# Influence of Applied Loads on Free Vibrations of Functionally Graded Material Plate-Shell Panels
## 1 Introduction  
Plate/shell panels have applications in a wide range of engineering situations, such as aeronautics and aerospace structural systems, among many others. Structures made of composite materials have been widely used to satisfy high performance demands. In such structures, stress singularities may occur at the interface between two different materials. In functionally graded structures, the smooth and continuous variation of the properties from one surface to the other eliminates abrupt changes in the stress and displacement distributions. In addition, a ceramic phase with low thermal conductivity can resist a high thermal environment, while the metal phase is strong with a mechanical load.  
Thousands of studies of plate/shell panels composite structures have been published in the last five decades. More recently, a great number of works have also been published in functionally graded material (FGM) structures. Here, some representative works involving plate/shell panels and continuous variation of the material properties through the thickness direction, used to validate the static nonlinear and linear buckling and free vibrations analyses, are mentioned [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13]. The aim and the novelty of this paper is to study the influence of external applied loads, which give the panels a state of stress and a slightly different geometry in comparison with the initial panel on the free vibration analysis of the FGM plates-shells, considering the geometrically nonlinear behavior.  
[PERSON] and [PERSON] [1] addressed the large deflections of FGM plates/shells, [PERSON] and [PERSON] [2; 3] studied the free vibrations and geometrically nonlinear analysis of FGM shells, [PERSON] et al. [4] presented a formulation for small strains and large displacements, and [PERSON] and [PERSON] [5] performed a geometrically nonlinear analysis of FGM shells. [PERSON] et al. [6] presented the stability equations of FGM plates, while [PERSON] and [PERSON] [7] and [PERSON] et al. [8] addressed the response of FGM plates in bending and free vibrations. [PERSON] et al. [9], [PERSON] et al. [10; 11], and [PERSON] et al. [12] presented models for a geometricallynonlinear analysis of FGM plates and shells. [PERSON] et al. [13] studied the buckling of FGM plates and shells.  
In this paper, the influence of the applied loads on the natural frequencies of functionally graded material plate/shell panels subjected to general loadings is analyzed. The formulation considers the following: geometrically nonlinear deformation with the through-thickness stress distribution calculations, linear buckling, and free vibration analyses. A finite element model based on a higher order shear deformation theory using a non-conforming flat triangular plate/shell element with three nodes with eight degrees of freedom per node was adopted to study the plate/shell structures. This finite element model was revealed to be very efficient in the analysis of the proposed structures--it is easy to model the geometry, there is a low number of total degrees of freedom, there is good accuracy of the results, and the model is fast.  
The solutions of some illustrative examples were performed, and the results are presented and discussed.