A unified solution for the vibration analysis of FGM doubly-curved shells of revolution with arbitrary boundary conditions

This paper describes a unified solution for the vibration analysis of functionally graded material (FGM) doubly-curved shells of revolution with arbitrary boundary conditions. The solution is derived by means of the modified Fourier series method on the basis of the first order shear deformation shell theory considering the effects of the deepness terms. The material properties of the shells are assumed to vary continuously and smoothly along the normal direction according to general three-parameter power-law volume fraction functions. In summary, the energy functional of the shells is expressed as a function of five displacement components firstly. Then, each of the displacement components is expanded as a modified Fourier series. Finally, the solutions are obtained by using the variational operation. The convergence and accuracy of the solution are validated by comparing its results with those available in the literature. A variety of new vibration results for the circular toroidal, paraboloidal, hyperbolical, catenary, cycloidal and elliptical shells with classical and elastic boundary conditions as well as different geometric and material parameters are presented, which may serve as benchmark solution for future researches. Furthermore, the effects of the boundary conditions, shell geometric and material parameters on the frequencies are carried out.