Shock and Vibration

Volume 2015, Article ID 187956, 10 pages

http://dx.doi.org/10.1155/2015/187956

## Damage Identification of Bridge Based on Chebyshev Polynomial Fitting and Fuzzy Logic without Considering Baseline Model Parameters

Department of Transportation, Jilin University, No. 5988, Renmin Street, Changchun, Jilin 130025, China

Received 15 August 2012; Accepted 16 February 2013

Academic Editor: Reza Jazar

Copyright © 2015 Yu-Bo Jiao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The paper presents an effective approach for damage identification of bridge based on Chebyshev polynomial fitting and fuzzy logic systems without considering baseline model data. The modal curvature of damaged bridge can be obtained through central difference approximation based on displacement modal shape. Depending on the modal curvature of damaged structure, Chebyshev polynomial fitting is applied to acquire the curvature of undamaged one without considering baseline parameters. Therefore, modal curvature difference can be derived and used for damage localizing. Subsequently, the normalized modal curvature difference is treated as input variable of fuzzy logic systems for damage condition assessment. Numerical simulation on a simply supported bridge was carried out to demonstrate the feasibility of the proposed method.

#### 1. Introduction

Bridge structures have endured progressive deterioration due to vehicle load and aging of material under the effect of external environment. In addition, the frequent occurrence of natural disasters could also cause damage and further accelerate the deterioration of bridge condition. Therefore, the damage assessment of bridge has attracted great interest of many researchers [1, 2].

Vibration-based damage identification method has been widely applied in the past few decades [3, 4]. Theoretical background of this method is that modal parameters (frequency, mode shape, etc.) are functions of physical properties (stiffness, mass, etc.) of bridge. Therefore, the changes of modal characteristics can be treated as the damage indicators. The commonly used modal parameters for damage identification include natural frequency, mode shape, and their derivatives, such as mode shape curvature, modal strain energy, and modal flexibility [5–9].

Among these modal characteristics, natural frequency has no relation with measuring positions and can be measured most conveniently and accurately. However, the frequency-based method possesses several apparent drawbacks. For instance, it is inferior sensitive to minor damage, and it is easily affected by environment [10]. Additionally, the simply supported reinforced concrete bridges are spatial symmetric structures. Natural frequency cannot distinguish damage at symmetric locations in these symmetric structures.

Comparing with natural frequency, mode shape and its derivatives contain the spatial information with respect to location of damage, and they are found to be better indicators for damage identification. The modal curvature is recognized as a more advanced damage indicator which is calculated from the displacement mode shape and firstly proposed by Pandey et al. [7]. It is found that the absolute changes in the curvature mode shapes between undamaged and damaged structures can effectively locate the damage region for cantilever and simply supported beam model. The changes in the curvature mode shape increase with increasing size of damage. However, this method requires the baseline data from intact structures. This baseline model data can be obtained by finite element simulation. In actual simulation, the accuracy of finite element model is affected by temperature because the elastic modulus of concrete is temperature dependent [11]. However, the modulus of elasticity versus temperature curve for concrete is unclear and complicated [12, 13]. If their relationship is not reasonable to consider, it will lead to incorrect result for damage identification.

To avoid this difficulty, several researchers have conducted effective work for damage detection without baseline model data. Ratcliffe [14] proposed a modified Laplacian operator-based method for 1D beam, which determines the damage location through a cubic curve fitting for the modal data obtained only from the damaged structure. Wu and Law [15] proposed a damage localization method based on uniform load surface (ULS) curvature for 2D plates, which can identify the damage using only the modal characteristics of damaged state if a gapped-smoothing technique is applied. Yoon et al. [16] extended the 1D gapped-smoothing method to the 2D gapped smoothing method for the damage identification of plate. And the baseline data of undamaged structure are not needed. Zhong and Oyadiji [17] presented a novel approach for crack detection based on difference between two sets of detail coefficients obtained by stationary wavelet transform without requiring the modal data of undamaged beam as a baseline. Yoon et al. [18] extended the gapped smoothing method for identifying the location of structural damage in a beam by introducing the global fitting method, which uses the modal shape data of damaged structure with an assumption that the undamaged structure is homogeneous and uniform. Cao and Qiao [10] adopted a novel Laplacian scheme for modal curvature-based damage identification to improve the anti-noise ability of standard Laplace operator. Rodríguez et al. [19] presented a Baseline Stiffness Method (BSM) to locate and evaluate the magnitude of structural damage whose baseline state is unknown.

Modal curvature can realize the damage localization and qualitatively determine the damage degree. But it is not suitable for quantitative identification. Artificial neural networks (ANNs) have been utilized by many researchers to identify damage location and severity [20–25], as they can achieve the nonlinear mapping between the inputs and outputs from certain samples training. Ko et al. [20] developed a three-stage scheme for damage detection of the cable-stayed Kap Shui Mun Bridge. ANNs are used in the first and third stage for damage alarming and specific damaged member(s) identification. Sahin and Shenoi [21] presented a damage detection algorithm using a combination of global (changes in natural frequencies) and local (curvature mode shapes) vibration-based analysis data as input in ANNs for location and severity prediction of damage in beam-like structures. Lee et al. [22] proposed an ANNs-based damage detection method using the differences or the ratios of the mode shape components as the input variables, which can effectively consider the modeling errors in the baseline finite element model. Bakhary et al. [23] proposed a statistical approach which takes the effect of uncertainties into account in developing an ANNs model.

ANNs can be able to predict an output pattern when they recognize a given input pattern, but there exists an inherent inability to represent knowledge acquired by the network in an explicit form. That is the “black box” problem of ANNs [26, 27]. Fuzzy systems allow for easier understanding because they are expressed in terms of linguistic variables [28, 28]. And they are finding increasing use in structural damage identification. Ganguli et al. [29, 30] adopted the natural frequency and modal shape curvature as the input of fuzzy logic systems and genetic fuzzy logic systems for the damage detection of helicopter rotor blades. Zhao and Chen [31] proposed a method based on principal component analysis, modified mountain clustering method, descent method, and fuzzy logic systems for the damage detection of concrete bridges. Taha and Lucero [32] introduced new techniques based on Bayesian updating and fuzzy sets to improve pattern recognition and damage detection of structures.

In this study, a new acquisition method for modal curvature difference (MCD) is proposed. Firstly, the modal curvature of damaged structure can be calculated through central difference approximation based on mode shape data. Further, modal curvature of undamaged structure can be obtained by Chebyshev polynomial fitting based on the modal curvature at feature nodes of damaged structure. Therefore, MCD parameters can be calculated by use of modal curvatures before and after damage. Subsequently, a fuzzy logic system for damage condition assessment of bridge is constructed using normalized modal curvature difference (NMCD) as input variable. Numerical simulation of a simply supported beam bridge verifies the feasibility of the proposed method.

#### 2. Theoretical Background

##### 2.1. Modal Shape Curvature

The curvature of Euler-Bernoulli beam can be expressed by where is the curvature of displacement curve and and are the first and second derivatives of displacement function, respectively. Based on the assumption of small slope of beam displacement, the first derivative is close to zero; thus the denominator in (1) is equal to one. Therefore, the following expression can be used for calculating the curvature of beam:

Considering that central difference can obtain a more accurate approximation, its calculation results satisfy the project requirements [33]. Therefore, it is widely applied to estimate modal curvatures from the mode shapes in practice. It can be expressed by where represents modal curvature, represents the mode number, represents the corresponding node number, is the element length, and is the mode shape value for the th node of the th mode.

The modal curvature difference () is obtained by subtracting the undamaged modal curvature vector from the respective damaged one, given by where and are the modal curvatures for damaged and undamaged structure, respectively.

##### 2.2. Chebyshev Polynomial Fitting

The Chebyshev polynomials are a sequence of orthogonal polynomials which are related to de Moivre’s formula and which can be defined recursively. They are important in approximation theory because they can be used for polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge’s phenomenon and provides an approximation that is close to the polynomial of best approximation to a continuous function under the maximum norm [34].

###### 2.2.1. Definition of Chebyshev Polynomial

The -order Chebyshev polynomial can be defined by and its recursive formulas are as follows:

###### 2.2.2. Functional Expression Using Chebyshev Polynomial

An orthogonal group can be constructed through Chebyshev polynomials, and the approximating polynomial of function at independent variable interval can be expressed by where are the orthogonal polynomials group and are the corresponding coefficients.

###### 2.2.3. Curve Fitting Procedure Using Chebyshev Polynomial

*(**1) Construction of Normalized Interval*. If , it can be transformed into the normalized interval by
where , , and is the number of .

*(**2) Acquisition of Normalized Data Table.* Through the construction of normalized interval, the data table for can be transformed into .

*(**3) Access to Zero Point of **-Order Chebyshev Polynomial.* The zero point of -order Chebyshev polynomial can be calculated by
where is the degree of the polynomial. Generally, .

*(**4) Acquisition of ** Corresponding to Zero Point *. The corresponding to zero point in data table can be obtained through interpolation method.

*(**5) Determination of Coefficients *. The for orthogonal polynomials group can be calculated by

*(**6) Functional Relationship between ** and *. The data table can be expressed by the following function:

*(**7**) Fitting Error between ** and *. The Fitting error between and can be obtained by the following mean square error function:

*2.3. Fuzzy Logic Systems*

*2.3. Fuzzy Logic Systems*

*A fuzzy logic system (FLS) is an information processing system that simulates the fuzzy inference ability of human brain. It is able to simultaneously handle numerical data and linguistic knowledge. It is a nonlinear mapping of an input data vector into a scholar output. Fuzzy set is characterized by a membership function which takes on values in the interval . The Mamdani system and Takagi-Sugeno system are the most widely used fuzzy logic systems. In the Mamdani system, fuzzy sets are used both in the antecedents and in the consequents of the rules, while the consequents are represented by functions in Takagi-Sugeno system. In this paper, the Mamdani system is adopted to carry out the damage identification. The fuzzy logic controller is composed of fuzzification, fuzzy inference, and defuzzification as shown in Figure 1.*