# Influence of Applied Loads on Free Vibrations of Functionally Graded Material Plate-Shell Panels
## 6 Conclusions  
A finite element model for the free vibration analysis of functionally graded material plate/shell panels under mechanical loading is presented. The loading is achieved by an incremental-iterative process to consider the geometrically nonlinear deformation of the structures.  
The finite element model is based on [PERSON]'s third-order shear deformation theory applied to a non-conforming flat triangular plate/shell element with three nodes and eight degrees of freedom per node.  
From the present applications, it is observed that the FGM plate/shell structures have a smooth variation of stresses through the thickness, and the free vibrations are influenced by the applied loads.  
For the clamped and simply supported (with immovable edges) FGM plates, the uniaxial compressive pressure load leads to a decrease in the natural frequencies, while the uniform transverse pressure load leads to an increase in the frequencies. For the clamped FGM cylindrical panels, both types of loads lead to a decrease in the natural frequencies of the free vibration. In contrast, for the hinged FGM cylindrical panels, a uniform uniaxial compressive load leads to an increase in the first two frequencies, while the center point load and radial pressure load decrease the first three frequencies.  
From these observations, it can be concluded that loads applied to plate/shell FGM structures can have significant impact on the natural frequencies, depending on the magnitude of the loads, the type of the applied loads, the boundary conditions, and the gradient index p.  
In this work, only the gradient index p = 1 has been considered to build the numerical cases that show the influence of the applied loads on the fundamental frequency of plates and cylindrical panels (Figures 11c-f, 15a-c and 16a-c). It is observed that the stiffness is related to the frequency at each load level. In the case of the hinged cylindrical panel, the deformed shape seems to have an important role in influencing the frequency.  
In future work, in order to develop a more comprehensive study, the effect on the variation of natural frequencies of higher modes (as shown in Figure 11f) with an arbitrary p-index could be studied. The present work provides results that can be used as a benchmark for interested readers to compare with new or available alternative theories in the development of new finite element models for FGM plate and shell structures with general geometry.  
Conceptualization, J.S.M.; methodology, J.S.M.; software, J.S.M.; validation, J.S.M.; formal analysis, all authors; investigation, all authors; resources, all authors; data curation, J.S.M.; writing--original draft preparation, J.S.M. and V.F.C.; writing--review and editing, V.F.C.; supervision, C.M.S.; project administration, C.M.S. All authors have read and agreed to the published version of the manuscript.  
This work was funded by FCT, Fundacao para a Ciencia e Tecnologia, through IDMEC, under LAETA, project UIDB/50022/2020.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
The authors dedicate this paper to the memory of Professor [PERSON] (1942-2022), who sadly passed away on 2 February 2022. He was a world academic and distinguished researcher expert in the fields of characterization of material properties, multidisciplinary optimization, solid mechanics, finite element techniques, advanced composite materials, and computational methods. We extend our condolences to his wife and family, his colleagues at Riga Technical University, and his friends in Latvia.  
The authors declare no conflicts of interest.