## 3 Higher-Order Equivalent Layer-Wise Approach
\right] \tag{34}\]  
in which \(A_{1},A_{2}\) represent the Lame parameters defined in (6), and \(R_{1},R_{2}\) are the principal radii of curvature of the shell middle surface shown in (7). On the contrary, the matrix \(\mathbf{M}^{(\tau\eta)}\) collects the inertia terms \(I^{(\tau\eta)}\) for every order of kinematic expansion \(\tau,\eta=0,1,2,\ldots,N,N+1\) as follows  
\[\mathbf{M}^{(\tau\eta)}=\left[ \begin{array}{ccc}I^{(\tau\eta)}&0&0\\ 0&I^{(\tau\eta)}&0\\ 0&0&I^{(\tau\eta)}\end{array} \right] \tag{35}\]  
If \(\rho^{(k)}\) denotes the mass density of the \(k\)-th ply, the definition shown below is used to evaluate the inertia masses \(I^{(\tau\eta)}\) of the laminated composite in hand  
\[I^{(\tau\eta)}=\sum_{k=1}^{l}\int\limits_{\xi_{k}}^{\xi_{k+1}}\rho^{(k)}F_{\tau }F_{\eta}H_{1}H_{2}d\zeta \tag{36}\]  
It is clear that \(\ddot{\mathbf{u}}^{(\eta)}\) represents the generalized acceleration component vector defined as follows  
\[\ddot{\mathbf{u}}^{(\eta)}=\left[ \begin{array}{cc}\ddot{u}_{1}^{(\eta)}& \ddot{u}_{2}^{(\eta)}&\ddot{u}_{3}^{(\eta)}\end{array} \right]^{T} \tag{37}\]  
for \(\eta=0,1,2,\ldots,N,N+1\). By inserting definition (28) into the motion equation shown in (33), the governing system can be conveniently written as a function of the degrees of freedom of the problem. As a consequence, one gets  
\[\sum_{\eta=0}^{N+1}\mathbf{L}^{(\tau\eta)}\mathbf{u}^{(\eta)}=\sum_{\eta=0}^{ N+1}\mathbf{M}^{(\tau\eta)}\ddot{\mathbf{u}}^{(\eta)} \tag{38}\]  
where the so-called fundamental operator \(\mathbf{L}^{(\tau\eta)}=\mathbf{D}_{\Omega}^{*}\mathbf{A}^{(\tau\eta)} \mathbf{D}_{\Omega}\) is introduced. In matrix form, it assumes the following aspect  
\[\mathbf{L}^{(\tau\eta)}=\left[ \begin{array}{ccc}L_{11}^{(\tau\eta)}&L_{12}^{( \tau\eta)}&L_{13}^{(\tau\eta)}\\ L_{21}^{(\tau\eta)}&L_{22}^{(\tau\eta)}&L_{23}^{(\tau\eta)}\\ L_{31}^{(\tau\eta)}&L_{32}^{(\tau\eta)}&L_{33}^{(\tau\eta)}\end{array} \right] \tag{39}\]For conciseness purposes, the definition of each term \(L_{fg}^{(\tau\eta)}\), for \(f,g=1,2,3\), is omitted in the present paper. Nevertheless, the reader can find their complete meaning in the work by [PERSON] et al. [61]. It should be noticed that they are formally equal to the ones employed in the ESL approach. Relation (38) represents the fundamental nucleus of the ELW approach and it identify a set of three differential equations for each order of kinematic expansions. Thus, \(3(N+2)\) equations must be solved to obtain the solution of this structural problem. In order to fully characterize the problem at issue, the proper set of boundary conditions must be enforced along each external edge. For a clamped edge (denoted by letter \"C\"), these conditions are written in terms of displacements. On the contrary, specific prescription are given to the generalized stress resultants for a free edge (indicated by letter \"F\"). Let us consider Figure 1 to identify each external edge. In particular, if an edge is defined by \(\alpha_{1}=\alpha_{1}^{0}\) or \(\alpha_{1}=\alpha_{1}^{1}\), for \(\alpha_{2}^{0}\leq\alpha_{2}\leq\alpha_{2}^{1}\), the boundary conditions become  
\[ \begin{array}{cccc}\text{C}&\rightarrow&u_{1}^{(\tau)}=u_{2}^{(\tau)}=u_{3}^{ (\tau)}=0\\ \text{F}&\rightarrow&N_{1}^{(\tau)}=N_{12}^{(\tau)}=T_{1}^{(\tau)}=0\end{array}  \tag{40}\]  
These edges are the Southern (S) and the Northern (N) ones in Figure 1. On the other hand, if the coordinates of an edge are \(\alpha_{2}=\alpha_{2}^{0}\) or \(\alpha_{2}=\alpha_{2}^{1}\), for \(\alpha_{1}^{0}\leq\alpha_{1}\leq\alpha_{1}^{1}\), one gets  
\[ \begin{array}{cccc}\text{C}&\rightarrow&u_{1}^{(\tau)}=u_{2}^{(\tau)}=u_{3} ^{(\tau)}=0\\ \text{F}&\rightarrow&N_{21}^{(\tau)}=N_{2}^{(\tau)}=T_{2}^{(\tau)}=0\end{array}  \tag{41}\]  
This is clearly the case of the Western (W) and Eastern (E) edges. In the present paper, the boundary conditions are specified for the panel under consideration following the sequence WSEN.