# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 3 3D Exact Shell Model for Hygro-Elastic Stress Analysis
In each \(j\) mathematical layer, the parametric coefficients \(H_{\alpha}\) and \(H_{\beta}\) and the variable elastic and hygrometric coefficients for FGM layers can be exactly defined by using the \(z\) coordinate in the middle of each \(j\) layer. Therefore, coefficients \(A_{s}^{j}\) (s from 1 to 19) and \(L_{r}^{j}\) (\(r\) from 1 to 4) become constant terms in the compact form of the system of differential equations in \(z\) defined in Equations (48)-(50). Equations (48)-(50) indicate a system of three differential equations of second order in \(z\). The unknowns are the displacement and moisture content amplitudes and the associated derivatives calculated to \(z\). The derivatives in \(\alpha\) and \(\beta\) have already been calculated via the harmonic forms for displacements and moisture content previously introduced. In this system, decoupling the variables is possible: this means a separate quantification of the moisture content profile through the thickness direction \(z\), addressed in an appropriate section. Therefore, it becomes a known term. The consequence of these choices is that the system contains second-order differential equations only, in the displacement amplitudes \(U^{j}\), \(V^{j}\), \(W^{j}\) and their derivatives in \(z\). Redoubling the variables as proposed in [71,72] allows reducing the system into a first-order one in \(z\). In each \(j\) layer, the \(3\times 1\) vector of unknowns (\(U^{j}\), \(V^{j}\), \(W^{j}\)) becomes a \(6\times 1\) unknown vector (\(U^{j}\), \(V^{j}\), \(W^{j}\), \(U^{j}\), \(V^{j}\), \(W^{j}\)); superscript \({}^{\prime}\) indicates derivatives performed to \(z\) (also written as \(\frac{\partial}{\partial z}\)). Terms \(M^{j}\) and \(M^{j^{\prime}}\) can be considered as known terms because they can be opportunely calculated, as will be demonstrated in the next section:  
\[ \begin{bmatrix}A_{6}^{j}&0&0&0&0&0\\ 0&A_{12}^{j}&0&0&0&0\\ 0&0&A_{19}^{j}&0&0&0\\ 0&0&0&A_{6}^{j}&0&0\\ 0&0&0&A_{12}^{j}&0\\ 0&0&0&0&A_{19}^{j}\\ 0&0&0&0&A_{19}^{j}\\ \end{bmatrix}\begin{bmatrix}U^{j}\\ V^{j}\\ W^{j}\\ U^{j^{\prime}}\\ W^{j^{\prime}}\\ W^{j^{\prime}}\\ \end{bmatrix} ^{\prime}=\]  
\[ \begin{bmatrix}0&0&0&A_{6}^{j}&0&0\\ 0&0&0&0&A_{12}^{j}&0\\ 0&0&0&0&0&A_{19}^{j}\\ -A_{1}^{j}&-A_{2}^{j}&-A_{3}^{j}&-A_{4}^{j}&0&-A_{5}^{j}\\ -A_{7}^{j}&-A_{8}^{j}&-A_{9}^{j}&0&-A_{10}^{j}\\ -A_{13}^{j}&-A_{14}^{j}&-A_{15}^{j}&-A_{16}^{j}&-A_{17}^{j}&-A_{18}^{j}\\ \end{bmatrix}\begin{bmatrix}U^{j}\\ V^{j}\\ W^{j}\\ U^{j^{\prime}}\\ U^{j^{\prime}}\\ W^{j^{\prime}}\\ \end{bmatrix} + \begin{bmatrix}0&0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ -L_{1}^{j}&0&0&0&0\\ -L_{2}^{j}&0&0&0&0\\ -L_{4}^{j}&-L_{3}^{j}&0&0&0\\ \end{bmatrix}\begin{bmatrix}M^{j}\\ M^{j}\\ 0\\ 0\\ 0\\ 0\\ 0\\ \end{bmatrix} , \tag{51}\]  
By defining vectors \(\mathbf{U}^{j}=[U^{j}\ V^{j}\ W^{j}\ U^{j^{\prime}}\ V^{j^{\prime}}\ W^{j}]^{T}\), \(\mathbf{U}^{i^{\prime}}=\frac{\partial\mathbf{U}^{j}}{\partial z}\) and \(\mathbf{M}^{j^{\prime}}=[M^{j}\ M^{j^{\prime}}\ 0\ 0\ 0\ 0]^{T}\) (where \(T\) means the transpose of a vector), the following compact form is possible:  
\[\mathbf{D}^{j}\mathbf{U}^{j^{\prime}}=\mathbf{A}^{j}\mathbf{U}^{j}+\mathbf{L}^{j}\mathbf{M}^{j}\,, \tag{52}\]the above equation, thanks to the definitions \(\mathbf{A}^{*j}=\mathbf{D}^{j^{-1}}\mathbf{A}^{j}\) and \(\mathbf{L}^{*j}=\mathbf{D}^{j^{-1}}\mathbf{L}^{j}\), can be rewritten as  
\[\mathbf{U}^{j^{\prime}}=\mathbf{D}^{j^{-1}}\mathbf{A}^{j}\mathbf{U}^{j}+\mathbf{D}^{j^{-1}}\mathbf{L}^{j }\mathbf{M}^{j}\,, \tag{53}\]  
\[\mathbf{U}^{j^{\prime}}=\mathbf{A}^{*j}\mathbf{U}^{j}+\mathbf{L}^{*j}\mathbf{M}^{j}\,. \tag{54}\]  
The moisture content profile through the thickness direction \(z\) can be calculated using one of the three methods proposed in the previous section. This profile can be reconstructed using a linear approximation of the moisture content in each \(j\) mathematical layer.