# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 4 Results
### New Benchmarks
The differences  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\boldsymbol{a}/\boldsymbol{h}\) & \(\boldsymbol{2}\) & \(\boldsymbol{4}\) & \(\boldsymbol{10}\) & \(\boldsymbol{50}\) & \(\boldsymbol{100}\) \\ \hline \multicolumn{6}{c}{\(\mathcal{M}\) at \((\alpha=a/2,\beta=b/2;\hat{z}=4h/5)[-]\)} \\ \hline
3D(\(\mathcal{M}_{a}\)) & 0.8000 & 0.8000 & 0.8000 & 0.8000 & 0.8000 \\
3D(\(\mathcal{M}_{c},1D\)) & 0.5595 & 0.5595 & 0.5595 & 0.5595 & 0.5595 \\
3D(\(\mathcal{M}_{c},3D\)) & 0.4351 & 0.5197 & 0.5525 & 0.5592 & 0.5594 \\ \multicolumn{6}{c}{\(v\) at \((\alpha=a/2,\beta=0;\hat{z}=4h/5)\) [mm]} \\ \hline
3D(\(\mathcal{M}_{a}\)) & \(-2.094\) & \(-2.089\) & \(-2.087\) & \(-2.086\) & \(-2.086\) \\
3D(\(\mathcal{M}_{c},1D\)) & \(-1.454\) & \(-1.461\) & \(-1.462\) & \(-1.462\) & \(-1.462\) \\
3D(\(\mathcal{M}_{c},3D\)) & \(-1.148\) & \(-1.362\) & \(-1.445\) & \(-1.462\) & \(-1.462\) \\  & \(w\) at \((\alpha=a/2,\beta=b/2;\hat{z}=h)\) [mm] & & & \\ \hline
3D(\(\mathcal{M}_{a}\)) & 1.226 & 2.661 & 6.791 & 34.07 & 68.16 \\
3D(\(\mathcal{M}_{c},1D\)) & 1.103 & 2.405 & 6.144 & 30.83 & 61.68 \\
3D(\(\mathcal{M}_{c},3D\)) & 1.008 & 2.336 & 6.113 & 30.83 & 61.67 \\  & \(\sigma_{zz}\) at \((\alpha=a/2,\beta=b/2;\hat{z}=h/5)\) [kPa] & & & \\ \hline
3D(\(\mathcal{M}_{a}\)) & \(-5915\) & \(-982.6\) & \(-127.7\) & \(-4.879\) & \(-1.218\) \\
3D(\(\mathcal{M}_{c},1D\)) & \(-1197\) & 44.77 & 26.65 & 1.217 & 0.3054 \\
3D(\(\mathcal{M}_{c},3D\)) & \(515.5\) & 173.0 & 30.09 & 1.222 & 0.3057 \\  & \(\sigma_{aa}\) at \((\alpha=a/2,\beta=b/2;\hat{z}=h)\) [MPa] & & & \\ \hline
3D(\(\mathcal{M}_{a}\)) & \(-7.939\) & \(-34.86\) & \(-43.94\) & \(-45.68\) & \(-45.73\) \\
3D(\(\mathcal{M}_{c},1D\)) & \(-63.24\) & \(-3.08\) & \(-89.55\) & \(-90.74\) & \(-90.78\) \\
3D(\(\mathcal{M}_{c},3D\)) & \(-90.78\) & \(-90.95\) & \(-90.83\) & \(-90.79\) & \(-90.79\)  
in the three moisture content profiles are shown in Figure 5c,d for two different cylinders: a thick and a thin one. The discrepancies between the calculated and assumed fields are really pronounced; the 1D and 3D computed profiles do not significantly differ, which is even more true as the thickness ratio increases. Figure 6 shows the profiles of the tree displacement components: two stresses and a strain. There is continuity in all the plots: this qualifies the correct introduction of the continuity conditions and elastic/hygrometric properties grading. No external mechanical loads are applied, and this is coherent with the transverse stress values at the bottom and top surfaces, 0. Figure 5: Second benchmark: one-layered FGM (\(p=2\)) cylinder with an imposed moisture content on the top and bottom surfaces. The figures show the volume fraction of the ceramic phase, the diffusion coefficient, and the moisture content profiles for a thick and a thin structure through their thickness. (**a**) Volume fraction of the ceramic phase \(V_{c}\). (**b**) Diffusion coefficient \(\mathcal{D}\). (**c**) Moisture content profile of the \(R_{a}/h=2\) cylinder. (**d**) Moisture content profile of the \(R_{a}/h=100\) cylinder. Figure 6: Second benchmark: one-layered FGM closed cylinder with an imposed moisture content on the top and bottom surfaces. The results are calculated for a moderately thin (\(R_{\alpha}/h=5\)) structure via the 3D(\(\mathcal{M}_{c},3D\)) model. **(a)** Amplitude of \(u\) displacement component. **(b)** Amplitude of \(v\) displacement component. **(c)** Amplitude of \(w\) displacement component. **(d)** Amplitude of \(\sigma_{zz}\) stress component. **(e)** Amplitude of \(\sigma_{\beta\beta}\) stress component. **(f)** Amplitude of \(\gamma_{\alpha\beta}\) strain component. The third benchmark considers a cylindrical sandwich shell panel with an FGM core and simply-supported edges. The top and the bottom skin are in line with the FGM law: the top skin is ceramic as the top surface of the core is; at the same time, the bottom skin is metallic as the bottom surface of the core is.