# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 4 Results
### New Benchmarks
The assessments of the previous subsection validated the results of this new  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Present Solution**} \\ \multicolumn{1}{c}{} & \multicolumn{1}{c}{**NL \(\rightarrow\)**} & \multicolumn{1}{c}{**10**} & \multicolumn{1}{c}{**50**} & \multicolumn{1}{c}{**100**} & \multicolumn{1}{c}{**200**} & \multicolumn{1}{c}{**300**} & \multicolumn{1}{c}{**3D FEM**} \\ \hline \multicolumn{1}{c}{} & \multicolumn{4}{c}{\(R_{\beta}/h=50\)} \\ \hline \multirow{3}{*}{\(\widehat{w}\) at \(z=h\)} & 3D(\(\mathcal{M}_{a}\)) & 557.35 & 551.43 & 551.22 & 551.17 & 551.16 & \multirow{3}{*}{411.97} \\  & 3D(\(\mathcal{M}_{c},1D\)) & 440.34 & 423.22 & 422.60 & 422.44 & 422.41 & 411.97 \\  & 3D(\(\mathcal{M}_{c},3D\)) & 417.06 & 398.20 & 397.51 & 397.34 & 397.31 & \multirow{3}{*}{279.92} \\ \hline \multirow{3}{*}{\(\widehat{w}\) at \(z=h/2\)} & 3D(\(\mathcal{M}_{a}\)) & 399.25 & 393.43 & 393.22 & 393.16 & 393.15 & \multirow{3}{*}{279.52} \\  & 3D(\(\mathcal{M}_{c},1D\)) & 303.27 & 287.38 & 286.80 & 286.66 & 286.63 & 279.92 \\  & 3D(\(\mathcal{M}_{c},3D\)) & 285.49 & 268.05 & 267.42 & 267.26 & 267.23 & \multirow{3}{*}{131.17} \\ \hline \multirow{3}{*}{\(\widehat{w}\) at \(z=0\)} & 3D(\(\mathcal{M}_{a}\)) & 228.40 & 222.78 & 222.58 & 222.53 & 222.52 & \multirow{3}{*}{131.17} \\  & 3D(\(\mathcal{M}_{c},1D\)) & 148.66 & 133.28 & 132.72 & 132.58 & 132.55 & 131.17 \\  & 3D(\(\mathcal{M}_{c},3D\)) & 136.53 & 119.68 & 119.07 & 118.91 & 118.88 & \multirow{3}{*}{118.88} \\ \hline \multirow{3}{*}{\(\widehat{u}\) at \(z=h\)} & 3D(\(\mathcal{M}_{a}\)) & \(-654.1\) & \(-651.9\) & \(-651.8\) & \(-651.8\) & \(-651.8\) & \multirow{3}{*}{\(-651.8\)} \\  & 3D(\(\mathcal{M}_{c},1D\)) & \(-563.3\) & \(-555.6\) & \(-555.3\) & \(-555.2\) & \(-555.2\) & \(-558.01\) \\  & 3D(\(\mathcal{M}_{c},3D\)) & \(-539.7\) & \(-531.1\) & \(-530.8\) & \(-530.7\) & \(-530.7\) & \\ \hline \multirow{3}{*}{\(\widehat{u}\) at \(z=h/2\)} & 3D(\(\mathcal{M}_{a}\)) & \(-515.9\) & \(-515.5\) & \(-515.5\) & \(-515.5\) & \(-515.5\) & \multirow{3}{*}{\(-515.5\)} \\  & 3D(\(\mathcal{M}_{c},1D\)) & \(-456.3\) & \(-453.7\) & \(-453.6\) & \(-453.6\) & \(-453.6\) & \(-453.6\) & \(-435.01\) \\  & 3D(\(\mathcal{M}_{c},3D\)) & \(-438.5\) & \(-435.4\) & \(-435.3\) & \(-435.3\) & \(-435.3\) & \(-435.3\) & \\ \hline \multirow{3}{*}{\(\widehat{u}\) at \(z=0\)} & 3D(\(\mathcal{M}_{a}\)) & \(-424.7\) & \(-426.0\) & \(-426.0\) & \(-426.1\) & \(-426.1\) & \multirow{3}{*}{\(-426.1\)} \\  & 3D(\(\mathcal{M}_{c},1D\)) & \(-392.8\) & \(-394.8\) & \(-394.9\) & \(-394.9\) & \(-394.9\) & \(-380.50\) \\  & 3D(\(\mathcal{M}_{c},3D\)) & \(-379.6\) & \(-381.7\) & \(-381.8\) & \(-381.8\) & \(-381.8\) & \\ \hline \multicolumn{1}{c}{} & \multicolumn{4}{c}{\(R_{\beta}/h=1000\)} \\ \hline \multirow{3}{*}{\(\widehat{w}\) at \(z=h\)} & 3D(\(\mathcal{M}_{a}\)) & 11,564 & 11,545 & 11,544 & 11,544 & 11,544 & \multirow{3}{*}{10,128} \\  & 3D(\(\mathcal{M}_{c},1D\)) & 10,150 & 10,065 & 10,062 & 10,061 & 10,061 & 10,128 \\  & 3D(\(\mathcal{M}_{c},3D\)) & 10,149 & 10,064 & 10,061 & 10,060 & 10,060 & \\ \hline \multirow{3}{*}{\(\widehat{w}\) at \(z=h/2\)} & 3D(\(\mathcal{M}_{a}\)) & 11,558 & 11,539 & 11,538 & 11,538 & \multirow{3}{*}{10,123} \\  & 3D(\(\mathcal{M}_{c},1D\)) & 10,145 & 10,060 & 10,057 & 10,056 & 10,056 & 10,056 \\  & 3D(\(\mathcal{M}_{c},3D\)) & 10,144 & 10,059 & 10,056 & 10,055 & 10,055 & \\ \hline \multirow{3}{*}{\(\widehat{w}\) at \(z=0\)} & 3D(\(\mathcal{M}_{a}\)) & 11,551 & 11,532 & 11,531 & 11,531 & 11,531 & \multirow{3}{*}{10,123} \\  & 3D(\(\mathcal{M}_{c},1D\)) & 10,139 & 10,054 & 10,051 & 10,050 & 10,050 & 10,050 \\  & 3D(\(\mathcal{M}_{c},3D\)) & 10,138 & 10,053 & 10,050 & 10,049 & 10,049 & \\ \hline \multirow{3}{*}{\(\widehat{u}\) at \(z=h\)} & 3D(\(\mathcal{M}_{a}\)) & \(-589.0\) & \(-588.5\) & \(-588.5\) & \(-588.5\) & \(-588.5\) & \multirow{3}{*}{\(-520.37\)} \\  & 3D(\(\mathcal{M}_{c},1D\)) & \(-521.1\) & \(-518.0\) & \(-517.9\) & \(-517.8\) & \(-517.8\) & \(-517.8\) & \\  & 3D(\(\mathcal{M}_{c},3D\)) & \(-521.1\) & \(-517.9\) & \(-517.8\) & \(-517.8\) & \(-517.8\) & \\ \hline \multirow{3}{*}{\(\widehat{u}\) at \(z=h/2\)} & 3D(\(\mathcal{M}_{a}\)) & \(-407.4\) & \(-407.2\) & \(-407.2\) & \(-407.2\) & \(-407.2\) & \multirow{3}{*}{3D shell model when applied to FGM layers: the results converge and are exact when the order of expansion \(N=3\) for the exponential matrix is coupled with a minimum number of \(M=300\) mathematical layer for the through the thickness mechanical properties and curvature approximation.