Explicit dynamic response of damaged beams with application to uncertain and identification problems

The influence of damage in beam-like structures is a problem widely analysed in the literature in view of its importance towards the assessment of the residual carrying capacity and the evaluation of the inevitable changes of the dynamic properties. Usually, the dynamic properties and the forced vibrations of structures, cannot be explicitly inferred, particularly in the presence of damage, since the computation of the roots of a highly nonlinear determinantal equation is preliminarily required. Rather, explicit formulations are usually preferred for their ease to use and the possibility to perform parametric analyses, very useful in the case of uncertain parameters, and also to address inverse problems. In this paper, aiming at proposing an explicit formulation of the dynamic response of cracked beams, a model which makes use of the distribution theory and does not require the enforcement of continuity conditions at the cracked sections is employed. Approximated explicit expressions of the main modal parameters related to crack severities are here provided and the reliability of the proposed formulas are verified with reference solutions. The approximated modal parameters can then be used to obtain explicitly the forced vibrations of cracked beams. The proposed methodology is here adopted to address both direct and inverse problems. First, the variability of the dynamic response of cracked beams due to the presence of cracks with uncertain, but bounded, depths is investigated; then, upper and lower bounds of the response are evaluated by making use of both time and frequency domain analyses. In addition, the possibility to adopt the same explicit approach for the identification of damage intensities considering frequency measurements is explored.