Journal of Structures

Volume 2015, Article ID 976349, 16 pages

http://dx.doi.org/10.1155/2015/976349

## A Substructural Damage Identification Approach for Shear Structure Based on Changes in the First AR Model Coefficient Matrix

Department of System Design Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan

Received 15 April 2015; Accepted 1 June 2015

Academic Editor: Elsa de Sá Caetano

Copyright © 2015 Liu Mei 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

A substructural damage identification approach based on changes in the first AR model coefficient matrix is proposed in this paper to identify structural damage including its location and severity. Firstly, a substructure approach is adopted in the procedure to divide a complete structure into several substructures in order to significantly reduce the number of unknown parameters for each substructure so that damage identification processes can be independently conducted on each substructure. To establish a relation between changes in AR model coefficients and structural damage for each substructure, a theoretical derivation is presented. Thus the accelerations are fed into ARMAX models to determine the AR model coefficients for each substructure under undamaged and various damaged conditions, based on which changes in the first AR model coefficient matrix (CFAR) is obtained and adopted as the damage indicator for the proposed substructure damage identification approach. To better assess the performance of the proposed procedure, a numerical simulation and an experimental verification of the proposed approach are then carried out and the results show that the proposed procedure can successfully locate and quantify the damage in both simulation and laboratory experiment.

#### 1. Introduction

Research and development of structural health monitoring (SHM) are getting strong attention for evaluating and maintaining structural integrity of an aging building or a suffering structure against natural hazards such as large earthquakes and strong winds [1]. The main parts of the SHM in civil engineering are damage detection and localization, which are essential monitoring zones for structures after major events such as earthquakes.

Damage identification based on vibration data generated by SHM systems has been extensively studied for several decades and the literature on the subject is rather immense. One of the earliest damage indicators studied is the estimated modal property from system identification [2, 3], as it is directly related to structural physics according to classical dynamics theory. The basic idea is that modal parameters, such as frequencies, mode shapes, and modal damping, are functions of the physical properties of the structure (mass, stiffness, and damping). Therefore changes in the physical properties will cause changes in the modal properties. Thus they are often referred to as mode-based methods. In 1979, Cawley and Adams [4] proposed frequency shifts as a damage indicator to detect damage in composite materials. Doebling and Farrar [5] considered the changes in the frequencies and mode shapes as the damage indicator of a bridge. A comprehensive summarization of damage identification techniques using changes in natural frequencies was presented by Doebling et al. in [6].

However, the modal parameters are proved to be insensitive to local damage for the fact that they are global properties of the structure while damage is a local phenomenon. Additionally, traditional mode-based methods, especially those that operate in time domain, are usually computationally intensive to implement and physical or finite-element models are necessarily required for those methods. In order to overcome these obstacles, time-series-based methods become another important category within the broader family of vibration-based methods for damage identification purposes. Comparing with the traditional mode-based methods, although time-series-based methods are also concerned with numerical modeling, they are more flexible because they are data-based rather than physics-based, which can use various damage features that do not necessarily have an explicit physical meaning [7]. Among those time-series-based methods, the autoregressive (AR) model, the autoregressive with exogenous input (ARX) model or the autoregressive moving average with exogenous input (ARMAX) model is mathematical structures that can be used to formulate various data-driven damage features. By adopting one of the standard algorithms, AR/ARX/ARMAX model parameters can be estimated from input-output datasets very efficiently. According to the specific feature extraction process, the damage features generated from these models can be classified into two categories: model coefficients based and model residual based. There are a lot of model coefficients based techniques that have been investigated during the past decades. Sohn and Farrar [8] proposed statistical pattern recognition methodology in which the recorded dynamic signals were modeled by adopting the AR time-series models and then classified from either undamaged or damaged systems by statistically examining changes in AR coefficients. Nair et al. [9] proposed a sensitive damage feature by only using the first three AR coefficients of the ARMA model. By looking into various changes in coefficients of the vector seasonal autoregressive integrated moving average (ARIMA) model, Omenzetter and Brownjohn [10] proposed a health monitoring algorithm for bridge structure by analyzing the time histories of static strain data. de Lautour and Omenzetter [11] adopted an artificial neural network (ANN) to detect the extent of the damage by feeding the AR coefficients as input features into it. On the other hand, model residual based methods have also been receiving attention and a large amount of related research has been documented and published. Fanning and Carden [12] proposed a statistical process control approach to detect damage by using the mean and variance of the residuals of the AR model to form the statistical process control charts. Mattson and Pandit [13] chose the standard deviation of the residual of the vector AR (VAR) model as the damage-sensitive index.

Though numerous damage identification methods mentioned above are already developed for SHM systems, most of them are not feasible or practical for large-scale civil structures due to the challenges such as high equipment costs, long setup time, difficulties in cabling, and the long computation time. To overcome these problems, some researchers have been using the substructure method for local damage identification of the large-scale structures. Koh et al. [14] are considered to be the first to present the concept of substructure identification. In their approach, the extended Kalman filter (EKF) was used as the numerical tool to identify unknown structural parameters. Yun and Lee [15] applied the sequential prediction error method to estimate unknown parameters of each substructure with noisy measurements. Park et al. [16] proposed structural damage identification methods based on the relative changes in localized flexibility properties, which are obtained either by applying a decomposition procedure to an experimentally determined global flexibility matrix or by processing the output signals of a vibration test in a substructure-by-substructure manner. Tee et al. [17] proposed a substructure identification method considering both first-order and second-order models. Koh and Shankar [18] proposed a substructure identification approach in frequency-domain without the need of interface measurement. Several new substructure methods have been developed in recent years. Xing and Mita [19] proposed a substructure approach to divide a complete structure into several substructures in order to significantly reduce the number of unknown parameters for each substructure so that damage identification processes can be independently conducted on each substructure. Hou et al. [20] proposed and experimentally studied a substructure isolation method which can be applied for local structural health monitoring and damage identification by virtually isolating the substructure from a large and complex global structure into a simple, small, and independent structure. Kuwabara et al. [21] proposed a damage identification method for high-rise buildings which is devised to find the story shear and bending stiffness of a specific story from the floor accelerations just above and below the specific story. Zhang et al. [22] presented a loop substructure identification method to estimate the parameters of any story in a shear structure using the cross power spectral densities (CPSD) of structural responses. Lee and Eun [23] presented a model-based substructuring method in which the damaged substructure is detected by tracing the distribution of the constraint forces at the nodes between the partitioned substructures and the local damage is found by the displacement curvature of the isolated substructure.

However, there are still several obstacles to application of the existing methods in practice. Some of these methods need a large amount of measured data and aim at the identification of the mass, stiffness, and damping matrices of the structure, thus too complicated to require long computation time. And some methods require the construction of structure models, but in many cases such models will be overly complex and possibly even impossible to obtain in reasonable time due to the complex nature of many realistic structural systems. Additionally, some other methods require the application of statistical pattern recognition method, in which training data sets are necessary from both the undamaged and damaged states of a system for the statistical modeling. This kind of methods is always impractical for the fact that it is extremely difficult to obtain the training data sets from various damaged states of the structure especially when it is in constant use. Finally, for some others, noise immunity problems of the methodologies are not verified, which may possibly cause some difficulties in the case of practical application. Thus in this paper, a substructural approach based on changes in the first AR model coefficient matrix has been proposed for local damage identification in shear structures. Firstly, a substructure algorithm is used to divide a complete structure into several substructures, each of which shares a common form of the equation of motion. Then the equation of motion for each substructure is rewritten in terms of ARX model with different inputs and outputs. In what follows, it is derived theoretically that the elements of changes in the first AR model coefficient matrix (hereafter will be termed as CFAR) corresponding to the output DOFs adjacent to the damaged location are proportional to the stiffness reduction in the structure, indicating the damage location and severity. Thus the accelerations are fed into autoregressive-moving average with exogenous inputs (ARMAX) models to determine the AR model coefficients for each substructure under undamaged and various damaged conditions, based on which the CFAR is obtained and adopted as the damage indicator for the proposed substructure damage identification approach afterwards. After presenting the proposed approach, a numerical simulation and an experimental verification were conducted and the results were presented to show the feasibility and stability of the proposed methodology. Finally, conclusions and expectations were discussed.

#### 2. Proposed Method

To illustrate the concept of substructuring, consider a shear building which is represented by a lumped mass system as shown in Figure 1(a). The dynamic equation of motion for the complete structure iswhere , , and are the mass, damping, and stiffness matrices, respectively, , , and are the dynamic response vectors of displacement, velocity, and acceleration relative to the ground, respectively, is an unit vector (), and is the ground acceleration.