# 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
In addition, it is \begin{table}
\begin{tabular}{c c c c c} \hline  & & & & \\ \hline
**Mode** & **Plate Unloaded** & **p\({}_{\mathbf{x}}\) = 2 MPa** & **p\({}_{\mathbf{z}}\) = 20 kPa** & **p\({}_{\mathbf{x}}\) = 2 MPa + p\({}_{\mathbf{z}}\) = 20 kPa** \\ \hline
1 & 142.325 & 141.442 & 144.252 & 143.185 \\
2 & 291.271 & 290.132 & 292.259 & 290.786 \\
3 & 292.066 & 291.098 & 293.134 & 292.019 \\
4 & 432.917 & 431.828 & 433.395 & 432.224 \\ \hline  & & & & \\ \hline
**Mode** & **Plate Unloaded** & **p\({}_{\mathbf{x}}\) = 4 MPa** & **p\({}_{\mathbf{z}}\) = 40 kPa** & **p\({}_{\mathbf{x}}\) = 4 MPa + p\({}_{\mathbf{z}}\) = 40 kPa** \\ \hline
1 & 142.325 & 140.552 & 149.526 & 147.391 \\
2 & 291.271 & 288.854 & 295.034 & 291.357 \\
3 & 292.066 & 290.042 & 295.806 & 293.672 \\
4 & 432.917 & 430.741 & 435.552 & 432.328 \\ \hline  & & & & \\ \hline
**Mode** & **Plate Unloaded** & **p\({}_{\mathbf{x}}\) = 10 MPa** & **p\({}_{\mathbf{z}}\) = 60 kPa** & **p\({}_{\mathbf{x}}\) = 10 MPa + p\({}_{\mathbf{z}}\) = 60 kPa** \\ \hline
1 & 142.325 & 137.843 & 157.010 & 154.310 \\
2 & 291.271 & 284.797 & 299.130 & 292.997 \\
3 & 292.066 & 287.177 & 299.768 & 296.824 \\
4 & 432.917 & 427.461 & 438.402 & 433.154 \\ \hline \end{tabular}
\end{table} 
Table 4: (**a**) Natural frequencies [Hz] for a fully clamped plate under different types of loading (p = 1). (**b**) Natural frequencies [Hz] for a fully clamped plate under different types of loading (p = 1). (**c**) Natural frequencies [Hz] for a fully clamped plate under different types of loading (p = 1). Figure 10: Variation of the non-dimensional frequency \(\overline{\omega}\) of the simply supported plate versus the gradient index p.  
shown that the size of the decreasing/increasing of frequency is directly connected with the increase of the loading. By analyzing the stiffness parameter for the plates, it can be verified that for the simply supported plate with axial pressure, \(\mathrm{p_{x^{\prime}}}\) the stiffness parameter decreases, and for a uniform transverse pressure, \(\mathrm{p_{z}}\), the stiffness parameter increases. For the clamped plate with axial pressure, \(\mathrm{p_{x}}\), the stiffness parameter decreases, and for a uniform transverse pressure, \(\mathrm{p_{z}}\), the stiffness parameter increases. As for the cases of center point load and transverse pressure load, in Figure 11a,b, the load-displacement paths for the case of axial pressure loads applied to clamped and simply supported plates are presented. For the case of the clamped plates, an imperfect initial plate is considered, which consists of, in general, an initial center displacement \(w_{\mathrm{c}}^{0}=h/100\). Figure 11: (**a**) Load-displacement path for the clamped plate. (**b**) Load-displacement path for the SSSS plate. (**c**) Clamped plate under \(\mathrm{p_{x}}\). (**d**) Clamped plate under \(\mathrm{p_{z}}\). (**e**) Simply supported plate under \(\mathrm{p_{x}}\). (**f**) Simply supported plate under \(\mathrm{p_{z}}\). From Figure 11a, it is observed that the minimum stiffness occurs for the load level of 138 MPa for the clamped plate, corresponding to the nonlinear buckling, and from Figure 11b, an inflection point near the load level of 35 MPa is observed. Considering the gradient index p = 1.0, the variation of the fundamental frequency versus applied type load is shown in Figure 11c,d for the case of the clamped plate. The variations match accordingly with Tables 3 and 4. Proceeding in the same way, the influence of magnitude and the type of applied loads on the fundamental frequency of simply supported plates with all the edges restrained in u and v displacements is shown in Figure 11e,f. The free vibration frequencies of the first and second modes are influenced by the magnitude and type of loading, as shown in Figure 11f, for the case of the simply supported plate. From Figure 11c-f, it can be verified that the variation of the fundamental frequency of the simply supported and clamped plates decreases as the stiffness of plate decreases under increasing axial pressure.