Simplified Formulation of Solution for Beams on Winkler Foundation allowing Discontinuities due to Loads and Constraints

Abstract

The paper discusses material for a course in Structural Mechanics addressed to second-Year Civil Engineering students. The response of beams on a Winkler foundation characterized by discontinuities in both the displacements (deflections and/or slopes) and forces (internal forces and/or loads) is studied. In particular, a simplified formulation for the solution of the discontinuous differential equation governing this problem is given. In some cases, the formulation is able to give the exact solution in a closed form. This is made possible through the use of the generalized functions, such as the well-known Unit Step Function and the Dirac delta function. The cases of discontinuities due to both loads and constraints are treated. The method of presentation of this material is by lecture. The time required to cover the arguments is 2 to 3 hours with 1 to 2 hours of revision. The lecture must be given after the classical beam theory has been covered. The new formulation presented in this paper will demonstrate that some particular aspects of mathematician analysis many he used to advantage to simplify an important problem in structural mechanics.