# 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
By a uniform pressure on the curved side, from \(\rm p_{y}=0\) to a final \(\rm p_{y}=25.0\); 50.0 MPa; 2. By a center point load, from \(\rm P_{c}=0\) to a final \(\rm P_{c}=20.0\); 40.0 kN, or by an external radial pressure, form \(\rm p_{r}=0\) to a final \(\rm p_{r}=2\) MPa; 3. By a uniform pressure on the curved side, from \(\rm p_{y}=0\) to a final \(\rm p_{y}=25.0\); 50.0 kPa, combined with a center point load, from \(\rm P_{c}=0\) to a final \(\rm P_{c}=20.0\); 40.0 kN, or combined with an external radial pressure, from \(\rm p_{r}=0\) to a final \(\rm p_{r}=1\) MPa; 2 MPa. The corresponding results are shown in Tables 8 a,b and 9 a,b. From Table 8 a,b, the following is observed:  
1. The uniform compressive load, the center point load, and the radial pressure load decrease the first three natural frequencies. 2. When any two types of loads are applied, the first three frequencies decrease much more. 3. It is shown that the frequencies decrease more with the increase of the loadings. By comparing Table 9 a,b, it is observed that the external radial pressure decreases the natural frequencies more than the center point load and uniform compressive load, as well as their combinations. Analyzing the stiffness parameter for the cylindrical panels, it can be verified that for the cylindrical hinged panel, subject to an axial pressure on the free curved edge, \(\rm p_{y}\), the stiffness parameter decreases. For a central point load, \(\rm P_{c}\), the stiffness parameter decreases, and for a uniform radial pressure, \(\rm p_{r}\), the stiffness parameter decreases. For the clamped panel subject to an axial pressure on the curved edge, \(\rm p_{y}\), the stiffness parameter decreases. For a central point load, \(\rm P_{c}\), the stiffness parameter decreases, and for a uniform radial pressure, \(\rm p_{r}\), the stiffness parameter decreases. Considering the gradient index \(\rm p=1.0\), the variation of the fundamental frequency versus applied type load and boundary conditions is shown in Figure 15a-c for the case of hinged cylindrical panels, and Figure 16a-c for the case of clamped cylindrical panels. For the case of the hinged panel subjected to a center point load or radial pressure, the stiffness decreases, thus the fundamental frequency decrease is very low when the center point load is very close to the limit load. Observing Figure 15c, the fundamental frequency of the hinged shell panel increases under the axial pressure load, although the stiffness decreases, which can be attributed to the effect of the deformed shape on the frequency. For the uniaxial pressure, linear buckling has been achieved at 279.6 MPa. However, using the nonlinear analysis, the maximum transverse displacement is obtained at 200 MPa. At this pressure, the panel has the highest natural frequency. Figure 16: (**a**) Clamped cylindrical panel. (**b**) Clamped cylindrical panel. (**c**) Clamped cylindrical panel. Inflection point at 140 kN. Inflection point at 2.25 MPa. Figure 15: (**a**) Hinged cylindrical panel. (**b**) Hinged cylindrical panel. Limit load: 89.92 KN. Limit load: 0.55 MPa. (**c**) Hinged cylindrical panel. For the axial compression load applied to the clamped cylindrical panel, shown in Figure 16a, the stiffness decreases and the fundamental frequency decreases under axial pressure load until the inflection point. For the same clamped cylindrical panel subjected to a center point load or a radial pressure load, the stiffness decreases, thus the frequency decreases until the inflection point.