## 5 Numerical Examples and Results
### Thermal Buckling Analysis
Furthermore, the critical buckling temperatures in the same instance are higher for nonlinear than the linear temperature change through the plate thickness. Figures 4b and 5b illustrate the impact of ratio a/h on the critical buckling temperature for case of linear and nonlinear temperature change through the plate thickness. This diagram exhibits the values of \(p=0\), \(1\), \(5\), \(10\) and it is evident that the most prominent curve is the one whose value is \(p=0\), while the remaining curves lie over each other when the value of ratio is a/h \(>15\). The impact of the elastic foundation on the critical buckling temperature for case of linear and nonlinear temperature change through the plate thickness is shown in Figure 6a,b. It should be underlined that results are presented for the shape function is No.10 because of its analytical integration. Numerical integration does not have to be employed. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(P\)} & \multirow{2}{*}{**Source**} & \multicolumn{5}{c}{\(\Delta\mathbf{t_{cr}}\)} \\ \cline{3-8}  & & \(\mathbf{k_{0}=0}\), \(\mathbf{k_{1}=0}\) & \(\mathbf{k_{0}=10}\) & \(\mathbf{k_{1}=0}\) & \(\mathbf{k_{0}=10}\), \(\mathbf{k_{1}=10}\) \\ \cline{3-8}  & & a/h = 10 & a/h = 20 & a/h = 10 & a/h = 20 & a/h = 10 & a/h = 20 \\ \hline \multirow{8}{*}{10} & [55] & 2.0971 & 0.5433 & 2.2923 & 0.5921 & 6.1448 & 1.5552 \\  & Present study & 2.0978 & 0.5434 & 2.2929 & 0.5922 & 6.1455 & 1.5553 \\  & No. 1 & 2.0976 & 0.5433 & 2.2928 & 0.5921 & 6.1453 & 1.5553 \\  & No. 2 & 2.0976 & 0.5433 & 2.2928 & 0.5921 & 6.1453 & 1.5553 \\  & No. 3 & 2.0971 & 0.5433 & 2.2923 & 0.5921 & 6.1448 & 1.5552 \\  & No. 4 & 2.101 & 0.5436 & 2.2961 & 0.5924 & 6.1487 & 1.5555 \\  & No. 5 & 2.0976 & 0.5433 & 2.2928 & 0.5921 & 6.1453 & 1.5553 \\  & No. 6 & 2.1002 & 0.5435 & 2.2954 & 0.5923 & 6.1479 & 1.5554 \\  & No. 7 & 2.0973 & 0.5433 & 2.2925 & 0.5921 & 6.145 & 1.5552 \\  & No. 8 & 2.0973 & 0.5433 & 2.2925 & 0.5921 & 6.145 & 1.5552 \\  & No. 9 & 2.0977 & 0.5433 & 2.2928 & 0.5921 & 6.145 & 1.5552 \\  & No. 10 & 2.0977 & 0.5433 & 2.2928 & 0.5921 & 6.1454 & 1.5553 \\  & No. 11 & 2.1005 & 0.5435 & 2.2927 & 0.5923 & 6.1482 & 1.5555 \\  & No. 12 & 2.0972 & 0.5433 & 2.2924 & 0.5921 & 6.1449 & 1.5552 \\  & No. 13 & 2.0975 & 0.5433 & 2.2927 & 0.5921 & 6.1452 & 1.5553 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._  
Figure 4: Impact of the ratio a/b and a/h on the critical buckling temperature \(\Delta\mathbf{t_{cr}}\) for case of linear temperature change across plate thickness: (**a**) a/h = 10, \(\mathbf{k_{0}=10}\), \(\mathbf{k_{1}=10}\); (**b**) a/b = 1, \(\mathbf{k_{0}=10}\), \(\mathbf{k_{1}=10}\). The numerical calculations were done to figure out the influence of every elastic foundation parameter (k\({}_{0}\) and k\({}_{1}\)). This was achieved by varying one of the parameters and setting the other one as a constant. Similarly to previous cases, the analysis is conducted for a linear and nonlinear rise in temperature and the findings are illustrated in Figures 7a and 8a (fixed value k\({}_{1}\) and variation of k\({}_{0}\)) and also in Figures 7b and 8b (fixed value k\({}_{0}\) and variation of k\({}_{1}\)). The Figures provide information about the curves that represent critical buckling temperature changes and their rapid rise that occurs because of the change of coefficient k\({}_{1}\), rather than when coefficient k\({}_{0}\) changes. Another important observation is that the curve with the most rapid rise has the value \(p=10\). Figure 5: Impact of the ratio a/b and a/h on the critical buckling temperature \(\Delta\)\({}_{\rm cr}\) for case of nonlinear temperature change across plate thickness: (a) a/h = 10, s = 3, k\({}_{0}\) = 10, k\({}_{1}\) = 10; (b) a/b = 1, s = 3, k\({}_{0}\) = 10, k\({}_{1}\) = 10. Figure 6: Impact of the power law index p and parameter s on the critical buckling temperature \(\Delta\)\({}_{\rm cr}\) for case of linear and nonlinear temperature change across plate thickness: (a) a/h = 10, a/b = 1, s = 1; (b) a/h = 10, a/b = 1, s = 3.