# Closed Form Solutions for Thermal Buckling of Functionally Graded Rectangular Thin Plates
## 4 Numerical Results and Discussion
Due to the small difference between the temperature distribution of the linear temperature field and nonlinear temperature field, \(\Delta T_{\rm cr}\) under these two temperature fields is close. Furthermore, \(\Delta T_{\rm cr}\) under a uniform temperature field is obviously lower than that under the other two temperature fields, so a uniform temperature field affects \(\Delta T_{\rm cr}\) more significantly than the other two. Table 12 presents \(\Delta T_{\rm cr}\) of Type-I FGM thin plate with different initial temperatures \(T_{0}\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(n\) & **Method** & _a/b = 1_ & _a/b = 2_ & _a/b = 3_ & _a/b = 4_ & _a/b = 5_ \\ \hline \multirow{4}{*}{1} & Present & 7.9426 & 19.8516 & 39.6876 & 67.4319 & 103.0568 \\  & CPT [16] & 7.9437 & 19.8594 & 39.7188 & 67.5220 & 103.2690 \\  & FPT [16] & 7.9400 & 19.8358 & 39.6248 & 67.2506 & 102.6355 \\  & TSDT [16] & 7.9400 & 19.8358 & 39.6248 & 67.2506 & 102.6356 \\  & SSDT [16] & 7.9400 & 19.8359 & 39.6249 & 67.2510 & 102.6365 \\ \hline \multirow{4}{*}{2} & Present & 7.0415 & 17.5995 & 35.1851 & 59.7816 & 91.3644 \\  & CPT [16] & 7.0426 & 17.6065 & 35.2130 & 59.8621 & 91.5538 \\  & FPT [16] & 7.0392 & 17.5853 & 35.1285 & 59.6184 & 91.9850 \\  & TSDT [16] & 7.0390 & 17.5840 & 35.1234 & 59.6037 & 90.9508 \\  & SSDT [16] & 7.0390 & 17.5840 & 35.1233 & 59.6034 & 90.9501 \\ \hline \multirow{4}{*}{5} & Present & 7.2645 & 18.1563 & 36.2964 & 61.6659 & 94.2363 \\  & CPT [16] & 7.2657 & 18.1642 & 36.3285 & 61.7585 & 94.4542 \\  & FPT [16] & 7.2615 & 18.1380 & 36.2236 & 61.4559 & 93.7481 \\  & TSDT [16] & 7.2606 & 18.1327 & 36.2025 & 61.3951 & 93.6069 \\  & SSDT [16] & 7.2606 & 18.1324 & 36.2014 & 61.3921 & 93.5999 \\ \hline \multirow{4}{*}{10} & Present & 7.4679 & 18.6645 & 37.3115 & 63.3884 & 96.8646 \\  & CPT [16] & 7.4692 & 18.6731 & 37.3463 & 63.4888 & 97.1005 \\ \cline{1-1}  & FPT [16] & 7.4644 & 18.6427 & 37.2246 & 63.1378 & 96.2820 \\ \cline{1-1}  & TSDT [16] & 7.4634 & 18.6366 & 37.2006 & 63.0687 & 96.1213 \\ \cline{1-1}  & SSDT [16] & 7.4634 & 18.6365 & 37.2001 & 63.0673 & 96.1183 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Critical buckling temperature \(\Delta T_{\rm cr}\) of SSSS type-II FGM plate with different \(a/b\) under a uniform temperature field (\(a/h=100\)). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & _n = 0_ & _n = 0.5_ & _n = 1_ & _n = 2_ & _n = 5_ & _n = 10_ \\ \hline Present & 32.355 & 18.332 & 15.032 & 13.326 & 13.748 & 14.133 \\ Kiani [5] & 33.628 & 19.053 & 15.623 & 13.850 & 14.289 & 14.690 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Critical buckling temperature \(\Delta T_{\rm cr}\) of CCCC type-II FGM rectangular plate under a uniform temperature field (\(a/h=100\), \(a/b=2\)). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**B.C.** & **T.F.** & _a/h = 20_ & _a/h = 30_ & _a/h = 40_ & _a/h = 50_ & _a/h = 60_ & _a/h = 70_ \\ \hline \multirow{3}{*}{CSCS} & uniform & 547.3748 & 283.8704 & 171.2913 & 113.7826 & 80.7593 & 60.1576 \\  & linear & 1133.6559 & 581.2390 & 352.4874 & 235.3624 & 167.7132 & 125.2884 \\  & nonlinear & 1182.5836 & 607.9852 & 369.1144 & 246.6213 & 175.8071 & 131.3707 \\ \hline \multirow{3}{*}{CSCC} & uniform & 742.2713 & 401.3746 & 247.6355 & 166.7565 & 119.4051 & 89.4729 \\  & linear & 1594.4982 & 822.6864 & 507.5211 & 343.2718 & 246.8452 & 185.6015 \\  & nonlinear & 1655.7434 & 859.5887 & 531.0572 & 359.4812 & 258.6364 & 194.5373 \\ \hline \multirow{3}{*}{CCCC} & uniform & 1013.9656 & 602.6029 & 384.3119 & 264.3487 & 192.0395 & 145.3645 \\  & linear & 2230.3048 & 1256.9658 & 787.2933 & 541.4994 & 394.6294 & 299.7612 \\ \cline{1-1}  & nonlinear & 2275.4643 & 1310.1371 & 822.7439 & 566.5211 & 413.1558 & 313.9882 \\ \hline \hline \end{tabular} Note: ‘T.F.’ represents Temperature Field.  
\end{table} 
Table 10: Critical buckling temperature \(\Delta T_{\rm cr}\) of type-I FGM rectangular thin plate with different \(a/h\) and various B.C. (\(a/b=2\), \(n=2\), \(T_{0}=300\) K).