# 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
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & & & & \\ \hline \multirow{2}{*}{**Index p**} & \multirow{2}{*}{**Frequency**} & \multirow{2}{*}{**Champed**} & \multirow{2}{*}{**Clamped**} & \multirow{2}{*}{**Py = 25 MPa**} & \multirow{2}{*}{**Camped**} \\  & & & & & \\ \hline \multirow{2}{*}{0.5} & 1 & 530.354 & 525.088 & 517.085 & 512.525 \\  & 2 & 910.949 & 906.395 & 902.049 & 897.850 \\  & 3 & 950.677 & 940.616 & 944.500 & 935.275 \\ \hline \multirow{2}{*}{1.0} & 1 & 523.455 & 518.643 & 509.337 & 504.472 \\  & 2 & 899.626 & 894.811 & 890.297 & 885.837 \\  & 3 & 938.582 & 927.958 & 932.117 & 922.348 \\ \hline \multirow{2}{*}{5.0} & 1 & 515.247 & 509.832 & 500.414 & 494.868 \\  & 2 & 892.631 & 887.328 & 882.877 & 877.858 \\  & 3 & 929.081 & 917.333 & 922.329 & 911.439 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: (**a**) Natural frequencies [Hz] for clamped FGM cylindrical panel, considering different gradient indexes and loading types. (**b**)Natural frequencies [Hz] for clamped FGM cylindrical panel, considering different gradient indexes and loading types. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & & & & \\ \hline \multirow{2}{*}{**Index p**} & \multirow{2}{*}{**Frequency**} & \multirow{2}{*}{**Clamped Unloaded**} & \multirow{2}{*}{**Clamped py = 25 MPa**} & \multirow{2}{*}{**Clamped Pr = 1 MPa**} & \multirow{2}{*}{**Clamped Py = 25 MPa**} \\  & & & & & +Pr = 1 MPa \\ \hline \multirow{4}{*}{0.5} & 1 & 530.354 & 525.746 & 487.428 & 483.450 \\  & 2 & 910.949 & 906.135 & 878.406 & 875.414 \\  & 3 & 950.677 & 940.051 & 928.638 & 921.602 \\ \hline \multirow{4}{*}{1.0} & 1 & 523.455 & 518.375 & 478.441 & 474.085 \\  & 2 & 899.626 & 894.541 & 865.624 & 862.355 \\  & 3 & 938.582 & 927.362 & 915.678 & 908.076 \\ \hline \multirow{4}{*}{5.0} & 1 & 515.247 & 509.531 & 468.468 & 463.246 \\  & 2 & 892.631 & 886.980 & 857.417 & 853.489 \\  & 3 & 929.081 & 916.673 & 905.549 & 896.673 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: (a) Natural frequencies [Hz] for clamped FGM cylindrical panel, considering different gradient indexes and loading types. (b) Natural frequencies [Hz] for clamped FGM cylindrical panel, considering different gradient indexes and loading types. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & & & & \\ \hline \multirow{2}{*}{**Index p**} & \multirow{2}{*}{**Frequency**} & \multirow{2}{*}{**Clamped Unloaded**} & \multirow{2}{*}{**Clamped py = 50 MPa**} & \multirow{2}{*}{**Clamped Pr = 2 MPa**} & \multirow{2}{*}{**Clamped Py = 50 MPa**} \\  & & & & & +Pr = 2 MPa \\ \hline \multirow{4}{*}{0.5} & 1 & 530.354 & 521.489 & 453.350 & 442.269 \\  & 2 & 910.949 & 901.862 & 849.881 & 842.178 \\  & 3 & 950.677 & 930.474 & 914.233 & 895.260 \\ \hline \multirow{4}{*}{1.0} & 1 & 523.455 & 513.874 & 445.354 & 432.959 \\  & 2 & 899.626 & 890.029 & 837.117 & 828.118 \\  & 3 & 938.582 & 917.243 & 902.145 & 880.988 \\ \hline \multirow{4}{*}{5.0} & 1 & 515.247 & 504.466 & 439.559 & 423.800 \\  & 2 & 892.631 & 881.968 & 830.819 & 819.155 \\  & 3 & 929.081 & 905.471 & 894.629 & 869.139 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: _Cont._(c) By a uniform pressure on the curved side, from \(\rm p_{y}=0\) to a final \(\rm p_{y}=20.0\); 40.0 MPa, combined with a center point load, from \(\rm P_{c}=0\) to a final \(\rm P_{c}=20.0\); 40.0 kN. The geometrically non-linear behavior is considered, and an incremental-iterative process is used. The results obtained with the present model are shown in Table 7 a,b. From these tables, the following can be observed:  
1. The uniform compressive load minimally increases the first two natural frequencies but minimally decreases the third natural frequency. 2. The transverse center point load decreases the first three natural frequencies. 3. When both types of loads are applied, the first two frequencies decrease, but less than when only the center point load is applied. 4. It is shown that the frequencies decrease/increase more with the increase of the loadings. 2. Clamped FGM panel loaded: 1.