# Influence of Applied Loads on Free Vibrations of Functionally Graded Material Plate-Shell Panels
## 5 Applications  
First, a few numerical applications are presented, which serve for model validation against alternative results published by other researchers in different analyses, and also allow us to calculate load-displacement paths, which will give the information about the normal and shear stresses, limit points, and inflection points. The later applications, which address the aim of this work, show the non-linear analysis by using Equations (22) and (23) to obtain the influence of the applied loading at several load levels on the free vibrations of the FGM plates and shells. The updated geometry and the tangential stiffness \([\mathbf{K}_{L}+\mathbf{K}_{\sigma}]\) matrix at each load level is calculated, and the eigenvalue problem in Equation (30) is solved for that loading condition to obtain the natural frequencies.  
Nonlinear Analysis of a Simply Supported and Clamped FGM Square Plate under Uniform Transverse Pressure  
Simply supported (SS) and clamped (CL) FGM square plates subjected to a uniform pressure load are considered in this first numerical application. The geometry is defined by the side dimension a = 1.0 m and the thickness h = 0.01 m. Three gradient indexes are considered: p = 0.5, p = 1, and p = 2. The plate, discretized through the thickness using 20 virtual layers, is made of constituents zirconia and aluminum (\(E_{c}\) = 151 \(\times\) 10\({}^{9}\) Pa, \(\upsilon_{c}\)= 0.3, \(\rho_{\mathrm{c}}\) = 5700 kg/m\({}^{3}\), \(E_{m}\) =70 \(\times\) 10\({}^{9}\) Pa, \(\
u_{m}\) = 0.3, \(\rho_{\mathrm{m}}\) = 2702 kg/m\({}^{3}\)). An incremental-iterative loading is applied, and the load-displacement paths obtained for the center of the plate are presented in Figure 3. This analysis had been validated in [PERSON] et al. [11], although for different geometry and material properties. It is observed that the displacements increase when the gradient index increases. This happens because \(E_{c}\) > \(E_{m}\), and as p increases, the volume fraction of ceramic decreases.  
For the last load increment, considering the simply supported plate, the through-the-thickness distributions of the total normal and transverse shear stresses at the Gauss point nearest to the center point of the plate are shown in Figures 4 and 5 for two different gradient indexes. From these figures, the influence of the gradient index p in the distributions of the stresses can be observed because the volume fractions of the constituents are related with the gradient index, and as expected, there are no abrupt changes in the stresses, in contrast with the structures made of composite materials. Thus, the results obtained for displacements, stresses, linear buckling, and natural frequencies of the plate-shell panels are affected by the gradient index p.  
In this work, the variation through the thickness of FGM is approximated considering a certain number of virtual layers (20 layers in the present work). All the quantities are obtained at the middle of each layer, including the stresses. The upper and lower surfaces correspond to \(\Xi/h\) = 0.5 and \(\Xi/h\) = \(-\)0.5, respectively, where the shear stresses are zero. However, as mentioned above, the first and last virtual layers are obtained at the middle of these layers, i.e., \(\Xi/h\) = 0.475 and \(\Xi/h\) = \(-\)0.475.  
Figure 4: Distribution of total normal stress \(\sigma_{\rm x}\) across the thickness.  
Figure 5: Distribution of total transverse shear stress \(\tau_{\rm xz}\) across the thickness.  
Figure 3: Load-displacement paths for different gradient indexes.