damage diagnosis algorithms using statistical pattern ...wg007jn8560/tr 180...3 wavelet-based damage...

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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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 2.7: Results of damage detection using DSFacc,1 for 50 gal bi-directional



xiii






















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


for 100 gal uni-directional random excitation for undamaged and damaged

cases .................................................................................................................44

xiv


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


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 3.17: 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

xix

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


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


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.


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


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.


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


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.


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


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


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.


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.


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


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


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


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


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.


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


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.


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


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.


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:


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)


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


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,


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


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.


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,


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


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


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.


32


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


33


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


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]



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]


35


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


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]



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]


36



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]



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]



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]


37


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]



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]



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]


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


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


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 .


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


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)


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.


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.


has the maximum value at the damage location.


is higher for the higher floors.


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


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


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


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).


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


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


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


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


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


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


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


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


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.


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):


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


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.


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


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


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.


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.


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


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.


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


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


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.


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


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


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


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


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.


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


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.


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.


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


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)


82

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.


83

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


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).


85

Ground 2 3 4 Roof0

5

10

15

20

25

DS

F2

Floor

SLEDLEMCECLE


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


86

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


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


87

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


88

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


89

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


90

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


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.


92

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


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


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


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


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.


99

Figure 4.1: Summary of the proposed framework


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


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


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


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

qq

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


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.


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.


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


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,


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.


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


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.


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


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


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


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


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.


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


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


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


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


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


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


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.


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


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


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


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.


129

Figure 5.1: Summary of the data compression using the K-SVD algorithm


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)


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


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)


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.


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)


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


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


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)


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)


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.


137

0 1 2 3 4 5 60

5

10

15

20

25

30

35

Nat

ural

fre

quen

cy (

Hz)

Mode #


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


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


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


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.


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


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


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


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.


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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