{\partial\alpha}+ \frac{\partial u^{k}}{\partial z}-\frac{u^{k}  
These relations are written for a spherical shell with constant radii of curvature [27; 28]. It is easy to see how the relations degenerate into the ones for cylinders, cylindrical shells and plates when one or both the parametric coefficients are equal to 1 (\(R_{\alpha}\) or \(R_{\beta}\) equals infinite). The displacement form of the constitutive equations may be obtained introducing in Equation (2) the geometrical relations of Equations (4)-(9):  
\[\sigma^{k}_{\alpha\alpha} = \frac{C^{k}_{11}(z)}{H_{\alpha}(z)}u^{k}_{\alpha}+\frac{C^{k}_{11} (z)}{H_{\alpha}(z)R_{\alpha}}w^{k}+\frac{C^{k}_{12}(z)}{H_{\beta}(z)}v^{k}_{, \beta}+\frac{C^{k}_{12}(z)}{H_{\beta}(z)R_{\beta}}w^{k}+C^{k}_{13}(z)w^{k}_{,z}\;, \tag{10}\] \[\sigma^{k}_{\beta\beta} = \frac{C^{k}_{12}(z)}{H_{\alpha}(z)}u^{k}_{\alpha}+\frac{C^{k}_{12 }(z)}{H_{\alpha}(z)R_{\alpha}}w^{k}+\frac{C^{k}_{22}(z)}{H_{\beta}(z)}v^{k}_{, \beta}+\frac{C^{k}_{22}(z)}{H_{\beta}(z)R_{\beta}}w^{k}+C^{k}_{23}(z)w^{k}_{,z }\;,\] (11) \[\sigma^{k}_{zz} = \frac{C^{k}_{13}(z)}{H_{\alpha}(z)}u^{k}_{\alpha}+\frac{C^{k}_{13 }(z)}{H_{\alpha}(z)R_{\alpha}}w^{k}+\frac{C^{k}_{23}(z)}{H_{\beta}(z)}v^{k}_{, \beta}+\frac{C^{k}_{23}(z)}{H_{\beta}(z)R_{\beta}}w^{k}+C^{k}_{33}(z)w^{k}_{,z }\;,\] (12) \[\sigma^{k}_{\beta z} = \frac{C^{k}_{44}(z)}{H_{\beta}(z)}w^{k}_{,\beta}+C^{k}_{44}(z)v^{ k}_{,z}-\frac{C^{k}_{44}(z)}{H_{\beta}(z)R_{\beta}}v^{k}\;,\] (13) \[H_{\alpha}=(1+\frac{z}{R_{\alpha}})=(1+\frac{\tilde{z}-h/2}{R_{\alpha}})\,\ \ \ H_{\beta}=(1+\frac{z}{R_{\beta}})=(1+\frac{\tilde{z}-h/2}{R_{\beta}})\,\ \ \ H_{z}=1. \tag{3}\]0 (14) \[H_{\alpha}=(1+\frac{z}{R_{\alpha}})=(1+\frac{\tilde{z}-h/2}{R_{\alpha}})\,\ \ \ H_{\beta}=(1+\frac{z}{R_{\beta}})=(1+\frac{\tilde{z}-h/2}{R_{\beta}})\,\ \ \ H_{z}=1. \tag{3}\]1  
In Equations (10)-(15), the subscripts \({}_{\alpha}\), \({}_{\beta}\) and \({}_{,z}\) have been introduced to compact the related partial derivative symbols \(\frac{\partial}{\partial\alpha}\), \(\frac{\partial}{\partial\beta}\) and \(\frac{\partial}{\partial z}\). All the coefficients that multiply the displacements and their derivatives are a function of the thickness coordinate \(z\), even when both the radii of curvature are infinite. \(H_{\alpha}\) and \(H_{\beta}\) depend on \(z\) when the relative curvature is concerned, the elastic coefficients depend on \(z\) because of the presence of FGM layers. These features are overcome assuming the real structure divided into a certain number M of mathematical layers. For each mathematical layer \(j\), the elastic and the parametric coefficients may be calculated in its middle point, with both the radii of curvature evaluated in the middle surface \(\Omega_{0}\) of the whole structure. The number of mathematical layers to be introduced is the main subject of study of the present paper, and this number depends on several parameters. If \(j\) is the index that identifies the global mathematical layers, Equations (10)-(15) can be rewritten for a generic layer \(j\) using constant coefficients:  
\[H_{\alpha}=(1+\frac{z}{R_{\alpha}})=(1+\frac{\tilde{z}-h/2}{R_{\alpha}})\,\ \ \ H_{\beta}=(1+\frac{z}{R_{\beta}})=(1+\frac{\tilde{z}-h/2}{R_{\beta}})\,\ \ \ H_{z}=1. \tag{3}\]2 Reference system and notations for the generic shell element.