## 2 3D Exact Static Analysis of FGM Structures  
The formulation of the proposed 3D static solution has been developed in mixed orthogonal curvilinear coordinates. All the details concerning the geometry and the reference system may be found in Figure 1. A generic spherical shell is shown; the reference system \((\alpha,\beta,z)\) is mixed curvilinear and orthogonal. The origin is located in a corner with \(z\) being normal to the middle surface \(\Omega_{0}\), heading towards the upper surface and \(\alpha\) and \(\beta\) are defined on it. \(z\) varies from \(-h/2\) to \(h/2\) being \(h\) the total thickness of the shell. \(z\) is a secondary thickness coordinates. It has its origin at the bottom of the shell, and thus varies from 0 to \(h\). Figure 2 gives a clear definition of both reference coordinates. In the present paper, functionally graded materials (FGMs), which change mechanical properties only through the thickness direction \(z\), have been taken into account. The three-dimensional linear elastic constitutive equations, written for shells embedding isotropic FGMs, must take into consideration the dependence of the elastic coefficients from the thickness coordinate \(z\). For each k physical layer that constitutes the shell, the link between the stress vector \(\mathbf{\sigma}^{k}=(\sigma^{k}_{\alpha\alpha},\sigma^{k}_{\beta\beta},\sigma^{k}_{ zz},\sigma^{k}_{\beta z},\sigma^{k}_{\alpha zz},\sigma^{k}_{\alpha\beta})\) and the strain vector \(\mathbf{\epsilon}^{k}=(\epsilon^{k}_{\alpha\alpha},\epsilon^{k}_{\beta\beta}, \epsilon^{k}_{zz},\gamma^{k}_{\beta z},\gamma^{k}_{\alpha z},\gamma^{k}_{ \alpha\beta})\) is given by the \(6\times 6\) matrix \(\mathbf{C}^{k}(z)\):  
\[\mathbf{C}^{k}(z)= \begin{bmatrix}C^{k}_{11}(z)&C^{k}_{12}(z)&C^{k}_{13}(z)&0&0&0\\ C^{k}_{12}(z)&C^{k}_{22}(z)&C^{k}_{23}(z)&0&0&0\\ C^{k}_{13}(z)&C^{k}_{23}(z)&C^{k}_{33}(z)&0&0&0\\ 0&0&0&C^{k}_{44}(z)&0&0\\ 0&0&0&0&C^{k}_{55}(z)&0\\ 0&0&0&0&0&C^{k}_{66}(z)\end{bmatrix} \,, \tag{1}\]  
the compact form of constitutive relations is:  
\[\mathbf{\sigma}^{k}=\mathbf{C}^{k}(z)\mathbf{\epsilon}^{k}\,. \tag{2}\]  
The output of the present formulation is given in terms of displacements and their derivatives made with respect to the thickness coordinate \(z\). In order to obtain constitutive equations in displacement form, the geometrical relations must be introduced. When shells with constant radii of curvature are considered, the following parametric coefficients can be defined:  
\[H_{\alpha}=(1+\frac{z}{R_{\alpha}})=(1+\frac{\tilde{z}-h/2}{R_{\alpha}})\,\ \ \ H_{\beta}=(1+\frac{z}{R_{\beta}})=(1+\frac{\tilde{z}-h/2}{R_{\beta}})\,\ \ \ H_{z}=1. \tag{3}\]  
\(R_{\alpha}\) and \(R_{\beta}\) are the radii of curvature in \(\alpha\) and \(\beta\) directions, respectively. \(\tilde{z}\) and \(z\) are the two thickness coordinates defined above. \(H_{z}=1\) because no curvature exists along the \(z\)-direction. \(H_{\alpha}\) and \(H_{\beta}\) are the crucial points as their introduction allows for writing a general formulation valid for spherical shells with two finite radii of curvature. This formulation can be easily extended to single-curvature structures and plates. When the considered geometry is a cylinder or a cylindrical panel, \(R_{\alpha}\) or \(R_{\beta}\) is infinite, which implies \(H_{\alpha}=1\) or \(H_{\beta}=1\). In the same way, when both the radii of curvature are infinite (plate geometry), \(H_{\alpha}=H_{\beta}=1\).