A modified Fourier–Ritz approach for free vibration analysis of laminated functionally graded shallow shells with general boundary conditions

• A unified vibration analysis method is developed for laminated functionally graded shallow shells.
• The method is universally applicable to general boundary conditions.
• New results for shells with elastically restrained edges are presented.
• The effects of material profile and geometrical parameters on the frequency of the shells are investigated.

This paper presents a modified Fourier–Ritz approach for free vibration analysis of laminated functionally graded shallow shells with general boundary conditions in the framework of first-order shear deformation shallow shell theory. The displacement and rotation components of the shells are represented by the modified Fourier series consisted of standard Fourier cosine series and several closed-form auxiliary functions introduced to ensure and accelerate the convergence of the series representation. The material properties are assumed to vary continuously through the thickness according to power-law distribution. Four common types of sandwich functionally graded shallow shells are studied. The bi-layered and single-layered functionally graded shallow shells are obtained as special cases of sandwich shells. By setting groups of boundary springs and assigning corresponding stiffness constants to the springs, different boundary conditions including free, clamped, simply supported, elastic boundaries and their combinations are considered. A comprehensive investigation concerning the free vibration of the laminated functionally graded shallow shells with completely free and simply-supported edges is given. The results show that the present method enables rapid convergence, high reliability and accuracy. Numerous new vibration results for shallow shells with different material distributions, lamination schemes and elastic restraints are provided. Some mode shapes of the shallow shells are depicted. Parameter studies illustrate that changes of boundary conditions, material types, thickness schemes and power-law exponents will affect obviously the free vibration characteristic of the shallow shells. In contrast to most existing techniques, the current method can be universally applicable to a variety of boundary conditions including all the classical cases, elastic restraints and their combinations without the need of making any change to the solution procedure.