## 5 Case Study
### Calculus of the Bending Rigidity of the Plate  
The analytical calculation of the global stiffness of the FGP with relation (10) has the advantage that it does not use the concept of a multilayer plate but has the disadvantage that the Poisson coefficient value is considered constant. The analytical calculation of the global stiffness of the FGP with relations (2) to (5) has the advantage that it can take into account the variation of Poisson's ratio (in steps, as for all parameters of the FGP), but has the disadvantage that it has no connection with the number of layers. Table 4 presents the values of bending rigidity for different values of Poisson's ratio, as follows (in Table 4): ceramic material (\(v_{\it{cram}}\)), average value of those two materials (\(v_{\it{mad}}\)), average values depending on power coefficient \(k\) of the layer values (\(v\)), constant value of 0.30 (\(v_{\it{const}}\)) and aluminum material (\(v_{\it{alum}}\)). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{10}{c}{\(k\)} \\ \cline{2-10}  & **0.05** & **0.10** & **0.25** & **0.50** & **1.00** & **2.00** & **4.00** & **6.00** & **8.00** \\ \hline \(v_{\it{cram}}\) [-] & 0.220 & 0.220 & 0.220 & 0.220 & 0.220 & 0.220 & 0.220 & 0.220 & 0.220 \\ \(D\) [Nm] & 2021.87 & 1928.13 & 1710.98 & 1484.42 & 1261.03 & 1087.29 & 938.37 & 847.36 & 781.93 \\ \hline \(v_{\it{mad}}\) [-] & 0.275 & 0.275 & 0.275 & 0.275 & 0.275 & 0.275 & 0.275 & 0.275 & 0.275 \\ \(D\) [Nm] & 2081.42 & 1984.91 & 1761.37 & 1528.14 & 1298.17 & 1119.31 & 966.01 & 872.32 & 804.96 \\ \hline \(v\) [-] & 0.286 & 0.285 & 0.284 & 0.283 & 0.282 & 0.281 & 0.280 & 0.279 & 0.279 \\ \(D\) [Nm] & 2095.33 & 1997.09 & 1771.15 & 1535.68 & 1303.74 & 1123.34 & 968.78 & 874.15 & 806.65 \\ \hline \(v_{\it{const}}\) [-] & 0.300 & 0.300 & 0.300 & 0.300 & 0.300 & 0.300 & 0.300 & 0.300 & 0.300 \\ \(D\) [Nm] & 2114.30 & 2016.27 & 1789.20 & 1552.28 & 1318.68 & 1137.00 & 981.27 & 886.10 & 817.67 \\ \hline \(v_{\it{alum}}\) [-] & 0.330 & 0.330 & 0.330 & 0.330 & 0.330 & 0.330 & 0.330 & 0.330 & 0.330 \\ \(D\) [Nm] & 2159.14 & 2059.03 & 1827.15 & 1585.2 & 1346.65 & 1161.11 & 1002.1 & 904.89 & 835.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The analytical values of bending rigidity by direct integration. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{10}{c}{\(k\)_Values_} \\ \cline{2-10}
**Layer No. ** & **0.05** & **0.10** & **0.25** & **0.50** & **1.00** & **2.00** & **4.00** & **6.00** & **8.00** \\ \hline
1 & 327,785.62 & 284,365.90 & 193,266.96 & 119,015.30 & 77,750.00 & 70,193.75 & 70,000.12 & 70,00  
For the constant value of Poisson's ratio, \(\
u=0.30\), the following results presented in Table 5 were obtained, for the analytical calculus under concept of multilayered plate. Analyzing the results from Tables 4 and 5, it is found that the bending rigidity of FGPs varies with the power coefficient \(k\) of the material law, with the number of layers (when using the concept of myltstrate plate) and with the Poisson's ratio \(\
u\). The graphic representations in Figure 8, in addition to the fact that they suggestively and correctly present the way of bending rigidity variation, allow us to find analytical variation curves, which can provide accurate data for any values of the parameters in the respective range. The curves in Figure 8 also show us that the bending rigidity value is slightly influenced by the Poisson's ratio, Figure 8a and especially by the number of layers, Figure 8b. Moreover, in many works the influence of the Poisson's ratio variation is neglected and a constant value of 0.30 is used. The mode of variation of the FGP bending rigidity versus the power coefficient \(k\), calculated based on the concept of multilayer plate and on the analytical calculus by direct integration, is presented in Figure 9a. In Figure 9b, the errors between those curves are graphically presented. The use of the multilayer plate concept requires the adoption of a number of layers that ensure the necessary accuracy of all results.