## 2 Definition of the Geometry  
As highlighted by [PERSON] in his book [103], each shell structure is a three-dimensional body which is bounded by two close surfaces, whose distance measured along the normal direction defines the thickness \(h\) of the shell. The middle surface is clearly equidistant from these two external surfaces and can be taken as reference domain for the governing equations. Its importance is evident, since its define the shape of the structure under consideration and rules the mechanical behavior of a shell structure if a two-dimensional structural model is considered. By means of the differential geometry, several kinds of shells can be studied in a general and unified manner. For instance, the same theoretical approach is used to define the geometry of doubly-curved, singly-curved and degenerate shells (or plates). With reference to Figure 1, in which a generic shell element is depicted, the position vector \(\mathbf{R}(\alpha_{1},\alpha_{2},\zeta)\) is introduced to identify each point of the three-dimensional medium as follows  
\[\mathbf{R}(\alpha_{1},\alpha_{2},\zeta)=\mathbf{r}(\alpha_{1},\alpha_{2})+ \frac{h(\alpha_{1},\alpha_{2})}{2}z\mathbf{n}(\alpha_{1},\alpha_{2}) \tag{1}\]  
where the dimensionless quantity \(z=2\zeta/h\in[-1,1]\) denotes the distance of the point \(P\) from its projection \(P^{\prime}\) on the reference domain. On the other hand, \(\mathbf{r}(\alpha_{1},\alpha_{2})\) is the position vector that describe the middle surface, which assumes different meanings according to the investigated geometry. A curvilinear orthogonal coordinates system \(O^{\prime}\alpha_{1}\alpha_{2}\zeta\) must be defined on the shell middle surface. In particular, the coordinates \(\alpha_{1}\), \(\alpha_{2}\) denote the lines of main curvature of the middle surface, whereas \(\zeta\) is the normal direction that can be specified by the following outward unit normal vector \(\mathbf{n}(\alpha_{1},\alpha_{2})\)  
Figure 1: Representation of a generic doubly-curved shell element, edge identification and lamination scheme. \[\mathbf{n}=\frac{\mathbf{r}_{1}\times\mathbf{r}_{2}}{|\mathbf{r}_{1}\times\mathbf{ r}_{2}|} \tag{2}\]  
where the symbol \"\(\times\)\" represents the vector product. For conciseness purposes, the notation \(\mathbf{r}_{i}=\partial\mathbf{r}/\partial\alpha_{i}\), for \(i=1,2\), is introduced. For the sake of completeness, it should be highlighted that the curvilinear coordinates \(\alpha_{1}\), \(\alpha_{2}\) vary according to the considered structure. For example, they assume the following meaning \(\alpha_{1}=\varphi,\alpha_{2}=\vartheta\) for a doubly-curved shell of revolution, whereas the notation \(\alpha_{1}=\varphi,\alpha_{2}=y\) is introduced for a singly-curved shell of translation. In the case of a flat panel, such as a rectangular plate, they can be assumed equal to \(\alpha_{1}=x,\alpha_{2}=y\). This aspect is to underline the effectiveness of the differential geometry. In general, the position vector of any reference surface \(\mathbf{r}(\alpha_{1},\alpha_{2})\) is expressed as  
\[\mathbf{r}(\alpha_{1},\alpha_{2})=f_{1}(\alpha_{1},\alpha_{2})\mathbf{e}_{1}+ f_{2}(\alpha_{1},\alpha_{2})\mathbf{e}_{2}+f_{3}(\alpha_{1},\alpha_{2})\mathbf{e}_{3} \tag{3}\]  
where \(f_{i}(\alpha_{1},\alpha_{2})\), for \(i=1,2,3\) are arbitrary functions needed to describe the surface under consideration. On the other hand, the symbols \(\mathbf{e}_{i}\), for \(i=1,2,3\), denote the unit vectors of axes of the global reference system \(\mathcal{O}^{\prime}x_{1}x_{2}x_{3}\) as depicted in Figure 1. In the following sections, the position vector \(\mathbf{r}(\alpha_{1},\alpha_{2})\) will be specified for the various shell structures under consideration. Further details concerning these features, as well as a wide range of shell structures, can be found in the recent books by [PERSON] et al. [2; 3]. In general, the curvilinear coordinates in hand must be bounded by specific values to define a finite domain.