## 4 Numerical Results
\begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**Items**} & \multirow{2}{*}{\(\overline{\overline{w}}\) and \(\overline{\sigma}_{x}\) with respect to the grid density (\(a/h=10\), (3,3,2) hierarchic model).} & \multicolumn{4}{c}{**Grid Density**} \\ \cline{2-7}  & 5\(\times\)5 & 7\(\times\)7 & 11\(\times\)11 & 15\(\times\)15 & 19\(\times\)19 \\ \hline \multirow{2}{*}{\(\overline{\overline{w}}\)} & 0.41165 & 0.45044 & 0.47284 & 0.47943 & 0.48220 \\  & (\(-\)14.631\%) & (\(-\)6.586\%) & (\(-\)1.941\%) & (\(-\)0.574\%) & (Ref.) \\ \hline \multirow{2}{*}{\(\overline{\sigma}_{x}\)(\(\times\)10\({}^{-1}\))} & 1.7186 & 2.9076 & 3.6306 & 3.7449 & 3.7819 \\  & (\(-\)54.557\%) & (\(-\)13.118\%) & (\(-\)4.001\%) & (\(-\)0.978\%) & (Ref.) \\ \hline \end{tabular}
\end{table} 
Table 1: Variation in \(\overline{w}\) and \(\overline{\sigma}_{x}\) with respect to the grid density (\(a/h=10\), (3,3,2) hierarchic model). \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{_a/h_} & \multirow{2}{*}{\(\overline{n}\)} & \multirow{2}{*}{\(\overline{e}\)} & \multicolumn{4}{c}{**Even**} & \multicolumn{2}{c}{**Uneven**} \\ \cline{4-7}  & & & **Present** & **Ref. [36]** & **Present** & **Ref. [36]** \\ \hline \multirow{4}{*}{10} & & 0.0 & 0.71929 & 0.71361 & 0.71929 & 0.71361 \\  & & 0.2 & 0.89159 & 0.88950 & 0.77523 & 0.81751 \\  & & 0.4 & 1.18520 & 1.18947 & 0.97001 & 0.95489 \\ \cline{2-7}  & & 0.0 & 0.93275 & 0.92873 & 0.93275 & 0.92873 \\  & & 0.2 & 1.28790 & 1.29241 & 1.14190 & 1.13392 \\  & & 0.4 & 2.23613 & 2.29216 & 1.55447 & 1.58372 \\ \hline \multirow{4}{*}{20} & & 0.0 & 0.68489 & 0.69209 & 0.68489 & 0.69209 \\  & & 0.2 & 0.84938 & 0.86177 & 0.74850 & 0.79069 \\  & & 0.4 & 1.13005 & 1.15510 & 0.92341 & 0.92626 \\ \cline{1-1} \cline{2-7}  & & 0.0 & 0.88691 & 0.89968 & 0.88691 & 0.89968 \\ \cline{1-1}  & & 0.2 & 1.22532 & 1.25611 & 1.09321 & 1.09915 \\ \cline{1-1}  & & 0.4 & 2.12631 & 2.24360 & 1.45637 & 1.44418 \\ \hline \end{tabular}
\end{table} 
Table 2: The calibrated central deflections \(\overline{w}\) of simply supported square FG porous plate. Next, the static bending and the free vibration of FG porous plates were parametrically examined for the major parameters. Figure 4a shows the variation of calibrated central deflection to the porosity parameter \(e\) for the even and uneven porosity distributions when the \(w/t\) ratio \(a/h\) is 20. It is seen that the central deflection of even porosity distribution is larger than that of uneven porosity distribution, because not only the total amount of porosity is larger at the even porosity distribution but also the mid-surface-biased uneven porosity distribution provides the higher plate bending stiffness than the even porosity distribution. Meanwhile, the calibrated central deflection increases in proportion either to the ceramic index \(n\) or to the porosity parameter \(e\). Thisis because the plate bending stiffness becomes lower as either of the two indices becomes larger. This increase trend in \(\overline{w}\) with increasing the porosity parameter \(e\) is less apparent at the uneven porosity distribution, because the porosity is biased towards the mid-surface while the ceramic having the higher elastic modulus becomes more concentrated near the top surface proportional to the ceramic index. Figure 4b shows the variation in \(\overline{w}\) to the porosity parameter for different \(w/t\) ratios \(a/h\), where the \(\overline{w}\) uniformly decreases in proportion to \(a/h\) because the central deflection is calibrated with the thickness, as given in Equation (22). Meanwhile, the relative difference in \(\overline{w}\) between two porosity distributions is not shown to be affected by the value of \(a/h\) because the thickness-wise distributions of metal and ceramic are not changed, even when the value of \(a/h\) is changed. Figure 5a represents the variation of \(\overline{w}\) to the porosity parameter \(e\) for different boundary conditions when the values of \(a/h\) and \(n\) are 20 and 0.3.