On applications of generalized functions to beam bending problems

Abstract

Using a mathematical approach, this paper seeks an efficient solution to the problem of beams bending under singular loading conditions and having various jump discontinuities. For two instances, the boundary-value problem that describes beam bending cannot be written in the space of classical functions. In the first instance, the beam is under singular loading conditions, such as point forces and moments, and in the second instance, the dependent variable(s) and its derivatives have jump discontinuities. In the most general case, we consider both instances. First, we study singular loading conditions and present a theorem by which the equivalent distributed force of a general class of singular loading conditions can be found. As a consequence of obtaining the equivalent distributed force of a distributed moment, we find a mathematical explanation for the corner condition in classical plate theory. While plate theory is not the focus of this paper, this explanation is interesting. Then beams with various jump discontinuities are considered. When beams have jump discontinuities the form of the governing differential equations changes. We find the governing differential equations in the space of generalized functions. It is shown that for Euler–Bernoulli beams with jump discontinuities the operator of the differential equation remains unchanged, only the force term changes so that delta function and its distributional derivatives appear within it. But for Timoshenko beams with jump discontinuities, in addition to changes in the force terms, the operator of one of the governing differential equations changes. We then propose a new method for solving these equations. This method which we term the auxiliary beam method, is to solve the governing differential equations not in the space of generalized functions but rather to solve them by means of solving equivalent boundary-value problems in the space of classical functions. The auxiliary beam method reduces the number of differential equations and at the same time obviates the need to solve these differential equations in the space of generalized functions which can be more difficult.