# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 3 3D Exact Shell Model for Hygro-Elastic Stress Analysis
By using Equations (61) and (62), Equation (60) is modified as  
\[\mathbf{U}^{j}(h^{j})=\mathbf{A}^{**j}\mathbf{U}^{j}(0)+\mathbf{L}^{*sj}\,, \tag{63}\]  
where \(\mathbf{U}^{j}_{t}\) indicates \(\mathbf{U}^{j}(h^{j})\) and it is defined at the top \(t\) of each \(j\) layer, and \(\mathbf{U}^{j}_{b}\) indicates \(\mathbf{U}^{j}(0)\) and it is defined at the bottom of each \(j\) layer. In this way, Equation (63) is defined as  
\[\mathbf{U}^{j}_{t}=\mathbf{A}^{**j}\mathbf{U}^{j}_{b}+\mathbf{L}^{**j}\,. \tag{64}\]  
Using Equation (64) allows to connect displacements and their derivatives in \(z\) defined at the top of the \(j\) mathematical layer with the same variables defined at the bottom of the same \(j\) layer. The general three-dimensional shell model is developed using a layer-wise approach. Inter-laminar continuity conditions in displacements and transverse stresses must be defined at each interface between the two adjacent mathematical layers. The inter-laminar continuity conditions for displacements come through congruence hypotheses:  
\[u^{j}_{b}=u^{j-1}_{t}\,,\,\,\,v^{j}_{b}=v^{j-1}_{t}\,,\,\,\,w^{j}_{b}=w^{j-1}_ {t}\,. \tag{65}\]  
The inter-laminar continuity conditions for transverse shear and normal transverse stresses are defined employing equilibrium hypotheses:  
\[\sigma^{j}_{zz_{b}}=\sigma^{j-1}_{zz_{t}}\,,\,\,\,\sigma^{j}_{az_{b}}=\sigma^{ j-1}_{az_{t}}\,,\,\,\,\sigma^{j}_{\beta z_{b}}=\sigma^{j-1}_{\beta z_{t}}\,. \tag{66}\]  
The formulation of this solution requires rewriting the inter-laminar continuity conditions, Equations (65) and (66), in a displacement form. Achieving this task is possible by recalling the constitutive Equations (39)-(44) and the harmonic form of the variables of the problem, Equations (1) and (31)-(33). This procedure allows writing the inter-laminar continuity condition in the amplitude displacements and their derivatives to the thickness coordinate. Then, compacting the notation is possible, recalling the vectors \(\mathbf{U}^{j}\) and \(\mathbf{M}^{j}\), and introducing a pair of transfer matrices. The procedure is similar to that employed in [25]; exception made for an additional coefficient multiplying the moisture content amplitude:  
\[ \begin{bmatrix}U\\ V\\ W\\ U^{\prime}\\ V^{\prime}\\ W^{\prime}\end{bmatrix} _{b}^{j}= \begin{bmatrix}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ T_{1}&0&T_{2}&T_{3}&0&0\\ 0&T_{4}&T_{5}&0&T_{6}&0\\ T_{7}&T_{8}&T_{9}&0&0&T_{10}\end{bmatrix} _{t}^{j-1,j} \begin{bmatrix}U\\ V\\ W\\ U^{\prime}\\ V^{\prime}\\ W^{\prime}\end{bmatrix} _{t}^{j-1}+ \begin{bmatrix}0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ T_{11}&0&0&0&0&0\end{bmatrix} _{t}^{j-1,j}\begin{bmatrix}M\\ M\\ 0\\The first three rows of Equation (67) denote the displacement continuity equations; the last three are the continuity conditions for stresses. The compact form of Equation (67) follows:  
\[\mathbf{U}_{b}^{j}=\mathbf{T}_{U}^{j-1,j}\mathbf{U}_{t}^{j-1}+\mathbf{T}_{M}^{j-1,j}\mathbf{M}_{t}^{ j-1}\,. \tag{68}\]  
Equation (68) expresses the link between the displacements and their \(z\) derivatives, calculated at the bottom of the \(j\)th layer, with their corresponding values plus the moisture content (and its \(z\) derivative), at the top of the \((j-1)\)th layer. The harmonic form implemented in all the variables of the problem automatically satisfies the simply supported boundary conditions, here reported for exhaustiveness:  
\[w=v=0,\ \sigma_{\alpha\alpha}=0\quad\text{ for }\quad\alpha=0,a\,, \tag{69}\] \[w=u=0,\ \sigma_{\beta\beta}=0\quad\text{ for }\quad\beta=0,b \tag{70}\]  
This solution extends the model already seen in [25] for the static analysis of shells subjected to static loads. In that context, mechanical loads can act on the external top and bottom surfaces, with components defined in the three directions \(\alpha\), \(\beta\), and \(z\).