Greens Function for an Infinite Elastic Plateon Winklers Foundation

In this technical note, an infinite thick plate on Winklerʹs foundation is studied. The effect of shear between the plate and the foundation on the deflection and the stresses is analyzed. It is assumed that the foundation has a stiffness k (the force needed to produce a unit displacement per area) and reacts in compression as well as tension. The effect of a concentrated normal unit force is investigated. The solution is based on the Airy stress function formulation. The Hankel transformation is employed to solve the biharmonic equation for the stress function. The transformed solution is subject to the transformed boundary conditions and the application of the inversion theorem leads to the final solution in the integral form. Judiciously selected dimensionless parameters make this solution relatively simple. Numerical results are obtained for some values of those parameters. In particular, the following two special cases are studied: (1) deflections of a relatively thin plate are compared to the results obtained by Timoshenko and Woinowsky-Krieger, giving an excellent correlation; and (2) with the thickness becoming infinite, the solution of Boussinesqʹs problem is readily recovered.