Statistical structural health monitoring methodologies and applications

2.3.1 Structural Health Monitoring Using Autoregressive Coefficients Based Hotelling’s T2 Control Chart: without 2.3.2 Structural Health Monitoring Using Autoregressive Coefficients Bas

STATISTICAL STRUCTURAL HEALTH MONITORING:

METHODOLOGIES AND APPLICATIONS

WANG ZENGRONG

NATIONAL UNIVERSITY OF SINGAPORE

2008

STATISTICAL STRUCTURAL HEALTH MONITORING:

METHODOLOGIES AND APPLICATIONS

WANG ZENGRONG

(B.Eng., XJTU)

A THESIS SUBMITTED

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF CIVIL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2008

To my parents

ACKNOWLEDGEMENTS

I would like to express my cordial gratitude to my supervisor, Associate Professor Ong Khim Chye Gary Throughout my graduate study, I have been greatly benefiting from his informative guidance and invaluable suggestions, without which this research would not have been possible Compelling justifications for my gratitude also come from his constant encouragement and unscheduled help, with which this research goes towards a successful completion It is his thoroughness, professionalism and commitment that make my graduate study a rewarding experience

I would like to appreciate Associate Professor Mohamed Maalej for his edifying instructions during the early days of my graduate study I would also like to appreciate Professor Quek Ser Tong for his constructive comments and for giving his insights in Hilbert-Huang transform

Many thanks are also due to my fellow graduate students, both inside and outside the Department of Civil Engineering, for the good time spent together

Finally, the financial support of the Research Scholarship and the state-of-the-art research facilities provided by the University are acknowledged

TABLE OF CONTENTS

Dedication ii Acknowledgements iii

Summary vii

CHAPTER ONE

INTRODUCTION 1

1.2.2 Statistical Techniques for Vibration Characteristics Based

SHM 3

CHAPTER TWO

MULTIVARIATE EXTENSION OF THE STRUCTURAL HEALTH

MONITORING SCHEME USING AUTOREGRESSIVE

COEFFICIENTS BASED STATISTICAL PROCESS CONTROL

2.2.2 Monitoring of Vibration Response Data Characteristics Based

on Multivariate Statistical Process Control 112.2.3 Effects of the Autocorrelation in the Characteristics Data 15

2.3.1 Structural Health Monitoring Using Autoregressive

Coefficients Based Hotelling’s T2 Control Chart: without

2.3.2 Structural Health Monitoring Using Autoregressive

Coefficients Based Hotelling’s T2 Control Chart: with

CHAPTER THREE

IMPROVING STRUCTURAL DAMAGE DETECTION SENSITIVITY

USING AUTOREGRESSIVE-MODEL-INCORPORATING

MULTIVARIATE EXPONENTIALLY WEIGHTED MOVING

3.2 Autoregressive-Model-Incorporating Multivariate Exponentially

Weighted Moving Average Control Chart for Structural Damage

4.2 Representation and Monitoring of Vibration Response Data: A

4.3.1 Numerical Example of a Five-Story Shear Frame 92

4.3.3 Experimental Example of the I-40 Bridge Benchmark 97

CHAPTER FIVE

A NONPARAMETRIC STATISTICAL FRAMEWORK FOR

5.2 Formulation of the Nonparametric Statistical Structural Health

DATA OF THE DAMAGE INDICATOR PROFILES FOR THE

HYPERBOLIC PARABOLOID ROOF SHELL CONSIDERED IN

APPENDIX B

SUMMARY

This research is concerned with statistical structural health monitoring (SHM) First, an innovative SHM scheme based on time series analysis and multivariate statistical process control (MSPC) techniques is presented The scheme consists of two major procedures, viz vibration response data representation and characteristics monitoring First, a series of autoregressive (AR) models is fitted to the response time histories of a structure to be monitored Representing the health condition of the structure, the coefficients of these AR models are extracted to form a set of multivariate data known

as vibration response data characteristics Hotelling’s T2 control chart is then applied to

monitor these characteristics obtained As an MSPC tool, Hotelling’s T2 control chart has the capacity of simultaneously monitoring the multivariate characteristics data without having to neglect the inherent relation between the components of the data The efficacy of the proposed SHM scheme is demonstrated by numerically simulated acceleration time histories based on a progressively damaged reinforced concrete (RC) frame, either with or without addressing the autocorrelation in the characteristics data

The results are compared with those obtained by using univariate Shewhart X control

chart to show the advantages of the proposed scheme in terms of the sensitivity of the defined damage indicator with respect to damage severity A parametric study is also included to investigate the effects of the number of data points used for AR model fitting, the order of AR models and the number and locations of sensors on the proposed scheme, as well as to further illustrate its potential as a promising SHM approach

To further improve structural damage detection sensitivity, a scheme using autoregressive-model-incorporating multivariate exponentially weighted moving average (MEWMA) control chart is presented This scheme comprises the procedures based on the undamaged or reference state of the structure being monitored and those based on its damaged or current state In the procedures based on the reference state, sets of multivariate data are formulated by a series of AR model fitting, and these data are then subjected to MEWMA control chart analysis to establish a benchmark damage indicator The damage indicator obtained in the procedures based on the current state is compared with the benchmark for the purpose of structural damage detection The autocorrelation in the multivariate data is addressed, and special procedures to allow

for the uncertainty involved in process parameter estimation as well as those for control limit determination are proposed for structural damage detection application A numerically simulated case study is used to verify the efficacy of the proposed scheme and to show its advantages A parametric study is also included to study the effects of some parameters and to demonstrate the robustness of the scheme against parameter selection

The issue of structural damage detection is then addressed through an innovative multivariate statistical approach By invoking principal component analysis (PCA), the vibration responses acquired from the structure being monitored are represented by the multivariate data of the sample principal component coefficients (PCCs) A damage indicator is then defined based on a MEWMA control chart analysis formulation, involving special procedures to allow for the effects of the estimated parameters and to determine the upper control limit (UCL) in the control chart analysis for structural damage detection applications Also, a data shuffling procedure is proposed to address the autocorrelation probably present in the obtained sample PCCs This multivariate statistical structural damage detection scheme can be applied to either the time domain responses or the frequency domain responses The efficacy and advantages of the scheme is demonstrated by the numerical examples of a five-story shear frame and a shear wall as well as the experimental example of the I-40 Bridge benchmark

Finally, a nonparametric statistical framework for SHM is presented Vibration response data are first represented by the coefficients of a series of fitted AR models in the time domain or by the averages of binned power spectral density (PSD) estimates

in the frequency domain Three types of statistical hypotheses are then formulated and tested by nonparametric techniques to monitor these characteristics Specifically, two-sample Kolmogorov-Smirnov test, Mann-Whitney test and Mood test are used in this

study For each type of hypothesis formulation, a function of the resulting P-values is

used to define a damage indicator profile (DIP) whereby damage locations are identified The highlight of this framework is that, due to its nonparametric nature, it does not require a particular functional form for the underlying population of an extracted vibration response data characteristic Two numerically simulated case studies, i.e., a 20-degree-of-freedom system and a hyperbolic paraboloid roof shell, demonstrate the efficacy of the proposed nonparametric SHM framework Multiple damage locations are also considered in the case studies

LIST OF TABLES

Table 2.1 Total number of outliers and average sensitivity based on

multivariate Hotelling’s T2 control chart and those based on

univariate Shewhart X control chart (without addressing the

autocorrelation) 26

Table 2.2 Total number of outliers and average sensitivity based on

multivariate Hotelling’s T2 control chart and those based on

univariate Shewhart X control chart (with addressing the

autocorrelation) 27

Table 2.3 Total number of outliers and average sensitivity based on

Hotelling’s T2 control chart for different numbers of data points

used for AR model fitting (with addressing the autocorrelation) 28

Table 2.4 Total number of outliers and average sensitivity based on

Hotelling’s T2 control chart for different orders of AR models

Table 2.5 Total number of outliers and average sensitivity based on

Hotelling’s T2 control chart for different numbers of sensors

(with addressing the autocorrelation) 30

Table 2.6 Total number of outliers and average sensitivity based on

Hotelling’s T2 control chart for different locations of sensors

(with addressing the autocorrelation) 31

Table 3.1 Values of the damage indicator in the false alarm test 62

Table 3.2 Values of the damage indicator based on MEWMA control chart

and those of the damage indicators based on other control charts 63

Table 3.3 Values of the damage indicator based on MEWMA control chart

Table 3.4 Values of the damage indicator based on MEWMA control chart

for different numbers of data points used for AR model fitting 65

Table 3.5 Values of the damage indicator based on MEWMA control chart

Table 4.1 Results for the numerical example of a five-story shear frame 100

Table 4.2 Results for the numerical example of a shear wall 101

Table 4.3 Results for the experimental example of the I-40 Bridge

benchmark 102

Table 5.1 DIPs based on the fitted AR coefficients for the 20-DOF system 130

Table 5.2 DIPs based on the PSD estimates for the 20-DOF system 131

Table A.1 DIPs based on the fitted AR coefficients for the hyperbolic

Table A.2 DIPs based on the PSD estimates for the hyperbolic paraboloid

LIST OF FIGURES

Fig 2.1 Illustration of the two-bay-and-two-story RC frame in (a) the

reference state and (b) the current state, and (c) the

corresponding finite element discretization (not drawn to scale) 32

Fig 2.2 Examples of the Hotelling’s T2 control charts obtained without

addressing the autocorrelation: the Hotelling’s T2 control charts

for the 3rd AR coefficient based on Acceleration Data Set 1 at

(a) Damage State 0 and (b) Damage State 3, respectively 34

Fig 2.3 Comparison of the SHM scheme based on multivariate

Hotelling’s T2 control chart and that based on univariate

Shewhart X control chart (without addressing the autocorrelation): (a), (c), (e) total numbers of outliers

corresponding to different damage states for Acceleration Data

Set 1 to 3, respectively, and (b), (d), (f) sensitivities of the total

number of outliers for Acceleration Data Set 1 to 3, respectively 35

Fig 2.4 Sample ACFs of the coefficients of the AR models fitted to the

acceleration time histories recorded at (a) Node 3, (b) 6, (c) 23

and (d) 26 in the x-direction for Acceleration Data Set 1 at

Damage State 0 (without addressing the autocorrelation) 38

Fig 2.5 Examples of the Hotelling’s T2 control charts obtained with

addressing the autocorrelation: the Hotelling’s T2 control charts

for the 5th AR coefficient based on Acceleration Data Set 5 at

(a) Damage State 0 and (b) Damage State 3, respectively 39

Fig 2.6 Comparison of the SHM scheme based on multivariate

Hotelling’s T2 control chart and that based on univariate

Shewhart X control chart (with addressing the autocorrelation):

(a), (c), (e) total numbers of outliers corresponding to different

damage states for Acceleration Data Set 4 to 6, respectively, and

(b), (d), (f) sensitivities of the total number of outliers for

Acceleration Data Set 4 to 6, respectively 40

Fig 2.7 Sample ACFs of the coefficients of the AR models fitted to the

acceleration time histories recorded at (a) Node 3, (b) 6, (c) 23

and (d) 26 in the x-direction for Acceleration Data Set 5 at

Damage State 0 (with addressing the autocorrelation) 43

Fig 2.8 Relative average sensitivities based on modal frequencies, MAC,

COMAC and the proposed damage indicator, respectively 44

Fig 3.1 Adequacy check: (a) normal probability plot and (b) sample

ACF plot of the selected first coefficients of the AR models

fitted to the acceleration time histories recorded at Node 23 in

Fig 3.2 Examples of the MEWMA control charts: the MEWMA control

charts for the selected 5th AR coefficients based on (a) the first

set of acceleration time histories in the reference state in

benchmark establishment, (b) the first set of acceleration time

histories in the false alarm test, (c) the second set of acceleration

time histories in the false alarm test, (d) Damage State 1, (e)

Damage State 2, and (f) Damage State 3 (Data Group 1) 68

Fig 3.3 Comparison of the structural damage detection scheme based on

MEWMA control charts and the schemes based on other control

charts: (a) Data Group 1, (b) Data Group 2 and (c) Data Group 3 71

Fig 3.4 Effects of the AR model order: (a) Data Group 1, (b) Data Group

Fig 3.5 Effects of the number of data points used for AR model fitting:

(a) Data Group 1, (b) Data Group 2 and (c) Data Group 3 77

Fig 3.6 Effects of the smoothing parameter λ : (a) Data Group 1, (b)

Fig 4.2 Data checking: normal probability plots of (a) time domain

based data and (b) frequency domain based data; sample ACF

plots of time domain based data (c) before and (d) after data

shuffling; and sample ACF plots of frequency domain based data

(e) before and (f) after data shuffling 104

Fig 4.3 Examples of the resulting MEWMA control charts using time

domain based data in the cases of (a) PSF, (b) PSF0 and (c)

PSF2, and those using frequency domain based data in the cases

Fig 4.4 Finite element modeling of the shear wall 110

Fig 4.5 Examples of the resulting MEWMA control charts in the cases

of (a) SW and (b) SW2 for Combination 1, and those in the

cases of (c) SW and (d) SW2 for Combination 3 111

Fig 4.6 Schematic representation of the test configuration of the I-40

Fig 4.7 Examples of the resulting MEWMA control charts in the cases

of (a) I40B and (b) I40B1 for Combination 1, and those in the

cases of (c) I40B and (d) I40B1 for Combination 3 114

Fig 5.2 Examples of the vibration response data characteristics at DOF

10 and DOF 19: (a) 1st fitted AR coefficients (reference state);

(b) averages of the PSD estimates in the 1st frequency bin

(reference state); (c) 1st fitted AR coefficients (TDS2); and (d)

averages of the PSD estimates in the 1st frequency bin (TDS2) 133

Fig 5.3 DIPs for TDS1 based on (a) AR coefficients and Formulation

III, and (b) PSD estimates and Formulation III; those for TDS2

based on (c) AR coefficients and Formulation II, and (d) PSD

estimates and Formulation III; and those for TDS3 based on (e)

AR coefficients and Formulation I, and (f) PSD estimates and

Fig 5.4 A hyperbolic paraboloid roof shell: (a) Three-dimensional view;

Fig 5.5 Examples of the vibration response data characteristics at Point

C and Point I: (a) 1st fitted AR coefficients (reference state); (b)

averages of the PSD estimates in the 10th frequency bin

(reference state); (c) 1st fitted AR coefficients (HPRS2); and (d)

averages of the PSD estimates in the 10th frequency bin

Fig 5.6 For HPRS1, DIPs based on Formulation I using (a) AR

coefficients and (b) PSD estimates; those based on Formulation

II using (c) AR coefficients and (d) PSD estimates; and those

based on Formulation III using (e) AR coefficients and (f) PSD

Fig 5.7 For HPRS2, DIPs based on Formulation I using (a) AR

coefficients and (b) PSD estimates; those based on Formulation

II using (c) AR coefficients and (d) PSD estimates; and those

based on Formulation III using (e) AR coefficients and (f) PSD

Fig 5.8 For HPRS3, DIPs based on Formulation I using (a) AR

coefficients and (b) PSD estimates; those based on Formulation

II using (c) AR coefficients and (d) PSD estimates; and those

based on Formulation III using (e) AR coefficients and (f) PSD

LIST OF ABBREVIATIONS

The following abbreviations are used in this thesis:

ACF = autocorrelation function;

AIC = Akaike’s information criterion;

AR = autoregressive;

ARMA = autoregressive moving average;

ARL = average run length;

CDF = cumulative distribution function;

COMAC = coordinate modal assurance criterion;

DFT = discrete Fourier transform;

DIP = damage indicator profile;

DOF = degree of freedom;

DSF = damage sensitive feature;

EWMA = exponentially weighted moving average;

FFT = fast Fourier transform;

FRF = frequency response function;

GMM = Gaussian mixture model;

MAC = modal assurance criterion;

MEWMA = multivariate exponentially weighted moving average;

MSPC = multivariate statistical process control;

PCA = principal component analysis;

PCC = principal component coefficient;

PSD = power spectral density;

RC = reinforced concrete;

RMS = root mean square;

SNR = signal-to-noise ratio;

SHM = structural health monitoring;

SPC = statistical process control;

UCL = upper control limit;

VAR = vector autoregressive; and

VARMA = vector autoregressive moving average

d = number of the points falling outside the control limits of the

Hotelling’s T2 control chart for the ith AR coefficient at the lth

damage state;

, ,n Mrq Mr n 1

Fα − − + = upper 100α percentage point of the F distribution with n

numerator degrees of freedom and Mrq−Mr− + denominator n 1degrees of freedom;

M = number of data windows in each column of zkl;

m = number of data points in each column of zkl;

N = number of damaged states;

n = number of output DOFs;

p = order of the AR models;

q = subgroup size for the control chart analysis;

r = number of sets of response time histories recorded at each

measurement location at a damage state;

i

S = average of the sample covariance matrices for the ith AR

coefficient, and is calculated based on the subgroups taken from the undamaged, or statistically in-control state;

s = number of data points in each data window;

2

jil

T = for the jth subgroup, the statistic plotted on the Hotelling’s T2

control chart based on il;

i UCL = upper control limit for the ith AR coefficient;

kl

z = kth set of response time histories at the lth damage state;

ijkl

z = ith response value in the kth time history at the jth measurement

location for the lth damage state;

α = probability that the test statistic falls out of the control limits when

the process is in control;

i = average of the sample mean vectors for the ith AR coefficient, and

is calculated based on the subgroups taken from the undamaged,

or statistically in-control state;

il = ith coefficients of the AR models at the lth damage state;

jil = sample mean vector of the jth subgroup for the ith AR coefficient

at the lth damage state; and

jkil

φ = ith coefficient of the AR model fitted to the response time history

in the jth time window at the kth measurement location for the lth damage state

The following symbols are used in Chapter Three:

D = damage indicator;

0

D = benchmark damage indicator;

i

H = UCL of the MEWMA control chart for the ith AR coefficient in

the current state;

il

H = UCL of the MEWMA control chart for the ith AR coefficient

corresponding to the lth set of response time histories in the reference state;

( , + )( )

Ia ∞ ⋅ = indicator function of (a, +∞);

M = number of data windows in each column of z or kl z ; k

m = number of data points in each column of z or kl z ; k

N = number of sets of response time histories recorded by each sensor

in the reference state;

n = number of output DOFs;

p = order of the AR models;

R = parameter used in the sampling of the fitted AR coefficients;

r = number of response time histories in a response time history set

recorded by a sensor;

s = number of data points in each data window;

TOL = tolerance used in the reference-state calibration;

z = kth response time history matrix in the lth set of time histories in

the reference state;

ijk

z = ith data point in the kth response time history recorded by the jth

sensor in the current state;

ijkl

z = ith data point of the kth response time history in the lth set of time

histories recorded by the jth sensor in the reference state;

α = calibration parameter used to determine the UCLs in the reference

state;

λ = smoothing parameter of a MEWMA control chart;

i = the mean vector of the reference-state selected ith AR coefficients

ˆi = estimate of i;

ki

μ = mean of the selected ith coefficient of the AR model fitted to the

response time histories recorded by the kth sensor in the reference

i = matrix storing the ith fitted AR coefficients in the current state;

il = matrix storing the selected ith fitted AR coefficients for the lth set

of time histories in the reference state;

il = matrix storing the ith fitted AR coefficients for the lth set of time

histories in the reference state;

j i⋅ = vector storing the jth selected ith fitted AR coefficients in the

current state;

j il⋅ = vector storing the jth selected ith fitted AR coefficients for the lth

set of time histories in the reference state;

jkil

φ = selected ith coefficient of the AR model fitted to the data points in

the jth time window for the lth set of time histories recorded by

the kth sensor in the reference state; and

jkil

φ = ith coefficient of the AR model fitted to the data points in the jth

time window for the lth set of time histories recorded by the kth sensor in the reference state

The following symbols are used in Chapter Four:

H = UCL for the multivariate data of the selected and shuffled sample PCCs

for the ith principal component;

il

H = UCL for the multivariate data of the selected and shuffled sample PCCs

for the ith principal component based on the lth set of the vibration

response data obtained in the reference state;

il

H = lth set of the magnitudes of the FRFs corresponding to the ith excitation

and obtained in the reference state;

n′′ = number of selected PCCs for an individual principal component used to

formulate the damage indicator;

P = selected and shuffled sample PCCs for the ith principal component in

the current state;

il

P = lth set of selected and shuffled sample PCCs for the ith principal

component in the reference state;

W = points plotted on the MEWMA control chart constructed for the PCCs

for the ith principal component in the current state;

2

jil

W = points plotted on the lth MEWMA control chart constructed for the

PCCs for the ith principal component in the reference state;

Z = generic form of the time domain data and the frequency domain data in

the current state;

ˆi = estimates of the mean vector for the multivariate data of the selected and

shuffled sample PCCs for the ith principal components; and

A , A = acceleration r time histories in the current state, and those in

the reference state;

B = backward shift operator;

ca , i ra = acceleration time history at the ith output DOF in the i

current state, and that in the reference state;

g ⋅ , gII( )⋅ , gIII( )⋅ = predefined functions for DIPs based on Formulation I, II

and III, respectively;

0

H , H = null hypothesis and alternative hypothesis, respectively; 1

c

I , I = number of data points in each acceleration time history in r

the current state, and that in the reference state;

c

M , M = number of time windows for each acceleration time r

history in the current state, and that in the reference state;

c

M ′ , M ′ = number of time windows for each acceleration time r

history in the current state, and that in the reference state for AR model based formulation;

c

M ′′ , M ′′ = number of time windows for each acceleration time r

history in the current state, and that in the reference state for PSD estimate based formulation;

N = number of PSD-estimate data points in each frequency

bin;

n = number of output DOFs;

IP , jl IIP , jl IIIP = P-values of the hypothesis tests based on jl rz and kjl cz , kjl

respectively corresponding to Formulation I, II and III;

p′ = order of the AR models fitted to the data points in the time

cS , kj rS = estimated PSDs based on the data points in the kth time kj

window of the acceleration time history at the jth output DOF in the current state, and those in the reference state;

cS , ′kj rS = estimated PSDs corresponding to the frequency range of ′kj

interest based on the data points in the kth time window of the acceleration time history at the jth output DOF in the current state, and those in the reference state;

cSkj, rSkj = average of the PSD-estimate data points based on the data

points in the kth time window of the acceleration time history at the jth output DOF in the current state, and that

in the reference state;

cS kjl, rS kjl = average of the PSD-estimate data points in the lth

frequency bin based on the data points in the kth time window of the acceleration time history at the jth output DOF in the current state, and that in the reference state;

s′ = number of data points in each time window for AR model

based formulation;

s′′ = number of data points in each time window for PSD

estimate based formulation;

cZ , jl rZ = populations from which jl cz kjl and rz kjl are taken,

respectively;

cz , kjl rz = lth data point of the vibration response data characteristic kjl

based on the data points in the kth time window of the acceleration time history at the jth output DOF in the current state, and that in the reference state;

cεij, rεij = realizations of white noise processes with zero means in

the current state, and those in the reference state;

I, II, III = parameter vectors in gI( )⋅ , gII( )⋅ and gIII( )⋅ ,

respectively;

jl

θ = shift parameter in Formulation II hypotheses, or scale

factor in Formulation III hypotheses;

μ = mean of the underlying random process corresponding to

the acceleration data points in a time window;

( )

cΦkj ⋅ , rΦkj( )⋅ = function of B used to define AR models in the current

state, and that in the reference state; and

cφkjl, rφkjl = lth coefficient of the AR model fitted to the data points in

the kth time window of the acceleration time history at the

jth output DOF in the current state, and that in the

reference state

Specifically, for civil engineering applications, traditional local damage detection techniques such as acoustic methods, magnetic field methods, etc suffer from the

1

CHAPTER ONE

limitation that the damage vicinity of a structure needs to be located a priori, which is

often impractical because of the large size of monitored structures Another difficulty for local damage detection techniques lies in the fact that the critical regions of civil engineering structures are usually covered up by non-structural members or equipment

so that they are difficult, if not impossible, to reach Also, disturbance to the regular operations of the structures being monitored for vibration based damage detection schemes are less As a result, SHM based on vibration characteristics is found to be especially useful in the civil engineering community

1.2 Literature Review

1.2.1 Vibration Characteristics Based SHM

Over the years a large variety of SHM schemes have been proposed Salawu (1997) reviewed the damage detection methods using modal frequencies Methods based on mode shapes (e.g., Pandey et al 1991; Lam et al 1998), frequency response functions (FRFs) (e.g., Fanning and Carden 2003), flexibility (e.g., Pandey and Biswas 1994; Bernal 2002), as well as advanced signal processing techniques (e.g., Hou et al 2000; Quek et al 2003) were also developed Correspondingly, a range of structure types have been investigated, including frames (e.g., Johnson et al 2000; Dyke et al 2003; Johnson et al 2004), bridges (e.g., Farrar and Jauregui 1998a, b; Koh and Dyke 2007),

a full-scale retrofitted building (Nayeri et al 2007), etc Detailed reviews of the relevant literature were documented by Doebling et al (1996), Sohn et al (2003), and Carden and Fanning (2004)

Vibration characteristics based SHM can be implemented either in the frequency domain or in the time domain Modal parameter identification techniques play an important role in frequency domain SHM approaches, in which modal frequencies,

2

CHAPTER ONE

mode shapes or mode shape curvatures, etc are chosen as damage sensitive features (DSFs) (Salawu 1997; Doebling et al 1996; Sohn et al 2003) Although they were initiated decades ago, SHM schemes using frequency domain data have been attracting on-going research efforts Schemes were also constructed based on transmissibilities (Worden 1997; Worden et al 2000), frequency response functions (Schulz et al 1998; Lee and Shin 2002), etc With the development of signal analysis techniques (e.g Hilbert-Huang transform (Huang et al 1998; Huang et al 1999)), some recently proposed modal parameter identification and SHM methods have already been documented (Yang et al 2003; Yang et al 2004; Ong et al 2008) Besides, Kullaa (2003) used both univariate and multivariate control charts to statistically monitor the changes of modal parameters Yet, some frequency domain SHM methods, especially those using only modal frequencies, have relatively low sensitivity with respect to damage severity in several applications, requiring either high signal-to-noise ratio measurement data or at least moderate damage levels This has been demonstrated by Farrar et al (1994) and Ong et al (2006), among other researchers As an alternative to frequency domain SHM methods and as a time domain SHM approach, the coefficients of the autoregressive (AR) models fitted to the measured response time histories of a structure have recently been chosen as DSFs (Sohn et al 2000a; Nair et

al 2006) The rationale behind this choice is that variation in physical parameters (e.g., stiffness parameters) indicating damage presence manifests itself by variation in the coefficients of the fitted AR models

1.2.2 Statistical Techniques for Vibration Characteristics Based SHM

Recently, with the development in sophisticated and readily available hardware for data acquisition and analysis, and, more importantly, with the capability of addressing

3

CHAPTER ONE

the essentially involved uncertainty quantitatively, SHM using probabilistic and statistical techniques has been drawing significant interests in the civil engineering research community

Vanik et al (2000), Yuen et al (2004), and Ching and Beck (2004) proposed a Bayesian probabilistic framework for structural health monitoring based on the probabilistic model updating scheme developed by Beck and Katafygiotis (1998), and Katafygiotis and Beck (1998) Worden (1997) and Worden et al (2000) proposed a pattern recognition scheme based on transmissibility functions and squared Mahalanobis distance Nair et al (2006) proposed statistical damage detection and localization algorithm where time series models are fitted to formulate a DSF and damage localization indices The differences in the means of DSFs are checked by

invoking t-test Giraldo et al (2006) constructed a statistical SHM scheme whereby

varying environmental conditions can be allowed for Nair and Kiremidjian (2007) addressed the SHM issues by Gaussian mixture models (GMMs)

To quantitatively address the uncertainty involved in the measured response data,

Sohn et al (2000a) applied Shewhart X control chart, a statistical process control

(SPC) tool, to the AR coefficients and successfully detected the anomalies associated with the damaged structures The same SPC tool has also been used by Fugate et al (2001) to monitor the residuals defined as the differences between the AR model predicted response histories and the measured ones As a univariate SPC tool, however,

Shewhart X control chart has the limitation that it can only monitor one variable at a

time For civil engineering applications, it is not uncommon that multivariate data need

to be examined Indeed, with the rapid development of the hardware for data acquisition systems, simultaneously measuring dynamic responses at different locations of a structure has become quite affordable Although univariate control charts

4

CHAPTER ONE

could be applied independently to each component of the multivariate data, misleading results may be obtained in some cases due to failing to allow for the inherent relation among the components of the multivariate data (Montgomery 2005) In this regard, the potential of multivariate statistical process control (MSPC) techniques in SHM applications needs to be investigated

The effective implementation of the SPC based SHM schemes depends on the establishment of the representative characteristics of the structural states Also, for practical applications, parameters necessary for the construction of control charts are usually unknown and therefore have to be estimated based on the reference state of the monitored structure, or the statistically in-control process Poor estimation of these unknown parameters may distort the performance of the control charts (Jensen et al 2006; Zamba and Hawkins 2006) The intimation follows that less complicated data representation techniques may be more advantageous in terms of the computational efforts involved in the reference-state calibration For instance, vector autoregressive (VAR) models, though having the capacity of modeling the intrinsic relation between components of multivariate data, might be less preferable to AR models from a reference-state-calibration point of view Consequently, to make use of the information contained in the multivariate measurements, a feasible approach as reported in this study is to incorporate the principal component analysis (PCA), a dimension reduction technique, into the control chart based structural damage detection formulation (Wang and Ong 2007b) This is to arrive at a multivariate statistical approach to the damage detection and health monitoring of structures, and to broaden the applicability of the formerly developed formulation Flexibility of domain choice (i.e., time domain data

or frequency domain data) achieved by the proposed scheme is also investigated

5

CHAPTER ONE

1.3 Objective and Scope of Work

As reviewed in Section 1.2, compared with the large number of deterministic ones, the research work in developing probabilistic and statistical SHM is, in general, limited Accordingly, with the main objective as formulating statistical SHM schemes, the present research scope is

(a) to propose a damage indicator through multivariate extension to account for multiple response time histories and to achieve better damage detection sensitivity;

(b) to address the autocorrelation possibly present in the characteristics of the vibration response data;

(c) to improve the sensitivity of the proposed damage indicator with special procedures for determination of the additionally introduced parameters in reference-state calibration;

(d) to conduct parametric studies to investigate the effects of the parameters involved as well as to verify the robustness of the proposed damage indicators; (e) to provide a multivariate statistical damage detection scheme with vibration response data represented either in the time domain or in the frequency domain; (f) to represent vibration response data more efficiently by incorporating dimension reduction techniques; and

(g) to construct a nonparametric statistical SHM framework

1.4 Organization of Thesis

The thesis comprises six chapters as follows:

(a) Chapter One gives research background, reviews related literature, delineates the research objectives and scope, and describe the thesis organization;

6

CHAPTER TWO

2.1 Introduction

Sohn et al (1999; 2000b) and Fugate et al (2001) addressed the structural damage detection issues using SPC techniques Specifically, they applied the control chart analysis to the residual errors obtained from the fitted AR models and successfully detected the anomalies associated with structural damage The efficacy of the technique to damage detection for telecommunication masts was demonstrated by Fanning and Carden (2001) Also, Kulla (2003) monitored the identified modal parameters using various univariate and multivariate control charts In this study, with the coefficients of the AR models respectively fitted to the simultaneously measured acceleration time histories at different locations of a structure chosen as damage

sensitive features, a novel SHM scheme is proposed in which Hotelling’s T2 control chart, an MSPC tool, is applied to monitor the multivariate data of the fitted AR coefficients Also, as an extension of the SHM procedures using AR coefficients based univariate control chart (Sohn et al 2000a), the advantages of the proposed MSPC based method are discussed through the sensitivity study included herein

CHAPTER TWO

In the subsequent sections, detailed procedures of the proposed SHM method are presented first, followed by a special discussion of the autocorrelation effects and the corresponding approach to deal with the autocorrelation A numerically simulated example of a reinforced concrete (RC) frame subjected to lateral loads is then used to demonstrate the efficacy and advantages of the proposed method, and the effects of some parameters included in the proposed method are also studied

2.2 Vibration Response Data Representation and Characteristics Monitoring

The proposed SHM scheme comprises two major procedures: vibration response data representation and characteristics monitoring In the procedure of vibration data representation, a series of AR models is fitted to the response time histories of the monitored structure, and the coefficients of the AR models are extracted, resulting in several sets of multivariate data Characterizing the health condition of the structure, these sets of multivariate data are then monitored by a MSPC tool known as

Hotelling’s T2 control chart

2.2.1 Representation of Vibration Response Data Based on Time Series Analysis

For a stochastic process Z t( ), an AR model of order p can be written as

CHAPTER TWO

are μ, φi (i=1, 2, …, p) and 2

a

σ , and are estimated in this study based on the

Yule-Walker method (Box et al 1994) The order p can be determined using one of the

readily available model selection criteria such as Akaike’s information criterion (AIC) (Box et al 1994)

For a structure which N increasing levels of damage are progressively introduced

to, there are a total of N+1 damage states (including the undamaged state) Suppose that the response time histories of the monitored structure subjected to random excitations in each damage state are measured at n locations with m data points in each response history, and that, for each individual damage state of the structure, r time histories are recorded at each measurement location Let z ijkl (i=1, 2, …, ;m

1, 2, , ;

j= … n 1, k= 2, …, r; and l=0, 1, …, N) denote the ith response value in the kth time history at the jth measurement location for the lth damage state Thus for the lth damage state of the monitored structure, the entire response time histories can

be written as the following r matrices:

divided into M time windows, each containing s data points This division procedure

is the same as the procedure used by Sohn et al (2000a) except that the division is now

CHAPTER TWO

repeatedly applied to every measured response time history, resulting in a total of Mnr

time windows Upon finishing the division procedure, a series of AR models of order

p can be fitted to the time windows respectively, and the corresponding AR

coefficients are extracted and recorded as

Hotelling’s T2 control charts

2.2.2 Monitoring of Vibration Response Data Characteristics Based on Multivariate Statistical Process Control

Once the vibration response data are represented by sets of the AR coefficients il(i=1, 2, …, ;p and l=0, 1, …, N), Hotelling’s T2 control charts can be applied to

implement the MSPC Principles of Hotelling’s T2 control chart can be found in the related statistics literature (e.g., Montgomery 2005) In fact, based on the ith AR

coefficients for the undamaged state stored in i0, which are assumed in this study to

be from a statistically in-control process, and noting that a large value of Mr can

where 1, i= 2, , … p ; q is the subgroup size for the control chart analysis; α is the

probability that the test statistic falls out of the control limits when the process is in control; Fα, ,n Mrq Mr n− − +1 is the upper 100α percentage point of the F distribution with

n numerator degrees of freedom and Mrq−Mr− + denominator degrees of n 1freedom; UCL i is the upper control limit for the ith AR coefficient; and LCL i is the

lower control limit for the ith AR coefficient For control chart analysis, the multivariate data of the ith AR coefficient at the lth damage state il are first divided

into a series of subgroups of size q Then for the jth (j=1, 2, …, Mr q) subgroup,

the statistic plotted on the Hotelling’s T2 control chart is

where i=1, 2, …, p; l =0, 1, …, N; i and Si are the average of the sample mean

vectors and the average of the sample covariance matrices for the ith AR coefficient,

respectively, and are calculated based on the subgroups taken from the undamaged, or statistically in-control state; and jil is the sample mean vector of the jth subgroup for

the ith AR coefficient at the lth damage state i, S and i jil are obtained from the following equations:

where l=0, 1, …, N Note that, based on the principles of the control chart analysis,

an unusual increase in the total number of outliers does not directly indicate that the monitored structure has been damaged It only means that the monitored characteristic has been shifted from a statistically in-control state to an out-of-control one due to the

existence of some assignable causes Typical assignable causes other than structural

damage for the current applications may include, for example, malfunctions in the experimental data acquisition system used and environmental factors The proposed SHM scheme is therefore based on the assumption that these non-structural-damage assignable causes can be reasonably excluded or neglected The same assumption was also made by some other researchers (Sohn et al 2000a; Fugate et al 2001) in developing SHM methods using univariate SPC tools

CHAPTER TWO

2.2.3 Effects of the Autocorrelation in the Characteristics Data

For the applications of Hotelling’s T2 control chart, two assumptions are made that the samples are taken from a population that has a multivariate normal distribution, and that the successive samples are independent over time As documented in some statistical literature (Mason et al 1997; Stoumbos et al 2000), although the effects of

practical nonnormal data on the performance of Hotelling’s T2 control chart are alleviated due to sample means included in the test statistic, in some cases much more false alarms can result from autocorrelated data In the current study, a straightforward method of sampling less frequently (Montgomery 2005) is used to address the autocorrelation in the data, or specifically, the fitted AR coefficients characterizing the health condition of the monitored structure are first sampled before they are subjected

to MSPC procedures

2.3 Case Study

To verify the proposed SHM scheme, an example of a two-bay-and-two-story reinforced concrete frame with a span of 1.8 m and a story height of 1.5 m is studied through numerically simulated data Section dimensions of the frame are 130 mm ×

220 mm for beams and 130 mm × 180 mm for columns The undamaged frame is schematically shown in Fig 2.1(a), and modeled by the finite element method as a linear elastic structure in which the Young’s modulus and the density are taken as 28 GPa and 2450 kg/m3, respectively Plane frame elements based on Euler-Bernoulli beam theory are used to discretize the structure, resulting in a total of 30 elements Fig 2.1(c) shows the finite element mesh and the node numbering In each of the progressively damaged states considered, damage is simulated as the simultaneous reduction of the Young’s modulus in both the end parts of each beam and column,