Natural frequencies of plates with square holes when subjected to in-plane uniaxial, biaxial or shear loading

Abstract

The finite element method is used to determine the natural frequencies of flat square plates containing eccentrically located square holes. The plates are subjected to in-plane uniaxial, biaxial or uniformly distributed shear along the four outer edges. These edges are either simply supported or clamped. In order to evaluate the stiffness and mass matrices, the non-conforming rectangular displacement element is used to model the out-of-plane behaviour of the plate. The in-plane stress distribution within the plate, required in the analysis, is determined by using a rectangular finite element having the only essential degrees of freedom at each of the four corner nodes. The element displacement functions are based on assumed strains and satisfy the exact requirements of strain free rigid-body modes of displacements. The variation of the natural frequency with the size and location of the hole is first investigated in the absence of any in-plane stresses. This analysis is then repeated for a series of values of the applied in-plane stresses. When uniform shear is applied, tension and compression zones are produced in the plate and hence the effect of locating the hole in each of the regions is also investigated. The values of the applied in-plane stresses ranged up to the point that would cause the plates to buckle. In this way a comprehensive set of results can be obtained.