, Volume 54, Issue 9, pp 1433–1449 | Cite as

Dynamics of beams with uncertain crack depth: stochastic versus interval analysis

  • Roberta SantoroEmail author
  • Giuseppe Muscolino
Stochastics and Probability in Engineering Mechanics


The present paper deals with the evaluation of the structural response of single-cracked beam-like structures under deterministic time variant excitations assuming the crack depth as an uncertain parameter. Taking into account the unavoidable uncertainty affecting also the damage characteristics in practical applications, the crack depth is modelled by both a stochastic and an uncertain-but-bounded variable. It follows that the structural beam response becomes a stochastic process or an interval function, respectively. In scientific literature for this kind of uncertainties the statistics as well as the bounds of the structural response are usually evaluated by applying the perturbation approach, whose accuracy is valid only for very small value of uncertainty. Aim of this paper is to provide an alternative procedure developed in the frequency domain: the starting point is the application of the so-called rational series expansion, recently proposed to derive an approximate explicit expression of the frequency response function. The accuracy of the present method is confirmed by analyzing a damaged prismatic cantilever steel beam subjected to an impulsive load. The results in terms of statistics as weel as bounds of the displacement beam tip are reported and compared with the Monte Carlo simulation and the combinatorial vertex method. The effects of the two models for the uncertain crack depth on the dynamic response are also compared in terms of interval bounds and the so-called confidence intervals provided by the stochastic analysis.


Damaged beams Uncertain crack depth Frequency domain response 



The support of Italian MIUR under the Grant PRIN-2015, Project No 2015JW9NJT entitled “Advanced mechanical modelling of new materials and structures for the solution of 2020 Horizon challenges”, is gratefully acknowledged.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of EngineeringUniversity of MessinaMessinaItaly

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