## 3 Definition of the Geometry  
Let us consider a generic doubly-curved shell element with variable thickness as the one schematically depicted in Figure 2. It is completely identified by means of the position vector \(\mathbf{R}(\alpha_{1},\alpha_{2},\zeta)\) defined as follows  
\[\mathbf{R}(\alpha_{1},\alpha_{2},\zeta)=\mathbf{r}(\alpha_{1},\alpha_{2})+ \frac{h(\alpha_{1},\alpha_{2})}{2}z\mathbf{n}(\alpha_{1},\alpha_{2}) \tag{9}\]  
where \(z=2\zeta/h(\alpha_{1},\alpha_{2})\), for \(z\in[-1,1]\), is a non-dimensional parameter which represents the distance between the generic point \(P\) within the three-dimensional shell body and its projection \(P^{\prime}\), located on the middle surface. It should be noted that \(O^{\prime}\alpha_{1}\alpha_{2}\zeta\) denotes the local reference coordinate system of the shell element. In particular, \(\alpha_{1},\alpha_{2}\) are the principal curvilinear coordinates of the shell middle surface, whereas \(\zeta\) specify the coordinate along the normal directions, which is identified by the  
Figure 1: Stacking sequence of a laminated composite structure with variable thickness made of several Functionally Graded Materials (FGM) or isotropic layers. outward unit normal vector \(\mathbf{n}(\alpha_{1},\alpha_{2})\). The following boundary values must be specified to define the three-dimensional domain of the shell. \[\alpha_{1}^{0}\leq\alpha_{1}\leq\alpha_{1}^{1} \tag{10}\]  
\[\alpha_{2}^{0}\leq\alpha_{2}\leq\alpha_{2}^{1} \tag{11}\]  
\[-\frac{h(\alpha_{1},\alpha_{2})}{2}\leq\zeta\leq\frac{h(\alpha_{1},\alpha_{2}) }{2} \tag{12}\]  
From the definition shown in Equation (9), it is clear that the thickness of the shell \(h(\alpha_{1},\alpha_{2})\) is arbitrary and can vary point by point. It should be recalled that its overall value is given by the sum of the thickness of each ply if the structure is made of several layers (Figure 1). Thus, one gets  
\[h(\alpha_{1},\alpha_{2})=\sum_{k=1}^{l}h_{k}(\alpha_{1},\alpha_{2}) \tag{13}\]  
where \(l\) stands for the number of layers. In the present paper, the thickness variation assumes the following aspect  
\[h(\alpha_{1},\alpha_{2})=h_{0}(1+f(\alpha_{1})+g(\alpha_{2})) \tag{14}\]  
in which \(h_{0}\) is the thickness reference value, whereas \(f(\alpha_{1}),g(\alpha_{2})\) are arbitrary functions that describe the thickness profile along the principal curvilinear coordinates \(\alpha_{1},\alpha_{2}\). It should be pointed out that there is no restriction on the choice of these functions, as illustrated in details in the work by [PERSON] et al. [104]. Nevertheless, they must represent a smooth variation and their effect has to involve each layer. Thus, the parameters \(h_{0,k}\), for \(k=1,2,...,l\), are introduced to specify the thickness reference value of each lamina. For completeness purposes, the reader can refer to the work by [PERSON] et al. [104] for a general treatise on the variable thickness. In connection with Equation (9),  
Figure 2: Generic doubly-curved shell element with variable thickness: edge identification, global and local reference coordinate system representation. \(\mathbf{r}(\alpha_{1},\alpha_{2})\) is the vector that allows to identify each point of the shell reference surface, which corresponds to the middle surface of the structure. In general, its expression can be written as follows  
\[\mathbf{r}(\alpha_{1},\alpha_{2})=f_{1}(\alpha_{1},\alpha_{2})\mathbf{e}_{1}+f_ {2}(\alpha_{1},\alpha_{2})\mathbf{e}_{2}+f_{3}(\alpha_{1},\alpha_{2})\mathbf{e }_{3} \tag{15}\]  
in which \(f_{i}(\alpha_{1},\alpha_{2})\), for \(i=1,2,3\), are generic functions that depend on the shell middle surface, whereas \(\mathbf{e}_{i}\), for \(i=1,2,3\), are the unit vectors which denote the main axes of the global reference system \(O^{\prime}x_{1}x_{2}x_{3}\) shown in \[\mathbf{R}(\alpha_{1},\alpha_{2},\zeta)=\mathbf{r}(\alpha_{1},\alpha_{2})+ \frac{h(\alpha_{1},\alpha_{2})}{2}z\mathbf{n}(\alpha_{1},\alpha_{2}) \tag{9}\]0 Equation (15) is fundamental to evaluate each geometric quantity related to the shell under consideration.