Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 284368 | 12 pages | https://doi.org/10.1155/2009/284368

A Deflection-Based Bridge Diagnosis Method

Academic Editor: Angelo Luongo
Received09 Nov 2008
Revised08 Mar 2009
Accepted09 Jun 2009
Published13 Aug 2009

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 real-life 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 damage-sensitive 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 time-series 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 real-life bridge is subject to varying environmental and operational conditions such as traffic, temperature, humidity, wind, and solar-radiation. These environmental effects also cause change in any well-defined 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 cable-stayed bridges and continuous rigid-frame 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 time-effective diagnosis algorithm. The proposed method is then applied to the simulated diagnosis of a real-life 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 real-life bridge system. In Section 4, a diagnosis algorithm is presented, which is then applied to identify those abnormal parts of a real-life 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 edge-labeled graph is a graph with edges being labeled with intervals. An edge-labeled graph can be represented by an edge-labeled drawing or by a table. For instance, the edge-labeled graph , where

can be represented by an edge-labeled drawing (Figure 4) or by a table (Table 1).



.74 .77

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 product-moment 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 H-based -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 H-based -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 H-based -deflection correlation graph of the bridge is defined as an edge-labeled 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 cable-suspension bridge across the Yangtze River, which is about  kilometers in length, and has an Asia-longest 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.



.982 .999.935 .985.915 .976.864 .948−.954−.891−.967−.915.883 .967.923.970.930.974
.913 .974.884 .962−.964−.907−.972−.924.908 .979.936.978.938.976
.950 .980
.935 .983.920 .968−.943−.851−.947−.860−.931−.833.845.943.848.946
..938 .980
.977 .994
.984.998.938.980.891 .977−.957−.883−.953−.869
.915.971.887 .974−.965−.898−.963−.884
.770 .857
.937.980.947.987
.902 .986
.984.999

Experiment. Set and choose , the BUILD_DC algorithm created a graph, which is shown in Table 3.



.986 .998.942 .985.923 .972.867 .946−.954−.899−.964−.919.891 .965.928.970.926.964
.919 .971.870 .949−.952−.902−.974 −.926.902 .961.937.979.944.976
.952 .979
.941 .986.916 .965−.954−.870−.933−.857−.921−.835.858.948.848.935
..945 .979
.979 .996
.987.999.941.977.909 .981−.959−.893−.942−.880
.910.967.905 .973−.953−.892−.964−.890
.788 .868
.937.977.955.986
.915 .986
.985.999

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 three-stage 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 deflection-based bridge diagnosis method, and we have justified the potential utility of this method by applying it to the simulated diagnosis of a real-life 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.

INPUT: : the set of all deflection checkpoints on a bridge;
: a set of deflection data acquired in the infancy of the bridge;
OUTPUT: : an graph of the bridge.
( 1)initially, is empty;
( 2)choose a real number , ;
the meaning of is clear according to statement 8;
( 3)choose two integers and ;
the meanings of and are clear according to statements ;
(4)for every pair of checkpoints, and ,
sentences calculate sample correlated coefficients between and ;
(5)randomly select pairs of length- sequences from , one for , and the other for ;
(6)for each pair of selected sequences, calculate a sample correlated coefficient;
sentence gets an initial estimate of ;
(7)find the smallest interval so that the sample correlated coefficients obtained
    previously fall in it with frequency ;
sentence gets a final estimate of by eliminating the randomness of the
    initial estimate;
(8) ; ;
     sentence adds a legal edge in ;
(9)if , then add in the edge with label ;

INPUT: : a graph of a bridge;
: a set of recent data;
: a real number, .
OUTPUT: : the collection of all -significantly abnormal cliques.
( 1)initially, is empty;
( 2)choose two integers and ;
( 3)for every pair of checkpoints, and ,
sentences calculate sample correlated coefficients between and ;
(4)randomly select pairs of length- sequences from , one for , and the other for ;
(5)for each pair of selected sequences, calculate a sample correlated coefficient;
sentence calculates the frequency at which these sample correlated coefficients fall in
   ;
(6)num the number of these sample correlated coefficients that fall in ;
sentence performs diagnosis; */
(7)if , then add and to ;
.

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 (NCET-05-0759).

References

  1. F. N. Catbas, M. Gul, and J. L. Burkett, “Conceptual damage-sensitive features for structural health monitoring: laboratory and field demonstrations,” Mechanical Systems and Signal Processing, vol. 22, no. 7, pp. 1650–1669, 2008. View at: Publisher Site | Google Scholar
  2. F. N. Catbas, M. Susoy, and D. M. Frangopol, “Structural health monitoring and reliability estimation: long span truss bridge application with environmental monitoring data,” Engineering Structures, vol. 30, no. 9, pp. 2347–2359, 2008. View at: Publisher Site | Google Scholar
  3. A. Deraemaeker, E. Reynders, G. De Roeck, and J. Kullaa, “Vibration-based structural health monitoring using output-only measurements under changing environment,” Mechanical Systems and Signal Processing, vol. 22, no. 1, pp. 34–56, 2008. View at: Publisher Site | Google Scholar
  4. A.-M. Yan, G. Kerschen, P. De Boe, and J.-C. Golinval, “Structural damage diagnosis under varying environmental conditions—part I: a linear analysis,” Mechanical Systems and Signal Processing, vol. 19, no. 4, pp. 847–864, 2005. View at: Publisher Site | Google Scholar
  5. A.-M. Yan, G. Kerschen, P. De Boe, and J.-C. Golinval, “Structural damage diagnosis under varying environmental conditions—part II: local PCA for non-linear cases,” Mechanical Systems and Signal Processing, vol. 19, no. 4, pp. 865–880, 2005. View at: Publisher Site | Google Scholar
  6. B. H. Koh and S. J. Dyke, “Structural health monitoring for flexible bridge structures using correlation and sensitivity of modal data,” Computers and Structures, vol. 85, no. 3-4, pp. 117–130, 2007. View at: Publisher Site | Google Scholar
  7. J. W. Lee, J. D. Kim, C. B. Yun, J. H. Yi, and J. M. Shim, “Health-monitoring method for bridges under ordinary traffic loadings,” Journal of Sound and Vibration, vol. 257, no. 2, pp. 247–264, 2002. View at: Publisher Site | Google Scholar
  8. J. J. Lee and C. B. Yun, “Damage diagnosis of steel girder bridges using ambient vibration data,” Engineering Structures, vol. 28, no. 6, pp. 912–925, 2006. View at: Publisher Site | Google Scholar
  9. Q. W. Zhang, “Statistical damage identification for bridges using ambient vibration data,” Computers and Structures, vol. 85, no. 7-8, pp. 476–485, 2007. View at: Publisher Site | Google Scholar
  10. E. P. Carden and J. M. W. Brownjohn, “ARMA modelled time-series classification for structural health monitoring of civil infrastructure,” Mechanical Systems and Signal Processing, vol. 22, no. 2, pp. 295–314, 2008. View at: Publisher Site | Google Scholar
  11. Y. Cheng and H. G. Melhem, “Monitoring bridge health using fuzzy case-based reasoning,” Advanced Engineering Informatics, vol. 19, no. 4, pp. 299–315, 2005. View at: Publisher Site | Google Scholar
  12. Z. Zhao and C. Chen, “A fuzzy system for concrete bridge damage diagnosis,” Computers and Structures, vol. 80, no. 7-8, pp. 629–641, 2002. View at: Publisher Site | Google Scholar
  13. G. Fu and A. G. Moosa, “Structural damage diagnosis using high resolution images,” Structural Safety, vol. 23, no. 4, pp. 281–295, 2001. View at: Publisher Site | Google Scholar
  14. O. García, J. Vehí, J. C. Matos, A. A. Henriques, and J. R. Casas, “Structural assessment under uncertain parameters via interval analysis,” Journal of Computational and Applied Mathematics, vol. 218, no. 1, pp. 43–52, 2008. View at: Publisher Site | Google Scholar
  15. J. M. Ko, Z. G. Sun, and Y. Q. Ni, “Multi-stage identification scheme for detecting damage in cable-stayed Kap Shui Mun Bridge,” Engineering Structures, vol. 24, no. 7, pp. 857–868, 2002. View at: Publisher Site | Google Scholar
  16. J. J. Lee, J. W. Lee, J. H. Yi, C. B. Yun, and H. Y. Jung, “Neural networks-based damage detection of bridges considering errors in baseline finite element models,” Journal of Sound and Vibration, vol. 280, no. 3, pp. 555–578, 2005. View at: Publisher Site | Google Scholar
  17. J. M. Ko and Y. Q. Ni, “Technology developments in structural health monitoring of large-scale bridges,” Engineering Structures, vol. 27, no. 12, pp. 1715–1725, 2005. View at: Publisher Site | Google Scholar
  18. A. R. Webb, Statistical Pattern Recognition, John Wiley & Sons, Chichester, UK, 2nd edition, 2002. View at: MathSciNet
  19. R. Diestel, Graph Theory, vol. 173 of Graduate Texts in Mathematics, Springer, Berlin, Germany, 3rd edition, 2005. View at: MathSciNet
  20. J. D. Hamilton, Time Series Analysis, Princeton University Press, Princeton, NJ, USA, 1994. View at: MathSciNet

Copyright © 2009 Xiaofan Yang 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.


More related articles

4052 Views | 489 Downloads | 0 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.