## 9 Numerical Results and Discussion
### Bending  
The bending behavior of a simply supported CNTRC nanobeam was investigated for various cases and compared with the results known in literature. In Figure 7, dimensionless deflections were depicted for \(l=1\) for different values of the nonlocal parameter \(ea\), \(ea=2\) (red), \(ea=1\) (blue), and \(ea=1/2\) (green), 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 if \(l<ea\), then the dimensionless deflection was higher than the dimensionless deflection obtained by classical theory (\(l=ea=1\)), and vice versa for \(l>ea\), the dimensionless deflection corresponded to the classical case. We observed that if the material length scale parameter \(l\) was smaller than the nonlocal parameter \(ea\), the CNTRC nanobeam exhibited a stiffness-softening effect [6]. Considering Figure 6, the error for the case \(ea=2\) was \(1.071\%\), \(ea=1\) was \(0.871\%\), and \(ea=1/2\) was \(0.698\%\). In Table 4, the maximum normalized deflection CNRT nanobeam for data given in [10] is presented: \(L/h=10\), \(b=4\) m, and \(K_{1}=0.1\) for the uniform load and nonlocal parameter \(l=\sqrt{2}\) nm, with respect to some values of the volume fraction parameter \(V_{CNT}\). We deduced that if the values of the parameters \(V_{CNT}\) increased, then the maximum deflection of a simply supported CNTRC nanobeam decreased, and our results were nearly identical to results obtained in [10]. The obtained results were very close to those presented in [10], being the maximum deviation observed, \(0.667\%\). This result also showed that by increasing \(V_{CNT}\), the deflection of simply supported CNTRC nanobeams decreased. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline  & & & \multicolumn{4}{c}{\(L/h=10\)} & \multicolumn{4}{c}{\(L/h=20\)} \\ \cline{3-13} _ea_ & **Beam** & \multicolumn{2}{c}{\(l=0\) nm} & \multicolumn{2}{c}{\(l=1\) nm} & \multicolumn{2}{c}{\(l=2\) nm} & \multicolumn{2}{c}{\(l=0\) nm} & \multicolumn{2}{c}{\(l=1\) nm} & \multicolumn{2}{c}{\(l=2\) nm} \\ \cline{3-13} \multirow{-13}{*}{**nm**} & \multirow{-1}{*}{**Theory**} & **Data** & **Data** & **Data** & **Data** & **Data** & **Data** & **Data** & **Data** & **Present** & **from** & **Present** & **from** & **Present** & **from** \\  & & **from** & **Present** & **from** & **[6]** & **[6]** & **[6]** & **[6]** & **[6]** & **[6]** & **[6]** & **[6]** & **[6]** & **[6]** & **[6]** \\ \hline \multirow{3}{*}{0} & E-B & 9.8696 & 9.7981 & 10.8437 & 10.8314 & 13.7606 & 13.7592 & 9.8696 & 9.8677 & 10.1837 & 10.1788 & 10.8437 & 10.8612 & 16.1124 \\  & T & 9.6227 & 9.6012 & 10.5724 & 10.5724 & 13.4716 & 13.4814 & 9.8062 & 9.7983 & 10.0487 & 10.0501 & 10.5746 & 10.5728 & 15.8327 \\  & E-B & 8.9830 & 8.9637 & 9.8696 & 9.8685 & 12.5194 & 12.5207 & 9.6320 & 9.6291 & 9.8696 & 9.8689 & 10.5826 & 10.5835 & 15.8574 \\  & T & 8.7583 & 8.7492 & 9.6222 & 9.6256 & 12.2159 & 12.2189 & 9.5706 & 9.5718 & 9.8067 & 9.7983 & 10.5151 & 10.5172 & 15.8991 \\  & E-B & 7.0761 & 7.0612 & 7.7741 & 7.7784 & 9.8696 & 9.8674 & 8.9830 & 8.9841 & 9.2047 & 9.2107 & 9.8696 & 9.8685 & 15.8784 \\  & T & 6.8990 & 6.8124 & 7.5800 & 7.6111 & 9.6222 & 9.6273 & 8.9258 & 8.9277 & 9.1460 & 9.1511 & 9.8067 & 9.8071 & 15.8144 \\ \hline \end{tabular}
\end{table} 
Table 3: Comparison of critical buckling loads for CNTR, Euler–Bernoulli beam, and Timoshenko beam for different values of \(ea\), \(l\), and \(L/h\). Figure 7: Dimensionless deflection with respect to variable \(x/L\) for \(L/h=10\) and for the simply supported nanobeam: \(l=1\), \(ea=2\) (red), \(ea=1\) (blue), and \(ea=1/2\) (green). Table 5 presents dimensionless displacements of the UD-beam [3] without NSGT by varying the slenderness ratio _L/h_ and volume fraction coefficient \(V_{CNT}\). It can be seen that if the slenderness ratio and volume fraction coefficient increased, then the dimensionless displacement decreased.