Benchmark solution for free vibration analysis of transversely isotropic thick rectangular plates

Abstract

In this research, by using displacement potential functions, the exact solution is presented for free vibration analysis of simply supported rectangular transversely isotropic plates with constant thickness. The displacement components of the plate are written in terms of displacement potential functions, and the governing differential equations are derived by the substitution of the displacement components in Navier’s equations. The two governing partial differential equations of fourth and second order are solved using the separation of variables method and satisfying the exact boundary conditions. Having no simplifying assumptions for the strain or stress distribution in the plate thickness, the obtained results in this paper are applicable to any arbitrary plate thickness with no limitation on its thickness ratio. Due to absence of available research on thick transversely isotropic plates, the obtained results are compared with other analytical works for thin and moderately thick plates and the finite element method for thick plates, showing remarkable agreement. Accurate natural frequencies are presented for different ranges of thickness ratio and aspect ratio and different materials. It is observed that with increasing thickness the ratio and aspect ratio of the plate, the non-dimensional natural frequencies decrease and increase, respectively. In addition, comparative results of isotropic material and four different transversely isotropic materials also illustrate that the shear modulus in transverse direction has important influence on the vibrational behavior of plates.