Explicit Assessment of the Forced Vibration of Multi-cracked Beams with Uncertain Damage Intensity

Damage in beams can strongly affect their dynamic properties and consequently their response. On the other hand, the correct evaluation of the damage intensity is a problem of difficult solution. For these reasons, exploring the variability of the dynamic response in damaged beams considering the damage intensities as uncertain parameter is a useful contribution to assess the safety of structures. Classic approaches require to treat damage as concentrated rotational springs, where continuity conditions have to be enforced, and to perform re-analyses varying the damage intensities in a desired range. In this paper, two main improvements to this procedure are provided. The first contribution is the use of an efficient solution which employs generalised functions to compute the dynamic properties of a damaged beam for a limited number of damage configurations; such a method avoids the enforcement of continuity conditions at the cracked sections. Then, an explicit continuous expression of such dynamic properties as a function of the damage intensities is proposed by employing the Sherman and Morrison formula. The proposed explicit approach is verified and applied for the evaluation of the dynamic properties of damaged beams; then, via modal superposition, the variability of the response associated to time histories is assessed as well as frequency-response curves for single and multi-cracked beams subjected to forced vibrations.