Abstract
By exploiting the “correlation of deflections” phenomena, we propose a new bridge diagnosis method. First, we introduce the notion of deflection correlation (DC) graphs and propose a method for building a DC graph. Second, we present a new algorithm for locating the abnormal cliques. Finally, we demonstrate the potential utility of the new method by applying it to the simulated diagnosis of a reallife bridge.
1. Introduction
Structural health monitoring (SHM) means the process of monitoring the condition of a structure and detecting the damages occurring in the structure over the time. SHM is a highly interdisciplinary area of research. In particular, the detection of damages in structures involves multiple disciplines such as statistics, pattern recognition, and algorithm design.
In the past three decades, SHM has been applied to the health monitoring of grand bridges. In such applications, it is desired that the early stage damage in a bridge to be detected by examining changes in its measured responses. A large number of methods have been proposed for detecting the abnormal behaviors occurring in bridges. Now, let us make a survey on the state of the art of this area. Catbas et al. [1] suggested to evaluate bridge condition using two damagesensitive features. Catbas et al. [2], Deraemaeker et al. [3], and Yan et al. [4, 5] studied how to diagnose structural damages under varying environmental conditions. Koh and Dyke [6] designed a bridge evaluation scheme by making use of the correlation of modal parameters. Lee et al. [7] presented a bridge detection method under loading. Lee and Yun [8] and Zhang [9] propose schemes of bridge detection using ambient vibration data. In addition, some advanced techniques, such as the timeseries classification [10], the fuzzy reasoning [11, 12], the image processing [13], the interval analysis [14], the multistage identification [15], and the neural network [16], were employed for the bridge detection purpose.
A reallife bridge is subject to varying environmental and operational conditions such as traffic, temperature, humidity, wind, and solarradiation. These environmental effects also cause change in any welldefined pattern, which may mask the change caused by structural damage or by sensor failure [17]. So, the authors believe that the best approach is not to use a single feature but a suite of validated features depending on the structure and damage type. In this context, new effective bridge diagnosis methods need to be developed.
Deflections are one of the most important physical quantities characterizing the change of a bridge. Commonly, a number of deflection checkpoints on the beam of a bridge are specified and, for each checkpoint, a dedicated sensor is assigned to measure its deflection values periodically. For our diagnosis purpose, a bridge will be regarded as being composed of a number of cliques, where each clique is composed of a deflection checkpoint and its dedicated sensor. This paper addresses the following diagnosis problem.
Starting with a set of measurements of the deflection data, identify those cliques that behave abnormally.
Once a diagnosis result is achieved, each clique that has been diagnosed as abnormal will need a separate examination to see whether the checkpoint is damaged or the sensor fails to work properly.
The diagnosis method to be proposed is inspired by the “correlation of deflections” phenomenon, which is explained as follows. Some types of bridges, especially cablestayed bridges and continuous rigidframe bridges, have integral beams. Consider such a bridge model with four deflection checkpoints, , , , and on the beam plus an coordinate system (see Figure 1). Depending on the traffic and environmental conditions, the beam may bend down or bend up. In the former case, P and Q should assume positive deflection values simultaneously, while R and S should take on negative deflection values simultaneously (see Figure 2). In the latter case, the converse is true (see Figure 3).
We will treat the aforementioned bridge diagnosis problem following the idea of pattern recognition [18] and exploiting the above mentioned “correlation of deflections” phenomenon. Our method is composed of two phases.
Phase I. Starting with a set of deflection measurements that were previously acquired when the bridge was in its healthy status, establish a reference pattern of the bridge.
Phase II. Using a set of recently acquired deflection data, a current pattern of this bridge is obtained. By comparing the current pattern with the reference pattern, identify a set of abnormal cliques.
For our purpose, a weighted graph, known as the deflection correlation graph (DC graph, for short), is defined, which will be taken as the reference pattern. A method is proposed for building a DC graph of a bridge. On this basis, we present a timeeffective diagnosis algorithm. The proposed method is then applied to the simulated diagnosis of a reallife bridge, where the cliques with distorted deflection data are identified correctly. This exhibits the potential utility of this method.
The subsequent materials are organized this way. Section 2 introduces some fundamental knowledge. Section 3 defines the DC graph of a bridge, and describes a method of building a DC graph, which is used to form two DC graphs of a reallife bridge system. In Section 4, a diagnosis algorithm is presented, which is then applied to identify those abnormal parts of a reallife bridge provided some deflection data are distorted. Finally, Section 5 briefly summarizes the work of this paper and indicates some issues that are worth further study.
2. Fundamentals
A graph is defined as an ordered pair of sets, , where elements in are referred as nodes, elements in are referred to as edges, and every edge is a set of two distinct nodes [19]. Graphs are a class of mathematical models for characterizing binary relations between objects.
An edgelabeled graph is a graph with edges being labeled with intervals. An edgelabeled graph can be represented by an edgelabeled drawing or by a table. For instance, the edgelabeled graph , where
can be represented by an edgelabeled drawing (Figure 4) or by a table (Table 1).
Consider a pair of random variables, and . Let denote their correlation coefficient. That is,
If we have a series of measurements of and written as and where , then the Pearson productmoment correlation coefficient, which is defined as
can be used to estimate the correlation coefficient of and . The Pearson correlation coefficient is also known as the sample correlation coefficient.
3. Deflection Correlation Graph
For clarity, we call the sensor dedicated to deflection checkpoint as sensor , and we call the corresponding clique as clique .
3.1. Definition of a Deflection Correlation Graph
For clique , let denote the measured deflection value at checkpoint at time . Then, forms a time series [20]. Let denote the measurement of . In this paper, we assume that the following reasonable hypothesis holds.
Suppose clique P is healthy. Then, is a weak stationary process [20].In the case that cliques and are healthy, it follows from hypothesis that the correlation coefficients assume a common value for all values of . In what follows, we will use the symbol to denote this common value. Clearly, characterizes an internal connection between the two checkpoints and, hence, is useful for diagnosis purpose. Unfortunately, the precise value of is not practically available.
Given a collection of historical data . For every pair of checkpoints, and , we can fetch a number of pieces of the form and, for each such piece, we can calculate a Pearson correlation coefficient according to the formula
Thereby, we can get a smallest interval in which at least % Pearson correlation coefficients fall. This interval can also characterize the internal connection between the two checkpoints. Now, let us introduce the following definition.
Definition 3.1. Given a collection of historical data for a bridge. Consider a pair of checkpoints, and , on a bridge. Let . (1)An Hbased deflection correlation interval (DC interval) of and , denoted , is defined as an interval in which falls with probability . For diagnosis purpose, it is appropriate to take .(2)An Hbased deflection correlation degree (DC degree) of and is defined as Clearly, . The nearer to one is close, the more strongly correlated and will be with each other.
Definition 3.2. Given a collection of historical data for a bridge. Let . An Hbased deflection correlation graph of the bridge is defined as an edgelabeled graph , where is the set of all cliques of the bridge,
For diagnosis purpose, we will take Figure 1 gives an exemplar graph.
In its infancy, a bridge can be regarded as healthy. So, the deflection data acquired during the infancy can be utilized to build a graph of a bridge. This graph will be taken as the reference pattern.
3.2. Construction of a Deflection Correlation Graph
Based on the previous discussions, let us describe a method for creating a deflection correlation graph.
3.3. An Example
The Chongqing MaSangXi Grand Bridge is a cablesuspension bridge across the Yangtze River, which is about kilometers in length, and has an Asialongest main span of meters (Figure 5). The monitoring system for this bridge holds deflection checkpoints numbered as (Figure 6) and has employed the photoelectronic imaging technique to enhance the measurement precision.
Since the MaSangXi grand bridge was put into use in , the deflection data on each checkpoint have been successively acquired at an interval of minutes. For the purpose of simulated diagnosis, we selected a set of complete deflection records from these historical data.
In our simulation experiments, We tried combinations of the values of the five parameters, and . Below are the reports of two such experiments.
Experiment. Set and choose , the BUILD_DC algorithm built a graph, which is shown in Table 2.
Experiment. Set and choose , the BUILD_DC algorithm created a graph, which is shown in Table 3.
4. A Bridge Diagnosis Method
In this section, we will propose a bridge diagnosis algorithm based on the reference pattern previously defined.
4.1. Abnormal Cliques
First, let us describe a kind of abnormal behaviors of cliques.
Definition 4.1. Consider a bridge. Let be a collection of infancy deflection data. Let be an based graph of a bridge. Let be a collection of recent data. Let . A clique is significantly abnormal if there is another clique such that with frequency .
It can be seen that the larger is, the more seriously abnormal a clique will be.
4.2. Description of the Diagnosis Algorithm
We are ready to present a diagnosis algorithm for bridges.
4.3. An Application
By applying the new diagnosis method, we performed simulated diagnosis of the MaSangXi grand bridge for times. Below is the threestage report of one of these diagnosis processes.
Stage I. Get a reference pattern. We choose the graph given in Table 3 as our reference pattern.
Stage II. Create an abnormal pattern. Starting with the data collection given in Subsection 3.3, we create a collection of distorted deflection data according to the following four rules.
Rule 1. Replace the data on checkpoints with the Gaussian noise with zero mean and a standard deviation of .
Rule 2. Replace the deflection data on checkpoints with the Rayleigh noise with zero mean and a standard deviation of .
Rule 3. Replace the deflection data on checkpoints with the Rician noise with zero mean and a standard deviation of .
Rule 4. Keep intact all of the other deflection data.
Stage III. Perform diagnosis. By setting , and executing algorithm DIAG_BRIDGE on we get the set of significantly abnormal cliques, , which does contain the three cliques with distorted data.
Among the experiments, there are up to times for which all of the distorted cliques were successfully isolated. This justifies the potential utility of the proposed diagnosis algorithm.
5. Conclusions
We have proposed a deflectionbased bridge diagnosis method, and we have justified the potential utility of this method by applying it to the simulated diagnosis of a reallife bridge.
Toward this direction, some issues are worth further study. First, the proposed diagnosis method needs to be improved because it may identify some healthy nodes as being “abnormal.” Second, it is important to make clear how the diagnosis outcome is affected by the parameters involved in the algorithm. Finally, it is necessary to incorporate some other important factors (say, temperature, strain) into the diagnosis scheme.


Acknowledgments
This work is supported by 863 Program of China (2006AA04Z433), Natural Science Foundation of China (10771227), and Program for New Century Excellent Talent of Educational Ministry of China (NCET050759).