# 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 c} \hline \hline  & \multicolumn{6}{c}{**Present Solution**} & \multicolumn{1}{c}{**3D FEM**} & **Ref. 3D [73]** \\  & \(\mathbf{NL\to}\) & \(\mathbf{10}\) & \(\mathbf{50}\) & \(\mathbf{100}\) & \(\mathbf{200}\) & \(\mathbf{300}\) & \(\mathbf{300}\) & \(\mathbf{300}\) \\ \hline  & \multicolumn{6}{c}{\(a/h=4\)} & & & & \\ \hline \(\hat{w}\) at \(z=h\) & \(3\)D(\(\theta_{a}\)) & \(4.2943\) & \(4.2842\) & \(4.2838\) & \(4.2837\) & \(4.2836\) & & \\ \(\hat{w}\) at \(z=h\) & \(3\)D(\(\theta_{c},1D\)) & \(3.3331\) & \(3.2130\) & \(3.2083\) & \(3.2071\) & \(3.2068\) & \(3.067\) & \(3.043\) \\  & \(3\)D(\(\theta_{c},3D\)) & \(3.1791\) & \(3.0482\) & \(3.0431\) & \(3.0418\) & \(3.0415\) & & \\ \hline \(\hat{\sigma}_{\text{ax}}\) at \(z=0\) & \(3\)D(\(\theta_{a}\)) & \(288.99\) & \(290.12\) & \(290.21\) & \(290.23\) & \(290.24\) & & \\ \(\hat{\sigma}_{\text{ax}}\) at \(z=0\) & \(3\)D(\(\theta_{c},1D\)) & \(-27.47\) & \(-22.16\) & \(-21.92\) & \(-21.85\) & \(-21.84\) & \(-75.57\) & \(-73.53\) \\  & \(3\)D(\(\theta_{c},3D\)) & \(-81.37\) & \(-73.99\) & \(-73.67\) & \(-73.58\) & \(-73.57\) & & \\ \hline  & \multicolumn{6}{c}{\(a/h=10\)} & & & & \\ \hline \(\hat{u}\) at \(z=h/2\) & \(3\)D(\(\theta_{a}\)) & \(-1.3433\) & \(-1.3417\) & \(-1.3417\) & \(-1.3417\) & & \\ \(\hat{u}\) at \(z=h/2\) & \(3\)D(\(\theta_{c},1D\)) & \(-0.8287\) & \(-0.8026\) & \(-0.8016\) & \(-0.8014\) & \(-0.8014\) & \(-0.7910\) & \(-0.7862\) \\  & \(3\)D(\(\theta_{c},3D\)) & \(-0.8134\) & \(-0.7871\) & \(-0.7861\) & \(-0.7859\) & \(-0.7858\) & & \\ \hline \(\hat{\sigma}_{\text{ax}}\) at \(z=h\) & \(3\)D(\(\theta_{a}\)) & \(-599.6\) & \(-490.7\) & \(-476.4\) & \(-469.2\) & \(-466.8\) & & \\ \(\hat{\sigma}_{\text{ax}}\) at \(z=h\) & \(3\)D(\(\theta_{c},1D\)) & \(-1102\) & \(-1021\) & \(-1006\) & \(-999.3\) & \(-996.7\) & \(-1058\) & \(-1006\) \\  & \(3\)D(\(\theta_{c},3D\)) & \(-1117\) & \(-1036\) & \(-1021\) & \(-1014\) & \(-1011\) & & \\ \hline  & \multicolumn{6}{c}{\(a/h=50\)} & & & \\ \hline \(\hat{w}\) at \(z=h/2\) & \(3\)D(\(\theta_{a}\)) & \(35.97\) & \(35.81\) & \(35.80\) & \(35.80\) & \(35.80\) & & \\ \(\hat{w}\) at \(z=h/2\) & \(3\)D(\(\theta_{c},1D\)) & \(29.64\) & \(28.50\) & \(28.46\) & \(28.45\) & \(28.45\) & \(28.57\) & \(28.45\) \\  & \(3\)D(\(\theta_{c},3D\)) & \(29.64\) & \(28.50\) & \(28.45\) & \(28.44\) & \(28.44\) & & \\ \hline  & \(3\)D(\(\theta_{a}\)) & \(-759.3\) & \(-730.7\) & \(-726.8\) & \(-724.9\) & \(-724.2\) & & \\ \(\hat{\sigma}_{\text{ax}}\) at \(z=h/2\) & \(3\)D(\(\theta_{c},1D\)) & \(-269.9\) & \(-255.3\) & \(-253.5\) & \(-252.5\) & \(-252.2\) & \(-250.4\) & \(-251.2\) \\  & \(3\)D(\(\theta_{c},3D\)) & \(-269.3\) & \(-254.7\) & \(-252.9\) & \(-252.0\) & \(-251.7\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: First assessment. One-layered FGM square plate (\(a/b=1\)), featuring different thickness ratios. The volume fraction power law considers \(p=2\) as the exponent. An external sovra-temperature field acts on the top (\(\theta_{t}=+1\) K) and bottom (\(\theta_{b}=0\) K) surfaces; \(m=n=1\). The reference solution is the asymptotic method of [PERSON] and [PERSON] [73], considering a 3D temperature profile along with thickness direction. A 3D FE model is also assessed. The results of the present solution are obtained with \(N=3\) and for a varying number of mathematical layers NL. shell model correctly handles the thermomechanical analysis of FGM shells. Furthermore, it simultaneously assessed the 3D FE model, which will be helpful in the following assessment to validate the hygromechanical analysis. The fourth assessment is meant to validate the hygro-elastic part of the 3D solution for shells embedding an FGM layer. To this end, it considers the previous test case as a reference and removes the thermal field in favor of a hygrometric one. Consequently, it focuses on a simply-supported one-layered FGM square shell.