## 5 Numerical Investigations and Remarks
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & & \(\overline{\mathbf{Z}}\)=100 & \(\overline{\mathbf{Z}}\)=300 & \(\overline{\mathbf{Z}}\)=500 \\ \hline \multirow{4}{*}{\(V^{*}_{CNT}=0.12\)} & \multirow{2}{*}{UD} & Shen [22] & 42.68(1,10) \({}^{*}\) & 23.40(1,8) & 18.18(1,7) \\  & & Present & 42.82(1,10) & 23.45(1,8) & 18.51(1,7) \\ \cline{2-5}  & \multirow{2}{*}{FG-X} & Shen [22] & 49.48(1,10) & 25.60(1,8) & 19.91(1,7) \\  & & Present & 48.65(1,10) & 25.37(1,8) & 19.78(1,7) \\ \hline \multirow{4}{*}{\(V^{*}_{CNT}=0.17\)} & \multirow{2}{*}{UD} & Shen [22] & 70.06(1,10) & 38.98(1,8) & 30.81(1,7) \\  & & Present & 70.28(1,10) & 39.06(1,8) & 30.85(1,7) \\ \cline{2-5}  & \multirow{2}{*}{FG-X} & Shen [22] & 82.76(1,10) & 43.83(1,8) & 34.15(1,7) \\  & & Present & 81.44(1,10) & 43.47(1,8) & 33.94(1,7) \\ \hline \multirow{4}{*}{\(V^{*}_{CNT}=0.28\)} & \multirow{2}{*}{UD} & Shen [22] & 84.34(1,10) & 45.11(1,8) & 35.56(1,7) \\  & & Present & 84.67(1,10) & 45.22(1,8) & 35.62(1,7) \\ \cline{1-1} \cline{2-5}  & \multirow{2}{*}{FG-X} & Shen [22] & 109.42(1,10) & 55.84(1,8) & 43.24(1,7) \\ \cline{1-1}  & & Present & 108.68(1,10) & 56.01(1,8) & 43.44(1,7) \\ \hline \hline \end{tabular} \({}^{*}\) the numbers in the parentheses denote the buckling mode \((m,n)\).  
\end{table} 
Table 1: Comparisons of critical buckling loads of non-corrugated FG-CNTRC cylindrical shells subjected to radial loads (in kPa) (\(h=0.001\) m, \(R/h=100\), \(T=300\) K, \(\overline{\mathbf{Z}}=\frac{l^{2}}{8h}\)). The results for a UD-CNTRC cylindrical shell are also listed in direct comparison. It can be seen that the buckling loads of FG-X type shells are larger and the buckling loads of FG-O type shells are smaller than those of the UD-CNTRC cylindrical shells. Especially, the buckling loads of corrugated cylindrical shells are greater than those of non-corrugated shells. The critical buckling loads of spiral stiffened shells are larger than those of ring corrugated shells and they significantly increase when the wave number of corrugation of a loop coil increases. The maximal critical buckling loads are reached at the very large wave number of corrugation of a loop coil. Figures 2 and 3 present the postbuckling load-deflection curves for ring and spiral corrugated CNTRC cylindrical shells for three cases of distribution (UD, X and O distributions) subjected to radial loads. The results for UD-CNTRC cylindrical shells are also included for direct comparison. Clearly, the postbuckling curve of FG-X distribution shells is highest, the postbuckling curve of FG-O distribution shells is lowest, and the postbuckling curve of the UD-CNTRC cylindrical shell is in the middle of those of FG-X and FG-O distribution shells. \begin{table}
\begin{tabular}{l c c c} \hline  & **UD** & **FG-X** & **FG-O** \\ \hline Round corrugated cylindrical shells \(V^{*}_{CNT}=0.12\), \(r=0.0075\) m, \(d=0.003\) m. & & \\ Non corrugated & 0.0207(1,5) \({}^{*}\) & 0.0267(1,5) & 0.0124(1,6) \\ Ring corrugated & 0.1668(1,1) & 0.1732(1,1) & 0.1625(1,1) \\ \(N=1\) & 0.1673(1,1) & 0.1737(1,1) & 0.1630(1,1) \\ \(N=3\) & 0.1716(1,1) & 0.1782(1,1) & 0.1672(1,1) \\ \(N=5\) & 0.1803(1,1) & 0.1874(1,1) & 0.1756(1,1) \\ Maximal critical buckling load & 0.8079(1,3,59) \({}^{**}\) & 0.8083(1,3,59) & 0.8076(1,3,59) \\ \hline Trapezoidal corrugated cylindrical shells \(V^{*}_{CNT}=0.12\), \(c=0.015\) m, \(f=0.0075\) m, \(\alpha=\pi/3\) & \\ Non corrugated & 0.0116(1,6) & 0.0160(1,6) & 0.0073(1,6) \\ Ring corrugated & 0.1419(1,2) & 0.1422(1,2) & 0.1417(1,2) \\ \(N=1\) & 0.1419(1,2) & 0.1423(1,2) & 0.1417(1,2) \\ \(N=3\) & 0.1424(1,2) & 0.1428(1,2) & 0.1422(1,2) \\ \(N=5\) & 0.1434(1,2) & 0.1437(1,2) & 0.1432(1,2) \\ Maximal critical buckling load & 0.3544(1,3,59) & 0.3548(1,3,59) & 0.3541(1,3,58) \\ \hline \end{tabular} \({}^{*}\)the numbers in the parentheses denote the buckling mode \((m,n)\); \({}^{**}\)the numbers in the parentheses denote the buckling mode and wave number of corrugation of a loop coil \((m,n,N)\).  
\end{table}