An approximate solution for vibrations of uniform and stepped functionally graded spherical cap based on Ritz method

Based on Ritz method, the vibration approximate solutions of uniform and stepped functionally graded (FG) spherical cap are carried out in this paper. The first-order shear deformation theory (FSDT) is used to derive energy expression. The selections of displacement functions are based on domain decomposition approach, in which the unified Jacobi polynomials are introduced to represent the displacement functions component along axial direction. In addition, the standard Fourier series still denote the displacement functions component along circumferential direction. The various boundary conditions are simulated by applying spring stiffness method. Then the Ritz method is employed to obtain the final results. The solutions of the same condition are compared with those obtained by finite element method (FEM) and published literatures to validate the present method. The results exhibit that the current approach has the advantages of high solution accuracy and fast convergence. On this basis, the numerical results concerning the effects of geometric parameters and boundary conditions on the vibration responses of the structure are also considered.