Keywords: cracked beams, Euler-Bernoulli beam theory, damaged structures, finite element analysis, Timoshenko beam theory.

This paper presents a computationally efficient two-node multi-cracked beam (MCB) element for the analysis of planar frame structures with concentrated damage subjected to static and dynamic loading. The proposed model follows the discrete spring (DS) representation of (linear elastic) always-open cracks, in which the beam is fully articulated at the position of each crack, and a set of axial, rotational and transverse elastic springs takes into account the residual stiffness of the damaged section, while axial, bending and shear deformations are considered in the undamaged regions of the beam between two consecutive singularities (i.e. the end nodes and the cracked sections).

The (physically consistent) flexibility modelling of concentrated damage [1] is adopted to derive the exact closed-form expressions for the deformed shape and internal forces of a beam with an arbitrary number, position and severity of cracks, subjected to static axial and transverse loading. By exploiting the generalised Hooke's law and the action-reaction principle, these results are used to determine the dimensionless stiffness matrix and the array of equivalent nodal forces. The consistent mass matrix is also computed, by adopting the same shape functions to represent the inertial forces on the MCB element.

Unlike the conventional DS models, which require the beam to be split at the position of each crack, with two finite element nodes added to both sides of each crack, the proposed MCB element embeds the effects of the concentrated damage without enlarging the size of the finite element model. It is shown that the exact static solution is retrieved independently of the mesh (i.e. the exact structural response can be obtained with a single MCB element per member), while a faster convergence is achieved for dynamic applications. This is confirmed with two numerical examples, which also demonstrate the improved performance of the proposed MCB element in comparison with the approximate local stiffness reduction (LSR) approach, very often used for problems of damage detection.

It must be noted that the proposed MCB element is equivalent to the analogous finite element formulation recently proposed by other researchers [2], in which however the (physically inconsistent) rigidity modelling of concentrated damage has been used.

1

A. Palmeri, A. Cicirello, "Physically-based Diracs delta functions in the static analysis of multi-cracked Euler-Bernoulli and Timoshenko beams", International Journal of Solids and Structures, 48, 2184-2195, 2011. doi:10.1016/j.ijsolstr.2011.03.024

2

S. Caddemi, I. Caliò, D. Rapiacavoli, "The influence of concentrated damage in the dynamic behaviour of framed structures", L'Ingegneria Sismica in Italia, ANIDIS 201, 2011.

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