Geometrically Nonlinear Vibrations of Functionally Graded Shallow Shells

Abstract

An original method for investigation of geometrically nonlinear vibrations of functionally graded shallow shells and plates with complex planform is presented. Shells under consideration are made from a composite of ceramics and metal. Power law of volume fraction distribution of materials through thickness is chosen. Mathematical statement is implemented in the framework of the refined geometrically nonlinear theory of the shallow shells of the first order (Timoshenko type). The proposed approach combines the application of the Rfunctions theory (RFM), variational Ritz method, procedure by Bubnov-Galerkin and Runge-Kutta method. Due to use of this combined algorithm it is possible to reduce the initial nonlinear system of motion equations with partial derivatives to a nonlinear system of ordinary differential equations. Investigation task of functionally graded shallow shells with arbitrary planform and different types of boundary conditions is carried out by the proposed method. Test problems and numerical results have been presented for one-mode approximation in time. In future, the developed method may be extended to investigation of geometrically nonlinear forced vibrations of functionally graded shallow shells with complex planform.