On applications of generalized functions to beam bending problems

Using a mathematical approach, this paper seeks an efficient solution to th e problem of beams bending under singular loading conditions and having var ious jump discontinuities. For two instances, the boundary-value problem th at describes beam bending cannot be written in the space of classical funct ions. In the first instance, the beam is under singular loading conditions, such as point forces and moments, and in the second instance, the dependen t variable(s) and its derivatives have jump discontinuities. In the most ge neral case, we consider both instances. First, we study singular loading co nditions and present a theorem by which the equivalent distributed force of a general class of singular loading conditions can be found. As a conseque nce of obtaining the equivalent distributed force of a distributed moment, we find a mathematical explanation for the corner condition in classical pl ate theory. While plate theory is not the focus of this paper, this explana tion is interesting. Then beams with various jump discontinuities are consi dered. When beams have jump discontinuities the form of the governing diffe rential equations changes. We find the governing differential equations in the space of generalized functions. It is shown that for Euler-Bernoulli be ams with jump discontinuities the operator of the differential equation rem ains 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 operat or of one of the governing differential equations changes. We then propose a new method for solving these equations. This method which we term the aux iliary beam method, is to solve the governing differential equations not in the space of generalized functions but rather to solve them by means of so lving equivalent boundary-value problems in the space of classical function s. The auxiliary beam method reduces the number of differential equations a nd at the same time obviates the need to solve these differential equations in the space of generalized functions which can be more difficult. (C) 200 0 Elsevier Science Ltd. All rights reserved.