On Euler–Bernoulli discontinuous beam solutions via uniform-beam Green’s functions

Abstract

The bending problem of Euler–Bernoulli discontinuous beams is dealt with. The purpose is to show that uniform-beam Green’s functions can be used to build efficient solutions for beams with internal discontinuities due to along-axis constraints and flexural-stiffness jumps. Specifically, upon deriving the equilibrium equation in the space of generalized functions, first it is seen that the original bending problem may be recast as linear superposition of a principal and an auxiliary bending problem, both involving a uniform reference beam and homogeneous boundary conditions. Then, based on the Green’s functions of the reference beam, closed-form solutions are developed for the principal beam response, while the auxiliary beam response is obtained by solving, in general, ( r + 2 s) algebraic equations written at the discontinuity locations, being r the number of discontinuities due to along-axis constraints, and s the number of flexural-stiffness jumps. In this manner, an appreciable reduction of computational effort is achieved as compared to alternative analytical solutions in the literature.