Elsevier

Computers & Structures

Volume 88, Issues 3–4, February 2010, Pages 154-164
Computers & Structures

Influence of statistical errors on damage detection based on structural flexibility and mode shape curvature

https://doi.org/10.1016/j.compstruc.2009.08.017Get rights and content

Abstract

Damage detection procedures based on measured natural frequencies and building structure mode shapes are discussed in this paper. Modal curvature and structural flexibility approaches are tested. Attention is paid to the modal identification errors that influence damage detection. This problem is studied using a computer simulation of a simple supported beam. For practical cases, the peak picking method and its statistical errors are considered. In order to distinguish between true and false damage detection results, the Absolute Damage Index is proposed. Finally, the complexity of the problem in the case of a real building structure is presented.

Introduction

Structural Health Monitoring (SHM) based on structural modal data, such as natural frequencies and mode shapes, is a rapidly developing field since the relation between natural frequencies and local defects has been reported [1]. Over the last 30 years, several damage detection methods have been formulated. The parallel development of electronics and measurement equipment made it possible to implement SHM theory in practice. The SHM methods that make use of measured modal data are effective and safe for construction. Vibration measurements are usually carried out under natural conditions, without applying additional forces, so the process is fully non-destructive for the structure. The philosophy and the possibilities of modern SHM have been discussed, e.g., in publication [2].

Historically, a basis for further research was given in publications [3], [4]. In [3], the authors have proved that local defects affect the mode shapes and some appropriate analysis, e.g., analysis of the axis curvature, enables us to find diagnostic information. Also, in [4], the authors show a relationship between a local defect and structural flexibility, which can be represented by the flexibility matrix that is created by synthesising natural frequencies and natural modes. Both the curvature and the flexibility approaches are currently used for damage detection in real engineering problems because they have some advantages, such as reliability and simplicity of application, and they do not require the mode shape function to be specified – only the coordinates in selected structural points are sufficient. Moreover, these points can form an irregular net.

The efficacy of the methods is reported by various authors, e.g., in papers [5], [6], [7]. In [5], a specially formulated damage index allows us to determine the damage position. The efficacy has been proved in the case of the Kap Shui Mun Bridge. In [6], an optimisation problem is formulated. It matches experimental and analytical flexibility matrices in order to locate the damage. In [7], on the other hand, another damage measure based on the maximum eigenvalue of the flexibility matrix is proposed. The reliability of the approach is tested in computer simulations of a simple supported beam and a concrete reinforced beam. A report of the flexibility method applied to a real case is presented in [8]. 6-m-Long reinforced concrete beams were subjected to tests. Cracks spread over a distance, and the flexibility method did not appear to be as effective in this case; only the damage appearance is stated, not its location.

The discussed damage detection methods are based on natural frequencies and mode shapes, which suffer from measurement errors in practice. The errors can potentially influence the damage identification result. Despite this fact, researchers rarely take measurement scatters into account. In a few papers, some hints are given on how to deal with the errors. For example, in [6], damage identification accuracy is controlled by calculating three errors connected with natural frequencies, the amplitude correlation coefficient and the system response in the time domain. In [7], the authors have proved that it is necessary to average the values of the damage measure calculated for the erroneous modal data in order to obtain a true result.

In this article, the influence of errors on the damage indices defined in the flexibility and modal curvature approaches is studied. The goal is to work out an efficient procedure for real engineering cases in which modal characteristics, such as natural frequencies, and mode shapes are measured in situ. The modal characteristics are affected by two kinds of errors: measurement errors and the errors resulting from the modal identification method. The measurement errors can be avoided, or at least reduced, as the researcher gains experience (e.g., [9]). The errors connected with modal identification method are another issue.

From a practical point of view, it is convenient for the modal identification to rely on measurements made during ambient excitations. Thus, the dynamic tests are performed during natural construction operation. Currently, there are several methods of modal identification on the basis of ambient vibrations. The algorithms are grouped and labelled as Operation Modal Analysis (OMA) methods [10]. A comparative study of their efficacy in evaluating bridge dynamic properties from experimental data is presented in [11]. The authors compare peak picking (PP), frequency domain decomposition (FDD), rational fraction polynomial (RFP) and subspace identification (SI) methods. A theoretical background of those methods can be found, for example, in [12], [13], [14].

In the research presented in this paper, the peak picking method has been applied, primarily because the statistical errors can be calculated according to formulae specified by the authors of the method [12].

Section snippets

Modal curvature method

Pandey et al. are the first to report on the relation between the mode shapes and the local structural damage [3]. The proposed damage indicator is formulated as the absolute difference between the mode shape curvatures of the damaged and the intact structure. It is proved to be effective in computer simulations of beams. Other researchers use this method and report its efficacy with respect to more complex structures (e.g., [5]). The author of this paper has tested this method in computer

Statistical errors of experimental modal identification in the peak picking method

As stated in Section 1, the experimentally determined natural frequencies and mode shapes are affected by statistical errors related to the identification method. It is assumed that the measurement errors are small and can be neglected.

In order to identify the building structure modal characteristics, the author uses the peak picking method. The approach is discussed in detail in [12]. The basic functions of the peak picking method are the auto-spectra Gpp(f) and the cross-spectra Gpq(f)

Damage indices in the curvature and flexibility approaches for erroneous modal characteristics

It is obvious in the literature that normalised damage indices provide quantitative information about the extent of the damage. The normalised indices for the curvature and the flexibility approaches are as follows:ZC(i)=WC(i)-W¯CσC,ZF(i)=WF(i)-W¯FσF,where W¯CandW¯F denote the mean values of WC(i) and WF(i), and σC and σF are the standard deviations of the indices. Each normalised index represents the standard deviation multiplicity, which is the difference between each index W(i) and the mean

A case study – a simply supported beam

The influence of natural frequency and mode shape scatters on damage position identification is analysed with computer simulations of a simply supported beam. It is a model of a double T steel beam, commonly used in engineering practice. The main moments of inertia differ greatly in this kind of cross-section; thus, there is an ability to compare the damage index efficacy in a more or a less stiff direction of the cross-section. The geometry of the beam is presented in Fig. 2. Young’s modulus

Case study – tower of the Vistula Mounting Fortress in Gdańsk

In this section, a trial to identify the most damaged region of a historic building is described. The structure of interest is the tower of the Vistula Mounting Fortress from the 15th century (Fig. 8). The tower was erected as both a lighthouse and a defensive tower. For centuries, the building was encased by a strong fortress surrounded by a moat. Warfare and fires destroyed the tower several times. The latest disaster took place in 1953, when 70% of the tower, which had been weakened in the

Conclusions

The complexity of damage identification in real engineering structures is discussed in the paper. Some practical conclusions may be mentioned. First, if modal characteristics are used for structural damage detection, the modal identification error should be taken into account because it can influence the result. The presented analysis has proved that neglecting modal characteristic errors in damage detection procedures can lead to false results. Furthermore, when erroneous modal data are the

Acknowledgements

This work has been supported by the Centre for Urban Construction and Rehabilitation CURE, founded by the 5th EU Framework Programme, acting at the Gdańsk University of Technology in Poland. This support is kindly acknowledged.

The author expresses great gratitude to Professor Czesław Szymczak and to the late Professor Henryk Walukiewicz for their scientific guidance.

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