Vibrations of moderately thick rectangular plates in terms of a set of static Timoshenko beam functions

Abstract

In this paper, a set of static Timoshenko beam functions is developed as the admissible functions to study the free vibrations of moderately thick rectangular plates using the Rayleigh-Ritz method. This set of beam functions is made up of the static solutions of a Timoshenko beam under a series of sinusoidal distributed loads. The beam is considered to be a unit width strip taken from the rectangular plate in a direction parallel to the edges of the plate. In addition, the geometric boundary conditions of the plate are exactly satisfied in this set of beam functions, and the effect of the shear correction factor on the admissible functions of the plate is also taken into account. It can be seen that the method is sound in theory and no complicated mathematical knowledge is needed. Each of the beam functions is only a third-order polynomial plus a sine function or a cosine function. Furthermore, a change of the boundary conditions of the plate only results in a change of the coefficients of the polynomial. The method is very simple and a unified computational program can be given for the plates with arbitrary boundary conditions and thickness. Comparison and convergency studies demonstrate the correctness and the accuracy of the method. It can be shown that using a small number of terms of the static Timoshenko beam functions can give rather accurate results for all cases. Finally, the effect of thickness-span ratio on the eigenfrequency parameters of Mindlin rectangular plates is studied in detail.