# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 4 Results
### New Benchmarks
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{R_{\alpha}/h}\) & \(\mathbf{2}\) & \(\mathbf{4}\) & \(\mathbf{10}\) & \(\mathbf{50}\) & \(\mathbf{100}\) \\ \hline \multicolumn{6}{c}{\(\mathcal{M}\) at \((\alpha=a/2,\beta=b/2;z=h/2)[-]\)} \\ \hline
3D(\(\mathcal{M}_{a}\)) & 0.5000 & 0.5000 & 0.5000 & 0.5000 & 0.5000 \\
3D(\(\mathcal{M}_{c},1D\)) & 0.2087 & 0.2087 & 0.2087 & 0.2087 & 0.2087 \\
3D(\(\mathcal{M}_{c},3D\)) & 0.1014 & 0.1672 & 0.2008 & 0.2083 & 0.2086 \\  & \(u\) at \((\alpha=a/2,\beta=0;z=3h/4)\) [\(10^{-3}\) mm] & & & \\ \hline
3D(\(\mathcal{M}_{a}\)) & \(-16.26\) & \(-15.23\) & \(-11.79\) & \(11.86\) & \(41.40\) \\
3D(\(\mathcal{M}_{c},1D\)) & \(-10.53\) & \(-9.736\) & \(-6.559\) & \(15.20\) & \(42.34\) \\
3D(\(\mathcal{M}_{c},3D\)) & \(-7.942\) & \(-8.844\) & \(-6.416\) & \(15.20\) & \(42.34\) \\  & \(w\) at \((\alpha=a/2,\beta=b/2;z=h/2)\) [\(10^{-5}\) mm] & & & \\ \hline
3D(\(\mathcal{M}_{a}\)) & 0.5350 & 1.184 & 3.310 & 17.49 & 35.22 \\
3D(\(\mathcal{M}_{c},1D\)) & 0.5000 & 1.152 & 3.125 & 16.16 & 32.44 \\
3D(\(\mathcal{M}_{c},3D\)) & 0.4660 & 1.116 & 3.102 & 16.16 & 32.44 \\  & \(\sigma_{zz}\) at \((\alpha=a/2,\beta=b/2;z=h/4)\) [kPa] & & & \\ \hline
3D(\(\mathcal{M}_{a}\)) & \(-108.2\) & \(-7.415\) & \(0.3735\) & \(0.2473\) & \(0.1354\) \\
3D(\(\mathcal{M}_{c},1D\)) & 0.6797 & 5.001 & \(-1.110\) & \(-0.5867\) & \(-0.3135\) \\
3D(\(\mathcal{M}_{c},3D\)) & 30.91 & 6.099 & \(-1.162\) & \(-0.5875\) & \(-0.3136\) \\  & \(\sigma_{\beta\beta}\) at \((\alpha=0,\beta=0;\tilde{z}=0)\) [kPa] & & & \\ \hline
3D(\(\mathcal{M}_{a}\)) & 298.9 & 142.1 & 87.89 & 81.26 & 81.57 \\
3D(\(\mathcal{M}_{c},1D\)) & \(-3.571\) & \(-92.06\) & \(-116.7\) & \(-116.9\) & \(-116.3\) \\
3D(\(\mathcal{M}_{c},3D\)) & \(-77.09\) & \(-112.7\) & \(-120.0\) & \(-117.0\) & \(-116.3\) \\  & \(\gamma_{az}\) at \((\alpha=a/2,\beta=0;\tilde{z}=h/3)\) [\(10^{-8}\)] & & & \\ \hline
3D(\(\mathcal{M}_{a}\)) & \(-191.6\) & \(-51.36\) & \(-11.08\) & \(-1.956\) & \(-0.9857\) \\
3D(\(\mathcal{M}_{c},1D\)) & 0.9706 & 46.64 & 26.65 & 5.584 & 2.790 \\
3D(\(\mathcal{M}_{c},3D\)) & 81.83 & 61.98 & 27.77 & 5.593 & 2.791 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Third benchmark: cylindrical sandwich shell panel with an FGM core subject to an external moisture content applied to the top and bottom surfaces. The results of all the 3D models consider \(N=3\) and \(NL=300\). the problem as the moisture content amplitude derives from [PERSON]'s law of diffusion. The results coming from 3D(\(\mathcal{M}_{a}\)) are always wrong because the moisture content evaluation is inaccurate. Considerable differences are present between the calculated and assumed fields. The moisture content profiles of Figure 9c,d once again demonstrate that the 1D and 3D moisture fields get closer in thin structures; despite the thickness, they always differ from the assumed profile, which completely disregards the physics of the problem. This reflects on the results in terms of displacements, strains, and stresses: the differences are high, and 3D(\(\mathcal{M}_{a}\)) does not provide a reasonable estimate. 3D(\(\mathcal{M}_{c},1D\)) provides acceptable results, but only when the shell is sufficiently thin. As in the previous cases, three displacement components, two stresses, and a strain are shown in their entirety along with the thickness direction. Figure 10 further qualifies the correct introduction of the continuity conditions, elastic/hygrometric properties grading, and mechanical boundary conditions. The transverse stresses \(\sigma_{\beta z}\) and \(\sigma_{zz}\) satisfy the external mechanical boundary conditions: they equal 0 at both the top and the bottom surfaces as no external load acts on them. All the quantities are continuous throughout the thickness; this qualifies the division into fictitious layers: they are thin enough to describe the mechanical properties evolution with continuity. Figure 9: Fourth benchmark: spherical sandwich shell panel featuring a FGM (\(p=0.5\)) core with an imposed moisture content on the top and bottom surfaces.