# Influence of Applied Loads on Free Vibrations of Functionally Graded Material Plate-Shell Panels
## 5 Applications
### Critical Loads of a Clamped FGM Square Plate under Uniaxial In-Plane Applied Load  
The linear buckling loads are obtained first in order to know which values should be considered for the calculations of the free vibrations under uniaxial in-plane applied load.  
For the purpose of comparison, a square (a \(\times\) a) FGM plate with all the edges movable and clamped, made of a mixture of zirconia and aluminum (Al/ZrO\({}_{2}\)), is considered. The material properties for the zirconia and aluminum are, respectively, \(\mathrm{E_{c}}\) = 151.0 \(\times\) 109 N/m\({}^{2}\), \(\
u_{c}\) = 0.3, \(\mathrm{E_{m}}\) = 70.0 \(\times\) 109 N/m\({}^{2}\), and \(\
u_{m}\) = 0.3; the side dimensions are a = 0.2 m and thickness H = 0.01 m. The FGM plates are under uniaxial compression, and the power law index-p ranges from full ceramic to metal. The critical buckling parameters, defined as \(\lambda_{cr}=N_{xx}\ a^{2}/D_{0}\), with \(D_{0}=E_{m}\ h^{3}/12\big{(}1-\
u^{2}\big{)}\), are shown in Table 1. From this table, an excellent agreement between the results (for the first mode) obtained is observed with the present model (PM) compared with those obtained by [PERSON] et al. [21] using the MITC4 model and by [PERSON] et al. [13]. For benchmarking purposes, the eigenvalues of some higher modes are also presented. Again, as \(\mathrm{E_{c}}\) > \(\mathrm{E_{m}}\), there is a decrease in the buckling loads when the p-index increases.  
For comparison purposes, a square FGM plate with all edges movable and clamped is considered, but now with H = 0.01 m and sides with lengths of a = 1.0 m. The material properties are the following: \(E_{c}\) = 380.0 \(\times\) 109 N/m\({}^{2}\), \(\
u_{c}\) = 0.3, \(E_{m}\) = 70.0 \(\times\) 109 N/m\({}^{2}\), \(\
u_{m}\) = 0.3. The FGM plates are under uniaxial compression, and the power law index-p ranges from full ceramic to metal. The critical buckling parameter is now defined as \(\lambda_{cr}=N_{xx}\ a^{2}/E_{c}H^{3}\), and the results obtained with the present model (PM) are compared in Table 2 with the alternative solutions for the first mode, obtained analytically by [PERSON] et al. [22] using the first-order shear deformation theory. Figure 9 shows the variation of the non-dimensional buckling loads with the gradient index p.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & & & & **Power Law Index** & & \\ \hline
**Mode Source** & **Ceramic** & **p = 0.2** & **p = 0.5** & **p = 1.0** & **p = 2.0** & **p = 5.0** & **Metal** \\ \hline
1 PM & 203.833 & 181.190 & 160.204 & 142.854 & 130.327 & 120.066 & 94.492 \\
1 Ref. [21] & 203.497 & 180.331 & 159.529 & 142.533 & 130.342 & 120.471 & - \\
1 Ref. [13] & 202.738 & - & 158.889 & 141.962 & - & 119.773 & - \\
2 PM & 230.012 & 204.622 & 180.998 & 161.502 & 146.936 & 135.029 & 106.628 \\
3 PM & 372.296 & 331.653 & 293.574 & 261.502 & 237.458 & 217.276 & 172.587 \\
4 PM & 473.132 & 421.549 & 373.224 & 332.490 & 301.889 & 276.107 & 219.332 \\ \hline \end{tabular}
\end{table}
Table 1: Non-dimensional buckling loads for plates with movable and clamped edges and different gradient indexes, with a/H = 20.  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & & & **Power Law Index** & & \\ \hline
**Mode Source** & **Ceramic** & **p = 1.0** & **p = 2.0** & **p = 5.0** & **Metal** \\ \hline
1 PM & 9.187 & 4.606 & 3.571 & 3.020 & 1.690 \\
1 Ref. [22] & 9.158 & 4.618 & 3.579 & 3.034 & - \\
2 PM & 10.688 & 5.342 & 4.155 & 3.493 & 1.969 \\
3 PM & 18.099 & 9.047 & 7.036 & 5.908 & 3.334 \\ \hline \end{tabular}
\end{table}
Table 2: Non-dimensional buckling loads for square plates with clamped edges and different gradient indexes, with a/H = 100.