Generalized solutions of beams with jump discontinuities on elastic foundations

The bending solutions of the Euler-Bernoulli and the Timoshenko beams with material and geometric discontinuities are developed in the space of genera lized functions. Unlike the classical solutions of discontinuous beams, whi ch are expressed in terms of multiple expressions that are valid in differe nt regions of the beam, the generalized solutions are expressed in terms of a single expression on the entire domain. It is shown that the boundary-va lue problems describing the bending of beams with jump discontinuities on d iscontinuous elastic foundations have more compact forms in the space of ge neralized functions than they do in the space of classical functions. Also, fewer continuity conditions need to be satisfied if the problem is formula ted in the space of generalized functions. It is demonstrated that using th e theory of distributions (i.e. generalized functions) makes finding analyt ical solutions for this class of problems more efficient compared to the tr aditional methods, and, in some cases, the theory of distributions can lead to interesting qualitative results. Examples are presented to show the eff iciency of using the theory of generalized functions.