Free vibration analysis of laminated composite shallow shells with general elastic boundaries

A unified and exact solution method is developed for the free vibration analysis of composite laminated shallow shells with general elastic boundary conditions, a class of problem which is of practical interest and fundamental importance but rarely attempted in the literature. Under the current framework, each of the shell displacements, regardless of boundary conditions, is expanded as a standard Fourier cosine series supplemented with closed-form auxiliary functions introduced to eliminate all the relevant discontinuities with the displacement and its derivatives at the edges and accelerate the convergence of series representations. Mathematically, such series expansions are capable of representing any functions (including the exact displacement solutions). Rayleigh–Ritz procedure is employed to obtain the exact solution based on the energy functions of the shell. The current method can be universally applicable to a variety of boundary conditions including all the classical cases, elastic restraints and their combinations. The excellent accuracy and reliability of the current solutions are validated by numerical examples, and the effects of boundary restraining stiffnesses and lamination schemes on frequency parameters are illustrated. New results for various shell curvatures including spherical, cylindrical and hyperbolical paraboloidal shells with elastically restrained edges are presented, which may serve as benchmark solutions.