# Buckling Analysis of CNTRC Curved Sandwich Nanobeams in Thermal Environment
## 5 Results and Discussion
### Parametric Study
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \(\alpha\) & \multicolumn{4}{c}{**SS**} & \multicolumn{4}{c}{**CC**} & \multicolumn{4}{c}{**CS**} \\ \cline{3-11}  & & \(V_{\text{cnt}}^{*}\)=12 & \(V_{\text{cnt}}^{*}\)=17 & \(V_{\text{cnt}}^{*}\)=28 & \(V_{\text{cnt}}^{*}\)=12 & \(V_{\text{cnt}}^{*}\)=17 & \(V_{\text{cnt}}^{*}\)=28 & \(V_{\text{cnt}}^{*}\)=12 & \(V_{\text{cnt}}^{*}\)=17 & \(V_{\text{cnt}}^{*}\)=28 \\ \hline \multirow{4}{*}{UD} & \(\pi/4\) & 74.6585 & 100.3288 & 139.9486 & 339.9366 & 442.8754 & 611.3111 & 212.5558 & 278.2913 & 381.2890 \\  & \(\pi/3\) & 41.9954 & 56.4350 & 78.7211 & 257.7059 & 343.7374 & 492.5488 & 149.4863 & 199.1205 & 281.3863 \\  & \(\pi/2\) & 18.6646 & 25.0822 & 34.9871 & 198.9635 & 272.9172 & 407.7090 & 104.4352 & 142.5681 & 210.0248 \\  & \(2\pi/3\) & 10.4989 & 14.1087 & 19.6803 & 178.3950 & 248.1198 & 378.0018 & 88.6652 & 122.7722 & 185.0451 \\ \hline \multirow{4}{*}{FG-X} & \(\pi/4\) & 74.8276 & 100.6190 & 140.6169 & 340.6800 & 444.0969 & 613.9162 & 213.0218 & 279.0764 & 383.0227 \\  & \(\pi/3\) & 42.0905 & 56.5982 & 79.0970 & 258.1393 & 344.4732 & 494.1862 & 149.7553 & 199.5840 & 282.4389 \\  & \(\pi/2\) & 18.7069 & 25.1548 & 35.1542 & 199.1755 & 273.3061 & 408.6553 & 104.5634 & 142.8020 & 210.5910 \\  & \(2\pi/3\) & 10.5226 & 14.1496 & 19.7743 & 178.5296 & 248.3875 & 378.7063 & 88.7442 & 122.9258 & 185.4411 \\ \hline \multirow{4}{*}{FG-O} & \(\pi/4\) & 74.5146 & 100.1307 & 139.7518 & 339.3283 & 442.1373 & 610.8484 & 212.1712 & 277.8003 & 380.9295 \\  & \(\pi/3\) & 41.9144 & 56.3235 & 78.6104 & 257.3788 & 343.3707 & 492.4602 & 149.2768 & 198.8661 & 281.2614 \\ \cline{1-1}  & \(\pi/2\) & 18.6286 & 25.0327 & 34.9379 & 198.8371 & 272.8157 & 407.8875 & 104.3507 & 142.4828 & 210.0675 \\ \cline{1-1}  & \(2\pi/3\) & 10.4786 & 14.0809 & 19.6526 & 178.3388 & 248.1110 & 378.2735 & 88.6244 & 122.7461 & 185.1463 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Effect of opening angle on the dimensionless buckling load of curved sandwich beam (\(0^{\circ}/90^{\circ}/c/90^{\circ}/0^{\circ}\)) (\(h_{c}/h_{f}=4\), \(h=L/10\), \(T=300\) K.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & & & & & & & & \\ \cline{3-11} \multirow{2}{*}{\(\mu\)} & \multicolumn{4}{c}{**T = 300 K**} & \multicolumn{4}{c}{**T = 500 K**} & \multicolumn{4}{c}{**T = 700 K**} \\ \cline{2-11}  & \(\lambda\) & **4** & **6** & **8** & **4** & **6** & **8** & **4** & **6** & **8** \\ \hline
0 & 0 & 81.8686 & 73.5399 & 66.3319 & 61.1267 & 56.6032 & 52.3027 & 19.0  
Figure 3 also depicts the buckling response for a SS (\(0^{\circ}/90^{\circ}/c/90^{\circ}/0^{\circ}\)) beam versus the thickness ratio, \(L/h\), while varying the opening angles. All the plots in Figure 3 feature a monotone decreasing behavior for increasing values of \(L/h\), reaching a plateau for \(L/h\geq 30\). Note also that an increased opening angle value decreases significantly the buckling load of the structure for each fixed value of \(L/h\). In Figure 4 the critical buckling load versus the opening angle is illustrated, taking into account the core-to-face sheet thickness ratio variation. A clear reduction of the beam stiffness with an increased core layer can be observed for each fixed opening angle, which is even more pronounced for lower values of the opening angles. Figure 5 also shows the double effect of the core-to-face sheet thickness ratio and CNT volume fraction on the dimensionless buckling load, with a clear shift of the curve upwards for increasing values of \(V_{cnt}\). The highest critical buckling load is reached for a volume fraction \(V_{cnt}=28\), where the lowest stability is observed for \(V_{cnt}=12\). The impact of the thermal environment on critical buckling load is visible in Figure 6, where an increased temperature value leads to a clear reduction in the buckling load for all the selected boundary conditions because of the thermal dependence of the mechanical properties of the materials.