The fictitious foundation approach to vibration analysis of plates with interior point supports

Abstract

A new approach to the continuum representation of interior point supports in the vibration analysis of plates is proposed. The plate is assumed to be supported by a fictitious non-uniform Winkler foundation. The flexibility distribution of the Winkler foundation, represented by a flexibility function, is such that it has a zero value at the point support locations but assumes large values, resulting in negligible restraint over the rest of the plate domain. The improvization of the point supports as a special case of the flexibility distribution of the elastic foundation allows one to solve the problem in a single domain in contrast to the multiple domain solutions available in the literature. The application of this approach, namely the fictitious foundation approach, is demonstrated by considering the free vibration of a simply supported rectangular plate with interior point supports. Also presented is a single domain application of the impulse function approach to plates with interior point supports, involving the representation of the concentrated reactions at the point support locations by a double Fourier series expansion of the impulse function. The results obtained by using these single domain approaches are compared with those obtained by using the multiple domain flexibility function approach, presented in an earlier study by the authors. The new approach presented in this paper, namely the fictitious foundation approach, for the simulation of interior point supports in plates provides a method for absorbing the discrete discontinuities in the force field, in the interior of the plate, into the governing differential equation and obtaining a continuum representation of the solution in a single domain of the plate.