Euler–Bernoulli beams with multiple singularities in the flexural stiffness

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:…”

Closed-form solutions for Euler–Bernoulli arbitrary discontinuous beams

“…(8), the symbols R n (x − y) = (x − y) n H (x − y)/n! denote the n-th order generalized integral of the unitstep function H (x − y), as in Ref.…”

“…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).…”

“…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.…”

On the Bending Analysis of Multi-Cracked Slender Beams with Continuous Height Variations

“…Afterwards, closed-form solutions for the static transverse displacements and stiffness matrix of a beam's finite element having an arbitrary number of transverse cracks have been derived at by several authors by implementing different mathematical methods. The Dirac delta function was implemented twice: by Biondi and Caddemi [3] in regard to the rigidity, and by Palmeri and Cicirello [4] in regard to the flexibility. Sequential solutions of coupled differential equations were implemented by Skrinar [5], while Skrinar and Pliberšek [6] implemented the principle of virtual work.…”