# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 4 Results
### Assessments
Therefore, this assessment verified that the 3D shell model correctly handles the thermomechanical analysis of FGM plates. Furthermore, it simultaneously assessed the 3D FE model, which will be helpful in the following assessment to validate the hygromechanical analysis. The second assessment is meant to validate the hygroelastic part of the 3D solution for plates 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 plate; the in-plane dimensions are \(a=b=100\) m, the thickness varies to obtain different thickness ratios (\(a/h=4\), 10, 50). The FGM layer relies on the same metallic and ceramic constituents, whose volume fraction follows the same power-law with \(p=2\). An external hygrometric field acts on the top (\(M_{t}=1.0\%\)) and bottom (\(M_{b}=0.5\%\)) surfaces; it has a harmonic form, with half-wave numbers \(m=n=1\). The reference results are obtained through the same 3D FE model of the previous assessment, in which the hygrometric field replaces the thermal one. Its previous validation allows considering it as a reliable source for reference results. Table 2 proposes a pair of results for each thickness ratio at different coordinates along with direction \(z\) in terms of a displacement, \(w\) or \(u\), and an in-plane shear component, \(\sigma_{\text{az}}\). Consistent with the previous test case, the results show that the 3D shell model always gives comparable results with the 3D FE model, despite the thickness ratio and the considered variable, when the number of mathematical layers is sufficiently high. \(NL=300\), coupled with an order of expansion \(N=3\) for the exponential matrix, always delivers the correct results. Therefore, this assessment confirmed the capabilities of the 3D shell model in handling the hygromechanical analysis of FGM plates. The third assessment considers a simply-supported one-layered FGM cylindrical shell. Again, the dimensions of the reference surface are fixed, as in the previous cases; they are \(a=1\) m and \(b=\frac{a}{3}R_{\beta}\), with \(R_{\beta}=10\) m. This test case also considers different thickness ratios \(R_{\beta}=50,1000\)) to evaluate their influence on the performance of the solution. The constituents of the FGM layer are the same as those defined at the beginning of Section 4 and considered in the previous assessments. The power law is also the same, with \(p=2\). The top and bottom external surfaces are subjected to an external sovra-temperature field with amplitudes \(\theta_{t}=+1\) K and \(\theta_{b}=0\) K. The half-wave numbers of the thermal field are \(m=n=1\). The reference solution is a refined 2D layer-wise solution based on the Unified Formulation [74], which considers a 3D temperature profile along with thickness direction. Table 3 proposes six results for each thickness ratio: the transverse displacement w and an in-plane displacement, evaluated at three different coordinates along with direction \(z\). The table also assesses a 3D FE model, solved through the Nasiran solver, which helps evaluate the hygroelastic analysis of the following assessment. IsoMesh meshed the geometry of each shell with 3D HEX8 elements; the mesh did not change with the thickness ratio and included 25 elements in the thickness direction and 30 in both the in-plane ones. Solid elements were necessary to define the mechanical properties evolution in the thickness direction and build significant thermal and mechanical variables profiles. The results show that the 3D shell model always coincides with the quasi-3D method [74], despite the thickness ratio and the considered variable, when the number of mathematical layers is sufficiently high. \(NL=300\), coupled with an order of expansion \(N=3\) for the exponential matrix, always delivers the correct results. Therefore, this assessment verified that the 3D