Department of Civil and Environmental Engineering
Stanford University
Report No. 180 June 2013
Damage Diagnosis Algorithms using Statistical Pattern Recognition for Civil Structures Subjected to Earthquakes
By
Hae Young Nohand
Anne S. Kiremidjian
The John A. Blume Earthquake Engineering Center was established to promote research and education in earthquake engineering. Through its activities our understanding of earthquakes and their effects on mankind’s facilities and structures is improving. The Center conducts research, provides instruction, publishes reports and articles, conducts seminar and conferences, and provides financial support for students. The Center is named for Dr. John A. Blume, a well-known consulting engineer and Stanford alumnus.
Address:
The John A. Blume Earthquake Engineering Center Department of Civil and Environmental Engineering Stanford University Stanford CA 94305-4020
(650) 723-4150 (650) 725-9755 (fax) [email protected] http://blume.stanford.edu
©2013 The John A. Blume Earthquake Engineering Center
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Abstract
In order to prevent catastrophic failure and reduce maintenance costs, the demands for the
automated monitoring of the performance and safety of civil structures have increased
significantly in the past few decades. In particular, there has been extensive research in
the development of wireless structural health monitoring systems, which enable dense
installation of sensors on structural systems with low installation and maintenance costs.
The main challenge of these wireless sensing units is to reduce the amount of data that
need to be transmitted wirelessly because the wireless data transmission is the major
source of power consumption. This dissertation introduces various damage diagnosis
algorithms that use statistical pattern recognition methods at sensor level. Therefore,
these algorithms do not require massive transmission of data, and thus are particularly
beneficial for use in wireless sensing units. Although damage diagnosis algorithms for
structural health monitoring have existed for several decades, statistical pattern
recognition techniques have been applied in this field only in the past decade. This
approach is receiving increasing recognition for its computational efficiency, which is
required when embedding such algorithms in wireless sensing units. These algorithms
can use either stationary ambient vibration responses before and after the damage or non-
stationary strong motion responses such as earthquake responses.
In the first part of this dissertation, three algorithms are introduced for damage diagnosis
using ambient vibration responses. Each vibration response is modeled as a time-series
with distinct parameters, which are closely related to the structural parameters. Damage
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diagnosis is performed by classifying the combinations of these parameters into damage
states using three statistical pattern recognition methods. The algorithms are validated
using the experimental data obtained from the benchmark structure of the National Center
for Research on Earthquake Engineering (NCREE) in Taipei, Taiwan, and the results
show that these algorithms can detect damage while more improvement is necessary for
damage localization.
The second part of the dissertation introduces a wavelet-based damage diagnosis
algorithm that uses non-stationary strong motion responses. Wavelet energies of each
response are extracted from various frequencies at different instances, and three damage
sensitive features are defined on the basis of the extracted wavelet energies. These
features are probabilistically mapped to damage states using fragility functions. The
framework to develop these wavelet damage sensitive feature-based fragility functions is
also discussed. The efficiency and robustness of the damage sensitive features are
validated using the two sets of experimental data: 30% scaled reinforced concrete bridge
column tests in Reno, Nevada, and 1:8 scale model of a four-story steel special moment-
resisting frame tests at the State University of New York at Buffalo. The performance of
the fragility functions to classify damage is validated using the numerically simulated
data obtained from the analytical model of the four-story steel special moment-resisting
frame. The results show that the wavelet-based features are closely related to structural
damage and the fragility functions can efficiently classify the damage state from the
features.
The last part of the dissertation discusses a data compression method using a sparse
representation algorithm. This method constructs a set of bases to represent each
structural response as their weighted sum. By creating an over-complete set of bases, the
responses can be represented using a few number of bases (i.e., sparse representation).
This method can reduce the amount of data to transmit and save the power consumption
of the wireless sensing units. This method enables the entire transmission of response
data to a server computer, and more sophisticated analysis of the data can be performed
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in global level. The method is validated using the white noise experimental data collected
from the four-story steel special moment-resisting frame tests at the State University of
New York at Buffalo, and significant compression ratio is achieved for upper floors while
maintain the information.
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Acknowledgments
This research was supported primarily by the Samsung Scholarship and the John A.
Blume Research Fellowship. Additional funding was provided by the National Science
Foundation – Civil, Mechanical, and Manufacturing Innovation Grant No. 0800932, and
the National Science Foundation – George E. Brown, Jr. Network for Earthquake
Engineering Simulation Research Grant No. 15BBK16379. The support provided by
these organizations is greatly appreciated.
This report was originally published as the Ph.D. dissertation of the first author. The
authors would like to thank Kincho H. Law, Jack W. Baker, Ram Rajagopal, Helmut
Krawinkler, and Gregory G. Deierlein for their insightful comments and constructive
feedback on the manuscript. The authors would also like to acknowledge Professor C-H.
Loh (National Taiwan University) and the National Center for Research on Earthquake
Engineering (NCREE) for providing the data collected from the Taiwanese benchmark
experiment conducted at the NCEE, Taipei, Taiwan; Dr. Hoon Choi (URS Corp.),
Professor M.“Saiid” Saiidi (University of Nevada, Reno) and Dr. Paul Somerville (URS
Corp.) for providing the data collected from the reinforced concrete bridge column
experiment conducted at the University of Nevada, Reno; and Professor Dimitrios G.
Lignos (Mcgill University) and Professor Krawinkler (Stanford University) for providing
the data collected from the four-story steel special moment-resisting frame experiment
conducted at the State University of New York, Buffalo.
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ix
Table of Contents
Abstract iv
Acknowledgments vii
List of Tables xii
List of Figures xv
1 Introduction 1
1.1 Motivation .........................................................................................................1
1.2 Objectives..........................................................................................................4
1.3 Overview ...........................................................................................................5
2 Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration
Data 7
2.1 Introduction .......................................................................................................8
2.2 Description of Damage Diagnosis Algorithms ...............................................10
2.2.1 Data Conditioning .................................................................................12
2.2.2 Time-Series Modeling of Vibration Data ..............................................14
2.2.3 Damage Diagnosis Algorithms ..............................................................19
2.2.3.1 Algorithm 1: AR Model with Hypothesis Tests ......................19
2.2.3.2 Algorithm 2: AR Model with Gaussian Mixture Models ........21
2.2.3.3 Algorithm 3: AR Model with Information Criteria .................24
2.3 Application of the Damage Diagnosis Algorithms to Experimental Data
Using the Taiwanese Benchmark Structure ....................................................28
x
2.3.1 Description of Experiment ....................................................................28
2.3.2 Results and Discussion ..........................................................................29
2.3.2.1 Algorithm 1: AR Model with Hypothesis Tests ......................29
2.3.2.2 Algorithm 2: AR Model with Gaussian Mixture Models ........41
2.3.2.3 Algorithm 3: AR Model with Information Criteria .................46
2.4 Conclusions .....................................................................................................50
3 Wavelet-Based Damage Sensitive Features for Structural Damage
Diagnosis Using Strong Motion Data 54
3.1 Introduction .....................................................................................................55
3.2 Development of Wavelet-Based Damage Sensitive Features .........................58
3.2.1 Wavelet Transformation and Wavelet Energies ....................................58
3.2.2 Definition of Damage Sensitive Features ..............................................66
3.2.2.1 DSF1 ........................................................................................66
3.2.2.2 DSF2 ........................................................................................67
3.2.2.3 DSF3 ........................................................................................69
3.3 Application of the Wavelet-Based Damage Sensitive Features to
Experimental data ...........................................................................................72
3.3.1 Reinforced Concrete Bridge Column Experiment ................................72
3.3.1.1 Description of Experiment ......................................................72
3.3.1.2 Results and Discussion ............................................................75
3.3.2 Four-Story Steel Moment-Resisting Frame Experiment .......................79
3.3.2.1 Description of Experiment ......................................................79
3.3.2.2 Results and discussion .............................................................82
3.3.3 Analytical Model of the Four-Story Steel Special Moment-Resisting
Frame Analysis ......................................................................................86
3.4 Conclusions .....................................................................................................90
4 Development of Fragility Functions as a Damage Classification/Prediction
Method Using a Wavelet-Based Damage Sensitive Feature 93
4.1 Introduction .....................................................................................................94
xi
4.2 Framework for Developing Fragility Functions Based on a Damage
Sensitive Feature .............................................................................................97
4.2.1 Data Collection: Structural Responses and Damage States ..................98
4.2.2 Feature Extraction: Wavelet-Based Damage Sensitive Feature from
Structural Responses as an Indicator of Damage State .......................100
4.2.3 Damage Classification/Prediction Model Development .....................100
4.3 Application of the Framework to Simulated Data Using an Analytical
Model of the Four-Story Steel Special Moment-Resisting Frame ................106
4.3.1 Description of the Analytical Model ...................................................107
4.3.2 Development of Fragility Functions for Different Damage States .....108
4.3.3 Results .................................................................................................111
4.3.4 Discussion ............................................................................................121
4.4 Conclusions ...................................................................................................123
5 Application of a Sparse Representation Method Using K-SVD Algorithm to
Data Compression of Experimental Ambient Vibration Data for SHM 125
5.1 Introduction ...................................................................................................126
5.2 Description of Data Compression Method....................................................127
5.2.1 Data Compression Using K-SVD Algorithm ......................................128
5.2.2 Data Transmission and Reconstruction ...............................................132
5.3 Application of the Data Compression Algorithm to Experimental Data
Using the Four-Story Steel Special Moment-Resisting Frame .....................133
5.3.1 Description of Experiment ..................................................................133
5.3.2 Results and Discussion ........................................................................134
5.4 Conclusions ...................................................................................................138
6 Summary, Conclusions, and Future Work 139
6.1 Summary and conclusions ............................................................................140
6.2 Future work ...................................................................................................146
xii
List of Tables
Number Page
Table 2.1: Number of data obtained and used for the analysis ..........................................12
Table 2.2: Stability of the first AR coefficient ..................................................................17
Table 2.3: Results of damage detection using DSFacc,1 for 60 gal unidirectional
random excitation for DP 1 using the point estimate and CI of
DSF, undamaged - DSF, DP1 ....................................................................................34
Table 2.4: Results of damage detection using DSFacc,1 for 60 gal unidirectional
random excitation for DP 2 using the point estimate and CI of
DSF, undamaged - DSF, DP1 ....................................................................................34
Table 2.5: Results of damage detection using DSFacc,1 for 100 gal unidirectional
random excitation for DP 1 using the point estimate and CI of
DSF, undamaged - DSF, DP1 ....................................................................................34
Table 2.6: Results of damage detection using DSFacc,1 for 100 gal unidirectional
random excitation for DP 2 using the point estimate and CI of
DSF, undamaged - DSF, DP2 ....................................................................................35
Table 2.7: Results of damage detection using DSFacc,1 for 50 gal bi-directional
random excitation for DP 1 using the point estimate and CI of
DSF, undamaged - DSF, DP1 ....................................................................................35
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Table 2.8: Results of damage detection using DSFacc,1 for 50 gal bi-directional
random excitation for DP 2 using the point estimate and CI of
DSF, undamaged - DSF, DP2 ....................................................................................35
Table 2.9: Results of damage detection using DSFacc,2 for 60 gal unidirectional
random excitation for DP 1 using the point estimate and CI of
DSF, undamaged - DSF, DP1 ....................................................................................36
Table 2.10: Results of damage detection using DSFacc,2 for 60 gal unidirectional
random excitation for DP 2 using the point estimate and CI of
DSF, undamaged - DSF, DP1 ....................................................................................36
Table 2.11: Results of damage detection using DSFacc,2 for 100 gal unidirectional
random excitation for DP 1 using the point estimate and CI of
DSF, undamaged - DSF, DP1 ....................................................................................36
Table 2.12: Results of damage detection using DSFacc,2 for 100 gal unidirectional
random excitation for DP 2 using the point estimate and CI of
DSF, undamaged - DSF, DP2 ....................................................................................37
Table 2.13: Results of damage detection using DSFacc,2 for 50 gal bi-directional
random excitation for DP 1 using the point estimate and CI of
DSF, undamaged - DSF, DP1 ....................................................................................37
Table 2.14: Results of damage detection using DSFacc,2 for 50 gal bi-directional
random excitation for DP 2 using the point estimate and CI of
DSF, undamaged - DSF, DP2 ....................................................................................37
Table 2.15: Results of mean values of various distance measures from the strain data
for 60 gal uni-directional random excitation for undamaged and damaged
cases .................................................................................................................44
Table 2.16: Results of mean values of various distance measures from the strain data
for 100 gal uni-directional random excitation for undamaged and damaged
cases .................................................................................................................44
xiv
Table 2.17: Results of mean values of various distance measures from the strain data
for 50 gal bi-directional random excitation for undamaged and damaged
cases .................................................................................................................44
Table 2.18: Parameters of Gaussian distributions for generating data ..............................47
Table 2.19: Damage patterns for the Z24 bridge ...............................................................49
Table 3.1: Scaling factor, Input RMS value, and Description of Damage for each DP
of the bridge column experiment .....................................................................74
Table 3.2: Distribution of DSF1 for no damage state cases ...............................................89
Table 4.1: Summary of three methods for probabilistic mapping between the DSF and
the SDR ..........................................................................................................101
Table 4.2: Advantages of kernel density estimation in comparison to data binning
method............................................................................................................104
Table 4.3: Performance of the DSF-based fragility functions for estimating a damage
state ................................................................................................................114
Table 4.4: The conditional mean and the standard deviation of the SDR given DSF .....116
Table 4.5: Advantages and disadvantages of different fragility functions ......................121
Table 5.1: The coefficient of determination (R2) values ..................................................135
xv
List of Figures
Number Page
Figure 2.1: Examples of corrupted data .............................................................................13
Figure 2.2: Variation of AIC with model order .................................................................15
Figure 2.3: Benchmark structure and sensor locations used at the NCREE test ...............16
Figure 2.4: Residuals diagnostics: (a) Variation of residuals with time; (b) Normal
probability plot of the residuals; (c) Variation of ACF with lag......................18
Figure 2.5: Photograph of the cut flanges of the column ...................................................29
Figure 2.6: Variation of DSF with damage: (a) Variation of DSFacc,2 for acceleration
data; (b) Variation of DSFstr for strain data .....................................................30
Figure 2.7: Point estimates of dDSFuDSF ,, ˆˆ using DSFacc,1: (a) 60 gal unidirectional
random excitation, X direction data result; (b) 60 gal unidirectional
random excitation, Y direction data result; (c) 100 gal unidirectional
random excitation, Y direction data result; (d) 50 gal bidirectional random
excitation, X direction data result; (e) 50 gal bidirectional random
excitation, Y direction data result ....................................................................32
Figure 2.8: Point estimates of dDSFuDSF ,, ˆˆ using DSFacc,2: (a) 60 gal unidirectional
random excitation, X direction data result; (b) 60 gal unidirectional
random excitation, Y direction data result; (c) 100 gal unidirectional
random excitation, Y direction data result; (d) 50 gal bidirectional random
xvi
excitation, X direction data result; (e) 50 gal bidirectional random
excitation, Y direction data result ....................................................................33
Figure 2.9: DM for acceleration data from bidirectional random excitation .....................39
Figure 2.10: Confidence intervals of the DSFstr for 50 gal bidirectional random
excitation ..........................................................................................................40
Figure 2.11: Δ1 for the acceleration data from 60 gal unidirectional random excitation-
X direction result for column (a) .....................................................................42
Figure 2.12: Δ1 for the acceleration data from 100 gal unidirectional random
excitation-Y direction result for column (b) ....................................................42
Figure 2.13: Δ1 for the acceleration data from 50 gal bidirectional random excitation-
Y direction result for column (b) .....................................................................43
Figure 2.14: Mean values of Δ1 for the strain data from 60 gal unidirectional random
excitation ..........................................................................................................45
Figure 2.15: Δ1 for the strain data from 100 gal unidirectional random excitation ...........45
Figure 2.16: Δ1 for the strain data from 50 gal bidirectional random excitation ...............46
Figure 2.17: Estimated optimum number of clusters for simulated data ...........................47
Figure 2.18: Estimated optimum number of clusters for the Taiwanese benchmark
structure............................................................................................................48
Figure 2.19: Estimated optimum number of clusters for the Z24 bridge data ...................49
Figure 2.20: True and estimated cluster means .................................................................50
Figure 3.1: Morlet wavelet basis function ........................................................................60
Figure 3.2: Wavelet coefficients of the acceleration response at the top of the bridge
column for different DPs: (a) DP 1; (b) DP 3; (c) DP 5; (d) DP 7; (e) DP 9;
(f) DP 11; (g) DP 13.........................................................................................63
Figure 3.3: Wavelet coefficients of acceleration responses at the roof of the four-
story steel moment-resisting frame for different DPs: (a) DP 1; (b) DP 2;
(c) DP 3; (d) DP 4 ............................................................................................64
Figure 3.4: Eshift(b) for the bridge column experiment for different DPs: (a) DP 1; (b)
DP 3; (c) DP 5; (d) DP 7; (e) DP 9; (f) DP 11; (g) DP 13 ...............................67
xvii
Figure 3.5: Cumulative sum of Eshift(b) for the bridge column experiment for different
DPs: (a) DP 1; (b) DP 3; (c) DP 5; (d) DP 7; (e) DP 9; (f) DP 11; (g) DP
13......................................................................................................................69
Figure 3.6: Shaking table test setup for the bridge column experiment (Modified
from Choi et al. 2007) ......................................................................................73
Figure 3.7: Variation of wavelet entropies for the bridge column experiment for
different values of ω0: (a) WEt; (b) WEs; (c) WEts ...........................................75
Figure 3.8: Instantaneous frequency for the bridge column experiment: (a) for
different values of ω0 at DP 1; (b) for different DPs at ω0 = 17 rad/s .............76
Figure 3.9: Variation of DSF values for the bridge column experiment for different
ω0: (a) DSF1; (b) DSF2; (c) DSF3 ....................................................................77
Figure 3.10: Four-story steel moment-resisting frame: (a) after the completion of
erection on the shake table (Modified from Lignos et al. 2008); (b) after
collapse ............................................................................................................80
Figure 3.11: Story drift ratio histories at various levels of input ground motion
intensity for the four-story steel moment-resisting frame experiment: (a)
first story; (b) second story; (c) third story; (d) fourth story (Modified from
Lignos and Krawinkler, 2009) .........................................................................81
Figure 3.12: Variation of entropies for the four-story steel moment-resisting frame
experiment for different values of ω0: (a) WEt; (b) WEs; (c) WEts ...................82
Figure 3.13: DSF1 for the four-story steel moment-resisting frame experiment ..............83
Figure 3.14: Eshift(b) for the four-story steel moment-resisting frame for different DPs:
(a) DP 1; (b) DP 2; (c) DP 3; (d) DP 4.............................................................84
Figure 3.15: DSF2 for the four-story steel moment-resisting frame experiment ..............85
Figure 3.16: DSF3 for the four-story steel moment-resisting frame experiment ..............86
Figure 3.17: DSF1 for two different input ground motions: (a) the 1989 Loma Prieta
Earthquake; (b) the 1994 Northridge Earthquake ............................................88
Figure 3.18: Scatter plot of DSF1 and maximum story drift ratio for the analytical
model of the four-story steel moment-resisting frame .....................................89
xviii
Figure 4.1: Summary of the proposed framework ............................................................99
Figure 4.2: Loma Prieta earthquake ground acceleration recorded at the Agnews state
hospital ...........................................................................................................109
Figure 4.3: Wavelet coefficients for the roof acceleration history of the four-story
SMRF subjected to scaled Loma Prieta earthquake motions: (a) Sa(T1,
2%) = 0.25g; (b) Sa(T1, 2%) = 0.5g; (c) Sa(T1, 2%) = 0.75g; (d) Sa(T1,
2%) = 1.125g ..................................................................................................109
Figure 4.4: DSF for various intensities of scaled Loma Prieta earthquake ground
motion recorded at the Agnews state hospital ...............................................110
Figure 4.5: Scatter plot of DSF versus SDR: (a) story 1; (b) story 2; (c) story 3; (d)
story 4.............................................................................................................111
Figure 4.6: Fragility functions of the four-story steel SMRF: (a) story 1; (b) story 2;
(c) story 3; (d) story 4 ....................................................................................112
Figure 4.7: Global fragility functions for the four-story steel SMRF ..............................113
Figure 4.8: Probability of being in each damage state for the four-story steel SMRF ....113
Figure 4.9: (a) Scatter plot of DSF versus maximum SDR; (b) Condition probability
density function of maximum SDR given DSF using the two-dimensional
kernel for the four-story steel SMRF .............................................................115
Figure 4.10: (a) Scatter plot of DSF versus maximum SDR and the conditional mean
and standard deviation; (b) Condition probability density function of
maximum SDR given DSF for the four-story steel SMRF ............................116
Figure 4.11: Conditional mean and standard deviation of SDR given DSF for the
four-story steel SMRF....................................................................................118
Figure 4.12: Scatter plots:(a) Sa(T1, 2%) versus SDR; (b) PFA versus SDR; (c)
wavelet-based DSF versus SDR ....................................................................119
Figure 4.13: Collapse Fragility functions based on: (a) Sa(T1, 2%); (b) the wavelet-
based DSF ......................................................................................................120
Figure 5.1: Summary of the data compression using the K-SVD algorithm ...................129
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Figure 5.2: The results of data reconstruction using the K-SVD algorithm for the roof
acceleration response at DP 1: (a) time-histories of original data and
reconstructed data; (b) time-history of representation error; (c) scatter plot
of original data vs. reconstructed data; (d) power spectrum density of
original data and reconstructed data ..............................................................134
Figure 5.3: The results of data reconstruction using the K-SVD algorithm for the
ground acceleration at DP 9: (a) time-histories of original data and
reconstructed data; (b) time-history of representation error; (c) scatter plot
of original data vs. reconstructed data; (d) power spectrum density of
original data and reconstructed data ..............................................................136
Figure 5.4: Modal analysis results for the original data and the reconstructed data at
DP 2 ...............................................................................................................137
Figure 5.5: Normalized RMSE versus compression ratio for the roof acceleration
response at DP 1.............................................................................................137
CHAPTER 1. Introduction
1
Chapter 1
Introduction
1.1 Motivation
During their lifecycles, structures are subjected to various loads, including everyday
loads caused by temperature change, traffic, and corrosion, and extreme loads, such as
earthquakes and hurricanes. Since everyday loads usually cause gradual deterioration to
structures, regular damage diagnosis and maintenance need to be performed to ensure the
structural safety. On the other hand, because extreme loads often result in more severe
damage, immediate assessment of structural damage after an event is essential to expedite
the emergency response and prevent further losses and injuries. Similarly, decisions on
repair and rehabilitation after such events can be greatly facilitated with information on
the type, location, and extent of damage. Currently, damage diagnosis is often achieved
by visual inspection by professionals. Human inspection, however, can be costly in time
and money, perilous in certain situations, inconsistent, limited to only surface
inspections, and inappropriate for immediate assessment of structures over large areas.
Therefore, we need to develop an automated system that efficiently and reliably
diagnoses structural damage. Consequently, extensive research has been conducted on
CHAPTER 1. Introduction
2
structural health monitoring (SHM) in the structural engineering community. With recent
developments in sensor technologies and wireless communication systems along with
advances in damage diagnosis algorithms, SHM has brought us closer to the realization
of structural damage assessment without conducting visual inspections.
SHM consists of damage diagnosis and prognosis, where damage diagnosis involves
detection, localization, and quantification of structural damage, and damage prognosis
involves the prediction of the remaining life of the structure (Rytter, 1993). One approach
to diagnosing damage is to use vibration-based methods, which detect damage by
observing changes in dynamic characteristics of a structure extracted from its vibration
responses. The vibration-based diagnosis is conducted using model-based methods or
non-model based methods (Doebling et al., 1996). The model-based methods, such as
system identification, utilize a pre-defined structural model and monitor the change in
structural parameters, such as stiffness, damping coefficient, natural frequency, and mode
shapes to identify damage. These methods are intuitive and have been well studied
(Ghanem and Shinozuka, 1995; Beck et al., 1994; Alvin and Park, 1994; Pandey and
Biswas, 1994; Doebling, 1996; Yun and Bahng, 2000; Koh et al., 2003; Beck and
Jennings, 2007), but they are computationally expensive, assume a detailed knowledge
about the structure, and often require vibration information from multiple locations in the
structure. Moreover, those structural parameters that current methods monitor are global
in nature. In other words, they reflect the overall behavior of the structure, and as a result,
they are not sensitive to minor local damage. In contrast, non-model based methods, such
as statistical pattern recognition methods, extract a damage sensitive feature (DSF) or
identify structural parameters from structural response measurements using signal
processing techniques without requiring detailed prior-knowledge about the structure, and
determine the damage state of the structure using pattern classification schemes (Sohn
and Farrar, 2001; Sohn et al., 2001). This DSF contains information on the damage state
of the structure, and the value of the DSF changes depending on the damage state of the
structure. The values of the DSF are mapped to different damage states using statistical
classification methods. Although these methods are less intuitive, they are
CHAPTER 1. Introduction
3
computationally efficient. With the recent development of autonomous sensing units and
wireless communications, structures can be densely instrumented to monitor their
response before and after and/or during an earthquake (Straser and Kiremidjian, 1998;
Lynch et al., 2004; Xu et al., 2004; Lynch and Loh, 2006; Kim et al., 2007). In addition,
the installation and maintenance cost of these units can be significantly reduced
(Doebling et al., 1996; Sohn et al., 2003). The statistical pattern recognition methods are
particularly suitable for embedding in the wireless sensor units because of their
computational efficiency.
Although statistical pattern recognition methods have been applied to a wide range of
fields for several decades, including biology, artificial intelligence, psychology, and
finance, they have been applied to structural health monitoring only in the past decade.
Various statistical pattern recognition methods are summarized by Duda et al. (2000),
Jain et al. (2000), and Fukunada (1990). Their applications to monitoring civil structures
involve several challenges due to the complex nature of civil structures. Because civil
structures often have a complicated geometry, which is composed of non-homogeneous
materials, there exist numerous failure modes and load redistribution mechanisms. The
effect of environmental and loading conditions on the vibration measurements from
structures is another major ambiguity for damage diagnosis.
Damage diagnosis algorithms using statistical pattern classification methods for civil
structures consist of three steps: data collection, feature extraction, and damage
classification (Nair et al., 2007). In the first step, sensor units are deployed at strategic
locations in a structure, and structural responses are collected from them periodically or
during/after extreme events. Then, damage sensitive features are extracted from these
structural responses. In the final step, the damage state of the structure is determined
using pattern recognition methods. These algorithms can be put into two categories
according to the data they use: ambient vibration responses before and after the damage,
or strong motion responses. The algorithms using ambient vibration responses include
time-series based analysis developed by, for example, Sohn et al. (2001) and Nair et al.
CHAPTER 1. Introduction
4
(2006). The algorithms using non-stationary strong motion responses include wavelet
analysis, which is widely used for studying non-stationary data (Hou et al., 2000;
Ghanem and Romeo, 2000; Kijewski and Kareem, 2003; Hera and Hou, 2004; Nair and
Kiremidjian, 2007; Noh et al., 2011). In particular, Nair and Kiremidjian (2007) and Noh
et al. (2011) introduced wavelet-based DSFs that can be used for a damage diagnosis
algorithm. This study investigated algorithms for both types of data.
1.2 Objectives
The main objective of this study is to develop damage diagnosis algorithms using
statistical pattern recognition methods that can efficiently and reliably assess the
structural damage of civil structures subjected to earthquakes without performing human
inspections. In particular, the detailed objectives of this study are as follows:
1. To develop damage diagnosis algorithms for wireless sensor units using acceleration
and strain responses to ambient vibration before and after damage and to validate
these algorithms using experimental data.
2. To develop damage diagnosis algorithms using acceleration responses to strong
motions during earthquakes and validate these algorithms using both experimental
and simulated data.
For validation purposes, we used experimental data from laboratory tests with controlled
damage states, as well as validated simulated data. Although the design and conduct of
the experiments are not in the scope of this study, each experiment is summarized to
provide the information necessary for understanding the validation results. While it
would be preferable to validate the algorithms with field data, repeated measurements of
structural responses at various damage states are extremely costly, if not impossible, in
the field. Also, for this study we assumed that the effects of various environmental
CHAPTER 1. Introduction
5
conditions are negligible for the current validations, but further investigation of the
environmental effects and validation of the robustness of the algorithms is necessary for
their practical application.
1.3 Overview
The subsequent chapters are organized as follows:
Chapter 2 introduces time-series based damage diagnosis algorithms using ambient
vibration measurements before and after damage. These algorithms define damage
sensitive features using autoregressive models and classify the damage state using
various methods such as hypothesis tests with t-statistics and Gaussian Mixture
models with Mahalanobis distances and information criteria. To validate the
algorithms, they are applied to a set of experimental data collected from a series of
shaking table tests of the Taiwanese benchmark structure conducted at the National
Center for Earthquake Engineering at Taipei, Taiwan.
Chapter 3 introduces wavelet-based damage sensitive features developed for non-
stationary acceleration responses recorded during the strong motion of earthquakes. It
also presents the relationship between these damage sensitive features and the
physical parameters of structures, such as mass, stiffness, damping ratio, and mode
shapes. Then, the performance of the DSFs was validated using two sets of
experimental data: a reinforced concrete bridge pier test conducted at the University
of Nevada, Reno, and a four-story steel moment-resisting frame test performed at the
State University of New York, Buffalo. In addition, the sensitivity of the proposed
DSFs with respect to different input ground motions is investigated using a
experimentally validated analytical model of the four-story steel moment-resisting
frame.
CHAPTER 1. Introduction
6
Chapter 4 describes a framework for developing fragility functions as a damage
classification and prediction model using wavelet-based damage sensitive features.
The framework is applied to a set of simulated data obtained from a validated
analytical model of the four-story steel moment-resisting frame introduced in Chapter
3 and subjected to 40 ground motions scaled to various intensities.
Chapter 5 provides a data compression method for reducing the data transmission rate
of wireless sensing units. This method has been developed for stationary ambient
vibration measurements. Chapter 5 also shows the efficiency of the compression
method using the experimental dataset of the Taiwanese benchmark structure.
Finally, chapter 6 discusses conclusions and future plans for the extension of current
work.
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
7
Chapter 2
Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
This chapter introduces three time-series based damage diagnosis algorithms and their
application to the benchmark experimental data from the National Center for Research on
Earthquake Engineering (NCREE) in Taipei, Taiwan. For this study, both acceleration
and strain data are analyzed. The algorithms first model the data as autoregressive (AR)
processes and then define damage sensitive features (DSF) and feature vectors in terms of
the first three AR coefficients. In the first algorithm, hypothesis tests using the t-statistic
are applied to evaluate the damage state. A damage measure (DM) is defined to measure
the damage extent. The results show that the DSFs from the acceleration data can detect
damage while the DSF from the strain data can localize the damage. The DM can be used
for damage quantification. In the second algorithm, a Gaussian mixture model (GMM) is
used to model the feature vector, and the Mahalanobis distance is defined to measure
damage extent. Additional distance measures are defined and applied to quantify damage.
The results show that damage measures can detect, quantify, and localize the damage for
the high intensity and the bidirectional loading cases. Finally, the third algorithm uses
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
8
various information criteria to identify the number of damage scenarios in the mixture of
the feature vectors extracted from the structure in various damage states. This algorithm
is first applied to a set of simulated data to verify its performance, and then applied to a
set of experimental data and a set of field data collected from the structures subjected to
systematically increasing extent of damage.
2.1 Introduction
Over the past decade, statistical pattern recognition methods of structural damage
diagnosis have been successfully applied to both simulated and experimental data. The
data that are available for validation and calibration of these methods, however, have
been very limited. Recognizing the need for additional data from experiments whereby
damage is introduced in a systematic and controlled way has led to a series of laboratory
tests.
This chapter describes three time-series based damage diagnosis algorithms and their
applications to the data obtained from the benchmark experiment conducted at the
National Center for Research on Earthquake Engineering (NCREE) in Taipei, Taiwan
(Lynch et al., 2006). The algorithm is based on the premise that structural damage will
change the vibration response of the structure. While previous validation and calibration
tests with this algorithm have involved only acceleration measurements, we used both
acceleration and strain data to evaluate the robustness of the algorithm. The algorithm
involves the modeling of vibration and strain data as autoregressive (AR) processes. We
use the first three AR coefficients of the model to define the feature vectors that serve as
the diagnostic tool for damage identification. As shown by Nair et al. (2006), these
coefficients are directly related to the mode shapes and frequencies of the structure and
thus can be used to capture changes in these structural properties.
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
9
Time-series modeling has been widely applied to vibration-based structural health
monitoring for modeling vibration data from various damage states. Sohn et al. (2001)
and Sohn and Farrar (2001) developed a two-state prediction model by combining AR
and auto-regressive with exogenous inputs (ARX) models. They fitted an AR model to
the reference data closest to a newly obtained data and then fitted an ARX model to the
reference data using the residual of the AR model as exogenous inputs. With this method,
the ratio of standard deviations of the residual errors from the ARX model of the
reference data and the new data is defined as a damage sensitive feature. Mattson and
Pandit (2006) also used the standard deviation of the residual of the AR model of
vibration data as a feature. In another study, Omenzetter and Brownjohn (2006)
investigated the coefficients of the autoregressive integrated moving average (ARIMA)
model of strain data for bridge monitoring. Similarly, Nair et al. (2006) modeled the
ambient vibration response of a structure as an AR process, defined damage sensitive
features as functions of the AR coefficients, and classified them into different damage
states using hypothesis tests. In addition, Nair et al. (2007) defined the AR coefficients as
a feature vector and applied a Gaussian mixture model (GMM) to the feature vectors.
Then they used Mahalanobis distance and gap statistics to detect damage. Both papers by
Nair et al. validated the algorithms using the simulated data obtained from the ASCE
benchmark structure. Taking a different approach, Zheng and Mita (2007) used the
distance between ARMA models to detect damage and validated its effectiveness using a
simulated data.
The three algorithms introduced in this chapter use feature vectors based on the AR
coefficients for damage diagnosis as follows. In the first algorithm, which is extended
from the work by Nair et al. (2006), damage sensitive features (DSFs) are defined as
functions of the first three AR coefficients. Then it applies hypothesis tests using the t-
statistic to discriminate a damaged state from an undamaged one. A damage measure
(DM) is defined in terms of the mean values and the standard deviation of the DSFs to
measure the extent of the damage. The second algorithm, which is extended from the
developments of Nair and Kiremidjian (2007), uses a GMM to model the feature vectors
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
10
and compute the distance between the mixtures to diagnose damage. To obtain an
efficient distance measure, we defined various distance measures including the
Mahalanobis distance, which is the Euclidean distance between the mixtures weighted
with respect to the inverse covariance matrix, and tested their performance using the
experimental data. In the third algorithm, the damage diagnosis is achieved by identifying
the number of clusters in the set of feature vectors using various information criteria. The
application results show that the algorithms are able to identify damage in majority of the
cases. These measures, however, do not always well represent damage extent and
location.
Section 2.2 explains the three time-series based algorithms in greater detail. Section 2.3
describes the NCREE experimental benchmark test and presents the results of the
application of the three algorithms. Finally, section 2.4 summarizes the chapter and
provides conclusions.
2.2 Description of Damage Diagnosis Algorithms
Three damage diagnosis algorithms are developed based on the AR model for damage
identification, quantification, and localization. For this purpose, AR coefficients are
computed using the data obtained from the undamaged and damaged structure and
analyze these sets of AR coefficients. In the damage diagnosis algorithms, DSFs are
defined as functions of the AR coefficients. To compare the data collected from the
structure in different damage states, the change in the values of DSFs are investigated as
damage takes place. Thus, the selection of the DSF is very important in part because it
has to reflect the physical properties of the structure. Nair et al. (2006) developed a
relationship between the structural modes and frequencies and the first three AR
coefficients and demonstrated that indeed these coefficients are suitable for damage
discrimination. The algorithms consist of three steps: (i) time series modeling of vibration
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
11
data, (ii) damage sensitive feature extraction, and (iii) statistical classification in order to
predict damage.
Modeling of the vibration data includes the removal of trends, obtaining the optimal
model order, and checking the assumptions of the residuals of the AR model. To extract
the appropriate DSF, the algorithms first fit an AR model to the data and investigate
different forms of the DSF. Then they apply statistical analysis on the DSF values from
the pre and post damage data in order to detect changes in damage states and test the
significance of that change using various statistical techniques.
After all preprocessing has been completed, the algorithms are performed in the
following steps:
1. Obtain data from an undamaged structure, from sensor i, denoted as xi(t) (i = 1,…, N),
where N is the number of sensors. Segment the data xi into chunks of finite duration,
xij(t) (j = 1,…, M), where M is the number of chunks. Populate a database with these
baseline data.
2. Standardize and normalize the data xij(t) to remove all trends and environmental
conditions to obtain txij~ .
3. Obtain data from a potentially damaged structure for the same sensor, denoted as zi(t),
(i = 1,…, N). As in the previous steps, segment zi(t) into zij(t) (j = 1,…, M) and then
standardize and normalize it to obtain tzij~ .
4. Fit an AR model to the data txij~ and tzij
~ for all i and j.
5. For each sensor i, define and compute the statistics of the DSFs for each chunk in the
pre- and post-damage data.
6. Classify the damage state of the structure using various statistical techniques.
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
12
The three algorithms share the steps 1 through 4 and apply separate methods for the steps
5 and 6. Section 2.2.1 presents data conditioning that corresponds to steps 1 through 3,
and section 2.2.2 explains the AR model and residual analysis that relates to step 4.
Finally section 2.2.3 introduces the DSFs and classification methods for each algorithm.
2.2.1 Data Conditioning
The first step of the analysis is to examine the data and eliminate those that are corrupted
or that do not satisfy the requirements for AR modeling. The data corruption could be due
to faulty instrumentation, changes in environmental conditions, faulty sensors, or faulty
transmission via cabling or via the wireless transmission, among many possibilities. For
example, data that are erratic or non-stationary or that do not appear to correspond to
structural behavior need to be discarded. Figure 2.1 (a) and (b) show examples of
corrupted or inappropriate data. In Figure 2.1 (a), the data are non-stationary and show
increasing amplitude. In Figure 2.1 (b) the data have sudden jumps, which implies that
the sensor may be unstable or loose. The data corruption is determined by judgment.
Table 2.1 shows the total number of structural response data obtained from the
experiment and the number of data used for the analysis for each case.
Table 2.1: Number of data obtained and used for the analysis 60 gal
X-direction unidirectional
random excitation
100 gal X-direction
unidirectional random
excitation
50 gal XY-direction bidirectional
random excitation
Acceleration Data
Direction X Y X Y X Y Total 6 6 6 6 6 6
Uncorrupted* 4 4 0 5 5 6
Strain Data (Z-direction)
Total 40 40 40 Uncorrupted* 35 38 39
* Only the uncorrupted data are used for the analysis.
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
13
0 10 20 30 40 50 60 70-0.2
-0.1
0
0.1
0.2
Time (s)
Acc
eler
atio
n (g
)
(a)
0 10 20 30 40 50 60 70 80 90 100-0.2
-0.1
0
0.1
0.2
Time (s)
Acc
eler
atio
n (g
)
(b)
Figure 2.1: Examples of corrupted data
The next step is to segment the data that appear to be appropriately collected by the
sensors to form chunks. Then we can standardize and normalize each of these chunks as
follows:
ij
ijijij
txtx
~(2.1)
where xij(t) is the structural response data (acceleration or strain) obtained from sensor i
and chunk j, and ij and ij are the mean and standard deviation of the data, xij(t),
respectively. The tilde in the notation is dropped from here on for simplicity. Subtracting
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
14
the mean value standardizes the data to the same amplitudes, and dividing by the standard
deviation normalizes the data to reduce the effect of local variability. This procedure
enables us to compare data from different sensor locations, loading conditions (i.e.,
different magnitudes and directions of loads), and/or environmental conditions.
2.2.2 Time-Series Modeling of Vibration Data
After standardizing and normalizing the data, the next step is to fit the time series models
to these data. If the input motion is available, we can apply the ARX model to fit the data.
If the input motion is not available, which is more likely in practice for ambient vibration
responses, the AR model can be used because it does not require input data. The Burg
and least-squares algorithms are applied to obtain the AR and ARX coefficients,
respectively (Brockwell and Davis, 2002).
The ARX model is given as
p
k
q
kijijkijkij tktyktxtx
1 1(2.2)
where xij(t) is the normalized acceleration data; yij(t) is the normalized input data; k and
k are the kth AR and exogenous input coefficient, respectively; p and q are the model
orders of the AR and the exogenous input processes, respectively; and ij(t) is the residual
term. Similarly, the AR model is
p
kijijkij tktxtx
1(2.3)
Note that the AR model is an ARX model with the exogenous input model order q =0.
To select the optimal order for the AR and the ARX models, the Akaike Information
Criteria (AIC) is applied. Figure 2.2 shows the variation of the AIC values for different
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
15
values of p in the ARX model applied to the experimental data explained in Section 2.3,
and each line corresponds to different input orders, q, of the ARX model. It should be
noted that when q is zero, the ARX model reduces to the simple AR model. We can
observe that as the input model order varies from q = 0 to 4, the AIC values are similar to
those of AR model of order p. Thus, the AR model is selected because it is the simplest
model that captures the characteristics of the data. For the analysis of this experimental
data, an optimal AR model order of 7 is chosen for both acceleration and strain data.
3 4 5 6 7 8 9 10-6.9
-6.8
-6.7
-6.6
-6.5
-6.4
-6.3
Output order p
AIC
q=0q=1q=2q=3q=4
Figure 2.2: Variation of AIC with model order
After the model order selection, the next step is to examine the stability of the AR
coefficients with respect to the chunk size (number of points in a segment) of the data
obtained from the test. First the acceleration and strain data are divided into chunks of the
same size and then compute the means and variances of the first AR coefficients. We can
then repeat this procedure using different chunk sizes and investigate several sets of data
in order to determine the optimal chunk size. Table 2.2 shows the result of this procedure
for the strain data obtained from sensor 1 as identified in Figure 2.3. The chunk sizes are
selected such that the coefficient of variation (COV) of the AR coefficients is less than
0.1. The chunk size is gradually increases from 100 to 1500 and the statistics for each set
are computed. From Table 2.2 we can observe that the mean value hardly changes with
increased chunk size, but the standard deviation decreases as expected. For chunk sizes
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
16
larger than 200, the COV is less than 0.05 (or 5%), which is considered an acceptable
variation for the purposes of this analysis. Thus, a chunk size of 200 points is used for the
analysis.
Figure 2.3: Benchmark structure and sensor locations used at the NCREE test
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
17
Table 2.2: Stability of the first AR coefficient Points. per
chunk 100 200 500 1000 1500
Mean -1.5219 -1.5287 -1.5388 -1.5471 -1.5520 Std.
deviation 0.0874 0.0811 0.0692 0.0578 0.0271
COV 0.0574 0.0531 0.0376 0.0279 0.0174
Following the AR modeling, we need to examine the residuals to determine if they are
independent and identically distributed as required for AR modeling. For this purpose,
the followings are investigated: the variation of residuals with time, their quantile-
quantile plot with a standard normal distribution, and their autocorrelation function with
respect to the lag. Figure 2.4 (a) shows the variation of the residuals within a chunk of
data for one example of response data. We can see that no trend exists, thus indicating
homoskedasticity in the residuals. Figure 2.4 (b) is the quantile-quantile plot of the
residuals with respect to a normal distribution. The residuals follow the straight line
closely and start deviating at the ends, which indicates that a slight deviation from
normality exists only at the tails. Figure 2.4 (c) shows the autocorrelation function (ACF)
of the residuals. Since the ACF values are small and decrease fast with lag, we can
conclude that the residuals are stationary. Thus, all conditions for fitting an AR model
are satisfied for this data set. Similar tests are performed for all the other data to ensure
the validity of the AR modeling.
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
18
0 50 100 150 200-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Sample
Err
or (
g)
(a)
-3 -2 -1 0 1 2 3-0.06
-0.04
-0.02
0
0.02
0.04
0.06
Normal theoretical quantiles
Dat
a qu
antil
es
(b)
0 50 100 150 200-0.02
0
0.02
0.04
0.06
0.08
0.1
Lag
AC
F
(c)
Figure 2.4: Residuals diagnostics: (a) Variation of residuals with time; (b) Normal probability plot of the residuals; (c) Variation of ACF with lag
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
19
2.2.3 Damage Diagnosis Algorithms
2.2.3.1 Algorithm 1: AR Model with Hypothesis Tests
The first algorithm involves damage sensitive features using the first three AR
coefficients and statistically classifies a damage state using the t-test. It first defines and
computes the statistics of the DSFs for each chunk in the pre- and post-damage data and
then computes the mean and pooled variance of the DSFs in order to determine the
statistical significance in the differences between the means of the DSFs from the pre-
and post-damage data using the t-test. In order to measure the damage extent, this
algorithm defines and computes the DM for each sensor data.
Feature Extraction
After extensive investigation of various combinations of AR coefficients as candidates
for DSF, it was observed that the first three coefficients are most sensitive to damage. As
stated in the introduction, Nair et al. (2006) have shown that these coefficients are
functionally related to the natural frequencies and mode shapes of the structure in
question. On the basis of this observation, two DSFs for the acceleration data, DSFacc,
and one DSF for the strain data, DSFstr, are defined as follows:
1str
1acc,2
23
22
21
1acc,1
DSF
DSF
DSF
(2.4)
where i is the ith AR coefficient. The DSFs are computed for the pre- and post-damage
data from all the sensors and loadings according to Equation (2.4). Then, the t-statistic
using the means and the variances of the pre- and post-damage DSF values from sensor i
are computed for the t-test to determine the significance of the difference between them.
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
20
Then the DM is defined in terms of the mean and the variance of the DSFs to quantify the
difference between the DSFs for the undamaged case and those for the damaged case as
follows:
2
uDSF,
2dDSF,uDSF, )(
DMS
(2.5)
where DSF,d and DSF,u are the mean values of the DSFs obtained from the damaged and
undamaged case, respectively, and S2DSF,u is the sample variance of the DSF from the
undamaged case.
Classification Using Hypothesis Tests
The hypothesis test to determine the statistical significance of the difference between
DSF, d and DSF, u is set up as follows:
dDSF,uDSF,1
dDSF,uDSF,0
:
:
H
H(2.6)
where H0 and H1 are the null and alternate hypotheses, respectively. H0 represents the
undamaged condition, and H1 represents the damaged condition. The significance level of
the test is set at 0.05. The hypothesis in Equation (2.6) is called a two-sided alternative.
To test the above hypothesis, the t-statistic (Rice, 1999) is used. The damage decision is
made by examining the point estimate and the confidence interval (CI) of the difference
between DSF, d and DSF, u. The t-statistic is defined as follows:
mns
t11
ˆˆ dDSF,uDSF,
(2.7)
where m and n are the number of samples obtained from DSFs from the damaged and
undamaged cases, respectively; and s is the pooled sample variance, given as
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
21
2
11 2dDSF,
2uDSF,2
nm
SmSns (2.8)
where S2DSF,d is the sample variance of the DSFs from the damaged case. The confidence
interval for the difference in DSF, u and DSF, d is
mn
stCI nm
11
2ˆˆ 2dDSF,uDSF,
(2.9)
where tn+m-2(/2) is the value of the t-distribution with n+m-2 degrees of freedom
obtained at /2.
2.2.3.2 Algorithm 2: AR Model with Gaussian Mixture Models
The second damage diagnosis algorithm defines a damage sensitive feature vector using
the first three AR coefficients, models the feature vectors using a GMM, and quantifies
damage using the Mahalanobis distance. On the basis of the assumption that the pre- and
post-damage data are from different Gaussian distributions, the algorithm computes the
parameters of the GMM, the mean vector and the covariance matrix. Then it measures the
Mahalanobis distance between the clusters to determine the damage state.
Feature Extraction
This algorithm defines the feature vector as the first three AR coefficients as follows:
3
2
1
torFeatureVec
(2.10)
The first three AR coefficients are chosen because they contain most of the information
in the data.
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
22
Then, the algorithm models the feature vectors from the pre- and post-damage data as
separate Gaussian clusters. A multivariate GMM with K clusters has the following form:
);()(
1:1 iii
K
iL φf Xx
(2.11)
where X is the collection of L training feature vectors, iφ is a Gaussian vector with the
mean vector iμ and the covariance matrix iΣ , and i is the non-negative cluster weight
for each class (Nair et al., 2006). The assumption that the feature vectors of the pre- and
post-damage data are from different Gaussian clusters determines the number of clusters
and the cluster weights for each cluster. The rest of the unknown parameters of GMM,
iμ and iΣ , can be calculated in the following way:
1
))((1
1
l
l
Tijij
l
ji
j
l
ji
μxμxΣ
xμ
(2.12)
where l is the number of the feature vectors in the cluster i, and jx is the feature vector j
in the cluster. The Gaussian cluster of the pre-damage data is used as a baseline, and that
of the post-damage data is compared with the baseline using the Mahalanobis distance in
order to determine the damage state.
Classification Using Mahalanobis Distance
In general, the Mahalanobis distance is a distance measure between two random vectors
of the same distribution. It represents the dissimilarity between the two vectors, given the
covariance of the components of the random vectors, and can be quantified as follows:
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
23
)()();,( 1 yxΣyxΣyx T (2.13)
where Σ is a covariance matrix. In this analysis, the vectors x and y correspond to the
mean values of the feature vectors obtained from the baseline and the damaged data, uμ
and dμ respectively.
We defined distance measures similar to the Mahalanobis distance but assumed that the
feature vectors of the pre- and post-damage data are from two different distributions. The
Mahalanobis distance assumes that both feature vectors obtained from pre- and post-
damage data share the same covariance matrix, which is not necessarily true in practice.
Thus, several different distance measures are defined on the basis of different covariance
matrices as follows in order to determine their correlation to the various damage levels:
)()();,( 11 dddu μμΣμμΣμμ
uuT
u (2.14a)
)()();,( 1
2 dddu μμΣμμΣμμ uud
Tu (2.14b)
)()();,( 13 dududu μμΣμμΣμμ
duT (2.14c)
1 1 1 1
2 2 2 24
1 11 1 2 2
( , ; ) ( ) ( )
2
Tu d u d
T T Tu d d u
u d u d u d
u u d d u d
μ μ Σ Σ μ Σ μ Σ μ Σ μ
μ Σ μ μ Σ μ μ Σ Σ μ
(2.14d)
uu
dududu
μΣμ
μμΣμμΣμμ
1
1
5
)()();,(
uT
uT
(2.14e)
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
24
where ,1
))((1
l
x Tujj
l
ju
μμxΣ
u
,1
))((1
l
y Tjdj
l
jd
dμyμΣ
,1
))((1
l
Tjj
l
jud
du μyμxΣ
1
))()))((()((1
l
Tjjjj
l
jdu
dudu μμyxμμyxΣ ,
l is the number of the feature vectors in the mixture, and jx and jy are the feature
vectors j of the undamaged and damaged data, respectively. The distance measure Δ1
follows the fundamental definition of the Mahalanobis distance under the assumption that
the distribution of the post-damage feature vectors y is similar to that of the pre-damage
feature vectors x and the covariance matrix of the pre-damage state is used to weigh the
distance between two vectors. The next distance measure, Δ2 uses the covariance matrix
of x and y as the representative covariance matrix of the distributions of x and y. Thus, it
includes the uncertainty of both the pre- and post-damage data. The measure Δ3 uses the
covariance matrix of the difference between the pre-damage feature vector x and the
post-damage feature vector y to weigh the distance. The measure Δ4 is derived from the
definition of the Mahalanobis distance except that the covariance matrices of x and y are
assumed to be different. The measure Δ5 is the distance measure that Nair and
Kiremidjian (2007) developed, which is the normalized Mahalanobis distance. Using the
definitions above, we can compute the distances between the pre- and post-damage
feature vectors. For reference, the distance measures between the pre-damage feature
vectors are also computed. In order to compute them, the feature vectors for the
undamaged case are divided into two groups by separating the even samples and the odd
samples. The distance between the two groups is theoretically zero. These results are
discussed in section 2.3.2.
2.2.3.3 Algorithm 3: AR Model with Information Criteria
The third algorithm defines the DSF as the first three AR coefficients and determines the
damage state by computing the optimal number of clusters in the mixture of DSFs from
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
25
the pre- and post-damage data. The number of clusters in the mixed set of DSFs is
identified as the number of damage states present in the data mixture. In general,
determining the optimum number of clusters present in a given data set is a difficult
problem. With this algorithm, we investigate methods using different information criteria
for clustering DSF values into each damage state. The hypothesis is that there will be one
cluster of DSFs for each damage scenario plus the baseline case. Then those methods are
applied to both simulated data and experimental data in order to compare their
performance. Among the investigated information criteria, the criterion developed by
Olivier et al. (1999) performed most accurately for estimating the number of damage
states in the data set.
Feature Extraction
In this algorithm, the first three AR coefficients are used as a DSF, and the mixture of
DSFs from various damage states is modeled as a multivariate GMM with K clusters.
Then, the information criteria are applied to determine K. This mixture model is the
density composed of a weighted sum of K cluster densities. The parameters of the model
(mixture weights, the mean and covariance matrix of each mixture) are fit to the data
using the Expectation-Maximization (EM) algorithm (Tibshirani et al., 2001). This is an
iterative algorithm that seeks to stabilize the values of the model parameters. One way of
determining the model parameters is to maximize the log-likelihood function. First, the
expectation of membership in each cluster is calculated for each data point (E step). Then,
the model parameters are updated on the basis of the maximum likelihood (M step). This
process is repeated until the change of the value of log-likelihood at each iteration is less
than some fixed tolerance. The first step of the algorithm is to initialize the model
parameters. In this analysis, the k-means algorithm is used to estimate the initial
parameters. The k-means algorithm, however, assumes that the number of clusters, K, is
known. The next section explains how to estimate the optimum number of clusters.
One possible problem with using the multivariate Gaussian mixture model in this kind of
study may come from the fact that the true distribution of AR coefficients is not Gaussian,
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
26
and that each point may not be independent of the others. However, independence is not
an issue for our case because the input acceleration data are assumed to be random
ambient vibrations. Optimally, the signal would be white noise; however, the field data
are not perfectly stationary. Since the purpose of the algorithm is to simply identify the
presence of multiple clusters (and therefore damage), Gaussian distribution is not a bad
assumption. Olivier et al. (1999) showed that their φβ information criterion is capable of
identifying the optimal number of clusters in non-Gaussian data sets.
Classification Using Information Criteria
A good estimator for the optimal number of clusters must balance closely fitting the
model to the data and reducing the complexity of the model. As the number of estimated
clusters increases, the model will fit the data more closely, up to the trivial case where
each point itself is represented by one cluster. To prevent over-fitting, a penalty term is
added to the optimization function, which is a function of the number of clusters
estimated. The information criteria considered in this section are based on the likelihood
function of the model plus some penalty term. We can calculate the optimal number of
clusters by minimizing the information criteria over k, where k is the number of estimated
clusters. The information criteria that is investigated in this section include the Akaike
information criterion (AIC) with two and three penalty factors, referred to as AIC and
AIC3, respectively; the minimum description length (MDL); and the information
criterion proposed by Olivier et al. (1999). They are discussed in greater detail as follows.
The Akaike information criterion (AIC) is given by Akaike (1973) as
kk 2likelihoodlog2)(AIC (2.15)
where k is the number of estimated clusters. However, as the likelihoodlog2 value
becomes larger, the penalty term becomes increasingly insignificant. For large data sets,
there is a high probability that the log-likelihood values will grow large. Bozdogan
(1993) argued that the penalty factor of 2 is not correct for finite mixture models
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
27
according to the asymptotic distribution of the likelihood ratio for comparing two models
with different parameters. Instead, he suggested using 3 as the penalty factor. This
information criterion is referred to as AIC3.
Rissanen (1978) proposed the idea of using the minimum description length (MDL) for
model selection. As the estimation error, he used the length of the Shannon-Fano code for
the data, which implies the complexity of the data given a model in information theory.
Shannon-Fano coding is a method of producing a code that is not a prefix of any other
code in the system on the basis of the data and their probabilities. The idea is to choose
the model that results in the minimum description length (or the minimum of the
combination of the model complexity and the estimation error). He used nklog as the
penalty term, where n is the number of data. He assumed that nlog represents the
precision for each parameter of any given model.
Olivier et al. (1999) developed a new information criterion, φβ, which is given as
))log(log(
parametersgiven hood
-likelilog lconditiona2 nkn(k)
(2.16)
where n is the total number of points in the data set and 0 < β < 1 is a parameter that we
input to scale the information criterion. If β is too large, the φβ criterion excessively
penalizes for complexity and underestimates the number of clusters. If β is too small, the
criterion overestimates the number of clusters. Olivier et al. (1999) derived desirable
lower and upper bounds for β, which are
)log(
))log(log(1
)log(
))log(log(
n
n
n
n (2.17)
They showed that this φβ criterion effectively estimates the number of clusters present in
simulated data. Although choosing an appropriate β is not trivial, this criterion is
effective in definitively showing a minimum value for the optimal number of clusters.
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
28
2.3 Application of the Damage Diagnosis
Algorithms to Experimental Data Using the
Taiwanese Benchmark Structure
2.3.1 Description of Experiment
A series of shake table tests of a three-story single-bay steel structure was designed and
performed by the National Center for Research on Earthquake Engineering (NCREE) in
Taipei, Taiwan, in order to provide information about a controlled damage occurrence on
a structure. Figure 2.3 shows the test structure deployed with various sensors at different
locations. The inter-story height of the benchmark structure is 3m. Floor dimensions at
every story are 3m by 2m, and each floor mass is 6 tons. The beams and the columns
consist of H150x150x7x10 steel I-beams, and each beam-column joint is designed as a
bolted connection (Lynch et al., 2006).
Various excitations on the benchmark structure were applied through a shaking table as
ground motions. The applied random excitations have maximum amplitudes of 60 gal
(cm/sec2) intensity in the X-direction, 100 gal intensity in the X-direction, and 50 gal
intensity in the XY-direction. In addition, the tests selected strong ground motion
excitations from the records of earthquakes in El Centro, California, 1994; Kobe, Japan,
1995; and Chi-Chi, Taiwan, 1999. Both wired and wireless sensors, including
accelerometers and strain gauges, were installed. This chapter uses the data from only the
wired sensors for the analysis. The data sampling frequency was 200 Hz. In Figure 2.3, A
and S represent the accelerometer and strain gauges, respectively. All the measurements
are used for the analysis including both types of sensors (A and S) at all the sensor
locations and for both the uni-directional and bi-directional random excitations; however,
some of the data were corrupted and could not be used. For each excitation with 50 gal,
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
29
60 gal, and 100 gal intensities, acceleration data were collected from 12 different
locations, and strain data were collected from 40 different locations. Table 2.1 shows all
the collected data and identifies the numbers of corrupted and usable data sets for the
analysis.
Figure 2.5: Photograph of the cut flanges of the column
In order to compare the structural behavior in a normal condition to that in damaged
conditions, two damage patterns (DPs) were introduced during the experiment. The first
DP involves the flange width reduction by 26.67% at the lower part of column 1 (shown
in Figure 2.3) at the ground floor (shown in Figure 2.5). The second damage case has the
flanges of both columns 1 and 2 cut the same amount near the ground floor. Hereafter,
these DPs are referred to as DP 1 and DP 2, respectively. Figure 2.5 shows the damage
applied to the base columns.
2.3.2 Results and Discussion
2.3.2.1 Algorithm 1: AR Model with Hypothesis Tests
Acceleration Data Analysis
To illustrate the algorithm, this section first presents the results for selected sensor
locations. Figure 2.6 shows the variation of the DSFs for the successful cases of damage
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
30
detection. Figure 2.6 (a) is the plot of DSFacc,2 for acceleration data AY1b (as identified
in Figure 2.3) for random excitation in the X-direction having a peak acceleration of 100
gal. Figure 2.6 (b) illustrates the results for strain sensor S3 (as shown in Figure 2.3) for
random bidirectional excitation. In both cases, the mean of the DSFs for the undamaged
case is less than that of DP 1, which is in turn smaller than that of DP 2.
0 50 100 150-2
-1.8
-1.6
-1.4
-1.2
-1
Acceleration samples
DS
Fac
c
UndamagedDP 1DP 2
DSF,
DP1
DSF,
undamaged
DSF,
DP2
(a)
0 50 100 150 200 250-1.8
-1.6
-1.4
-1.2
-1
Strain samples
DS
Fst
r
UndamagedDP 1DP 2
DSF,
undamaged
DSF,
DP1
DSF,
DP2
(b)
Figure 2.6: Variation of DSF with damage: (a) Variation of DSFacc,2 for acceleration data; (b) Variation of DSFstr for strain data
Tables 2.3 to 2.14 show the results of the t-test for the AR model and the confidence
intervals for the difference in the mean values of the damaged and undamaged cases. In
analyzing the acceleration data, we first considered the two DSFacc defined by Equation
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
31
(2.4). It was found that DSFacc,2 works well for damage detection, but does not effectively
distinguish between DP 1 and DP 2. DSFacc,1 results in smaller confidence intervals but is
able to distinguish DP 1 from DP 2. Tables 2.3 and 2.4 show the damage detection results
using DSFacc,1 for the unidirectional excitation with a peak amplitude of 60 gals, Tables
2.5 and 2.6 show the damage detection results for the unidirectional excitation with a
peak amplitude of 100 gals, and Tables 2.7 and 2.8 show the results for the bi-directional
excitation with a peak amplitude of 50 gals.
Tables 2.9 through 2.14 show the results using DSFacc,2 for the unidirectional excitation
with a peak amplitude of 60 gals and 100 gals and for the bi-directional excitation with a
peak amplitude of 50 gals. The sensor locations are listed in the first column of the tables.
Figure 2.7 shows the point estimations of dDSFuDSF ,, ˆˆ using DSFacc,1, and Figure 2.8
shows those using DSFacc,2.
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
32
Figure 2.7: Point estimates of dDSFuDSF ,, ˆˆ using DSFacc,1: (a) 60 gal unidirectional
random excitation, X direction data result; (b) 60 gal unidirectional random excitation, Y direction data result; (c) 100 gal unidirectional random excitation, Y direction data result; (d) 50 gal bidirectional random excitation, X direction data result; (e) 50 gal bidirectional
random excitation, Y direction data result
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
33
Figure 2.8: Point estimates of dDSFuDSF ,, ˆˆ using DSFacc,2: (a) 60 gal unidirectional
random excitation, X direction data result; (b) 60 gal unidirectional random excitation, Y direction data result; (c) 100 gal unidirectional random excitation, Y direction data result; (d) 50 gal bidirectional random excitation, X direction data result; (e) 50 gal bidirectional
random excitation, Y direction data result
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
34
Table 2.3: Results of damage detection using DSFacc,1 for 60 gal unidirectional random excitation for DP 1 using the point estimate and CI of DSF, undamaged - DSF, DP1
Sensor No.
Damage Decision
Point Estimate
Confidence Interval
A1a H0 0.0010 [-0.0107, 0.0127] A3a H0 -0.0010 [-0.0141, 0.0121] A1b H0 0.0092 [-0.0013, 0.0196] A3b H0 0.0011 [-0.0105, 0.0126]
AY3a H0 0.0108 [-0.0185, 0.0400] AY1b H0 -0.0014 [-0.0095, 0.0068] AY2b H0 0.0257 [-0.0039, 0.0553] AY3b H0 0.0114 [-0.0188, 0.0416]
Table 2.4: Results of damage detection using DSFacc,1 for 60 gal unidirectional random
excitation for DP 2 using the point estimate and CI of DSF, undamaged - DSF, DP1 Sensor
No. Damage Decision
Point Estimate
Confidence Interval
A1a H1 0.0223 [0.0105, 0.0340] A3a H0 0.0002 [-0.0137, 0.0141] A1b H1 0.0196 [0.0102, 0.0289] A3b H0 0.0101 [-0.0010, 0.0211]
AY3a H1 -0.0433 [-0.0755, -0.0111] AY1b H1 0.0203 [0.0115, 0.0290] AY2b H0 -0.0300 [-0.0603, 0.0004] AY3b H0 -0.0227 [-0.0552, 0.0098]
Table 2.5: Results of damage detection using DSFacc,1 for 100 gal unidirectional random
excitation for DP 1 using the point estimate and CI of DSF, undamaged - DSF, DP1 Sensor
No. Damage Decision
Point Estimate
Confidence Interval
AY2a H1 -0.0224 [-0.0329, -0.0118] AY3a H1 -0.0428 [-0.0622, -0.0234] AY1b H1 0.0185 [0.0120, 0.0249] AY2b H1 -0.0253 [-0.0345, -0.0160] AY3b H1 -0.0554 [-0.0773, -0.0334]
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
35
Table 2.6: Results of damage detection using DSFacc,1 for 100 gal unidirectional random excitation for DP 2 using the point estimate and CI of DSF, undamaged - DSF, DP2
Sensor No.
Damage Decision
Point Estimate
Confidence Interval
AY2a H1 -0.0538 [-0.0687, -0.0388] AY3a H1 -0.0668 [-0.0877, -0.0458] AY1b H1 0.0429 [0.0367, 0.0491] AY2b H1 -0.0665 [-0.0825, -0.0505] AY3b H1 -0.1038 [-0.1288, -0.0787]
Table 2.7: Results of damage detection using DSFacc,1 for 50 gal bi-directional random
excitation for DP 1 using the point estimate and CI of DSF, undamaged - DSF, DP1 Sensor
No. Damage Decision
Point Estimate
Confidence Interval
A1a H1 0.0299 [0.0211, 0.0387] A2a H1 0.0079 [0.0020, 0.0137] A3a H1 -0.0134 [-0.0218, -0.0050] A1b H1 0.0223 [0.0123, 0.0322] A3b H0 -0.0050 [-0.0137, 0.0038]
AY1a H1 0.0206 [0.0134, 0.0278] AY2a H1 0.0102 [0.0041, 0.0163] AY3a H0 0.0048 [-0.0015, 0.0111] AY1b H1 0.0315 [0.0238, 0.0392] AY2b H0 0.0041 [-0.0025, 0.0107] AY3b H0 0.0001 [-0.0058, 0.0060]
Table 2.8: Results of damage detection using DSFacc,1 for 50 gal bi-directional random
excitation for DP 2 using the point estimate and CI of DSF, undamaged - DSF, DP2 Sensor
No. Damage Decision
Point Estimate
Confidence Interval
A1a H1 0.0452 [0.0364, 0.0540] A2a H0 0.0045 [-0.0016, 0.0105] A3a H0 -0.0079 [-0.0164, 0.0006] A1b H1 0.0445 [0.0341, 0.0549] A3b H0 -0.0058 [-0.0138, 0.0023]
AY1a H1 0.0257 [0.0182, 0.0331] AY2a H1 0.0180 [0.0118, 0.0241] AY3a H0 0.0051 [-0.0016, 0.0118] AY1b H1 0.0441 [0.0353, 0.0529] AY2b H0 0.0053 [-0.0018, 0.0123] AY3b H0 0.0034 [-0.0031, 0.0098]
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
36
Table 2.9: Results of damage detection using DSFacc,2 for 60 gal unidirectional random
excitation for DP 1 using the point estimate and CI of DSF, undamaged - DSF, DP1 Sensor
No. Damage Decision
Point Estimate
Confidence Interval
A1a H1 0.0569 [0.0029, 0.1109] A3a H0 0.0123 [-0.0218, 0.0463] A1b H0 -0.0111 [-0.0582, 0.0361] A3b H0 -0.0019 [-0.0369, 0.0331]
AY3a H1 0.0590 [0.0129, 0.1050] AY1b H1 0.0730 [0.0192, 0.1268] AY2b H1 0.0636 [0.0076, 0.1195] AY3b H1 0.0490 [0.0028, 0.0951]
Table 2.10: Results of damage detection using DSFacc,2 for 60 gal unidirectional random
excitation for DP 2 using the point estimate and CI of DSF, undamaged - DSF, DP1 Sensor
No. Damage Decision
Point Estimate
Confidence Interval
A1a H0 -0.0481 [-0.1008, 0.0046] A3a H0 0.0143 [-0.0220, 0.0506] A1b H1 -0.0725 [-0.1157, -0.0292] A3b H0 0.0159 [-0.0191, 0.0509]
AY3a H0 -0.0227 [-0.0653, 0.02] AY1b H1 -0.0857 [-0.1371, -0.0342] AY2b H1 -0.0607 [-0.1118, -0.0096] AY3b H0 -0.0386 [-0.0800, 0.0029]
Table 2.11: Results of damage detection using DSFacc,2 for 100 gal unidirectional random
excitation for DP 1 using the point estimate and CI of DSF, undamaged - DSF, DP1 Sensor
No. Damage Decision
Point Estimate
Confidence Interval
AY2a H1 0.1665 [0.1301, 0.2030] AY3a H1 0.1234 [0.0842, 0.1626] AY1b H1 0.1603 [0.1268, 0.1937] AY2b H1 0.1301 [0.0925, 0.1676] AY3b H1 0.1532 [0.1133, 0.1932]
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
37
Table 2.12: Results of damage detection using DSFacc,2 for 100 gal unidirectional random excitation for DP 2 using the point estimate and CI of DSF, undamaged - DSF, DP2
Sensor No.
Damage Decision
Point Estimate
Confidence Interval
AY2a H1 0.2989 [0.2578, 0.3400] AY3a H1 0.2093 [0.1713, 0.2473] AY1b H1 0.2926 [0.2566, 0.3287] AY2b H1 0.2379 [0.1962, 0.2797] AY3b H1 0.2645 [0.2275, 0.3016]
Table 2.13: Results of damage detection using DSFacc,2 for 50 gal bi-directional random
excitation for DP 1 using the point estimate and CI of DSF, undamaged - DSF, DP1 Sensor
No. Damage Decision
Point Estimate
Confidence Interval
A1a H0 0.0119 [-0.0193, 0.0432] A2a H1 -0.1111 [-0.1465, -0.0758] A3a H0 -0.0181 [-0.0473, 0.0109] A1b H0 0.0248 [-0.0093, 0.0588] A3b H0 0.0078 [-0.0224, 0.0381]
AY1a H0 -0.0307 [-0.0624, 0.0009] AY2a H1 -0.0601 [-0.0925, -0.0277] AY3a H0 -0.0334 [-0.0713, 0.0034] AY1b H1 -0.0434 [-0.0736, -0.0131] AY2b H1 -0.0429 [-0.0751, -0.0107] AY3b H0 -0.0041 [-0.0394, 0.0311]
Table 2.14: Results of damage detection using DSFacc,2 for 50 gal bi-directional random
excitation for DP 2 using the point estimate and CI of DSF, undamaged - DSF, DP2 Sensor
No. Damage Decision
Point Estimate
Confidence Interval
A1a H1 -0.0745 [-0.1042, -0.0448] A2a H1 -0.12843 [-0.1623, -0.0946] A3a H1 -0.0489 [-0.0786, -0.0193] A1b H1 -0.0742 [-0.1059, -0.0424] A3b H1 -0.0285 [-0.0566, -0.0003]
AY1a H1 -0.0872 [-0.1214, -0.0529] AY2a H1 -0.1412 [-0.1744, -0.1079] AY3a H1 -0.0497 [-0.0890, -0.0103] AY1b H1 -0.1107 [-0.1464, -0.0749] AY2b H1 -0.0804 [-0.1171, -0.0437] AY3b H0 -0.0166 [-0.0523, 0.0190]
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
38
The analysis of the DSFacc,1 has yielded the following observations:
1. In the case of unidirectional random excitation with peak amplitude of 60 gals, DP 1
is not detected at any of the sensor locations while DP 2 is detected at 4 out of 8
sensor locations using the t-statistic. As shown in Figures 2.7 (a) and (b), the point
estimates of dDSF,uDSF, ˆˆ for DP 2 are larger than those for DP 1 except at 1 out of
8 sensor locations.
2. When the 100 gal unidirectional random excitation is applied, both DP 1 and DP 2 are
detected at all of the sensor locations using t-statistic. Also, the point estimates of
dDSF,uDSF, ˆˆ for DP 2 are larger than those for DP 1, as shown in Figure 2.7 (c).
Thus, DSFacc,1 could potentially be used for developing a damage extent measure;
however, testing with additional data will be needed before it can be applied widely.
3. In the case of bidirectional random excitation with a peak amplitude of 50 gals, DP 1
is detected at 7 out of 11 sensor locations, and DP 2 is detected at 5 out of 11 sensor
locations. As shown in Figure 2.7 (d) and (e), the point estimates of dDSF,uDSF, ˆˆ
for DP 2 are larger than those for DP 1 except at 2 out of 11 sensor locations.
Furthermore, the analysis shows the largest point estimates to be closest to the
damaged area.
For DSFacc,2, following observations have been found:
1. When the unidirectional random excitation with peak acceleration of 60 gal is applied
to the structure, DP 1 is detected at 5 out of 8 sensor locations while DP 2 is detected
at 3 out of 8 sensor locations using t-statistic.
2. For the unidirectional random excitation with peak acceleration of 100 gal, damage
was detected at all sensor locations. While this observation is encouraging in that it
enables us to identify damage, this DSF appears to be non-informative for damage
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
39
localization purposes. As shown in Figure 2.8 (c) the point estimates of
dDSF,uDSF, ˆˆ for DP 2 are larger than those for DP 1.
3. When the 50 gal bidirectional random excitation is applied, DP 1 is detected at 4 out
of 11 sensor locations, and DP 2 is detected at 10 out of 11 sensor locations. As
shown in Figure 2.8 (d) and (e), the point estimates of dDSF,uDSF, ˆˆ for DP 2 are
larger than those for DP 1.
Figure 2.9 shows the damage measure DM, using DSFacc,1, at each sensor for DP 1 and
DP 2. The values of DM for DP 2 are higher than those for DP 1 demonstrating that this
definition of DM appears to be sufficient to identify the relative magnitude of damage
and can thus enable the tracking of damage growth. Additional testing and analysis,
however, are necessary to fully support this claim.
A1a A2a A3a A1b A3b AY1a AY2a AY3a AY1b AY2b AY3b0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Sensors
DM
DP2DP1
Figure 2.9: DM for acceleration data from bidirectional random excitation
From the analysis described in this section, we can conclude that the difference between
the mean of DSFs for the undamaged and damaged structure increases when base
excitation with larger peak acceleration is applied and more severe damage is introduced
to the structure. For low-intensity excitation, such as the 60 gal unidirectional and 50 gal
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
40
bi-directional input motions, damage is identified near the lower floors in close proximity
to where indeed damage was introduced. However, testing with various minor damage
patterns and real data needs to be carried out to further validate the algorithm.
Strain Data Analysis
Figure 2.10 shows the values of the confidence intervals (CIs) of the difference between
the mean values of DSF of the strain data for the bidirectional random excitation with a
peak acceleration of 50 gals.
0 6 12 18 24 30 36 40-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
Sensors
CI
DP1DP2
Figure 2.10: Confidence intervals of the DSFstr for 50 gal bidirectional random excitation
As observed in Figure 2.3, strain sensors S1-S6 are at the location of the cut on column 1
for DP 1. Similarly for DP 2, strain sensors S1-S6 and S13-S18 are at the location of the
cut on columns 1 and 2 respectively.
From the analysis of strain data, the following observations are made:
1. In the case of DP 1, strain sensors S1 and S3 to S6 have larger values of CIs as
compared to other sensors. These sensors are close to the damaged area correctly
pointing to the damage occurrence and its location.
2. High values of CI are also observed for strain sensors S1, S3-S6 and S13-S18 in the
case of DP 2. Again these sensors are close to the damaged region on the structure .
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
41
3. The CIs for the higher floors are consistently lower than those at lower floors (Figure
2.10). Thus localization of damage could potentially be achieved using the strain
measurements. Again, additional studies and test cases need to be investigated to
further reinforce this observation.
2.3.2.2 Algorithm 2: AR Model with Gaussian Mixture Models
Acceleration Data Analysis
Figure 2.11 shows Δ1 of the acceleration data for the unidirectional random excitation (60
gal) for damaged and undamaged cases. We can observe that the undamaged cases and
the damaged cases are hardly distinguishable. The magnitudes of the distance measure
from pre- and post-damage data are very close to one another, which indicates that the
change in vibration response due to the damage is not significant. It is possible that
unidirectional random excitation with the peak acceleration of 60 gal is not strong enough
for us to detect the damage. An examination of the root-mean-square (RMS) value of the
response data resulted in values as high as 0.7 mg while the noise level of the recording
instrument was 0.4 mg. Therefore, the instrument noise level was too high for reliable
damage analysis.
When applying the second algorithm to the acceleration data for the unidirectional
random excitation (100 gal), it was observed that Δ1, Δ2, Δ3, and Δ5 have larger quantities
at all the sensor locations for DP 2 than for DP 1, which are in turn larger than for the
undamaged case. Δ4 has larger quantities for DP 2 than for DP 1, but those for the
undamaged case are not always less than for the damaged cases. Of all the measures
listed in Equation (2.14), Δ4 least resembles a distance measure between the damaged and
undamaged coefficients. As a result, Δ4 is not considered to be a suitable measure of
damage. The distance measures are maximum for the second floor data, which is the
closest floor from the damage, and minimum for the third floor. The reason the distance
measure for the roof is higher than for the third floor might be the influence of higher
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
42
modes that may be excited because of the asymmetry introduced by the damage at the
base of the structure. Figure 2.12 illustrates the distance measure Δ1 of the acceleration
data for the unidirectional random excitation (100 gal) for the damaged and undamaged
cases.
AX1a AX3a0
0.5
1
1.5
2
Sensors
1
1,undamaged
1,DP1
1,DP2
Figure 2.11: Δ1 for the acceleration data from 60 gal unidirectional random excitation-X
direction result for column (a)
AY1b AY2b AY3b 0
1
2
3
4
5
6
7
Sensors
1
1,undamaged
1,DP1
1,DP2
Figure 2.12: Δ1 for the acceleration data from 100 gal unidirectional random excitation-Y
direction result for column (b)
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
43
AY1b AY2b AY3b 0
1
2
3
4
Sensors
1
1,undamaged
1,DP1
1,DP2
Figure 2.13: Δ1 for the acceleration data from 50 gal bidirectional random excitation-Y
direction result for column (b)
Figure 2.13 shows the distance measure Δ1 of the acceleration data for the bidirectional
random excitation (50 gal) for the damaged and undamaged cases. We can observe that
the damaged cases have higher distance measures than the undamaged cases, but it is
hard to distinguish between DP 1 and DP 2. It is possible that bidirectional random
excitation with the peak acceleration of 50 gal is not strong enough for us to distinguish
the two different damage patterns using this algorithm.
Strain Data Analysis
Tables 2.15 to 2.17 show the mean values of the distance measures of the strain data.
Table 2.15 shows that the distance measures for the strain data for the unidirectional
random excitation of 60 gal can distinguish the damaged cases from the undamaged case.
It also shows that DP 1 has higher distance measures than DP 2, although DP 2 is a more
severe damage case. These results again point to possible excessive noise in the response
motions. Furthermore, the response motions are measured in the z-direction and would be
most affected by input motions that are in the y-direction. The 60 gal motion is applied in
the x-direction causing very small strains, and consequently the values may be masked by
possible noise in the sensors.
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
44
Table 2.15: Results of mean values of various distance measures from the strain data for 60 gal uni-directional random excitation for undamaged and damaged cases
Mean(Δ1) Mean(Δ2) Mean(Δ3) Mean(Δ4) Mean(Δ5) Undamaged 0.3003 0.7963 0.2653 3.4532 0.0114
DP1 1.9740 6.6970 1.5145 5.2080 0.0751 DP2 1.0650 3.3727 0.8110 4.5245 0.0413
Table 2.16: Results of mean values of various distance measures from the strain data for
100 gal uni-directional random excitation for undamaged and damaged cases Mean(Δ1) Mean(Δ2) Mean(Δ3) Mean(Δ4) Mean(Δ5)
Undamaged 0.3468 0.8944 0.2632 8.8976 0.0095 DP1 0.9205 1.7466 0.9628 6.7689 0.0254 DP2 1.9404 6.8201 1.5201 10.1990 0.0502
Table 2.17: Results of mean values of various distance measures from the strain data for
50 gal bi-directional random excitation for undamaged and damaged cases Mean(Δ1) Mean(Δ2) Mean(Δ3) Mean(Δ4) Mean(Δ5)
Undamaged 0.3559 0.8911 0.2477 8.6000 0.0074 DP1 0.7098 1.3376 0.7450 18.4979 0.0143 DP2 2.0007 6.2339 1.5005 15.6551 0.0391
Tables 2.16 and 2.17 show that the distance measures for the strain data for the
unidirectional random excitation of 100 gal and those for the bidirectional random
excitation of 50 gal have smaller values for the undamaged cases than DP 1, which are in
turn smaller than DP 2. Figures 2.14 to 2.16 show the mean values of the distance
measure Δ1 of the strain data at each location. The following observations are made from
these figures:
1. For the 100 gal and 50 gal peak acceleration excitation, the distance measure for DP 2
has higher values than that for DP 1.
2. For the 100 gal and 50 gal peak acceleration excitation, the distance measure for DP 1
has the maximum value at the damage location.
3. For the 100 gal and 50 gal peak acceleration excitation, the distance measure for DP 2
is higher for the higher floors.
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
45
4. For the 60 gal peak acceleration excitation, the distance measure for DP 1 is higher
than that for DP 2.
5. For the 60 gal peak acceleration excitation, the distance measure for DP 1 is higher
for the higher floors.
0 6 12 18 24 30 3638 400
1
2
3
4
5
6
7
Sensors
1
mean1,undamaged
mean1,DP1
mean1,DP2
Figure 2.14: Mean values of Δ1 for the strain data from 60 gal unidirectional random
excitation
0 6 12 18 24 30 3638 400
1
2
3
4
Sensors
1
mean1,undamaged
mean1,DP1
mean1,DP2
Figure 2.15: Δ1 for the strain data from 100 gal unidirectional random excitation
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
46
0 6 12 18 24 30 3638 400
1
2
3
4
Sensors
1
mean1,undamaged
mean1,DP1
mean1,DP2
Figure 2.16: Δ1 for the strain data from 50 gal bidirectional random excitation
2.3.2.3 Algorithm 3: AR Model with Information Criteria
To validate Algorithm 3, three sets of data are used: one set of simulated data,
experimental data, and field data. First, the numerical simulation is performed in order to
duplicate the results of the numerical simulation performed by Olivier et al. (1999). The
experimental data is from the Taiwanese benchmark structure introduced in section 2.3.1.
Finally, the field data are collected from the Z24 Bridge in Switzerland, which was
subjected to settlement in one pier. The performance of the φβ information criterion is
compared with the four other criteria, -2 (log-likelihood), AIC, AIC3, and MDL in order
to detect the presence of damage in a mixed data set of AR coefficients extracted from
pre- and post-damage acceleration measurements. In addition, the effect of different
values of β on the performance of the φβ criterion is also investigated over a wide range
of estimated clusters. To create a mixed data set, the AR coefficients from the undamaged
state and various damage scenarios are mixed in one data set. By minimizing the
information criteria as described in section 2.2.3.3, the optimal number of mixtures is
estimated.
The simulated data are generated from three of two-dimensional Gaussian distributions
with the parameters given in Table 2.18. The results are shown in Figure 2.17. The
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
47
criteria φβ,min, φβ,avg, and φβ,max correspond to φβ criteria with different β values – lower
bound, average of lower and upper bound, and upper bound, respectively. For
comparison, the results for -2 (log-likelihood) values are also shown in Figure 2.17. For
the most part, the results are similar to Olivier et al.’s simulation. However, these results
find the MDL criterion less effective. In general, the φβ criterion with β = 0.3 or 0.4
produced the best results. The results show that the φβ criteria are found to estimate the
number of clusters without over-parameterizing the data, which is consistent with the
results by Olivier et al. (1999).
Table 2.18: Parameters of Gaussian distributions for generating data Number of samples Mean Variance
300 5
0
1 0
0 1
300 0
0
5 2
2 3
300 -5
0
1 0
0 1
Figure 2.17: Estimated optimum number of clusters for simulated data
The second set of data is from the Taiwanese benchmark structure subjected to the bi-
directional excitation with peak amplitude of 50 gals. Figure 2.17 shows the results for
the various information criteria over cluster estimates of 1 through 5 for the strain data
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
48
collected from the bottom of the column on the first story. The result of AIC is very
similar to that of AIC3, thus omitted in the figure for simplicity. We can observe that -
2 (log-likelihood), AIC3, MDL, and φβ,min overestimate the number of clusters while
φβ,max underestimates it. The criterion φβ,avg results in the estimation of the optimum
number of clusters to be three, which is the correct number of different damage patterns
for this experiment.
Figure 2.18: Estimated optimum number of clusters for the Taiwanese benchmark
structure
The field data are collected from the Z24 Bridge, which was an overpass that spanned the
A1 Berne-Zurich motorway. Before its demolition, the bridge was subjected to a number
of controlled damage scenarios, during which acceleration data was collected from a
large number of accelerometers at 100 Hz sampling rate. The damage patterns of concern
in this study are a series of controlled settlements of one bridge pier as shown in Table
2.19. The data used in this analysis are the first three AR coefficients computed from an
acceleration signal from a sensor above the pier that was subjected to settlement in the
vertical direction. A more detailed description of the structural details of the bridge and
details of the test can be found in Wenzel (1997-1998).
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
49
Table 2.19: Damage patterns for the Z24 bridge Damage Pattern Description
Baseline No settlement DP 1 20 mm settlement DP 2 40 mm settlement DP 3 80 mm settlement DP 4 95 mm settlement
Figure 2.19 shows the value of various information criteria over cluster estimates of 1
through 8. The AIC3, MDL, and φβ with β set at its lower bound are not significantly
different than the -2 (log-likelihood), and all of them overestimate the number of
clusters. On the other hand, setting β at its upper bound penalizes additional complexity
too harshly and, as a result, underestimates the number of clusters. However, using an
average of the upper and lower bounds for β for φβ results in a minimum value at 5
clusters, which is the true number of different damage scenarios.
Figure 2.19: Estimated optimum number of clusters for the Z24 bridge data
Estimating the correct number of clusters in a data mixture is useful; however, it is
important to verify that the Gaussian populations estimated by the EM algorithm match
well with the actual data set. For this verification, the EM algorithm is applied to the Z24
bridge data, and to simplify the graphical presentation, the results are taken in two
dimensional AR coefficient space, dropping out the third coefficient. According to the
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
50
results in Figure 2.19, the EM algorithm is applied to the data mixture assuming that
there exist five clusters in the data mixture. The estimated cluster means are compared to
the actual damage scenario population means in Figure 2.20. AR1 and AR2 correspond to
the first and the second AR coefficients, respectively. As expected, when sample
populations are far apart, the model fits the data well. However, when sample populations
are very close to each other, the model has trouble fitting clusters to those populations.
This result also explains why the values of information criterion φβ do not change much
for the number of clusters 3 or above in Figure 2.19.
Figure 2.20: True and estimated cluster means
2.4 Conclusions
This chapter has presented the results of applying three time-series based damage
diagnosis algorithms (Nair et al., 2006; Nair and Kiremidjian, 2007; Noh and Kiremidjian,
2011) to the experimental data obtained from the benchmark structure of the National
Taiwan University. Both acceleration and strain data from the wired system were
analyzed. The vibration and strain data were modeled as autoregressive (AR) and
autoregressive time series with exogenous input (ARX). We found that the AR model is
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
51
sufficient to capture the characteristics of both the acceleration and strain measurements
and thus adopted it for damage discrimination. Before applying the AR model to the data,
the stability of the AR coefficients was investigated for different size of data. It was
found that the first AR coefficient changes by less than or about 5% with data samples of
200 or more. In addition, the residuals of the time-series model are also investigated to
check for stationarity and Gaussianity, and both the acceleration and the strain data are
found to be stationary and Gaussian. In this process, corrupted data are identified and
eliminated from the analysis. The uncorrupted data were then used in the subsequent
damage analyses.
Several damage sensitive features are investigated for damage diagnosis, and among
them the first three AR coefficients are used to define the feature vector. In the first
algorithm, a damage sensitive feature (DSF) is defined as a function of the first three AR
coefficients for the acceleration, and the first AR coefficient is used for the strain data.
Differences in the mean values of the DSF before and after damage indicate that there is
damage in the structure, and the t-test is used to evaluate the statistical significance of
that difference. In addition, a damage measure DM that is based on the mean and
variances of the DSFs is introduced, and we found that the DM can be directly correlated
to the amount of damage in this simple application. In the second algorithm, a Gaussian
mixture model (GMM) is used to characterize the feature vector. Damage diagnosis is
achieved by determining the distance between the mixtures. To quantify damage extent,
various distance measures are used including the Mahalanobis distance, which is defined
as the Euclidean distance between the mixtures weighted with respect to the inverse
covariance matrix. The third algorithm uses the first three AR coefficients as a feature
vector and detects damage by identifying the number of clusters in the mixture of feature
vectors using various information criteria. Four information criteria, including Akaike
Information Criteria (AIC), AIC3, minimum description length (MDL), and Olivier et
al.’s φβ criterion, are investigated for identifying the optimum number of clusters in the
mixture of feature vectors from various damage states. The mixture is modeled as a
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
52
multivariate Gaussian mixture model with k clusters, and then, the information criteria
are applied to determine k.
The results from the first algorithm presented in this chapter show that the DSFacc,2 can be
used for damage detection; DSFacc,1 and/or DM can be used for damage extent, and
DSFstr can be used for damage localization. The results from the second algorithm show
that the Mahalanobis distances for acceleration data and strain data can detect damage for
100 gal and 50 gal peak acceleration excitation, but not for 60 gal peak acceleration
excitation. It is likely that unidirectional random excitation with the peak acceleration of
60 gal is not strong enough for us to detect the damage because the noise level of the
accelerometers used to measure structural response is of the same order as the RMS value
of the measurements. In addition, the Mahalanobis distances for acceleration data can be
used to localize damage, while the mean values of the distance measures of the strain data
appear to be well correlated to damage extent.
The results of the third algorithm show that Olivier et al.’s φβ criterion works noticeably
better than other similar information criteria in identifying optimal number of clusters for
all three sets of data. However, identifying a suitable β parameter is important for the
performance of the φβ criterion. We found that taking an average of the upper and lower
bound is a good starting point for β, and this method works well in identifying the
number of clusters. A number of issues for the area of damage diagnosis remain as
follows. First, a good next step would be to check how well the mixture model fits the
damage scenarios. Specifically, given that our estimator correctly fits the number of
clusters, we can compute how well each cluster fits to a damage scenario because we
know which scenario each data point originates from. If the clusters are scattered across
multiple scenarios, then it may not make sense to use this method for damage diagnosis.
Secondly, given that this method can detect damage when it occurs, it needs to be seen
how robust this method is in avoiding false positives. Another way of phrasing this is to
determine under what circumstances AR coefficients might migrate when damage is not
present in the structure (i.e. due to temperature, strength and frequency of input ground
CHAPTER 2. Time-Series Based Damage Diagnosis Algorithm Using Ambient Vibration Data
53
motion, etc.). Third, this method is computationally-intensive because it is an iterative
method running over a large number of cluster estimates. For the purposes of single
device remote sensing, this method will be very difficult to implement. Nevertheless,
assuming that these issues can be adequately resolved, this method can be an initial step
in identifying damage in a structure. Once the number of clusters is estimated, further
analysis can be performed to compute the extent of damage.
Although the initial results of the analysis are promising, more testing needs to be
performed. These additional tests should involve varying degrees of damage, loading
conditions, environmental conditions such as temperature and humidity, as well as
different sequences of damage occurrences. In addition, different damage locations on the
structure should be considered, and damage sequences should also be investigated. It is
only after extensive experimentation and field testing with calibration that these models
can be widely applied. Nevertheless, the results presented in this chapter are encouraging
and represent promising initial step towards achieving this goal.
Chapter 3
Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
This chapter introduces three wavelet-based damage sensitive features (DSF) that use
structural responses recorded during the strong motion of an earthquake to diagnose
structural damage. Since earthquake motion is non-stationary, previously introduced
methods based on autoregressive models are not suitable for application. On the other
hand, the wavelet transform represents data as a weighted sum of time-localized waves,
and thus appropriate to model the non-stationary earthquake responses. These wavelet-
based DSFs are defined as functions of wavelet energies at particular scales and at
specific times. The first DSF (DSF1) indicates how the wavelet energy at the natural
frequency of the undamaged structure changes as the damage progresses in the structure.
The second DSF (DSF2) indicates how much the wavelet energy is spread out in time.
The third DSF (DSF3) reflects how slowly the wavelet energy decays with time. In order
to evaluate their performance, these DSFs were extracted from the data collected from
two physical experiments conducted on shake tables. The results show that as the damage
extent increases, the values of DSF1 decrease, and the values of DSF2 and DSF3 increase.
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
55
Thus, these DSFs can be used to diagnose structural damage. The robustness of these
DSFs is also shown in this chapter using a set of simulated data obtained from an
analytical model of a four-story steel moment-resisting frame subjected to forty ground
motions scaled to various intensities.
3.1 Introduction
In order to assess damage immediately after an earthquake using the recordings from the
strong motion, it is necessary to develop an algorithm that directly utilizes these
recordings. Currently, there are no methods that enable us to diagnose structural damage
immediately after an earthquake using the structural responses from that earthquake.
Previous work on wireless damage diagnosis algorithms, including the work introduced
in Chapter 2, has focused on the use of ambient vibrations obtained before and
immediately after the occurrence of an extreme event such as an earthquake (Sohn and
Farrar, 2001; Nair et al., 2006; Farrar and Worden, 2007). Because these previous
damage diagnosis algorithms use ambient vibration measurements that are stationary,
they cannot be used with earthquake motions which are non-stationary. For this purpose,
we have developed a new method that uses wavelet energies of input ground motions and
structural acceleration responses to detect damage in the structure from strong earthquake
motion. The main advantage of this approach is that it captures the non-stationary
character of both earthquake ground motions and structural responses by using the
wavelet transform. While the diagnostic algorithm proposed in this chapter may be
particularly suitable for embedding in wireless monitoring systems, it is not restricted to
implementation for such systems and can easily be applied to wired systems.
Early work using wavelet analysis for structural health monitoring has been carried out
from several different perspectives. From a system identification perspective, Basu and
Gupta (1997) applied wavelet analysis to obtain the spectral moments and peak structural
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
56
responses of multi-degree-of-freedom (MDOF) systems subjected to non-stationary
seismic excitations. Ghanem and Romeo (2000) also represented the equation of motion
in terms of a wavelet basis and solved the inverse problem for time-varying system
parameter estimation. Similarly, Kijewski and Kareem (2003) used wavelet analysis for
system identification, and Basu (2005) and Joseph and Minh-Nghi (2005) used wavelet
analysis to identify stiffness degradation and damping, respectively, from the equation of
motion. A number of papers discuss using modified Littlewood-Paley wavelet packets to
identify modal parameters of MDOF systems, such as natural frequencies, mode shapes,
and associated modal damping ratios, using ambient vibration responses (Basu and Gupta,
2000; Chakraborty et al., 2006; Basu et al., 2008). From a signal processing perspective,
Staszewski (1998) used wavelet analysis combined with various methods such as
thresholding and quantization for data compression and feature selection for fault
detection. Hou et al. (2000) and Hera and Hou (2004) detected sudden changes in
acceleration time-histories using a discrete wavelet transform; however, their application
is limited to the ambient vibration data obtained from the ASCE benchmark structure.
Goggins et al. (2006) investigated the degree of correlation between wavelet coefficients
from ground excitation and building floor responses and reported that the correlation
decreases as the structural behavior changes from linear to non-linear. Although this
observation is valuable, Goggins et al. did not offer a classification scheme to relate the
correlation value to a damage state. Later, Curadelli et al. (2007) extracted the
instantaneous frequency and the damping coefficient from free vibration responses and
successfully showed that damage can be detected with these parameters. Spanos et al.
(2007) also extracted instantaneous frequency from inelastic seismic structural responses
and showed that wavelet analysis can capture the evolution of frequency contents through
the instantaneous frequencies and can potentially detect global damage.
In this chapter, we introduce a new damage diagnosis model that uses the Morlet wavelet
to characterize the response motion of a structure subjected to earthquake ground motion.
The method is based on monitoring the damage sensitive features (DSFs) that reflect
changes in the structure due to progressive damage. The main objective of the study this
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
57
chapter describes is to define several DSFs and test their performance to determine their
ability to characterize damage. These DSFs are robust to the variations in the input
ground motions, and this framework can be applied to different types of structures. For
this purpose, we have introduced three DSFs as functions of wavelet energies at a
particular scale (Escale(a)) and at a particular time (Eshift(b)). The wavelet energy at a
particular scale was first introduced by Nair and Kiremidjian (2007) for damage
diagnosis using ambient vibration data, and wavelet energy at a particular time was
developed by Noh and Kiremidjian (2009).
The three DSFs have been tested to determine their ability and sensitivity to diagnose
damage using acceleration response data from two shake table experiments were utilized.
In the first experiment, a 30 % scale model of a reinforced concrete bridge column was
subjected to different levels of ground motion intensity at the Network for Earthquake
Engineering Simulation (NEES) facility at the University of Nevada, Reno (Choi, 2007).
The second experiment was conducted with a 1:8 scale model of a four-story steel
moment-resisting frame at the NEES facility in the State University of New York at
Buffalo (Lignos et al. 2008; Lignos and Krawinkler, 2009). Then, an analytical model of
the second experiment structure was used to simulate acceleration responses of the
structure subjected to different input ground motions in order to test the robustness and
the sensitivity of the DSFs with varying input ground motions. The results of the
experimental verification show that the values of these DSFs migrate as the extent of
damage increases. Thus, these DSFs can be used as an indicator of damage in the
structure. The DSFs are also robust to different input ground motions. Further
development of classification schemes to map these DSFs to damage states is necessary
to complete the damage diagnosis algorithm, and this work is presented in Chapter 4.
This chapter first introduces the wavelet-based DSFs (section 3.2) and shows the
relationship between these DSFs and physical parameters of the structure. Then, the
paper presents the validation of their performance using two sets of experimental data
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
58
and a set of simulated data (section 3.3). Finally, conclusions are given in the last section
(3.4).
3.2 Development of Wavelet-Based Damage
Sensitive Features
For this analysis, acceleration responses of a structure and the ground acceleration were
collected during an earthquake at each sensor location, for example, each floor. We first
standardized each acceleration time-history by subtracting its mean. Then the wavelet
transform of the acceleration responses and the corresponding wavelet energies are
computed. These wavelet energies indicate how the vibration energy of the acceleration
response is distributed in time and frequency. On the basis of these wavelet energies, we
can define three DSFs to indicate structural damage. The first DSF (DSF1) indicates
damage by quantifying how much energy is lost in the acceleration response at the
proximity of the natural frequency of the undamaged structure. The second and the third
DSFs indicate damage by quantifying how slowly the energy of the acceleration response
decays. The two sections that follow describe the procedure to compute these DSFs.
More importantly, they investigate the relationship between the DSFs and physical
parameters of the structural system in order to justify behavior of the DSFs.
3.2.1 Wavelet Transformation and Wavelet Energies
The continuous wavelet transform (CWT) of a function f(t)L2(), where L2() is the
set of square integrable functions, represents the function or the time-history f(t) as a sum
of dilated (by the scale parameter a) and time-shifted (by the shift parameter b) wavelets.
Since wavelets are localized waves that span a finite time duration, CWT can represent
time-varying characteristics of f(t). It is mathematically defined (Mallat, 1999) as
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
59
dt
a
bt
atfbaWf
ψ*1
)(),( (3.1)
where (t)L2() is called the mother wavelet and * represents a complex conjugate.
The mother wavelet (t) is dilated by various scale parameters a and translated by shift
parameters b to create basis functions called daughter wavelets. Here, the scale is
inversely related to the frequency of the wavelet. These basis functions are convoluted
with f(t), to compute the wavelet coefficients Wf(a,b). The time-history measurement f(t)
is sampled at discrete points in the time domain with a constant interval of sΔt , and the
shift parameter b is taken at those discrete points. For simplicity, those discrete points, 1
sΔt , 2 sΔt , …, K sΔt , are referred to as b = 1, 2, 3, …, K, where K is the number of
data points in the measurement. For this analysis, the Morlet wavelet is used as the
mother wavelet since its shape resembles earthquake pulses (Nair and Kiremidjian, 2007).
The Morlet wavelet was originally introduced in order to analyze seismic recordings
(Morlet et al., 1982; Goupillaud et al., 1984). Since then, it has been used in various
applications including mechanical fault diagnosis (Lin and Qu, 2000; Lin and Zuo, 2003;
Vass and Cristalli, 2005) and system identification (Lardies et al. 2004; Kijewski and
Kareem, 2003) because of its pulse-like shape and the mathematical properties that make
it suitable for localized harmonic analysis. The Morlet wavelet is a special case of the
Gabor wavelet, which has the best time-frequency resolution, in other words, the smallest
Heisenberg box (Mallat, 1999; Hong and Kim, 2004). Thus, the Morlet wavelet also has
the smallest Heisenberg box and consequently is best suited for this study. The analytical
expression for the Morlet wavelet is
2
2
0)(x
xj eexψ
(3.2)
where ω0 reflects the trade-off between time and frequency resolutions. The coefficient
ω0 ≥ 5 is chosen to satisfy the admissibility condition, and only the real part of the Morlet
wavelet is used for computational simplicity. Figure 3.1 illustrates the Morlet wavelet.
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
60
Figure 3.1: Morlet wavelet basis function
In Equation (3.2), ω0 determines the center frequency of the Morlet wavelet. The Fourier
transform of the mother wavelet, (t), has the maximum amplitude at the center
frequency, ω0. For this study, we use the daughter wavelet whose pseudo frequency
matches the natural frequency of the structure. The pseudo frequency of the daughter
wavelet is the frequency where its Fourier transform has the maximum amplitude and is
defined as the center frequency ω0 divided by its scale a. Since we choose the scale in
such a way that the pseudo frequency of the daughter wavelet matches the natural
frequency of the structure, we do not have to worry about the center frequency of the
original mother wavelet. However, ω0 also determines time and frequency resolutions of
the daughter wavelet with a particular pseudo frequency, which can affect the results of
the analysis. A larger value of ω0 results in a smaller frequency resolution while a smaller
value of ω0 results in a smaller time resolution for a daughter wavelet with a particular
pseudo frequency. Applying the notion of root-mean-square (RMS) duration, the time
(Δt) and frequency (Δf) resolutions of the wavelet function at scale a are as follows (Chui,
1992):
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
61
2
aΔt (3.3)
22
1
aΔf
(3.4)
Equation (3.5) represents the relationship between the center frequency of the Morlet
wavelet (2
00 f ) and the scale â, which is the scale of the daughter wavelet whose
pseudo frequency corresponds to the natural frequency of the original undamaged
structure (fn). Using this equation, the time and frequency resolutions of the daughter
wavelet with scale â can be represented as functions of f0 as follows:
nf
fa 0ˆ (3.5)
20
nf
fΔt (3.6)
22 0f
fΔf n
(3.7)
The effective window sizes or bandwidths of a wavelet in the time and frequency domain
are 2Δt and 2Δf. More details can be found in Kijewski and Kareem (2003).
In order to find the optimal wavelet basis, the Shannon entropy of the scalogram is often
used (Coifman and Wickerhauser, 1992; Zhuang and Baras, 1994; Rosso et al., 2001;
Lardies et al., 2004; Hong and Kim, 2004). In information theory the Shannon entropy
represents the amount of the information or the length of the code necessary to convey
the information. It is also often used as a measure of energy concentration. A low value
of entropy corresponds to a high concentration of energy; thus the basis with lower
entropy implies that the shape of this basis matches the shape of the measurement better
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
62
than other bases with higher entropy. We can define the three wavelet entropies of time
(WEt), scale (WEs), and time and scale (WEts) as follows:
, ' 2 , '
' 1
log ( )K
t t b t bb
WE p p
(3.8)
, ' 2 , 'log ( )
a
s s a s aa S
WE p p
(3.9)
, ', ' 2 , ', '
' 1 '
log ( )a
K
ts ts b a ts b ab a S
WE p p
(3.10)
where Sa is a set of scales used for the entropy analysis,
b a s
a s
btΔtbaWf
ΔtbaWfp
2
2
',),(
)',(,
a b s
b s
asΔtbaWf
ΔtbaWfp
2
2
',),(
),'(,
b a s
sabts
ΔtbaWf
ΔtbaWfp
2
2
',',),(
)','(, is the
multiplication of two scalar values, and || is the absolute value of the quantity. For each
set of data, an optimal value of ω0 is determined such that the wavelet entropies for the
acceleration responses of the undamaged structure are minimized.
In order to develop the DSFs we first examined the pattern of wavelet coefficients from
different earthquake responses. Figures 3.2 and 3.3 show the variations of the wavelet
coefficients computed from the structural responses for various damage patterns (DPs)
obtained from the two experiments mentioned in the introduction, the details of which are
explained in the next section. DPs are numbered in increasing order of the damage extent.
These figures show that as the intensity of the input motion increases, the peaks or ridges
of the wavelet coefficients shift both in time and in scale. These changes in the pattern of
wavelet coefficients correlate well with the damage status of the structure.
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
63
5 10 15 20 25
12
5 10 15 20 25 12
5 10 15 20 25 12
Sca
le
5 10 15 20 25 12
5 10 15 20 25 12
5 10 15 20 25 12
Time (s)5 10 15 20 25
12
50
100
150
200
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Figure 3.2: Wavelet coefficients of the acceleration response at the top of the bridge
column for different DPs: (a) DP 1; (b) DP 3; (c) DP 5; (d) DP 7; (e) DP 9; (f) DP 11; (g) DP 13
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
64
5 10 15 20 25
1
2
5 10 15 20 25
1
2
5 10 15 20 25
1
2
Time (s)
Sca
le
5 10 15 20 25
1
2 50
100
150
200
(a)
(b)
(c)
(d)
Figure 3.3: Wavelet coefficients of acceleration responses at the roof of the four-story steel moment-resisting frame for different DPs: (a) DP 1; (b) DP 2; (c) DP 3; (d) DP 4
In order to quantify the shift of peaks in scale, we can use the wavelet energy at scale a
(Escale(a)) defined by Nair and Kiremidjian (2007):
2
1)( ),(
K
bsascale ΔtbaWfE (3.11)
The square of the norm of the wavelet coefficients is called the scalogram, referred to as
wavelet energy at scale a and time-shift b. Thus, Escale(a) is the sum of all the wavelet
energies over time at scale a. As mentioned before, the scale chosen is denoted as â,
which corresponds to the natural frequency of the undamaged structure. It is assumed that
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
65
â is known prior to the analysis from white noise tests or structural design specifications.
Figures 3.2 and 3.3 show that Escale(a) is maximum at the natural frequency of the
structure for low levels of damage. As the damage extent increases, the peaks of the
wavelet coefficients shift up in scale. Hence the changes in the structure appear to be
manifested as a decrease in Escale(a) computed at the scale â. This decrease can be
explained by the fact that as the damage progresses, the structural vibration loses high
frequency components due to loss of stiffness. Nair and Kiremidjian (2007) proved that
the Escale(a) of the acceleration responses at higher scales depends on structural parameters
such as mode shapes, stiffness and damping coefficients, and seismic masses. Thus,
Escale(a) is well correlated with the damage extent of the structure, assuming that the
damaged structure is an equivalent linear system with reduced stiffness.
The shift of the peaks of wavelet coefficients in time shown in Figure 3.2 and 3.3 is then
quantified by the time history of the wavelet energy at time-shift b (Eshift(b)) (Noh and
Kiremidjian, 2009). The Eshift(b) is defined as the sum of the scalogram over the scale at
time-shift b and is given by
2
)( ),(
Sa
sbshift ΔtbaWfE (3.12)
For this algorithm, S is defined as
S = â, 2â (3.13)
where â is the same scale chosen to compute the Escale(a). The reason for choosing S as
defined in Equation (3.13) is that most of the wavelet energies are concentrated at these
scales and the energies at other scales are possible sources of noise. The Eshift(b) represents
the distribution of vibration energy over time. Figures 3.2 and 3.3 show that as the
intensity of damage increases, the wavelet coefficients decay more slowly with time.
Hence, the time-history of the Eshift(b) also decays more slowly for more severely damaged
cases. Based on the Eshift(b), DSF2 and DSF3 are formulated in the next section.
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
66
3.2.2 Definition of Damage Sensitive Features
This section defines three damage sensitive features using the Escale(a) and the Eshift(b) as
indicators of structural damage and investigates the sensitivity of these features to
structural damage.
3.2.2.1 DSF1
DSF1, which is a function of Escale(a), is defined as
tot
ascale
E
EDSF )ˆ(
1 (3.14)
where â is the scale that corresponds to the natural frequency of the undamaged structure,
and Etot is the total wavelet energy of the acceleration response. We chose the
normalization method such that when two non-damaging ground motions with different
amplitudes are applied to a structure, the values of DSF1 from the structural responses are
identical. In this way, the DSF1 can be robustly applied to different amplitudes of ground
motion responses, and the values are comparable. Due to the normalization, the value of
the DSF1 varies between 0 and 1. We tested two different methods to compute Etot - the
sum of the values of Escale(a) at all dyadic scales, and the sum of the values of Escale(a) at
the natural frequency (equivalent to â) and at the half of the natural frequency (equivalent
to 2â). The sum of energies at dyadic scales is equivalent to the signal energy (Mallat,
1999), but we observed that using the sum of energies at only two scales near the natural
frequency gives a more accurate result. A possible explanation is that adding all the
energies at all dyadic scales can also accumulate the noise in the measurement while
picking only a few scales is equivalent to applying an efficient noise filter.
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
67
0 5 10 15 20 25 300
0.10.2
0 5 10 15 20 25 30012
0 5 10 15 20 25 30012
0 5 10 15 20 25 30012
Esh
ift(b
)
0 5 10 15 20 25 30012
0 5 10 15 20 25 30012
0 5 10 15 20 25 30012
Time (s)
(a)
(b)
(c)
(f)
(g)
(e)
(d)
t05
t95
Figure 3.4: Eshift(b) for the bridge column experiment for different DPs: (a) DP 1; (b) DP
3; (c) DP 5; (d) DP 7; (e) DP 9; (f) DP 11; (g) DP 13
3.2.2.2 DSF2
In order to quantify the shift of the wavelet coefficient peaks in time, we defined the
effective time of vibration (ETV) for each acceleration time-history. We then computed
the percentage of the sum of the Eshift(b) outside this ETV as DSF2. The ETV is defined as
the time between t05 and t95 where t05 and t95 are the times when the cumulative sums of
the Eshift(b) for the input ground motion are 5% and 95% of the total sum of the Eshift(b),
respectively. In other words, the ETV is the time when strong ground motion occurs. This
idea of the ETV is equivalent to the definition of the 90% cumulative duration of strong
ground motion, which is the interval between the times at which 5% and 95% of the total
energy has been reached (Trifunac and Brady, 1975). The energy refers to the integral of
the squared acceleration recordings, which is similar to the Arias integral (Arias, 1970).
Kramer summarized different approaches to define the duration of strong motion using an
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
68
acceleration recording (1996). We calculated the ETV from each input ground motion,
and then computed the DSF2 for the corresponding structural responses.
Figure 3.4 shows an example of the time histories of the Eshift(b) for an increasing extent
of damage as well as the locations of t05 and t95 (indicated by dash lines) from the bridge
column experiment. The locations of t05 and t95 are same for all the time histories in
Figure 3.4 because the input ground motions are scaled versions of one earthquake record.
Figure 3.4 shows that the time histories of Eshift(b) decay down more slowly as the
intensity of the input motion increases. This observation is consistent with the time shift
of the peaks of the wavelet coefficients in Figure 3.2. Furthermore, Figure 3.5 shows the
cumulative sum of the Eshift(b) for input and output responses for different DPs. Since the
input motions are scaled versions of the same record, the shapes of the cumulative sum of
the input motions are almost identical to each other. The cumulative sum of the output
responses, however, decreases in magnitude relative to that of the input and also in slope
as the damage progresses. In order to define this relationship between the Eshift(b) and the
damage, the DSF2 quantifies how much the Eshift(b) of the acceleration response is spread
out in comparison to the Eshift(b) of the input ground motion. It also shows what portion of
the wavelet energy occurred outside the strong motion. If the time-history of the Eshift(b)
for an acceleration response decays more slowly than that of the input ground motion, the
DSF2 value will be higher than 0.1. This percentage will increase as the damage
progresses in the structure because the time history of the Eshift(b) will decay more slowly.
For actual earthquake ground motions the duration of the strong motion should also
increase with larger amplitude. As a result, the response is likely to be even more widely
spread out than that shown in Figure 3.4. Thus, we can expect that the DSF2 will have
similar increasing trends as the duration of the ground motion increases. The downside of
utilizing the DSF2 is that we need the information of the input ground motion to compute
the ETV. Thus, the communication between sensor units or between sensor units and a
server computer should be available in both directions for us to compute the DSF2. It
should be noted that the necessary information from the input ground motion is only two
values, t05 and t95, and not the entire time-history measurement.
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
69
0 10 20 300
5
10
15
Time (s)
Cum
. sum
of E
shift
(b)
0 10 20 300
50
100
150
200
Time (s)C
um. s
um o
f Esh
ift(b
)
0 10 20 300
500
1000
Time (s)
Cum
. sum
of E
shift
(b)
0 10 20 300
1000
2000
3000
Time (s)
Cum
. sum
of E
shift
(b)
0 10 20 300
2000
4000
6000
Time (s)
Cum
. sum
of E
shift
(b)
0 10 20 300
5000
10000
Time (s)
Cum
. sum
of E
shift
(b)
0 10 20 300
5000
10000
15000
Time (s)
Cum
. sum
of E
shift
(b)
InputOutput
(a) (b) (c) (d)
(e) (f) (g) Figure 3.5: Cumulative sum of Eshift(b) for the bridge column experiment for different
DPs: (a) DP 1; (b) DP 3; (c) DP 5; (d) DP 7; (e) DP 9; (f) DP 11; (g) DP 13
3.2.2.3 DSF3
In order to define DSF3, we first define the center of Eshift(b) (CE) as the first moment (or
the centroid) of the time history of Eshift(b) after t95, and it is given as
95
/)(
/)(
95
95 tE
ΔtbE
CEK
ttbbshift
K
ttbsbshift
s
s
(3.15)
where K is the number of data points in the measurement. The CE is a measure of how
slowly the wavelet energy decays in time after the strong ground motion. As the damage
progresses, the time history of the Eshift(b) decays more slowly, as shown in Figure 3.4,
which results in a larger value of the CE. In order to standardize the value of the CE, we
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
70
normalized the CE of each structural response by the CE of its input ground motion. The
DSF3 is defined as this normalized CE. Similar to the computation of DSF2, that of DSF3
also requires the input ground motion recording for the normalization.
We proved the relationship between the CE and the extent of damage for the single
degree of freedom (SDOF) system as shown below. Nair and Kiremidjian (2007) showed
the relationship between the scalogram and the structural parameters to be as follows:
)(),(2 bXAbaWf (3.16)
where ,
)1(2
)21(exp2
20
2222
nn a
aA
,)()(2 *qGpG
,)(
)(1
2
1
2
d
qGdpG
and )2exp( nX . The parameter n is the natural frequency, ξ is the damping ratio,
0 is the coefficient of the Morlet wavelet, G(s) is the Fourier transform of the input
ground motion, 21, nnjqp , )2exp( 01 nad , and 21 nd is the
damped natural frequency. Because the amplitude of the input ground motion is small
after t95, the response motion is less affected by the non-stationary large amplitude input
motion. Hence, the structural response will stay within the linear region, which is the
main assumption for obtaining the relationship shown in Equation (3.16) (see Nair and
Kiremidjian, 2007).
Using Equations (3.15) and (3.16), the CE for an acceleration response of a SDOF system
can be derived as
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
71
)'()1(
112
'
2
1
1)1)(1(')1(
2
1
2
1
2
1
2
1'
2
1
1'
'2
)1(
2
1
1
1
95
/)(
/)(
95
95
YLLY
LYLY
L
YYYYL
LY
LY
LY
LY
L
YL
cYLL
tE
ΔtbE
CE
L
LL
L
c
c
L
c
c
K
Δttbbshift
K
Δttbsbshift
s
s
(3.17)
where ,1/95 sttKL ,sΔtXY and ))1/(2exp(' 95 sn Δtt . We made
the approximation 0LY because Y < 1 and L >>1. As the extent of damage increases,
the natural frequency decreases, and as a result, Y increases. In Equation (3.17), an
increase in Y results in an increase of the numerator and a decrease of the denominator,
which in turn causes the CE to increase. The sensitivity of the CE with respect to Y is
calculated as
22
22
)'()1(
)1()1(2)3'2(
2
'
YLLY
LLYLLYLL
Y
CE
(3.18)
It is notable that this sensitivity has the order of 2
1
Y. Since Y < 1, the sensitivity of the
CE with respect to Y is sufficiently large.
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
72
3.3 Application of the Wavelet-Based Damage
Sensitive Features to Experimental data
In order to validate the performance of the DSFs, we applied them to two sets of
experimental data presented in the following sections. The first experiment was
conducted at the University of Nevada, Reno, and involved a reinforced concrete bridge
column subjected to the scaled 1994 Northridge earthquake ground motions of increasing
intensity. For the second experiment, which was conducted at the State University of
New York at Buffalo, a four-story steel moment-resisting frame was subjected to the
scaled 1994 Northridge earthquake ground motions of increasing intensity. Acceleration
responses of the structures are collected and the corresponding damage states of the
structures are observed during the experiments. Then, the DSFs are extracted from those
data in order to validate their sensitivity to structural damage. The robustness of these
DSFs to different input ground motions is also shown using a set of simulated data
obtained from an analytical model of a four-story steel moment-resisting frame subjected
to forty ground motions scaled to various intensities. The description of the two
experiments and the results are discussed in section 3.3.1 and 3.3.2, respectively, and the
results from the numerical simulation are presented in section 3.3.3.
3.3.1 Reinforced Concrete Bridge Column Experiment
3.3.1.1 Description of Experiment
The reinforced concrete bridge column experiment, which was designed according to the
2004 Caltrans Seismic Design Criteria version 1.3, was performed at the NEES facility at
the University of Nevada, Reno (Choi et al. 2007). The height of the column was 3009.9
mm (108.5 in), and the diameter of the specimen was 35.56 mm (14 in). 22 #4 grade 60
rebars are used for longitudinal reinforcement, and galvanized steel wire with the
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
73
diameter of 6.36 mm (0.25 in) is used for transverse reinforcement with 25.44 mm (1.0
in) of pitch. Figure 3.6 shows the bridge column and the experiment setup. We used the
acceleration measurements at the top and the bottom of the column, which were obtained
during the experiment.
Figure 3.6: Shaking table test setup for the bridge column experiment (Modified from
Choi et al. 2007)
For the experiment, the specimen was centered on the shaking table, and the footing was
assumed to be fixed at the base. Two 121.92 121.92 243.84 cm (4 4 8 ft) concrete
blocks each weighing 88.964 kN (20 kips) were used as the inertia mass. These blocks
were connected to the top of the specimen to simulate inertia forces. A steel spreader
beam was bolted to the top of the column head to provide an axial load of 275.790 kN
(62 kips) to the column. Figure 3.6 shows the experimental setup. The column was
subjected to a series of scaled ground motions with increasing intensity. The ground
motion was the fault normal component of the 1994 Northridge earthquake recorded at
the Rinaldi station. The amplitude of the input ground acceleration was scaled by
increasing factors of 0.05, 0.10 … 1.65 as shown in Table 3.1. These tests are referred to
as damage pattern (DP) 1, 2 … 13 hereafter. Table 3.1 also shows the RMS values of the
each input ground motions and the description of damage for each DP. Most of the
damage was concentrated at the bottom of the column. The input motion was highly
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
74
asymmetric in the two loading directions due to the asymmetric velocity pulse, which is
common for near-fault ground motions. Thus, the responses of the column were also
asymmetric. Because the pulse contains most of the energy of the earthquake motion, it
causes large residual displacement to the column in one direction. According to Choi et
al. (2007), the column behaved elastically for DP 1 through DP 4. Most of flexural cracks
formed after DP 4. The first rebar yielding occurred at DP 5, and concrete spalling
occurred at the column base during DP 6. At DP 9, the direction of the residual
displacement changed because the column period elongated due to damage and became
close to the period of the return pulse in the input record. At DP 11, spiral exposure
occurred, and the residual displacement became visible (52 mm). Longitudinal bar
exposure was observed at DP 12 with residual displacement of 145 mm (5.69 in). At the
final DP 13, extensive spalling occurred with more exposure of rebar, the residual
displacement was 339 mm (13.36 in), and the drift ratio was 15 %.
Table 3.1: Scaling factor, Input RMS value, and Description of Damage for each DP of the bridge column experiment
DP Scaling Factor
Input RMS (g)
Description of Damage
1 0.05 0.0053 no damage 2 0.10 0.0105 small cracks on south side 3 0.20 0.0210 more cracks on both sides 4 0.30 0.0313 more cracks 5 0.45 0.0448 NA 6 0.60 0.0576 spalling on south side 7 0.75 0.0696 more spalling, cracks open wider 8 0.90 0.0827 more spalling, cracks open wider 9 1.05 0.0963 spalling on north side 10 1.20 0.1094 5 spirals exposure 11 1.35 0.1224 more spirals exposure 12 1.50 0.1352 longitudinal bar exposure 13 1.65 0.1478 bar exposure on both sides
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
75
3.3.1.2 Results and Discussion
In order to find the optimal ω0 value, we investigated wavelet entropies for ω0 between 5
and 30 rad/s. The ω0 value is limited to 30 rad/s because of the time resolution (the time
span of the daughter wavelet whose pseudo frequency matches the natural frequency of
the structure becomes longer than the strong motion duration). For this set of data, WEt is
minimized at ω0 = 6 (2Δt = 0.88 seconds, 2Δf = 0.36 Hz), and WEs is minimized at ω0 =
17 rad/s (2Δt = 2.53 seconds, 2Δf = 0.13 Hz). Considering both time and frequency, WEts
is minimum at ω0 = 9 rad/s (2Δt = 1.35 seconds, 2Δf = 0.23 Hz). Figure 3.7 shows these
results. Note that the variations of WEt and WEts are relatively small for ω0 between 5 and
17 rad/s. Although ω0 = 5 rad/s is not the optimal solution for all three entropies, the
difference between the minimum value of each entropy and the entropy at ω0 = 5 rad/s is
small (less than 0.5).
10 20 308.5
9
9.5
10
10.5
0 (rad/s)
WE
t
10 20 304
4.5
5
5.5
6
0 (rad/s)
WE
s
10 20 3015
15.5
16
16.5
17
0 (rad/s)
WE
ts
(a) (b) (c)
Figure 3.7: Variation of wavelet entropies for the bridge column experiment for different values of ω0: (a) WEt; (b) WEs; (c) WEts
Since there are three different values of optimal ω0, the DSFs are computed using all
those ω0 values including 5 rad/s in order to investigate the sensitivity of the DSF values
as ω0 changes. We found that the instantaneous frequency is sensitive to the choice of
ω0. The instantaneous frequency at time t is defined as the pseudo frequency of the
daughter wavelet that results in the maximum value of the scalogram among all the scales
at that particular instant of time t. Thus, the instantaneous frequency is the dominant
frequency of the time-history measurement at time t. We averaged the instantaneous
frequencies between t05 and t95 and computed the average instantaneous frequency of DP
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
76
1 for ω0 between 5 and 30 rad/s. This result is shown in Figure 3.8 (a). The average
instantaneous frequency approaches 1.5 Hz as ω0 increases, and the value is stable for ω0
> 17 rad/s. This result shows that ω0 for the optimum value of the WEs results in reliable
estimation of the instantaneous frequency with the smallest time resolution. We
computed the instantaneous frequencies of all DPs ω0 = 17 rad/s, and this result is shown
in Figure 3.8 (b). The average instantaneous frequency of DP 1 is 1.5 Hz. An inspection
of the Fourier spectra of this acceleration response shows that the dominant frequency
with maximum spectra is close to 1.2 Hz.
5 10 15 20 25 30
0.8
1
1.2
1.4
1.6
0 (rad/s)
Inst
anta
neou
s fr
eque
ncy
(Hz)
(a)
1 2 3 4 5 6 7 8 9 10 11 12 13
0.8
1
1.2
1.4
1.6
DP
Inst
anta
neou
s fr
eque
ncy
(Hz)
(b) Figure 3.8: Instantaneous frequency for the bridge column experiment: (a) for different
values of ω0 at DP 1; (b) for different DPs at ω0 = 17 rad/s
Figure 3.9 (a), (b), and (c) show the results for the DSFs using different values of ω0. As
we can expect from the small variation of the WEs, the values of the DSF1 are similar for
different values of ω0. For the DSF2 and the DSF3, the results are sensitive to the value of
ω0. As ω0 increases, the DSF values lose the increasing trend with the increasing damage.
Thus, using ω0 = 5 rad/s is an appropriate choice for the analysis of this set of data.
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
77
1 2 3 4 5 6 7 8 9 10 11 12 13 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
DP
DS
F1
w0 = 5 rad/s
w0 = 6 rad/s
w0 = 9 rad/s
w0 = 17 rad/s
1 2 3 4 5 6 7 8 9 10 11 12 13 0
10
20
30
40
50
60
70
80
90
100
DPD
SF
2
w0 = 5 rad/s
w0 = 6 rad/s
w0 = 9 rad/s
w0 = 17 rad/s
(a) (b)
1 2 3 4 5 6 7 8 9 10 11 12 13 0
1
2
3
4
5
6
DP
DS
F3
w0 = 5 rad/s
w0 = 6 rad/s
w0 = 9 rad/s
w0 = 17 rad/s
(c)
Figure 3.9: Variation of DSF values for the bridge column experiment for different ω0: (a) DSF1; (b) DSF2; (c) DSF3
On the basis of the Morlet wavelet analysis with ω0 = 5 rad/s, scale 0.54 corresponds to
the natural frequency of the undamaged column. Thus, we chose â as 0.54 for all DPs.
We also investigated the Escale(a) values at all scales for the acceleration response at DP 1
and found that the Escale(a) is the largest at scale 0.54. As the damage extent increases,
however, the scale where the largest Escale(a) value occurs shifts from scale 0.54 to scale
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
78
1.08 (lower frequency). This indicates that the dominant scale of the acceleration
response increases as the damage progresses, and the change of dominant scale is
reflected as the change in the DSF1 value. The dominant scale (or frequency) can also be
obtained from ambient vibration data if available.
Figure 3.9 (a) shows the results of the DSF1. The DSF1 is normalized by the sum of two
Escale(a) values at scales 0.54 and 1.08. Thus, it represents the proportion of wavelet
energy at scale 0.54 and how the energy shifts to the higher scale as the damage
progresses. Figure 3.9 (a) shows that when the column is not damaged the DSF1 value is
over 0.9, and as the damage progresses, the DSF1 value decreases, which implies that the
proportion of the Escale(a) at scale 0.54 (energy at higher frequency) decreases while it
increases at scale 1.08 (energy at lower frequency). According to Choi et al. (2007), most
of the flexural cracks were formed and opened up wider after DP 4. Spalling of concrete
started at DP 5, and the strain exceeded its yield strain at DP 4. This is well reflected in
the results of the DSF1 that the value of the DSF1 is close to 1 up to DP 4 and starts
decreasing significantly afterwards.
Figure 3.9 (b) shows the results of the DSF2. The value of the DSF2 is close to 10-20%
for lower DPs, and it increases as the damage increases. These results are similar to the
results of the DSF1; the DSF2 value is small up to DP 4, and the value starts increasing
for damage larger than DP 5. The result implies that as the damage progresses the
response motion decays more slowly, which can be correlated to period elongation due to
stiffness degradation.
Figure 3.9(c) shows the value of the DSF3, which increases as the damage progresses
indicating that the Eshift(b) decays more slowly with increased column damage. The value
of the DSF3 starts increasing after DP 1 and remains constant after DP 4. Based on the
bridge column experimental data, the DSF3 is sensitive to small damage such as cracking,
while the DSF1 and the DSF2 are more sensitive to more severe damage such as spalling.
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
79
3.3.2 Four-Story Steel Moment-Resisting Frame
Experiment
3.3.2.1 Description of Experiment
The second experiment was a series of shake table tests of a 1:8 scale model for a four-
story steel moment-resisting frame with reduced beam moment connections designed
according to current seismic provisions (IBC 2003, AISC-07-05, FEMA 350). The
experiment was conducted at the NEES facility at the State University of New York at
Buffalo (Lignos et al. 2008; Lignos and Krawinkler, 2009). The structure is shown in
Figure 3.10 (a) after completion of the erection process on the shake table. The primary
interest is on the model frame shown on the left.
The structure was subjected to the 1994 Northridge earthquake ground motion recorded
at Canoga Park station. The testing sequence that was executed for the structure included
a service level earthquake (SLE, 40% of the unscaled record), a design level earthquake
(DLE, 100% of the unscaled record), a maximum considered earthquake (MCE, 150% of
the unscaled record), and a collapse level earthquake (CLE, 190% of the unscaled record).
These scaled intensities of the ground motion are referred to as DP 1, 2, 3, and 4.
According to elastic modal identification from white noise tests, the structure had a
predominant period 1T =0.45 seconds. For each DP, accelerations at each floor including
the ground motion and the roof response were measured.
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
80
(a)
(b)
Figure 3.10: Four-story steel moment-resisting frame: (a) after the completion of erection on the shake table (Modified from Lignos et al. 2008); (b) after collapse
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
81
(a)
(b)
(c)
(d)
Figure 3.11: Story drift ratio histories at various levels of input ground motion intensity for the four-story steel moment-resisting frame experiment: (a) first story; (b) second story; (c) third story; (d) fourth story (Modified from Lignos and Krawinkler, 2009)
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
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Figure 3.11 shows the story drift ratio (SDR) of the structure at various intensities from
elastic behavior up to collapse. During the SLE the structure remained elastic. During the
DLE the structure reached a maximum SDR of about 1.6% with the inelastic action
observed at the column base and first floor beams. Localized damage, such as local
buckling of the flange plates, was not noticeable. During the MCE the frame reached a
maximum SDR of about 5% with plastic deformation evident from local buckling of the
plates that represented plastic hinge elements of the frame (see Lignos and Krawinkler,
2009). During the CLE test the frame experienced a maximum SDR of about 13% with a
full first three story collapse mechanism. Figure 3.10 (b) shows the frame after collapse.
10 20 309
9.5
10
10.5
11
0 (rad/s)
WE
t
10 20 308
8.5
9
9.5
10
0 (rad/s)
WE
s
10 20 3017
17.5
18
18.5
19
0 (rad/s)
WE
ts
(a) (b) (c)
Figure 3.12: Variation of entropies for the four-story steel moment-resisting frame experiment for different values of ω0: (a) WEt; (b) WEs; (c) WEts
3.3.2.2 Results and discussion
We computed the wavelet entropies for ω0 between 5 and 30 rad/s, and the results are
shown in Figure 3.12. The WEt is minimized at ω0 = 5 (2Δt = 0.57 seconds, 2Δf = 0.55
Hz), and the WEs is minimized at ω0 = 30 rad/s (2Δt = 3.40 seconds, 2Δf = 0.094 Hz).
Considering both time and frequency, the WEts is minimum at ω0 = 8 rad/s (2Δt = 0.88
seconds, 2Δf = 0.36 Hz). The variations of the WEt and the WEts are relatively small for
ω0 between 5 and 30 rad/s as shown in the Figure 3.12. The DSFs are computed for three
ω0 values, and similar to the results in the first experiment, the DSF1 values are not
sensitive to the value of ω0 while the DSF2 and the DSF3 performs best with ω0 = 5 rad/s.
Thus, ω0 = 5 rad/s is used for this set of data.
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
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Ground 2 3 4 Roof0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
DS
F1
Floor
SLEDLEMCECLE
Figure 3.13: DSF1 for the four-story steel moment-resisting frame experiment
Figure 3.13 illustrates the DSF1 normalized with respect to the sum of two values of
Escale(a) at scale 0.406 and 0.812. The results are presented for each DP at each floor.
Scale 0.406 corresponds to the pseudo frequency of 2.0 Hz, which is the first natural
frequency of the undamaged frame. Because the natural frequency of the test frame is
known based on the white noise tests (Lignos et al. 2008), we can use this scale to
compute the DSFs. We observed that the Escale(a) at scale 0.406 has the largest wavelet
energy at DP 1 among all the scales. At the ground level, the values of the DSF1 do not
vary much as the damage progresses because there is no damage at the “ground” level
(no damage on the shake table). The same figure illustrates that for the upper floors, the
DSF1 decreases as the intensity of the input ground motion increases. With the increasing
level of damage, the wavelet energy reduces at scale 0.406, which corresponds to the first
natural frequency of the undamaged frame, and increases at scale 0.812, which
corresponds to a frequency lower than the natural frequency of the undamaged frame.
This change in energy at scale 0.812 can be measured by 11 DSF . In Figure 3.13, the
DSF1 at the second floor does not change much between DP 3 (MCE) and DP 4 (CLE).
This happens because the first story is damaged during DP 1 through DP 3, reaching a
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
84
plastic rotation of about 5%. After this plastic deformation limit, all the beams and the
base of the first story columns start deteriorating in strength. This is reflected in the DSF1
as the small change of values between DP 3 and DP 4 at the second floor.
0 5 10 15 20 250
0.5
1
(a)
0 5 10 15 20 250
5
(b)
0 5 10 15 20 250
5
(c)
0 5 10 15 20 250
5
(d)
Time (s)
Esh
ift(b
) for
Roo
f
Figure 3.14: Eshift(b) for the four-story steel moment-resisting frame for different DPs: (a)
DP 1; (b) DP 2; (c) DP 3; (d) DP 4
The time history of the Eshift(b) at the roof for each DP is illustrated in Figure 3.14. It is
shown in the figure that the time histories of the Eshift(b) spread out further as the intensity
of the input ground motion increases and damage progresses. Unlike Figure 3.4 where the
time histories of the Eshift(b) spread out both to the left of t05 and to the right of t95, the time
histories obtained from the steel frame spread out only to the right of t95. This happens
primarily because the concrete bridge column develops cracks even at the early stages of
a strong vibration where the amplitude is still small, i.e. we have period elongation
because of cracking. For the steel frame this is not the case since its behavior is elastic
when the amplitude of the vibration is small. Moreover, based on white noise tests that
were conducted after each damage level, no stiffness degradation was observed in the test
frame prior to collapse (Lignos et al. 2009).
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
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Ground 2 3 4 Roof0
5
10
15
20
25
DS
F2
Floor
SLEDLEMCECLE
Figure 3.15: DSF2 for the four-story steel moment-resisting frame experiment
Figure 3.15 shows the DSF2, which quantifies how much the time history of Eshift(b) is
spread out in time. The DSF2 is computed using the scales 0.406 and 0.812. At the
ground floor the values of the DSF2 are 10 % at all DPs since the ETV is calculated based
on the ground motion. At all the other floors the value of the DSF2 increases as the
damage in the frame progresses. The increase of the values at the second floor between
DP 3 (MCE) and DP 4 (CLE) is small for the same reason that we mentioned earlier for
the DSF1. The value of DSF2 at the second floor is the largest among all the floors for
each DP. This might be correlated to the fact that the first story was damaged first and
had the largest damage extent.
Figure 3.16 shows the results of the DSF3. The damage in the frame causes the time
history of the Eshift(b) to decay more slowly, which results in larger values of the DSF3. It
is similar to the results of the DSF2 that the values of the DSF3 at the ground floor are 1
for all DPs, and the DSF3 at all the other floors have increasing values as the damage
progresses. Also, the value of the DSF3 at the second floor is the largest among all the
floors for each DP, but the differences between the values of the DSF3 at different floors
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
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are smaller than those observed for the DSF2. The relationships between the values of the
DSFs at different floors have to be studied carefully since the measured accelerations are
global in nature resulting in complex interactions between structural parameters at each
floor.
Ground 2 3 4 Roof0
0.5
1
1.5
DS
F3
Floor
SLEDLEMCECLE
Figure 3.16: DSF3 for the four-story steel moment-resisting frame experiment
3.3.3 Analytical Model of the Four-Story Steel Special
Moment-Resisting Frame Analysis
An analytical model of the four-story steel moment-resisting frame described in section
3.3.2.1 is utilized to provide acceleration responses of the frame in order to examine the
sensitivity of the proposed DSFs with respect to different input ground motions. Details
about this analytical model, which account for component deterioration, can be found in
Lignos and Krawinkler (2009). The sensitivities of all the DSFs have been examined for
381 input ground motions, which are scaled to various intensities from forty original
ground motion recordings in order to investigate the effect of varying excitation profiles
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
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and amplitudes of input ground motion on the ability of the DSFs to predict damage. Two
aspects of the effectiveness of the DSFs are investigated. The first is the consistency of
values of the DSFs in the absence of damage, and the second is the change in values of
the DSFs with respect to varying degrees of structural damage. The forty ground motions
are selected from large magnitude earthquakes (6.5 < M < 7.0) recorded at sites that are
13 to 40 km from the rupture zone (Medina and Krawinkler, 2003). Each ground motion
is scaled several times in order to capture several damage states up to and including
collapse. Historically forty records seem to be adequate for statistical analysis of
structural responses (Vamvatsikos and Cornell, 2002). These ground motions are used as
inputs to the analytical model of the structure, and the DSFs are computed from the
acceleration responses of the roof DSFs. The damage states were estimated from the
maximum SDR of the structure. For this study, five damage states are defined as follows:
no damage (within the elastic limit) (0% ≤ SDR < 1%), slight damage (1% ≤ SDR < 2%),
moderate damage (2% ≤ SDR < 3%), severe damage (3% ≤ SDR < 6%), and collapse
(6% ≤ SDR). The threshold SDR values are chosen to represent different damage states
based on current practice (FEMA 440, FEMA 356).
Figure 3.17 (a) and (b) show how the DSF1 values change for scaled versions of two
illustrative ground motions, the 1989 Loma Prieta earthquake recorded at the Capitola
station and the 1994 Northridge earthquake recorded at the Northridge-17645 Saticoy
street station. For no damage state cases (DP 1 and DP 2), the value of the DSF1 is
between 0.9 and 1, and as the damage increases the value of the DSF1 decreases to below
0.4. In order to further illustrate the consistency of the DSF1 for the no damage state, the
distribution of the DSF1 values for the no damage state is first computed as shown in
Table 3.2. 79 scaled ground motions of the forty recordings resulted in the no damage
state. Table 3.2 shows that the values of DSF1 stay between 0.87 and 1 for 94% of the
cases. Secondly, the change of the DSF1 values with respect to structural damage is
shown in Figure 3.18. Figure 3.18 shows the scatter plot of the DSF1 values and the
corresponding maximum SDR in semi-log scale and the linear fit for the data for various
degrees of structural damage, represented as maximum SDR. We can observe that the
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
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DSF1 values decrease exponentially as the damage extent of the structure increases. The
correlation coefficient of the DSF1 and the log of the maximum SDR is 0.87. Therefore,
this DSF1 is strongly correlated with the SDR and can be used to estimate the damage
state of the structure accurately. According to this analysis using the analytical model, the
DSF1 is robust to the variations in the input ground motions. Similar results are observed
with DSF2 and DSF3.
1 2 3 4 5 6 7 8 90
0.2
0.4
0.6
0.8
1
DP
DS
F1
(a)
1 2 3 4 5 6 7 8 9 10 110
0.2
0.4
0.6
0.8
1
DP
DS
F1
(b)
Figure 3.17: DSF1 for two different input ground motions: (a) the 1989 Loma Prieta Earthquake; (b) the 1994 Northridge Earthquake
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
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Figure 3.18: Scatter plot of DSF1 and maximum story drift ratio for the analytical model
of the four-story steel moment-resisting frame
Table 3.2: Distribution of DSF1 for no damage state cases Range Count
0.37 – 0.43 1 0.43 – 0.49 2 0.49 – 0.55 0 0.55 – 0.62 0 0.62 – 0.68 0 0.68 – 0.74 0 0.74 – 0.80 2 0.80 – 0.87 0 0.87 – 0.93 26 0.92 – 1.00 48
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
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3.4 Conclusions
Three damage sensitive features (DSF) using the continuous wavelet transform of
earthquake responses are developed and applied to two sets of experimental data and one
set of simulated data. The experimental datasets are obtained from recent shake table
experiments of a 30% scaled model of a reinforced concrete bridge column and a 1:8
scale model of a four-story steel moment-resisting frame. The simulated dataset is
obtained from an analytical model of the four-story steel moment-resisting frame that is
used for the second experiment. The continuous Morlet wavelet transform is applied to
the acceleration response of the structure during the strong ground motion, and the
wavelet energies at a particular scale and at a particular time are defined based on the
wavelet coefficients. Then three DSFs are developed as functions of the wavelet energies
for structural damage diagnosis. DSF1 measures how the wavelet energy at the natural
frequency of the undamaged structure changes as the damage progresses in the structure.
According to the results from the experimental data, the DSF1 value decreases as the
damage extent increases. This is because the wavelet energy reduces at the scale
corresponding to the first natural frequency of the undamaged structure with the
increasing levels of damage. DSF2 measures how much the wavelet energy spread out in
time and DSF3 measures how slowly it decays. The values of the DSF2 and the DSF3 both
increase as the damage extent increases.
For the computation of the three DSFs, different levels of information are required. The
DSF1 can be calculated using only the structural response while the computation of the
DSF2 and the DSF3 requires both the structural response and the information from the
input ground motion. Thus, the communication between the sensor units at different
locations has to be set up and synchronized in order to compute the DSF2 and the DSF3
for damage diagnosis. It should be noted that this communication between the sensor
units do not have a large demand in power because the information that needs to be
transferred is only a few numbers such as t05, t95, and CE extracted from the input ground
CHAPTER 3. Wavelet-Based Damage Sensitive Features for Structural Damage Diagnosis Using Strong Motion Data
91
motions, and not the entire time-history measurements. The power consumption is an
important issue when damage diagnosis algorithms are embedded in wireless sensing
units, which are power limited. Because their major source of power consumption is
wireless data transmission, the damage diagnosis algorithm based on the proposed DSFs
is particularly suitable to be embedded in the wireless sensing units. The computational
efficiency of the algorithm, however, also needs to be tested for embedding the algorithm
into wireless sensing units. The algorithm can be also embedded in wired sensing units.
The three DSFs have different sensitivities to various levels of damage according to the
results of the applications. The value of the DSF3 changes more for lower levels of
damage than for more severe levels of damage. On the other hand, the values of the DSF1
and the DSF2 show more changes for larger damage. Thus, the DSF3 is more sensitive to
smaller levels of damage and the DSF1 and the DSF2 are more sensitive to larger levels of
damage. Therefore, a combination of these DSFs may be required for robust damage
diagnosis.
In addition to the experimental results, an analytical model of the four-story steel frame is
developed, and the DSFs are tested for sensitivity and robustness with structural response
data obtained from forty earthquake ground motions covering a wide range of magnitudes
and distances. The results of these sensitivity analyses showed again that the DSFs are
directly correlated to damage states defined through story drift ratio limits, and the DSF
values are robust to the input ground motions.
In order to apply the method to other types of structures or various other ground motions,
additional testing will be necessary. Moreover, it would be desirable to test the algorithm
with data collected from field experiments where noise and other environmental
conditions may show to be a factor. A key advantage of using the DSFs to diagnose
damage, however, is that these DSFs can be computed directly from the acceleration
recording at each sensor location on the structure and do not rely on the computation of
the story drifts, which is always a challenging process.
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In practical applications, it is unlikely to have reference values of these DSFs
corresponding to each damage state. Thus, a pre-defined system needs to map the values
of these DSFs to different damage states of the structure when an earthquake occurs. For
this purpose, the framework to build fragility functions that define the probabilistic
relationship between these DSFs and the damage state of the structure is introduced in
Chapter 4 to complete the damage diagnosis algorithm.
In summary, we developed three damage sensitive features (DSFs) using wavelet
energies and theoretically derived the relationship between these wavelet energies and
structural parameters that are important for damage characterization. The performance of
these DSFs was tested using two sets of experimental data. Further sensitivities and
robustness of the DSFs were evaluated using the analytical model of the four-story frame
subjected to 381 scaled ground motions. Both the numerical and experimental results
systematically showed excellent correlation between the damaged estimated by the DSFs
and the observed damage. These results demonstrate that the DSFs presented in this
chapter are a good candidate for use in automated damage diagnosis following large
earthquakes. To associate the values of the DSFs with damage states, we developed a
probabilistic classification method, which is presented in Chapter 4.
93
Chapter 4
Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
This chapter discusses the new framework to develop fragility functions to classify and/or
predict structural damage using the wavelet-based damage sensitive feature (DSF)
introduced in Chapter 3. Fragility functions are commonly used in performance-based
earthquake engineering (PBEE) for predicting the damage state of a structure subjected to
an earthquake. The prediction is based on the intensity of the ground motion at a given
hazard level. This process often involves estimating the damage as a function of
structural response, such as the story drift ratio and the peak floor absolute acceleration.
Structural displacements are, however, difficult to estimate in practice. In contrast, the
wavelet-based DSF can be easily estimated from recorded structural response. In the
framework discussed in this chapter, the structure is subjected to multiple earthquake
loadings, and the structural absolute acceleration response is obtained during the
earthquake excitations using an analytical model of a structure. From this structural
acceleration response, the wavelet-based DSF corresponding to each time history is
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
94
extracted. This information is in turn used to develop fragility functions that predict the
probabilities of the structure being in various damage states given the value of the DSF.
These fragility functions can either be used in SHM to classify the damage state of the
structure following an earthquake or used in PBEE as a prediction model for structural
behavior. The performance of the proposed framework was demonstrated and validated
with a set of numerically simulated data for a four-story steel moment-resisting frame
designed according to current seismic provisions. The results show that the damage state
of the frame is predicted with less variance using the fragility functions derived from the
wavelet-based DSF than it is with fragility functions derived from alternate acceleration-
based measures, such as the spectral acceleration at the first mode period of the structure
and the peak floor acceleration. Therefore, the fragility functions based on the wavelet-
based DSF can be used as a probabilistic damage classification model in the field of
SHM and an alternative damage prediction model in the field of PBEE.
4.1 Introduction
In order to ensure the minimum safety of structures, building codes have enforced several
restrictions in the design and maintenance process of civil structures, but it has become
difficult to apply a consistent design code to all cases as structural design becomes
diverse and new construction technologies emerge. As a result, performance-based
earthquake engineering (PBEE) has received increasing attention among structural
engineering researchers and practitioners (Ghobarah, 2001; Krawinkler and Miranda,
2004). The main goal of PBEE is to predict the performance of a structure subjected to
earthquakes in a probabilistic manner and to design accordingly in order to achieve
selected performance objectives (SEAOC, 1995; Ghobarah, 2001; Porter et al., 2007).
PBEE is a methodology in which the design criteria are expressed in terms of achieving
performance objectives when the structure is subjected to various levels of seismic hazard
(Ghobarah, 2001). In the conventional PBEE framework, four parameters are used to
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
95
compute annual loss rate of a structure due to earthquakes. The first parameter is intensity
measure (IM), which quantifies the intensity of an earthquake ground motion, such as the
peak ground acceleration and the spectral acceleration at the first mode period. The
second parameter is engineering demand parameter (EDP), which represents the
structural response to the earthquake, such as the peak story drift ratio or the absolute
floor acceleration. The third is damage measure (DM), which describes the discrete
physical damage state of the structure, such as cracking and spalling. The last is decision
variable (DV), which relates to the actual loss, such as casualties, downtime, and
monetary loss (Singhal and Kiremidjian, 1995; Ibarra et al., 2002; Ibarra et al., 2005;
Vamvatsikos and Cornell, 2002; Medina and Krawinkler, 2003; Zareian and Krawinkler,
2006; Zareian and Krawinkler, 2007; Porter et al., 2007; Haselton and Deierlein, 2007).
The PBEE probabilistically assesses the performance of the structure by defining the
conditional probabilities of the parameters, progressively from the IM to the DV. A
fragility function is used to map an IM or an EDP to a DM in order to predict the
probability of the structure being in a specific damage state, such as slight, moderate, or
severe damage given a certain intensity of an earthquake. However, many uncertainties
are involved with using these measures due to inelastic structural behavior and variations
in the time-histories of ground motions.
This chapter introduces a framework that combines the knowledge from SHM and PBEE
in order to develop fragility functions using a wavelet-based DSF introduced in Chapter
3. These fragility functions can be utilized in two ways. In SHM, they can be used to
classify the damage state of a structure after an earthquake on the basis of the DSF.
Naeim et al. (2006), Ibarra and Krawinkler (2005), Zareian and Krawinkler (2007),
Lignos and Krawinkler (2009), and FEMA P695 utilized fragility functions for damage
assessment using the story drift ratio (SDR) as an EDP. This SDR is a commonly used
EDP for identifying structural damage and the measure most commonly used to relate
ground motion intensity to structural damage (BSS Council, 1997; Ghobarah, 2001; Liu,
2004; Ramirez and Miranda, 2008; FEMA 440). In practice, obtaining a direct and
accurate evaluation of structural drift is difficult and expensive to automate because of
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
96
the cost of displacement sensors and the need for reference points. In contrast, the
wavelet-based DSF is computed from the acceleration response of a structure, which can
be measured directly, and, unlike other measurements, errors due to integration of the
absolute acceleration are minimized. Therefore, we can use the wavelet-based DSF to
estimate the SDR, which in turn can be used for damage classification. From a PBEE
perspective, the proposed fragility functions can predict structural damage using the
wavelet-based DSF as an EDP. The advantage of the proposed framework is that it
utilizes fragility functions based on the absolute acceleration measurements. This is
particularly beneficial for post-event assessment because the DSF computed from
structural response measurements is more strongly related to the structural damage than
other conventional IMs or simple acceleration-based EDPs, such as the spectral
acceleration and the peak floor acceleration (see section 4.3.3). The DSF can replace IMs
to predict drift-based EDPs or replace simple acceleration-based EDPs to directly predict
structural damage. In addition, fragility functions based on the DSF can be used to
compute the annual loss rate of a structure as a part of the PBEE process (Ramirez and
Miranda, 2008). Computation of the annual loss, however, is not within the scope of this
study.
The framework presented here consists of first obtaining the structural response and the
resulting damage state from a nonlinear response history analysis of an analytical model
or from available information of the instrumented structure of interest. We then extract
the DSF from each structural response using wavelet analysis and define the probabilistic
mapping between the DSF and the damage state. Three methods for computing the
probabilistic mapping are introduced. For validation, the framework was applied to the
simulated data obtained from the analytical model of the four-story steel special moment-
resisting frame subjected to a set of scaled earthquake ground motions (Lignos and
Krawinkler, 2009). The performance of the DSF for damage diagnosis is compared with
that of the spectral acceleration and the peak floor acceleration, and the results show that
the DSF can estimate the SDR with less variance than the other measures. It is important
to note that fragility functions obtained from an analytical model can be periodically
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
97
updated as measurements are taken and the structural model is updated. The proposed
framework can be used with wireless or wired structural monitoring systems to improve
the state of knowledge about estimating the damage state of a structure. Moreover, the
framework incorporates recent advances in analytical models that describe the structural
response, damage diagnosis algorithms using statistical signal processing methods, and
statistical parameter estimation techniques.
This chapter is organized as follows. Section 4.2 introduces the framework for building
the fragility functions using the wavelet based DSF. Section 4.3 then explains how this
framework was validated using an analytical model of a four-story steel moment-resisting
frame, part of an office building designed in Los Angeles according to current seismic
provisions. Finally, section 4.4 presents conclusions.
4.2 Framework for Developing Fragility Functions
Based on a Damage Sensitive Feature
A framework for developing fragility functions for structures subjected to earthquake
ground motions has been developed using a damage sensitive feature (DSF). The DSF is
computed from each floor absolute acceleration response and is used as a indicator of
structural damage. The framework consists of three steps: (1) collecting absolute
acceleration response data and corresponding damage state from a structure subjected to
various intensities of seismic loading; (2) extracting a specific feature from these data
using appropriate statistical pattern recognition methods; and (3) developing a damage
classification/prediction model by constructing fratility functions, which maps the
specific feature to a potential damage state of the structure. It is assumed that, as with
PBEE, a reliable analytical model of a structure or information from an instrumented
building, which is sufficient for developing such a model, is available. The three steps of
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
98
the framework are summarized in Figure 4.1, and more details of the procedure are
described in the following sections.
4.2.1 Data Collection: Structural Responses and Damage
States
In the first step of the framework (Figure 4.1), an analytical model of a structural system
is subjected to a set of ground motions using incremental dynamic analysis (IDA)
(Vamvatsikos and Cornell, 2002). The IDA involves a set of ground motions, which is
scaled to various intensities and applied to a structure to evaluate its seismic performance
under various intensities of loading. The set of ground motions can be selected by
numerous methods (see Katsanos et al., 2009), and each ground motion is scaled by a set
of scale factors. Since the statistical techniques are used in the framework as discussed in
the following sections, we need to have an adequate amount of unbiased structural
response data for various damage states. Historically, 40 records seem to be adequate to
compute statistics for different engineering demand parameters (EDPs) of a structural
system (Vamvatsikos and Cornell, 2002). In this framework, the wavelet-based DSF1,
which is introduced in Chapter 3, is used as a measure of seismic performance of a
structure; thus, the absolute acceleration response of each floor of the structure, from
which the wavelet-based DSF is extracted (see section 4.2.2), is collected during each
ground motion excitation. The corresponding maximum story drift ratio (SDR) for each
story is also obtained from the same model in order to determine the damage states of the
structure (see section 4.2.3). The SDR at each story is computed as the maximum drift
(displacement) difference between the floor and the ceiling normalized by the height of
the story. To evaluate the performance of the wavelet-based DSF as an indicator of
structural damage, the values of the DSF are compared to the values of SDR. This
relationship between the DSF and SDR is defined as the fragility function, as discussed in
section 4.2.3.
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
99
Figure 4.1: Summary of the proposed framework
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
100
4.2.2 Feature Extraction: Wavelet-Based Damage
Sensitive Feature from Structural Responses as an
Indicator of Damage State
In the second step depicted in Figure 4.1, the wavelet-based DSF1, described in Chapter
3, is extracted from the individual absolute acceleration histories for each floor of a
structural system. Here the wavelet-based DSF is used because the structural response to
earthquake strong motions has time-varing characteristics. Wavelet analysis is suitable
for studying time-varying characteristics of non-stationary signals such as earthquake
responses because it represents the signal as a sum of dilated and time-shifted wavelets
that are localized in time. Other methods, such as auto-regressive time-series analysis and
Fourier analysis, cannot be used in this case because they assume that the signals are
stationary. Before the wavelet transform is applied, each acceleration response is
standardized by subtracting the mean of the response to offset different initial conditions
of the measurements. In order to directly correlate increases in damage to increases in the
value of the DSF1, the DSF1 is redefined as one minus the original DSF1, and it is
referred to as DSF hereafter for simplicity. Note that the DSF value varies between 0
(when there is no damage) and 1 (when the structure is severely damaged).
4.2.3 Damage Classification/Prediction Model
Development
The final step of the framework (see Figure 4.1) is to develop a damage
classification/prediction model that probabilistically maps the DSF to the damage state by
constructing fragility functions based on the wavelet-based DSF. Fragility functions
provide the conditional probability of being or exceeding each damage state given the
value of DSF; thus can be used to classify or predict the damage state of the structure
from the absolute acceleration measurements. The fragility functions are first empirically
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
101
computed using kernel density estimation from the values of SDR and DSF at each story,
which are collected and computed in the previous steps. A kernel is a symmetric
weighting function used for non-parametric estimation, and the kernel density estimation
makes non-parametric estimations of the probability density function from noisy
observation based on the kernel (Wand, 1995). Then, a cumulative distribution function
(CDF) is fitted to the empirical fragility functions. Two alternative methods to provide
more detailed information about the conditional probability of the SDR given the DSF
are also presented in this section. One method uses two-dimensional kernel to directly
estimate each conditional probability, and the other one uses one-dimensional kernel to
estimate the mean and the variance of the conditional density function. Then we can fit a
probability density function (PDF) to the empirical conditional probability distribution.
These three methods are summarized in Table 4.1.
Table 4.1: Summary of three methods for probabilistic mapping between the DSF and the SDR
Methods Outcome Advantages One-dimensional Gaussian kernel for the DSF and the
beta CDF fitting
Fragility function (Prob(DS≥DSi|DSF=dsf))
Beneficial when the damage states are clearly defined in
terms of the SDR Two-dimensional
Gaussian kernel for the DSF and the SDR and the
lognormal PDF fitting
Conditional probability of the SDR given the DSF
(Prob(SDR=sdr|DSF=dsf)) Beneficial when the damages states are not clearly defined The conditional probability
can be computed for any SDR value.
One-dimensional Gaussian kernel for the DSF and the
lognormal PDF fitting
Conditional mean and standard deviation (μSDR|DSF, σSDR|DSF)
Conditional probability of the SDR given the DSF
(Prob(SDR=sdr|DSF=dsf)) The fragility functions can be computed for each story of the structure using the DSF
computed from individual story responses, or can be computed for the entire structure
using the DSF from the roof absolute acceleration responses and the maximum SDR
among all the stories. The fragility functions for each story can be used for more detailed
diagnosis of damage at a specific story of a structure. The global fragility functions can
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
102
be used after an earthquake to quickly assess the overall damage of a structure. The
overall assessment of a structure would be particularly useful when multiple structures
have to be assessed in a timely manner. In this section, the procedure is described for
computing fragility functions for each story separately, but similar procedure can be
applied to compute the fragility functions for the overall damage.
Damage states (DS) are discrete variables most often defined as ‘no damage,’ ‘slight
damage,’ ‘moderate damage,’ and ‘severe damage.’ In this study, each damage state
(DSi) covers a range of SDR values. Thus, a set of threshold values of SDR is specified
for each damage state DSi as follows:
SDR if
SDR if
SDR if
State Damage
1
211
100
nnn SDRSDRDS
SDRSDRDS
SDRSDRDS
(4.1)
where SDRis are monotonically increasing threshold values for increasing is, and n is the
number of damage states. Similarly, FEMA 356 uses four different damage states,
namely, operational, immediate occupancy, life safety, and collapse prevention. Using
this definition of damage states, the fragility function can be defined as follows:
dsfSDR
dsfDSdsfG
i
ii
DSFSDRProb
DSFDSProb)((4.2)
where Gi(dsf) is the fragility function for being or exceeding damage state i given a value
of the DSF, dsf.
Typically an empirical fragility function for each damage state described above is
computed using data binning. From the numerical simulation and the structural damage
diagnosis algorithm, pairs of DSF and SDR values, dsfi, sdri , are computed for
acceleration responses at each floor. Data binning is then used to segregate DSF values
into each bin and count the number of pairs whose SDR values belong to each set of DSi
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
103
within the bin (Porter et al., 2007). Alternatively, we can apply the kernel density
estimation using the following equation:
nh
xxK
hxX
n
i
i
1
)(1
)(Prob.
(4.3)
where xis are n realizations of the random variable X, K is a kernel, and h is a smoothing
parameter or the bandwidth of the kernel K. By the definition of the conditional
probability, Gi(dsf) in Equation (4.2) can be rewritten as
dsf
dsfSDRdsfG i
i
DSFProb
DSF,SDRProb )(
. (4.4)
Substituting the Equation (4.3) into the Equation (4.4), the empirical fragility function for
being or exceeding damage state i for the DSF value of dsf, Ĝi(dsf), is defined as
N
q
q
N
p
p
ip
N
q
q
N
p
p
ip
N
N
ppip
i
h
dsfdsfK
h
dsfdsfKSDRsdrI
h
dsfdsfK
hN
h
dsfdsfK
hSDRsdrI
N
dsfdsfIN
dsfdsfISDRsdrIN
dsfG
1
1
1
1
1
1
)(
)()(
)(11
)(1
)(1
)(1
)()(1
)(ˆ
(4.5)
Where N is the number of dsfi, sdri pairs, and I(x) is an indicator function that is 1 if x
is true and 0 otherwise. The kernel assigns a different weight for each pair of DSF and
SDR values. We use the kernel K(x) whose weight is higher for the x values near 0. Using
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
104
a rectangular kernel with height 1 is equivalent to the conventional data binning methods.
Equation (4.5) estimates the probability of the structure being or exceeding each damage
state DSi when the value of the DSF is dsf by using all the pairs of DSF and SDR values.
The use of all the pairs leads to a smoother representation of the fragility functions, Ĝis,
than does the data binning method. The advantages of using the kernel density estimation
instead of the data binning method are summarized in Table 4.2.
Table 4.2: Advantages of kernel density estimation in comparison to data binning method Kernel Density Estimation Data Binning Method
All the available dsfi, sdri pairs are used with different weights to estimate Ĝi(dsf).
Thus, this method reduces problems caused by lack of dsfi, sdri data or biased sampling.
Lack of data or biased data can result in some bins with no data - Ĝi(dsf) values cannot be defined for these
bins. Ĝi(dsf) values can be computed at all the values of DSF, thus resulting in a dense representation
of Ĝi(dsf).
Ĝi(dsf) values can be computed only at each bin, resulting in a sparse
representation of Ĝi(dsf). The resulting fragility function is a smooth
curve The resulting fragility function is not
always smooth. A conventional CDF is then fitted to the empirically computed fragility functions. The
advantages of fitting a conventional CDF are as follows: (1) the function is completely
described by a few parameters; (2) the function is continuous, thus defined for all
possible DSF values (no interpolation is necessary); and (3) the function increases
monotonically. The lognormal CDF is used in conventional fragility functions, but other
functions, such as the beta CDF and the truncated normal CDF, can also be used
depending on the data. In general, the CDF that minimizes the fitting error, such as a
root-mean-square error (RMSE), is selected. Several CDFs of interest are fitted to the
data using a nonlinear least-square method, and the CDF that has the smallest RMSE is
chosen. In this study, the beta CDF was proven to have the smaller fitting error compared
to the lognormal CDF and the truncated normal CDF. The error was the smallest for the
beta CDF mostly because the wavelet-based DSF has a value between 0 and 1, and the
beta CDF is defined only between 0 and 1.
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
105
Alternatively, we can estimate the conditional probability of the SDR given the DSF
using a two-dimensional kernel as follows:
N
q
q
N
p
pp
N
q
q
N
p
pp
h
dsfdsfK
h
sdrsdr
h
dsfdsfK
h
h
dsfdsfK
hN
h
sdrsdr
h
dsfdsfK
hhNdsfsdr
1 1
1 212
1 11
1 2121
)(
),(1
)(11
),(11
)DSFSDR(obPr
(4.6)
where K(x, y) is a two-dimensional kernel centered at (x, y). This equation follows
directly from the definition of the conditional probability and the kernel density
estimation in Equation (4.3). If the two-dimensional kernel can be factorized into K(dsf)
and K(sdr), then the Equation (4.6) can be rewritten as
N
q
q
N
p
pp
h
dsfdsfK
h
h
sdrsdrK
hh
dsfdsfK
hdsfsdr
1 11
1 2211
)(1
)(1
)(1
)DSFSDR(obPr
(4.7)
For the same reason explained above, we can fit a conventional PDF to this empirical
conditional probability distribution by minimizing the RMSE. The lognormal distribution
is appropriate for this conditional probability because the SDR values are bounded by
zero on the lower side. The advantage of this method is that we do not need to discretize
the range of the SDR into specific DSs. Instead, we can directly compute the conditional
probability of the SDR given the DSF without computing the cumulative conditional
distribution. In addition, this method considers the uncertainty in both the DSF and the
SDR measurements unlike the previous method that considers the uncertainty of only the
DSF by using the one-dimensional kernel.
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
106
The second alternative method is to estimate the mean and the variance of the SDR given
the DSF (μSDR|DSF, and σ2SDR|DSF, respectively) and then fit a PDF. In other words, we can
obtain the conditional probability distribution of the SDR given the DSF. This method is
particularly useful when damage states are not clearly defined by the SDR or when the
conditional density function of the SDR needs to be convoluted with other conditional
density function for further risk analysis. The estimates of the conditional mean, DSFSDR ,
and the conditional variance, DSFSDR2 , for the DSF value of dsf can be computed using a
kernel as follows:
n
n
m
mm
dsf
h
dsfdsfK
h
dsfdsfKsdr
)(
)(ˆ
DSFSDR
(4.8)
n
n
m
mdsfDSFSDRm
dsf
h
dsfdsfK
h
dsfdsfKsdr
m
)(
)(ˆˆ
2
DSFSDR2
(4.9)
Once the mean and the variance are computed, the lognormal distribution function is used
to fit the conditional distribution of the SDR given the DSF by the method of moments.
4.3 Application of the Framework to Simulated
Data Using an Analytical Model of the Four-
Story Steel Special Moment-Resisting Frame
The framework for building fragility functions based on the DSF that is described in the
previous section was validated using a set of numerically simulated data from a four-
story two-bay steel special moment-resisting frame (SMRF). This frame is a perimeter
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
107
lateral resisting system of an office building designed in Los Angeles based on current
seismic provisions such as the IBC and the AISC. The connections of the SMRF are
reduced beam section (RBS) and have been designed in accordance with FEMA 350. An
analytical model of this frame has been developed and validated experimentally up to
collapse (see Lignos and Krawinkler, 2009). Component deterioration was simulated in
the same model with the modified Ibarra-Krawinkler model that can simulate up to four
modes of deterioration as explained by Lignos and Krawinkler (2009). The analytical
model of the structure was subjected to a set of 40 ground motions scaled to various
intensities, and absolute acceleration time-histories at each floor were obtained. The
unscaled ground motions in this set have large magnitude (6.5 < M < 7.0) and distances
from the rupture zone of 13 km < R < 40 km (see Medina and Krawinkler, 2003). The
median of the acceleration spectrum of the unscaled motions matches the design level
acceleration spectrum for the area in which the office building is designed. Hence, the
ground motion set is a suitable representative one for the location of the structure. The
response of the SMRF was evaluated up to collapse using IDA, where the spectral
acceleration at the first mode period (Sa(T1, 2%)) was used as an intensity measure of the
ground motion. The first mode period of the four-story SMRF is 1.32 second. The
absolute acceleration responses were collected for each level of intensity, and the
wavelet-based damage diagnosis algorithm described in section 4.2.2 was applied to the
data to extract the DSF. Based on the DSF, fragility functions are computed for damage
assessment of the four-story SMRF.
4.3.1 Description of the Analytical Model
The analytical model of the four-story steel SMRF is developed in DRAIN-2DX analysis
program (Prakash et al., 1993) with elastic beam column elements and deteriorating
springs at their ends that follow a bilinear hysteretic response. This model simulates
critical component deterioration modes such as strength, post-capping strength, and
unloading stiffness deterioration due to cyclic loading (see Lignos and Krawinkler,
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
108
2009). Deterioration parameters for the components were extracted from a steel
component database for deterioration modeling (Lignos and Krawinkler, 2007, 2009).
Both the component analytical model and the DRAIN-2DX software have been validated
using a series of shaking table tests of a scaled model of the four-story SMRF. These tests
were conducted at the Network for Earthquake Engineering Simulation (NEES) facility at
the State University of New York at Buffalo (Lignos and Krawinkler, 2009; Lignos et al.,
2011). Lignos et al. (2011) showed that the analytical model successfully reproduced the
dynamic response of the test specimen. The model was able to simulate strength and
stiffness deterioration of the steel beams and columns of the structure.
4.3.2 Development of Fragility Functions for Different
Damage States
Wavelet transform was applied to absolute acceleration time-histories of each floor of the
structure. Figure 4.2 shows an example of the ground acceleration from Loma Prieta
earthquake motion recorded at the Agnews state hospital, and Figure 4.3 shows the
wavelet coefficients for the roof absolute acceleration responses of the structure subjected
to various intensities of the Loma Prieta earthquake motion. In Figure 4.3, the horizontal
axis is the time shift parameter b, and the vertical axis is the scaling parameter a. The
absolute values of the wavelet coefficients are represented by different colors – the
brighter colors imply higher values of the wavelet coefficients, as shown in the color-bar,
and thus represent higher concentration of energy. As the intensity of the input motion
increases, the peaks of the wavelet coefficients shift both in time and in scale. As
explained in Chapter 3, the higher the intensity of the input ground motion, the larger the
severity of structural damage is. Thus, the changes in the pattern of the wavelet
coefficients can be used as a good indicator of the damage extent of the steel frame.
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
109
0 10 20 30 40-0.2
-0.1
0
0.1
0.2
Time (s)
Acc
eler
atio
n (
g)
Figure 4.2: Loma Prieta earthquake ground acceleration recorded at the Agnews state
hospital
Figure 4.4 shows the values of DSF for various intensities of the Loma Prieta earthquake
ground motion recorded at the Agnews state hospital shown in Figure 4.2. This figure
shows that as damage progresses in the structure, the DSF values have a general
increasing trend. Thus, it is demonstrated empirically that the DSF can be a good
indicator of structural damage.
0 10 20 30 40 50 0
2
4
0 10 20 30 40 50 0
2
4
Time (s)
0 10 20 30 40 50 0
2
4
Sca
le 0 10 20 30 40 50 0
2
4
50
100
150
200
(b)
(c)
(d)
(a)
Figure 4.3: Wavelet coefficients for the roof acceleration history of the four-story SMRF subjected to scaled Loma Prieta earthquake motions: (a) Sa(T1, 2%) = 0.25g; (b) Sa(T1,
2%) = 0.5g; (c) Sa(T1, 2%) = 0.75g; (d) Sa(T1, 2%) = 1.125g
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
110
0.05 0.10 0.25 0.50 0.75 1.00 1.13 1.19 1.220
0.2
0.4
0.6
0.8
1
Sa(T1, 2%) (g)
DS
F
Figure 4.4: DSF for various intensities of scaled Loma Prieta earthquake ground motion
recorded at the Agnews state hospital To develop the fragility functions, five damage states are defined for the SMRF in terms
of SDR at each story. These five damage states, DS0, DS1, …, DS4, correspond to no
damage (i.e., within the elastic limit) (0% ≤ SDR < 1%), slight damage (1% ≤ SDR <
2%), moderate damage (2% ≤ SDR < 3%), severe damage (3% ≤ SDR < 6%), and
collapse (6% ≤ SDR), respectively. The threshold values of SDR are selected as
representative values to describe different damage states based on current practice
(FEMA 356 and FEMA 440).
A pair of DSF and SDR, dsfi, sdri , was computed for the individual floor absolute
acceleration response for each ground motion excitation. Figure 4.5 shows the
distribution of dsfi, sdri pairs from all the ground motion excitations for each story.
This figure shows that DSF and SDR are well correlated. The correlation coefficient (ρ)
of the pairs for stories 1 to 4 are also shown in the figure. The standard Gaussian function
is defined as the kernel K in Equation (4.5), and the smoothing parameter (or bandwidth)
of 0.2 is used. The Gaussian kernel is a powerful kernel widely used in pattern
recognition (Evalgelista, 2007). The bandwidth was chosen to be the smallest value that
made the empirical fragility functions increase monotonically.
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111
0 0.2 0.4 0.6 0.8 1
100
101
DSF
SD
R (
%)
0 0.2 0.4 0.6 0.8 1
100
101
DSF
SD
R (
%)
0 0.2 0.4 0.6 0.8 1
100
101
DSF
SD
R (
%)
0 0.2 0.4 0.6 0.8 1
100
101
DSF
SD
R (
%)
= 0.699
= 0.871 = 0.831
= 0.844
(b)(a)
(c) (d) Figure 4.5: Scatter plot of DSF versus SDR: (a) story 1; (b) story 2; (c) story 3; (d) story
4
4.3.3 Results
Fragility functions for each story of the four-story steel SMRF are shown in Figure 4.6.
Each fragility function represents the probability that the damage state at each story is
equal to or greater than DSi. The fragility function for DS0 is one (G0(dsf) = 1 for all dsf).
Equation (4.5) was used to compute the empirical fragility functions for the individual
story based on the acceleration measurements at the ceiling of the story of interest. The
beta CDF was fitted to the empirical fragility functions using non-linear least-squares
fitting based on the Gauss-Newton method (Hartley, 1961). Since the original beta CDF
would always be 1 when the DSF is 1, the original beta CDF was scaled by the maximum
value of the empirical fragility function, Ĝi(·), before fitting. Figure 4.6 allows us to
estimate at which stories we should expect damage to form for the four-story steel
SMRF. For instance, when a DSF equals 0.75 for the first story the probability of having
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
112
moderate or more severe damage to the first story of the structure is more than 60%. This
correlates well with the information presented in Figure 4.5 because for a DSF value of
0.75 the story drift ratios on average are more than 2%.
Figure 4.6: Fragility functions of the four-story steel SMRF: (a) story 1; (b) story 2; (c)
story 3; (d) story 4 Traditionally, in PBEE a fragility function is computed based on the maximum SDR
among all the stories in order to conduct a global assessment of damage for a given
structure. Similarly, Figure 4.7 shows global fragility functions for the structure using the
DSF of the roof acceleration responses as an EDP and the maximum SDR among all the
stories as a measure of damage. This figure gives a sense of overall structural damage for
DSF values. The DSF and SDR pairs used for computing the global fragility functions in
Figure 4.7 have a higher correlation coefficient than those pairs used for the fragility
functions for each story. Thus, the fragility functions for the entire structure can identify
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
113
the overall damage with smaller error. The reason for the global fragility functions to
have the smaller error can be that the absolute acceleration measurements reflect the
global behavior of the structure, thus the damage in the structure at any location can
affect the value of the DSF that is computed from the roof acceleration response.
Figure 4.7: Global fragility functions for the four-story steel SMRF
Figure 4.8: Probability of being in each damage state for the four-story steel SMRF
The probability of being at each damage state DSi (see Figure 4.8) can be computed based
on the difference between two adjacent fragility functions for DSi and DSi+1 from Figure
4.7. In order to determine the damage state of a structure given a DSF value, a weighted
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
114
average of all the possible damage states can be used from Figure 4.8. Note that for the
four-story steel SMRF, when the DSF value is close to 0.5, the probabilities of being in
different damage states are similar to each other and damage assessment is more
ambiguous than when the DSF is closer to 0 or 1. Table 4.3 shows the performance of the
DSF-based fragility functions for estimating damage states. Five sets of fragility
functions are evaluated – four sets for each story and one set for global. Each row of the
table shows the percentage of obtaining a correct damage state using the fragility
functions with no error (exact), ± 1 level of DS error, and ± 2 levels of DS error,
respectively. ± 1 level of DS error implies, for example, that DS1, DS2, or DS3 is obtained
using the fragility functions when the correct damage state is DS2. Collapse (DS4) is
considered as severe damage (i.e., part of DS3) in this result because of its sensitive
nature (see more discussion about collapse later in this section). As expected, the fragility
functions for the overall structure have higher chance of estimating the damage state
correctly than the fragility functions for each story. The fragility functions for the overall
structure can estimate the damage state exactly for 75% of the data and estimate with ± 1
level of DS error for 98% of the data. The first story has the lowest chance of estimating
the damage state exactly (51%) and with ± 1 level of DS error (85%).
Table 4.3: Performance of the DSF-based fragility functions for estimating a damage state
Story 1 Story 2 Story 3 Story 4 Global Exact 51 % 56 % 62 % 56 % 75 %
Within ± 1 DS 85 % 97 % 98 % 96 % 98 % Within ± 2 DS 95 % 99.7% 100 % 100 % 99.7 %
Using the two-dimensional Gaussian kernels, the conditional probability of the SDR
given the DSF is computed for various DSF values. To compute the conditional
probability, we used the Gaussian kernels with the bandwidth (or the standard deviation)
of 0.1 for the DSF and 0.0106 for the SDR, which are Silverman’s optimum bandwidth
(h) for the Gaussian kernel (Silverman, 1986). It is given as
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
115
5
1
ˆ06.1
nh (4.10)
where is the sample standard deviation, and n is the number of samples. Figure 4.9
shows the scatter plot of the DSF from the roof acceleration responses and the maximum
SDR among all the stories and the conditional density functions fitted to the lognormal
PDF for several DSF values.
(a) (b)
Figure 4.9: (a) Scatter plot of DSF versus maximum SDR; (b) Condition probability density function of maximum SDR given DSF using the two-dimensional kernel for the
four-story steel SMRF
Alternatively, the conditional mean and the standard deviation of the SDR given the DSF
were first computed using the Gaussian kernel, and then the lognormal distribution was
fitted to the data using the method of moments. Figure 4.10 (a) shows the scatter plot of
the DSF from the roof acceleration responses and the maximum SDR among all the
stories and the conditional mean and the standard deviation of the SDR given the DSF,
represented by the solid and the dotted lines, respectively. To compute DSFSDR and
DSFSDR , we used the Gaussian kernels with the bandwidth of 0.1 for the DSF as before.
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
116
Figure 4.10 (b) shows the conditional density functions of the SDR given the DSF values
of 0.2, 0.5, and 0.8 using the lognormal distribution. Both the mean and the variance of
the SDR increase as the DSF increases. This indicates that the DSF is positively
correlated to the SDR, and the DSF can estimate lower levels of damage more
confidently than more severe levels. Table 4.4 and Figure 4.11 shows DSFSDR and
DSFSDR for various DSF values in more detail.
(a) (b)
Figure 4.10: (a) Scatter plot of DSF versus maximum SDR and the conditional mean and standard deviation; (b) Condition probability density function of maximum SDR given
DSF for the four-story steel SMRF
Table 4.4: The conditional mean and the standard deviation of the SDR given DSF
DSF DSFSDR DSFSDR
0.009 0.664 0.405 0.030 0.683 0.413 0.062 0.719 0.428 0.091 0.760 0.443 0.122 0.818 0.463 0.149 0.884 0.484 0.180 0.987 0.515 0.213 1.139 0.566
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
117
0.239 1.304 0.631 0.275 1.595 0.780 0.300 1.842 0.935 0.331 2.188 1.179 0.371 2.667 1.536 0.396 2.977 1.759 0.404 3.071 1.824 0.413 3.180 1.898 0.426 3.348 2.007 0.447 3.609 2.165 0.457 3.730 2.234 0.465 3.834 2.289 0.478 3.994 2.371 0.496 4.213 2.474 0.503 4.304 2.513 0.511 4.401 2.552 0.522 4.524 2.600 0.530 4.626 2.636 0.538 4.713 2.665 0.552 4.881 2.715 0.558 4.941 2.731 0.572 5.093 2.766 0.581 5.192 2.785 0.590 5.281 2.799 0.602 5.399 2.812 0.610 5.470 2.817 0.625 5.608 2.820 0.648 5.786 2.807 0.669 5.937 2.778 0.701 6.120 2.718 0.723 6.226 2.669 0.750 6.337 2.609 0.775 6.422 2.558 0.801 6.497 2.510 0.824 6.550 2.474 0.850 6.600 2.436 0.875 6.634 2.403 0.900 6.656 2.371 0.923 6.667 2.343 0.936 6.669 2.328
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118
0 0.2 0.4 0.6 0.8 10
2
4
6
DSF
SD
R (
%)
SDR|DSF
SDR|DSF
Figure 4.11: Conditional mean and standard deviation of SDR given DSF for the four-
story steel SMRF
The performance of the wavelet-based DSF in damage diagnosis is compared with that of
the Sa(T1, 2%) and the peak roof acceleration (PRA), which are conventional measures
used to predict the damage state of a structure in PBEE. In order to compare the three
measures, Sa(T1, 2%), PRA, and DSF, the coefficient of determination (R2) (Rao, 1973)
was used. Figures 4.11 (a), (b), and (c) show the scatter plots of SDR--Sa(T1, 2%), SDR--
PRA, and SDR--DSF, respectively. A moving average of each data set was also computed
and is shown as a solid line in the same figures. The size of the window for the moving
average was set to be 40 because this is the number of the ground motions used in this
analysis. The fact that the moving average curves are not smooth or monotonically
increasing is not a significant problem in this case because we just use the curves to
compare the performance of the three measures (not to estimate SDR from the three
different measures). Using the moving average model, the R2 value was computed as
follows (Rao, 1973):
ii
iii
yy
yyR
2
2
2
)(
)ˆ(1
(4.11)
where yi is the ith y-axis data (sdri in this application), ŷi is the estimated value of yi using
the moving average, and y is the mean of yis. The R2 value represents the proportion of
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
119
variability in the data estimated by the moving average model for the DSF and Sa(T1,
2%), that is, R2 can be used to compare how well each of the two measures can estimate
the SDR. Based on Figure 4.12, the DSF has a R2 value of 0.70 compared to 0.62 when
Sa(T1, 2%) is used and to 0.63 when PRA is used. This happens because the DSF takes
advantage of the entire absolute acceleration histories of the four-story steel SMRF based
on its nonlinear response compared to traditional Sa(T1, 2%), which is based on the
elastic system, and the PRA, which is based on only one value in the acceleration history.
From the same figure, a similar observation is obtained regarding dispersions, if we
compare the vertical dispersion of the SDR given the value of the three measures. The
observation is that the DSF can estimate the SDR with smaller variance compared to the
other two measures, and thus the DSF-based fragility functions can be more effective in
prediction of the damage state of a given structure. It is recognized that in order to
validate the performance of the DSF-based fragility functions compared to traditional
intensity measures, such as Sa(T1, 2%) and PRA, the proposed framework should be
tested with various types of structures.
0 1 21
2
3
4
5
6
Sa(T1, 2%) (g)
SD
R (
%)
0 1 21
2
3
4
5
6
Peak Roof Acceleration (g)
SD
R (
%)
0 0.5 11
2
3
4
5
6
DSF
SD
R (
%)
(a) (b) (c) Figure 4.12: Scatter plots:(a) Sa(T1, 2%) versus SDR; (b) PFA versus SDR; (c) wavelet-
based DSF versus SDR
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
120
Figure 4.13 shows the probability of collapse of the structure conditioned on the values of
the Sa(T1, 2%) (Figure 4.13 (a)) and the wavelet-based DSF (Figure 4.13 (b)). A
lognormal CDF was fitted to the empirical CDFs, and the coefficient of variation (COV)
of the fitted lognormal CDF was used as an indicator of a better measure of median
collapse capacity. A median collapse capacity using Sa(T1, 2%) is 1.3g, and that using
DSF is 0.8. The standard deviations normalized with respect to the mean values of each
measure (COVs) are 0.37 and 0.17 for Sa(T1, 2%) and DSF, respectively. Therefore, the
DSF can predict the probability of collapse with smaller error than Sa(T1, 2%) can;
however, according to recent experimental data, acceleration measurements are sensitive
near collapse (see Lignos and Krawinkler, 2009, and Suita et al., 2009), and the DSF-
based fragility functions to estimate the probability of collapse of a structure subjected to
ground motions need to be validated more carefully in the future.
0 1 2 30
0.2
0.4
0.6
0.8
1
1.2
Sa (T1, 2%) (g)
Pro
bab
ility
of
Co
llap
se
(a)
0 0.5 10
0.2
0.4
0.6
0.8
1
1.2
DSF
Pro
bab
ility
of
Co
llap
se
(b)
Empirical CDFLognormal Fit
Empirical CDFLognormal Fit
Median : 1.3 gCOV: 0.37
Median : 0.8COV : 0.17
Figure 4.13: Collapse Fragility functions based on: (a) Sa(T1, 2%); (b) the wavelet-based
DSF
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
121
Table 4.5: Advantages and disadvantages of different fragility functions Methodology Advantages Disadvantages
IM → EDP → DM IMs are easy to obtain.
IMs are not as strongly related to drift-based EDP
or structural damage as DSF.
DSF → drift-based EDP → DM
DSFs are easy to obtain and more strongly
related to damage than other acceleration-based
EDPs.
Structure needs to be instrumented for damage
assessment.
Drift-based EDP → DM Drift-based EDPs are
strongly related to structural damage.
Drift-based EDPs are difficult and expensive to
measure accurately.
4.3.4 Discussion
This section discusses the advantages and some of the remaining challenges of the
proposed framework for earthquake damage assessment of structures. The main
advantage of the proposed method is that fragility functions in terms of the wavelet-based
DSF can be developed before an earthquake event occurs and periodically updated by
using information on structural parameters extracted from ambient vibration data
obtained from the as-built structure or from response analysis of data resulting from small
earthquakes. Another important aspect of the approach is that direct measurement of
acceleration responses are used to estimate the drift-based EDP or the potential damage
states of the structure with a corresponding probability of each damage state. Directly
estimating the damage state from drift-based EDPs, such as SDR and maximum drift, is
more reliable and accurate; however, accurate measurement of drift is often expensive to
obtain. Estimating the damage state from IMs, such as spectral acceleration and peak
ground acceleration, or conventional acceleration-based EDPs, such as peak floor
acceleration, is not as reliable as estimating from the DSF as discussed in section 4.3.3.
Table 4.5 summarizes the advantages and disadvantages of various fragility functions
including the one based on the DSF. Although our observations are based on the analysis
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
122
of steel structures, the method is general and can be readily adapted to other structures
provided appropriate analytical model can be developed for the structure.
For purposes of practical applications, it is important that both the damage diagnosis
algorithm and the absolute acceleration measurements are reliable particularly when the
structure exhibits large deformations. Reliability of acceleration measurements can be
achieved through repeated testing and calibration of the instrumentation providing these
accelerations. Reliability of the damage algorithm can be greatly improved with
additional laboratory testing and future field observations. While there have been
numerous instrumented structures subjected to earthquakes in the past several decades,
the instrumentation in such buildings is usually very sparse, and the damage is poorly
documented to provide appropriate testing of the algorithm. Recent experiments
conducted under the National Science Foundation (NSF) program for the NEES has
provided the opportunity for systematic testing of variety of structures to different
damage states. As these data become available the algorithm will be further validated and
updated as needed. Such testing and validation will greatly increase the accuracy of the
predicted damage state using a DSF-based fragility function. This study presents the first
confirmation of the algorithm and is intended to serve as a proof of concept. As the data
become more readily available, the accuracy of the analytical model that is used to
compute the absolute acceleration time-histories could also be greatly improved resulting
in more reliable fragility functions.
In general, fragility functions using the wavelet-based DSF can be generated for variety
of generic building types. These generic fragilities can then be modified as information
on a particular structure becomes available resulting in structure-specific functions to be
used as part of a damage assessment scheme.
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
123
4.4 Conclusions
This chapter presents a new framework that combines concepts from performance-based
earthquake engineering (PBEE) and structural health monitoring (SHM) to compute
fragility functions for steel structures as a damage assessment/prediction model using a
wavelet-based damage sensitive feature (DSF) introduced in Chapter 3. The analytical
formulations that relate the DSF to structural parameters is also given in Chapter 3, which
provides the theoretical foundation for using the wavelet-based parameters. The DSF-
based fragility functions can be used as an alternative damage prediction model in the
field of PBEE for drift-based EDPs or DMs and a probabilistic damage classification
model for DSF in the field of SHM. The proposed framework is based on information
retrieved from an extensive set of structural responses extracted from an analytical model
of a structure subjected to a set of ground motions utilizing incremental dynamic analysis.
The wavelet-based DSF is then computed based on the floor absolute acceleration time-
histories, and different damage states are defined based on the maximum story drift ratio
that gives DSF an engineering meaning. The probabilistic relationship between the
damage states and the DSF is computed as fragility functions, which provide the
conditional probability of the structure being or exceeding a particular damage state given
the value of the DSF. Three different methods are presented for computing the fragility
functions for several damage states, the conditional probability of the SDR given the
DSF, and their conditional mean and standard deviation. These fragility functions can be
computed for each story separately or for the entire structure to assess the overall damage
state.
The framework is validated using a set of numerically simulated data from a four-story
steel special moment-resisting frame subjected to various intensities of 40 different
ground motions. The results show that the DSF-based fragility functions can predict the
damage state of the frame and the median collapse capacity with less variance than the
fragility functions derived from alternate acceleration-based conventional measures, such
CHAPTER 4. Development of Fragility Functions as a Damage Classification/Prediction Method Using a Wavelet-Based Damage Sensitive Feature
124
as the spectral acceleration at the first mode period and the peak roof acceleration. In
addition, the fragility functions for each story can provide the damage location by story
while the global fragility functions can assess the overall structural damage state with
smaller error. Thus, the global fragility functions would be suitable for a preliminary
assessment of a structure immediately after an earthquake while the fragility functions for
each story would be useful for more detailed analysis of damage.
The fragility functions are computed using a particular wavelet-based DSF in the study;
however, the framework can potentially be used with any valid DSF that can reliably
estimate the damage state of a structure. Further verification and testing of its damage
assessment capabilities need to be performed as additional data for different types of
structures become available, and the general form of the fragility functions for a group of
similar types of structures can be explored. It is also necessary to investigate the
feasibility of implementing this damage classification method using the DSF-based
fragility functions on a wireless structural health monitoring system. The DSF-based
damage assessment framework presented in this chapter, however, serves to introduce the
method and as a proof of concept with verification based on currently available
laboratory test data.
125
Chapter 5
Application of a Sparse Representation Method Using K-SVD Algorithm to Data Compression of Experimental Ambient Vibration Data for SHM
This chapter introduces a data compression method using the K-SVD algorithm and its
application to experimental ambient vibration data for structural health monitoring
(SHM) purposes. Because many damage diagnosis algorithms that use system
identification require vibration measurements of multiple locations, it is necessary to
transmit long threads of data. In wireless sensor networks for SHM, however, data
transmission is often a major source of battery consumption. Therefore, reducing the
amount of data to transmit can significantly lengthen the battery life and reduce
maintenance cost. The K-SVD algorithm was originally developed in information theory
for sparse signal representation. This algorithm creates an optimal over-complete set of
bases, referred to as a dictionary, using singular value decomposition (SVD) and
represents the data as sparse linear combinations of these bases using the orthogonal
CHAPTER 5. Application of a Sparse Representation Method Using K-SVD Algorithm to Data Compression of Experimental Ambient Vibration Data for SHM
126
matching pursuit (OMP) algorithm. Since ambient vibration data are stationary, we can
segment them and represent each segment sparsely. Then only the dictionary and the
sparse vectors of the coefficients need to be transmitted wirelessly for restoration of the
original data. We applied this method to ambient vibration data measured from a four-
story steel special moment-resisting frame (SMRF). The results show that the method can
compress the data efficiently and restore the data with very little error.
5.1 Introduction
In recent years, there has been increasing interest in structural health monitoring (SHM)
using wireless sensors (Straser and Kiremidjian, 1998). One of the challenges of
implementing wireless sensors is reducing its battery power consumption. Because the
major source of power consumption is wireless data transmission, efficient and reliable
data compression techniques can significantly reduce the maintenance cost of the
wireless monitoring systems, and thus various efforts have been made in this area (Lynch
et al., 2003; Caffrey et al., 2004; Sazonov et al., 2004). Data compression methods have
been developed in various fields including image processing, information theory, and
computer science (Sayood, 2000; Salomon, 2007). Data compression is also closely
related to machine learning because compression involves extraction of information from
the history of data, which in turn can be used for the prediction of data.
This chapter introduces a data compression method using sparse representations based on
the K-SVD algorithm and applies this method to ambient vibration measurements for
SHM purposes. Sparse representation methods have been successfully applied to
compression, de-noising, feature extraction, and so on in various fields (Marcellin et al.,
2000, Starck et al., 2003, Gastaud and Starck, 2004). These methods involve the
representation of signals using an over-complete dictionary. The K-SVD algorithm is an
iterative method that generalizes the K-means clustering process (Aharon, 2006). During
CHAPTER 5. Application of a Sparse Representation Method Using K-SVD Algorithm to Data Compression of Experimental Ambient Vibration Data for SHM
127
each iteration, it designs and updates an optimal dictionary that contains prototype signal-
atoms (or bases) based on the training data and then represents the signal sparsely on the
basis of the dictionary using any pursuit algorithm. In other words, we can represent data
as a linear combination of only a few atoms of the dictionary using this algorithm. We
apply this idea to SHM for the purpose of compressing stationary ambient vibration
measurements in order to reduce the communication burden of wireless sensors. We first
segment the data and compute the sparse representation of each segment separately. Then
transmitting only the sparse representation and the dictionary enables us to restore the
data. We validated this method using a set of experimental white noise data collected
from a shake-table test of a four-story steel special moment-resisting frame (SMRF)
conducted at the State University of New York, Buffalo, introduced in Chapter 3.
The chapter is organized as follows. Section 5.2 explains the data compression and
reconstruction methods using the K-SVD algorithm, and section 5.3 describes its
application to experimental data and the results. Finally, section 5.4 presents conclusions
and future work.
5.2 Description of Data Compression Method
The complete data transmission process using the compression method based on the K-
SVD algorithm consists of four steps: (1) collecting structural vibration data, (2)
compressing the data using the K-SVD algorithm; (3) transmitting the compressed data
wirelessly; and (4) reconstructing the data. This process is summarized in Figure 5.1. The
focus of this study is on the second step, compression of the data, and the other steps are
explained briefly. In the first step, the collected vibration data is segmented into small
sections. The length of each segment needs to be long enough to contain the dynamic
characteristics of the structure, but longer segments result in a larger size of dictionary,
which increases the computation and transmission demands. In the second step, the data
CHAPTER 5. Application of a Sparse Representation Method Using K-SVD Algorithm to Data Compression of Experimental Ambient Vibration Data for SHM
128
compression method using the K-SVD algorithm first initializes the over-complete set of
bases, or the dictionary, and then repeats the following two steps: sparsely representing
the data using the dictionary and updating each atom of the dictionary. The dictionary can
be initialized by either random vectors or randomly selected data segments. The bases
represent the hidden structure of the acceleration measurements, so the acceleration
measurements can be represented as the weighted sum of the bases. By constructing an
over-complete set, the acceleration measurements can be represented using only a few
number of bases. In other words, we can afford to have more customized bases by
increasing the number of bases in the set. Then, the acceleration measurements are
represented sparsely by selecting the bases that are similar (or close) to themselves. These
two steps are repeated until the results converge or for a set number of iterations. The
third step involves wireless transmission of the dictionary and the sparse representation of
the data. The compression rate is computed in section 5.2.2. In the final step, the
vibration data can be reconstructed using a simple matrix multiplication of the
transmitted dictionary and the sparse representation of the data.
5.2.1 Data Compression Using K-SVD Algorithm
The K-SVD algorithm searches for a sparse representation of data using an over-
complete set of bases. The mathematical representation of this objective is (Aharon et al.,
2006)
2
,min
FD XY DX subject to 00ix T for all i (5.1)
where F
is the Frobenius norm, Y n NR is a set of N training data, D n KR is an over-
complete set of K n bases, where each column corresponds to a base, X K NR is the
sparse representation coefficients of Y with respect to D, xi is the ith column of X, 0 is
the l0 norm that indicates the number of non-zero elements of a vector, and T0 is a pre-
determined sparsity of X.
CHAPTER 5. Application of a Sparse Representation Method Using K-SVD Algorithm to Data Compression of Experimental Ambient Vibration Data for SHM
129
Figure 5.1: Summary of the data compression using the K-SVD algorithm
CHAPTER 5. Application of a Sparse Representation Method Using K-SVD Algorithm to Data Compression of Experimental Ambient Vibration Data for SHM
130
In order to obtain an optimal solution, the K-SVD algorithm repeats the following two
steps: sparse coding and dictionary update (Aharon et al., 2006). The first step involves
the decomposition of the data with respect to the current dictionary. In other words, the
sparse coding computes the representation coefficients X that satisfy the objective
function. For the first iteration, the dictionary can be initialized as a random matrix with
normalized columns or as a matrix whose columns are the normalized first K training
data. Because the exact solution of the sparse coding is numerically infeasible to compute
(Davis et al., 1997), pursuit algorithms, such as the matching pursuit (MP), the
orthogonal matching pursuit (OMP), the basis pursuit (BP), and the focal
underdetermined system solver (FOCUSS), are often used to search for an approximate
solution for X. For small T0, they provide a good approximation to the true solution
(Aharon et al., 2006). Because the K-SVD algorithm decouples the sparse coding step
from the dictionary updating step, it has the flexibility to work with any pursuit algorithm
and in turn allow us to choose the algorithm according to our needs and constraints. For
our application, we use the OMP algorithm to find X because of its simplicity and
efficiency (Pati et al., 1993). The second step involves using the singular value
decomposition (SVD) to update each atom of D as well as the non-zero elements of X
that use this atom. Updating both a dictionary atom and the corresponding coefficients at
the same time accelerates the convergence because the update of the next atom of the
dictionary will be based on the more relevant coefficients. According to Aharon et al.
(2006), the effect of this simultaneous updating is equivalent to the leap from gradient
descent to Gauss-Seidel methods in optimization. We repeat these two steps until the
objective function (1) is below a pre-determined threshold.
In the first step, we solve the optimization problem (1) assuming that D is fixed. Using
the following expansion of the objective function
2 2
21
N
i iFi
Y DX y Dx
(5.2)
CHAPTER 5. Application of a Sparse Representation Method Using K-SVD Algorithm to Data Compression of Experimental Ambient Vibration Data for SHM
131
where yi is the ith column of Y, we can decouple the objective function (5.1) to N
problems as follows:
2
2min
ii i
xy Dx subject to 00ix T for all i. (5.3)
The OMP algorithm is a simple greedy algorithm that solves each of these N problems
using simple inner-products between the data and the dictionary atoms (Pati et al., 1993).
This algorithm sequentially chooses each atom of the dictionary that maximizes the
inner-product with yi until it has chosen T0 atoms. Then it computes the coefficients xi
using the least-square solutions.
The next step updates each atom of the dictionary and the corresponding non-zero
elements of X, simultaneously. We first assume that X and D are fixed except for the one
column of the dictionary in question (dk) and the corresponding coefficients, the kth row
of X (xTk). Then the objective function (1) can be rewritten as (Aharon et al., 2006)
2
2
1
2
2
KT
j jFj F
T Tj j k k
j kF
Tk k k F
Y DX Y d x
Y d x d x
E d x
(5.4)
where Ek corresponds to the representation error of N training data excluding the kth atom
of the dictionary. The SVD can find the new dk and xTk that minimize the functions in (4),
but in order to maintain the sparsity of X, we constrain the updates on xTk to be only on
the non-zero elements, using the following procedure. First let wk be the set of indices of
the non-zero elements of xTk and define Ωk
kN wR as the matrix with ones at (wk(i), i)
and zeroes elsewhere. Then Tkx =xT
kΩk becomes the row of non-zero elements of xTk, and
kY =YΩk corresponds to a subset of training data that currently uses dk for the sparse
CHAPTER 5. Application of a Sparse Representation Method Using K-SVD Algorithm to Data Compression of Experimental Ambient Vibration Data for SHM
132
representation. Similarly, kE =EkΩk only includes the error from YΩk. Now we can restrict
the updates on xTk to be only on the non-zero elements by minimizing the following
function instead of (4):
2 2T Tk k k k k k k k FF
E d x E d x . (5.5)
The SVD of kE =EkΩk =USVT provides the new dk and Tkx that minimize (5) as the first
column of U and the first column of V multiplied by S(1,1), respectively. Similar to the
K-means, which computes the mean of each cluster K times in order to update the
parameters, the K-SVD algorithm computes SVD K times to update the entire dictionary.
5.2.2 Data Transmission and Reconstruction
After obtaining the sparse representation of the data, wireless sensors transmit the
dictionary D and the representation coefficients X. Since X is a sparse matrix, the sensors
need to transmit only the non-zero elements of X and their locations as well as the
number of non-zero elements, T0. The compression ratio of this algorithm is
02n K T N
n N
. If N K , the compression ratio approaches 02T
n.
We can reconstruct the data using the following linear relationship:
Y DX (5.6)
CHAPTER 5. Application of a Sparse Representation Method Using K-SVD Algorithm to Data Compression of Experimental Ambient Vibration Data for SHM
133
5.3 Application of the Data Compression
Algorithm to Experimental Data Using the
Four-Story Steel Special Moment-Resisting
Frame
5.3.1 Description of Experiment
The data compression method based on the K-SVD algorithm is applied to the
experimental ambient vibration data collected from the four-story steel SMRF test
introduced in Chapter 3. As described in section 3.3.2.1, the frame is subjected to the
1994 Northridge earthquake ground motion recorded at Canoga Park station. The testing
sequence included a service level earthquake (SLE, 40% of the unscaled record), a design
level earthquake (DLE, 100% of the unscaled record), a maximum considered earthquake
(MCE, 150% of the unscaled record), and a collapse level earthquake (CLE, 190% of the
unscaled record). For system identification and damage diagnosis purposes, the frame is
also subjected to white noise excitations between each earthquake loading. Five white
noise excitations are applied before SLE, and four white noise excitations are applied
after each earthquake loading. These nine excitations are referred as damage pattern (DP)
1, 2, 3, …, 9, hereafter. According to elastic modal identification from white noise tests,
the frame had a predominant period of 0.45 second in undamaged state. The acceleration
measurements are collected at each floor, and the sampling rate is 128 Hz. The data
compression method is applied to these white noise excitation responses.
CHAPTER 5. Application of a Sparse Representation Method Using K-SVD Algorithm to Data Compression of Experimental Ambient Vibration Data for SHM
134
0 1 2 3 4 5
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Time (s)
Acc
eler
atio
n (g
)
(a)
Original dataReconstructed data
0 1 2 3 4 50
0.02
0.04
0.06
0.08
0.1
Time (s)
Rep
rese
ntat
ion
erro
r (g
)
(b)
-0.2 -0.1 0 0.1 0.2
-0.1
-0.05
0
0.05
0.1
0.15
Original data (g)
Rec
onst
ruct
ed d
ata
(g)
= 0.98938R2 = 0.97885
(c)
0 2 4 6 8 100
0.05
0.1
Frequency (Hz)
PS
D
(d)
Original dataReconstructed data
Figure 5.2: The results of data reconstruction using the K-SVD algorithm for the roof
acceleration response at DP 1: (a) time-histories of original data and reconstructed data; (b) time-history of representation error; (c) scatter plot of original data vs. reconstructed
data; (d) power spectrum density of original data and reconstructed data
5.3.2 Results and Discussion
Figures 5.2 (a)-(d) show an example of the reconstructed data results for the roof
acceleration response at DP 1. We chose 50 70 as the size of the dictionary matrix D
and 3 as T0. We performed similar analyses for all the data. The results we have
presented, however, clearly reflect the overall performance of the algorithm. Figure 5.2
(a) shows that the time-history of the reconstructed data is a good approximation of the
original data. Similarly, Figure 5.2 (b) shows that their representation error, which is the
absolute value of the difference between the original data and the reconstructed data, is
very small. Figure 5.2 (c) shows the scatter plot of the original data and the reconstructed
CHAPTER 5. Application of a Sparse Representation Method Using K-SVD Algorithm to Data Compression of Experimental Ambient Vibration Data for SHM
135
data and quantifies the reconstruction performance using the correlation coefficient (ρ)
and the coefficient of determination (R2) values. The ρ value represents the linear
dependence between two sets of data, and the R2 value represents the proportion of
variability in the data recovered from the sparse representation (Rao, 1973). The R2 value
is also equivalent to the mean square error normalized by the mean square value of the
original signal and subtracted from one. Table 5.1 summarizes the coefficient of
determination (R2) values for all the measurements at all the DPs. We can observe that
lower floors have smaller R2 values than upper floors and severe DPs have smaller R2
values than moderate DPs. A possible explanation for this observation is that the
responses of upper floors tend to be smoother because the structure acts like a filter to the
input excitation. Therefore, we can represent the smoother responses of the upper floor
with a small number of bases more accurately than the lower floor responses, which are
noisier. In addition, severe damage to the structure introduces non-linear behavior, which
makes the sparse representation difficult. Figure 5.2 (d) is the power spectral density of
the original and the reconstructed data. Similarly, Figure 5.3 (a)-(d) show the
reconstructed data results for the ground acceleration at DP 9. As we can expect from the
low R2 value for this case, shown in Table 5.1, the magnitude of the representation error
in Figure 5.3 (b) is much larger than that in Figure 5.2 (b).
Table 5.1: The coefficient of determination (R2) values DP 1 DP 2 DP 3 DP 4 DP 5 DP 6 DP 7 DP 8 DP 9
Ground 0.772 0.911 0.547 0.437 0.510 0.432 0.422 0.515 0.430 Floor 2 0.903 0.955 0.841 0.787 0.742 0.732 0.757 0.737 0.690 Floor 3 0.926 0.962 0.886 0.840 0.809 0.813 0.765 0.741 0.700 Floor 4 0.954 0.975 0.917 0.899 0.870 0.879 0.844 0.805 0.786
Roof 0.979 0.991 0.967 0.946 0.948 0.946 0.929 0.905 0.878
CHAPTER 5. Application of a Sparse Representation Method Using K-SVD Algorithm to Data Compression of Experimental Ambient Vibration Data for SHM
136
0 1 2 3 4 5
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Time (s)
Acc
eler
atio
n (g
)
(a)
Original dataReconstructed data
0 1 2 3 4 50
0.02
0.04
0.06
0.08
0.1
Time (s)
Rep
rese
ntat
ion
erro
r (g
)
(b)
-0.2 -0.1 0 0.1 0.2
-0.1
-0.05
0
0.05
0.1
0.15
Original data (g)
Rec
onst
ruct
ed d
ata
(g)
= 0.65606R2 = 0.43033
(c)
0 10 20 30 40 500
0.01
0.02
0.03
Frequency (Hz)
PS
D
(d)
Original dataReconstructed data
Figure 5.3: The results of data reconstruction using the K-SVD algorithm for the ground acceleration at DP 9: (a) time-histories of original data and reconstructed data; (b) time-history of representation error; (c) scatter plot of original data vs. reconstructed data; (d)
power spectrum density of original data and reconstructed data
In order to verify that the reconstructed data preserve the structural information, we
applied modal analysis to both the original and the reconstructed data and compared the
natural frequencies extracted from them. Figure 5.4 shows that the five natural
frequencies computed from the original data are very similar to those from the
reconstructed data for DP 2. Although we need to further test the performance of the
reconstructed data in order to validate that the reconstructed data can replace the original
data, the presented results show an encouraging potential for this data compression
algorithm.
CHAPTER 5. Application of a Sparse Representation Method Using K-SVD Algorithm to Data Compression of Experimental Ambient Vibration Data for SHM
137
0 1 2 3 4 5 60
5
10
15
20
25
30
35
Nat
ural
fre
quen
cy (
Hz)
Mode #
Original dataReconstructed data
Figure 5.4: Modal analysis results for the original data and the reconstructed data at DP 2
Figure 5.5 shows the normalized root-mean-square error (RMSE) between the original
data and the reconstructed data for different compression ratio (n
T02 ) using the roof
acceleration response at DP 1. The error was computed from the RMSE normalized by
the root-mean-square value of the original data. The data were compressed using a series
of T0s varying from 1 to 25, which correspond to the compression ratios from 0.04 to 1.
We can clearly observe the trade-off between the error and the compression ratio from
the decrease in the normalized error with the increase in the compression ratio. For the
error of 5% and 10%, the compression ratio is about 40% and 20%, respectively.
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Compression ratio
Nor
mal
ized
RM
SE
Figure 5.5: Normalized RMSE versus compression ratio for the roof acceleration
response at DP 1
CHAPTER 5. Application of a Sparse Representation Method Using K-SVD Algorithm to Data Compression of Experimental Ambient Vibration Data for SHM
138
5.4 Conclusions
The data compression method that this chapter introduces using the K-SVD algorithm has
proven that it can efficiently compress the structural ambient vibration data and restore
them with small errors, and is therefore promising for the application to wireless sensor
networks for structural health monitoring (SHM) purposes. The K-SVD algorithm
iterates the process of designing and updating an over-complete set of prototype signal-
atoms, referred to as a dictionary, and then sparsely represents the data with respect to the
current dictionary. For sparse representation, we use the orthogonal matching pursuit
(OMP) algorithm because of its simplicity and efficiency. To update the dictionary, the
singular value decomposition (SVD) is applied for each atom of the dictionary
sequentially. The sparse representation coefficients are updated with the dictionary in
order to accelerate the convergence. These two steps are repeated until the representation
error reaches a pre-determined threshold. After all the data are represented sparsely, the
wireless sensors that monitor structures can transmit the dictionary and the sparse
representation coefficients to the central computer for restoration of the data. The K-SVD
algorithm enables us to reduce the size of the data to transmit wirelessly and in turn,
reduce the battery power consumption. We tested this algorithm using the experimental
ambient vibration data collected from the shake table test of a four-story steel special
moment-resisting frame. The results show that the algorithm can compress the
acceleration data efficiently and restore them with small errors. These results are
encouraging with respect to future uses of the proposed data compression method for
SHM purposes. Further analyses to validate the quality of the reconstructed data, to
investigate the computational complexity of the method in order to evaluate the
feasibility of its implementation in wireless sensors, and to use the sparse representation
as the damage sensitive feature for structural damage diagnosis algorithm are in progress.
139
Chapter 6
Summary, Conclusions, and Future Work
The development of structural health monitoring systems, which allows automated
monitoring of structural conditions in an efficient and reliable way, can significantly
enhance the sustainability of structures by reducing maintenance costs and preventing
catastrophic failure. This research area requires a multi-disciplinary approach that
encompasses structural engineering, sensor technology, wireless communication, signal
processing, and statistical analysis. The focus of this dissertation is the development of
vibration-based structural damage diagnosis algorithms using statistical pattern
recognition methods. The vibration-based algorithms diagnose structural damage on the
basis of the premise that structural damage will change its dynamic characteristics, which
in turn is reflect to the vibration response of the structure. Although structural health
monitoring methods have existed for several decades, statistical pattern recognition
techniques have been applied in this field only in the past decade. This approach is
receiving increasing recognition, particularly for its computational efficiency, which is
required when embedding such algorithms in wireless sensing units. These algorithms
can use either stationary ambient vibration responses before and after the damage or non-
stationary strong motion responses. Chapter 2 introduces time-series based algorithms
CHAPTER 6. Summary, Conclusions, and Future Work
140
that utilize ambient vibration data, and Chapter 3 and 4 introduce wavelet-based
algorithms for non-stationary earthquake response data. Finally Chapter 5 presents a data
compression method using a sparse representation algorithm called K-SVD, which is
useful for reducing the power consumption of wireless sensing units.
6.1 Summary and conclusions
Chapter 2 introduces three time-series based damage diagnosis algorithms that utilize
ambient vibration responses of a structure, both acceleration and strain, and their
applications to the experimental data obtained from the benchmark structure of the
National Center for Research on Earthquake Engineering (NCREE) in Taipei, Taiwan.
These algorithms use autoregressive (AR) models and various classification methods,
such as t-statistics, Gaussian mixture models, and information criteria. In previous
research, such algorithms have been validated by simulated and experimental data, but
the data from systematic and well-controlled experiments are very limited. Recognizing
the need for validation, additional experimental data from a series of laboratory tests in
Taipei, Taiwan, are acquired, and the algorithms are validated using those data sets. In
this chapter, strain data are also analyzed in order to identify local damage. Although
most prior research has focused on acceleration data, which reflects the global response
of a structure, the strain data results suggest that strain can provide more localized
information about damage.
In the first algorithm, a damage sensitive feature (DSF) is defined as a function of the
first three AR coefficients for the acceleration, and the first AR coefficient is used for the
strain data. DSFs, which are extracted from structural response data, contain information
about the damage state of a structure. Differences in the mean values of the DSF before
and after damage indicate that there is damage in the structure, and the t-test is used to
evaluate the statistical significance of that difference. In addition, a damage measure DM
CHAPTER 6. Summary, Conclusions, and Future Work
141
that is defined from the mean and variances of the DSFs is introduced, and we found that
the DM can be directly correlated to the amount of damage in this simple application. In
the second algorithm, a Gaussian mixture model (GMM) is used to characterize the
feature vector. Damage diagnosis is achieved by determining the distance between the
mixtures. To quantify damage extent, various distance measures are used including the
Mahalanobis distance, which is defined as the Euclidean distance between the mixtures
weighted with respect to the inverse covariance matrix. The third algorithm uses the first
three AR coefficients as a feature vector and detects damage by identifying the number of
clusters in the mixture of feature vectors using various information criteria. Four
information criteria, including Akaike Information Criteria (AIC), AIC3, minimum
description length (MDL), and Olivier et al.’s φβ criterion, are investigated for identifying
the optimum number of clusters in the mixture of feature vectors from various damage
states. The mixture is modeled as a multivariate Gaussian mixture model with k clusters,
and then, the information criteria are applied to determine k.
The results from the first algorithm show that the DSFacc,2 can be used for damage
detection; DSFacc,1 and/or DM can be used for damage extent, and DSFstr can be used for
damage localization. The results from the second algorithm show that the Mahalanobis
distances for acceleration data and strain data can detect damage for 100 gal and 50 gal
peak acceleration excitation, but not for 60 gal peak acceleration excitation. It is likely
that unidirectional random excitation with the peak acceleration of 60 gal is not strong
enough for us to detect the damage because the noise level of the accelerometers used to
measure structural response is of the same order as the root-mean-square of the
measurements. In addition, the Mahalanobis distances for acceleration data can be used to
localize damage, while the mean values of the distance measures of the strain data appear
to be well correlated to damage extent. The results of the third algorithm shows that
Olivier et al.’s φβ criterion works noticeably better than other similar information criteria
in identifying optimal number of clusters for all three sets of data. However, identifying a
suitable β parameter is important for the performance of the φβ criterion. We found that
CHAPTER 6. Summary, Conclusions, and Future Work
142
taking an average of the upper and lower bound is a good starting point for β, and this
method works well in identifying the number of clusters.
Chapter 3 introduces three DSFs using wavelet analysis for acceleration responses to
non-stationary earthquake motions, shows the relationship between these wavelet
energies and structural parameters that are important for damage characterization, and
validates the performance of these DSFs using experimental data. Earlier statistical
pattern recognition methods were primarily developed for stationary ambient vibration
responses. Structural responses to an earthquake, however, are evolutionary in nature,
and thus, previously used methods, such as the auto-regressive model, do not apply
because they assume a stationary response. Hence, several wavelet-based damage
sensitive features are introduced to capture the time-varying characteristic of structural
responses to earthquakes. Wavelet analysis is appropriate to model the non-stationary
earthquake responses because the wavelet transform represents data as a weighted sum of
wavelets which are short duration waves, and thus localized in both time and frequency.
While there has been research on using wavelets to analyze non-stationary signals, this
work is the first to define damage sensitive features in order to design damage diagnosis
algorithms and to validate their performance using experimental data. For this purpose,
the two data sets collected from the following shake table tests are used: 30% scaled
reinforced concrete bridge column tests in Reno, Nevada, and 1:8 scale model of a four-
story steel special moment-resisting frame tests at the State University of New York at
Buffalo. Further sensitivities of the DSFs were evaluated by subjecting the analytical
model of the four-story frame to 40 ground motions. A distinct advantage of my damage
sensitive features is that they enable damage diagnosis to be directly performed using
strong motion data, eliminating the need to take additional ambient vibration
measurements after an earthquake.
To develop the DSFs, the continuous wavelet transform is applied to the acceleration
response of the structure during the strong ground motion, and the wavelet energies at a
particular scale and at a particular time are defined based on the wavelet coefficients.
CHAPTER 6. Summary, Conclusions, and Future Work
143
Then, the three DSFs are developed as functions of these wavelet energies. DSF1
measures how the wavelet energy at the natural frequency of the undamaged structure
changes as the damage progresses in the structure. Based on the results from
experimental data, DSF1 decreases as the damage extent increases. This is because the
wavelet energy reduces at the scale corresponding to the first natural frequency of the
undamaged structure with the increasing levels of damage. DSF2 measures how much the
wavelet energy spread out in time and DSF3 measures how slowly it decays. The values
of DSF2 and DSF3 both increase as the damage extent increases.
The three DSFs have different sensitivities to various levels of damage according to the
results of the applications. The values of DSF3 change more for lower levels of damage
than for more severe levels of damage. On the other hand, the values of DSF1 and DSF2
show more changes for larger damages. Thus, DSF3 is more sensitive to smaller levels of
damage and DSF1 and DSF2 are more sensitive to larger levels of damage. Therefore, a
combination of these DSFs may be required for robust damage diagnosis. In addition to
the experimental results, the results of the sensitivity analyses using an analytical model
show that the DSFs are directly correlated to damage states defined through story drift
ratio limits, and the DSF values are robust to the input ground motions.
Chapter 4 introduces the framework to build fragility functions that define the
probabilistic relationship between these DSFs and the damage state of the structure. In
practical applications, it is unlikely to have reference values of these DSFs corresponding
to each damage state. Thus, a pre-defined system is needed to compare the values of
these DSFs when an earthquake occurs in order to estimate the damage state of the
structure. For this purpose, this framework combines concepts from Performance Based
Earthquake Engineering (PBEE) and Structural Health Monitoring (SHM) to compute
fragility functions as a damage assessment/prediction model using the wavelet-based
DSF1. The DSF-based fragility functions can be used as an alternative damage prediction
model in the field of PBEE and a probabilistic damage classification model for DSF in
the field of SHM. The proposed framework is based on information retrieved from an
CHAPTER 6. Summary, Conclusions, and Future Work
144
extensive set of structural responses extracted from an analytical model of a structure.
Incremental Dynamic Analysis is utilized using a set of earthquake ground motions. The
wavelet-based DSF is then computed based on the floor absolute acceleration histories,
and different damage states are defined based on the maximum story drift ratio that gives
DSF an engineering meaning. The probabilistic relationship between the damage states
and the DSF is computed as fragility functions, which provide the conditional probability
of the structure exceeding a particular damage state given the value of DSF. These
fragility functions can be computed for each story separately or for the entire structure to
assess the overall damage state.
The framework is validated using a set of numerically simulated data from a four-story
steel special moment-resisting frame subjected to various intensities of 40 different
ground motions. The results show that the DSF-based fragility functions can predict the
damage state of the frame and the median collapse capacity with less variance than the
fragility functions derived from alternate acceleration-based conventional measures, such
as spectral acceleration and peak roof acceleration. In addition, the fragility functions for
each story can provide the damage location by story while the global fragility functions
can assess the overall structural damage state with smaller error. Thus, the global fragility
functions would be suitable for a preliminary assessment of a structure immediately after
an earthquake while the fragility functions for each story would be useful for more
detailed analysis of damage.
The fragility functions are computed using a particular wavelet-based DSF in the study;
however, the framework can potentially be used with any valid DSF that can reliably
estimate the damage state of a structure. Further verification and testing of its damage
assessment capabilities need to be performed as additional data for different types of
structures become available, and the general form of the fragility functions for a group of
similar types of structures can be explored. It is also necessary to investigate the
feasibility of implementing this damage classification method using the DSF-based
fragility functions on wireless structural health monitoring system. The DSF-based
CHAPTER 6. Summary, Conclusions, and Future Work
145
damage assessment framework presented in this chapter, however, serves to introduce the
method and as a proof of concept with verification based on currently available
laboratory test data.
Chapter 5 introduces the data compression method that uses the K-SVD algorithm and
shows that it can efficiently compress the structural ambient vibration data and restore
them with small errors, and is therefore promising for the application to wireless sensor
networks for structural health monitoring purposes. This method will reduce the amount
of data to transmit and enable the entire transmission of response data to a server
computer, and as a result, more sophisticated analysis of data will be possible during the
use of wireless sensing units. Such analysis is particularly difficult with wireless sensing
units because the radios used in these units have very low data transmission rates, and
sending large amounts of data has thus been difficult due to excessive power
consumption.
The K-SVD algorithm iterates the process of designing and updating an over-complete
set of prototype signal-atoms, referred to as a dictionary, and then sparsely represents the
data with respect to the current dictionary. For sparse representation, we use the
orthogonal matching pursuit (OMP) algorithm because of its simplicity and efficiency.
To update the dictionary, the singular value decomposition (SVD) is applied for each
atom of the dictionary sequentially. The sparse representation coefficients are updated
with the dictionary in order to accelerate the convergence. These two steps are repeated
until the representation error reaches a pre-determined threshold. After all the data are
represented sparsely, the wireless sensors that monitor structures can transmit the
dictionary and the sparse representation coefficients to the central computer for
restoration of the data.
We tested this algorithm using the experimental ambient vibration data collected from the
shake table test of a four-story steel moment resisting frame. The results show that the
algorithm can compress the acceleration data efficiently and restore them with small
errors. These results are encouraging with respect to future uses of the proposed data
CHAPTER 6. Summary, Conclusions, and Future Work
146
compression method for structural health monitoring purposes. Further analyses to
validate the quality of the reconstructed data, to investigate the computational complexity
of the method in order to evaluate the feasibility of its implementation in wireless
sensors, and to use the sparse representation as the damage sensitive feature for structural
damage diagnosis algorithm are in progress.
The main contributions of this research are as follows:
Various damage diagnosis algorithms are validated using experimental data.
Damage sensitive features are developed for non-stationary strong motion data.
Methodology from the performance-based earthquake engineering is applied to
damage classification for structural health monitoring, and vice versa.
Sparse representation algorithm is applied for data compression of structural
vibration responses.
6.2 Future work
The following research directions will be explored in the future to further enhance the
developed damage diagnosis algorithms:
Implement the algorithms into wireless sensing units and validate their performance.
For this purpose, discrete wavelet transform or wavelet packet transform can be
applied for computational efficiency and investigate the consequences.
Develop fragility functions for structures of various types, sizes, and materials and
explore methods to develop generalized fragility functions.
CHAPTER 6. Summary, Conclusions, and Future Work
147
Consider the effects of various environments, live loads, and non-structural
components.
Apply data fusion and other statistical methods to combine different types of
measurements and spatially distributed measurements.
Identify different types of damage, such as fatigue damage diagnosis and prognosis,
fracture, and pre-stress/post-tension tendon failure, by combining physical models
and statistical analysis.
Develop multi-layer smart monitoring systems that utilize new types of sensing, such
as mobile/robotic sensing and remote sensing, and use local sensing and diagnosis
combined with global diagnosis and risk analysis.
148
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