## 3 Results and Discussion
### Verification Cases  
#### 3.1.1 Case 1--Maximum Transverse Deflection of Plates  
The first verification study considers an isotropic and homogeneous square simply supported plate submitted to a uniform transversely distributed load \(p_{z}\). A set of edge-to-thickness ratios, \((a/h)\), was considered, and the analyses were performed considering double-symmetry with a 4 \(\times\) 4 mesh. The results are presented in the non-dimensional form, \(\overline{w}=\frac{Et^{3}w}{p_{z}at}\), in Table 1 for a set of edge-to-thickness ratios and compared with the results of other authors. Table 1 also shows the relative deviations between the present model's results and the three reference results. The relative deviations are calculated as \(dev(\%)\ =\ \left(\overline{w}-\overline{w}_{ref}\right)/\overline{w}_{ref} \times 100\), where \(\overline{w}_{ref}\) stands for the reference non-dimensional deflection.  
As can be observed, the results present a very good agreement with the alternative authors, particularly with ([PERSON], 1991) [45], whose model is based on the Mindlin-Reissner theory and was implemented through a triangular plate bending element with thirty-five degrees of freedom. Regarding the agreement with the remaining references, there is also a good agreement, except for the thick beams where a greater deviation is observed, although only below 5%.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**a/h** & Present \(\mathbf{w}\) & **[PERSON] [45]** & \(\mathbf{w}\) & **[PERSON] et al. [46]** & \(\mathbf{w}\) & **[PERSON] and [PERSON] [47]** \\ \hline
4 & 0.05899 & 0.05873 & 0.05631 & 0.05656 \\
5 & 0.05369 & 0.05356 & 0.05201 & 0.05360 \\
10 & 0.04661 & 0.04666 & 0.04627 & 0.04632 \\
20 & 0.04484 & 0.04494 & 0.04483 & 0.04486 \\ \hline
**a/h** & **-** & _dev (\%)_[45] & _dev (\%)_[46] & _dev (\%)_[47] \\ \hline
4 & - & 0.44 & 4.76 & 4.30 \\
5 & - & 0.24 & 3.23 & 0.17 \\
10 & - & \(-\)0.11 & 0.73 & 0.63 \\
20 & - & \(-\)0.22 & 0.02 & \(-\)0.04 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Non-dimensional maximum transverse deflection \(\overline{w}\) and relative deviations (%).  
1.2 Case 2--Shear Correction Factors for Sandwich Structure with Outer Layers Made of Functionally Graded Materials  
The second verification study addresses the determination of shear correction factors for a sandwich structure. This sandwich structure has a metallic core, and the outer layers are made of a metal-ceramic functionally graded material. All the materials involved have a Poisson ratio of 0.3. In this case, the FGM is ruled by the known power-law volume fraction, characterized by an exponent that functions as an adjustment parameter of the materials' mixture evolution through the thickness. The results obtained are presented in Table 2.  
The results obtained in the present study are fully coincident with those obtained by ([PERSON] et al., 2008) [48]. In consequence, the relative deviations, calculated as \(dev(\%)=\left(k-k_{ref}\right)/k_{ref}\times 100\), where \(k_{ref}\) stand for the reference shear correction factor and are null for all the cases in Table 2. Hence, for the sake of simplicity, Table 2 presents only the values obtained for each combination of \(E_{c}/E_{m}\) and N.  
This coincidence is observed for different degrees of dissimilarity of the constituent materials' Young's modulus (\(E_{c}/E_{m}\)) and different mixtures' evolution, as denoted by the set of power-law exponents (N) considered.