Damage identification in Euler frames

Abstract

The present paper examines the effect of local damage in elastic frames composed by Euler beams. The direct problems of undamaged and damaged frames are studied using the Wittrick–Williams algorithm. The related inverse problems of identifying material parameters in the undamaged structure and damage parameters, i.e. location and intensity, in the damaged frame are studied using procedures based on the comparison of numerical and pseudo-experimental natural frequencies. It is shown that the material parameters can be reliably determined and, with reference to the damaged frame, important considerations on the number of frequencies required in order to evaluate damage parameters are outlined. The sensitivity of the proposed procedure to experimental errors is also studied.

Introduction

Structural damage consists in a loss of stiffness inducing variations of both static and dynamic responses with respect to the undamaged structure. To avoid structural failure, it is required that damage is detected in its early stage, which, in the last decades has lead to the development of damage identification techniques, requiring the solution to an inverse problem that combines a model of the structure to measured response quantities [1]. A wealth of techniques is reported in the literature, based for instance on the variation of dynamic characteristics, such as natural frequencies, mode shapes, dynamic flexibilities, or static quantities, such as displacements or strains induced by applied loads [2], [3], [4], [5]. A recent and up to date overview on the subject of damage identification and the wider issue of structural health monitoring can be found in [6]. Particular attention has been devoted to the model of damage, which is critical whenever a relationship between the damage model and the real crack depth is sought. Numerous attempts to quantify local defects are reported in the literature and a comprehensive survey of crack modelling approaches is reported in [7]. In short, the effect of a notch on the structure’s flexibility can be represented by a local flexibility whose magnitude can be estimated by experimentation or by using fracture mechanics methods [3], [8], or as an alternative, by one-dimensional continuum theories [9], among which, more modern ones are based on the principles of fracture mechanics [10], [11]. A discussion on these issues can be found in [12].

The present paper examines the problem of identification of local or concentrated damage, that is a notch, in elastic one-storey frames. The damage consists of a notch that reduces the height of the cross section at a given abscissa x_d and is modelled by a reduction in the rigidity of the beam. This reduction can be described by means of an appropriate rotational spring with stiffness k_ϕ [3]. Hence, the damage parameters to be identified are k_ϕ and x_d only. This approach differs from the frequently used approach of damage identification, which is tackled as a stiffness reconstruction problem with a great number of unknowns. Here, only a small number of unknowns are considered, which is a computationally efficient strategy providing results with physical meaning [5].

The effect of damage is studied with reference to the variation induced in the natural frequencies of vibration, that are evaluated, for both undamaged and damaged structures, by means of an analytical method based on properties of the dynamic stiffness matrix, i.e. the Wittrick–Williams algorithm [13], [14], [15]. Other approaches are possible, for instance using compliance matrices, as in [16]. The direct problem of the undamaged frame has been studied obtaining an explicit expression of the ith dimensionless frequency parameter $a_i^4={\omega }_i^4{\mathit{mL}}^4/\mathit{EI}$ with respect to the geometrical and mechanical properties of the model, that is: ith natural frequency ω_i distributed mass m per unit length, Young’s modulus of the material E, moment of inertia I and length L. Assuming that I and L are known, the ratio E/m can be identified by means of the minimization of an objective function which measures the differences between analytical and measured frequencies. Once the mechanical parameters of the model have been reliably updated, the variation of the frequencies of the damaged frame are studied as a function of the damage parameters, i.e. dimensionless location l = x_d/L and rigidity of the rotational spring k_ϕ. For a known ω_i of the damaged frame, for each possible l, one value of stiffness k_ϕ exists which corresponds to a value of the ith natural frequency equal to ω_i. Therefore, for each frequency, a curve k_ϕ(l) is obtained, defining k_ϕ for all the possible positions of the notch. This analysis can be used to examine the uniqueness of the solution to the inverse problem. In fact, the curves k_ϕ(l) obtained for different ω_i cross at the abscissa where the notch is located, providing the solution to the inverse problem. This fact allows the determination of the minimum number of frequencies required to obtain a unique solution to the inverse problem, which is then solved by the minimization of an objective function measuring the differences between analytical and measured variations of natural frequencies in the undamaged and damaged states [5]. Different damage configurations are considered to assess the reliability of this procedure. The sensitivity to instrumental errors of the identification technique presented is studied for both the undamaged and damaged frame [17].