Nonlinear Free Vibration of Functionally Graded Shallow Shells with Variable Thickness Resting on Elastic Foundation

The chapter considers the practice of important problem about investigation of the geometrically nonlinear free vibration of FGM shallow shells of the variable thickness. The shells under consideration are supported by elastic foundation and can have arbitrary shape in the plan. The mathematical formulation of the problem was done in the framework of the refined shear deformation theory of the first order. To calculate the effective characteristics of the material, the simple power law is applied. The thickness variation of the shallow shells is carried out according to the given law, such as, linear, parabolic or another law. Elastic foundation is described by Pasternak’s model. The proposed approach is based on the application of the R-functions theory, variational Ritz’s method and Bubnov-Galerkin procedure. Nonlinear frequencies and backbone curves are defined due to solution of the nonlinear ordinary differential equation, obtained as a result of using eigen functions of the linear problem and solution of some auxiliary boundary problem. The comparisons of the obtained results with known ones for a special case of the shells and plates are fulfilled. Dynamic analysis for the panels and plates with linear and parabolic thickness for the different constituent materials of FGM and parameters of elastic foundation is carried out. The effect of materials and thickness of the shell on the natural frequencies and backbone curves is shown.