## 5 Numerical Results
These results could be interestingly applied for practical design purposes. In addition, we evaluate the effect of some mechanical and geometry parameters on the thermal buckling load \(\Delta T_{cr}\) of CNTCR stiffened cylindrical shells reinforced by FG wavy CNTs, and subjected to a uniform temperature rise. More specifically, we study the possible effects on the critical thermal buckling load related to different distributions of CNTs within the material, to the presence of ring and stringer stiffeners, to different aspect ratios \(A_{R}\), as well as to different waviness indexes \(w\) and volume fractions \(V_{CNT}^{*}\). A first investigation considers a cylindrical shell with \(L/R=1\), and \(R/h=200\), which is made by TI and TD material properties. The numerical results obtained in terms of thermal buckling load are listed in Tables 3 and 4 under the assumptions of: TI and TD material properties, respectively, complete absence of ring and stringer stiffeners (i.e., \(N_{r}=N_{s}=0\)), \(A_{R}\) varying from 50 up to 1000, and varying waviness index (i.e., \(w=0\), and \(w=0.425\)). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Profiles**} & \multirow{2}{*}{\(A_{R}\)} & \multicolumn{2}{c}{\(V_{\text{CNT}}^{*}=\mathbf{0.12}\)} & \multicolumn{2}{c}{\(V_{\text{CNT}}^{*}=\mathbf{0.28}\)} \\ \cline{3-6}  & & \(\mathbf{w}=\mathbf{0}\) & \(\mathbf{w}=\mathbf{0.425}\) & \(\mathbf{w}=\mathbf{0}\) & \(\mathbf{w}=\mathbf{0.425}\) \\ \hline \multirow{4}{*}{\(\wedge\)-profile} & \(A_{R}=50\) & 355.6195 & 409.7686 & 346.9656 & 446.356 \\  & \(A_{R}=100\) & 342.6400 & 407.8564 & 338.9678 & 447.5867 \\  & \(A_{R}=500\) & 336.1531 & 404.8622 & 334.6581 & 448.6354 \\  & \(A_{R}=1000\) & 335.5539 & 404.3232 & 333.9651 & 448.7409 \\ \hline \multirow{4}{*}{\(X\)-profile} & \(A_{R}=50\) & 375.9555 & 413.1924 & 366.8814 & 450.1936 \\  & \(A_{R}=100\) & 356.2644 & 409.7816 & 351.3718 & 449.1031 \\ \cline{1-1}  & \(A_{R}=500\) & 344.5911 & 404.1034 & 343.0042 & 446.3097 \\ \cline{1-1}  & \(A_{R}=1000\) & 343.5005 & 403.1505 & 342.1739 & 445.7818 \\ \hline \multirow{4}{*}{\(O\)-profile} & \(A_{R}=50\) & 356.0734 & 406.7710 & 343.8665 & 431.8363 \\  & \(A_{R}=100\) & 341.6468 & 404.8307 & 335.4600 & 432.9287 \\ \cline{1-1}  & \(A_{R}=500\) & 334.4003 & 402.2474 & 331.3018 & 432.8269 \\ \cline{1-1}  & \(A_{R}=1000\) & 333.7368 & 401.8458 & 330.8940 & 432.7693 \\ \hline \multirow{4}{*}{\(U\)-profile} & \(A_{R}=50\) & 367.6717 & 410.8309 & 358.5940 & 437.8143 \\  & \(A_{R}=100\) & 352.3555 & 408.3589 & 344.8158 & 437.9515 \\ \cline{1-1}  & \(A_{R}=500\) & 341.3101 & 405.0617 & 336.4925 & 437.7304 \\ \cline{1-1}  & \(A_{R}=1000\) & 340.1902 & 404.5425 & 335.6926 & 437.6768 \\ \hline \multirow{4}{*}{\(\text{Sigmoidal-profile}\)} & \(A_{R}=50\) & 384.5763 & 401.4333 & 375.2713 & 412.1084 \\ \cline{1-1}  & \(A_{R}=100\) & 366.8624 & 397.5881 & 357.3245 & 411.5575 \\ \cline{1-1}  & \(A_{R}=500\) & 354.0331 & 391.1778 & 347.7359 & 408.1837 \\ \cline{1-1}  & \(A_{R}=1000\) & 352.8324 & 390.1325 & 346.8347 & 407.5554 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Critical thermal buckling load for different distributions of the CNTs within the structure made of TI material properties considering \(L/R=1,R/h=200,N_{r}=0,N_{s}=0\). As visible from Tables 3 and 4, it is worth noticing that the thermal buckling load of nanocomposites with straight CNTs (i.e., \(w=0\)), features a monotone decrease for increasing values of \(V_{CNT}^{*}\) and \(A_{R}\). Differently, nanocomposites with wavy CNTs (i.e., \(w=0.425\)) exhibit an increasing buckling temperature for an increasing \(V_{CNT}^{*}\) and a decreasing \(A_{R}\). A clear increase in the buckling temperature is visible moving from straight CNTs to wavy CNTs, or equivalently by comparing the results for \(w=0\), and \(w=0.425\), as well as for fixed values of \(A_{R}\), \(V_{CNT}^{*}\) and fixed profiles. A meaningful effect on the thermal buckling behavior of the structure is also related to the different distributions of CNTs within the nanocomposite.