## 9 Numerical Results and Discussion
### Buckling
The effects of the nonlocal parameter (\(ea\)), the material length scale parameter (\(l\)), and the slenderness ratio \(L/h\) for Euler-Bernoulli and Timoshenko beams on the critical buckling loads are presented in Table 3.  
From Table 3, by comparing the results obtained in [6] and the present work, we deduced that the critical buckling loads increased if the nonlocal parameter \(l\) increased, while the critical buckling load increased if the nonlocal parameter \(ea\) increased. Also, it was clear that the difference between the critical buckling loads of the Euler-Bernoulli beam and the Timoshenko beam was insignificant. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(V_{CNT}\)} & \multirow{2}{*}{\(ea\)} & \multicolumn{2}{c}{**UD-Beam**} & \multicolumn{2}{c}{**X-Beam**} & \multicolumn{2}{c}{**O-Beam**} & \multicolumn{2}{c}{**O-Beam**} \\ \cline{3-10}  & & \multicolumn{1}{c}{**Data from [3]**} & \multicolumn{1}{c}{**Present**} & \multicolumn{1}{c}{**Data from [3]**} & \multicolumn{1}{c}{**Present**} & \multicolumn{1}{c}{**Data from [3]**} & \multicolumn{1}{c}{**Present**} & \multicolumn{1}{c}{**Present**} \\ \hline  & 0 & 17.2929 & 17.4913 & 20.3867 & 20.4104 & 12.1347 & 12.2512 & 16.7435 \\
0.12 & \(\sqrt{2}\) & 15.8712 & 15.1837 & 19.0337 & 19.1444 & 10.8132 & 10.7845 & 14.9782 \\  & 2 & 14.6655 & 14.6948 & 17.8492 & 17.9104 & 9.7513 & 9.7385 & 13.8572 \\ \hline  & 0 & 27.2229 & 27.2778 & 32.7365 & 32.9613 & 18.6032 & 18.5788 & 23.5721 \\
0.17 & \(\sqrt{2}\) & 24.8201 & 24.9941 & 30.3636 & 30.4593 & 16.4684 & 16.4004 & 21.2437 \\  & 2 & 22.8078 & 22.8813 & 28.3115 & 28.3784 & 14.7732 & 14.6982 & 19.7845 \\ \hline  & 0 & 36.0119 & 36.2174 & 43.0629 & 43.1557 & 26.5113 & 26.6231 & 32.8132 \\
0.28 & \(\sqrt{2}\) & 33.3770 & 33.4011 & 40.5028 & 40.6134 & 23.7818 & 23.8113 & 28.9435 \\  & 2 & 31.1014 & 31.1334 & 38.2304 & 38.4517 & 21.5618 & 21.6289 & 26.4389 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The normalized buckling loads for different values of the volume fraction parameter. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**UD-Beam**} & \multicolumn{3}{c}{**O-Beam**} & \multicolumn{3}{c}{**X-Beam**} \\ \cline{2-10}  & \multicolumn{1}{c}{**Data from [3]**} & \multicolumn{1}{c}{**Data from [3]**} & \multicolumn{1}{c}{**Present**} & \multicolumn{1}{c}{**Data from [3]**} & \multicolumn{1}{c}{**Data from [3]**} & \multicolumn{1}{c}{**Present**} & \multicolumn{1}{c}{**Data from [3]**} & \multicolumn{1}{c}{**Present**} & \multicolumn{1}{c}{**Present**} \\ \hline  & 0.132 & 0.0986 & 0.0996 & 0.0604 & 0.05877 & 0.059173 & 0.1367 & 0.128833 & 0.130742 & 0.17832 \\
0.17 & - & 0.15155 & 0.15267 & - & 0.087704 & 0.085132 & - & 0.199495 & 0.197371 & 0.23751 \\
0.28 & - & 0.22005 & 0.21973 & - & 0.155755 & 0.15614 & - & 0.289646 & 0.28743 & 0.33112 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of critical buckling values of the CNTRC Timoshenko nanobeam for classical theory. Figure 6: Variation of the dimensionless buckling load with respect to the slenderness ratio for \(b/h=1\), \(l=0.2\), and \(V_{CNT}=0.12\) for different distribution forms: X-beam (red), UD-beam (green), and O-beam (blue). In all cases, the values of critical buckling loads for the nonlinear case were higher in comparison to the values of critical buckling loads for the linear case, accounting for the same type of nanobeam.