A computational approach for free vibration of non-uniform flexural–shear plates

Abstract

The concept and an analytical model of a flexural–shear plate are proposed in this paper. The governing differential equation for vibration of such a plate with variably distributed mass and stiffness considering the effects of damping and axial forces is established. Using appropriate transformations, the equation is reduced to a Bessel's equation or an Euler's equation or a differential equation with constant coefficients by selecting suitable expressions, such as power functions and exponential functions, for the distribution of mass and stiffness along the height of the plate. The exact solutions of one-step non-uniform flexural–shear plates are derived and used to obtain the frequency equations of multi-step non-uniform flexural–shear plates. The new exact approach which combines the transfer matrix method and closed form solutions of one-step flexural–shear plates are proposed for free vibration analysis of multi-step non-uniform flexural–shear plates. The new transfer matrix and the resulting frequency equations are presented. Numerical examples demonstrate that the calculated natural frequencies and mode shapes of tall buildings with narrow rectangular plane configuration which are treated as flexural–shear plates for vibration analysis are in good agreement with the corresponding experimental data.