The use of generalised functions in the discontinuous beam bending differential equations

Abstract

This paper discusses materials for a course in Strength of Materials and Mechanics of Solids, addressed to students of second-year Mechanical Engineering, Ocean Engineering, Civil Engineering and Aerospace Engineering. In particular, the presence of discontinuities in the beam-bending differential equations is considered. This problem is solved by the use of the generalised functions, among which the best known is the Dirac delta function. In particular Macaulay's approach, which uses these functions when discontinuous mechanical loads are present is here extended to the cases in which discontinuous external loads are present, giving discontinuities on displacements and rotations. Moreover, the cases in which natural and essential constraints are along the beant axis, giving different kinds of discontinuities, are presented. This extension shows the same easy applicability and the same practical advantages of the Macaulay's approach, always reducing to one the differential equations to be solved in order to find the displacements law. Moreover the formulation is given in a uniform way for any kind of discontinuity appearing in the beam. The rhode of presentation of this material is by lecture and is run as a regular course. Hours required to cover the arguments are 3 to 4 with 2 to 3 revision hours. This lecture must be held after that the classical beam bending problem has been treated. The new aspects presented in this paper hopefully help the students to find important connections between some aspects of mathematician analysis and an important problem of applied mechanics.