3D exact and 2D generalized differential quadrature models for free vibration analysis of functionally graded plates and cylinders

The paper proposes a comparison between a three-dimensional (3D) exact solution and several two-dimensional (2D) numerical solutions. Numerical methods include classical 2D finite elements (FEs), and classical and refined 2D generalized differential quadrature (GDQ) solutions. The free vibration analysis of two different configurations of functionally graded material (FGM) plates and cylinders is proposed. The first configuration considers a one-layered FGM structure. The second one is a sandwich configuration with external classical skins and an internal FGM core. Low and high order frequencies are analyzed for thick and thin simply supported structures. Vibration modes are investigated to make a comparison between results obtained via the 2D numerical methods and those obtained by means of the 3D exact solution. The 3D exact solution is based on the differential equations of equilibrium written in general orthogonal curvilinear coordinates. This exact method is based on a layer-wise approach where the continuity of displacements and transverse shear/normal stresses is imposed at the interfaces between the layers of the structure. The 2D finite element results are obtained by means of a well-known commercial FE code. Classical and refined 2D GDQ models are based on a generalized unified approach which considers both equivalent single layer and layer-wise theories. The differences between 2D numerical solutions and 3D exact solutions depend on the considered mode, the order of frequency, the thickness ratio of the structure, the geometry, the embedded material and the lamination sequence.