## 9 Numerical Results and Discussion
### Buckling  
In the following, we considered the values of the parameters [19]\(ea=l=1\) nm, \(L/h=10\), \(b=h=5\) mm, \(E=210\)\(\times\)\(10^{9}\), and \(V=0.24\). In Figure 4, the dimensionless buckling load \(\stackrel{{\sim}}{{P}}_{0}=\frac{P_{0}L^{2}}{ET}\) with respect to the small-scale parameter \(ea\) is depicted, where the solid line was plotted based on data from [13], and the dashed line was plotted based on present results. Figure 4: Dimensionless buckling load for Euler–Bernoulli (green) and Timoshenko beams (red), with respect to small-scale parameter \(ea\), for \(E=210\)\(\times\)\(10^{9}\), \(v=0.24\), \(h=b=5\) nm, and \(L=100\) h.  
It can be seen that the dimensionless buckling load decreased once the small-scale parameter increased. The buckling load for the Euler-Bernoulli beam was higher than the buckling load for the Timoshenko beam. On the other hand, the results obtained in this paper were nearly identical to the results presented in [13]. (The maximum difference was 0.73%). Figure 5 shows the influence of the slenderness ratio (\(L/h\)) on the dimensionless buckling loads for a simply supported CNTRC nanobeam for \(l=1\), \(h=1\) nm, \(v=0.3\), and \(\overline{w}=\frac{100 wEl}{q}\) and for the three cases: \(ea=1/2\) (red), \(ea=1\) (green), and \(ea=2\) (blue), where the solid line was plotted based on data from [6], and the dashed line was plotted based on present results. It was observed that the dimensionless buckling load was lower than the ones given by classical theory (\(l=ea\)), when \(l<ea\). If \(l>ea\), then the dimensionless buckling load was higher than the ones given by classical theory. Also, for \(l<ea\), the dimensionless buckling load increased with the increasing of the slenderness ratio, and for \(l>ea\), the dimensionless buckling load decreased with the increasing of the slenderness ratio. An excellent agreement was observed between the results obtained in this paper and results obtained in [6]. (The maximum difference was 0.302%.) We remark that for \(l<ea\), the nanobeam was softened, while for \(l>ea\), the nanobeam was hardened, becoming easy to deform; also, the size effects were more significant for the lower values of the slenderness ratio (\(L/h<30\)). In Figure 6, the variation of the dimensionless buckling load is presented with respect to the slenderness ratio for UD-beam (green), X-beam (red), and O-beam (blue) for \(ea=\sqrt{2}\) nm and \(b=h\), where the solid line was plotted based on data from [10], and the dashed line was plotted based on present results. It was clear that if \(L/h\) increased, then the values of the dimensionless buckling loads increased at the beginning, and finally, it stabilized at a constant value corresponding to the known results of the Euler-Bernoulli beam [10]. In Table 1, the normalized buckling load of the CNTRC nanobeam is presented for the classical theory (\(ea=0\)) and nonlocal theory for \(ea=\sqrt{2}\) and \(ea=2\) for different values of the volume fraction parameter \(V_{CNT}\). From Table 1, we can deduce that the results obtained by classical theory for the buckling values were higher than the results obtained for nonlocal theories. Also, if the nonlocal parameter \(ea\) increased, then the values of the normalized buckling load decreased. The highest increase in the normalized buckling loads was more pronounced in the X-beam in comparison with the UD-beam and O-beam. Figure 5: Dimensionless buckling load with respect to slenderness ratio for \(l=1\). Effect of nonlocal parameter \(ea\) for \(E=210\times 10^{9}\), \(v=0.24\), \(h=b=5\) nm, and \(L=100\)h: \(ea=1/2\) (red), \(ea=1\) (green), and \(ea=2\) (blue). In the particular case of the Timoshenko beam, for \(L/h=15\), the present solution based on NSGT agreed with the buckling results presented in [3; 38], as shown in Table 2.  
From Table 2, it followed that the values of the critical buckling load for the CNTRC Timoshenko beam increased, while the volume fraction parameter \(V_{CNT}\) for any type of beam decreased.