Mathematical Problems in Engineering

Volume 2018, Article ID 6752456, 13 pages

https://doi.org/10.1155/2018/6752456

## Multidamage Detection of Bridges Using Rough Set Theory and Naive-Bayes Classifier

School of Resources and Civil Engineering, Northeastern University, Shenyang 110819, China

Correspondence should be addressed to Li Liang; moc.361@uen-ll

Received 10 December 2017; Revised 3 April 2018; Accepted 10 April 2018; Published 27 May 2018

Academic Editor: Filippo Ubertini

Copyright © 2018 Shuang Sun 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

This paper is intended to introduce a two-stage detection method to solve the multidamage problem in bridges. Vibration analysis is conducted to acquire the dynamic fingerprints which are regarded as information sources. Bayesian fusion is used to integrate these sources and preliminarily locate the damage. Then, the RSNB method which combines rough set theory and Naive-Bayes classifier is proposed to simplify the sample dimensions and fuse the remaining attributes for damage extent detection. A numerical simulation of a real structure, the Sishui Bridge in Shenyang, China, is conducted to validate the effectiveness of the proposed detection method. Data fusion based method is compared with single-valued index method at the damage localization stage. The proposed RSNB method is compared with the Back Propagation Neural Network (BPNN) method at the damage qualification stage. The results show that the proposed two-stage damage detection method has better performances in regard to transparency, accuracy, efficiency, noise robustness, and stability. Furthermore, an ambient excitation modal test was carried out on the bridge to obtain the vibration responses and assess the damage condition with the proposed method. This novel approach is applicable for early damage detection and provides a basis for bridge management and maintenance.

#### 1. Introduction

Rapidly aging bridges are a serious problem in many countries. Bridges deteriorate gradually due to environmental effects and traffic loads, making them commonly structurally deficient over the course of their service life [1]. The early detection of damage is very significant to ensure bridge health and integrity, prevent catastrophic failure, and prolong their service life.

The vibration based damage detection method, as a kind of nondestructive evaluation technique, can be used to detect damage in bridges. Its basic principle is that damage causes changes in the bridge’s structural properties which modify the vibration characteristics [2]. There are two general approaches of vibration based damage detection. First is the model updating approach, which involves updating certain parameters to ensure agreement between the experimentally measured modal parameters and the intact finite element model. The primary drawback of this method is that, due to the inevitable errors in modeling assumption and measurement, many models would reasonably match observations, which might lead to the wrong model identification [3]. The second most popular approach is the dynamic fingerprint method. Dynamic fingerprints are functions of the structural physical properties (mass, stiffness, and damping) and modal parameters (natural frequencies and mode shapes). Structural condition can be assessed through comparisons of the tested and intact responses. Although dynamic fingerprints are damage-sensitive they are limited in terms of application to civil infrastructures because the responses extracted from field sensors are readily influenced by environmental noise [4].

Recently, data fusion based damage detection method has attracted much attention [5–8]. Structural vibration responses are regarded as multiple sources and then transmitted to the fusion system. It can effectively solve the problems such as limited useful data and weak identification accuracy generally occurring in the field vibration data. Neural network, as a common data fusion technique, has been widely utilized by many researchers to identify structural damage. It mimics human cognitive processes to establish relations between input vibration responses and output damage condition. Papers that discuss neural networks on structural damage detection include, Yeung and Smith [8], Mehrjoo et al. [9], Jiang et al. [10], Min et al. [11], Shu et al. [12], Ugalde et al. [13], and Abdeljaber and Avci [14]. The basic and most important issue of this method is the training of a multilayer feed-forward neural network. A large number of neurons are needed to favor a better approximation, inevitably leading to a more complex system and higher computational cost. In neural networks, the mathematical approximation to describe the input and output relations is a black-box process. The disadvantage is the limited insights into the development of the model, which are hardly understandable by the decision maker [10].

In this work, we propose the incorporation of several fusion theories with recent developments in artificial intelligence to establish a transparent and efficient damage detection system. The proposed detection process begins by locating the damage through Bayesian fusion; then, damage extent is classified by RSNB method with the combination of rough set theory and Naive-Bayes classifier. To fully illustrate the detection performance of the proposed method, the damage location detection results by Bayesian fusion are compared with those by single-valued dynamic fingerprint. Damage extent detection by RSNB is compared with BPNN in terms of transparency, accuracy, efficiency, robustness, and stability. The overall organization of this paper is as follows: in Section 2, dynamic fingerprints are calculated to acquire damage information. In Section 3, the Bayesian fusion method is employed in order to achieve damage localization. In Section 4, a novel method called RSNB which makes use of rough set theory and Naive-Bayes classifier is presented for damage qualification. In Section 5, an example of application of the proposed method is provided. Conclusions are drawn in Section 5.

#### 2. Dynamic Fingerprints

The structural damage can be detected by comparing the changes of dynamic fingerprint between the intact and damaged states at the measuring points.

##### 2.1. Modal Flexibility Curvature (MFC)

If any damage exists in a bridge, the stiffness will be reduced and cause a change of the flexibility because the structural stiffness matrix and flexibility matrix are reciprocal inversion matrices [15]. The flexibility matrix can be expressed as follows:where is the th frequency; is the th mode shape vector; is the number of degree of freedoms (DOFs).

Modal flexibility difference can be written as follows: where , are the intact and damaged modal flexibility matrices, respectively.

The maximum absolute values in each column of are chosen to form a row matrix:

Then the modal flexibility curvature is calculated by a central differentiation procedure. It can be expressed as the following [16]:where is the th element in the row matrix ; is the distance between two adjacent nodes.

##### 2.2. Curvature Mode Shapes Difference (CMSD)

Curvature mode shapes can be obtained from the mode shape displacement [17].where is the modal displacement at node of the th order; is the distance between two adjacent nodes.

The structural curvature mode shape difference of each order is given by the following:where and are the intact and damaged CMS, respectively.

##### 2.3. Uniform Load Surface Curvature Difference (ULSCD)

Uniform load surface (ULS) is the derivative of modal flexibility [18]. For a linear system, the modal deflection at node under uniform unit load all over the structure can be approximated as the following:where , represent the node numbers; is the modal displacement at node of the th order; is the number of mode orders.

Based on the second-order difference principle, uniform load surface curvature can be given by the following:where is the distance between two adjacent nodes.

Uniform load surface curvature difference is defined as the following:where , are the intact and damaged ULSC, respectively.

#### 3. Two-Stage Damage Detection System

In this paper, a data fusion based damage detection system is set up. Dynamic fingerprints of every measurement point are sent to the fusion center. Then, a global decision is obtained to locate the damage. The fingerprints of the damaged objects are further used to detect the damage extent. The two-stage structural damage detection system is shown in Figure 1, and the main procedures are listed as follows: