Closed form solutions of Euler–Bernoulli beams with singularities

Non-linear vibration of Euler-Bernoulli beams

“…Then, in the case of a cantilevered beam subjected to a single moment at its free end, the difference between the linear and the nonlinear theories based on both the mathematical curvature and the physical curvature was shown. Biondi and Caddemi [8] studied the problem of the integration of the static governing equations of the uniform Euler-Bernoulli beams with discontinuities, considering the flexural stiffness and slope discontinuities.…”

Closed-form solutions for Euler–Bernoulli arbitrary discontinuous beams

“…2 are used to solve arbitrary discontinuous beams under static loads. For generality, dimensionless piecewise-continuous loading functions 21), denote arbitrary continuous functions for a which a fourth-order and a third-order primitive are assumed to exist, respectively, as generally encountered in engineering applications [4,7,8].…”

“…A strategy to avoid enforcing internal conditions has been later devised by Biondi and Caddemi [7,8]. Based on the theory of generalized functions, their solution is formulated for stepped beams with internal springs and involves enforcing the 4 B.C.…”

“…It has been applied by Caddemi and coworkers to identification problems [9][10][11][12] and equilibrium stability problems [13]. The method by Biondi and Caddemi, however, requires the first-and the second-order primitives of the loading functions to be continuous, respectively, at the deflection discontinuity locations [8] and at the rotation discontinuity locations [7]. For this reason, it cannot be applied to beams where, at a given location, may occur a shear-force + deflection discontinuity (due to an along-axis roller support or an external point force + internal translational springs) and/or a bending-moment + rotation discontinuity (due to an along-axis rotational support or an external point moment + internal rotational springs).…”

The Hu–Washizu variational principle for the identification of imperfections in beams

“…Moreover, the presented procedure will be shown to be suitable for identification of concentrated damages that is a subject to which a great interest has been devoted in the specific literature [3,7,8,10,11,14,15,24,25]. Recently, a model of the Euler-Bernoulli beam with singularities able to model the presence of concentrated damages has been proposed [20,[29][30][31]. According to the latter approach, the concentrated damages are cracks treated as singularities modelled by means of Dirac's delta distributions (x − x 0i ) superimposed onto a uniform inertia moment I 0 of the undamaged cross section as follows:…”