The influence of concentrated damages on the dynamic behaviour of framed structures

Structural health monitoring techniques are often based on static or dynamic response-based damage detection methods since the occurrence of damage can alter both the static and the dynamic behaviour of structures. The first needed step for successfully solving an inverse damage identification problem, appears to be the definition of a reliable model of the damaged structures and the evaluation of an accurate solution of the direct problem. In this paper a new powerful approach, aimed to evaluate the frequencies and the vibration modes of frame structures in presence of concentrated damage, is presented. This approach is based on a closed form solution, previously derived by the authors, relative to the vibration modes of a beam with an arbitrary number of concentrated damage represented by means of internal elastic hinges. The exact explicit solution possesses the same analytical structure of the undamaged beam, being a function of four integration constants only, regardless of the number of the damaged cross sections. On the basis of the explicit expressions provided for the eigenmodes, the exact dynamic stiffness matrix of the multi-cracked Euler-Bernoulli beam is derived. The knowledge of the exact dynamic stiffness matrix of the multi-cracked beam, as a function of the end degrees of freedom only, represents a fundamental result since it allows the direct evaluation of the global dynamic stiffness matrix of frame structures in presence of an arbitrary number of concentrated damages along its members. The great advantage of the proposed approach, with respect to the classical approach based on the knowledge of the dynamic stiffness matrix of the undamaged beam, is that the degrees of freedom of the overall frame structure are exactly the same of the equivalent undamaged structure irrespective of the number of the concentrated damages. This fact represents an advantage both from a computational cost and an implementation effort. Moreover, it facilitates the numerical investigation relative to the evolution of damage in complex frame structures when subjected to dynamic loadings.