Closed form solutions of Euler–Bernoulli beams with singularities

The problem of the integration of the static governing equations of the uniform Euler–Bernoulli beam with discontinuities is studied. In particular, two types of discontinuities have been considered: flexural stiffness and slope discontinuities. Both the above mentioned discontinuities have been modeled as singularities of the flexural stiffness bymeans of superimposition of suitable distributions (generalized functions) to a uniform one dimensional field. Closed form solutions of governing differential equation, requiring the knowledge of the boundaryconditions only, are proposed, and no continuityconditions are enforced at intermediate cross-sections where discontinuities are shown. The continuityconditions are in fact embedded in the flexural stiffness model and are automaticallyaccounted for bythe proposed integration procedure. Finally, the proposed closed form solution for the cases of slope discontinuity is compared with the solution of a beam having an internal hinge with rotational spring reproducing the slope discontinuity.