## 2 Theoretical Model
### Shell Formulation
\begin{array}{l}\overline{B}^{(k)}_{nm}=\overline{C}^{(k)}_{nm}\ \ \ \ \text{for}\ \ \ n,m=1,2,3,6\\ \overline{B}^{(k)}_{nm}=x\overline{C}^{(k)}_{nm}\ \ \ \text{for}\ \ \ n,m=4,5\end{array}  \tag{22}\]  
where \(\kappa\) is the shear correction factor, the value of which will be specified in the notation representative of the structural model through a proper superscript. This model assumes that each layer of the laminated composite is made of linearly elastic and homogenous materials. The plies are perfectly joined together, and both orthotropic and isotropic mediums can be taken into account. It should be recalled that the elastic constants \(\overline{C}^{(k)}_{nm}\) are evaluated in the geometric reference system \(O^{\prime}\alpha_{1}\alpha_{2}\zeta\) through proper transformations that take into account the orientation of the material [46; 99]. If the \(k\)-th layer is orthotropic, the stress-strain relation is given as  
\[ \begin{array}{l}\left[\begin{array}{c}\sigma^{(k)}_{1}\\ \sigma^{(k)}_{2}\\ \tau^{(k)}_{12}\\ \tau^{(k)}_{1n}\\ \tau^{(k)}_{2n}\\ \sigma^{(k)}_{n}\end{array}\right]=\left[\begin{array}{cccccc}\overline{C}^{ (k)}_{11}&\overline{C}^{(k)}_{12}&\overline{C}^{(k)}_{16}&0&0&\overline{C}^{( k)}_{13}\\ \overline{C}^{(k)}_{12}&\overline{C}^{(k)}_{22}&\overline{C}^{(k)}_{26}&0&0& \overline{C}^{(k)}_{23}\\ \overline{C}^{(k)}_{16}&\overline{C}^{(k)}_{26}&\overline{C}^{(k)}_{66}&0&0& \overline{C}^{(k)}_{36}\\ 0&0&0&\overline{C}^{(k)}_{44}&\overline{C}^{(k)}_{45}&0\\ 0&0&0&\overline{C}^{(k)}_{45}&\overline{C}^{(k)}_{55}&0\\ \overline{C}^{(k)}_{13}&\overline{C}^{(k)}_{23}&\overline{C}^{(k)}_{36}&0&0& \overline{C}^{(k)}_{33}\end{array}\right]\left[\begin{array}{c}\epsilon^{(k)} _{1}\\ \epsilon^{(k)}_{2}\\ \gamma^{(k)}_{12}\\ \gamma^{(k)}_{1n}\\ \gamma^{(k)}_{2n}\\ \varepsilon^{(k)}_{n}\end{array}\right]\end{array}  \tag{23}\]  
in which the elastic constants \(\overline{C}^{(k)}_{nm}\) can be related to the nine independent engineering constants of the material--\(E^{(k)}_{1},E^{(k)}_{2},E^{(k)}_{3},G^{(k)}_{12},G^{(k)}_{13},G^{(k)}_{23},\)\(\
u^{(k)}_{12},\
u^{(k)}_{13},\
u^{(k)}_{23}\)--as specified in the book by [PERSON] et al. [99]. It should be recalled that the hypothesis of plane stress is needed in lower-order theories, such as the Reissner-Mindlin theory or First-order Shear Deformation Theory (FSDT). When needed, the reduced elastic coefficients \(\overline{Q}^{(k)}_{nm}\) are used. The meaning of these coefficients is explained in detail in [99]. On the other hand, the elastic foundation, studied in this paper, is approximated by applying uniformly distributed springs at the top and/or bottom surfaces of the shell (Figure 2). The stiffnessesof the springs are denoted by \(k_{1f}^{(\pm)}\), \(k_{2f}^{(\pm)}\), and \(k_{3f}^{(\pm)}\) along the three coordinate directions \(\alpha_{1}\), \(\alpha_{2}\), and \(\zeta\), respectively. Superscripts \((+)\) or \((-)\) denote that these springs are located on the top or the bottom surface of the shell, respectively. The elastic foundation is modeled according to the Winkler-Pasternak elastic foundation theory. In this paper, only the elastic foundations with the uniform thickness \(h_{f}^{(\pm)}\) are considered. Further information about this theory can be found in the papers [126; 127; 128; 135] and in the book [99]. Forces produced by the springs along the three coordinate directions can be written in the form of generalized external load vector of the foundation \(\mathbf{q}_{f}^{(\tau)}\) as  
\[\mathbf{q}_{f}^{(\tau)}=-\sum_{\eta=0}^{N+1}\mathbf{L}_{f}^{(\tau\eta)} \mathbf{u}^{(\eta)}-\sum_{\eta=0}^{N+1}\mathbf{M}_{f}^{(\tau\eta)}\bar{\mathbf{ u}}^{(\eta)} \tag{24}\]  
for \(\tau=0,1,2,\ldots,N,N+1\), where the operators \(\mathbf{L}_{f}^{(\tau\eta)}\) and \(\mathbf{M}_{f}^{(\tau\eta)}\) are the stiffness matrix and inertia matrix, respectively, of the linear elastic foundation.