## 5 Numerical Investigations and Remarks

Table 2: Critical buckling loads of non-corrugated and corrugated FG-CNTRC cylindrical shells (in MPa). Figure 2: Effect of carbon nanotube (CNT) reinforcement type on the postbuckling curve of ring corrugated cylindrical shells (trapezoidal corrugation, \(V^{*}_{CNT}=0.12\), \(c=0.015\) m, \(f=0.0075\) m, \(\alpha=\pi/3\)). Tables 3 and 4 show the effect of geometrical properties of corrugation and CNT volume fraction \(V_{CNT}^{*}\) on the critical buckling load of round and trapezoidal corrugated cylindrical shells, respectively. As can be seen, a very large effect of CNT reinforcement on the critical buckling of shells is recognized. The critical buckling load increases when the volume fraction of CNT \(V_{CNT}^{*}\) increases. The effect of geometrical parameters of round and trapezoidal corrugation on the critical buckling load of spiral corrugated FG-CNTRC cylindrical shells is also presented in these tables. Clearly, the critical buckling loads of corrugated cylindrical shells increase when the geometrical parameters of corrugations increase. Figure 4 shows the effect of CNT volume fraction \(V_{\text{CNT}}^{*}=0.12\); \(0.17\) and \(0.28\) on the postbuckling curve of ring corrugated UD-CNTRC cylindrical shells and corresponding non-corrugated UD-CNTRC cylindrical shells subjected to radial loads. As can be seen the increase of CNT volume fraction \(V_{\text{CNT}}^{*}\) yields an increase of the critical buckling load and postbuckling strength of shells. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(V_{\text{CNT}}^{*}\) & \(c\) **(m)** & \(f\)**(m)** & \
\begin{tabular}{c} **Non** \\ **Corrugated** \\ \end{tabular} &
\begin{tabular}{c} **Ring** \\ **Corrugated** \\ \end{tabular} & \multicolumn{2}{c}{**Spiral Corrugated**} \\ \cline{4-7}  & & & **Corrugated** & **Corrugated** & \(N\) **= 1** & \(N\) **= 3** & \(N\) **= 5** \\ \hline  & 0.01 & 0.005 & 0.0009(1,9) & 0.0048(1,3) & 0.0048(1,3) & 0.0048(1,3) & 0.0048(1,3) & 0.0048(1,3) \\
0 & 0.015 & 0.0075 & 0.0009(1,9) & 0.0072(1,2) & 0.0072(1,2) & 0.0073(1,2) & 0.0074(1,2) \\  & 0.02 & 0.01 & 0.0009(1,9) & 0.0091(1,2) & 0.0091(1,2) & 0.0092(1,2) & 0.0093(1,2) \\ \hline  & 0.01 & 0.005 & 0.0116(1,6) & 0.0688(1,2) & 0.0688(1,2) & 0.0689(1,2) & 0.0691(1,2) \\
0.12 & 0.015 & 0.0075 & 0.0116(1,6) & 0.1419(1,2) & 0.1419(1,2) & 0.1424(1,2) & 0.1434(1,2) \\  & 0.02 & 0.01 & 0.0116(1,6) & 0.1741(1,1) & 0.1752(1,1) & 0.1837(1,1) & 0.2018(1,1) \\ \hline  & 0.01 & 0.005 & 0.0175(1,6) & 0.1018(1,2) & 0.1018(1,2) & 0.1020(1,2) & 0.1024(1,2) \\
0.17 & 0.015 & 0.0075 & 0.0175(1,6) & 0.2074(1,2) & 0.2075(1,2) & 0.2082(1,2) & 0.2097(1,2) \\  & 0.02 & 0.01 & 0.0175(1,6) & 0.2796(1,1) & 0.2813(1,1) & 0.2953(1,1) & 0.3248(1,1) \\ \hline  & 0.01 & 0.005 & 0.0262(1,6) & 0.1592(1,2) & 0.1592(1,2) & 0.1595(1,2) & 0.1600(1,2) \\
0.28 & 0.015 & 0.0075 & 0.0262(1,6) & 0.3337(1,2) & 0.3338(1,2) & 0.3350(1,2) & 0.3373(1,2) \\  & 0.02 & 0.01 & 0.0262(1,6) & 0.3586(1,1) & 0.3607(1,1) & 0.3778(1,1) & 0.4138(1,1) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Effects of geometrical parameters of corrugation and CNT volume fraction \(V_{CNT}^{*}\) on the critical buckling of UD-CNTRC cylindrical shells (trapezoidal corrugation, \(\alpha\) = \(\pi\)/3) (in MPa). Figure 4: _Cont._  
A comparison of the postbuckling curve of ring and spiral corrugated UD-CNTRC cylindrical shells in two cases, round corrugation and trapezoidal corrugation, is presented in Figures 5 and 6. Spiral corrugation is considered with the wave number of corrugation of a loop coil \(N=5\) and \(N=10\). Clearly, the wave number of corrugation of a loop coil strongly influences the postbuckling behavior of cylindrical shells and an usual change is obtained when the wave number of corrugation of a loop coil changes. Figure 4: Effect of CNT volume fraction \(V_{\text{CNT}}^{*}\) on the postbuckling curve of ring corrugated UD-CNTRC cylindrical shells and corresponding non-corrugated UD-CNTRC cylindrical shells (trapezoidal corrugation, \(c=0.01\) m, \(f=0.005\) m, \(\alpha=\pi/3\)).