## 4 Numerical Results
C, S and F stand for clamped, simply supported and free, and the combined four charac-ters indicate a set of boundary conditions specific to the sides 1, 2, 3 and 4 of the FG porous plate shown in Figure 3a. It is observed that the lowest central deflection occurs at CCCC and the highest one is seen at SFSF. The second and third in the magni-tude of \(\overline{w}\) are SSSS and SCSC, because the stronger the boundary condition is, the higher the plate bending stiffness is. Meanwhile, the relative difference in \(\overline{w}\) between two porosity distributions is not seen to be affected by the boundary condition type. Figure 5b compares the variations of \(\overline{w}\) between three cosine-type porosity distributions, where the sym shows the lowest level and the unsym 2 shows a lower level than the unsym 1, respectively. The FG plate with the sym porosity distribution is stiffer than the FG plates with unsym 1 and 2 porosity distributions because the porosity in the sym distribution is concentrated at the mid-surface. Meanwhile, the bending stiffness of FG plates having the unsym 2 porosity distribution is higher than that of FG plates with the unsym 1 porosity distribution. This is because the porosity in the former distribution is biased towards the top surface, but the ceramic with the higher elastic modulus is concentrated at the bottom surface when the ceramic index \(n\) is 0.3. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{_a/h_} & \multirow{2}{*}{\(n\)} & \multirow{2}{*}{\(e\)} & \multicolumn{2}{c}{**Even**} & \multicolumn{2}{c}{**Uneven**} \\ \cline{3-6}  & & & **Present** & **Ref. [36]** & **Present** & **Ref. [36]** \\ \hline \multirow{4}{*}{10} & \multirow{4}{*}{0.5} & 0.0 & 0.09702 & 0.09763 & 0.09702 & 0.09763 \\  & & 0.2 & 0.09682 & 0.09719 & 0.09896 & 0.10013 \\  & & 0.4 & 0.09598 & 0.09592 & 0.10120 & 0.10216 \\ \cline{2-6}  & \multirow{4}{*}{1.0} & 0.0 & 0.08767 & 0.08796 & 0.08767 & 0.08796 \\  & & 0.2 & 0.08350 & 0.08324 & 0.08863 & 0.08859 \\  & & 0.4 & 0.07331 & 0.07194 & 0.08893 & 0.08991 \\ \hline \multirow{4}{*}{20} & \multirow{4}{*}{0.5} & 0.0 & 0.02503 & 0.02475 & 0.02503 & 0.02476 \\  & & 0.2 & 0.02497 & 0.02463 & 0.02555 & 0.02542 \\  & & 0.4 & 0.02475 & 0.02429 & 0.02615 & 0.02606 \\ \cline{2-6}  & \multirow{4}{*}{1.0} & 0.0 & 0.02262 & 0.02230 & 0.02262 & 0.02230 \\  & & 0.2 & 0.02154 & 0.02110 & 0.02282 & 0.02271 \\ \cline{1-1}  & & 0.4 & 0.01893 & 0.01821 & 0.02299 & 0.02288 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The calibrated fundamental frequencies \(\overline{\omega}_{(1,1)}\) of simply supported square FG porous plate. Figure 6a compares the variation in \(\overline{\sigma}_{x}\) to the porosity parameter \(e\) for different porosity distributions. The peak value of \(\overline{\sigma}_{x}\) occurs at the top of plate for all the three distributions and the unsym 1 distribution shows the highest level. The \(\overline{\sigma}_{x}\) of unsym 2 distribution is lower than one of sym distribution, and furthermore it decreases in proportion to the porosity parameter \(e\). This is because the elastic modulus at the top for the unsym 2 distribution is lower than one for the sym, and its decrease with increasing the value of \(e\) is larger than the increase of \(\overline{w}\) proportional to \(e\). Figure 6b compares the variations of \(\overline{\tau}_{xz}\) to the value of \(e\) for three different porosity distributions. The peak of \(\overline{\tau}_{xz}\) appears near the neutral surface regardless of the porosity distribution, and the highest level occurs at the unsym 1 distribution. The calibrated shear stresses of unsym 2 and sym distributions decrease proportionally to \(e\) because the decrease inshear modulus with increasing \(e\) is larger than the increase in \(\overline{w}\) with the value of \(e\). Figure 4: Dependence of the calibrated central deflection on the porosity parameter: (**a**) for different ceramic indices \((a/h=20)\), (**b**) for different width–thickness ratios \((n=0.3)\).