# Influence of Applied Loads on Free Vibrations of Functionally Graded Material Plate-Shell Panels
## 5 Applications
### Free Vibrations of a Simply Supported FGM Square Plate  
The free vibrations of a simply supported FGM square plate are now investigated. The material properties for the zirconia and aluminum are, respectively, the following: \(\mathrm{E_{c}=151.0\times 109~{}N/m^{2}}\), \(\upsilon_{c}=0.3\), \(\rho_{c}=5700~{}\mathrm{kg/m^{3}}\), and \(\mathrm{E_{m}=70.0\times 109~{}N/m^{2}}\), \(\upsilon_{m}=0.3\), \(\rho_{m}=2702~{}\mathrm{kg/m^{3}}\). The side dimension is \(a=0.05~{}\mathrm{m}\), and the thickness is H = 0.01 m. The non-dimensional frequencies \(\overline{\omega}=\omega h\sqrt{\rho_{m}/E_{m}}\) obtained with the present model (PM) are presented in Table 3 and compared with the alternative solutions obtained by [PERSON] et al. [6] using a NURBS-based isogeometric approach and an HSDT model. For pure ceramic and pure metal, the results are very similar. The discrepancies between the results for the FGM plates are attributed to the different homogenization considered by [PERSON] et al. [6] following the Mori-Tanaka scheme, while the present model uses the rule of mixtures. The corresponding graphical representation of the non-dimensional frequencies obtained by the present model is presented in Figure 10 for vibration modes 1 and 2. The next investigation is to achieve how the free vibrations of an FGM plate with all the edges clamped and a power law index \(\mathrm{p}=1.0\) are affected by two types of loading: the in-plane loading and transverse loading. The material properties are as follows: \(\mathrm{E_{c}=380.0\times 109~{}N/m^{2}}\), \(\upsilon_{c}=0.3\), \(\rho_{m}=3000~{}\mathrm{kg/m^{3}}\), \(\mathrm{E_{m}=70.0\times 109~{}N/m^{2}}\), \(\upsilon_{m}=0.3\), \(\rho_{m}=2702~{}\mathrm{kg/m^{3}}\). The geometry is defined by \(a=1.0~{}\mathrm{m}\) and H = 0.01 m. The results obtained are given in Table 4a-c. First, the free vibrations are calculated for the unloaded plate, and afterwards, for the three cases of loading:  
1. Uniaxial in-plane uniform load from \(\mathrm{p_{x}=0}\) to a final \(\mathrm{p_{x}=2.0}\), 4.0, and 10.0 MPa, and the geometrically nonlinear behavior is considered;
2. Transverse uniform load from \(\mathrm{p_{z}=0}\) to a final \(\mathrm{p_{z}=20.0}\), 40.0, and 60.0 kPa;  
Figure 9: Variation of the non-dimensional buckling loads versus the gradient index \(\mathrm{p}\) for a clamped plate with immovable edges. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Mode** & **Source** & **Ceramic** & **p = 1.0** & **p = 2.0** & **Metal** \\ \hline \multirow{3}{*}{1} & PM & 0.2462 & 0.2275 & 0.2254 & 0.2112 \\  & Ref. [6] & 0.2461 & 0.2185 & 0.2190 & 0.2113 \\  & PM & 0.4539 & 0.4342 & 0.4237 & 0.3899 \\  & Ref. [6] & 0.4539 & 0.4118 & 0.4039 & 0.3897 \\  & PM & 0.4539 & 0.4342 & 0.4237 & 0.3899 \\  & Ref. [6] & 0.4539 & 0.4118 & 0.4039 & 0.3897 \\  & PM & 0.5379 & 0.5013 & 0.4930 & 0.4620 \\  & Ref. [6] & 0.5385 & 0.4794 & 0.4768 & 0.4623 \\  & PM & 0.5379 & 0.5010 & 0.4930 & 0.4620 \\  & Ref. [6] & 0.5385 & 0.4794 & 0.4768 & 0.4623 \\  & PM & 0.5379 & 0.5010 & 0.4930 & 0.4620 \\  & Ref. [6] & 0.5385 & 0.4794 & 0.4768 & 0.4623 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Non-dimensional frequencies \(\overline{\omega}\) for simply supported plate with different gradient indexes. Here, \(p_{z}\) is uniformly distributed pressure applied at the lower surface of the plate with the positive direction of z. \(p_{x}\) is pressure applied at the edge \(x=0\) with the positive direction of x axis, in the usual way. In this context, \(P_{c}\) will be a concentrated load applied to the center of the plate aligned with the positive direction of axis z. From Table 4 a-c, it is seen that the uniform compressive loading decreases the natural frequencies, while the uniform transverse pressure increases the natural frequencies. In addition, it is observed when both types of loads are applied, the first frequency increases, but less so than when only the transverse pressure is applied, and the other frequencies increase or decrease depending on the values of two types of loadings.