## 5 Numerical Results
A sigmoidal profile seems to yield to the most conservative results, at least for wavy CNTs, both for TI and TD material properties, whereas the highest results are reached for an asymmetric \(\wedge\)-distribution. Based on a comparative evaluation of the results in Tables 3 and 4, it is clearly visible as TD material properties yield always to more conservative buckling temperatures, for the same distributions, as well as for fixed \(A_{R}\), \(w\), and \(V_{CNT}^{*}\). This underlines the great importance of considering the actual geometry and distribution of CNTs within a nanocomposite structure. A second parametric investigation considers the influence of the \(R/h\) geometrical ratio on the thermal buckling load, for different CNT profiles, and the double condition of TI and TD material properties. Figures 3 and 4 show the numerical plots of \(\Delta T_{cr}\) vs. \(R/h\), as obtained for a fixed \(L/R=2\), \(V_{CN}^{*}=0.17\), \(A_{R}=1000\), and for two different waviness indexes \(w=0\), and \(w=0.425\), respectively. As visible from Figures 3 and 4, all the curves obtained for TD materials properties lie always below those ones based on TI material properties, independent of the selected distribution. Moreover, the critical thermal buckling load \(\Delta T_{cr}\) decreases monotonically for increasing \(R/h\) ratios, reaching the lowest estimations for a symmetric \(O-\) distribution or a sigmoidal distribution, depending on whether straight (\(w=0\)) or wavy (\(w=0.425\)) CNTs are dispersed within the nanocomposite. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Profiles**} & \multirow{2}{*}{\(A_{R}\)} & \multicolumn{2}{c}{\(V_{CNT}^{*}=0.12\)} & \multicolumn{2}{c}{\(V_{CNT}^{*}=0.28\)} \\ \cline{3-6}  & & \(w=0\) & \(w=0.425\) & \(w=0\) & \(w=0.425\) \\ \hline \multirow{4}{*}{\(\wedge\)-profile} & \(A_{R}=50\) & 350.5315 & 401.0361 & 342.4973 & 433.3668 \\  & \(A_{R}=100\) & 338.9949 & 398.8109 & 335.5588 & 433.8537 \\  & \(A_{R}=500\) & 333.2777 & 395.5772 & 331.5408 & 434.1682 \\  & \(A_{R}=1000\) & 332.7525 & 395.0126 & 330.9463 & 434.1723 \\ \hline \multirow{4}{*}{\(X\)-profile} & \(A_{R}=50\) & 367.6339 & 403.9401 & 358.4562 & 436.5301 \\  & \(A_{R}=100\) & 350.0884 & 400.3097 & 345.5374 & 434.9102 \\  & \(A_{R}=500\) & 340.1044 & 394.5435 & 338.5318 & 431.6631 \\  & \(A_{R}=1000\) & 339.1756 & 393.5870 & 337.8357 & 431.0726 \\ \hline \multirow{4}{*}{\(O\)-profile} & \(A_{R}=50\) & 350.8708 & 398.4375 & 339.8652 & 420.952 \\  & \(A_{R}=100\) & 338.0843 & 396.1992 & 332.5311 & 421.3494 \\  & \(A_{R}=500\) & 331.6981 & 393.3625 & 328.9072 & 420.6853 \\  & \(A_{R}=1000\) & 331.1157 & 392.9302 & 328.5520 & 420.5504 \\ \hline \multirow{4}{*}{\(U\)-profile} & \(A_{R}=50\) & 360.7792 & 401.5390 & 352.3864 & 425.8502 \\  & \(A_{R}=100\) & 347.2884 & 398.8054 & 340.2270 & 425.4899 \\  & \(A_{R}=500\) & 337.3306 & 395.3611 & 333.1242 & 424.8383 \\  & \(A_{R}=1000\) & 336.3705 & 394.8309 & 332.4403 & 424.7290 \\ \hline \multirow{4}{*}{Sigmoidal-profile} & \(A_{R}=50\) & 376.4948 & 394.1399 & 366.8533 & 403.9861 \\  & \(A_{R}=100\) & 359.9199 & 390.0668 & 351.1715 & 402.9769 \\ \cline{1-1}  & \(A_{R}=500\) & 348.5575 & 383.5699 & 342.9259 & 399.2144 \\ \cline{1-1}  & \(A_{R}=1000\) & 347.5143 & 382.5242 & 342.1573 & 398.5467 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Critical thermal buckling load for different distributions of the CNTs within the structure made of TD properties considering \(L/R=1\), \(R/h=200\), \(N_{r}=0\), \(N_{s}=0\). Figure 5a-e also represent the variation of the thermal buckling load \(\Delta T_{cr}\) vs. \(R/h\) for different distributions of straight CNTs within the thickness and for increasing aspect ratios \(A_{R}\) from 50 up to 1000. A clear reduction of the thermal buckling load is always reached for increasing aspect ratios, such that, for each given distribution and aspect ratio \(A_{R}=50\) we obtain the highest curves, while obtaining the lowest estimations for \(A_{R}=1000\).