# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 4 Results
### Assessments
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multirow{2}{*}{\(\mathbf{NL}\rightarrow\)} & \multirow{2}{*}{\(\mathbf{10}\)} & \multicolumn{2}{c}{**Present Solution**} & \multirow{2}{*}{\(\mathbf{200}\)} & \multirow{2}{*}{\(\mathbf{300}\)} & \multirow{2}{*}{\(\mathbf{3D}\) FEM} \\  & & & & & & & \\ \hline \multicolumn{8}{c}{\(a/h=4\)} \\ \hline \(\widehat{w}\) at \(z=h\) & \(3\)D(\(\mathcal{M}_{a}\)) & \(227.5\) & \(225.7\) & \(225.6\) & \(225.6\) & \(225.6\) & \\ \(\widehat{w}\) at \(z=h\) & \(3\)D(\(\mathcal{M}_{c},1D\)) & \(177.9\) & \(171.1\) & \(170.8\) & \(170.7\) & \(170.7\) & \(143.3\) \\  & \(3\)D(\(\mathcal{M}_{c},3D\)) & \(151.0\) & \(141.9\) & \(141.6\) & \(141.5\) & \(141.5\) & \\ \hline \(\partial_{xa}\) at \(z=0\) & \(3\)D(\(\mathcal{M}_{a}\)) & \(-32\),\(309\) & \(-32\),\(391\) & \(-32\),\(383\) & \(-32\),\(380\) & \(-32\),\(380\) & \\ \(3\)D(\(\mathcal{M}_{c},1D\)) & \(-53\),\(262\) & \(-53\),\(245\) & \(-53\),\(234\) & \(-53\),\(230\) & \(-53\),\(230\) & \(-53\),\(230\) & \(-751\),\(662\) \\  & \(3\)D(\(\mathcal{M}_{c},3D\)) & \(-75\),\(532\) & \(-75\),\(561\) & \(-75\),\(551\) & \(-75\),\(548\) & \(-75\),\(548\) & \\ \hline \multicolumn{8}{c}{\(a/h=10\)} \\ \hline \(\widehat{u}\) at \(z=h/2\) & \(3\)D(\(\mathcal{M}_{a}\)) & \(-262.4\) & \(-262.1\) & \(-262.1\) & \(-262.1\) & \(-262.1\) & \\ \(\widehat{u}\) at \(z=h/2\) & \(3\)D(\(\mathcal{M}_{c},1D\)) & \(-231.9\) & \(-230.5\) & \(-230.4\) & \(-230.4\) & \(-230.4\) & \(-226.0\) \\  & \(3\)D(\(\mathcal{M}_{c},3D\)) & \(-227.3\) & \(-225.8\) & \(-225.7\) & \(-225.7\) & \(-225.7\) & \\ \hline \(\partial_{xa}\) at \(z=h\) & \(3\)D(\(\mathcal{M}_{a}\)) & \(-55\),\(689\) & \(-35\),\(256\) & \(-32\),\(575\) & \(-31\),\(224\) & \(-30\),\(772\) & \\ \(3\)D(\(\mathcal{M}_{c},1D\)) & \(-83\),\(669\) & \(-64\),\(410\) & \(-61\),\(712\) & \(-60\),\(336\) & \(-59\),\(873\) & \(-63\),\(825\) \\  & \(3\)D(\(\mathcal{M}_{c},3D\)) & \(-87\),\(250\) & \(-68\),\(062\) & \(-65\),\(359\) & \(-63\),\(979\) & \(-63\),\(515\) & \\ \hline \multicolumn{8}{c}{\(a/h=50\)} \\ \hline \(\widehat{u}\) at \(z=0\) & \(3\)D(\(\mathcal{M}_{a}\)) & \(-257.3\) & \(-258.1\) & \(-258.1\) & \(-258.1\) & \(-258.1\) & \\ \(\widehat{u}\) at \(z=0\) & \(3\)D(\(\mathcal{M}_{c},1D\)) & \(-236.0\) & \(-237.0\) & \(-237.0\) & \(-237.0\) & \(-237.0\) & \(-236.8\) \\  & \(3\)D(\(\mathcal{M}_{c},3D\)) & \(-235.9\) & \(-236.8\) & \(-236.8\) & \(-236.9\) & \(-236.9\) & \\ \hline \(\partial_{xa}\) at \(z=h/2\) & \(3\)D(\(\mathcal{M}_{a}\)) & \(-145\),\(561\) & \(-137\),\(645\) & \(-136\),\(589\) & \(-136\),\(056\) & \(-135\),\(878\) & \\ \(\widehat{v}_{xa}\) at \(z=h/2\) & \(3\)D(\(\mathcal{M}_{c},1D\)) & \(-114\),\(235\) & \(-107\),\(530\) & \(-106\),\(648\) & \(-106\),\(203\) & \(-106\),\(054\) & \(-105\),\(149\) \\  & \(3\)D(\(\mathcal{M}_{c},3D\)) & \(-114\),\(035\) & \(-107\),\(338\) & \(-106\),\(457\) & \(-106\),\(013\) & \(-105\),\(865\) & \\ \hline \end{tabular}
\end{table} 
Table 2: Second assessment. One-layered FGM square plate (\(a/b=1\)), featuring different thickness ratios. The geometry, materials, and FGM power law are the same as the first assessment. The thermal load is substituted by an external moisture content acting on the top (\(M_{t}=1.0\%\)) and bottom (\(M_{b}=0.5\%\)) surfaces; \(m=n=1\). The reference solution is the 3D FE model, already validated in the previous assessment. The results of the present solution are obtained with \(N=3\) and for a varying number of mathematical layers NL. confirmed the capabilities of the 3D shell model in handling the hygromechanical analysis of FGM shells. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & & \multicolumn{6}{c}{**Present Solution**} & \multicolumn{1}{c}{**3D FEM**} & \multicolumn{1}{c}{**Ref. [74]**} \\  & **NL \(\rightarrow\)** & **10** & **50** & **100** & **200** & **300** & **300** & **Ref.