# 3D Stress Analysis of Multilayered Functionally Graded Plates and Shells under Moisture Conditions
## 4 Results
### Assessments  
The present solution handles several geometries and different load cases. As discussed, it can study plates, cylinders, cylindrical and spherical shells under mechanical, thermal, and hygrometric load. Furthermore, it is not limited to isotropic monolayer structures, but it also handles orthotropic and multi-layered lamination schemes, and it can go up to layers embedding FGM. However, confident use of the model is possible only after validated against established solutions already offered in the literature. The authors did not find hygro-elastic results from exact 3D solutions in the literature which were applied to FGMs. For this reason, the validation process of the model is built by separately validating its different sections against the results provided by other researchers, exploiting the parallel of the moisture content field and the thermal field, and complementing this process with the assistance of 3D FE (Finite Element) models. A static 3D FE model is solved through the Nastran solver SOL101. IsoMesh meshed the geometry of each plate 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, hygrometric, and mechanical variables profiles. Depending on the geometry, the coordinate system of the model has one, two, or none curvilinear coordinate. For consistency with the analytical model, the displacement-related boundary conditions are defined on the lateral surfaces of the structure following the harmonic form hypotheses: a pair of displacement coordinates is set 0 on each surface. First, the harmonic thermal/hygrometric field is introduced in an equation, following Equation (1). It is applied to the top and bottom surfaces of the structure; the preprocessor automatically calculates the field values at each node of the external surfaces through their coordinates. The first run solves the thermal part of the problem and returns the temperature (or moisture content) at each node. This field is then applied to a further (and identical) FE model, which solves the elastic part. The thermal, hygrometric, and mechanical properties of the FGM layer are given imagining to split the structures into a number of fictitious layers coinciding with the number of elements in the thickness direction. The properties of each fictitious layer are calculated at its midpoint, following Equations (83)-(85). Needless to say, they are not exact solutions, but they can guide in benchmarking the proposed model. The first assessment considers a simply-supported one-layered FGM square plate. It investigates different thickness ratios (\(a/h=4\), 10, 50) in plates with in-plane dimensions of \(a=b=100\) m. The FGM layer relies on a metallic constituent, and a ceramic one, whose mechanical, thermal, and hygrometric properties are defined at the beginning of Section 4. 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. The thermal field has a harmonic form, with half-wave numbers \(m=n=1\). The reference solution is the asymptotic method of [PERSON] and [PERSON] (2007), which considers a 3D temperature profile along with thickness direction. Table 1 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, \(\sigma_{\text{az}}\). The results show that the 3D shell model always coincides with [PERSON] and [PERSON]'s asymptotic method, 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.