Free vibration and stability of functionally graded shallow shells according to a 2D higher-order deformation theory

Natural frequencies and buckling stresses of shallow shells made of functionally graded materials (FGMs) are analyzed by taking into account the effects of transverse shear and normal deformations, and rotatory inertia. The modulus of elasticity of shells is assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional (2D) higher-order theory for rectangular functionally graded (FG) shallow shells is derived through Hamilton’s principle. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of FG shallow shells with simply supported edges. Three types of simply supported shallow shells with positive, zero and negative Gaussian curvature are considered. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency and also buckling stress are examined in detail. Critical buckling stresses of FG shells subjected to in-plane stresses are also obtained and a relation between the buckling stress and natural frequency of simply supported FG shells without in-plane stresses is presented. The modal transverse stresses have been obtained by integrating the three-dimensional (3D) equations of motion in the thickness direction with satisfying the surface boundary conditions of a shell. The present numerical results are also verified by satisfying the energy balance of external and internal works are considered to be sufficient with respect to the accuracy of solutions. It is noticed that the present 2D higher-order approximate theories can predict accurately the natural frequencies and buckling stresses of simply supported FG shallow shells.