# Closed Form Solutions for Thermal Buckling of Functionally Graded Rectangular Thin Plates
## 4 Numerical Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**B.C.** & \(n\) & _a/h = 20_ & _a/h = 30_ & _a/h = 40_ & _a/h = 50_ & _a/h = 60_ & _a/h = 70_ & _a/h = 80_ \\ \hline \multirow{3}{*}{CSCS} & 1 & 583.7802 & 306.6174 & 186.0370 & 123.9484 & 88.1339 & 65.7278 & 50.8301 \\  & 2 & 547.3748 & 283.8704 & 171.2913 & 113.7826 & 80.7593 & 60.1576 & 46.4854 \\  & 5 & 517.0574 & 264.5071 & 158.8189 & 105.2290 & 74.5765 & 55.4993 & 42.8584 \\ \hline \multirow{3}{*}{CSCC} & 1 & 783.2272 & 431.1364 & 267.9431 & 181.1536 & 130.0341 & 97.5959 & 75.8125 \\  & 2 & 742.2713 & 401.3746 & 247.6355 & 166.7565 & 119.4051 & 89.4729 & 69.4245 \\  & 5 & 713.3132 & 376.0498 & 230.3741 & 154.5835 & 110.4567 & 82.6559 & 64.0761 \\ \hline \multirow{3}{*}{CCCC} & 1 & 1057.5407 & 640.8390 & 413.1423 & 285.7974 & 208.3541 & 158.0883 & 123.7707 \\  & 2 & 1013.9656 & 602.6029 & 384.3119 & 264.3487 & 192.0395 & 145.3645 & 113.6185 \\ \cline{1-1}  & 5 & 1030.1122 & 571.4283 & 359.7632 & 246.1036 & 178.2185 & 134.6260 & 105.0764 \\ \hline \hline \end{tabular} Note: ‘B.C.’ stands for Boundary Condition.  
\end{table} 
Table 8: Critical buckling temperature \(\Delta T_{\rm cr}\) of type-I FGM rectangular thin plate with different \(a/h\) and various B.C. under a uniform temperature field (\(a/b=2\), \(T_{0}=300\) K). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**B.C. ** & \(n\) & _a/h = 20_ & _a/h = 30_ & _a/h = 40_ & _a/h = 50_ & _a/h = 60_ & _a/h = 70_ \\ \hline \multirow{3}{*}{CSCS} & 1 & 583.7802 & 306.6174 & 186.0370 & 123.9484 & 88.1339 & 65.7278 & 50.8301 \\  & 2 & 547.3748 & 283.8704 & 171.2913 & 113.7826 & 80.7593 & 60.1576 & 46.4854 \\  & 5 & 517.0574 & 264.5071 & 158.8189 & 105.2290 & 74.5765 & 55.4993 & 42.8584 \\ \hline \multirow{3}{*}{CSCC} & 1 & 783.2272 & 431.1364 & 267.9431 & 181.1536 & 130.0341 & 97.5959 & 75.8125 \\  & 2 & 742.2713 & 401.3746 & 247.6355 & 166.7565 & 119.4051 & 89.4729 & 69.4245 \\  & 5 & 713.3132 & 376.0498 & 230.3741 & 154.5835 & 110.4567 & 82.6559 In addition, the critical temperatures are also plotted in Figures 3-6 for different thickness-length ratios \(h/a\), power law indexes \(n\), length-width ratios \(a/b\) and initial temperatures \(T_{0}\) for type-I FGM plate. Figure 3 shows the relation between \(\Delta T_{\rm cr}\) and \(h/a\) for type-I FGM plate. The critical buckling temperature \(\Delta T_{\rm cr}\) increases as \(h/a\) increases since the thicker plates have larger bending rigidities than the thinner ones. One can also see that the critical buckling temperature of rectangular CCCC plate is the highest, whereas that of the CSCS plate is the lowest; this is because the CCCC plate has the highest bending rigidity compared with the other two cases. Furthermore, the uniform temperature field influences \(\Delta T_{\rm cr}\) more remarkably than the other two temperature fields. Figure 4 demonstrates that \(\Delta T_{\rm cr}\) decreases as \(n\) decreases. It can be concluded from Equation (16) that the volume fraction of metal increases as \(n\) increases, and metallic materials have a lower critical buckling temperature than ceramic materials; this is because the thermal expansion coefficient \(\alpha\) of ceramic materials is less than that of metallic materials. Figure 5 shows the relation between \(\Delta T_{\rm cr}\) and \(a/b\) for different \(n\). It is observed that, with \(a/b\) increasing, \(\Delta T_{\rm cr}\) increases gradually whatever \(n\) is; this indicates that the longer plates have a higher critical buckling temperature for different \(n\). Homogeneous isotropic ceramic plate has the highest \(\Delta T_{\rm cr}\) and metallic plate has the lowest, and the volume fraction of the metal component increases as \(n\) increases. Figure 6 indicates that \(\Delta T_{\rm cr}\) of type-I FGM plate decreases almost linearly as \(T_{0}\) increases. For temperature-independent material, the \(\Delta T_{\rm cr}\) stays at a constant value, thus \(T_{\rm cr}\) has the same increment as that of \(T_{0}\).