Vibration of tapered Mindlin plates in terms of static Timoshenko beam functions

Abstract

In this paper, the free vibrations of rectangular Mindlin plates with variable thickness in one or two directions are investigated. The thickness variation of the plate is continuous and can be represented by a power function of the rectangular co-ordinates. A wide range of tapered rectangular plates can be described by giving various index values to the power function. Two sets of new admissible functions are developed, respectively, to approximate the flexural displacement and the angle of rotation due to bending of the plate. The eigenfrequency equation is obtained by using the Rayleigh–Ritz method. The complete solutions of displacement and angle of rotation due to bending for a tapered Timoshenko beam (a strip taken from the tapered Mindlin plate in some direction) under a Taylor series of static load have been derived, which are used as the admissible functions of the rectangular Mindlin plates with taper thickness in one or two directions. Unlike conventional admissible functions which are independent of the thickness variation of the plate, the static Timoshenko beam functions presented in this paper are closely connected with the thickness variation of the plate so that higher accuracy and more rapid convergence can be expected. Some numerical results are furnished for both truncated Mindlin plates and sharp-ended Mindlin plates. On the basis of convergence study and comparison with available results in literature, it is shown that the first few eigenfrequencies can be obtained with quite satisfactory accuracy by using only a small number of terms of the static Timoshenko beam functions.