Statistical Modeling Methods for Structural Damage Identification

Early damage detection is very important for in-time repair and retrofit of civil structures. The past years have seen a growing interest in automatic/programmable damage detection techniques using measurements from structural monitoring because of its potential economic benefit for maintenance practices. Recently, statistical pattern recognition (SPC) has been brought forward as a promising way to realize this goal. The essence of SPC is to use well-defined concepts in statistics for data classification, which is an intuitive way to account for the inherent uncertainty in the data acquisition process. In this paper, two types of statistical damage detection algorithm are described and applied to the acceleration responses collected from a scaled two-bay steel frame subjected to random excitation. Damage is introduced into this specimen by replacing two members near the right joint with more flexible ones. The influence coefficient method for damage detection is based on correlation analysis between responses measured at two different locations, while the autoregressive (AR) modeling method evaluates the change in coefficients and residual characteristics of the AR model estimated from response from only one sensor node. Influence coefficients here are basically linear regression coefficients of one node’s response on another, and AR model, as indicated by its name, regress the current response on those from several time steps back. Both techniques have successfully identified the damage existence in the steel frame. It is shown that the amount of change occurred in the influence coefficient values from different node combinations can be used to localize damage. For the AR modeling method, the autocorrelation function of residuals at each sensing unit is proved to be a more effective damage location indicator than Mahalanobis distance of model coefficients at each sensing unit in this case. INTRODUCTION In recent years, data-driven structural health monitoring (SHM) has been actively investigated by the civil engineering community as an important tool for future infrastructure maintenance.[2] The availability of affordable sensing and data transmission/storage systems has made continuous structural monitoring physically feasible [11]. In addition, the occasional failures of important transportation infrastructures (e.g. Minnesota I-35W) have attracted such ample media coverage and public attention that more effective structural assessment and repair/retrofit methods are much sought after to ensure public safety. The development of data-driven SHM will lead to automatic structural diagnosis, which will hopefully be less biased and less expensive than traditional human inspection methods. Generally, data-driven structural health monitoring consists of two steps: periodically sampling data from the structure being monitored and compressing the large quantity of data for possible structural damage information. Most existing structural monitoring systems measures only vibration responses, as they are the easiest to obtain. A paradigm for vibration-based damage detection has been proposed by Farrar et al. [3], where this procedure in divided into four stages: 1) Operational evaluation; 2) Data acquisition and cleansing; 3) feature selection and data compression, and 4) statistical model development/decision making. The first two stages take care of data collection details, while the last two are about decision making based on the data collected. For a long time, step 4) of this paradigm has been overlooked; a lot of literatures can be found on feature extraction, but in comparison few researches address the problem of statistical damage threshold construction. Realizing this problem, a series of studies has been conducted by Farrar and his colleagues on hypothesis testing [6] for damage threshold determination, which is a part of their statistical pattern recognition (SPC)[5] scheme for structural damage identification. SPC is a family of methods that performs data/pattern classification using statistical analysis concepts. This approach concerns both feature extraction and threshold evaluation, is well suited to process large amount of data, and automatically accommodates the inherent uncertainty of the signal acquisition process inside its framework. To obtain an objective evaluation of its potential for damage identification, however, results from its application to various types of structures are needed. Also, to understand the pros and cons of different statistical approaches they need to be compared and contrasted. In this paper, two statistical damage detection methods will be used on acceleration measurements collected from a two-bay steel frame; the influence coefficients method is based on regression between the responses from two nodes, and the autoregressive(AR) [1] modeling method evaluates the change in estimated model coefficients and residuals. Data-driven thresholds from change point analysis and cross validation [4] will be used for the features extracted from these two methods. The organization of this paper is as follows; Section 2 presents the influence coefficient method for structural damage identification, which is basically regression of one node’s response on another; Section 3 is dedicated to description of univariate autoregressive modeling and damage indices from the model parameters and residuals. Section 4 focus on the threshold evaluation schemes for the damage features using resampling techniques. In section 5, the damage detection algorithms are applied to acceleration measurements collected from a two-bay steel frame specimen, and the results are presented and compared. STATISTICAL METHOD 1: LINEAR REGRESSION BETWEEN RESPONSES FROM TWO SENSOR NODES If a linear structure is under static/quasi-static loading, then the structural response at any two locations should be linearly correlated: Here and indicate the response location, and is a time label. The correlation coefficient is determined by both the force distribution and structural stiffness properties. In practice, most structures are subjected to dynamic loads. However, if only a small part of the structure (with large stiffness and insignificant mass) is monitored, then it can be assumed that the local behavior could be captured by a static model. This pair-wise regression method has been applied to detect damage in a beam column specimen and a simulated model of the steel frame here [7, 8]. The feature extraction methodology can be summarized into the following steps; 1) for each pair of and , the correlation coefficients and and the corresponding residuals are evaluated using the least squares method (Fig.1); 2) to check the stability of estimation, two indices are calculated: accuracy factor and normalized estimation error ; 3) correlation coefficients from those node pairs with close to 1 and close to zero are selected as damage indicators. This approach is straightforward in principle and efficient for computation. Also, the change in the values of regression coefficients can reflect to some extent the location of damage. However, when the dynamic response content becomes prominent (i.e. regression is performed for two far-apart nodes), the algorithm performance will degrade. STATISTICAL METHOD 2: AUTOREGRESSIVE MODELING OF RESPONSE AT ONE LOCATION Univariate autoregressive model[1] is among the most widely applied time series analysis tools. It basically expresses the value of the signal at a certain time point as a linear combination of its previous values and a random error term : Here are the model coefficients, and is the model order. AR coefficients can be estimated from collected signals using one of the standard algorithms such like Yule-Walker and Burg, and model residuals/errors can then be obtained as the difference between model prediction and the real signal. Autoregressive damage detection algorithms can be based on either model coefficients or -0.05 0 0.05 -0.04 -0.02 0 0.02 0.04 ui u j data reg. line Figure 1 Regression for influence coefficients residual characteristics. In the remainder of this section, one example from either category will be presented. The Mahalanobis distance of AR coefficients It has been proved that if the signal is really an AR process, then any regular coefficients estimator from the signal is asymptotically unbiased and normally distributed with covariance matrix [1]. Therefore, a metric that represents the coeffcients’ deviation in the probability space of normal distribution seems a good choice of damage feature. Mahalanobis distance[10] is such a metric defined from the definition of multivariate normal distribution. The estimator of the Mahalanobis distance between a potential outlier vector and baseline sample set can be obtained as where is the average of the baseline sample feature vectors, and the estimated covariance matrix. When applying this method, the baseline signals are first segmented (often with large overlap) and for each segment an AR coefficient vector are estimated. Signals from current structural state are processed likewise and for each coefficient vector obtained its Mahalanobis distance to the baseline coefficients set will be computed. These Mahalanobis distance features are then compared with the Mahalanobis distances within baseline set. When the structural system is damaged, it is expected that the Mahalanobis distance feature for AR coefficients will increase significantly (Fig. 2). 0 50 100 150 200 250 300 350 400 450 0 20 40 60 80 100 120 140 160 segment num. sq ua re d M ah al an ob is d is ta nc e from healthy state signal 1 (baseline) from healthy state signal 2 from damaged state Figure 2 Mahalanobis distance for structural damage identification Q-statistic trace of AR model residuals If the structural condition has undergone changes, then the baseline model will no longer provide a good fit to the new signals collected. As the result, the residual series, instead of resembling a white noise, shall carry some identifiable patterns over time. Ljung and Box[1] proposed a statistic to measure the difference between the residual series and white noise: where is the sample size, is the number of lags, and is the autocorrelation at the lag. Since this statistic follows a distribution under the normality assumption of the input, a statistical test can be devised at significance level , by setting the rejection threshold at , which is the quantile of the distribution with degrees-of-freedom. The Q statistic trace is a plot of the Q statistic against the baseline AR model order (Fig. 2). For undamaged state signal, a downward trend is expected in the plot. Thus if the trace is oscillating, stopping decreasing at short or going upward as the model order increase, the system is recognized as damaged. 0 20 40 60 80 100 0 1 2 3 4 5 x 10 4 AR order Q st at is tic Q statistic trace undamaged state undamaged state (baseline) damaged state Figure 3 Ljung-Box statistic for structural damage identification THRESHOLD CONSTRUCTION METHODS After feature selection, appropriate damage threshold construction schemes are needed to reach a decision on the current state of the structure being monitored. If the feature distribution can be assumed, then statistical hypothesis testing could be used for damage identification. However, in most cases the feature distribution is unknown and cannot be approximated by an analytical function. As such, the damage threshold will be determined based on the data solely using one of the resampling techniques. Change point analysis using cumulative sum Change point analysis is used to find the point in a data sequence where the data characteristics change and the confidence level associated with this change point. It is an effective way of identifying thresholds and detecting subtle variations. A variety of approaches is available for performing change point analysis, such as cumulative sum, deviance reduction, and least squares. Here, the cumulative sum method [12] will be adopted for damage threshold construction for the influence coefficients features. The cumulative sum for a data sequence is calculated as below: 1) Subtract every value in this sequence by its mean . 2) Compute the cumulative sum at step by adding up the values occurred before and at . (i.e. ) 0 1 2 3 4 5 6 7 8 9 10 0 0.05 0.1 0.15 0.2 0.25 0.3 sample num cu lm ul at iv e su m Figure 4 the cumulative sum plot from the implementation in section 5. Experienced individuals can identify the change point directly as the place where the trend of the cumulative sum plot changes.(Fig.4) But as our aim is automatic damage threshold construction, a bootstrap analysis is performed; the original signal sequence is randomly permuted for times and each time the maximum absolute value of the cumulative sum of the new sequence is recorded. The threshold for the maximum absolute accumulative sum of a data sequence of constant properties is then set at the point above which values occurred. The significance level is set to 5% in all the applications in this paper. Cross validation As described in the previous section, the Mahalanobis distance feature is computed between a certain coefficients vector and a set of reference vectors. Its theoretical distribution is hard to derive and the feature itself often exhibit large fluctuations even within the healthy state. Thus, a cross-validation approach is devised for its damage threshold construction. It is essentially an empirical estimation of the feature distribution through recomputing the statistic for many a time by leaving out a certain portion of the observations, and can be viewed as a variation of standard bootstrapping technique [4]. ... segmentation ... segmentation AR coefficients estimation (e.g. Yule-Walker) AR coefficients for the left-out segment Mahalanobis distance Empirical distribution Threshold Figure 5 the ‘cross-one-out’ threshold evaluation method for Mahalanobis distance of model coefficients Fig. 5 is a flow chart of this approach. It can also be described in words as follows: First a segment is cut from the baseline signal at a random time point and reserved for testing, and sample segments of the same size are cut with a preset overlap from the remaining signal. The Mahalanobis distance between the AR model coefficients of left-out segment and those of the sample set is then computed and stored. This process is repeated for a large number of times and the value beyond which 5% of the tests occur is used as threshold in subsequent analysis. This method has already been applied in several previous researches [13, 14], and is shown to demonstrate superior performance to other threshold evaluation techniques such as hypothesis testing based on Gaussian assumption of model coefficients and Mont Carlo simulation [11, 13, 14]. EXPERIMENTAL RESULTS The aforementioned influence coefficients and AR modeling method, together with their respective threshold evaluation approaches, are applied to detect damage in a scaled two-bay steel frame constructed from steel tubes. (Fig 6) The structure, instrumented with 21 accelerometers, is excited from the left beam-column joint by an electro-dynamic shaker (Fig 7). To simulate a structural damage, the potion between sensor R1 and R3 and that between R4 and R6 are switched out by 20% less stiff tubes. For each structural scenario, five random vibration tests are performed and the acceleration responses collected are used as input to the algorithms. The first algorithm produces for each dataset influence coefficients, each from a particular node combination. To facilitate the subsequent decision making process, only those coefficients with average evaluation accuracy factor >0.9 and normalized estimation error are examined. Out of the 24 coefficients selected (Table xx), the cumulative-sum-based change point analysis successfully identified damage for 21 of them over the 10 tests. It is observed, however, that the 3 coefficients that failed to report damage are all from regression between distant nodes, which are by nature not quite reliable. Figure 7 a schematic plot of the girder mounted with accelerometers Figure 6 the two-bay steel frame Fig 7 displays the estimation results from 4 influence coefficients, all indicating damage except for the one showed in the lower right corner. One interesting fact noticed is that though the nodes 5 and 6 are nowhere near damage, their corresponding influence coefficient value still increased abruptly as a result of structural change. This is probably because these correlation coefficients reflect the ratio of structural vibration ‘shape function’ values evaluated at two different locations, and those combinations with most affected coefficients are determined by not only the damage location but also the structural layout. Yet still, generally there will be a higher chance of observing changes near the damage for a properly restrained regular structure . Also if an analytical model of the structure is available, the influence coefficients can be used as input to a model updating scheme to find out the damage location.