A unified solution for vibration analysis of functionally graded cylindrical, conical shells and annular plates with general boundary conditions

• A unified vibration analysis method is developed for thick functionally graded shells.
• The method is universally applicable to general boundary conditions.
• New results for shells with elastically restrained edges are presented.
• The effects of the volume fractions on the frequency of the shells are investigated.

In this paper, a unified solution method for free vibration analysis of functionally graded cylindrical, conical shells and annular plates with general boundary conditions is presented by using the first-order shear deformation theory and Rayleigh–Ritz procedure. The material properties of the structures are assumed to change continuously in the thickness direction according to the general four-parameter power-law distributions in terms of volume fractions of constituents. Each of displacements and rotations of those structures, regardless of boundary conditions, is expressed as a modified Fourier series, which is constructed as the linear superposition of a standard Fourier cosine series supplemented with auxiliary polynomial functions introduced to eliminate all the relevant discontinuities with the displacement and its derivatives at the edges and accelerate the convergence of series representations. The excellent accuracy and reliability of the current solutions are confirmed by comparing the present results with those available in the literatures, and numerous new results for functionally graded cylindrical, conical shells and annular plates with elastic boundary conditions are presented. The effects of boundary conditions and the material power-law distribution are also illustrated.