# Influence of Applied Loads on Free Vibrations of Functionally Graded Material Plate-Shell Panels
## 5 Applications
### Free Vibrations of Hinged and clamped FGM Cylindrical Panels  
First, for comparison purposes, we utilized a clamped FGM cylindrical panel with the following geometry: R/10, a/h = 100, E\({}_{\text{c}}\) = 380.0 \(\times\) 109 N/m\({}^{2}\), \(\
u_{c}=0.3\), \(\rho_{\text{c}}\) = 3000 kg/m\({}^{3}\), E\({}_{\text{m}}\) = 70.0 \(\times\) 109 N/m\({}^{2}\), \(\
u_{m}=0.3\), \(\rho_{\text{c}}\) = 2707 kg/m\({}^{3}\). The non-dimensional fundamental frequency is given by \(\overline{\omega}=\omega a^{2}\sqrt{\rho_{m}/D_{m}}\), with \(D_{m}=E_{m}H^{3}/12\left(1-\
u_{m}^{2}\right)\). The results obtained with the present model are presented and compared in Table 6, with good agreement, with the alternative solutions of [PERSON] and [PERSON] (2018) obtained using a higher-order formulation and a C\({}_{0}\) finite element and the solutions published by [PERSON] et al. (2019) obtained using a higher-order shear deformation theory and radial basis functions. The graphical representation of the non-dimensional fundamental frequency versus the p-index is presented in Figure 14.  
The natural vibrations of the hinged and clamped FGM cylindrical shells from Section 5.2, with the same geometry and material properties, are now analyzed. The first three natural frequencies are obtained for the initial geometry (unloaded panel) considering different gradient indexes p. Next, the cylindrical panel is loaded by three kinds of loads. The applied loads produced displacements--new geometry--and a stress distribution over the plate/shell structure, which depends on the gradient index p. The new geometry and the state of stresses, which influences the stiffness of the structure at each load level, both have an influence on the values of the natural frequencies. In Tables 7 a,b, 8 a,b and 9 a,b, the first three natural frequencies are shown for different gradient indexes. The graphic representation of the influence of applied loads on the fundamental frequency is shown in Figure 14 for the gradient index p = 1. The loadings are defined in Figure 6. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & & \multicolumn{3}{c}{**Power Law Index**} \\ \hline
**Mode** & **Source** & **Ceramic** & **p = 0.5** & **p = 1.0** \\ \hline  & PM & 102.807 & 86.285 & 77.186 \\
1 & Ref. (2018) & 102.923 & 87.545 & 77.077 \\  & Ref. (2018) & 102.787 & 85.478 & 77.638 \\ \hline  & & & & \\ \end{tabular}
\end{table} 
Table 6: Non-dimensional fundamental frequency \(\overline{\omega}\) for the clamped cylindrical panel with different gradient indexes. Figure 14: Variation of the non-dimensional frequency \(\overline{\omega}\) versus the gradient index p for a clamped cylindrical panel. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & & & & \\ \hline \hline \multirow{2}{*}{**Index p**} & \multirow{2}{*}{**Frequency**} & \multirow{2}{*}{**Hinged**} & \multirow{2}{*}{**Hinged**} & \multirow{2}{*}{**Py = 20.0 MPa**} & \multirow{2}{*}{**Hinged**} & \multirow{2}{*}{**Py = 20 MPa**} \\  & & & & & \\ \hline \multirow{2}{*}{0.5} & 1 & 313.274 & 316.566 & 295.679 & 300.890 \\  & 2 & 351.139 & 355.694 & 340.132 & 345.626 \\  & 3 & 479.608 & 479.055 & 465.256 & 465.001 \\ \hline \multirow{2}{*}{1.0} & 1 & 305.210 & 310.625 & 286.167 & 291.501 \\  & 2 & 343.077 & 348.121 & 331.984 & 337.146 \\  & 3 & 473.809 & 473.269 & 458.445 & 458.245 \\ \hline \multirow{2}{*}{5.0} & 1 & 297.077 & 302.268 & 275.787 & 281.273 \\  & 2 & 335.446 & 339.251 & 321.954 & 326.915 \\  & 3 & 471.027 & 470.347 & 454.304 & 454.022 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: (**a**) Natural frequencies [Hz] for hinged FGM cylindrical panel, considering different gradient indexes and loading types. (**b**) Natural frequencies [Hz] for hinged FGM cylindrical panel, considering different gradient indexes and loading types.