## 4 Numerical Results and Discussion
### Comparison and Validation
Further results of a systematic analysis are listed in Table 5, in terms of the first natural frequency of the FG curved nanobeam, for different patterns, different boundary conditions, and two opening angles. In this case, a total number of layers \(N_{L}=12\) is assumed, as in the previous case, while considering a weight fraction, \(g^{*}_{GPL}=1\%\), a non local parameter \(\mu=1\ nm\), and a null value of the foundation constants \(k_{1}=k_{2}=0\). Table 6 also lists the variation of the first three natural frequencies of the composite curved nanobeam reinforced with GPLs for different values of the foundation parameters \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(\frac{k}{k}\) & \(\omega_{t}\) & \(\mu^{2}\)=0(\(mm\))\({}^{2}\) & \(\mu^{2}\)=1(\(mm\))\({}^{2}\) & \(\mu^{2}\)=2(\(mm\))\({}^{2}\) & \(\mu^{2}\)=3(\(mm\))\({}^{2}\) & \(\mu^{2}\)=4(\(mm\))\({}^{2}\) \\ \hline \multirow{4}{*}{\(50\)} & & **Ref. [17]** & **Present** & **Ref. [17]** & **Present** & **Ref. [17]** & **Present** & **Ref. [17]** & **Present** & **Ref. [17]** & **Present** \\ \cline{2-10}  & & & **Ref. [40]** & **Present** & **Ref. [40]** & **Present** & **Ref. [40]** & **Present** \\ \hline \multirow{4}{*}{\(10\)} & \(0\) & \(8.2006\) & \(8.2141\) & \(25.8678\) & \(26.0937\) \\  & \(1\) & \(7.8236\) & \(7.8366\) & \(24.3792\) & \(24.4961\) \\  & \(2\) & \(7.4942\) & \(7.5057\) & \(23.1083\) & \(23.1601\) \\  & \(3\) & \(7.2032\) & \(7.2134\) & \(22.0084\) & \(22.0212\) \\  & \(4\) & \(6.9437\) & \(6.9527\) & \(21.0456\) & \(21.0353\) \\
20 & \(0\) & \(8.2922\) & \(8.2955\) & \(37.0381\) & \(37.0333\) \\  & \(1\) & \(7.9110\) & \(7.9138\) & \(34.7887\) & \(34.7932\) \\  & \(2\) & \(7.5779\) & \(7.5803\) & \(32.8611\) & \(32.9158\) \\  & \(3\) & \(7.2837\) & \(7.2858\) & \(31.1899\) & \(31.3129\) \\  & \(4\) & \(7.0213\) & \(7.0231\) & \(29.7263\) & \(29.9234\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparative evaluation between the proposed formulation and predictions from a previous study [57], in terms of first natural frequency for varying L/h ratios, nonlocal parameter \(\mu\), and boundary conditions. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\frac{L}{h}\)} & \multirow{2}{*}{\(\mu^{2}\) (\(mm\))\({}^{2}\)} & \multicolumn{3}{c}{**S-S Boundary**} & \multicolumn{3}{c}{**C-C Boundary**} \\  & & & **Conditions (\(\theta_{0}\) = \(\pi\)/3)** & **Conditions (\(\theta_{0}\) = \(\pi\)/6)** \\ \cline{3-8}  & & & **Ref. [40]** & **Present** & **Ref. [40]** & **Present** \\ \hline \multirow{4}{*}{\(10\)} & \(0\) & \(8.2006\) & \(8.2141\) & \(25.8678\) & \(26.0937\) \\  & & \(1\) & \(7.8236\) & \(7.8366\) & \(24.3792\) & \(24.4961\) \\  & & \(2\) & \(7.4942\) & \(7.5057\) & \(23.1083\) & \(23.1601\) \\  & & \(3\) & \(7.2032\) & \(7.2134\) & \(22.0084\) & \(22.0212\) \\  & & \(4\) & \(6.9437\) & \(6.9527\) & \(21.0456\) & \(21.0353\) \\
20 & \(0\) & \(8.2922\) & \(8.2955\) & \(37.0381\) & \(37.0333\) \\  & & \(1\) & \(7.9110\) & \(7.9138\) & \(34.7887\) & \(34.7932\) \\  & & \(2\) & \(7.5779\) & \(7.5803\) & \(32.8611\) & \(32.9158\) \\  & & \(3\) & \(7.2837\) & \(7.2858\) & \(31.1899\) & \(31.3129\) \\  & & \(4\) & \(7.0213\) & \(7.0231\) & \(29.7263\) & \(29.9234\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparative evaluation between the proposed formulation and predictions from a previous study [57], in terms of first natural frequency for varying L/h ratios, nonlocal parameter \(\mu\), and boundary conditions. \(k_{1},\ k_{2}\). The results in Table 6 refer to Pattern 2, where the following parameters are assumed: \(N_{L}\ =\ 12\), \(g^{*}_{GPL}\ =\ 1\%_{o}=1\ nm\), \(\theta_{0}\ =\ \frac{\pi}{3}\) Rad. As visible in Table 6, the natural frequency clearly increases for increasing values of the foundation parameters, \(k_{1},\ k_{2}\), due to increased stiffness of the foundation.