Closed-form solutions for Euler–Bernoulli arbitrary discontinuous beams

Abstract

Euler-Bernoulli arbitrary discontinuous beams acted upon by static loads are addressed. Based on appropriate Green's functions here derived in a closed form, the response variables are obtained: (a) for stepped beams with internal springs, as closed-form functions of the beam discontinuity parameters, without enforcing neither internal nor boundary conditions; (b) for stepped beams with internal springs and along-axis supports, as closed-form functions of the unknown reactions of the along-axis supports only, to be computed by enforcing pertinent conditions. A remarkable reduction in computational effort is achieved, in this manner, compared to competing methods in the literature.

Green’s function for uniform Euler–Bernoulli beams at resonant condition: Introduction of Fredholm Alternative Theorem

“…For instance, Failla [8] proposed a solution for Euler-Bernoulli arbitrary discontinuous beams using the static Green's function.…”

General finite element description for non-uniform and discontinuous beam elements

“…This problem will be tackled, in this paper, by resorting to the theory of generalized functions. This theory has been traditionally applied to discontinuous beam only [2,3,[17][18][19][20][21]. Herein it will be used to build, for non-uniform and discontinuous beams, the full set of response variables due to end nodal displacements as well as to in-span loads, either concentrated or distributed.…”

“…Static analysis of discontinuous beam elements has also attracted the interest of several researchers [2,3,[17][18][19][20][21]. For FE analysis purposes, the complementary energy-based method in ref.…”

Exact frequency response of bars with multiple dampers

“…In fact, the B.C. can still be considered as homogeneous, while the end dampers are modelled as internal dampers located at x 1 = 0 + and x n = L − [45]. Changes to consider a single damper at a given position are straightforward.…”