## 5 Numerical Illustrations
### Parametric Studies  
After validating the present approach, parametric studies are performed to examine the vibrational characteristics of the FG-GPLRC circular arch. The fundamental frequencies of the FG-GPLRC circular arch are tabulated in Tables 5-7. The results involve various GPL distribution patterns, different end conditions and geometric configures. The increase of fundamental frequency (\(\Omega_{\rm GPL}-\Omega_{\rm Epoxy}\))/\(\Omega_{\rm Epoxy}\times 100\) is given in the brackets with \(\Omega_{\rm GPL}\) and \(\Omega_{\rm Epoxy}\) being the fundamental frequencies of circular arch reinforced with and without GPLs, respectively. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Distribution**} & \multirow{2}{*}{\(p\)} & \multicolumn{3}{c}{**SS**} & \multicolumn{3}{c}{**CF**} & \multicolumn{3}{c}{**CC**} \\ \cline{3-10}  & & \(N=11\) & \(N=15\) & \(N=20\) & Exact & \(N=11\) & \(N=15\) & \(N=20\) & \(N=11\) & \(N=15\) & \(N=20\) \\ \hline UD & 10 & 0.0478 & 0.0478 & 0.0478 & 0.0478 & 0.0210 & 0.0210 & 0.0210 & 0.1260 & 0.1262 & 0.1262 \\  & 20 & 0.0478 & 0.0478 & 0.0478 & 0.0478 & 0.0210 & 0.0210 & 0.0210 & 0.1260 & 0.1262 & 0.1262 \\  & 50 & 0.0478 & 0.0478 & 0.0478 & 0.0478 & 0.0210 & 0.0210 & 0.0210 & 0.1260 & 0.1262 & 0.1262 \\ FG-X & 10 & 0.0551 & 0.0551 & 0.0551 & 0.0551 & 0.0245 & 0.0245 & 0.0245 & 0.1393 & 0.1395 & 0.1395 \\  & 20 & 0.0553 & 0.0553 & 0.0553 & 0.0553 & 0.0245 & 0.0245 & 0.0245 & 0.1393 & 0.1395 & 0.1395 \\  & 50 & 0.0553 & 0.0553 & 0.0553 & 0.0553 & 0.0245 & 0.0245 & 0.0245 & 0.1393 & 0.1395 & 0.1395 \\ FG-O & 10 & 0.0382 & 0.0382 & 0.0382 & 0.0382 & 0.0165 & 0.0165 & 0.0165 & 0.1017 & 0.1018 & 0.1018 \\  & 20 & 0.0379 & 0.0379 & 0.0379 & 0.0378 & 0.0165 & 0.0165 & 0.0165 & 0.1016 & 0.1017 & 0.1018 \\  & 50 & 0.0378 & 0.0378 & 0.0378 & 0.0165 & 0.0165 & 0.0165 & 0.1016 & 0.1017 & 0.1018 \\ FGA & 10 & 0.0439 & 0.0439 & 0.0439 & 0.0439 & 0.0188 & 0.0188 & 0.0188 & 0.1142 & 0.1143 & 0.1143 \\  & 20 & 0.0438 & 0.0438 & 0.0438 & 0.0188 & 0.0188 & 0.0188 & 0.1142 & 0.1143 & 0.1143 \\  & 50 & 0.0438 & 0.0438 & 0.0438 & 0.0188 & 0.0188 & 0.0188 & 0.1142 & 0.1143 & 0.1143 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Convergence study of the fundamental frequencies of FG-GPLRC circular arch. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**B.C.**} & \multirow{2}{*}{**Mode**} & \multicolumn{3}{c}{**\(E_{1}\)/\(E_{2}=15\)**} & \multicolumn{3}{c}{**\(E_{1}\)/\(E_{2}=40\)**} \\ \cline{3-6}  & & **Semi-Analytical** & **Ref. [37]** & **Semi-Analytical** & **Ref. [37]** \\ \hline S-S & 1 & 2.8805 & 2.8808 & 2.3032 & 2.3034 \\  & 2 & 14.0465 & 14.0479 & 10.3561 & 10.3571 \\  & 3 & 29.3568 & 29.3597 & 20.2660 & 20.2680 \\ C-C & 1 & 18.5866 & 18.5885 & 12.6764 & 12.6777 \\  & 2 & 26.9630 & 26.9657 & 19.7824 & 19.7844 \\  & 3 & 41.8072 & 41.8114 & 32.3538 & 32.3570 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparisons of natural frequencies for laminated circular arches (\(\theta_{0}=\pi/2\), \(L/H=10\)). The various GPLs distribution patterns are involved. It can be seen obviously that the fundamental frequency promotes impressively by adding more GPLs into the matrix. It can be concluded that the addition of GPL nanofillers improve the stiffness hence the fundamental frequency of the FG-GPLRC circular arch effectively. Again, for the same amount of GPL nanofillers, the pattern FG-X increases the fundamental frequency more powerfully compared with others. This is due to the fact that the normal stress is very high near the bottom and top surfaces of the arch and is small near its mid-plane. Such a dispersion of GPLs can take full advantage of the advanced material properties of GPL nanofillers to improve the arch stiffness. The fundamental frequency of FG-GPLRC circular arch against length-to-thickness ratio \(l_{\text{GPL}}/h_{\text{GPL}}\) together with various GPL patterns are depicted in Figure 6. Here only the S-S end condition is discussed due to the similarity.