## 5 Numerical Results  
The numerical results are compared with the main predictions from the available literature in order to demonstrate the performance of the proposed approach. Table 1 summarizes the results in terms of critical thermal load for an isotropic cylindrical shell with different thickness ratios \(h/R\) and subjected to a uniform temperature rise. A twofold material is assumed for numerical computations, as found in [71]; namely, a pure metal (i.e., an Aluminum) with Young's modulus of 70 GPa, Poisson's ratio 0.3, and coefficient of thermal expansion \(23\times 10^{-6}\), as well as a pure ceramic (i.e., an Alumina) with Young's modulus of 380 GPa, Poisson's ratio 0.3, and coefficient of thermal expansion \(7.4\times 10^{-6}\). As visible from Table 1, the results based on the proposed formulation resemble quite well those predicted in [70]. This demonstrates the accuracy of the GDQ-based solutions of the problem. More in detail, predictions by Mirzavand and Eslami [71] are always more conservative compared to our results, whereas a clear increase of the critical thermal load can be observed for increasing \(h/R\) ratios. Another parametric investigation is performed for a FGM cylindrical shell, where the outer surface is metal-rich, and the inner surface is ceramic-rich (Ti-6 Al-4V/ZrO\({}_{2}\)). The material properties in terms of Young's modulus and thermal expansion coefficient, are expressed as a nonlinear function of temperature, as suggested in [72]. Table 2 reports the thermal buckling loads \(T_{cr}\) (expressed in K) for the simply supported FGM cylindrical shells with different values of volume fraction index \(P\), and subjected to a uniform temperature rise. \begin{table}
\begin{tabular}{c c c c} \hline \(\mathbf{p}\) & **Method** & **TD** & **TI** \\ \hline  & Present & 354.5263 & 363.8471 \\
0.2 & [73] & 365.0509 & 378.5620 \\  & [74] & 353.7 & - \\ \hline  & Present & 360.8525 & 371.0475 \\
0.5 & [73] & 372.6016 & 387.4075 \\  & [74] & 360.119 & - \\ \hline  & Present & 369.8812 & 380.9365 \\
1 & [73] & 383.4589 & 399.5912 \\  & [74] & 368.785 & - \\ \hline  & Present & 384.4789 & 395.7673 \\
2 & [73] & 401.2534 & 417.9189 \\  & [74] & 383.244 & - \\ \hline  & Present & 395.9562 & 406.1519 \\
3 & [73] & 415.4941 & 430.7708 \\  & [74] & 394.319 & - \\ \hline  & Present & 413.3029 & 419.3226 \\
5 & [73] & 437.9041 & 447.0620 \\  & [74] & 411.669 & - \\ \hline \end{tabular}
\end{table} 
Table 2: Buckling temperatures \(T_{cr}\) (in K) as computed by the Generalized Differential Quadrature (GDQ) approach, compared to predictions by [73; 74] for a Functionally Graded Material (FGM) cylindrical shell, with temperature-dependent (TD) or temperature-independent (TI) material properties considering \(H=1/400\) and \(L^{2}/H=300\). \begin{table}
\begin{tabular}{c c c c} \hline \(\mathbf{h/R}\) & **Method** & **Pure Metal (Aluminum)** & **Pure Ceramic (Alumina)** \\ \hline
0.003 & Present & 55.1114 & 171.2921 \\
[71] & 52.1867 & 181.375 \\ \hline
0.005 & Present & 91.7741 & 285.2439 \\  & [71] & 90.8012 & 289.537 \\ \hline
0.007 & Present & 128.4551 & 399.2525 \\  & [71] & 124.447 & 402.683 \\ \hline
0.01 & Present & 182.6926 & 567.8284 \\  & [71] & 187.33 & 597.253 \\ \hline
0.012 & Present & 219.6345 & 682.6477 \\  & [71] & 216.007 & 688.039 \\ \hline
0.015 & Present & 273.1372 & 848.9400 \\  & [71] & 268.953 & 862.72 \\ \hline \end{tabular}
\end{table} 
Table 1: Thermal buckling \(\Delta T_{cr}\) as predicted by the present work and [71] for a uniform temperature rise and \(L/R=1\). The numerical analysis is repeated for TD and TI material properties, while evaluating the accuracy of our results with respect to the ones predicted in [73; 74]. Based on this comparative evaluation, our results shows a very good agreement with estimations by [73; 74], and fall always between a range of values whose lower limit is determined by [74] and the upper limit is determined by [73], respectively. As also visible in Table 2, the thermal buckling load seems to increase with the CNT volume fraction \(p\).