On nonuniform Euler–Bernoulli and Timoshenko beams with jump discontinuities: application of distribution theory

Abstract

In this article, bending of nonuniform Euler–Bernoulli and Timoshenko beams with jump discontinuities in the slope, deflection and mechanical properties are studied. The governing equations are obtained in the space of generalized functions, and the expression of its governing differential equations in terms of a single displacement function and a single rotation function is shown always to be possible. In contrast, for a nonuniform Euler–Bernoulli beam with jump discontinuities in slope and deflection and abrupt changes in flexural stiffness, the governing equation can be written in terms of a single displacement function only under certain conditions. It is observed that for most discontinuous nonuniform Euler–Bernoulli beams we cannot write the governing differential equation in terms of a single displacement function: usually, if there are n discontinuity points on a nonuniform Euler–Bernoulli beam, n+1 displacement functions appear in the governing equilibrium equation.