Damage identification in Euler frames
Highlights
► Effect of local damage in beam frames is studied by the Wittrick–Williams algorithm. ► The inverse problems of identifying material and damage parameters are considered. ► Procedures based on the comparison of frequencies are used. ► The minimum number of frequencies required to evaluate damage parameters is defined. ► 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 xd 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 kϕ [3]. Hence, the damage parameters to be identified are kϕ and xd 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 with respect to the geometrical and mechanical properties of the model, that is: ith natural frequency ωi 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 = xd/L and rigidity of the rotational spring kϕ. For a known ωi of the damaged frame, for each possible l, one value of stiffness kϕ exists which corresponds to a value of the ith natural frequency equal to ωi. Therefore, for each frequency, a curve kϕ(l) is obtained, defining kϕ 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 kϕ(l) obtained for different ωi 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].
Section snippets
The Wittrick–Williams algorithm
In this paragraph, the main characteristics of the Wittrick–Williams algorithm are briefly summarized. As it is well-known [18], the solution to the equation of motion of an Euler beam:can be expressed in the form: v(x, t) = ϕ(x)eiωt, where ϕ(x) is the mode of vibration and ω is the circular frequency. Introducing the non-dimensional variable ξ = x/L, the following equation of motion is obtained:The solution to Eq. (2) can be written as:
Direct problem
The considered frames are composed of two columns and a beam whose lengths are respectively Lc and Lb. The geometric and mechanical characteristics are indicated anywhere with the subscripts c for the columns and b for the beam. When the subscript is absent, reference is made to the generic structural element. Besides the undamaged state, the case in which local damage reduces the height of a given cross section from hU to hD is considered. It has been assumed that the width of the notch is so
Inverse problem
The inverse problem is solved by minimizing objective functions measuring the differences between analytical and measured response quantities. The parameters to be identified are the ratio E/m of the undamaged structure and the damage parameters kϕ and l. The objective functions are based on the variation of frequencies [5], which are quite reliable response quantities, easy to measure and requiring a small number of measurement points. A drawback of this approach is that it cannot discern
Sensitivity to experimental noise of the identification procedure
A real damage identification procedure is based on experimental data which are affected by unavoidable instrumental noise. In order to assess the performance of the proposed identification procedure when frequency measurements are affected by instrumental errors, the experimental frequencies are here modelled as random variables:where is the actual value of the ith natural frequency; Ri are random variable uniformly distributed between −1 and 1 with zero mean and independent
Conclusions
This paper addresses the problem of model updating and damage identification in an elastic Euler beam frame, by means of its of natural frequencies, which are calculated using the Wittrick–Williams algorithm. As a first step, in the undamaged frame, it is shown that the material parameters ratio E/mb can be reliably estimated minimizing an objective function based on the differences between numerical and experimental frequencies. The sensitivity to the instrumental error of this procedure is
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