Khac-Duy Nguyen M.E., B Duy Nguyen.pdf · 2018-05-28 · Khac-Duy Nguyen M.E., B.E A Thesis...
Transcript of Khac-Duy Nguyen M.E., B Duy Nguyen.pdf · 2018-05-28 · Khac-Duy Nguyen M.E., B.E A Thesis...
STRUCTURAL DAMAGE IDENTIFICATION
USING EXPERIMENTAL MODAL
PARAMETERS VIA CORRELATION
APPROACH
Khac-Duy Nguyen
M.E., B.E
A Thesis submitted in fulfilment of the requirements for the degree of
Doctor of Philosophy
School of Civil Engineering and Built Environment
Science and Engineering Faculty
Queensland University of Technology
2018
Structural Damage Identification using Experimental Modal Parameters via Correlation Approach Page i
© 2017 Khac-Duy Nguyen Page i
Keywords
Structural health monitoring, vibration-based, damage identification,
correlation approach, genetic algorithm, optimisation-based, modal strain energy,
geometric modal strain energy, sensitivity-weighted search space, benchmark
structures, spring-mass system, three story shear building, complex truss structure, I-
40 Bridge.
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© 2017 Khac-Duy Nguyen Page iii
Abstract
Damage identification is an important step within structural health monitoring
(SHM) for the assessment of the structural safety. To date, a large number of damage
identification algorithms have been proposed. Among those, methods using vibration
characteristics have been intensively investigated and widely applied to various
practical applications of our infrastructure. Feasibility of vibration-based damage
identification methods generally rely on vibration measurement and physical damage
detection models. However, the vibration features are normally measured with some
errors, and the damage identification models are inevitably established with some
differences compared to the real structures. Therefore, there is a demand to continue
discovering new vibration features which are highly sensitive to damage but little
affected by measurement and modelling uncertainties.
It is well-know that both modal strain energy and eigenvalue can be used as
damage indicators. While modal strain energy is more sensitive to elemental damage
but less accurately estimated, eigenvalue is measured with better accuracy but less
sensitive to damage. It is reasonably expected that combined use of them will give
more reliable damage identification results. This study presents new damage
identification algorithms based on modal strain energy-eigenvalue ratio (MSEE).
Firstly, a method using a simplified term of MSEE called geometric modal strain
energy-eigenvalue ratio (GMSEE) is developed. Damage is identified by optimizing
the correlation level between a measured GMSEE change vector and a numerical
one. The method requires only measured modal parameters (i.e., natural frequency
and mode shape) and geometric information of the structural element. Secondly, a
modification of this method is presented considering use of the full term of MSEE.
The modified method is capable for damage identification with fewer modes
compared with the original one although it requires both material and sectional
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properties of the structural element. Thirdly, in order to enhance the efficiency of the
proposed algorithms, an enhanced technique is proposed for the optimisation
procedure. The technique refers sensitivity-weighted search space which is able to
reduce the effect of measurement noise on low-sensitivity elements.
Performance of the proposed damage identification algorithms and enhanced
technique has been validated with a simple numerical model and some experimental
models of various scales from small to large. It has been found that the modal strain
energy-eigenvalue ratio is a sensitive damage indicator for large structures and the
enhanced technique significantly improve the effectiveness of the damage
identification methods proposed in this study. Results obtained in this study show the
high capability of the proposed damage identification algorithms for damage
identification where uncertainties associated with measurement and structural
identification are present. Also, the results show a great promise of the proposed
damage identification algorithms in practical applications for real structures. In
summary, this thesis provides an efficient damage identification scheme for
structural health monitoring of infrastructure.
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Table of Contents
Keywords........................................................................................................................................... i
Abstract ............................................................................................................................................iii
Table of Contents ............................................................................................................................. v
List of Figures ................................................................................................................................... ix
List of Tables .................................................................................................................................... xv
List of Abbreviations ...................................................................................................................... xvii
List of Symbols ................................................................................................................................ xix
Statement of Original Authorship .................................................................................................. xxiii
Acknowledgments ......................................................................................................................... xxv
CHAPTER 1: INTRODUCTION ............................................................................................................ 1
1.1 Background ............................................................................................................................ 1
1.2 Research Problems ................................................................................................................ 2
1.3 Research Aim and Objectives ................................................................................................. 3
1.4 Significance of the Study ........................................................................................................ 3
1.5 Scope of the Study ................................................................................................................. 4
1.6 Thesis Outline ........................................................................................................................ 4
CHAPTER 2: LITERATURE REVIEW .................................................................................................... 9
2.1 Introduction to Damage Identification .................................................................................... 9
2.2 Vibration-based Damage Identification ................................................................................ 11 2.2.1 Natural Frequency-based Damage Identification Methods ......................................... 11
2.3 Mode Shape-based Damage Identification Methods ............................................................. 13 2.3.1 Methods based on Direct Use of Mode Shape Change ............................................... 13 2.3.2 Modal Flexibility-based Methods ............................................................................... 14 2.3.3 Mode Shape Curvature-based Methods ..................................................................... 16 2.3.4 Modal Strain Energy-based Methods ......................................................................... 18
2.4 Signal-based Damage Identification Methods ....................................................................... 21
2.5 Incoporating Soft Computing Approaches for Damage Identification .................................... 23 2.5.1 Damage Identification incorporating Evolutionary Computation ................................ 23 2.5.1.1 Genetic Algorithm ..................................................................................................... 24 2.5.1.2 Other Evolutionary Computation Methods ................................................................ 26 2.5.2 Damage Identification using Artificial Neural Networks .............................................. 27
2.6 Practical Issues ..................................................................................................................... 29 2.6.1 Measurement Errors ................................................................................................. 29 2.6.2 Modelling Errors........................................................................................................ 30 2.6.3 Ill-Posed Problem ...................................................................................................... 31
2.7 Summary of Literature Review ............................................................................................. 32
CHAPTER 3: GEOMETRIC MODAL STRAIN ENERGY-EIGENVALUE RATIO (GMSEE) CORRELATION METHOD ........................................................................................................................................ 35
3.1 Conventional Correlation-based Damage Identification Methods ........................................... 35 3.1.1 Conventional Correlation-based Damage Identification Methods ............................... 35
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3.2 Theoretical Development of GMSEE Correlation Method ...................................................... 37 3.2.1 Sensitivity Analysis of GMSEE .................................................................................... 37 3.2.2 Locating damage using GMSEE change vector ............................................................ 40 3.2.3 Quantifying Damage .................................................................................................. 42 3.2.3.1 Quantifying Damage using Natural Frequency Change ............................................... 42 3.2.3.2 Quantifying Damage using GMSEE Change................................................................. 43
3.3 Summary of GMSEE Correlation Damage Identification Method ........................................... 43
3.4 idenfication of Correlative Damage Vector using Genetic Algorithm ..................................... 45
3.5 Numerical Verification.......................................................................................................... 46 3.5.1 Damage Identification with Noise-free Modal Data .................................................... 47 3.5.2 Damage Identification with Measurement Noise ....................................................... 50
3.6 Experimental Verification ..................................................................................................... 60 3.6.1 Case Study 1: LANL 8-DOF System.............................................................................. 60 3.6.2 Case Study 2: LANL Three-story Building Model ......................................................... 62
3.7 Conclusions .......................................................................................................................... 69
CHAPTER 4: MODAL STRAIN ENERGY-EIGENVALUE RATIO (MSEE) CORRELATION METHOD .......... 71
4.1 Sensitivity Analysis for Modal Strain Energy-Eigenvalue Ratio (MSEE) ................................... 71 4.1.1 Change in Elemental MSEE ........................................................................................ 71 4.1.2 Change in Total MSEE ................................................................................................ 72
4.2 Damage Identification using MSEE Change ........................................................................... 73
4.3 Summary of MSEE Correlation Damage Identification Method.............................................. 75
4.4 Numerical Verification.......................................................................................................... 76 4.4.1 Damage Identification with Noise-free Modal Data .................................................... 76 4.4.2 Damage Identification with Measurement Noise ....................................................... 78 4.4.2.1 Damage Identification Results using 8 modes ............................................................ 78 4.4.2.2 Damage Identification Results using 4 modes ............................................................ 86 4.4.3 Damage Identification with Modelling Errors ............................................................. 94
4.5 Experimental Verification ..................................................................................................... 99
4.6 Conclusions ........................................................................................................................ 103
CHAPTER 5: SENSITIVITY-WEIGHTED SEARCH SPACE FOR CORRELATION-BASED DAMAGE IDENTIFICATION ........................................................................................................................... 105
5.1 Theoretical Development of Sensitivity-Weighted Search Space ......................................... 105
5.2 Numerical Verification........................................................................................................ 107
5.3 Conclusions ........................................................................................................................ 110
CHAPTER 6: EXPERIMENTAL VALIDATION FOR A COMPLEX TRUSS STRUCTURE........................... 111
6.1 Introduction of Laboratory Through-Truss Bridge Model .................................................... 111
6.2 Experimental Setup ............................................................................................................ 114
6.3 Modal Extraction and Verification ...................................................................................... 119 6.3.1 Modal Extraction ..................................................................................................... 119 6.3.2 Modal Verification ................................................................................................... 127
6.4 Damage Identification for QUT Through-Truss Bridge Model .............................................. 131 6.4.1 Damage Identification using GMSEE Method ........................................................... 134 6.4.2 Damage Identification using MSEE Method.............................................................. 138
6.5 Conclusions ........................................................................................................................ 143
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CHAPTER 7: EXPERIMENTAL VALIDATION FOR A REAL BRIDGE .................................................... 147
7.1 Experiment Description and Modal Properties ................................................................... 147
7.2 Damage Identification using GMSEE Correlation Method.................................................... 150
7.3 Damage Identification using MSEE Correlation Method ...................................................... 155
7.4 Conclusions ........................................................................................................................ 159
CHAPTER 8: CONCLUSIONS AND FUTURE STUDIES ...................................................................... 161
8.1 Summary and Conclusions ................................................................................................. 161
8.2 Future Studies .................................................................................................................... 167
REFERENCES ................................................................................................................................. 169
APPENDICES ................................................................................................................................. 177
Appendix – A: Examples of Geometric Quantity of Stiffness Matrix................................................ 177
Appendix – B: Publications Derived from This Project .................................................................... 178
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List of Figures
Figure 2-1. Four diagnostic levels in damage identification process [adapted from
Rytter (1993)] ................................................................................................................... 10
Figure 3-1. Schematic of GMSEE correlation-based damage identification method ........... 44
Figure 3-2. Two-dimensional truss bridge (Rahai et al. 2007) ............................................ 46
Figure 3-3. Damage identification result for 2-D truss model: Case 1: D9 = 20% ............. 49
Figure 3-4. Damage identification result for 2-D truss model: Case 2: D4 = 20%,
D11 = 30% ...................................................................................................................... 49
Figure 3-5. Damage identification result for 2-D truss model: Case 3: D7 = 20%, D8
= 25%, D10 = 30% .......................................................................................................... 49
Figure 3-6. Detection probability results for 2-D truss model under noise: Case 1 .............. 53
Figure 3-7. Detection probability results for 2-D truss model under noise: Case 2 .............. 54
Figure 3-8. Detection probability results for 2-D truss model under noise: Case 3 .............. 55
Figure 3-9. Average damage extent results for 2-D truss model under noise: Case 1 ........... 57
Figure 3-10. Average damage extent results for 2-D truss model under noise: Case 2 ......... 58
Figure 3-11. Average damage extent results for 2-D truss model under noise: Case 3 ......... 59
Figure 3-12. LANL 8-DOF system (Duffey et al. 2001)..................................................... 61
Figure 3-13. Damage identification result for 8-DOF system by GMSEE correlation
method .............................................................................................................................. 62
Figure 3-14. 4-DOF three-story building from LANL ........................................................ 63
Figure 3-15. Basic dimensions of the three story building model ....................................... 64
Figure 3-16. Modal parameters of baseline model .............................................................. 66
Figure 3-17. Damage probability results for the shear building by GMSEE method ........... 67
Figure 3-18. Average damage extent results for the shear building by GMSEE method ...... 68
Figure 4-1. Schematic of MSEE correlation-based damage identification method .............. 75
Figure 4-2. Damage identification results for the 2-D truss model by MSE, GMSEE
and MSEE methods using first 4 modes............................................................................. 77
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Figure 4-3. Detection probability results by MSE, GMSEE and MSEE correlation
methods under high noise levels using 8 modes: Case 1 .................................................... 79
Figure 4-4. Detection probability results by MSE, GMSEE and MSEE correlation
methods under high noise levels using 8 modes: Case 2 .................................................... 80
Figure 4-5. Detection probability results by MSE, GMSEE and MSEE correlation
methods under high noise levels using 8 modes: Case 3 .................................................... 81
Figure 4-6. Average damage extent results by MSE, GMSEE and MSEE correlation
methods under high noise levels using 8 modes: Case 1 .................................................... 83
Figure 4-7. Average damage extent results by MSE, GMSEE and MSEE correlation
methods under high noise levels using 8 modes: Case 2 .................................................... 84
Figure 4-8. Average damage extent results by MSE, GMSEE and MSEE correlation
methods under high noise levels using 8 modes: Case 3 .................................................... 85
Figure 4-9. Detection probability results by MSE and MSEE correlation methods
under high noise levels using 4 modes: Case 1 .................................................................. 87
Figure 4-10. Detection probability results by MSE and MSEE correlation methods
under high noise levels using 4 modes: Case 2 .................................................................. 88
Figure 4-11. Detection probability results by MSE and MSEE correlation methods
under high noise levels using 4 modes: Case 3 .................................................................. 89
Figure 4-12. Average damage extent results by MSE and MSEE correlation methods
under high noise levels using 4 modes: Case 1 .................................................................. 91
Figure 4-13. Average damage extent results by MSE and MSEE correlation methods
under high noise levels using 4 modes: Case 2 .................................................................. 92
Figure 4-14. Average damage extent results by MSE and MSEE correlation methods
under high noise levels using 4 modes: Case 3 .................................................................. 93
Figure 4-15. Damage identification results for the 2-D truss model by MSE and MSEE
methods considering modelling errors: Case 1 ................................................................... 96
Figure 4-16. Damage identification results for the 2-D truss model by MSE and MSEE
methods considering modelling errors: Case 2 ................................................................... 97
Figure 4-17. Damage identification results for the 2-D truss model by MSE and MSEE
methods considering modelling errors: Case 3 ................................................................... 98
Figure 4-18. Damage identification for the 8-DOF system by MSEE method .................... 99
Figure 4-19. Detection probability results for the shear building by GMSEE and
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MSEE methods. .............................................................................................................. 101
Figure 4-20. Average damage extent results for the shear building by GMSEE and
MSEE methods ............................................................................................................... 102
Figure 5-1. Average MSEE sensitivity of the first 4 modes of the 2-D truss model ........... 108
Figure 5-2. Sensitivity-weighted search space of all elements of the 2-D truss model ....... 108
Figure 5-3. Damage identification results using MSEE method for a noised data in
Case 2 of the 2-D truss model .......................................................................................... 109
Figure 6-1. QUT steel through-truss bridge model ........................................................... 112
Figure 6-2. Dimensions of QUT steel through-truss bridge model .................................... 113
Figure 6-3 Joint connections at some typical positions ..................................................... 113
Figure 6-4. Sensor layouts for vibration measurement of the QUT through-truss bridge
model ........................................................................................................................ 115-116
Figure 6-5. Measured DOFs for the QUT through-truss bridge model .............................. 117
Figure 6-6 Sensors at some typical joints of the QUT through-truss bridge model ............ 117
Figure 6-7 Representative acceleration time-series of layout 1 of the QUT through-
truss bridge model ........................................................................................................... 118
Figure 6-8. Element numbering for the examined truss plane of the QUT through-truss
bridge model ................................................................................................................... 119
Figure 6-9 SVD diagram and the identified natural frequencies for the QUT through-
truss bridge model ........................................................................................................... 121
Figure 6-10 Modal strain energy of the unselected mode at 7Hz from two different
data sets .......................................................................................................................... 124
Figure 6-11 Modal strain energy of the unselected mode at 62.125Hz .............................. 124
Figure 6-12 Modal strain energy for the first selected mode (15.375Hz) from 2
different data sets ............................................................................................................ 125
Figure 6-13 Modal strain energy for the second selected mode (30.25Hz) from 2
different data sets ............................................................................................................ 125
Figure 6-14 Modal strain energy for the third selected mode (58.75Hz) from 2
different data sets ............................................................................................................ 126
Figure 6-15 Experimental mode shapes identified by FDD method .................................. 126
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Figure 6-16 FE model of the QUT through-truss bridge model .........................................127
Figure 6-17. Comparison of two FE models of a bottom chord for refining member’s
stiffness ...........................................................................................................................128
Figure 6-18. Three identical modes obtained from the FE model of the QUT through-
truss bridge structure ........................................................................................................130
Figure 6-19. Spring-in-series model of bolted truss element .............................................131
Figure 6-20. Illustration of damaged elements on the QUT through-truss bridge model
........................................................................................................................................133
Figure 6-21. Sensitivity-weighted search space of all elements of the examined plane
of the QUT through-truss bridge model ............................................................................134
Figure 6-22. Damage identification results for state 1-1 using GMSEE method with
sensitivity-weighted search space .....................................................................................136
Figure 6-23. Damage identification results for state 1-2 using GMSEE method with
sensitivity-weighted search space .....................................................................................136
Figure 6-24. Damage identification results for state 2-1 using GMSEE method with
sensitivity-weighted search space .....................................................................................136
Figure 6-25. Damage identification results for state 1-1 using GMSEE with
conventional search space ................................................................................................137
Figure 6-26. Damage identification results for state 1-2 using GMSEE with
conventional search space ................................................................................................137
Figure 6-27. Damage identification results for state 2-1 using GMSEE method with
conventional search space ................................................................................................138
Figure 6-28. Damage identification results for state 1-1 using MSEE method with
sensitivity-weighted search space .....................................................................................139
Figure 6-29. Damage identification results for state 1-2 using MSEE method with
sensitivity-weighted search space .....................................................................................139
Figure 6-30. Damage identification results for state 2-1 using MSEE method with
sensitivity-weighted search space .....................................................................................139
Figure 6-31. Damage identification results for state 1-1 using MSEE method with
conventional search space ................................................................................................141
Figure 6-32. Damage identification results for state 1-2 using MSEE method with
conventional search space ................................................................................................141
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Figure 6-33. Damage identification results for state 2-1 using MSEE method with
conventional search space ............................................................................................... 141
Figure 6-34. Damage identification results by traditional MSE method using only 3
experimental modes ........................................................................................................ 142
Figure 6-35. Damage identification results by traditional MSE method using 3
experimental modes together with 100 numerical modes ................................................. 143
Figure 7-1. I-40 Bridge over Rio Grande in Albuquerque, NM, USA (Farrar et al.
1994) .............................................................................................................................. 148
Figure 7-2. Sensor layout and location of damage ............................................................ 149
Figure 7-3. Four damage levels: cuts at the middle span of on the north plate girder ........ 149
Figure 7-4. Variation in stiffness around the crack ........................................................... 151
Figure 7-5. Result for damage case E-3 by GMSEE correlation method with
sensitivity-weighted search space .................................................................................... 153
Figure 7-6. Result for damage case E-4 by GMSEE correlation method with
sensitivity-weighted search space .................................................................................... 153
Figure 7-7. Result for damage case E-3 by GMSEE correlation method with
conventional search space ............................................................................................... 154
Figure 7-8. Result for damage case E-4 by GMSEE correlation method with
conventional search space ............................................................................................... 154
Figure 7-9. Result for damage case E-3 by MSEE correlation method with sensitivity-
weighted search space ..................................................................................................... 157
Figure 7-10. Result for damage case E-4 by MSEE correlation method with
sensitivity-weighted search space .................................................................................... 157
Figure 7-11. Result for damage case E-3 by MSEE correlation method with
conventional search space ............................................................................................... 158
Figure 7-12. Result for damage case E-4 by MSEE correlation method with
conventional search space ............................................................................................... 158
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List of Tables
Table 3-1. Damage scenarios for 2-D truss model .............................................................. 46
Table 3-2. Quantification of damage extent prediction error caused by measurement
noise ................................................................................................................................. 60
Table 3-3. Damage states of the three-story shear building model ...................................... 65
Table 4-1. Quantification of damage extent prediction error caused by measurement
noise using 8 modes .......................................................................................................... 86
Table 4-2. Quantification of damage extent prediction error caused by measurement
noise using 4 modes .......................................................................................................... 94
Table 6-1. Details of structural members of QUT steel through-truss bridge model .......... 114
Table 6-2. Summary of mode selection for the QUT through-truss bridge model.............. 123
Table 6-3. Comparison of natural frequencies obtained from experimental model and
FE model of the QUT through-truss bridge structure ....................................................... 130
Table 6-4. Damage scenarios for the QUT through-truss bridge structure ......................... 132
Table 6-5. Natural frequencies of the QUT through-truss bridge structure at
undamaged and damaged states ....................................................................................... 133
Table 6-6. Summary of damage identification results for the QUT through-truss bridge
model (with the use of sensitivity-weighted search space) ............................................... 140
Table 7-1. Natural frequencies of the I-40 Bridge ............................................................ 149
Table 7-2. MAC values after the introduction of damage ................................................. 150
Table 7-3. Equivalent stiffness reduction ......................................................................... 151
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List of Abbreviations
2D Two Dimensional
ANN Artificial Neural Network
AR Autoregressive
ARX Autoregressive with Exogenous Inputs
ACF Autocorrelation Function
CCF Cross-correlation Function
CDF Curvature Damage Factor
COMAC Coordinate Modal Assurance Criteria
CS Cuckoo Search
CWT Continuous Wavelet Transform
DOF Degree of Freedom
FDD Frequency Domain Decomposition
FE Finite Element
FRF Frequency Response Function
GA Genetic Algorithm
GMSE Geometric Modal Strain Energy
GMSEE Geometric Modal Strain Energy-Eigenvalue ratio
HS Harmony Search
MAC Modal Assurance Criteria
MDLAC Multiple Damage Location Assurance Criteria
ML-GA Multi-layer Genetic Algorithm
MSE Modal Strain Energy
MSEE Modal Strain Energy-Eigenvalue ratio
PCA Principal Component Analysis
PSO Particle Swarm Optimisation
RFC Relative Flexibility Change
SHM Structural Health Monitoring
STFT Short-time Fourier Transform
WT Wavelet Transform
VBDI Vibration-based Damage Identification
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List of Symbols
C Damage scaling coefficient
ijF Fraction of modal strain energy of the jth element for the ith mode
u
iiF Diagonal component of flexibility matrix before damage
d
iiF Diagonal component of flexibility matrix after damage
ijG Geometric modal strain energy-eigenvalue ratio (GMSEE) of the
jth element for the ith mode
H Girder height
K System stiffness matrix
jK Stiffness matrix of element j
0jK Geometric quantity of stiffness matrix of element j
M System mass matrix
MDLAC Multiple damage location assurance criterion
MSE MSE vector for all modes
iMSE MSE vector for mode i
,1iMSE First data set of the ith MSE data
,2iMSE Second data set of the ith MSE data
ijMSEE MSEE component of mode i and element j
MSEMAC Modal assurance criteria of MSE
N Number of MSE values in MSE vector
fS Sensitivity matrix of natural frequency
GMSEES Sensitivity matrix of GMSEE
GMSEEiS Sub-sensitivity matrix of GMSEE for mode i
GMSEEavgS Average sensitivity matrix of GMSEE
GMSEEuS Sensitivity matrix of GMSEE at undamaged state
GMSEEdS Sensitivity matrix of GMSEE at damaged state
MSEEavgS Average sensitivity matrix of MSEE
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MSEEuS Sensitivity matrix of MSEE at undamaged state
MSEEdS Sensitivity matrix of MSEE at damaged state
MSEEiS Sub-sensitivity matrix of MSEE for mode i
MSEEijS Sensitivity coefficient of MSEE for mode i and element j
meanjS Average sensitivity of element j to damage
iTMSEE Total MSEE of mode i
ijU Modal strain energy of the jth element for the ith mode
iU Total modal strain energy for the ith mode
ijU Geometric modal strain energy of the jth element for the ith mode
if Natural frequency of mode i
if Noised natural frequency of mode i
jk Stiffness constant of the jth element
k(x) Stiffness constant of the element at location x
joint-1k Tangential contact stiffness at the first end of a truss bar
joint-2k Tangential contact stiffness at the second end of a truss bar
bark Axial stiffness of a truss bar
ek Equivalent stiffness of a truss member
jl Length of element j
n number of elements
m number of modes
n Number of MSE values being greater than the mean value of
MSE vector
MSEp Percentage of MSE greater than its mean value
jx Damage extent variable of element j
j Damage index for element j
ij , f
i Independent random number in range of [-1, 1]
ΔG Measured GMSEE change vector
Δ iG Measured GMSEE change vector for mode i
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ΔMSEE Measured MSEE change vector
iΔMSEE Measured MSEE change vector for mode i
Δ iTMSEE Measured total MSEE change of mode i
δG Analytical GMSEE change vector
δ iG Analytical GMSEE change vector for mode i
δD Damage vector
jD Damage extent at element j
δMSEE Analytical MSEE change vector
iδMSEE Analytical MSEE change vector for mode i
iTMSEE Analytical total MSEE change of mode i
i Eigenvalue of mode i
u Mean value of damage extent at undamaged state
Damage extent threshold
u Standard deviation of damage extent at undamaged state
Noise level of mode shape
f Noise level of natural frequency
Φ Mode shape matrix
iΦ Mode shape vector of mode i
ij Mode shape component of mode i for element j
ij Noised mode shape component of mode i for element j
"( )i x Modal curvature at location x for mode i
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Statement of Original Authorship
The work contained in this thesis has not been previously submitted to meet
requirements for an award at this or any other higher education institution. To the
best of my knowledge and belief, the thesis contains no material previously
published or written by another person except where due reference is made.
Signature: QUT Verified Signature
Date: March 2018
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Acknowledgments
First of all, I would like to thank my principal supervisor, Prof. Tommy Chan
for his endless support and guidance of my research throughout my PhD study.
During my candidature, I have benefited from his enthusiasm, high sense of
responsibility and invaluable advice to my research. His logical, detailed and
constructive comments have helped me complete my research successfully. I also
thank him for being always available for me to keep in touch no matter where he is
(in Australia or overseas). He is my ultimate role model as researcher, advisor,
lecturer and mentor.
I would like to thank my first associate supervisor, Prof. David Thambiratnam
for his care, encouragement and advice on my study. Moreover, I appreciate his
careful editing for my papers, making them more readable as well as his willingness
to share his academic and life experience with me.
I would also like to thank Dr. Andy Nguyen as my second associate
supervisor. He has shared with me his fruitful experience on how to keep a PhD
study on track as well as helped me a lot with my study and life settlement since my
first day at QUT. I thank him for his constant support and advice on my research. I
have also learnt a lot from his expertness on modal testing and sensor technology as
well as his great passion for research.
Special thanks are extended to my M.S. supervisor, Prof. Jeong-Tae Kim at
Pukyong National University, South Korea. He has given me the beginning lessons
on structural dynamics, modal testing and analysis, structural health monitoring, and
damage identification. Without his guidance, I would not be able to achieve much in
my research project.
My experimental tests on the QUT through-truss bridge would not have been
possible without the support of Banyo staff members, especially Mr. Barry Hume.
His support made my tests become easier and more professional. I also greatly
acknowledge Mr. Craig Cowled for his design and fabrication of this truss bridge
model.
My sincere thanks also spread to Mehran Aflatooni, Tharindu Kodikara, Zhi
Xin Tan, Ngoc Thach Le, Shojaeddin Jamali, Parviz Moradi, Ziru Xiang, Manal
Structural Damage Identification using Experimental Modal Parameters via Correlation Approach Page xxvi
© 2017 Khac-Duy Nguyen Page xxvi
Hussin, Thisara Pathirage, Thuy Nguyen, Benyamin Monavari, Hossein Moravej and
other fellows in the SHM research team at QUT for their help, debates and feedbacks
on my work. It was my honour to be a member of such a friendly, dynamic and
knowledgeable group.
I would like to express my appreciation to Queensland University of
Technology for the scholarships that supported during my candidature, and also for
providing the research facilities and technical support. I would also like to thank Dr.
Chaminda Gallage and Dr. Jason Watson for being panel members of my final
seminar and for their recommendations for making this thesis better. I sincerely
thank QUT Research Student Centre and HDR Student Support Team (especially
Ms. Tiziana La Mendola) for their direct and indirect support during my PhD
candidature. My gratitude is also given to the Los Alamos National Laboratory, USA
for providing experimental data used in this research.
Last but not least, I would like to thank my parents for giving me the birth,
for their unconditional love and support during my study abroad. My strength to
complete this research through to the end is from them. I also thank all my relatives
and friends who have helped, concerned and loved me in millions of ways in the past
years.
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1Chapter 1: Introduction
1.1 BACKGROUND
Structural safety is an essential issue for all civil structures such as bridges,
buildings, offshore platforms and nuclear power plants. In many developed countries,
the budget for the maintenance of structures has been annually expended. Despite
these efforts, damage in these structures seems to be inevitable since bridges are
subjected to various extreme loadings and environmental impacts that might have
been underestimated in the design process. Moreover, many bridges which were
constructed long time ago are becoming aged and their load-carrying capability has
declined. Therefore, it is important to detect early any abnormal behaviour or
damage in structures so that their lives can be prolonged by taking appropriate
actions.
Recently, the structural health monitoring (SHM) technology has drawn a
serious attention from researchers as an economic, efficient and intelligent way for
the maintenance of structures. SHM refers to a non-destructive structural evaluation
process of tracking and assessing the safety and performance of the structure using
an on-structure sensing system (Aktan et al. 2000, Chan and Thambiratnam 2011).
Over the past three decades, a large amount of research focusing on the first function
of SHM (i.e., structural safety) has been conducted. It is very important to have a
reliable damage detection procedure because if damage is not detected correctly, it can
eventually lead to local failure of the structural elements and in consequence to the collapse
of the whole structure. Damage detection methods are often classified based on type of
structural features used, such as natural frequency, mode shape, displacement, strain,
curvature, flexibility, strain energy or damage identification techniques such as
damage index, feature correlation, neural networks, etc. Most of these approaches are
based on vibration measurement.
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Among these techniques, modal vector correlation-based methods have been
found to be effective for locating damage as well as estimating damage extent since
they integrate advanced optimisation techniques to solve the damage problem.
Several correlation methods have been developed using changes in either natural
frequency (Messina et al. 1998) or mode shape (Shi et al. 2000b) or elemental modal
strain energy (Wang et al. 2012). However, these methods still have some limitations
as highlighted as follows. While the frequency correlation method requires the
availability of a sufficient set of natural frequencies, the accuracy of the other
methods relies very much on the number of analytical mode shapes used and the
degree of matching between numerical models and real structures. Therefore, when
dealing with a large number of structural members in a complex structure, the
existing correlation-based methods may become less practical, less accurate and may
generate high degree of false detection. Also, solving techniques for the correlation-
based methods need to be improved when dealing with high measurement noise
usually associated with real structures.
1.2 RESEARCH PROBLEMS
From the above presentation, it is evident that developing an effective
damage identification method is essential for the maintenance of structures. Degrees
of false-positive and false-negative detection need to be reduced, and accuracy of
predicting the damage extent needs to be improved, especially when noise is
introduced in measurement. The following research problems are identified:
1. How to develop a vibration-based damage indicator that is practical, suitable
for identifying damage via the correlation technique, accurate for both single
and multiple damage cases and suitable in the presence of measurement and
modelling uncertainties?
2. How to improve the damage detection accuracy when dealing with
structures having large number of degrees of freedom compared to the
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number of modes that could be measured?
3. How to minimize false detection caused by high measurement noise?
1.3 RESEARCH AIM AND OBJECTIVES
The main aim of this study is to develop new vibration based damage
identification methods for civil structures by using correlation of modal parameters
together with optimisation approaches. The proposed method is aimed at improving
the accuracy of damage detection, reducing the degree of false detection and
increasing practicality. The objectives of this research are as follows:
1. To investigate what vibration parameters and what approaches to be selected
for the development of new damage identification scheme.
2. To develop a sensitivity-based and correlation-based method using vibration
parameter incorporating both natural frequency and mode shape terms. Also,
it is needed to develop a method to estimate damage extent using the
proposed vibration parameter.
3. To improve the proposed method for the case where only fewer modes can
be measured.
4. To develop a technique that can reduce false detection associated with
measurement noise.
1.4 SIGNIFICANCE OF THE STUDY
This research deals with the general problem of structural damage
identification including locating the damage and estimating its severity. The research
is significant because it will overcome the several of the existing issues to provide
safer structures. The issues are: (i) existing damage detection methods have troubles
with accuracy in locating the damage, estimating its severity and false alarms, (ii)
there are certain differences between all real structures and their numerical models,
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(iii) vibration parameters of structures are measured with a certain degree of error,
(iv) only few modal parameters of a structure can be measured and reliable for the
use of damage identification purpose; and (v) findings of this work may lead to
more understanding of the impact of structural damage on structural behaviour.
1.5 SCOPE OF THE STUDY
The methodology of this study is derived for the damage which results in a
reduction of structural stiffness. For real world applications, stiffness reduction is
generally used to represent other damage types such as crack, corrosion and material
degradation. The feasibility of the methodology is validated using a physical spring-
mass system, a multi-story building model, a plate-girder bridge and truss bridge
models. However, the methodology can also be applied to other kinds of civil
structures such as beam-type bridges and frame structures. It should be noted that
this study does not examine effects of environmental factors on vibration parameters
of structures as there is already a large amount of research on compensation for
changes in vibration characteristics due to changes in environmental conditions.
Those environmental effect studies can be used in conjunction with this study for
better damage identification.
1.6 THESIS OUTLINE
The research focuses of the development and application of damage
identification algorithms for structures. The contributions of the research are in the
development of damage identification methods using experimental modal
parameters, development of a technique dealing with structures consisting of a large
number of degrees of freedom, and verification of the proposed methods and the
technique for a large complicated truss structure. This section summarizes the
overview of the thesis on each of the remaining chapters.
Chapter 2 reviewed the recent literature on vibration-based damage
identification algorithms and application of soft computing approaches for the
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damage identification problem. From the review of vibration-based damage
identification algorithms, it is found that using both natural frequency and mode
shape information would give better damage results. Also, modal strain energy-based
methods have shown to be effective to locate damage and to estimate damage
severity. Modal strain energy and natural frequency are used as key parameters for
the damage identification algorithms developed in this research. Additionally, the
modal vector correlation-based technique is found to be effective to locate damage
and hence is employed to formulate objective functions in this research. From the
review of soft computing approaches, evolutionary optimisation techniques have
shown excellent performance in solving damage identification problems. One of
these algorithms, the genetic algorithm (GA) is utilized as a basic technique in the
damage identification framework developed in this research.
Chapter 3 and 4 describe the development of damage identification
algorithms. The algorithms are based on a new damage indicator called modal strain
energy-eigenvalue ratio. Chapter 3 provides detail of a correlation-based method
using geometric modal strain energy-eigenvalue ratio (GMSEE) and its validation
for simple numerical and experimental models. The sensitivity of GMSEE to
structural stiffness reduction is derived using measured modal parameters and
geometric information of structural elements such as length and/or thickness
depending upon structure types. Unscaled damage extent information can be
identified from optimizing the correlation level between the measured GMSEE
change vector and numerical GMSEE change vector. Herein, genetic algorithm (GA)
can be embedded as a search engine to find an optimal solution. Then, final damage
extent information is identified using a scale factor calculated from the total change
in GMSEE. Due to an assumption that the fractional modal strain energy does not
change after a damage episode, the method still requires a good number of measured
modes. The algorithm was validated for a numerical 2-D truss model and two simple
laboratory models.
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In Chapter 4, an improvement made for the method developed in Chapter 3 is
presented regarding the capability for damage identification with fewer modes.
Herein, change in modal strain energy-eigenvalue ratio (MSEE) is utilized instead of
change in geometric modal strain energy-eigenvalue ratio (GMSEE). A new vector
form consisting of individual MSEE of each member and total MSEE is used to
formulate the objective function showing the correlation level between the measured
MSEE change and analytical MSEE change. The new approach can reduce the effect
caused by the assumption made in the original method made in Chapter 3; and hence
can reduce the number of modes used. However, this method will require full
information of structural elements including both geometric and material properties.
In summary, both methods developed in Chapter 3 and 4 are useful depending upon
particular cases where either full information of structural elements or sufficient
information of vibration modes is provided.
Chapter 5 presents a technique to improve damage identification capability
for large structures where measurement noise associated with these structures is
significant. The technique is based on the observation that damage identification is
only reliable for elements with high vibration stress. A search technique considering
sensitivity of individual structural element is developed wherein the search space of
an element is weighted with its sensitivity. The hidden concept behind this technique
is that the importance of each element is treated unequally; the higher sensitivity the
element has, the broader its search space is. This technique is tested for the
numerical 2D truss model in condition of high measurement noise.
Chapter 6 describes the validation of the developed damage identification
algorithms for a complex truss model with 100 degrees of freedom assessed. First of
all, vibration tests were conducted using a wired sensing system with 18
accelerometers. A major number of DOFs were measured by multiple roving setups.
Vibration characteristic of unmeasured DOFs were obtained by simple interpolation
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from the measured ones. Subsequently, modal parameters extracted from the
measured responses were verified with a FE model. Then, some damage scenarios
were simulated on the experiment model by introducing bolt loose to selected truss
elements. Finally, damage identification for the truss model was performed using the
proposed damage indemnification algorithms.
Chapter 7 provides the validation of the developed damage identification
algorithms for a benchmark full-scale bridge. The validation study used existing
experimental data of the I-40 Bridge provided by the Los Alamos National
Laboratory. For higher resolution of damage identification results, mode shapes of
unmeasured locations were estimated using cubic interpolation technique. In order to
reduce fitting errors associated with high order curves of the whole mode shapes, the
interpolation task was performed individually for each span. Results showed the
potential of the developed damage identification algorithms for real applications.
Chapter 8 summarizes the findings and conclusions of the thesis as well as
recommendations for future work. This research presents the development of
damage identification algorithms for structures based on modal vector correlation
principal. A sensitivity-weighted search space concept is developed, which is
effective for large structures with high measurement noise. Validation has been
performed for several structures with various scales from small to large. The future
work includes the issues for improving the damage identification algorithms
developed in this thesis, as well as the potentials to extend the research to various
applications.
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Chapter 2: Literature Review Page 9
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2Chapter 2: Literature Review
This section first provides the concept of damage identification. Then it
reviews theory of existing vibration-based damage identification (VBDI) methods and
their applications. Herein, VBDI methods are categorized into three groups: 1)
methods using change in natural frequency, 2) methods using change in mode shape
and its secondary features such as modal curvature, modal flexibility and modal strain
energy, and 3) methods based on vibration signals. Some soft computing techniques
used to solve the damage identification problem using vibration features are introduced
in this section, including some evolutionary optimisation algorithms such as genetic
algorithm, particle swarm optimisation, harmony search and cuckoo search, and
Artificial Neural Networks. Some practical issues are also discussed in this chapter.
2.1 INTRODUCTION TO DAMAGE IDENTIFICATION
In a structural system, damage can be defined as changes to the material
and/or geometric properties, which adversely affects the current or future
performance of the system (Sohn et al. 2003). Damage is typically categorized into
linear and nonlinear. Linear damage refers to the case when the initially linear-elastic
structure behaves in a linear manner after the occurrence of damage. Conversely, the
behaviour of a linear-elastic structure becomes nonlinear after a nonlinear damage
has been introduced. For example, linear damage is associated with section corrosion
or material degradation, while nonlinear damage is associated with fatigue crack,
frozen bearings or loose connections. The damage identification algorithms in this
study are developed under the phenomena of linear damage. For simplification, any
damage introduced in this study is assumed as linear damage.
Damage identification in a structural system is a process of examining
changes in measured response of the system to detect, locate and characterize
damage in the system (Farrar and Doebling 1997). According to Rytter (1993), the
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damage identification process can be illustrated in four levels as shown in Fig.2-1.
Level 1 gives the information whether damage is present in the structure. Level 2
and 3 provides information about the location and the size of the damage. The most
sophisticated level is the prognosis of the remaining life, which requires a
comprehensive interpretation of the impact of the discovered damage on the
structure.
Figure 2-1. Four diagnostic levels in damage identification process [adapted from
Rytter (1993)]
Over the past three decades, a large amount of research on vibration-based
damage identification (VBDI) has been conducted due to their reliability,
practicability and cost effectiveness (Doebling et al. 1998, Wang and Chan 2009,
Fan and Qiao 2010). The basic concept of the VBDI is to examine the correlation
between the change in structural properties (e.g., mass, stiffness, damping or
boundary condition) and the change in modal parameters which can be obtained by
measuring dynamic responses of the structure. In the following sections, the
common VBDI algorithms are reviewed.
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2.2 VIBRATION-BASED DAMAGE IDENTIFICATION
2.2.1 Natural Frequency-based Damage Identification Methods
Natural frequency has been widely used for damage diagnosis by many
researchers. Natural frequency is a global parameter of structures so that the change
in local members would be reflected by the change in natural frequency. Using
natural frequency change as a damage indicator is very efficient since natural
frequency can be quickly extracted from a vibration signal at a single location. The
degree of natural frequency change is dependent on the position of the damage
relative to the mode shape (Rytter and Kirkegaard 1994). It has been reported that
the measurement of natural frequency is reliable (Salawu 1997, Doebling et al.
1998) and less affected by random error sources than other modal parameters (Farrat
et al. 1997, Doebling et al. 1997).
On the contrary, Doebling et al. (1996) criticized in his review that natural
frequency change has low sensitivity to damage so the corresponding detection
methods require either very precise measurements or large levels of damage. He
used the data obtained from the test on the I-40 Bridge (Farrar et al. 1994) to
demonstrate his opinion. However, changes in the natural frequencies of the bridge
due to environmental variations were not controlled. Also, significant reductions of
resonant frequencies (exceeding 5%) were observed after the bending stiffness of the
bridge cross section was reduced by 21%.
Gudmundson (1982) mathematically formulated the changes in natural
frequencies of beams due to cracks, notches or other small geometrical changes. It
has been found that the predictions for geometrical changes close to supports are as
not as good as for the changes in other positions. That is due to the fact that vibration
behaviours of the regions close to supports are influenced by the nonlinear stress
intensity. The method was proved to be accurate for up to 10% change in natural
frequency when damage occurred closely to supports. Apart from those cases, the
prediction of natural frequency changes due to geometrical changes was very
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accurate.
Kim et al. (2003a) developed a direct damage index method for beam like
structures using the ratio of fractional changes in frequency for two modes. The
method was found to be practical as it needs only the measured modal parameters
and stiffness matrix. Also, damage severity was also accurately predicted by
correlating the size of fatigue crack to the change in natural frequencies. However,
the method is capable for only single damage scenarios.
As one of the earliest works for the forward methods, Cawley and Adams
(1979) proposed an optimisation-based method using the ratio of frequency changes
in two modes together with sensitivity analysis for localizing damage and
quantifying damage. The method was found sensitive to detect damage location in
aluminum plates and cross-ply carbon-fiber-reinforced plastic plate all but with
erroneous prediction of damage amount.
Messina et al. (1998) proposed a forward method to locate damage and
estimate damage severity using the Multiple Damage Location Assurance Criterion
(MDLAC) that is a correlation factor between the change of a measured natural
frequency set and that of the corresponding numerical natural frequency set. Damage
location is identified by searching the numerical damage set which gives the largest
MDLAC. Also, the first order and second order sensitivity matrices have been
derived for damage extent estimation. It was found that the both methods could
estimate damage extent accurately, and the second order method with a much higher
calculation effort gave only a little improvement in accuracy.
In summary, the natural frequency-based methods are simple, cost efficient
and reliable. Relying on natural frequency change, the existence of damage, the
location and severity of damage can be accurately predicted. However, the methods
cannot distinguish damage at symmetric locations in a symmetric structure (Cawley
and Adams 1979, Shih 2009). It is found that the direct method can only deal with
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single damage cases. Although the correlation-based method proposed by Messina et
al. (1998) enables to detect multiple-damage, the method much relies on the
availability of a sufficient set of natural frequencies. That is due to the fact that the
change of natural frequencies in different multiple-damage scenarios might be the
same.
2.3 MODE SHAPE-BASED DAMAGE IDENTIFICATION METHODS
2.3.1 Methods based on Direct Use of Mode Shape Change
Mode shape is a modal parameter which represents the relative deformation
of a structure in modal domain. West (1986) probably presented the first systematic
use of mode shape information for detecting damage based on only measurements of
mode shape sets before and after a damage episode. The statistical term named
modal assurance criteria (MAC) was used to determine the degree of correlation
between modes before and after a damage episode:
( )
( )( )
2*
* *( , )
T
i j
T T
i i j j
MAC i j =Φ Φ
Φ Φ Φ Φ (2-1)
where iΦ is the ith mode shape vector of the structure in the undamaged state, and
*
jΦ is the jth mode shape vector of the structure in the damaged state. The MAC
takes on values from zero representing no correlation, to one representing full
correlation. The change in MAC across the different partitioning techniques is used
to localize the structural damage.
Since the MAC does not give spatial information of structural damage, it is
hard to determine the damage location. An improvement of the modal assurance
criterion is the coordinate modal assurance criterion (COMAC) proposed by Lieven
and Ewins (1988). The COMAC identifies spatially which degree-of-freedom (DOF)
contributes negatively to a low value of MAC; then the damage is located at the
position corresponding to that DOF. The COMAC is calculated as follows:
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( )2
*
1
2 *2
1 1
( )
m
ki ki
k
m md
ki ki
k k
COMAC k =
= =
=
(2-2)
where ki is the undamaged mode shape value of the kth DOF corresponding to the
ith mode; *
ki is the damaged mode shape value of the kth DOF corresponding to the
ith mode; and m is the highest mode number of interest.
Ko et al. (1994) proposed a method to locate damage by combining
sensitivity analysis, MAC calculation and COMAC calculation. Sensitivity analysis
is carried out to narrow the set of mode shapes used in MAC and COMAC analysis.
It was observed that the method can better detect the damage location by using the
selected modes rather than by using all the modes. However, damage location should
be assumed for the selection of mode shapes.
Inherited from the MDLAC method proposed by Messina et al. (1998), Shi et
al. (2000b) modified the MDLAC method by using mode shape changes instead of
natural frequency changes. They succeeded to detect single-damage and multiple-
damage cases in a truss structure. It was observed that the damage detection results
using the modified MDLAC contain less false-positive detection than those using the
original MDLAC with natural frequency changes. Also, the modified MDLAC
method requires much fewer modes than the original one. However, the accuracy of
the method relies on the number of analytical mode shapes used and the degree of
matching between the analytical mode shapes and the experimental mode shapes
including the measured and unmeasured ones.
2.3.2 Modal Flexibility-based Methods
Damage detection based on changes in modal flexibility was first proposed
by Pandey and Biswas (1994). The flexibility matrix is obtained from measured
natural frequencies and mass-normalized mode shapes as follows:
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2
1
1m1 T T
i i
i iω=
= =-F ΦΩ Φ ΦΦ (2-3)
where 1 2[ , ,..., ]n=Φ Φ Φ Φ is the mode shape matrix, iΦ is the ith mode shape,
2( )idiag= Ω is the diagonal matrix containing the eigen-frequencies, and m is the
number of modes. The damage location can be determined by comparing the
damaged flexibility to the undamaged flexibility. A merit of the method is that
damage can be estimated from a few of the lower vibration modes which can be
easily measured. That is due to the fact that the flexibility of a structure converges
rapidly with increasing frequency, and hence, a good estimation of the flexibility
matrix can be obtained from only a few lower modes. However, mode shapes are
required to be mass-normalized that requires an accurate mass matrix from a
numerical model. Another drawback of the method is that the change in flexibility
matrix itself does not provide the information of damage extent.
Zhao and DeWolf (1999) compared the sensitivities of natural frequencies,
mode shapes and modal flexibility to structural stiffness reduction for a five degree
spring-mass system. It was found that modal flexibility (which is a function of both
natural frequencies and mode shapes) is more sensitive to damage than either natural
frequencies or mode shapes.
Ni et al. (2008) applied the flexibility method to detect damage in the Ting
Kau cable stayed Bridge. They defined the relative flexibility change (RFC) index to
identify damage location:
u d
ii ii
i u
ii
F FRFC
F
−= (2-4)
where iRFC is the relative flexibility change at the ith DOF; u
iiF and d
iiF are the
diagonal component of the flexibilities matrix before and after damage, respectively.
It was found that the RFC performs well for locating single damage cases but may
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provide false-positive identification for multiple-damage cases. Also, the RFC index
performance is significantly reduced by introducing ambient conditions.
Shih (2009) combined the modal flexibility change and a modal strain
energy-based damage index to detect damaged elements in truss bridges using a few
lower modes. The method was found effective to locate single and multiple damage
cases. However, the method was only verified on a numerical truss bridge, in which
no systematic and measurement errors were considered. Also, simultaneous use of
two methods to assess damage in same elements may lead to conflict in the damage
decision and can result in more false-positive or false-negative detections.
Chen and Nagarajaiah (2007) developed a forward method based on modal
flexibility correlation. Damage is identified by minimizing the difference between an
analytical flexibility matrix and the measured flexibility matrix using sensitivity
analysis and Gauss-Newton optimisation algorithm. The method was verified by
simulation examples on a three-bay cantilever truss structure and a six-bay simply-
supported truss structure. It was found that the method is successful to detect damaged
elements, even if the identified modes are corrupted by light noise (1% for natural
frequencies and 3% for mode shapes). However, the authors noted that the accuracy of
the method would be affected by high noise level of the identified modes.
Koo et al. (2010) proposed another forward method for shear buildings by
using deflections obtained from modal flexibility. The method was found to be
successful to estimate the location and extent of damage by incorporating with an FE
model and an optimisation process. However, the feasibility of the method was
verified only by a numerical study and limited to shear buildings.
2.3.3 Mode Shape Curvature-based Methods
Mode shape curvature is the second derivative of mode shape, and therefore
it has a direct relationship with bending strain mode shape in beams, plates and
shells. Curvature-based methods are capable of locating damage but they are only
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suitable for flexural members. Pandey et al. (1991) first proposed the use of mode
shape curvature to locate the damage for a cantilever beam and a simply-supported
beam. Mode shape curvature can be computed from displacement mode shapes by a
central difference approximation as:
, 1 , , 1"
, 2
2i j i j i j
i j
jl
+ − − + = (2-5)
where ,i j is the component at node j of the ith displacement mode shape, and
jl is
the distance between two nodes. They demonstrated that absolute changes in mode
shape curvatures are useful to locate damage, and the intensity of damage is
proportional to the changes in mode shape curvatures.
Abdel Wahab and De Roeck (1999) applied the method of Pandey et al.
(1991) with a small modification making use of all measured modes to detect
damage in the pre-stressed concrete bridge Z24. The curvature damage factor (CDF)
was proposed to locate damage as follows:
" *"
1
1 m
j ij ij
i
CDFm =
= − (2-6)
where m is the total number of modes to be considered; "
ij and *"
ij is the
component at node j of the ith mode shape curvature of the intact structure and that of
the damaged structure, respectively. They found that when more than one fault exists
in the structure, it is impossible to locate damage by using only one single mode, but
the combination of all modes (e.g., CDF) gives a clear identification of these
locations.
Ni et al. (2000) developed a damage index using the mode shape curvature
change rate for damage detection. The mode shape curvature change rate for the ith
mode is calculated as:
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*" "
*" "( )
ij ij
i
ij ij
j
Index j −
= −
(2-7)
Then, a Z-value of mode shape curvature change rate for the ith mode is
obtained as:
( )
( )( )
i ii
i
index indexZ j
index
−=
(2-8)
where and represent the mean and the standard deviation of the index vector.
Based on the Neyman-Pearson criterion, a damage is identified at location j if
( ) 3iZ j . Numerical damage identification results for the Tsing Ma Bridge showed
that the Z-value diagrams obtained from longitudinal modal curvatures can correctly
indicate damaged segments of deck. However, several false positive detections were
also observed using the Z-value indicator. Also, the method was only verified with
single damage cases at noise-free condition.
2.3.4 Modal Strain Energy-based Methods
Modal strain energy (MSE) is known as a damage sensitivity parameter and
is widely used to formulate damage detection methods. The first damage detection
method based on change in MSE was developed by Stubbs et al. (1992). The ith MSE
of an arbitrary structure can be obtained as follows:
T
i i iK =Φ KΦ (2-9)
where K is the system stiffness matrix. The method relies on the assumption that the
change in elemental fraction of modal strain energy of the ith mode before and after
damage is negligible. The idea behind this assumption is that the eigenvalues of the
structure can be assumed to be linear to damage (Messina et al. 1998). It has been
reported by the authors that this assumption is reasonable for small damage but gives
significant error for large damage. . A damage index j of the ith element is obtained
as follows:
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* *
0
1
**
0
1
mT
i j i ij i
j mTji j i i
i
KE
EK
=
=
= =
Φ K Φ
Φ K Φ
(2-10)
where jE is the parameter representing material stiffness properties of the jth
element; 0jK is the geometric quantities of the stiffness matrix of the jth element; m
is the number of modes; the asterisk (*) denotes the damage state.
Kim and Stubbs (2002) modified the original damage index to overcome the
singularity problem associated with zero-modal strain energy of elements at nodal
points of a mode. The modified damage index is as follows:
* * * *
0 0
1 1
**
0 0
1 1
m nT T
i j i i k i ij i k
j m nT Tji j i i k i i
i k
KE
EK
= =
= =
+
= =
+
Φ K Φ Φ K Φ
Φ K Φ Φ K Φ
(2-11)
where n is the number of elements. They found that the modified damage index can
significantly reduce false-positive detection compared with the original one.
Kim and Stubbs (2002) also proposed an improved damage index utilizing
changes in both natural frequency and modal strain energy. The improved damage
index is calculated as follows:
* *
0
1
*
0
1
/
nT
i j ij i
j nTj
i i i j i
i
E
EK g nd
=
=
= =
+
Φ K Φ
Φ K Φ
(2-12)
where 2 2( / )i i ig ω ω= is the relative change in the ith eigenvalue; and nd is the
number of damage in the structure. It was found that damage severity is predicted
more accurately by using the improved damage index than by using the original and
the modified ones. However, the improved damage index method has several
limitations as: the number of damage locations must be known, these locations
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should be symmetric, and damage extents at these locations should be equivalent.
The performance of the method was actually proved only for this kind of damage.
Shi et al. (1998) used elemental modal strain energy change ratio as a
damage indicator. They found that this parameter is an efficient indicator in locating
structural damage. Shi et al. (2000a) further expand the method to quantify damage
extent by analyzing the sensitivity to local damage. They evaluated the method on an
FEM model of a fixed-supported beam and on a two-story frame structure model. It
was found that damage quantification is successful to quantify single damage with
7% noise but not good for multiple damage under the same noise level. Also, they
noted that the accuracy of damage extent estimation significantly relies on the
number of numerical modes used and modeling accuracy for higher modes.
Hu et al. (2006) developed a damage detection method based on cross modal
strain energy. Damage extent is calculated by a least square approach using both
changes in natural frequencies and mode shapes. The method was verified on a FEM
of a five story frame structure. It was observed that the method is able to detect small
damage (5% stiffness reduction) under mild noise environment. However, its
accuracy is significantly affected by high-noise measurements. Also, the method
faces with singularity problem which requires proper selections of modes.
Shih et al. (2009) studied on the application of MSE-based damage index
methods and found that their performance on damage quantification is only feasible
for single damage cases. Moreover, it was reported that the relatively small damage
is hard to be detected by these indices when multiple damage is introduce
(Wahalathantri et al. 2010).
Wang et al. (2012) proposed a new MDLAC using modal strain energy
change for damage detection. A mode selection algorithm was also proposed to
select the modes which are the most potential to indicate damage. The correlation
coefficient is calculated as follows:
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2
. ( )( )
. ( ). ( )
T
MSE
T TMDLAC
=
MSE δMSE δDδD
MSE MSE δMSE δD δMSE δD (2-13)
where 1 2[ , ,..., ,..., ]T
i m=MSE MSE MSE MSE MSE is the elemental modal strain
energy vector, iMSE is the elemental modal strain energy vector for the selected
mode ith, and m is the number of selected modes. The method was successful to
detect single damage on a lab-scale complicated steel truss. However, high degree of
false-positive detection was also observed in the damage detection result. Also, as
numerical mode shapes of many sampled FE models are calculated, a huge
computation resource must be used. Besides, this method relies much on the
accuracy of the established FE model compared to the real structure.
2.4 SIGNAL-BASED DAMAGE IDENTIFICATION METHODS
Different from modal-based damage detection methods that need the
extraction of modal information from measured signals, signal-based methods
directly use experimental signals or their features to determine damage information.
Sohn et al. (2000) developed an autoregressive (AR) model of time history
responses for damage detection purposes. In the damage detection procedure, time
series are compressed using principal component analysis (PCA) to reduce their
dimension prior to be used to construct the AR model. AR coefficients which are
considered as damage-sensitive features are estimated from smaller windows of the
compressed time signals. The multidimensional space AR coefficients are then
projected into 1D space vector using Bayes’ theorem since the sensitivities of the AR
coefficients to damage are different each other. The occurrence of damage is realized
through an outlier analysis known as X-bar control chart of the projected AR
coefficients. Feasibility of the method was verified by an experiment on concrete
columns. The occurrence of damage was successfully detected and false-positive
indication of damage was minimized.
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Sohn and Farrar (2001) extended the study of Sohn et al. (2000) for damage
location problems using a two-stage prediction model combining AR and
autoregressive with exogenous inputs (ARX) models. Spatial time series are first
used to construct the AR models. The errors between the measurements and the
predictions by the AR models are assumed to be mainly caused by the unknown
external inputs, and therefore, can be predicted by another AR models, so-called
ARX. The final errors between measurements and the predictions by the ARX
models are considered as the damage sensitive feature. The ratio of the final error in
a damage state to that in the undamaged state is used as a damage locating index. It
was found that the method is promising for automated continuous SHM systems
since it is simple and does not require much interaction with users.
Liu et al. (2009) developed a decentralized damage detection method called
ACF-CCF using the autocorrelation function (ACF) of individual sensors and the
cross-correlation function (CCF) of designed node pairs. The occurrence of damage
is realized by the change in ACF. Once damage is alerted, the location of damage can
be determined as at the region of the node pair on which the change in CCF is
observed. It is found that the method can accurately detect damage regions with
small degree of false-positive detection. From experimental results, Jayawardhana et
al. (2011) demonstrated that the ACF-CCF method is a better method compared with
the AR-ARX method regarding time consuming and damage detection accuracy.
Jayawardhana et al. (2013) applied a noise filtering technique, named discrete-
time Wiener filter, for damage alert and location. The idea of the Wiener filter-based
method is that the error obtained after filtering the signal at a damaged state is different
from that at the undamaged state. The CCF of the error is also applied to locate
damage, as similar with the means presented by Liu et al. (2009). It was found that the
method is effective to realize damage occurrence as well as to locate damaged regions
at noise-presented condition. However, the method requires the same operational and
environmental conditions before and after a damage episode.
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Wavelet transform-based methods are also effective signal based damage
detection methods. Wavelet transforms (WTs) are time-frequency transformation
tools based on windowing techniques with various time intervals. The tools can
simultaneously perform low frequency responses with high resolution using large
windows, and high frequency responses with low resolution using small windows.
Compared to the short-time Fourier Transform (STFT), WTs can provide more
accurately the location of a transient signal (observed in the high frequency diagram)
while the low frequency response is also obtained with equivalent accuracy. Hou et
al. (2000) showed that location of damage can be detected by spikes in the details of
the wavelet transform of a response data. Fan and Qiao (2009) applied a 2-D
continuous wavelet transform (CWT) on mode shapes for damage detection in plate-
type structures. It was found that the method is superior in noise immunity and
robust under limited sensor data.
2.5 INCOPORATING SOFT COMPUTING APPROACHES FOR DAMAGE IDENTIFICATION
In order to deal with more complicated damage situations in more complex
structures, it is essential to incorporate the existing damage algorithms with powerful
computation tools. Soft computing approaches can provide effective computation to
solve damage detection problems. The idea of soft computing was initiated by Zadeh
(1994). Different from hard computing, the soft computing is tolerance of
imprecision, uncertainty, partial truth and approximation. It aims at achieving close
resemblance with human like decision making. Soft computing is used when there is
a lack of information about a problem itself. The followings are several soft
computing approaches used for solving damage detection problems.
2.5.1 Damage Identification incorporating Evolutionary Computation
Evolutionary computation is a family of optimization algorithms inspired by
biological evolution. The evolutionary computation techniques include genetic
algorithm (GA), particle swarm optimisation (PSO), harmony search (HS), cuckoo
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search (CS), etc. Evolution computation techniques can be used to identify structural
damage by minimising the difference between the changes in experimental modal
parameters and those of a theoretical model. The following sections will focus on
applications of the evolutionary computation techniques on damage identification.
2.5.1.1 Genetic Algorithm
Genetic algorithm (GA) is one of the most widely used and powerful
optimisation tools for damage detection problems. The method is based on the
mechanism of biological evolution by the repetitive process of encoding, selection,
crossover and mutation (Goldberg 1989). In the method, a population of a structural
property such as structural stiffness is established for damaged structures. Each
member of the population is encoded into a binary string or a real number.
Subsequent generations are generated from those members based on the principle
“survival of the fittest”. Different from traditional optimisation techniques, GA
evaluates many solutions simultaneously, that potentially avoid convergence to a
non-global optimum.
Friswell et al. (1998) presented a combined genetic and sensitivity algorithm
for damage detection. In order to reduce the effect of systematic error, changes in
frequency from the undamaged to the damaged structure is considered instead of
natural frequency of the damaged structure. The GA is used to locate damage by
assuming that damage occurs at a single location and introducing into the objective
function a term considering other damage locations. The sensitivity analysis is then
used to estimate damage extent of the identified damaged elements. The method was
successful to locate damage for a numerical cantilever beam and an experimental
cantilever plate.
Hao and Xia (2002) investigated the performance of several damage objective
functions using three criteria, which are change in natural frequency, change in mode
shape and the combination of the two. They found that using change in natural
frequencies provides better damage prediction results than using change in mode
shapes. It was also observed that the performance of damage detection by combining
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two parameters is significantly affected by the relative weight of mode shape to
natural frequency.
Kim et al. (2007) examined the feasibility of using modal strain energy for
updating damage parameters using GA. They demonstrated that neither mode shape
nor modal strain energy can detect locations of saw cuts in a free-free beam model.
Also it has been reported that the combination of natural frequency and modal strain
energy is the most effective parameter for damage parameter updating.
Gomes and Silva (2008) compared GA-based and modal sensitivity-based
method in structural damage detection. They found that damaged members are
successfully located by both the modal sensitivity and GA with equivalent degrees of
accuracy. Regarding damage extent prediction, while the sensitivity-based method
has limitation on dealing with large amount of damage due to its assumptions and
simplification, the GA-based method which does not use such simplifications cannot
correctly converge to the actual damage extent.
Wang (2012) proposed a multi-layer genetic algorithm (ML-GA) method
using correlation of modal strain energy vector as the objective function. The method
is developed to overcome ill-conditioned problems that happen when the number of
damage parameters is very large. In the method, the damage parameter updating
process is divided into multiple layers. In the first layer, the damage parameters are
categorized into major groups and the optimisation is done for each group. In the
subsequent layers, the optimisation is done for a larger group with the starting point
inherited from the previous layers. The process ends up at the final layer where one
group includes all damage parameters. The method was successful in identifying the
damaged member of a complicated truss bridge model with significant reduction of
computation cost. However, high degree of false-positive detection was still
observed. Also, the damage extent prediction needs to be improved.
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2.5.1.2 Other Evolutionary Computation Methods
Kennedy and Eberhart (1995) developed a population-based global
optimisation technique, called Particle Swarm Optimisation (PSO). PSO is loosely
modeled on the basis of social behavior, such as bird flocking or fish schooling. The
technique has many similarities with GA such as the concepts of stochastic process,
crossover operation and fitness. However, instead of using genetic operators, the
individuals in a population are generated by cooperation and competition. PSO is
flying potential solution through hyperspace, accelerating toward “better” solutions.
Kang et al. (2012) attempted to apply PSO for solving structural damage detection
problems. The method was successful to locate damage and estimate damage extent
for a simply supported beam and a truss structure, even at 10% noise of mode shape
measurement. Seyedpoor (2012) proposed a two stage damage detection method
using modal strain energy damage index for locating damage and PSO for damage
extent estimation. The method showed the high efficiency on identifying the location
and extent of multiple structural damages for beams and truss structures.
Geem et al. (2001) developed a music-based meta-heuristic optimisation
algorithm, named Harmony Search (HS). It is inspired by the improvisation process
of musicians that searches for a perfect state of harmony. The process starts with an
initial random harmony memory which comprises of many harmonies in the form of
vectors. A new harmony is improvised from the harmony memory with a harmony
memory considering rate, together with a pitch adjusting process. That harmony will
replace the minimum harmony from the harmony memory if it is better than the
minimum one. Similar to GA, HS algorithm can solve continuous variable problems
as well as discrete variable problems. Differently from GA that considers only two
parent strings to produce new strings in the next generation, HS makes a new vector
after considering all existing vectors. Miguel et al. (2012) applied the HS algorithm
to structural damage detection problems. The HS algorithm was examined for three
criteria which are frequency and mode shape changes, dynamic residual force vector,
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modal flexibility change. It was found that damage location and damage extent of a
numerical cantilever beam is identified correctly even in noise condition of up to
5%.
Cuckoo Search (CS) is a promising evolutionary optimisation algorithm
recently developed by Yang and Deb (2009). The algorithm is inspired by the special
lifestyle of some cuckoo species that lay their eggs in the nests of other host birds of
other species. The reproduction procedure is based on the obligate brood parasitism,
which means the cuckoo eggs are not detected by the host birds and can become
mature cuckoos. In the algorithm, each egg in a nest represents a solution, and a
cuckoo egg represents a new solution. CS aims at replacing a not-so-good solution in
the nests by using the new potentially better solutions (cuckoo eggs). Hosseinzadeh
et al. (2014) attempted to use the CS algorithm for detecting damage in several
structure types such as beams, truss bridges and shear buildings. The results showed
that CS is an effective, robust and speedy optimisation method for structural damage
problems.
There has still been an argument on which algorithm is the best among GA,
PSO, CS and HS. It is very hard to make a comprehensive comparison as these
algorithms used different solver parameters; and for each algorithm, different
settings of solver parameters give different performances. Particularly for the
damage identification problem, it is observed that GA was the most widely used
technique and has also demonstrated its effectiveness for a variety of structures as
well as damage.
2.5.2 Damage Identification using Artificial Neural Networks
Artificial neural networks (ANNs) are computational models inspired by the
structure and functions of biological neural networks. An ANN comprised of a large
number of highly connected simple processing units, so-called neurons (Wu et al.
1992). The strength of the connections between the neurons is represented by
weights. ANNs are capable of machine learning and pattern recognition. One of the
powerful characteristics of ANNs is the capability of modeling non-linear
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relationships, and therefore, they can produce correct or nearly correct outputs in the
presence of incomplete inputs or in the presence of noise (Balageas et al. 2006,
Dackermann 2010).
In damage detection problems, ANNs are trained to recognize the structural
response features (frequencies, mode shapes, frequency response, etc.) of an
undamaged structure as well as those of the structure sustaining damage. Wu et al.
(1992) might be the pioneers introducing ANNs in structural damage detection. They
used Fourier spectrum portions as inputs of a single-layer perceptron neural network
and found that the ANN are capable of identifying damaged members and the extent
of the damage from patterns in the frequency response of the structure.
Pandey and Barai (1995) adopted a multi-layer perceptron ANN to identify
damage in a truss bridge. They concluded that the performance of the network with
two hidden layers is better than that with one single layer. It was also demonstrated
that only a few measurements are needed to train the network for damage
identification.
Ko et al. (2000) applied two special kinds of ANNs which are auto-
associative neural network and probabilistic neural network for the detection of
damage occurrence and location in the Tsing Ma Bridge. It was found that the
occurrence and location of damage can be detected only in low noise conditions.
Bandara et al. (2014) proposed a damage detection method by combining
ANNs, principal component analysis (PCA) and frequency response functions
(FRFs). FRFs are used as inputs of a neural network. PCA is used to reduce the size
of the FRF data inputting into the neural network. Numerical results for a two-story
framed structure demonstrated that the method can deal with single and multiple
damage cases in noise levels up to 10%. It was also found that two hidden layers are
sufficient for a neural network to deal with damage identification problems.
Most of the studies showed the excellent performance of ANNs when dealing
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with non-linear and noise-infested damage identification problems. Also,
incorporating ANNs do not require physical relationship between damage and
measured features. However, there are two critical reasons that limit the application
of ANNs. Firstly, the techniques require a large amount of training data for that their
verification has only been conducted for small structures. Secondly, the techniques
require a very accurate numerical model for generating training data since damage
simulation on real structures is prohibited.
2.6 PRACTICAL ISSUES
2.6.1 Measurement Errors
In practice, measurement errors can be caused by errors in measurement
equipment (e.g., sensors, data acquisition devices), system synchronization errors,
and calculation errors associated with modal extraction techniques, etc. The errors
existing in the measured data limit the successful use of damage identification
methods (Friswell et al. 1997). For instance, existence of measurement noise may
state certain less pronounced damaged elements undetectable (false-negative
detection), while identifying some undamaged structural elements as damaged
(false-positive detection) (Farrar and Doebling 1998).
Another source of measurement error is the change in environmental
conditions temperature, humidity, wind effect and traffic effect (Aktan et al. 1994,
Salawu 1997). It has been noted that the change by about 5% might be required to be
able to detect the damage in structures (Salawu 1997). In order to increase the
accuracy of damage detection results, many researchers have attempted to
discriminate the changes in modal parameters due to environmental effects from
those due to damage effect. Ni et al. (2005) modelled temperature effects on modal
frequencies of the Ting Kau Bridge using long-term structural health monitoring
data. The support vector machine (SVM) technique was applied to quantify the
effect of temperature on modal frequencies. It shows that the SVM models exhibit
good capabilities for mapping between the temperature and modal frequencies. Kim
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et al. (2003b, 2007) presented vibration-based damage monitoring scheme to give
warming of damage occurrence, damage location and damage severity under
temperature variation conditions using a pattern recognition technique. Peeters and
De Roeck (2001) utilized an ARX model to estimate the effect of temperature
change in natural frequencies and shown the possibility of damage identification
under temperature variation conditions. Cross et al. (2013) attempted to estimate the
effects of temperature, traffic load and wind speed to natural frequencies of the
Tamar Bridge. It was found that the model was accurate for the first mode but less
accurate for higher modes. Most of the previous studies showed that the fluctuation
in natural frequencies after compensated with the change in environmental factors
can be considered as random errors.
Effect of measurement error in damage identification has been studied by
many researchers. Some researchers evaluated measurement noise effect using
deterministic analysis (Shi et al. 2000, Ren and De Roeck 2002). However, this kind
of analysis is not very reliable as only one noised dataset is considered, and different
dataset will give different results. Some other researchers used statistical analysis to
evaluate noise effect (Xia et al. 2002; Xia and Hong, 2003). It has been shown that
the presence of noise reduces the detection probability of actual damage while
increases the probability of false-positive detection. An et al. (2006) suggested a
60%-threshold for detection probability to discriminate the damaged elements. This
threshold has been found appropriate to avoid both false negative and false positive
elements.
2.6.2 Modelling Errors
Many damage identification methods rely on the finite element (FE) model.
However, in practice, it is very hard to obtain an accurate FE model due to the
uncertainties in structural material properties, structural geometric properties,
boundary conditions, structural types, etc. Hence, the damage identification methods
will have great difficulty in distinguishing between the actual damage sites and false
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detection. One possible solution is to update the FE model of the structure to produce
a reliable model (Friswell and Mottershead 1995). Obviously, the quality of the
damage identification results is critically dependent upon the degree of matching
between the FE model and the actual model.
One possible solution is to update the FE model of the structure to produce a
reliable model. However, it was found that the difference between the actual modal
parameters and the updated ones were still significant, especially for large scale
structures. For example, errors of up to 11% in natural frequencies were observed for
the FE model of the Kap Shui Mun Bridge located in Hong Kong after model
updating (Zhang et al. 2000). Also, the updated FE model of the Saftu Link cable-
stayed bridge in Singapore had significant errors in natural frequencies of up to 9%
(Brownjohn and Xia 2000). Moreover, it is worth noting that the number of possible
structural parameters is significantly larger than the number of measured modes.
Therefore, the update of modal parameters does not guarantee the better FE model.
2.6.3 Ill-Posed Problem
The damage identification problem is often ill-posed due to calculation errors
or other uncertainties, which leads to non-uniqueness of the solutions of damage
location and severity. Salawu (1997) reported that the damage identification is only
reliable for elements with high strain energy since only very small change in modal
parameters will be a result of a very large change in structural stiffness of low-strain-
energy elements. The accuracy of damage prediction is higher for the damage
occurring at sections of high modal strain amplitude than for the one at sections of
low modal strain amplitude (Salane and Baldwin 1990). Hsu and Loh (2006)
conducted a damage identification study for a frame structure and reported about
abnormal results at the elements with MSE close to zero. In order to avoid these
false errors, they suggested a criterion for ignoring the elements with low level of
MSE. In another study for beam structures, Wahalathantri et al. (2012) showed some
false errors in the elements close to nodal points. They suggested multiplying the
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damage results by a modification function as a form of normalized modal strain
energy curve.
2.7 SUMMARY OF LITERATURE REVIEW
From the review of literature, the following summary has been drawn:
1) Both natural frequency and mode shape represent information of
structural characteristics. While natural frequency contains global
structural information, mode shape contains spatial structural
information. Use of natural frequencies has limitation on locating
damage but it can be generally measured with higher degree of accuracy.
Use of mode shapes is very suitable for damage identification but the
estimation of mode shapes is more sensitive to noise.
2) Modal strain energy is found to be more sensitive to damage than other
spatial modal parameters such as mode shape, modal curvature and
modal flexibility, but it is more affected by noise and incomplete data.
3) It has been shown that using changes in both natural frequency and mode
shape is more effective for damage identification than using only mode
shape or natural frequency. Use of both parameters would give better
results in case of measurement noise (mainly associated with
experimental mode shapes) and modelling errors (mainly associated with
analytical mode shapes).
4) Damage index-based methods are practical since they use only measured
modal parameters and simple computation to identify damage. However,
these methods are potential to false-positive as well as false-negative
detections.
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5) Correlation-based methods are found to be very effective to identify
damage in single and multiple damage cases. However, sensitivity
analysis of spatial modal parameters relies much on the matching
between numerical mode shapes and experimental ones, and mass-
normalized mode shapes are required. Therefore, errors in analytical
mode shapes may significantly affect the accuracy of the damage
detection methods based on sensitivity of modal strain energy.
6) It is shown that signal-based methods are simple and straightforward, but
cannot provide information of damage extents. Therefore, those methods
are suitable for online SHM and can be used as the first step in the
damage identification processes.
7) Using evolutionary optimisation techniques is an excellent way to solve
damage identification problem. Amongst the optimisation methods, GA
is the most widely used tool for damage detection because of its ease and
powerful performance. However, there are several problems that need to
be overcome. First, the computational process takes a long time to
converge. Second, if more elements are included in the damage search,
more local optima results associated with false detection can be
observed. Third, the selection of modal parameters for the objective
function may affect the accuracy of the damage detection.
8) ANNs is found to be excellent to solve non-linear and noise-infested
damage identification problems. Despite this, their practicality is very
limited since they require a large amount of training data and good
numerical models that accurately reflect the dynamic properties of the
real structures.
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9) Many existing damage identification methods have shown their
capability to cope with level 2 and level 3 of the damage identification
problem. However, these methods have mostly been validated with
numerical models or simple experimental structures. Only few studies
have been conducted for complex structures.
10) It has been shown that measurement and modelling uncertainties
significantly affect damage identification results. Also, calculation error
associated with low-strain-energy elements is another source of
uncertainty in damage identification.
Based on the above literature review, this research explores two effective
damage identification methods based on a new vibration parameter formulated from
modal strain energy and natural frequency. The correlation approach is selected to
formulate the damage identification problem. Also, genetic algorithm (GA) is
selected as an optimisation tool to provide the optimal solution of damage state. The
next chapter will present the development of a damage identification method using
geometric modal strain energy-eigenvalue ratio via correlation approach.
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3Chapter 3: Geometric Modal Strain Energy-
Eigenvalue Ratio (GMSEE) Correlation Method
This chapter describes the development of a correlation-based method using
geometric modal strain energy-eigenvalue ratio (GMSEE). The method utilizes
measured modal parameters (i.e., natural frequency and mode shape) and geometric
information of structural elements such as length and/or thickness depending upon
structure types. This method is able to identify damage location and to estimate
damage extent. The method does not requires modal information obtained from
numerical model; therefore, false detection caused by differences in numerical model
and experimental model can be reduced. This chapter summarizes the theoretical
development, numerical verification, and experimental validation for simple
laboratory models.
3.1 CONVENTIONAL CORRELATION-BASED DAMAGE IDENTIFICATION METHODS
3.1.1 Conventional Correlation-based Damage Identification Methods
Messina et al. (1998) developed a sensitivity- and statistical-based criterion
called multiple damage location assurance criterion (MDLAC) using change in
natural frequency to locate damage as well as to estimate the correlative damage
extent. With similar forms, Shi et al. (2000b) and Wang et al. (2012) employed the
formulation to detect damage using mode shape change and elemental modal strain
energy (MSE) change, respectively. The general formula of the criterion is as
follows:
( ) ( )
2T
Z
T T
. ( )MDLAC ( )
. . ( ) . ( )=
ΔZ δZ δDδD
ΔZ ΔZ ΔZ δD ΔZ δD (3-1)
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where Z represents for a modal feature such as natural frequency, mode shape or
MSE; ΔZ is the measured modal feature change vector; δZ is the analytical modal
feature change vector for a known relative damage extentδD . The damage sites can
be identified by searching the largest MDLAC value using an optimisation
algorithm. The following modal features can be used in Eq. (1) for identifying
damage.
Natural Frequency Change
If natural frequency is used, δZ can be calculated based on the following
sensitivity equation (Messina et al. 1998):
2
1.
8
T
i i k i
T
k i i i
f
D f
=
Φ K Φ
Φ MΦ (3-2)
where if is the analytical frequency change of the ith mode for a known damage size
kD at location k; iΦ is the ith mode shape; kK is the kth elemental stiffness matrix;
and M is the system mass matrix. For this method, the sizes of δZ and ΔZ are
equal to the number of measured modes.
Mode Shape Change (MSC)
If mode shape is used, the sensitivity of the ith mode shape to a known
damage size kD at element k is given by (Shi et al. 2000b):
1
( )Tn
i r k ir
rk r i
r iD =
= −
−
Φ Φ K ΦΦ (3-3)
where iΦ is the ith analytical mode shape change for a damage kD ; i is the ith
eigenvalue; and n is the number of numerical modes used in the calculation (n
should be equal to the total number of degrees-of-freedom (DOFs) of the system).
For the MSC correlation method, the sizes of δZ and ΔZ are equal to the product of
the number of measured modes and the number of degrees of freedom.
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Modal Strain Energy Change (MSEC)
If MSE is used, the sensitivity equation can be expressed as follows (Shi et
al. 2000a)
T
T
1
2 ( )n
ij r k ii j r
k r ir
Ur i
D =
= −
−
Φ K ΦΦ K Φ (3-4)
where ijU is the analytical MSE change of the jth element for the ith mode regarding a
damage kD at element k; and n is the number of numerical modes used in the
calculation. For the MSEC correlation method, the sizes of δZ and ΔZ are equal to
the product of the number of measured modes and the number of elements.
3.2 THEORETICAL DEVELOPMENT OF GMSEE CORRELATION METHOD
Despite their effectiveness on damage identification, the conventional
correlation-based methods still have some limitations as highlighted as follows. The
frequency-based correlation method is found effective for single damage cases, but
its accuracy is limited for multiple damage cases and its requirement on the
availability of a sufficient set of natural frequencies. On the other hand, spatial
modal-based correlation methods using mode shape or modal strain energy are found
more effective than the frequency-based method for multiple damage cases.
However, the accuracy of these methods relies very much on the number of
analytical mode shapes used and the degree of matching between numerical models
and experimental models which is not generally guaranteed for real structures. In
order to overcome the above limitations of the existing methods, a novel modal
vector correlation method is developed using measured modal parameters and
geometric information of structures.
3.2.1 Sensitivity Analysis of GMSEE
For a linear structure, the fractions of modal strain energies at undamaged
and damaged states are given as follows, respectively (Kim and Stubbs 1995):
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 38
© 2017 Khac-Duy Nguyen Page 38
ij
ij
i
UF
U= (3-5)
*d
d
ij ij
ij
i i
U UF
U U
+=
+ (3-6)
where ijF and *
ijF are respectively the fraction of MSE of the jth element for the ith
mode at undamaged and damaged states; Tij i j iU =Φ K Φ is the MSE of the jth
element for the ith mode at undamaged state; Ti i iU =Φ KΦ is the total MSE for the
ith mode at undamaged state; iΦ is the mode shape vector of the ith mode; jK is the jth
elemental stiffness matrix; and K is the system stiffness matrix; d ijU and d iU are
the changes in the jth elemental MSE and the total MSE for the ith mode.
It has been reported that the fractional MSE can be assumed to be unchanged
after damage ( *ij ijF F ) (Kim an Stubbs, 1995, Stubbs and Osegueda 1990). Thus,
Eqs. (3-5) and (3-6) can be combined to the following equation:
d
d
ij ij ij
i i i
U U U
U U U
+=
+ (3-7)
By expanding and rearranging Eq. (3-7), the following expression can be
obtained:
d
d 0iij ij
i
UU U
U− = (3-8)
By neglecting change of mass after damage, the term d /i iU U can be
expressed by the change in eigenvalue (Kim and Stubbs 2002) as follows:
d di i
i i
U
U
= (3-9)
where 2(2 )i if = is the eigenvalue of the ith mode, and d i is the change in the ith
eigenvalue. By substituting Eq. (3-9) into Eq. (3-8), we obtain:
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 39
© 2017 Khac-Duy Nguyen Page 39
d
d 0iij ij
i
U U
− = (3-10)
According to Kim and Stubbs (1995), by taking the rigidity term out of the
elemental stiffness matrix, the elemental MSE at undamaged and damaged states can
be expressed in terms of elemental geometric MSE as follows:
ijij jU k U= (3-11)
* d ( d )( d )ij ijij ij ij j jU U U k k U U= + = + + (3-12)
where T0ij i j iU =Φ K Φ is the geometric MSE of the jth element for the ith mode at
undamaged state; jk is the stiffness constant of the jth element at undamaged state;
0jK contains only the geometric quantity of the jth elemental stiffness matrix; d jk
and d ijU are the change in stiffness constant of the jth element and the change in
geometric MSE of the jth element for the ith mode. Note that, the stiffness constant of
an element is the resistance of the element against external forces, which is
proportional to material properties (e.g., elastic modulus) and cross sectional
properties (e.g., 2nd moment of area for beam elements, cross-sectional area for truss
elements or plate thickness for plate elements). Examples of geometric quantity of
stiffness matrix for some element types are provided in Appendix-A. By substituting
Eq. (3-11) and (3-12) into Eq. (3-10), we obtain:
( )( ) dd d 0i
ij ij ij ijj j j j
i
k k U U k U k U
+ + − − = (3-13)
By expanding, rearranging and neglecting high order terms, Eq. (3-13) leads
to:
d
d diij ij ijj j j
i
U k U k U k
− = − (3-14)
By dividing two sides of Eq. (3-14) by i and jk , the following equation is
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 40
© 2017 Khac-Duy Nguyen Page 40
obtained:
2
ddd ij ij jiij
i i ji
kU UU
k
− = − (3-15)
Equation (3-15) can be rewritten in term of change in ratio of geometric MSE
to eigenvalue (GMSEE) as follows:
T0
d d dij i j i
ij j j
i i
UG D D
= − = −
Φ K Φ
or
T0d
d
ijij i j i
j i i
G U
D = − = −
Φ K Φ (3-16)
where ijij iG U = is the GMSEE of the jth element and the ith mode; and
d d /j j jD k k= is the relative reduction of stiffness of element j.
3.2.2 Locating damage using GMSEE change vector
Following the principle of the correlation-based methods, the damage
identification problem can be transformed to an optimization problem searching for
the best correlation between the measured GMSEE change vector and the analytical
one. As an expansion of the Eq. (3-1), the MDLAC function for GMSEE is defined
as follows:
( ) ( )
2T
GMSEE
T T
Δ .δMDLAC (δ )
Δ .Δ . δ .δ=
G GD
G G G G (3-17)
where ΔG is the measured GMSEE change vector; and δG is the analytical
GMSEE change vector for a known damage vector δD . MDLAC values range from
0 to 1, indicating correlation level from no correlation to exact correlation between
the patterns of GMSEE changes. The damaged elements and their correlative damage
extents can be identified by searching the largest MDLAC value using an
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 41
© 2017 Khac-Duy Nguyen Page 41
optimisation algorithm. If m modes are used, the measured and analytical GMSEE
change vectors are given by the following expressions:
1Δ
...
Δ Δ
...
Δ
i
m
=
G
G G
G
(3-18)
1
...
...
i
m
=
δG
δG δG
δG
(3-19)
where Δ iG is the measured GMSEE change vector for the ith mode which can be
calculated directly from measured modal data and geometric stiffness matrix; δ iG is
the analytical GMSEE change vector for the ith mode and is calculated based on a
sensitivity matrix as follows:
1
1
2
2
d0 ... 0
d
d0 ... 0
dδ δ
... ... ... ...
d0 0 ...
d
i
i
i
in
n
GD
GD
GD
=
G D
or GMSEEδ δi i=G S D (3-20)
where GMSEEiS is the sub-sensitivity matrix of GMSEE for the ith mode; and n is the
number of structural elements. Values of the diagonal entries of the above matrix are
calculated by Eq. (3-16). For convenience, the analytical GMSEE change vector for
all measured modes described in Eq. (3-19) can be rewritten in the following
expression:
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 42
© 2017 Khac-Duy Nguyen Page 42
GMSEE1
GMSEE2
GMSEE
δ δ...
m
=
S
SG D
S
or GMSEEδ δ=G S D (3-21)
where GMSEES is the sensitivity matrix of GMSEE for all measured modes.
3.2.3 Quantifying Damage
As δD obtained from maximizing MDLAC value described in equation (3-
17) is a correlative vector, it means different scales of δD will give the same value of
MDLAC. Therefore, the damage scaling coefficient, C, such that .δC D gives the
actual damage extent in percentage, must be obtained. The followings present two
algorithms used to calculate this coefficient.
3.2.3.1 Quantifying Damage using Natural Frequency Change
Conventionally, the damage scaling coefficient can be obtained using changes
in measured natural frequencies with the first-order approximation as described in
Messina et al. (1998). In this study, their equation is little modified for using all the
measured natural frequencies as follows:
f
Δ
[ .δ ]C =
f
S D (3-22)
where Δf is the measured frequency change vector; and fS is the sensitivity matrix
of natural frequency in which its entries can be calculated as follows:
T
T.
2
i j ii i
j i i
f f
D
=
Φ K Φ
Φ KΦ (3-23)
where if is the analytical frequency change of the ith mode for a known damage size
jD at element j.
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 43
© 2017 Khac-Duy Nguyen Page 43
3.2.3.2 Quantifying Damage using GMSEE Change
Since GMSEE change is used for estimating damage location and correlative
damage extent, it would be more logical to use this parameter for damage extent
estimation. Based on the calculation of GMSEE sensitivity, the following equation
can be used for the prediction of the damage scaling coefficient, C:
GMSEEavg
Δ
[ .δ ]C =
G
S D (3-24)
where GMSEEavgS is the average sensitivity matrix obtained from the pre-damaged
sensitivity matrix GMSEEuS calculated with modal information at undamaged state, and
the post-damaged sensitivity matrix GMSEEdS calculated with modal information at
damaged state. It is worth noting that unlike the frequency sensitivity-based method,
the damage extent estimation method based on GMSEE sensitivity does not require
information of material and cross sectional properties which are generally obtained
with certain errors.
3.3 SUMMARY OF GMSEE CORRELATION DAMAGE IDENTIFICATION METHOD
Figure 3-1 illustrates the schematic of the proposed GMSEE correlation-
based damage identification method. Firstly, one set of vibration responses is
measured as a baseline (undamaged state) and another set is measured later to check
for its damage status (damaged state). Natural frequencies and mode shapes for each
state are extracted from the corresponding vibration response set. Secondly, a
damage detection model is established based on a numerical model (e.g., FEM,
Euler-Bernoulli beam, shear-beam, etc.) but including only necessary information
(e.g., element types and geometries) for the calculation of elemental geometric
stiffness matrices. The measured GMSEE change is then calculated based on the
experimental modal parameters and the elemental geometric stiffness matrices. The
sensitivity matrix of GMSEE is also calculated using the experimental modal
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 44
© 2017 Khac-Duy Nguyen Page 44
parameters and the elemental geometric stiffness matrices, as described in equations
(3-16), (3-20) and (3-21). Subsequently, an optimisation process is run to find the
correlative elemental stiffness reduction vector which gives the best correlation
between the analytical GMSEE change vector and the measured one. Note that the
analytical GMSEE change vector is calculated using the sensitivity matrix
established before. Lastly, the damage scaling coefficient C is calculated based on
either equation (3-22) or equation (3-24), and the estimated damage extent is
obtained.
Compared to other correlation-based methods, the proposed GMSEE
correlation-based method has some advantages as follows: 1) the calculation of
measured and analytical GMSEE change vectors does not require information of
material and sectional properties of elements, making it more practical as well as
avoiding numerical errors; 2) mode shapes are not required to be mass-normalized
which also reduces the effect of numerical errors on damage identification; and 3)
the method does not use numerical modal information; therefore, false detection
caused by differences in numerical and experimental modal parameters is eliminated.
Figure 3-1. Schematic of GMSEE correlation-based damage identification method.
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 45
© 2017 Khac-Duy Nguyen Page 45
3.4 IDENFICATION OF CORRELATIVE DAMAGE VECTOR USING GENETIC ALGORITHM
As stated in chapter 2.5.1.1, Genetic algorithm (GA) is one of the most
widely used and powerful optimisation tools for solving the damage identification
problems (Wang et al. 2012, Chou and Ghaboussi 2001, Hao and Xia 2002). It is
based on the mechanism of biological evolution by a repetitive process of
reproduction, crossover, mutation and selection (Goldberg 1989). In GA, a
population of candidate solutions is evolved toward better solutions. Each candidate
solution can be represented in a form of a string or chromosome. Selection is a
process of selecting a portion of the existing population to breed to a new generation.
A fitness-based process is employed to decide which individual solutions are
selected. The fitter solutions measured by a fitness function are typically selected.
Crossover is a process of taking parent solutions and producing a child solution from
them. Crossover can make a child solution with better fitness compared to its parents.
Mutation is a process of altering one or more gene values in a chromosome. Mutation
can change a candidate solution to a new one with better fitness.
The classic binary-coded GA transforms unknown variables to a binary string
(chromosome or representation). For the damage identification problem, we can
consider the damage extent in percentage and convert it to a binary number. It has
been reported that the binary-coded GA is efficient for finding the solutions.
However, it is not convenient to use as not only it requires the variables to be
encoded to a binary string but also it must decode the string into candidate solutions,
evaluate and return the resulting fitness back to the binary-coded string representing
the evaluated candidate solutions. Also, as the number of variables increase, the size
of the chromosome becomes very large and can lead to poor performance (Yi et al.
2009).
A more convenient encoding type, called real-coded GA, simply describes a
chromosome as a row of real values in which each value represents the candidate
solution of each variable. It does not require such mapping of variables-gene-
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 46
© 2017 Khac-Duy Nguyen Page 46
candidate solutions. Moreover, as the size of the chromosome is the same as the size
of the solution vector, the real-coded GA is capable of searching a large solution
domain. This study adapted the real-coded GA as a search engine for the correlative
damage vector.
3.5 NUMERICAL VERIFICATION
A 2-D numerical truss model presented in Rahai et al. (2007), as shown in
figure 3-2, is used to evaluate the feasibility of the proposed method. The finite
element (FE) model of the truss consists of 25 bar elements of various cross
sectional areas, and 21 active degrees of freedom (DOFs). More details of the
structure can be found in the reference cited above. Damage in the structure is
simulated as a stiffness reduction of individual elements. As shown in Table 3-1,
three damage scenarios are considered with different locations of damage, number of
damaged elements and damage severities.
Figure 3-2. Two-dimensional truss bridge [adapted from Rahai et al. 2007].
Table 3-1. Damage scenarios for 2-D truss model.
Damage
Scenario Damaged elements Cross-sectional area reduction
Case 1 9 20%
Case 2 4
11
20%
30%
Case 3
7
8
10
20%
25%
30%
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 47
© 2017 Khac-Duy Nguyen Page 47
3.5.1 Damage Identification with Noise-free Modal Data
Performance of the proposed GMSEE correlation-based method is compared
with the conventional correlation-based method using modal strain energy (MSE)
change (Wang et al. 2012). For the GMSEE correlation-based method, analytical and
measured GMSEE change vectors of the first 8 modes are calculated. Note that, the
sensitivity matrix of GMSEE is obtained using only the corresponding modes. For
the MSE correlation-based method, analytical and measured MSE change vectors of
the first 8 modes are also calculated. However, different from the GMSEE method,
the MSE sensitivity matrix is obtained by using 21 numerical modes, which are all
the modes of the FE model. This gives the best accuracy for the MSE sensitivity
estimation.
For each method, the correlative damage vector is determined by searching
the greatest MDLAC value using the genetic algorithm (GA). In this study, the GA
optimisation toolbox embedded in MATLAB software package (MATLAB 2012) is
utilized to solve the optimisation problem. The solver parameters are set as follows.
The number of variables is 25 corresponding to the total number of truss elements
The range of the variables is [-1; -1E-10] corresponding to the possible range of
damage extent. The population size should be set large enough in order to reach the
global optimum, but should not be very large as the convergence time will increase.
Herein the population size is set as 200 as of eight time of the number of dimensions
(i.e., 25). The crossover fraction rate does not need to be set very high due to that a
large population size has been defined. In this study, it is set as 0.5. The convergence
tolerance is used as the condition to stop the GA process. , and this parameter should
be small enough to avoid premature convergence. In this study, it is set as 1E-10 . As
this is a constrained optimisation problem, the adaptive feasible mutation function
integrated in the toolbox is used for generating mutated individuals.
Damage identification results by the two correlation-based methods are
illustrated in Figs. 3-3 to 3-5, in which the results of the GMSEE correlation method
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 48
© 2017 Khac-Duy Nguyen Page 48
using the frequency-based quantification technique (using Eq. 3-22) and GMSEE-
based quantification technique (using Eq. 3-24) are separated. For the single damage
case (Case 1), excellent results are obtained by both correlation methods. As shown
in Fig 3-3, it can be seen that both methods predict very similar results with accurate
damage location and accurate damage level. Also, both quantification techniques
used in the GMSEE method give the same prediction of damage extent. For the
double damage case (Case 2), although the MSE correlation method is successful in
locating the actual damaged elements, some false indications of damage are
observed (Figure 3-4). Moreover, the damage extents of the actual damaged
elements are quite underestimated which is 9.4% for element 4 and 23.3% for
element 11. Compared to the MSE correlation method, the GMSEE correlation
method gives better prediction for this damage case. The proposed method predicts
very well the locations of damage with no false indications. Regarding damage
extent estimation, the GMSEE method with any of the two quantification techniques
gives better accuracy for the small damage in element 4, and comparable accuracy
for the damage in element 11. For the triple damage case (Case 3), both correlation
methods correctly predict the locations of damaged elements (Fig 3-5). It is found
that the damage extents predicted by the MSE method are little more accurate than
those obtained by the GMSEE method. Regarding false identification, although both
methods generate some small false indications and those obtained from the GMSEE
method slightly higher possibility of damage, their damage levels are negligible. It is
also observed that the two quantification techniques using the GMSEE method give
very identical results. From this comparison, the proposed GMSEE method shows its
high capability of identifying damage with less modeling effort. Also, it can be
concluded that the GMSEE-based quantification technique can be used as an
alternative way for damage extent estimation in correlation-based methods.
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 49
© 2017 Khac-Duy Nguyen Page 49
Figure 3-3. Damage identification result for 2-D truss model: Case 1: D9 = 20%.
Figure 3-4. Damage identification result for 2-D truss model: Case 2: D4 = 20%,
D11 = 30%.
Figure 3-5. Damage identification result for 2-D truss model: Case 3: D7 = 20%,
D8 = 25%, D10 = 30%.
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
60
Element Number
Dam
age E
xte
nt (%
)
MSE Method
GMSEE Method, Frequency-based Quantification
GMSEE Method, GMSEE-based Quantification
Actual Damage
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
60
Element Number
Dam
age
Exte
nt (%
)
MSE Method
GMSEE Method, Frequency-based Quantification
GMSEE Method, GMSEE-based Quantification
Actual Damage
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
60
Element Number
Dam
age E
xte
nt (%
)
MSE Method
GMSEE Method, Frequency-based Quantification
GMSEE Method, GMSEE-based Quantification
Actual Damage
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 50
© 2017 Khac-Duy Nguyen Page 50
3.5.2 Damage Identification with Measurement Noise
In practice, measurement noise is unavoidable, and therefore, its effect on
damage identification performance should be considered. In this section, the
performance of the GMSEE correlation method under noise condition is examined
and compared with the results obtained by the MSE correlation method.
It was reported that natural frequency is measured with much higher
precision than mode shape (Doebling et al. 1998). Therefore, in this study, three
noise levels of 2% (noise level 1), 5% (noise level 2) and 7% (noise level 3) are
considered for mode shapes, and the corresponding noise levels for natural
frequencies are 0.5%, 1% and 1.25%. It has been reported that these noise levels are
reasonable for mode shapes and natural frequencies (Shi et al. 2000a, Shi et al.
2000b). Noise-contaminated mode shape and noise-contaminated natural frequency
can be simulated as follows (Ren and De Roeck 2002):
(1 )ij ij ij = + (3-25)
(1 )f f
i iif f = + (3-26)
where ij and ij are the mode shape components with noise and without noise,
respectively; if and if are the ith natural frequencies with and without noise,
respectively; and f are the noise levels of mode shapes and natural frequencies,
respectively; ij and f
i are independent random numbers in the range of [-1, 1]. In
order to evaluate the effect of different noise levels on damage identification
performance, the same sets of ij and f
i are used for different noise levels. Note
that, noise is added to modal parameters of both undamaged and damaged models.
Statistical damage identification following the procedure proposed by An et
al (2014) is performed to evaluate the robustness of the improved method under
noise condition. Firstly, at each noise level, 100 identification results are generated
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 51
© 2017 Khac-Duy Nguyen Page 51
for each damage case. Secondly, a damage extent threshold is determined by
examining 100 identification results of the undamaged state under noise.
Considering the predicted damage extents at the undamaged state are a normal
distribution, the following equation is used to calculate the damage extent threshold
with capability of preventing 90% of false detection (Xia et al. 2002):
1.3u u = + (3-27)
where is the damage extent threshold; u and u is respectively the mean value
and the standard deviation of the damage extent distribution of all elements for all
damage identification results at undamaged state. For this truss model, the damage
thresholds are found to be 1.3% for noise level 1, 2.7% for noise level 2, and 4.4%
for noise level 3. Finally, detection probability for each element is estimated by
taking the ratio of number of times that its damage extent exceeds the threshold to
the total number of identification results (i.e., 100). Note that, the GMSEE-based
quantification technique is used to obtain the damage scaling coefficient as it has
similar performance with the frequency-based quantification technique.
Detection probability results by the GMSEE correlation method are
illustrated in Figs. 3-6 to 3-8. Normally, decision can be made on the favour of the
trend with higher probability. That means an element with probability of more than
50% can be considered as damaged. In this study, to be more confident in decision
making, a probability threshold of 60% is used to decide whether or not an element
is damaged. It has been reported that 60% is an appropriate level to avoid both false
negative and false positive elements (An et al. 2014). It is shown that the detection
probabilities of the actual damaged elements in the three damage cases tend to
reduce when measurement noise increases. All actual damaged elements are
successfully detected by the proposed method at the first two noise levels. At the
third noise level, only good results are obtained for Case 3. For Case 2, although
actual damaged elements (elements 4 and 11) are successfully detected, their
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 52
© 2017 Khac-Duy Nguyen Page 52
detection probabilities are quite low. For the single damage case, the damage at
element 9 is not able to be detected at the highest noise level. It is also observed that
the false detection probabilities increase with the noise level. However, the
probabilities of the false elements do not exceed the probability threshold in all three
damage cases. For comparison purpose, results by the MSE correlation method are
also illustrated in Figs. 3-6 to 3-8. It is obvious that the detection probabilities of the
actual damaged elements by the MSE method are similar to those by the GMSEE
method in the first noise level. For Case 2, as noise increases, higher probabilities are
obtained for the actual damaged elements (elements 4 and 11) by using the MSE
method, especially at the highest noise level. However, for this damage case, the
false-positive detection probabilities obtained from the MSE method are
significantly higher than those obtained from the GMSEE method. For the single
damage case (Case 1), the proposed GMSEE method gives higher damage
probability for the actual damage (element 9). For Case 3, both methods have
comparable results when the measurement noise increases.
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 53
© 2017 Khac-Duy Nguyen Page 53
(a) Frequency noise = 0.5%, mode shape noise = 2%
(b) Frequency noise = 1%, mode shape noise = 5%
(c) Frequency noise = 1.25%, mode shape noise = 7%
Figure 3-6. Detection probability results for 2-D truss model under noise: Case 1
(actual damage: element 9)
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
De
tectio
n P
rob
ab
ility
(%
)
MSE Method
GMSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
De
tectio
n P
rob
ab
ility
(%
)
MSE Method
GMSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
De
tectio
n P
rob
ab
ility
(%
)
MSE Method
GMSEE Method
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 54
© 2017 Khac-Duy Nguyen Page 54
(a) Frequency noise = 0.5%, mode shape noise = 2%
(b) Frequency noise = 1%, mode shape noise = 5%
(c) Frequency noise = 1.25%, mode shape noise = 7%
Figure 3-7. Detection probability results for 2-D truss model under noise: Case 2
(actual damage: elements 4 and 11)
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
De
tectio
n P
rob
ab
ility
(%
)
MSE Method
GMSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
De
tectio
n P
rob
ab
ility
(%
)
MSE Method
GMSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
De
tectio
n P
rob
ab
ility
(%
)
MSE Method
GMSEE Method
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 55
© 2017 Khac-Duy Nguyen Page 55
(a) Frequency noise = 0.5%, mode shape noise = 2%
(b) Frequency noise = 1%, mode shape noise = 5%
(c) Frequency noise = 1.25%, mode shape noise = 7%
Figure 3-8. Detection probability results for 2-D truss model under noise: Case 3
(actual damage: elements 7, 8 and 10)
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
De
tectio
n P
rob
ab
ility
(%
)
MSE Method
GMSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
De
tectio
n P
rob
ab
ility
(%
)
MSE Method
GMSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
De
tectio
n P
rob
ab
ility
(%
)
MSE Method
GMSEE Method
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 56
© 2017 Khac-Duy Nguyen Page 56
It is also important to evaluate the effect of noise on damage quantification.
Figures 3-9 to 3-11 show the damage extent results by the GMSEE and MSE
correlation methods by taking the average of all identification results in each case. It
is shown that the actual damaged elements are better distinguished from the
undamaged ones by the proposed GMSEE method than the MSE method.
Particularly for Case 1, it is very hard to distinguish the damaged element (element
9) by the MSE method at the highest noise level while the proposed GMSEE method
is still able to show the damaged element. It is also found that the average damage
extents of the actual damaged elements by both methods become less accurate when
noise increases. Table 3-2 shows the errors of damage extents for the actual damaged
elements compared to the predictions at the noise-free condition. The results by the
GMSEE method have acceptable errors at the low noise level. At the second noise
level, the estimations are significantly affected for Case 1 and Case 3. At the third
noise level, the estimations are significantly affected for all three damage cases.
Despite that, the GMSEE method seems to be less affected by noise than the MSE
method. Except for the higher errors observed in some cases (element 10 in Case 3,
element 4 in Case 2 at high noise levels), most of the estimations by the GMSEE
method have smaller or similar levels of errors compared to those from the MSE
method. In particular, the GMSEE method gives much smaller errors for element 9
in Case 1, element 11 in Case 2 (at noise levels 1 and 2), and element 8 in Case 3.
Moreover, by taking the average errors for all damage cases, it is shown that the
results by the GMSEE method are generally better than those from the MSE method.
In summary, the proposed GMSEE correlation method can be considered as a robust
method when modal data is polluted by noise.
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 57
© 2017 Khac-Duy Nguyen Page 57
(a) Frequency noise = 0.5%, mode shape noise = 2%
(b) Frequency noise = 1%, mode shape noise = 5%
(c) Frequency noise = 1.25%, mode shape noise = 7%
Figure 3-9. Average damage extent results for 2-D truss model under noise: Case 1
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Ave
rage
Da
ma
ge
Exte
nt (%
)
MSE Method
GMSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Ave
rage
Da
ma
ge
Exte
nt (%
)
MSE Method
GMSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Ave
rage
Da
ma
ge
Exte
nt (%
)
MSE Method
GMSEE Method
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 58
© 2017 Khac-Duy Nguyen Page 58
(a) Frequency noise = 0.5%, mode shape noise = 2%
(b) Frequency noise = 1%, mode shape noise = 5%
(c) Frequency noise = 1.25%, mode shape noise = 7%
Figure 3-10. Average damage extent results for 2-D truss model under noise: Case 2
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Ave
rage
Da
ma
ge
Exte
nt (%
)
MSE Method
GMSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Ave
rage
Da
ma
ge
Exte
nt (%
)
MSE Method
GMSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Ave
rage
Da
ma
ge
Exte
nt (%
)
MSE Method
GMSEE Method
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 59
© 2017 Khac-Duy Nguyen Page 59
(a) Frequency noise = 0.5%, mode shape noise = 2%
(b) Frequency noise = 1%, mode shape noise = 5%
(c) Frequency noise = 1.25%, mode shape noise = 7%
Figure 3-11. Average damage extent results for 2-D truss model under noise: Case 3
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Ave
rage
Da
ma
ge
Exte
nt (%
)
MSE Method
GMSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Ave
rage
Da
ma
ge
Exte
nt (%
)
MSE Method
GMSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Ave
rage
Da
ma
ge
Exte
nt (%
)
MSE Method
GMSEE Method
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 60
© 2017 Khac-Duy Nguyen Page 60
Table 3-2. Quantification of damage extent prediction error caused by measurement
noise
Damage
scenario
Damaged
elements
Error by MSE correlation (%) Error by GMSEE correlation (%)
Noise 1 Noise 2 Noise 3 Noise 1 Noise 2 Noise 3
Case 1 9 7.16 13.58 19.34 3.79 9.34 13.16
Case 2 4
11
2.76
9.34
1.74
6.61
2.48
9.40
0.09
2.91
2.99
1.06
5.30
10.7
Case 3
7
8
10
0.04
2.61
0.08
2.20
5.23
7.04
6.50
10.80
13.10
0.94
0.32
3.82
3.05
1.62
11.00
0.10
5.00
14.50
Average error 3.67 6.07 10.27 1.98 4.84 8.13
Noise 1: Frequency noise = 0.5%, mode shape noise = 2%; Noise 2: Frequency noise = 1%,
mode shape noise = 5%; Noise 3: Frequency noise = 1.25%, mode shape noise = 7%.
3.6 EXPERIMENTAL VERIFICATION
It is important to evaluate the performance of the GMSEE correlation method
for experimental models. Herein, damage identification has been carried out for an 8
degree-of-freedom (DOF) system and a three-story shear building model. Both
experiments have been conducted by the Los Alamos National Laboratory (LANL),
USA. In the following experimental case studies, GMSEE change vector is used to
obtain the damage scaling coefficient as described in Eq. (3-24) since this technique
does not require information of structural stiffness.
3.6.1 Case Study 1: LANL 8-DOF System
Figure 3-12 shows the experimental set up of an 8-DOF system carried out by
the LANL to study the effectiveness of various vibration-based damage detection
methods. The system is constructed by a series of eight translating masses connected
by springs. Each mass is an aluminum disc with a center hole. The masses slide on a
highly polished steel rod that constrains them to move only in the translational
direction. At the undamaged state, all springs are identical and have a linear spring
constant. Damage is simulated by replacing the fifth spring with another spring
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 61
© 2017 Khac-Duy Nguyen Page 61
which has stiffness 14% lower than the original one. Horizontal acceleration
responses are measured at each mass, giving a total of eight measured DOFs. The
structure is excited by a random force generated by an electro-dynamic shaker.
Natural frequencies and mode shapes of the first three modes have been used for
damage identification. It is worth noting that the examined damage case is the most
challenging case compared to damages at the extreme ends of the system (i.e.,
springs 1 and 7), as mentioned in Duffey et al. (2001). They also reported that
damage at the level of 14% was more difficult to detect than that at lower and higher
levels.
Figure 3-13 shows the damage identification results obtained by using the
GMSEE correlation method. It is obvious that the damaged spring has been
successfully identified. The estimated severity (67.8%) is found to be quite different
from the simulation (14%). This difference could be caused by the change in friction
between the masses and the steel rod after the spring was replaced. It is well-known
that friction is a source of damping which affects vibration behavior of the structure.
This has been previously reported in the reference (Duffey et al. 2001). Also, no
false alarms have been observed in the damage results predicted by the proposed
method.
Figure 3-12. LANL 8-DOF system (Duffey et al. 2001)
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 62
© 2017 Khac-Duy Nguyen Page 62
Figure 3-13. Damage identification result for 8-DOF system by GMSEE correlation
method.
3.6.2 Case Study 2: LANL Three-story Building Model
Figure 3-14 shows the experimental set up of a 4-DOF shear building model
carried out by the LANL. The model consists of aluminium columns and plates
jointed by bolts with a rigid base sliding on rails that allows movement in the x-
direction only. At each floor, the top and bottom plates (30.5x30.5x2.5cm) are
connected by four aluminium columns (17.7x2.5x0.6cm), forming a 4-DOF system
(with 3 active DOFs). Also, there is a centre column (15.0x2.5x2.5cm) suspended
from the top floor, which is used to as a source of inducing nonlinear behaviours to
the system when it hit a bumper mounted on the second floor. The distance between
the suspended column and the bumper can be adjusted to vary the extent of the
nonlinearity. An electromagnetic shaker was used to give excitation to the base floor
of the system. Four accelerometers with a nominal sensitivity of 1000 mV/g were
mounted to the centre lines of four aluminium floor plates to measure vibration
response of the system in x-direction.
1 2 3 4 5 6 70
20
40
60
80
100
Element Number
Dam
age E
xte
nt (%
)
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 63
© 2017 Khac-Duy Nguyen Page 63
(a) Three-story frame structure (b) Bumper and suspended column
at top floor
Figure 3-14. 4-DOF three-story building from LANL (Figueiredo et al. 2009)
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 64
© 2017 Khac-Duy Nguyen Page 64
Figure 3-15. Basic dimensions of the three story builing model (Figueiredo et al.
2009)
A total of 7 states including the baseline state and the column damage states
are studied. The details of each state condition are provided in Table 3-3. For
example, state 2 refers to a 50% stiffness reduction in the column located between
the base and the first floor, in the intersection of plane B and D which is defined in
Fig. 3-15. For each state, floor’s equivalent damage extent is provided in Table 3-3,
which introduces the average reduction of stiffness of all four columns in each floor.
For example, 50% stiffness reduction in column 1BD (column BD in level 1) is
equivalent to 12.5% stiffness reduction in the first floor. It is worth noting that there
was no impact between the bumper and the suspended column during the excitation
to avoid nonlinear behaviour of the structure.
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 65
© 2017 Khac-Duy Nguyen Page 65
Table 3-3. Damage states of the three-story shear building model
State State condition Floor’s equivalent
stiffness reduction
State 1 Baseline condition 0
State 2 Column 1BD – 50% stiffness reduction 12.5%
State 3 Column 1AD + 1BD – 50% stiffness reduction 25%
State 4 Column 2BD – 50% stiffness reduction 12.5%
State 5 Column 2AD + 2BD – 50% stiffness reduction 25%
State 6 Column 3BD – 50% stiffness reduction 12.5%
State 7 Column 3AD + 3BD – 50% stiffness reduction 25%
Up to three vibration modes were extracted from the acceleration responses
using the frequency domain decomposition (FDD) approach. Mode shapes and
natural frequencies of the these modes for baseline condition are provided in Figure
3-16. In order to increase the difficulty level for the damage identification task, only
the first mode is used. Statistical damage identification was also conducted for this
experiment. For the undamaged state and each damaged state, 5 data sets of the first
vibration mode were extracted. As a result, 25 damage identification results can be
obtained for each damage state. Also, by cross-checking among the undamaged data
sets, 20 damage identification results can be obtained for the baseline state. From the
baseline results, a threshold of 10% was determined based on Eq. (3-27).
Figure 3-17 shows the damage identification results obtained by using the
GMSEE correlation method. As shown in Fig. 3-17(c)-(f), excellent results are
obtained by the GMSEE method for damage states 3-4 referring stiffness reduction
in floor 2 and damage states 5-6 referring stiffness reduction in floor 3. Detection
probability of the actual damage is found 100% in these damage states. Regarding
the damage states in the first floor, as shown in Fig. 3-17(a)-(b), the proposed
method is only able to detect damage confidently for the more severe damage case.
For state 1 with 12.5% damage in the first floor, the GMSEE method gives a low
damage probability (about 55%) for the actual damaged floor. For the greater
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 66
© 2017 Khac-Duy Nguyen Page 66
damage state in the first floor, the original method shows a higher detection
probability of about 65% for the first floor but there are some false errors of about
15% and 40% in the second and the third floors, respectively.
Average damage extent results are shown in Fig. 3-18. It can be seen that the
damage extents estimated by the proposed method match well with the actual
damage extents for the first two states, as shown in Fig. 3-18(a-b). Results for other
states show overestimation by about 10%-30%. However, these errors can be
considered to be acceptable for this experiment since the replacement of column
members might cause some model uncertainties such as change in joint stiffness.
Besides, the baseline condition might be changed after the damaged column was
replaced back with the intact column, but only one baseline state was examined for
all damage states. Regarding the change of damage extent, the GMSEE method is
able to indicate the increasing trend of damage extent in each floor shown in each
pair of the plots in Fig. 3-18: (a) and (b), (c) and (d), and (e) and (f).
(a) Mode 1 (30.87 Hz) (b) Mode 2 (53.55 Hz) (c) Mode 3 (71.19 Hz)
Figure 3-16. Modal parameters of baseline model
-1 -0.5 0 0.5 1Base
Floor 1
Floor 2
Floor 3
-1 -0.5 0 0.5 1Base
Floor 1
Floor 2
Floor 3
-1 -0.5 0 0.5 1Base
Floor 1
Floor 2
Floor 3
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 67
© 2017 Khac-Duy Nguyen Page 67
(a) State 2: damage at 1st floor (12.5%) (b) State 3: damage at 1st floor (25%)
(c) State 4: damage at 2nd floor (12.5%) (d) State 5: damage at 2nd floor (25%)
(e) State 6: damage at 3rd floor (12.5%) (f) State 7: damage at 3rd floor (25%)
Figure 3-17. Damage probability results for the shear building by GMSEE method
0 10 20 30 40 50 60 70 80 90 100
Floor 1
Floor 2
Floor 3
Detection Probability (%)
0 10 20 30 40 50 60 70 80 90 100
Floor 1
Floor 2
Floor 3
Detection Probability (%)
0 10 20 30 40 50 60 70 80 90 100
Floor 1
Floor 2
Floor 3
Detection Probability (%)
0 10 20 30 40 50 60 70 80 90 100
Floor 1
Floor 2
Floor 3
Detection Probability (%)
0 10 20 30 40 50 60 70 80 90 100
Floor 1
Floor 2
Floor 3
Detection Probability (%)
0 10 20 30 40 50 60 70 80 90 100
Floor 1
Floor 2
Floor 3
Detection Probability (%)
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 68
© 2017 Khac-Duy Nguyen Page 68
(a) State 2: damage at 1st floor (12.5%) (b) State 3: damage at 1st floor (25%)
(c) State 4: damage at 2nd floor (12.5%) (d) State 5: damage at 2nd floor (25%)
(e) State 6: damage at 3rd floor (12.5%) (f) State 7: damage at 3rd floor (25%)
Figure 3-18. Average damage extent results for the shear building by GMSEE method
0 10 20 30 40 50 60 70 80
Floor 1
Floor 2
Floor 3
Average Damage Extent (%)
GMSEE Method
Actual Damage
0 10 20 30 40 50 60 70 80
Floor 1
Floor 2
Floor 3
Average Damage Extent (%)
GMSEE Method
Actual Damage
0 10 20 30 40 50 60 70 80
Floor 1
Floor 2
Floor 3
Average Damage Extent (%)
GMSEE Method
Actual Damage
0 10 20 30 40 50 60 70 80
Floor 1
Floor 2
Floor 3
Average Damage Extent (%)
GMSEE Method
Actual Damage
0 10 20 30 40 50 60 70 80
Floor 1
Floor 2
Floor 3
Average Damage Extent (%)
GMSEE Method
Actual Damage
0 10 20 30 40 50 60 70 80
Floor 1
Floor 2
Floor 3
Average Damage Extent (%)
GMSEE Method
Actual Damage
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 69
© 2017 Khac-Duy Nguyen Page 69
3.7 CONCLUSIONS
The GMSEE correlation method was developed for locating damage and
estimating damage extent in structures. The performance of the method was verified
through numerical simulations and laboratory experiments. Results from the
numerical study showed that the proposed method could predict damage location and
damage extent with reasonable accuracy compared to the MSE correlation method.
Also, the GMSEE correlation demonstrated its robustness under noise conditions.
Performance of the proposed method was equivalent with the MSE correlation
method with regard to damage detection probability and false alarm probability.
Although the GMSEE method gave worse results for the double-damage case,
similar or better results were obtained by the proposed method for the single-damage
and the triple-damage cases. It is worth noting that the proposed GMSEE method is
more practical than the traditional MSE method because mode shapes are not
necessary to be mass-normalized, only geometric information of structural elements
is required, and only experimental modal information is required. Moreover, damage
quantification results from the proposed method were found to be less affected by
noise than those from the existing method. By applying this method to the
experimental models, the actual damaged elements were accurately located and their
damage extents were somewhat successfully quantified. Apart from the small
damage case in the first floor of the shear building model, other damage elements
were successfully located by the GMSEE method. Also, damage trends were
successfully detected by the proposed method.
Chapter 3: Geometric Modal Strain Energy-Eigenvalue Ratio (GMSEE) Correlation Method Page 70
© 2017 Khac-Duy Nguyen Page 70
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 71
© 2017 Khac-Duy Nguyen Page 71
4Chapter 4: Modal Strain Energy-Eigenvalue
Ratio (MSEE) Correlation Method
The GMSEE correlation method proposed in Chapter 3 is based on the
assumption that the fractional modal strain energy is unchanged after damage. In
order to dominate the errors caused from this assumption, a good number of modes
should be used for constructing the GMSEE change vector. This chapter describes a
modified version of the correlation method proposed in Chapter 3, which is able to
reduce fault detection caused by the above assumption. The modified method uses
modal strain energy-eigenvalue ratio (MSEE) instead of geometric modal strain
energy-eigenvalue ratio (GMSEE). Similar to the original method, the MSEE
correlation method does not require numerical modal information. However, the
method needs information of elements’ rigidity. This chapter summarizes the
theoretical development for the MSEE correlation method, numerical verification,
and experimental validation for simple laboratory models.
4.1 SENSITIVITY ANALYSIS FOR MODAL STRAIN ENERGY-EIGENVALUE RATIO (MSEE)
4.1.1 Change in Elemental MSEE
By multiplying two sides of Eq. (3-16) by the jth stiffness constant kj, we have
the change in modal strain energy-eigenvalue ratio (MSEE):
T
d d di j i MSEE
ij j ij j
i
MSEE D S D
= − = −Φ K Φ
(4-1)
where Tij i j i iMSEE =Φ K Φ is the MSEE of the jth element and the ith mode; and
MSEEijS is the sensitivity coefficient for element j.
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 72
© 2017 Khac-Duy Nguyen Page 72
4.1.2 Change in Total MSEE
It should be noted that Eq. (3-16) is a simplified expression of the following
equation by ignoring the change in fractional modal strain energy:
TT0
d d di j ii i
ij ij j
i j i
G F Dk
= −Φ K ΦΦ KΦ
(4-2)
By multiplying two sides of Eq. (4-2) by the jth stiffness constant kj, and
taking the summation for all elements, we have the change in total modal strain
energy-eigenvalue ratio:
TT
1 1 1
d d dn n n
i j ii iij ij j
j j ji i
MSEE F D = = =
= − Φ K ΦΦ KΦ
(4-3)
Considering the fact that the total change in fractional modal strain energy is
zero (1
d 0n
ij
j
F=
= ), Eq. (4-3) can be rewritten as follows:
T
1 1
d dn n
i j i MSEEi j ij j
j ji
TMSEE D S dD= =
= − = − Φ K Φ
(4-4)
where iTMSEE is the total modal strain energy-eigenvalue ratio (total MSEE) of
mode i and can be calculated from measured mode shape and eigenvalue as
Ti i i iTMSEE =Φ KΦ .
Different from the calculation for the elemental MSEE change in Eq. (4-1),
the equation for the total MSEE change is an exact expression without considering
the assumption that the fractional modal strain energy is unchanged. However, the
total MSEE is a global parameter which is less sensitive to stiffness reduction in each
member. Therefore, an appropriately combined use of these two parameters may help
to improve the damage prediction. The following section will present a combined use
of these two parameters for damage identification.
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 73
© 2017 Khac-Duy Nguyen Page 73
4.2 DAMAGE IDENTIFICATION USING MSEE CHANGE
For locating damage, the multiple damage location assurance criterion
(MDLAC) is modified for MSEE change vector as follows:
( ) ( )
2T
MSEE
T T
Δ .δMDLAC (δ )
Δ .Δ . δ .δ=
MSEE MSEED
MSEE MSEE MSEE MSEE (4-5)
where ΔMSEE is the measured MSEE change vector including the elemental MSEE
change vector and the total MSEE change; and δMSEE is the analytical MSEE
change vector for a known damage vector δD . These vectors can be expressed as
follows:
1
1Δ
...
Δ
...
Δ
i
i
m
m
TMSEE
TMSEE
TMSEE
=
ΔMSEE
ΔMSEEΔMSEE
ΔMSEE
(4-6)
1
1
...
...
i
i
m
m
TMSEE
TMSEE
TMSEE
=
δMSEE
δMSEEδMSEE
δMSEE
(4-7)
where Δ iMSEE is the measured elemental MSEE change vector for the ith mode
which can be calculated directly from measured modal data and elemental stiffness
matrix; iTMSEE is the measured total MSEE change vector for the ith mode which
can be calculated directly from measured modal data and system stiffness matrix;
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 74
© 2017 Khac-Duy Nguyen Page 74
δ iMSEE is the analytical elemental MSEE change vector for the ith mode; and
iTMSEE is the analytical total MSEE change vector for the ith mode which is
calculated by Eq. (4-4). Herein, δ iMSEE is calculated based on a sensitivity matrix
as follows:
1
1
2
2
d0 ... 0
d
d0 ... 0
dδ δ
... ... ... ...
d0 0 ...
d
i
i
i
in
n
MSEED
MSEED
MSEED
=
MSEE D
or MSEEδ δi i=MSEE S D (4-8)
where MSEEiS is the sub-sensitivity matrix of MSEE for the ith mode; and n is the
number of structural elements. Value of each diagonal entry of the above matrix is
calculated by Eq. (4-1).
After the correlative damage vector δD is identified by maximizing the
MDLAC function in Eq. (4-5), the damage extent can be obtained with the scaling
coefficient C calculated by the following expression:
MSEEavg
Δ
[ .δ ]C =
MSEE
S D (4-9)
where MSEEavgS is the average sensitivity matrix obtained from the pre-damaged
sensitivity matrix MSEEuS calculated with modal information at undamaged state, and
the post-damaged sensitivity matrix MSEEdS calculated with modal information at
damaged state. Herein, the undamaged stiffness matrix can be used to calculate
MSEEdS as the damaged stiffness matrix is unknown.
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 75
© 2017 Khac-Duy Nguyen Page 75
4.3 SUMMARY OF MSEE CORRELATION DAMAGE IDENTIFICATION METHOD
In summary, the MSEE correlation method incorporates two parameters
which are elemental MSEE and total MSEE. The sensitivity formula for the first
parameter is less accurate due to the assumption that the fractional elemental MSE is
unchanged after damage. Using change in the elemental MSEE is good for
identifying damage location as it directly evaluate the change in stiffness of each
element. However, it potentially generates false positive detection due to the errors
caused by the assumption, especially at low-sensitivity elements. Meanwhile, as the
change in total MSE is calculated without the above assumption, it can be used to
refine the prediction result in which the false positive detection is reduced.
Figure 4-1 illustrates the schematic of the proposed MSEE correlation-based
damage identification method. The procedure is very similar to the one in Fig. 3-1
for GMSEE correlation method. The main difference is that the damage detection
model requires not only types and geometries of the elements but also their material
and section properties. Also, for the MSEE-based damage identification procedure,
both elemental MSEE and total MSEE need to be calculated.
Figure 4-1. Schematic of MSEE correlation-based damage identification method.
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 76
© 2017 Khac-Duy Nguyen Page 76
4.4 NUMERICAL VERIFICATION
The 2-D numerical truss model as shown in Fig. 3-2 is used to evaluate the
feasibility of the proposed method. Three damage scenarios shown in Table 3-1 are
considered in this section. Also, the GA optimisation approach is used for searching
optimal solution of the correlative damage vector. The same GA configuration to the
one in Chapter 3 is defined for this verification.
4.4.1 Damage Identification with Noise-free Modal Data
This section will compare the damage identification results by the GMSEE
and the MSEE correlation method using fewer modes. Only the first 4 modes of the
truss model are used in this section. Damage identification results by the MSE,
GMSEE and MSEE correlation methods using the first 4 modes are illustrated in Fig.
4-2. As shown in Fig. 4-2, for all three damage cases, both the MSE and GMSEE
correlation methods are able to identify the actual damaged element with reasonable
accuracy. However, both methods generate some false positive damage
identification. On the other hand, it is shown in Fig. 4-2 that the modified method
using MSEE change vector give better results in which the actual damaged elements
are identified accurately and no false positive damage indication is observed. It is
worth noting that most of the damage extent results for the actual damage elements
by the three methods are comparable. Only damage extent of element 10 in Case 3 is
well overestimated by the GMSEE and MSEE methods compared to the traditional
MSE method but this is expectable as fewer modes are used and both proposed
methods rely on the assumption mentioned in Chapter 3.2.1 (i.e., *ij ijF F ) to simplify
the damage identification problem. From this comparison, the proposed MSEE
correlation method shows its higher capability of identifying damage compared to
the GMSEE correlation method. Compared to the traditional MSE correlation
method, the MSEE method can be considered more accurate regarding false
identification and more practical as it requires much less information for solving the
damage identification problem. However, there is still room for future improvement
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 77
© 2017 Khac-Duy Nguyen Page 77
as both GMSEE and MSEE overestimate the damage extent for severe damage when
fewer modes are used.
(a) Case 1: D9 = 20%
(b) Case 2: D4 = 20%, D11 = 30%
(c) Case 3: D7 = 20%, D8 = 25%, D10 = 30%
Figure 4-2. Damage identification results for the 2-D truss model by MSE, GMSEE
and MSEE methods using first 4 modes
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
60
Element Number
Dam
age
Exte
nt (%
)
MSE Method
GMSEE Method
MSEE Method
Actual Damage
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
60
Element Number
Dam
age
Exte
nt (%
)
MSE Method
GMSEE Method
MSEE Method
Actual Damage
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
60
Element Number
Dam
age
Exte
nt (%
)
MSE Method
GMSEE Method
MSEE Method
Actual Damage
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 78
© 2017 Khac-Duy Nguyen Page 78
4.4.2 Damage Identification with Measurement Noise
As the lowest noise level did not affect much on performance of the damage
identification methods, this section will evaluate the robustness of the MSEE
correlation method under the high noise levels (i.e., level 2: mode shape noise is 5%,
natural frequency noise is 1%; and level 3: mode shape noise is 7%, natural
frequency noise is 1.25%). Damage results by the MSEE correlation method using
the first 8 modes and the first 4 modes are compared. The same sets of noised modal
data as in Chapter 3 have been used. A similar approach for damage probability
analysis has also been applied. Again, a probability threshold of 60% is used to
decide whether or not an element is damaged.
4.4.2.1 Damage Identification Results using 8 modes
Detection probability results by the MSEE correlation method using the first
8 modes are illustrated in Figs. 4-3 to 4-5. For comparison, results of the MSE and
GMSEE methods are replotted in the same figures. It is shown that all actual
damaged elements are successfully detected by the MSEE method at the high noise
levels. For Case 1, the detection probability level for the actual damaged element
(element 9) is significantly higher than those obtained by the MSE and GMSEE
methods. Also, only the MSEE method is able to detect this damage at the highest
noise level. Other two methods show a probability of the actual damaged element
below the 60%-threshold. For the double damage case (Case 2), although the MSEE
method shows smaller detection probabilities for element 4 in noise level 2 and for
element 11 in noise level 3 compared to those of the MSE method, the differences
between them are not significant. The MSEE method can be considered to have
similar performance to the MSE method for this case. Compared to the original
GMSEE method, the MSEE method has much improved the results. As shown in
Fig. 4-4(b), the detection probabilities of element 4 and 11 have respectively
increased from 63% to 89% and from 68% to 98% by using the MSEE method. For
the third damage case, the MSEE method slightly improves the results of the original
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 79
© 2017 Khac-Duy Nguyen Page 79
GMSEE method. Also, results of the MSEE method are comparable with those of the
traditional MSE method. Regarding false positive detection, it is observed that the
detection probabilities of the undamaged elements by the MSEE method do not
exceed the 60%-threshold in all three damage cases. Also, false detection
probabilities by the MSEE method seem to be smaller than those obtained from the
other two methods.
(a) Noise level 2: frequency noise = 1%, mode shape noise = 5%
(b) Noise level 3: frequency noise = 1.25%, mode shape noise = 7%
Figure 4-3. Detection probability results by MSE, GMSEE and MSEE correlation
methods under high noise levels using 8 modes: Case 1
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
Dete
ction P
robab
ility
(%
)
MSE Method
GMSEE Method
MSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
Dete
ction P
robabili
ty (
%)
MSE Method
GMSEE Method
MSEE Method
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 80
© 2017 Khac-Duy Nguyen Page 80
(a) Noise level 2: frequency noise = 1%, mode shape noise = 5%
(b) Noise level 3: frequency noise = 1.25%, mode shape noise = 7%
Figure 4-4. Detection probability results by MSE, GMSEE and MSEE correlation
methods under high noise levels using 8 modes: Case 2
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
De
tectio
n P
robab
ility
(%
)
MSE Method
GMSEE Method
MSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
Dete
ction P
robabili
ty (
%)
MSE Method
GMSEE Method
MSEE Method
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 81
© 2017 Khac-Duy Nguyen Page 81
(a) Noise level 2: frequency noise = 1%, mode shape noise = 5%
(b) Noise level 3: frequency noise = 1.25%, mode shape noise = 7%
Figure 4-5. Detection probability results by MSE, GMSEE and MSEE correlation
methods under high noise levels using 8 modes: Case 3
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
De
tectio
n P
robab
ility
(%
)
MSE Method
GMSEE Method
MSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
Dete
ction P
robabili
ty (
%)
MSE Method
GMSEE Method
MSEE Method
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 82
© 2017 Khac-Duy Nguyen Page 82
Effect of noise on damage quantification is also illustrated in Figs. 4-6 to 4-8.
It is found that the actual damaged elements are more distinguished from the
undamaged ones by using the MSEE method compared to the results obtained by the
MSE and GMSEE methods, especially for Case 1. As shown in Fig. 4-6(b), the
MSEE method still shows a good prediction for the damage at element 9 under noise
level 3, whereas the traditional MSE method even cannot indicate element 9 as
damaged. Table 4-1 shows the errors of damage extents of the actual damaged
elements by the three methods compared to their predictions at the noise-free
condition. As shown in the table, the MSEE method seems to be less affected by
noise than the other two methods. Most of the predictions by the MSEE method
show less errors compared to those obtained from the other two methods. Some
exceptions are observed for element 4 at noise level 2, element 11 at noise level 2,
and element 7 at both noise levels. However, the differences for these cases are not
very noticeable. By taking the average errors for all damage cases, it is shown that
the results by the MSEE method are generally better than those from the MSE and
GMSEE methods.
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 83
© 2017 Khac-Duy Nguyen Page 83
(a) Noise level 2: frequency noise = 1%, mode shape noise = 5%
(b) Noise level 3: frequency noise = 1.25%, mode shape noise = 7%
Figure 4-6. Average damage extent results by MSE, GMSEE and MSEE correlation
methods under high noise levels using 8 modes: Case 1
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Ave
rage
Da
mag
e E
xte
nt (%
)
MSE Method
GMSEE Method
MSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Avera
ge D
am
age E
xte
nt (%
)
MSE Method
GMSEE Method
MSEE Method
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 84
© 2017 Khac-Duy Nguyen Page 84
(a) Noise level 2: frequency noise = 1%, mode shape noise = 5%
(b) Noise level 3: frequency noise = 1.25%, mode shape noise = 7%
Figure 4-7. Average damage extent results by MSE, GMSEE and MSEE correlation
methods under high noise levels using 8 modes: Case 2
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Ave
rage
Da
mag
e E
xte
nt (%
)
MSE Method
GMSEE Method
MSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Avera
ge D
am
age E
xte
nt (%
)
MSE Method
GMSEE Method
MSEE Method
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 85
© 2017 Khac-Duy Nguyen Page 85
(a) Noise level 2: frequency noise = 1%, mode shape noise = 5%
(b) Noise level 3: frequency noise = 1.25%, mode shape noise = 7%
Figure 4-8. Average damage extent results by MSE, GMSEE and MSEE correlation
methods under high noise levels using 8 modes: Case 3
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Ave
rage
Da
mag
e E
xte
nt (%
)
MSE Method
GMSEE Method
MSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Avera
ge D
am
age E
xte
nt (%
)
MSE Method
GMSEE Method
MSEE Method
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 86
© 2017 Khac-Duy Nguyen Page 86
Table 4-1. Quantification of damage extent prediction error caused by measurement
noise using 8 modes
Damage
scenario
Damaged
elements
MSE Error (%) GMSEE Error (%) MSEE Error (%)
Noise 2 Noise 3 Noise 2 Noise 3 Noise 2 Noise 3
Case 1 9 13.58 19.34 9.34 13.16 7.00 7.60
Case 2 4
11
1.74
6.61
2.48
9.40
2.99
1.06
5.30
10.7
4.60
2.20
2.1
4.1
Case 3
7
8
10
2.20
5.23
7.04
6.50
10.80
13.10
3.05
1.62
11.00
0.10
5.00
14.50
2.60
0.90
9.20
1.8
2.8
12.1
Average error 6.07 10.27 4.84 8.13 4.42 5.08
Noise 2: Frequency noise = 1%, mode shape noise = 5%; Noise 3: Frequency noise = 1.25%, mode shape noise = 7%.
4.4.2.2 Damage Identification Results using 4 modes
Damage detection probability results by the MSEE correlation method using
the first 4 modes are illustrated in Figs. 4-9 to 4-11. For comparison, results of the
traditional MSE correlation method are also plotted in Figs. 4-9 to 4-11. It should be
noted that damage identification by the GMSEE method is not performed since the
method does work very well with fewer modes in noise-free condition (Fig 4.2). It is
showed in Figs. 4-9 to 4.11 that the detection probabilities of the actual damaged
elements by the MSEE method reduce when fewer modes are used. In particular,
damage at element 4 of Case 2 is not detected at noise level 2, and showing slightly
beyond the threshold at noise level 3. However, these results are expected because
the sizes of the numerical and measured MSEE vectors are reduced that leads to
larger influence of noise to the correlation function. Despite the reduction in the
detection probability, most of the damaged elements are well detected. Moreover, the
detection probabilities of undamaged elements do not exceed the probability
threshold in all three damage cases. Compared to the results of the traditional MSE
method, the proposed MSEE method is outperforming in all three damage cases. The
MSEE method gives better detection probabilities for all the damaged elements. As
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 87
© 2017 Khac-Duy Nguyen Page 87
shown in Figs. 4-9 and 4-10, the MSE method not only fails to detect the two
damages at element 4 and 11 in Case 2 at noise level 3 but also fails to detect the
single damage of Case 1 at the two noise levels. Also, as shown in Fig. 4-11, false
detection of above 60% is observed at several elements such as elements 11, 15 and
23 at noise level 2, and elements 13 and 14 at noise level 3.
(a) Noise level 2: frequency noise = 1%, mode shape noise = 5%
(b) Noise level 3: frequency noise = 1.25%, mode shape noise = 7%
Figure 4-9. Detection probability results by MSE and MSEE correlation methods
under high noise levels using 4 modes: Case 1
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
Dete
ction
Pro
babili
ty (
%)
MSE Method
MSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
Dete
ction P
robabili
ty (
%)
MSE Method
MSEE Method
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 88
© 2017 Khac-Duy Nguyen Page 88
(a) Noise level 2: frequency noise = 1%, mode shape noise = 5%
(b) Noise level 3: frequency noise = 1.25%, mode shape noise = 7%
Figure 4-10. Detection probability results by MSE and MSEE correlation methods
under high noise levels using 4 modes: Case 2
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
De
tectio
n P
robab
ility
(%
)
MSE Method
MSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
Dete
ction P
robabili
ty (
%)
MSE Method
MSEE Method
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 89
© 2017 Khac-Duy Nguyen Page 89
(a) Noise level 2: frequency noise = 1%, mode shape noise = 5%
(b) Noise level 3: frequency noise = 1.25%, mode shape noise = 7%
Figure 4-11. Detection probability results by MSE and MSEE correlation methods
under high noise levels using 4 modes: Case 3
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
De
tectio
n P
robab
ility
(%
)
MSE Method
MSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
20
40
60
80
100
Element Number
Dete
ction P
robabili
ty (
%)
MSE Method
MSEE Method
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 90
© 2017 Khac-Duy Nguyen Page 90
Damage quantification results are shown in Figs. 4-12 to 4-14. Similar to the
results obtained with 8 modes, the average damage extent results by MSEE method
are more readily compared to the MSE method. The damaged elements are much
more distinguished from the undamaged ones by using the MSEE method. It is also
found that the damage extents of undamaged elements become greater when fewer
modes are used. This is expected as noise effect becomes greater as fewer modes are
used. Table 4-2 shows the errors of damage extents of the actual damaged elements
by the two methods compared to their predictions at the noise-free condition. Similar
to the results with 8 modes, the MSEE method seems to be less affected by noise
than the MSE method. Most of the predictions by the MSEE method show less errors
compared to those obtained from the other two methods. Only one exception is
observed for element 10 at noise level 3 where the error by the MSEE method is little
higher than that of the MSE method. Considering the average errors, the values
obtained by the MSEE method are significantly smaller than those from the MSE
method.
In summary, the improved MSEE correlation method can be considered as a
more robust method compared to the traditional MSE method as well as the original
GMSEE method when modal data is polluted by noise. Although performance of the
improved method reduces when fewer modes are used, it is still able to detect most
of the damage with good accuracy under noise of up to 1.25% in natural frequencies
and 7% in mode shapes.
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 91
© 2017 Khac-Duy Nguyen Page 91
(a) Noise level 2: frequency noise = 1%, mode shape noise = 5%
(b) Noise level 3: frequency noise = 1.25%, mode shape noise = 7%
Figure 4-12. Average damage extent results by MSE and MSEE correlation methods
under high noise levels using 4 modes: Case 1
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Ave
rage
Da
mag
e E
xte
nt (%
)
MSE Method
MSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Avera
ge D
am
age E
xte
nt (%
)
MSE Method
MSEE Method
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 92
© 2017 Khac-Duy Nguyen Page 92
(a) Noise level 2: frequency noise = 1%, mode shape noise = 5%
(b) Noise level 3: frequency noise = 1.25%, mode shape noise = 7%
Figure 4-13. Average damage extent results by MSE and MSEE correlation methods
under high noise levels using 4 modes: Case 2
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Ave
rage
Da
mag
e E
xte
nt (%
)
MSE Method
MSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Avera
ge D
am
age E
xte
nt (%
)
MSE Method
MSEE Method
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 93
© 2017 Khac-Duy Nguyen Page 93
(a) Noise level 2: frequency noise = 1%, mode shape noise = 5%
(b) Noise level 3: frequency noise = 1.25%, mode shape noise = 7%
Figure 4-14. Average damage extent results by MSE and MSEE correlation methods
under high noise levels using 4 modes: Case 3
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Ave
rage
Da
mag
e E
xte
nt (%
)
MSE Method
MSEE Method
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
Element Number
Avera
ge D
am
age E
xte
nt (%
)
MSE Method
MSEE Method
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 94
© 2017 Khac-Duy Nguyen Page 94
Table 4-2. Quantification of damage extent prediction error caused by measurement
noise using 4 modes
Damage
scenario
Damaged
elements
MSE Error (%) MSEE Error (%)
Noise 2 Noise 3 Noise 2 Noise 3
Case 1 9 18.00 18.29 8.30 10.30
Case 2 4
11
18.34
30.82
18.93
31.10
8.90
13.70
6.00
12.30
Case 3
7
8
10
6.20
8.70
15.00
6.10
11.20
14.20
3.00
0.50
13.60
1.50
4.00
15.40
Average error 16.18 16.64 8.00 8.25
Noise 2: Frequency noise = 1%, mode shape noise = 5%; Noise 3: Frequency noise = 1.25%, mode shape noise = 7%.
4.4.3 Damage Identification with Modelling Errors
One difference between the original GMSEE method and MSEE method is
that the MSEE method needs information of stiffness constants of all elements. This
section will evaluate the effect of modelling errors in stiffness constants on the
performance of the MSEE method. Results of the traditional MSE correlation
method are also generated for comparison, as this method also relies on numerical
model.
Elastic modulus of the FE model is assumed consisting of normally
distributed random errors with zero means and specific variances. The elastic
modulus of the errored FE model can be calculated as follows:
(1 )E Ej j jE E = + (4-10)
where jE is the elastic modulus of element j in the errored FE model; jE is elastic
modulus of element j in the correct FE model (or the structure that damage
identification is performed for); E is the error level of the elastic modulus; Ej is a
random number with zero mean and variance of 1. Two error levels are considered in
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 95
© 2017 Khac-Duy Nguyen Page 95
this study as 10% and 20%.
Damage identification results using 4 modes with modelling errors are shown
in Figs. 4-15 to 4-17. It is shown that the traditional MSE method is significantly
affected by modelling errors. The quantification accuracy for the actual damaged
elements is much reduced when modelling error increases. Also, false positive errors
become greater when higher modelling error is considered. Compared to the MSE
method, it can be seen that the MSEE method presented in this thesis is less affected
by modelling error. The error in damage quantification for the actual damaged
elements is small even with the modelling error of 20%. Moreover, almost no false
positive errors are generated by the MSEE method.
The results in this section demonstrates that the proposed method is more
practical compared to the traditional MSE method since a correct FE model is not
required. The reason for this is that the traditional MSE method relies not only on
structural information but also on numerical modal parameters, whereas the MSEE
method uses measured modal parameters directly.
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 96
© 2017 Khac-Duy Nguyen Page 96
(a) Results of MSE method
(b) Results of MSEE method
Figure 4-15. Damage identification results for the 2-D truss model by MSE and
MSEE methods considering modelling errors: Case 1
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
60
Element Number
Da
mag
e E
xte
nt (%
)
Modelling Error = 0%
Modelling Error = 10%
Modelling Error = 20%
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
60
Element Number
Dam
age E
xte
nt (%
)
Modelling Error = 0%
Modelling Error = 10%
Modelling Error = 20%
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 97
© 2017 Khac-Duy Nguyen Page 97
(a) Results of MSE method
(b) Results of MSEE method
Figure 4-16. Damage identification results for the 2-D truss model by MSE and
MSEE methods considering modelling errors: Case 2
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
60
Element Number
Dam
age E
xte
nt (%
)
Modelling Error = 0%
Modelling Error = 10%
Modelling Error = 20%
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
60
Element Number
Dam
age E
xte
nt
(%)
Modelling Error = 0%
Modelling Error = 10%
Modelling Error = 20%
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 98
© 2017 Khac-Duy Nguyen Page 98
(a) Results of MSE method
(b) Results of MSEE method
Figure 4-17. Damage identification results for the 2-D truss model by MSE and
MSEE methods considering modelling errors: Case 3
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
60
Element Number
Dam
age E
xte
nt (%
)
Modelling Error = 0%
Modelling Error = 10%
Modelling Error = 20%
1 3 5 7 9 11 13 15 17 19 21 23 250
10
20
30
40
50
60
Element Number
Dam
age E
xte
nt (%
)
Modelling Error = 0%
Modelling Error = 10%
Modelling Error = 20%
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 99
© 2017 Khac-Duy Nguyen Page 99
4.5 EXPERIMENTAL VERIFICATION
The experimental data of the 8-DOF system and the three-story shear
building model is used to experimentally validate the MSEE correlation method. For
the 8-DOF system, its only damage case is considered. For the three-story shear
building, all the six damage states are considered.
Figure 4-18 shows the damage identification results for the 8-DOF system
obtained by using the MSEE correlation method with the first three modes. Similar
to the results obtained from using the GMSEE method, the damaged spring has been
successfully identified and the estimated severity (68.8%) is close to the value
obtained with the GMSEE method (67.8%). Also, no false alarms have been
observed in the damage results predicted by the MSEE method.
Figure 4-18. Damage identification for the 8-DOF system by MSEE method
Figure 4-19 shows the damage probability results obtained by using the
original GMSEE and improved MSEE methods. As shown in Fig. 4-19(c)-(f),
similar results are obtained by both methods for damage states 3-6, showing the
1 2 3 4 5 6 70
20
40
60
80
100
Element Number
Dam
age E
xte
nt (%
)
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 100
© 2017 Khac-Duy Nguyen Page 100
detection probabilities of 100% for the actual damaged floors. Regarding the damage
states in the first floor, as shown in Fig. 4-19(a)-(b), the improved method has much
better results compared to the original GMSEE method. For state 1 with 12.5%
damage in the first floor, although both methods show low probability of damage in
the first floor, the MSEE method has greater value (about 70%) compared to the
GMSEE method’s result (about 55%). Also, the detection probability of the first
floor using the MSEE method is greater than the probability threshold (60%). For the
greater damage state in the first floor, as shown in Fig. 4-19(b), results obtained from
the MSEE method shows its outperformance of the original method, indicating
100% of detection probability in the first floor and 0% of detection probability for
other floors.
Average damage extent results of the six damage states are shown in Fig. 4-
20. It is found that the results by the MSEE are quite identical to those obtained with
the GMSEE method, except for state 2 where the MSEE method show better results
in which no damage extents observed at the 2nd and the 3rd floors. Similar to the
results of the GMSEE method, damage extents estimated by the MSEE method
match well to the actual damage extents for the first two states. Results of the MSEE
method for other states also show overestimation by about 10%-30%. It should be
noted again that these errors are understandable for this system considering some
uncertainties associated with the experiment. Regarding the change of damage
extent, the MSEE method was successful to indicate the increasing trend of damage
extent in each floor, similar to what are observed in the results of GMSEE method in
Chapter 3.
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 101
© 2017 Khac-Duy Nguyen Page 101
(a) State 2: damage at 1st floor (12.5%) (b) State 3: damage at 1st floor (25%)
(c) State 4: damage at 2nd floor (12.5%) (d) State 5: damage at 2nd floor (25%)
(e) State 6: damage at 3rd floor (12.5%) (f) State 7: damage at 3rd floor (25%)
Figure 4-19. Detection probability results for the shear building by GMSEE and MSEE
methods.
0 10 20 30 40 50 60 70 80 90 100
Floor 1
Floor 2
Floor 3
Detection Probability (%)
GMSEE Method
MSEE Method
0 10 20 30 40 50 60 70 80 90 100
Floor 1
Floor 2
Floor 3
Detection Probability (%)
GMSEE Method
MSEE Method
0 10 20 30 40 50 60 70 80 90 100
Floor 1
Floor 2
Floor 3
Detection Probability (%)
GMSEE Method
MSEE Method
0 10 20 30 40 50 60 70 80 90 100
Floor 1
Floor 2
Floor 3
Detection Probability (%)
GMSEE Method
MSEE Method
0 10 20 30 40 50 60 70 80 90 100
Floor 1
Floor 2
Floor 3
Detection Probability (%)
GMSEE Method
MSEE Method
0 10 20 30 40 50 60 70 80 90 100
Floor 1
Floor 2
Floor 3
Detection Probability (%)
GMSEE Method
MSEE Method
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 102
© 2017 Khac-Duy Nguyen Page 102
(a) State 2: damage at 1st floor (12.5%) (b) State 3: damage at 1st floor (25%)
(c) State 4: damage at 2nd floor (12.5%) (d) State 5: damage at 2nd floor (25%)
(e) State 6: damage at 3rd floor (12.5%) (f) State 7: damage at 3rd floor (25%)
Figure 4-20. Average damage extent results for the shear building by GMSEE and
MSEE methods.
0 10 20 30 40 50 60 70 80
Floor 1
Floor 2
Floor 3
Average Damage Extent (%)
GMSEE Method
MSEE Method
Actual Damage
0 10 20 30 40 50 60 70 80
Floor 1
Floor 2
Floor 3
Average Damage Extent (%)
GMSEE Method
MSEE Method
Actual Damage
0 10 20 30 40 50 60 70 80
Floor 1
Floor 2
Floor 3
Average Damage Extent (%)
GMSEE Method
MSEE Method
Actual Damage
0 10 20 30 40 50 60 70 80
Floor 1
Floor 2
Floor 3
Average Damage Extent (%)
GMSEE Method
MSEE Method
Actual Damage
0 10 20 30 40 50 60 70 80
Floor 1
Floor 2
Floor 3
Average Damage Extent (%)
GMSEE Method
MSEE Method
Actual Damage
0 10 20 30 40 50 60 70 80
Floor 1
Floor 2
Floor 3
Average Damage Extent (%)
GMSEE Method
MSEE Method
Actual Damage
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 103
© 2017 Khac-Duy Nguyen Page 103
4.6 CONCLUSIONS
The MSEE correlation method which is a modified version of the GMSEE
correlation method was developed for locating damage and estimating damage extent
in structures. The performance of the method was verified through numerical
simulations and laboratory experiments. Results from the numerical study showed
that the MSEE method could predict damage location and damage extent with better
accuracy compared to the GMSEE correlation method when fewer measured modes
are used. False detection is basically reduced with the MSEE method. Also, the
MSEE correlation demonstrated its robustness under noise conditions. Performance
of the MSEE correlation method was better compared to the GMSEE correlation
method with regard to damage detection probability. When fewer modes were used,
performance of the MSEE method reduced but it was still able to detect most of the
damage with good accuracy. In the condition of modelling errors, the results showed
small errors in the damage identification by the MSEE method. Compared to the
traditional MSE method, the MSEE method was much less affected by modelling
errors. By applying the MSEE method to the experimental models, the method also
made improvement in damage identification results compared to the original
GMSEE method.
In summary, the MSEE correlation method improves the damage
identification results compared to the GMSEE method. However, the MSEE requires
information of structural stiffness. Therefore, in the cases where many modes can be
measured and structural stiffness is not accurately measured, the GMSEE correlation
method would be preferred. In the case where only few modes can be measured and
the structural stiffness can be obtained properly, the MSEE correlation method would
be preferred. Compared to the traditional MSE method, the proposed MSEE is more
practical and more robust.
Despite the excellent results obtained from the MSEE method, it is found that
the damage probabilities and damage extents of some undamaged elements become
Chapter 4: Modal Strain Energy-Eigenvalue Ratio (MSEE) Correlation Method Page 104
© 2017 Khac-Duy Nguyen Page 104
greater when fewer modes are used. Therefore, it can be expected that false
identification will rise up when dealing with a more complicated structure or with
fewer modes. In the next chapter, an enhanced technique is developed to reduce false
identification caused by measurement noise.
Chapter 5: Sensitivity-Weighted Search Space for Correlation-based Damage Identification Page 105
© 2017 Khac-Duy Nguyen Page 105
5Chapter 5: Sensitivity-Weighted Search Space
for Correlation-based Damage Identification
This chapter presents a technique to improve the effectiveness of correlation-
based damage identification methods for large structures where measurement noise
associated with these structures is large. The technique refers sensitivity-weighted
search space which can reduce the fault detection caused by measurement noise. It is
based on the observation that damage identification is only reliable for elements with
high vibration stress. The search technique considers sensitivity of individual
structural element wherein the search space of an element is weighted with its
sensitivity. The developed technique is then tested for the numerical 2D truss model.
5.1 THEORETICAL DEVELOPMENT OF SENSITIVITY-WEIGHTED SEARCH SPACE
As stated in the literature review, the damage identification was found only
reliable for elements with high strain energy since only very small change in modal
parameters will be a result of a very large change in structural stiffness of low-strain-
energy elements. In order to overcome this problem, Wahalathantri et al. (2012)
suggested multiplying the damage results by a modification function as a form of
normalized modal strain energy curve. However, this technique is only suitable for
adjusting a damage location result. It is not suitable to multiply a damage extent
result by this curve as it will affect the true damage extent.
In this study, instead of adjusting the results with a modification function,
adjustment is applied to the search space. Conventionally, search spaces for all
elements are selected to be in a same range (e.g., from 0% to 100%) which represents
the possible range of damage severity for each element. In other words, the high-
sensitivity elements have the same range as the low-sensitivity elements. Therefore,
the conventional range scheme may generate some false detection, especially in the
Chapter 5: Sensitivity-Weighted Search Space for Correlation-based Damage Identification Page 106
© 2017 Khac-Duy Nguyen Page 106
conditions of that measurement noise is significant and/or the number of degrees of
freedom is much greater than the number of measured modes.
It is worth noting that the low-sensitivity elements contribute little to the
convergence of the objective function, and therefore, their importance should be
treated differently from high-sensitivity elements in the optimization process.
Considering the distribution of elemental MSE in all modes, a sensitivity-weighted
search space (SWSS) scheme is developed for the optimization-based forward
methods. As the sensitivity of elemental MSEE is in a form of MSE, it can be used to
modify the traditional search space. The range for each element is defined based on
its sensitivity as follows:
mean
meanδ 0;100% .
max( )
j
j
SD
S= (5-1)
where δ jD is the damage extent variable of the jth element, Sjmean is the mean MSEE
sensitivity of the jth element to damage considering all measured modes; max(Smean)
is the maximum value of the mean sensitivities. Using this technique, the importance
of an element is treated unequally with other elements. The elements with high
sensitivity have broader range, while the ones with low sensitivity have narrower
range. The idea behind this scheme is that the high-sensitivity elements are allowed
to vary more flexibly than the low-sensitivity elements; hence, the convergence of
the objective function is more likely affected by the high-sensitivity elements. It is
also worth noting again that the range of the damage extent variable does not
represent the range of the damage. The final damage extent is the product of the best
damage extent vector and a damage coefficient mentioned in Chapter 3 and Chapter
4.
Chapter 5: Sensitivity-Weighted Search Space for Correlation-based Damage Identification Page 107
© 2017 Khac-Duy Nguyen Page 107
5.2 NUMERICAL VERIFICATION
The 2-D truss model shown in Fig.3-2 is used to verify the feasibility of the
proposed search space in correlation-based damage identification. One of the data of
Case 2 under noise level 2 is used for the verification. Also, the MSEE correlation
method using the first 4 modes is used for damage identification. Figure 5-1 shows
the average MSEE sensitivity for all elements of the truss model. It can be seen that
the sensitivities of the elements 13-17 are very low and it is expected that false
detection can be observed at these elements. Using the sensitivity-weighting
technique, the search space is different for each element as shown in Fig. 5-2. The
broadest ranges are applied for elements 1 and 7 and the narrowest ranges are applied
for elements 13-17. The actual damaged elements 4 and 11 have moderate ranges of
[0; 0.37] and [0; 0.23], respectively.
Figure 5-3 shows damage identification results using the conventional search
space and the sensitivity-weighted search space. As shown in Fig. 5-3(a), the MSEE
method gives significant false detection at element 17 when using the conventional
search space. This falsely detected element is found to be one of the low-sensitivity
elements. On the other hand, as shown in Fig. 5-3(b), the MSEE method with the
sensitivity-weighted search space gives more readily results where the predicted
damage extent of element 17 is reduced significantly.
Chapter 5: Sensitivity-Weighted Search Space for Correlation-based Damage Identification Page 108
© 2017 Khac-Duy Nguyen Page 108
Figure 5-1. Average MSEE sensitivity of the first 4 modes of the 2-D truss model
Figure 5-2. Sensitivity-weighted search space of all elements of the 2-D truss model
0 5 10 15 20 250
10
20
30
40
Element Number
Avera
ge S
ensitiv
ity
0 5 10 15 20 250
0.2
0.4
0.6
0.8
1
Element Number
Searc
hin
g r
ange
Chapter 5: Sensitivity-Weighted Search Space for Correlation-based Damage Identification Page 109
© 2017 Khac-Duy Nguyen Page 109
(a) using conventional search space
(b) using sensitivity-weighted search space
Figure 5-3. Damage identification results using MSEE method for a noised data in
Case 2 of the 2-D truss model
0 5 10 15 20 250
10
20
30
40
50
Element Number
Dam
age E
xte
nt
(%)
0 5 10 15 20 250
10
20
30
40
50
Element Number
Dam
age E
xte
nt
(%)
Chapter 5: Sensitivity-Weighted Search Space for Correlation-based Damage Identification Page 110
© 2017 Khac-Duy Nguyen Page 110
5.3 CONCLUSIONS
An enhanced technique, called sensitivity-weighted search space, was
developed to improve the effectiveness of correlation-based damage identification
methods. The technique was validated for the numerical 2-D truss model. It is found
that the technique can reduce fault detection usually associated with low-sensitivity
elements. In the example of the 2-D truss model, false detection at element 17 with
low-sensitivity was significantly reduced by applying the sensitivity-weighted search
space. In the next chapter, this technique will be applied together with the proposed
correlation methods for damage identification of a much more complicated truss
structure.
Chapter 6: Experimental Validation for a Complex Truss Structure Page 111
© 2017 Khac-Duy Nguyen Page 111
6Chapter 6: Experimental Validation for a
Complex Truss Structure
This chapter presents the validation of the proposed damage identification
methods and enhanced techniques for a laboratory through-truss bridge model with a
large number of degrees of freedom. Firstly, the material properties and dimensions
of the bridge model are described. Also, an FE model of the bridge is presented with
the adjustment in cross section properties to better represent the behaviour of the
experimental model. Secondly, the experimental setup is described regarding sensor
layouts and data acquisition system. Thirdly, modal extraction and modal verification
results are presented. Finally, damage identification results for the bridge model are
presented.
6.1 INTRODUCTION OF LABORATORY THROUGH-TRUSS BRIDGE MODEL
As shown in Fig. 6-1, the steel through-truss bridge model was assembled at
Banyo Pilot Plant Precinct of Queensland University of Technology, as a part of a
previous PhD project on structural health monitoring (Cowled et al. 2015). The
structure is a 3-span cantilever truss bridge model with total length of 8.55m. The
height of the main frame is 1.8m and the width of the bridge is 0.9m. The truss has
20 bays, each of which is 0.45m in length except the bays at two ends with each
length of 0.225m. Detailed dimensions are illustrated in Fig. 6-2. The structure
consists of 198 nodes and 318 members of various cross sectional areas. The main
structural members including chords, webs, struts and beams are made of cold
formed mild steel with square/rectangular hollow sections. Meanwhile, the bracing
members are steel flat bars. Detailed cross section and material properties for all
members are listed in Table 6-1. The members in the two main planes are jointed
using bolt connection and steel gusset plates as shown in Fig. 6-3. The lateral struts
Chapter 6: Experimental Validation for a Complex Truss Structure Page 112
© 2017 Khac-Duy Nguyen Page 112
and beams are also bolted to the gusset plates and the braces are bolted directly to the
struts or beams. M6 bolts were used for most of the joints except at the joints of the
main frames where M8 bolts were used. As healthy condition, the M8 bolts were
fastened to 10Nm and M6 bolts were fastened to 4Nm using a torque wrench. A pin
in slotted hole was set at each far end of the bridge to simulate roller supports. A pin
in fitted hole was set at the bottom of each main frame to simulate hinge support.
Figure 6-1. QUT steel through-truss bridge model
Chapter 6: Experimental Validation for a Complex Truss Structure Page 113
© 2017 Khac-Duy Nguyen Page 113
Figure 6-2. Dimensions of QUT steel through-truss bridge model
(a) chord-chord joint (b) main frame-chord joint (c) brace-beam joint
Figure 6-3 Joint connections at some typical positions
Chapter 6: Experimental Validation for a Complex Truss Structure Page 114
© 2017 Khac-Duy Nguyen Page 114
Table 6-1. Details of structural members of QUT steel through-truss bridge model
Members Section type Dimension
(mm)
Young’s
Modulus
(GPa)
Mass
density
(kg/cm3)
Top and bottom
chords Square hollow 20x20x1.6
200 7.85x103
Diagonals Square hollow 20x20x1.6
Vertical webs (at
supports) Square hollow 30x30x3.0
Webs (others) Square hollow 20x20x1.6
Struts Square hollow 20x20x1.6
Beams Rectangular
hollow 50x25x2.0
Braces Flat bar 20x3.0
6.2 EXPERIMENTAL SETUP
Total 18 accelerometers including 14 PCB393B05 sensors with a nominal
sensitivity of 10V/g and 4 PCB393B04 sensors with a nominal sensitivity of 1V/g
were used to measure vibration response of one plane of the bridge model. The first
14 accelerometers labelled from S1 to S14 are PCB393B05 type and the rest labelled
from S15 to S18 are PCB393B04 type. A chassis NI cDAQ-9172 embedded with
five DSA modules NI-9234 with 4 channels for each was used to capture the signals
from the accelerometers. In order to achieve precise synchronization across different
modules, programming was made using LabVIEW to ensure that all the DSA
modules share one time base source (Nguyen et al. 2014).
Due to the above sensor shortage, a roving test method was designed to
capture the response of most of the DOFs in the examined plane of the truss model.
As shown in Fig. 6-4, six (6) sensor layouts were designed in which 17 sensors were
roved along the truss length and one sensor was used as the reference (i.e., sensor
S2). As modal strain energy of each element is calculated from the mode shape of 4
DOFs at its ends, redundant DOFs were measured for some important elements as
Chapter 6: Experimental Validation for a Complex Truss Structure Page 115
© 2017 Khac-Duy Nguyen Page 115
shown in Fig. 6-4(f) to reduce the uncertainty associated with the roving test. The
structure was excited by a hammer at the joint next to the mid span joint (i.e., the
joint of the 9th and 10th bays). The impact location was covered with a thick rubber
patch to minimize nonlinear interaction between the hammer and the structure. It
should be noted that roving sensors might change the mass of the system in different
layouts. However, as the damage identification methods use the change in modal
parameters before and after damage, the inaccuracy in modal estimation caused this
effect can be considered to induce small error. The sampling rate was set as 2048Hz
and the duration of measurement for each layout was set as 2 minutes. Totally,
vibration responses of 88 DOFs over 100 DOFs of the truss plane were measured.
Later, modal features of the unmeasured 12 DOFs have been estimated from the
measured ones using linear interpolation method. Figure 6-5 shows all the measured
DOFs of the truss bridge model. Figure 6-6 shows the sensor installation at some
typical locations. As an example, time-history acceleration measures of some
locations of the first layout are illustrated in Fig. 6-7. It is worth noting that the
number of elements to be examined in this research is 99 which is one of the largest
numbers of elements that have been considered so far. The elements are numbered
from 1 to 99 as shown in Fig. 6-8.
(a) Layout 1
Figure 6-4. Sensor layouts for vibration measurement of the QUT through-truss
bridge model
Chapter 6: Experimental Validation for a Complex Truss Structure Page 116
© 2017 Khac-Duy Nguyen Page 116
(b) Layout 2
(c) Layout 3
(d) Layout 4
(e) Layout 5
(f) Layout 6
Figure 6-4. Sensor layouts for vibration measurement of the QUT through-truss
bridge model
Chapter 6: Experimental Validation for a Complex Truss Structure Page 117
© 2017 Khac-Duy Nguyen Page 117
Figure 6-5. Measured DOFs for the QUT through-truss bridge model
(a) at joint of 2 inclined-top chords (b) at joint of inclined chord-horizontal
chord
(c) at top of a main frame (d) at joint of bottom chords
Figure 6-6 Sensors at some typical joints of the QUT through-truss bridge model
Chapter 6: Experimental Validation for a Complex Truss Structure Page 118
© 2017 Khac-Duy Nguyen Page 118
(a) Reference channel S2 (z-direction)
(b) Channel S10 (z-direction)
(c) Channel S11 (x-direction)
Figure 6-7 Representative acceleration time-series of layout 1 of the QUT through-
truss bridge model
0 20 40 60 80 100 120-0.1
-0.05
0
0.05
0.1
Time (sec)
Accele
ration A
mplit
ude (
g)
0 20 40 60 80 100 120-0.1
-0.05
0
0.05
0.1
Time (sec)
Accele
ration A
mplit
ude (
g)
0 20 40 60 80 100 120-0.1
-0.05
0
0.05
0.1
Time (sec)
Accele
ration A
mplit
ude (
g)
Chapter 6: Experimental Validation for a Complex Truss Structure Page 119
© 2017 Khac-Duy Nguyen Page 119
(a) Left half of the truss plane
(b) Right half of the truss plane
Figure 6-8. Element numbering for the examined truss plane of the QUT through-
truss bridge model
6.3 MODAL EXTRACTION AND VERIFICATION
6.3.1 Modal Extraction
The modal analysis software package ARTeMIS Extractor Pro version 5.3
developed by Structural Vibration Solution A/S was used to process vibration data
from the truss structure. The frequency domain decomposition (FDD) method
embedded in ARTeMIS was used to extract modal information such as natural
frequencies and mode shapes. For signal processing, the signals were resampled to
Chapter 6: Experimental Validation for a Complex Truss Structure Page 120
© 2017 Khac-Duy Nguyen Page 120
512 Hz that gives the new frequency range of interest to be 256 Hz. The number of
frequency point was set as 2048 that gives the frequency resolution to be 0.125 Hz. It
should be noted only the values from 0 to about the first half of the frequency range
are considered as they are more reliable for mode shape estimation. It is also worth
noting that the frequency resolution can be finer by increasing the number of
frequency point. However, this will make the singular value decomposition (SVD)
diagrams very noisy and it is very hard to pick the modes.
Figure 6-9 shows the SVD diagrams for the vibration data of all the test
layouts in the intact condition. Natural frequencies of the truss plane can be
identified from the peaks of the first SVD diagram and corresponding mode shapes
can be estimated. As shown in Fig. 6-9, there are many peaks, but not all of them can
be used for damage identification. Some peaks represent local modes due to local
vibration of individual elements. Some peaks are not stable due to the nonlinearity of
the structure or due to the uncertainties of the roving test (such as the reference
sensor is close to nodal point of these modes). In order to select appropriate modes
for damage identification, the following criteria are applied: 1) the mode must have
low complexity that represents for a true mode; 2) the mode must have a good
repeatability in modal strain energy for different data sets in a same structural
condition; and 3) the mode must represent global behaviour of the structure.
For the first criterion, the modes with complexity lower than 20 are selected.
For the second criterion, the modal assurance criteria of modal strain energy
(MACMSE) are calculated for the selected modes due to the first criterion, and then the
modes with MAC value greater than 95% are selected. The equation of MACMSE is as
follows:
( )
( )( )
2
,1 ,2
,1 ,1 ,2 ,2
( )
T
i i
MSE T T
i i i i
MAC i =MSE MSE
MSE MSE MSE MSE (6-1)
where ,1iMSE is the first data set of the ith MSE data of the structure, and ,2iMSE is
Chapter 6: Experimental Validation for a Complex Truss Structure Page 121
© 2017 Khac-Duy Nguyen Page 121
the second data set of the ith MSE data of the structure. For the third criterion, we
consider a good MSE distribution must contain a good number of high MSE values.
The quality of the MSE distribution can be evaluated by the ratio of number of MSE
values greater than the mean value over the total number of MSE values, as follows:
100%MSE
np
N
= (6-2)
where nis the number of MSE values being greater than the mean value of MSE
distribution and N is the number of MSE values (e.g., N = 99 for this case study). In
this study, the modes with at least 20% of MSE values greater than the mean value
(i.e., MSEp 20%) are selected. It is worth nothing that this criterion was set as a
result of trade-off between the number of modes and the quality of the MSE
distribution.
Figure 6-9 SVD diagram and the identified natural frequencies for the QUT through-
truss bridge model
Chapter 6: Experimental Validation for a Complex Truss Structure Page 122
© 2017 Khac-Duy Nguyen Page 122
Table 6-2 show the summary of modal characteristics of all the peaks selected
from the SVD diagram. It can be seen that only three modes (i.e., 15.375 Hz, 30.25
Hz and 58.75 Hz) satisfy all the three criteria above. These modes are respectively
marked as mode 1, mode 2 and mode 3 in the SVD diagram of Fig. 6-9. Figure 6-10
shows an example of modal strain energy of an unselected mode that does not satisfy
the repeatability and Fig. 6-11 shows an example for an unselected mode that does
not satisfy the global behaviour requirement. Figure 6-12 to 6-14 show modal strain
energies of the three selected modes (modes 1-3). It is obvious that these modes have
good repeatability and represent global behaviour. Figure 6-15 shows the mode
shapes associated with the identified modes. These modes can be in-plane bending
modes or torsional modes (with consideration of the other truss plane). In the later
section, an FE model will provide better understandings for these modes.
Chapter 6: Experimental Validation for a Complex Truss Structure Page 123
© 2017 Khac-Duy Nguyen Page 123
Table 6-2. Summary of mode selection for the QUT through-truss bridge model
Peak
frequency (Hz)
Complexity
(%)
Satisfactory of
low
complexity
Satisfactory of
repeatability
Satisfactory of
global behaviour
4.875 75.985 No - -
7.000 12.930 Yes No (0.42*) -
15.375 0.435 Yes Yes (0.99) Yes (22.20%**)
20.375 48.186 No - -
24.625 94.755 No - -
28.000 27.787 No - -
30.25 12.255 Yes Yes (0.98) Yes (24.24%)
32.375 46.763 No - -
34.625 28.484 No - -
38.875 83.598 No - -
45.375 38.959 No - -
49.875 93.862 No - -
52.125 56.000 No - -
53.750 34.909 No - -
58.750 12.628 Yes Yes (0.98) Yes (30.30%)
62.125 6.852 Yes Yes (1.00) No (15.15%)
64.250 13.685 Yes No (0.92) -
66.000 59.591 No - -
68.75 10.027 Yes Yes (0.99) No (18.18%)
75.125 59.760 No - -
77.625 27.000 No - -
81.500 87.996 No - -
90.625 70.432 No - -
96.125 65.088 No - -
98.625 64.536 No - -
102.125 58.582 No - -
105.625 57.229 No - -
110.750 45.934 No - -
*value in the parentheses represents MACMSE
**value in the parentheses represents MSEp
Chapter 6: Experimental Validation for a Complex Truss Structure Page 124
© 2017 Khac-Duy Nguyen Page 124
Figure 6-10 Modal strain energy of the unselected mode at 7Hz from two different data
sets
Figure 6-11 Modal strain energy of the unselected mode at 62.125Hz
0 10 20 30 40 50 60 70 80 90 1000
2
4
6
8
10
12x 10
4
Element Number
Modal S
train
Energ
y
Set 1
Set 2
0 10 20 30 40 50 60 70 80 90 1000
2
4
6
8
10x 10
5
Element Number
Modal S
train
Energ
y
Chapter 6: Experimental Validation for a Complex Truss Structure Page 125
© 2017 Khac-Duy Nguyen Page 125
Figure 6-12 Modal strain energy for the first selected mode (15.375Hz) from 2 different
data sets
Figure 6-13 Modal strain energy for the second selected mode (30.25Hz) from 2
different data sets
0 10 20 30 40 50 60 70 80 90 1000
2000
4000
6000
8000
10000
12000
14000
Element Number
Modal S
train
Energ
y
Set 1
Set 2
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5
6
7
8x 10
4
Element Number
Modal S
train
Energ
y
Set 1
Set 2
Chapter 6: Experimental Validation for a Complex Truss Structure Page 126
© 2017 Khac-Duy Nguyen Page 126
Figure 6-14 Modal strain energy for the third selected mode (58.75Hz) from 2 different
data sets
(a) Mode 1: 15.375 Hz
(b) Mode 2: 30.25 Hz
(c) Mode 3: 58.75 Hz
Figure 6-15 Experimental mode shapes identified by FDD method
0 10 20 30 40 50 60 70 80 90 1000
0.5
1
1.5
2x 10
5
Element Number
Modal S
train
Energ
y
Set 1
Set 2
Chapter 6: Experimental Validation for a Complex Truss Structure Page 127
© 2017 Khac-Duy Nguyen Page 127
6.3.2 Modal Verification
In order to validate the identified experimental modes, a finite element (FE)
model of the bridge was established using SAP2000 software package as shown in
Fig. 6-16. The main structural members including chords, webs and diagonals are
modelled as in-plane truss element, which allows in-plane rotation but constraints
out-of-plane rotation. The struts, beams and braces are modelled as normal truss
element which allows rotations in any plane. The left inner supports are modelled as
pins and the right inner supports are modelled as rollers. For the outer supports, they
are supposed to be rollers but it is observed from the experimental model that these
supports do not link very well to the superstructure. This is also evidenced from their
large displacements showed in the experimental mode shapes (Fig. 6-15). Therefore,
the outer supports are modelled as vertical springs with the same stiffness constant of
80 kN/m obtained from a trial and error process.
Figure 6-16 FE model of the QUT through-truss bridge model
Chapter 6: Experimental Validation for a Complex Truss Structure Page 128
© 2017 Khac-Duy Nguyen Page 128
As the main members are jointed at the gusset plates, their stiffness can be
reduced or increased. In order to calculate their true stiffness, two detailed FE models
of a bottom chord are established and compared. Details of these models are shown
in Fig. 6-17. The first model considers a half bar member fixed at one end.
Meanwhile, the second model considers a system of a gusset and a half bar member
which already exclude the length of the gusset. A same axial resultant force (i.e.,
200N in this study) is put at the end of the truss element in each model and axial
deformation is calculated for each FE model. It is found that the end deformation in
the first model (i.e., 1.803E-3 mm) is about 60% of that in the second model (i.e.,
3.04E-3 mm). That means the gusset make the overall axial stiffness of the truss
member reduce by about 60% (if it is considered fixed). Taking into account this
effect and by considering that bolt connection at the gusset also reduces the overall
axial stiffness of the member, the axial stiffness of the main members in the FE
model of the QUT through-truss bridge structure is nominally reduced by 50%
compared to the as-built stiffness.
(a) FE model considering only the truss
bar
(b) FE model considering truss bar and
gusset plate
Figure 6-17. Comparison of two FE models of a bottom chord for refining member’s
axial stiffness
Chapter 6: Experimental Validation for a Complex Truss Structure Page 129
© 2017 Khac-Duy Nguyen Page 129
Using the refined FE model, three identical modes can be obtained from the
FE model of the truss structure as shown in Fig. 6-18. It can be found that these
numerical modes match quite well with the modes extracted from the experimental
data. The first two modes represent in-plane bending behaviour and the third mode
represents torsional behaviour. As shown in the fourth column of Table 6-3,
differences in the natural frequencies between the experimental model and FE model
are very small, up to only 2%. However, as shown in the fifth column of Table 6-3,
mode shapes of the FE model have low correlation with those of the experimental
model, especially for mode 2. This may be due to the difference in structural
properties of individual elements between the FE model and the experimental model.
For the damage identification methods that heavily rely on FE model (e.g., MSE
correlation method), this FE model may need more refinement or updating in local
(elemental) level. Despite that, the above comparison demonstrates that the three
identified experimental modes are feasible modes and can be used for damage
identification of the proposed methods.
Chapter 6: Experimental Validation for a Complex Truss Structure Page 130
© 2017 Khac-Duy Nguyen Page 130
(a) Mode 1: 15.185 Hz (1st bending mode)
(b) Mode 2: 30.854 Hz (2nd bending mode)
(c) Mode 3: 59.480 Hz (1st torsional mode)
Figure 6-18. Three identical modes obtained from the FE model of the QUT through-
truss bridge structure
Table 6-3. Comparison of natural frequencies obtained from experimental model and
FE model of the QUT through-truss bridge structure
Mode order Natural frequency (Hz) Freq.
Difference (%) MAC
FEM Experiment
1 15.185 15.375 1.24 0.653
2 30.854 30.250 2.00 0.304
3 59.480 58.750 1.24 0.598
Chapter 6: Experimental Validation for a Complex Truss Structure Page 131
© 2017 Khac-Duy Nguyen Page 131
6.4 DAMAGE IDENTIFICATION FOR QUT THROUGH-TRUSS BRIDGE MODEL
Several damage scenarios were designed for the QUT through-truss bridge
model to examine the performance of the proposed correlation damage identification
methods and the enhanced optimisation technique. It is well known that bolt loose is
one of the popular damage types in truss structures. Stiffness of the whole element is
dependent on not only the truss bar but also the joint stiffness. Figure 6-18 shows a
physical model of a truss member, consisting of stiffness of the truss bar and stiffness
of the joints. The joint stiffness represents the tangential contact stiffness of the
bolts, and this value is proportional to contact pressure caused by bolt torque (Kartal
et al. 2011). The equivalent stiffness of the member can be expressed as follows:
joint-1 joint-2
1 1 1 1
e bark k k k= + + (6-3)
where joint-1k and
joint-2k refer to the joint stiffness values at each end of the truss bar;
bark is the axial stiffness of the bar itself. When all bolts are fully fastened (
joint-1 joint-2k k= = ), the equivalent stiffness of the member is equal to bark . When
some bolts are partially loosened, the equivalent stiffness will reduce. When all bolts
are fully loosened (joint-1 0k = or
joint-2 0k = ), the equivalent stiffness become vanish
or the member is totally failed. It is worth noting that the stiffness of the joint is also
affected by many other factors such as surface roughness, elasticities and contact
area; and these factors are hard to be controlled. Therefore, in this study, only the
existence of damage and the increasing trend of damage are considered.
Figure 6-19. Spring-in-series model of bolted truss element
Chapter 6: Experimental Validation for a Complex Truss Structure Page 132
© 2017 Khac-Duy Nguyen Page 132
Various bolt loose scenarios are examined in this study, as summarized in
Table 6-4. In Test 1, the damage identification methods are validated for their
capability of identifying a single damage (i.e., element 10) with different severity
levels. Meanwhile, in Test 2, their performance on identifying a multiple damage
case (i.e., elements 7 and 67) is verified. Figure 6-20 illustrates the positions of the
damaged elements considered in these tests. Table 6-5 summarizes the natural
frequencies of the truss bridge model for the two undamaged states and three
damaged states. It can be seen that the changes in natural frequencies are not very
noticeable. For the first test, only the first natural frequency slightly reduced after all
the bolts of element 10 were loosened to hand tightening. For the second test, only
slight change is observed in the natural frequency of mode 2. These small changes in
natural frequencies are reasonable considering the structure is very large and the
contribution of each individual member on the overall behaviour of the structure is
very small. To clarify this point, a damage of 20% in element 10 (similar to state 1-1)
is simulated in the FE model. The changes in the first three numerical frequencies are
very small of about 0.033 Hz (0.22%), 0.042 Hz (0.14%) and 0.048 Hz (0.09%),
respectively. These changes are even much smaller than the frequency resolution
(0.125) in the experimental study.
Table 6-4. Damage scenarios for the QUT through-truss bridge structure
Test State Description
Test 1
State 1-0 Undamaged
State 1-1 Damage at #10: bolts at one end loosen to hand tightening
State 1-2 Damage at #10: bolts at two ends loosen to hand tightening
Test 2
State 2-0 Undamaged: all bolts refastened
State 2-1 Damage at #7 and #67: bolts at two ends loosen to hand
tightening
Chapter 6: Experimental Validation for a Complex Truss Structure Page 133
© 2017 Khac-Duy Nguyen Page 133
(a) State 1-1 and state 1-2
(b) State 2-1
Figure 6-20. Illustration of damaged elements on the QUT through-truss bridge
model
Table 6-5. Natural frequencies of the QUT through-truss bridge structure at
undamaged and damaged states
Test State Natural frequency (Hz)
Mode 1 Mode 2 Mode 3
Test 1
State 1-0 15.375 30.250 58.750
State 1-1 15.375 30.250 58.750
State 1-2 15.250 30.250 58.750
Test 2 State 2-0 15.250 30.250 58.750
State 2-1 15.250 30.125 58.750
Figure 6-21 shows the search space for each element based on the sensitivity-
weighting technique described Chapter 5. Only some elements have wide ranges
with upper bound of over 0.5, such as elements 10, 20 and 69. A good number of
elements have medium ranges with the upper bound varying from 0.1 to 0.5, such as
elements 6, 36 and 65. And many elements have very narrow ranges with the upper
bound of under 0.1, such as elements 2, 3 and 11. For the location of the elements,
Chapter 6: Experimental Validation for a Complex Truss Structure Page 134
© 2017 Khac-Duy Nguyen Page 134
please refer to Fig. 6-8. The GA solver parameters are set as follows: the number of
variables is 99 corresponding to the total number of truss elements of the examined
plane, the range of the variables is [-1; -1E-10], the population size is 500 as of about
five times of the number of the dimension (i.e., 99), the crossover fraction rate and
the convergence tolerance are respectively set as 0.5 and 1E-10 as similar to the
values selected in Chapter 3. The adaptive feasible mutation function is used to
generate mutated individuals.
Figure 6-21. Sensitivity-weighted search space of all elements of the examined plane
of the QUT through-truss bridge model
6.4.1 Damage Identification using GMSEE Method
Damage identification results by the GMSEE correlation method are
illustrated in Fig. 6-22 to 6-24. It is important to set a damage threshold for decision
making of damage. In a previous study by Park et al. (2013), they chose 5% as a
damage threshold for damage identification of a simple 5-story shear building.
Considering the much larger number of elements and higher uncertainties expected
with the roving test, a threshold of 10% is used in this study for decision making
about the damaged elements. It is worth nothing that a threshold of 10% represents a
0 10 20 30 40 50 60 70 80 90 1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Element Number
Searc
hin
g r
ange
Chapter 6: Experimental Validation for a Complex Truss Structure Page 135
© 2017 Khac-Duy Nguyen Page 135
safety level of 90% where an element with 90% stiffness remaining is still
considered as undamaged. As shown in Fig. 6-22, the GMSEE method is successful
to predict the actual damaged element of state 1-1 (i.e., element 10). It is also shown
that the damage extents of all undamaged elements are below the damage threshold.
For the case of higher damage severity (state 1-2), as shown in Fig. 6-23, the
GMSEE method successfully identify the damaged element at 10 and also
successfully show the increasing trend in the damage severity of this element. It is
also obvious that the method falsely detects elements 8 and 99 as damage although
their damage extents are only lightly beyond the threshold. As these elements are
close to the actual damaged element (i.e, element 10), the induced damage might
change the orientation and/or force distribution of these elements. Therefore, for
maintenance, it is recommended to fix not only the element with high possibility of
damage but also its adjacent elements if their damage extents are beyond the
threshold. As shown in Fig. 6-24, the GMSEE method gives a good prediction for the
multiple damage case (state 2-1) in which the two damaged elements (i.e., elements 7
and 67) are well predicted and no significant false detection is observed.
As summary, the GMSEE method is successful to detect all the simulated
damage for the QUT through-truss bridge model with a large number of elements.
Despite some false detection in state 1-2, the GMSEE method can still be considered
as a reliable method as the number of falsely detected elements is accounted for only
2% of the total number of the elements.
Chapter 6: Experimental Validation for a Complex Truss Structure Page 136
© 2017 Khac-Duy Nguyen Page 136
Figure 6-22. Damage identification results for state 1-1 using GMSEE method with
sensitivity-weighted search space
Figure 6-23. Damage identification results for state 1-2 using GMSEE method with
sensitivity-weighted search space
Figure 6-24. Damage identification results for state 2-1 using GMSEE method with
sensitivity-weighted search space
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 1000
10
20
30
40
50
60
Element Number
Dam
age E
xte
nt
(%)
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 1000
10
20
30
40
50
60
Element Number
Dam
age E
xte
nt
(%)
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 1000
10
20
30
40
50
60
Element Number
Dam
age E
xte
nt
(%)
Chapter 6: Experimental Validation for a Complex Truss Structure Page 137
© 2017 Khac-Duy Nguyen Page 137
For comparison, the results obtained by the GMSEE method with
conventional search space are shown in Figs. 6-25 to 6-27. For the single damage
cases (states 1-1 and 1-2), the GMSEE method is still able to detect the actual
damaged element 10. However, it generates a lot of false elements, especially when
the damage severity increases. Also, some of the false elements have the same degree
of damage extents with the actual damaged element. For the multiple damage case
(state 2-1), the actual damaged elements 7 and 67 can be detected but again false
detection is observed at many other elements and some of them have similar damage
extent as the actual ones.
Figure 6-25. Damage identification results for state 1-1 using GMSEE with
conventional search space
Figure 6-26. Damage identification results for state 1-2 using GMSEE with
conventional search space
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 1000
10
20
30
40
50
60
Element Number
Dam
age E
xte
nt
(%)
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 1000
10
20
30
40
50
60
Element Number
Dam
age E
xte
nt
(%)
Chapter 6: Experimental Validation for a Complex Truss Structure Page 138
© 2017 Khac-Duy Nguyen Page 138
Figure 6-27. Damage identification results for state 2-1 using GMSEE method with
conventional search space
6.4.2 Damage Identification using MSEE Method
Damage identification results by the MSEE correlation method are illustrated
in Fig. 6-28 to 6-30. The threshold of 10% is also used for decision making about the
damaged elements. As shown in Figs. 6-28 and 6-29, the MSEE method is successful
to predict the actual damaged element of states 1-1 and 1-2 (i.e., element 10). Similar
to the results from the GMSEE method, the MSEE method successfully recognizes
the increasing trend in the damage severity of the actual damaged element.
Regarding false detection, it is found that the MSEE method generates less false
detection. The damage results look more distinguished between the damaged element
and undamaged elements, and only element 99 is falsely detected by the threshold
when larger damage is considered (i.e., state 1-2). For the multiple damage case
(state 2-1), the two actual damaged elements (i.e., elements 7 and 67) are
successfully predicted by the MSEE method with no significant false detection, as
shown in Fig. 6-30. It is again found that the damaged elements are more
distinguished from undamaged elements compared to the one obtained from the
GMSEE method.
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 1000
10
20
30
40
50
60
Element Number
Dam
age E
xte
nt
(%)
Chapter 6: Experimental Validation for a Complex Truss Structure Page 139
© 2017 Khac-Duy Nguyen Page 139
Figure 6-28. Damage identification results for state 1-1 using MSEE method with
sensitivity-weighted search space
Figure 6-29. Damage identification results for state 1-2 using MSEE method with
sensitivity-weighted search space
Figure 6-30. Damage identification results for state 2-1 using MSEE method with
sensitivity-weighted search space
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 1000
10
20
30
40
50
60
Element Number
Dam
age E
xte
nt
(%)
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 1000
10
20
30
40
50
60
70
Element Number
Dam
age E
xte
nt
(%)
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 1000
10
20
30
40
50
60
Element Number
Dam
age E
xte
nt
(%)
Chapter 6: Experimental Validation for a Complex Truss Structure Page 140
© 2017 Khac-Duy Nguyen Page 140
Table 6-6 summarizes the prediction results obtained by the both methods.
Compared to the GMSEE method, damage extents predicted by the MSEE method
are higher for all cases. Also, the MSEE method show better prediction regarding
false detection. For state 1-2, only one falsely detected element is found by the
MSEE method, compared to two by the GMSEE method. Also, for state 1-2, the
results obtained by the MSEE method are more distinguished between the actual
damaged element (i.e., element 10 with 61.3%) and the falsely detected element (i.e.,
element 99 with 15.2%).
Table 6-6. Summary of damage identification results for the QUT through-truss
bridge model (with the use of sensitivity-weighted search space)
State Actual damaged
elements
Predicted damaged elements
GMSEE method MSEE method
1-1 10 (left bolts loosen) 10 (18.5%*) 10 (26.3%*)
1-2 10 (all bolts loosen) 8 (11.6%), 10 (51%),
99 (17%)
10 (61.3%), 99 (15.2%)
2-1 7, 67 (all bolts loosen) 7 (22%), 67 (18.4%) 7 (27.1%), 67 (21.4%)
*Values in parentheses indicate predicted damage extent (%)
For comparison, the results obtained by the MSEE method with conventional
search space are shown in Figs. 6-31 to 6-33. For the single damage cases (states 1-1
and 1-2), the MSEE method is successful to identify the actual damaged element 10.
Although false detection is observed in some elements, the MSEE method with
conventional search space give significantly less false elements compared to the
GMSEE method. For the multiple damage case (state 2-1), the MSEE method still
can detect the actual damaged elements 7 and 67 but generates many false elements.
Chapter 6: Experimental Validation for a Complex Truss Structure Page 141
© 2017 Khac-Duy Nguyen Page 141
Figure 6-31. Damage identification results for state 1-1 using MSEE method with
conventional search space
Figure 6-32. Damage identification results for state 1-2 using MSEE method with
conventional search space
Figure 6-33 Damage identification results for state 2-1 using MSEE method with
conventional search space
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 1000
10
20
30
40
50
60
Element Number
Dam
age E
xte
nt
(%)
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 1000
10
20
30
40
50
60
70
Element Number
Dam
age E
xte
nt
(%)
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 1000
10
20
30
40
50
60
Element Number
Dam
age E
xte
nt
(%)
Chapter 6: Experimental Validation for a Complex Truss Structure Page 142
© 2017 Khac-Duy Nguyen Page 142
Damage identification results for the three damage states using the same
optimisation settings, but conducted by the traditional MSE correlation method, are
plotted in Figs. 6-34 and 6-35. The results in Fig. 6-34 are obtained by using only 3
experimental modes whereas the results in Fig. 6-35 are obtained with the additional
use of 100 numerical modes from the FE model. For more detail about the calculation
of MSE sensitivity, please refer Eq. 3-4. As shown in the figures, the method fails to
detect the actual damaged elements in all three states either with additional numerical
modes or not. As shown in Fig. 3-3, there are a significant number of false errors when
only 3 experimental modes are used, and therefore, it is impossible to point out the
actual damaged elements. The results by using additional 100 numerical modes are
more readily with significantly fewer false errors (Fig. 3-4). However, the prediction
misses all the actual damaged elements in all three damage states. Therefore, the direct
comparison to the results conducted by the GMSEE and MSEE methods indicates
noteworthy improvements of the proposed methods.
(a) State 1-1 (b) State 1-2
(c) State 2-1
Figure 6-34 Damage results by traditional MSE method using 3 experimental modes
0 10 20 30 40 50 60 70 80 90 1000
20
40
60
80
100
Element Number
Da
ma
ge
Exte
nt (%
)
0 10 20 30 40 50 60 70 80 90 1000
20
40
60
80
100
Element Number
Da
ma
ge
Exte
nt (%
)
0 10 20 30 40 50 60 70 80 90 1000
20
40
60
80
100
Element Number
Da
ma
ge
Exte
nt (%
)
Chapter 6: Experimental Validation for a Complex Truss Structure Page 143
© 2017 Khac-Duy Nguyen Page 143
(a) State 1-1 (b) State 1-2
(c) State 2-1
Figure 6-35 Damage identification results by traditional MSE method using 3
experimental modes together with 100 numerical modes.
6.5 CONCLUSIONS
This chapter presented the vibration tests on the QUT through-truss bridge
model and the verification results of the two proposed correlation methods, GMSEE
and MSEE developed in Chapter 3 and 4 respectively, along with the enhanced
technique, sensitivity-weighted search space developed in Chapter 5.
A roving strategy with six layouts was designed to capture the modal
response of one plane of the bridge model. Three modes were successfully extracted
from the acquired acceleration signals. The FE model was fine-tuned considering the
stiffness reduction of the main elements caused by the gusset plates. Then the
identified experimental modes were successfully verified with the modes obtained
from the FE model. The difference in natural frequencies between the experimental
model and the FE model is very small, up to 1.1%.
0 10 20 30 40 50 60 70 80 90 1000
20
40
60
80
100
Element Number
Da
ma
ge
Exte
nt (%
)
0 10 20 30 40 50 60 70 80 90 1000
20
40
60
80
100
Element Number
Da
ma
ge
Exte
nt (%
)
0 10 20 30 40 50 60 70 80 90 1000
20
40
60
80
100
Element Number
Da
ma
ge
Exte
nt (%
)
Chapter 6: Experimental Validation for a Complex Truss Structure Page 144
© 2017 Khac-Duy Nguyen Page 144
Several damage scenarios were simulated by loosening the bolts connecting
the elements and the gusset plates. The first two damage cases (states 1-1 and 1-2)
refer a single damage at element 10 with different damage extent levels by the
increase of the number of loosened bolts. The third damage case (state 2-1) refers a
multiple damage case in which elements 7 and 67 were damaged. It was found that
the natural frequencies did not change very much after damage was induced. This is
explainable as the structure is very large, and therefore, the contribution of each
element in the global behaviour of the structure is very small.
Damage identification by the GMSEE and MSEE correlation methods
incorporating with the sensitivity-weighting technique was performed. It was found
that both methods successfully detected the actual damaged elements in all the three
damage states. Also, both methods generated small rates of false detection which
were about 2% and 1% by the GMSEE and MSEE methods, respectively. The MSEE
method showed slightly clearer prediction in which damaged elements were more
distinguished from the undamaged elements.
For state 1-2 where damage at element 10 was severe, both methods showed
false detection at element 99 along with the accurate prediction at element 10. It is
noted that the falsely detected element 99 is adjacent to the actual damaged element
10. From this observation, it is recommended to check all the elements within the
region of the identified damaged element as the damage at an element may affect the
orientation of its adjacent members and/or the force distribution to them.
Besides, results obtained from the experiment demonstrated the advantage of
using the sensitivity-weighted search space for modal correlation methods.
Significant false detection was reduced by adopting the technique, especially for the
GMSEE method.
It can be also concluded that both methods developed in this study have better
performance compared to the traditional MSE correlation method for complicated
Chapter 6: Experimental Validation for a Complex Truss Structure Page 145
© 2017 Khac-Duy Nguyen Page 145
experimental structures where measurement noise and modelling error can be
significant. It was shown that the traditional method failed to detect any damage of
the QUT through truss bridge model.
Chapter 6: Experimental Validation for a Complex Truss Structure Page 146
© 2017 Khac-Duy Nguyen Page 146
Chapter 7: Experimental Validation for a Real Bridge Page 147
© 2017 Khac-Duy Nguyen Page 147
7Chapter 7: Experimental Validation for a
Real Bridge
This chapter presents the validation of the proposed damage identification
methods and enhanced techniques for a real bridge, the I-40 Bridge over Rio Grande
in Albuquerque, NM, USA. The experimental setup is first described and modal
properties are presented. Then damage identification results by the GMSEE and
MSEE correlation methods are presented.
7.1 EXPERIMENT DESCRIPTION AND MODAL PROPERTIES
As shown in Fig. 7-1, the I-40 Bridge is a slab-on-girder bridge constructed
in the early 1960s. It consists of three continuous spans with a total length of 129 m.
Damage identification experiments on the bridge were conducted by the LANL in
1993, as described in Farrar et al. (1994). A series of forced vibration tests was
performed, using a hydraulic shaker to generate a dynamic force onto the bridge.
Acceleration responses of the bridge were measured at 26 locations distributed along
two plate girders as shown in Fig. 7-2. Four levels of damage were introduced at the
mid-point of the middle span of the north plate girder by cutting the girder. The first
damage case E-1 consisted of a 61 cm long, 0.95 cm side cut through the web
centered at the mid-height of the web. The second case E-2 extends the initial cut to
the bottom of the web. The third case E-3 consisted of the cut in damage E-2 and an
additional halfway cut in the bottom flange from either side. In the last case E-4, the
flange was cut completely through. Details of the damage scenarios are shown in
Fig. 7-3. More details about the experiment can be found in Farrar et al. (1994).
Natural frequencies and modal assurance criterion (MAC) values of the first
six modes before and after the damage episodes are listed in Tables 7-1 and 7-2,
respectively. It is clearly seen that natural frequencies decrease a little for the third
Chapter 7: Experimental Validation for a Real Bridge Page 148
© 2017 Khac-Duy Nguyen Page 148
damage case (E-3) and drop significantly for the last case (E-4). On the other hand,
they are found to increase for the first and the second cuts (E-1 and E-2). This can be
explained as the change in structural properties due to changing environmental
conditions is more significant than the ones caused by these low-level damages.
Also, as shown in Table 7-2, it is found that mode shapes for the fourth damage case
(E-4) significantly change. For other damage cases, very little changes in mode
shapes are observed, except the fifth mode shape. As mode shapes are generally little
affected by environmental changes, this implies the first three damage cases caused
very small effects on dynamic characteristics of the bridge. Note that the large
changes in the fifth mode shape are basically due to measurement errors rather than
due to damage because the location of damage is right at the nodal point of this
mode shape. Based on the above observations, only the two most severe damage
cases (E-3 and E-4) are considered for the verification of the GMSEE and MSEE
correlation methods. Also, only the first two measured modes are used for the
damage identification as it was reported that the higher modes of this experiment are
not reliable (Farrar et al. 1994).
Figure 7-1. I-40 Bridge over Rio Grande in Albuquerque, NM, USA (Farrar et al.
1994)
Chapter 7: Experimental Validation for a Real Bridge Page 149
© 2017 Khac-Duy Nguyen Page 149
Figure 7-2. Sensor layout and location of damage (Farrar et al. 1994)
E-1 E-2 E-3 E-4
Figure 7-3. Four damage levels: cuts at the middle span of on the north plate girder
(Farrar et al. 1994)
Table 7-1. Natural frequencies of the I-40 Bridge
Case Mode number
1 2 3 4 5 6
Undamaged 2.48 2.95 3.49 4.08 4.17 4.64
E-1 2.51
(1.25)
2.99
(1.22)
3.57
(2.08)
4.12
(1.07)
4.21
(0.99)
4.69
(1.20)
E-2 2.52
(1.67)
2.99
(1.40)
3.51
(0.56)
4.10
(0.43)
4.20
(0.69)
4.66
(0.60)
E-3 2.46
(-0.71)
2.94
(-0.26)
3.48
(-0.40)
4.04
(-0.98)
4.14
(-0.69)
4.59
(-0.99)
E-4 2.27
(-8.45)
2.83
(-3.90)
3.49
(-0.19)
3.99
(-2.15)
4.15
(-0.47)
4.52
(-2.41)
*Values in parentheses indicate relative change (%) in natural frequency with
respect to the undamaged case.
Chapter 7: Experimental Validation for a Real Bridge Page 150
© 2017 Khac-Duy Nguyen Page 150
Table 7-2. MAC values after the introduction of damage
Case Mode number
1 2 3 4 5 6
E-1 0.997 0.999 0.998 0.994 0.976 0.998
E-2 0.995 0.997 1 0.989 0.991 1
E-3 0.998 0.998 0.999 0.990 0.975 0.998
E-4 0.863 0.894 0.997 0.947 0.940 0.969
7.2 DAMAGE IDENTIFICATION USING GMSEE CORRELATION METHOD
A damage detection model is established as an extended experimental model,
consisting of 72 Euler-Bernoulli beam elements with an average length of 3.58 m for
each element. Note that the model does not require the information of material and
cross sectional properties such as elastic modulus and 2nd moment of area. Model of
opened cracks studied by Christides and Barr (1984) is used to estimate stiffness
variation in vicinity of the crack in the I-40 Bridge. The stiffness can be considered
to be linearly reducing from the location of the crack, and the crack effective length
is equal to one and a half of the girder height (Sinha et al. 2002). A shown in Fig. 7-
4, the effective length of the crack of the I-40 Bridge is accounted for 4.845m for the
girder height of 3.23m. The equivalent stiffness reduction of the elements at the crack
can be obtained by averaging the stiffness reduction along the elements, as listed in
Table 7-3. It is clearly shown that only small damage was induced in the first two
crack scenarios, and only the last two crack scenarios made severe damage on the
cracked elements.
Cubic smoothing spline interpolation method is used to generate the refined
experimental mode shapes from the measured values. Herein, the refined mode
shapes are constructed separately for each span considering appropriate boundary
conditions. This is to reduce fitting error associated with high order curves of the
Chapter 7: Experimental Validation for a Real Bridge Page 151
© 2017 Khac-Duy Nguyen Page 151
whole mode shapes. Refined experimental modal curvatures are then estimated from
the refined experimental mode shapes and used to calculate geometric modal strain
energy, based on the Euler-Bernoulli beam theory as follows:
" 2[ ( )] dj
ij ilU x x= (7-1)
where "( )i x is the modal curvature at location x for the ith mode; and jl is the
length of the jth element.
Figure 7-4. Variation in stiffness around the crack
Table 7-3. Equivalent stiffness reduction
Damage case E-1 E-2 E-3 E-4
Stiffness reduction (%) 0.82 7.4 25.6 50
Figures 7-5 and 7-6 show damage identification results for damage cases E-3
and E-4 respectively, using the GMSEE method and sensitivity-weighting technique.
As shown in Fig. 7-5, it is possible to detect the actual damage location for case E-3
around sensor N7. The method also detects a false-positive region in the middle span
of the south girder. However, this false detection is acceptable as two girders are well
connected by a system of concrete deck, stringers and floor beams. It is also found
that the predicted damage extents for the actual damage members and the false ones
are very small, being less than 5%. Although this result is quite far from the induced
Chapter 7: Experimental Validation for a Real Bridge Page 152
© 2017 Khac-Duy Nguyen Page 152
damage (25.6%), it still well represents the little changes in natural frequencies and
mode shapes shown in Tables 7-1 and 7-2. Please note that the crack model shown in
Fig. 7-4 was developed for opened cracks but the crack in case E-3 was not well
opened. Also, the opened part of this crack is more likely representing a crack in
lateral bending mode. A model for this type of crack will need to be studied in future
to better represent its damage effect.
For the most severe damage case (E-4), as shown in Fig. 7-6, the GMSEE
correlation method shows excellent performance by accurately detecting the location
of damage without false alarms. The damage extent is predicted as about 45% which
is very close to the equivalent induced stiffness reduction (i.e., 50%). Compared to
the results obtained from the two popular methods, MSE damage index method and
the modal flexibility method, the proposed method shows similar performance in
locating damage. However, the two existing methods could not provide proper values
of damage extents. Details of the results obtained from these two methods can be
found elsewhere (Farrar and Jauregui, 1996).
For comparison, the results obtained by the GMSEE method with
conventional search space are shown in Figs. 7-7 to 7-8. As shown in Fig. 7-7, ones
are still able to detect the actual damage for case E-3 but there are many more false
locations obtained in the result such as at locations N5, N9, N13 and S5. For the
damage case E-4, the actual damage location can be realized but some false locations
are also observed at N5, N9 and S5 (Fig. 7-8).
Chapter 7: Experimental Validation for a Real Bridge Page 153
© 2017 Khac-Duy Nguyen Page 153
Figure 7-5. Result for damage case E-3 by GMSEE correlation method with
sensitivity-weighted search space
Figure 7-6. Result for damage case E-4 by GMSEE correlation method with
sensitivity-weighted search space
N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 N130
2
4
6
8
10
Dam
age E
xte
nt
(%)
North Girder
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S130
2
4
6
8
10
Sensor Location
Dam
age E
xte
nt
(%)
South Girder
N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 N130
20
40
60
Dam
age E
xte
nt
(%)
North Girder
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S130
20
40
60
Sensor Location
Dam
age E
xte
nt
(%)
South Girder
Chapter 7: Experimental Validation for a Real Bridge Page 154
© 2017 Khac-Duy Nguyen Page 154
Figure 7-7. Result for damage case E-3 by GMSEE correlation method with
conventional search space.
Figure 7-8. Result for damage case E-4 by GMSEE correlation method with
conventional search space.
N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 N130
2
4
6
8
10
Dam
age E
xte
nt
(%)
North Girder
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S130
2
4
6
8
10
Sensor Location
Dam
age E
xte
nt
(%)
South Girder
N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 N130
20
40
60
Dam
age E
xte
nt
(%)
North Girder
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S130
20
40
60
Sensor Location
Dam
age E
xte
nt
(%)
South Girder
Chapter 7: Experimental Validation for a Real Bridge Page 155
© 2017 Khac-Duy Nguyen Page 155
7.3 DAMAGE IDENTIFICATION USING MSEE CORRELATION METHOD
A similar damage detection model of 72 Euler-Bernoulli beam elements is
used for the MSEE method. Different from the damage detection model of the
GMSEE method, the one of the MSEE method requires the information of material
and cross sectional properties but these can be considered constant for all elements.
The same interpolation strategy as the one of the GMSEE method is used to generate
the refined experimental mode shapes from the measured values. Then refined
experimental modal curvatures are estimated and the modal strain energy is
calculated based on the Euler-Bernoulli beam theory as follows:
" 2[ ( )] d
j
ij ilU k(x) x x= (7-2)
where "( )i x is the modal curvature at location x for the ith mode; jl is the length of
the jth element; and k(x) is the stiffness constant for the element at location x.
Figures 7-9 and 7-10 show damage identification results for damage cases E-
3 and E-4 respectively, using the MSEE method and sensitivity-weighting technique.
As shown in Fig. 7-9, the MSEE method also shown a false-positive region in the
middle span of the south girder, however the region is narrower compared to the one
observed in the results of the GMSEE method. Besides, the damage location is well
located and the predicted region is narrower to the actual damage compared to the
results of the GMSEE method. Similar to the result of GMSEE method, damage
extents estimated by the MSEE method are less than 5%. For the most severe
damage case (E-4), as shown in Fig. 7-10, the MSEE correlation method shows
slightly better performance by more accurately locating the region of the damage.
The damage extent is well predicted as about 48% which is similar to the prediction
of the GMSEE method and close to the induced damage extent. It is worth noting
that, to the best of the candidate’s knowledge, this study is one of a few studies
which are able to give good estimations of damage extents for damage cases E-3 and
E-4. A study by Kim and Stubbs (2003) was also successful in quantifying damage
Chapter 7: Experimental Validation for a Real Bridge Page 156
© 2017 Khac-Duy Nguyen Page 156
for these cases by using frequency change, but they considered the first damage level
(E-1) as the baseline.
For comparison, the results obtained by the MSEE method with conventional
search space are shown in Figs. 7-11 to 7-12. As shown in Fig. 7-11, the MSEE
method is still successful to detect the actual damage location for the case E-3 but
there is one more false location is observed in between N8 and N9. For the most
severe damage case E-4, the actual damage location is well detected by the MSEE
method even the sensitivity-weighting technique is not employed (Fig. 7-12).
Chapter 7: Experimental Validation for a Real Bridge Page 157
© 2017 Khac-Duy Nguyen Page 157
Figure 7-9. Result for damage case E-3 by MSEE correlation method with
sensitivity-weighted search space
Figure 7-10. Result for damage case E-4 by MSEE correlation method with
sensitivity-weighted search space
N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 N130
2
4
6
8
10
Dam
age E
xte
nt
(%)
North Girder
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S130
2
4
6
8
10
Sensor Location
Dam
age E
xte
nt
(%)
South Girder
N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 N130
20
40
60
Dam
age E
xte
nt
(%)
North Girder
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S130
20
40
60
Sensor Location
Dam
age E
xte
nt
(%)
South Girder
Chapter 7: Experimental Validation for a Real Bridge Page 158
© 2017 Khac-Duy Nguyen Page 158
Figure 7-11. Result for damage case E-3 by MSEE correlation method with
conventional search space.
Figure 7-12. Result for damage case E-4 by MSEE correlation method with
conventional search space.
N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 N130
2
4
6
8
10
Dam
age E
xte
nt
(%)
North Girder
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S130
2
4
6
8
10
Sensor Location
Dam
age E
xte
nt
(%)
South Girder
N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 N130
20
40
60
Dam
age E
xte
nt
(%)
North Girder
S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S130
20
40
60
Sensor Location
Dam
age E
xte
nt
(%)
South Girder
Chapter 7: Experimental Validation for a Real Bridge Page 159
© 2017 Khac-Duy Nguyen Page 159
7.4 CONCLUSIONS
This chapter presented the verification of the two proposed correlation
methods, GMSEE and MSEE along with the sensitivity-weighted search space for a
real bridge, the I-40 Bridge in NM, USA.
The modal data were provided by the SHM research group at LANL, USA.
Modal properties of the first six modes were provided but only the first two modes
were used as the higher modes are not reliable. Also, the two most severe damage
levels, E-3 and E-4, were considered for the verification of the proposed methods.
Damage identification by the GMSEE and MSEE correlation methods were
performed. By integrating the sensitivity-weighting technique, both methods
successfully detected the actual damage location for the two damage scenarios. Also,
both methods generated a false detection at the mid span of the south girder for the
damage case E-3. However, this false detection is acceptable considering two girders
are well connected by a system of concrete deck, stringers and floor beams. Besides,
the MSEE method showed slightly better prediction in both damage cases. The
damage regions were more localized by the MSEE method rather than by the
GMSEE method.
Also, the sensitivity-weighted search space showed its important role in
reducing false-positive detection. When the sensitivity-weighted search space is not
incorporated, performance of both methods generally reduced. The actual damage
location could still be detected but more false locations were observed. It was also
found that the MSEE method performed better than the GMSEE method when using
the conventional search space. This is expected as the GMSEE method generally
requires more reliable modes to give good results.
From the experimental verification results, both GMSEE and MSEE
correlation methods demonstrated their excellent performance in damage
identification for the I-40 Bridge. It can be concluded that results obtained in this
Chapter 7: Experimental Validation for a Real Bridge Page 160
© 2017 Khac-Duy Nguyen Page 160
study are more reliable than the previous results obtained by Kim at Stubbs (2003)
where the first damage level (E-1) was used as baseline. In this study, both proposed
methods were able to give good estimations of damage for the last two damage cases
by considering the baseline as the undamaged state. Especially for case E-4, very
accurate predictions were obtained by the GMSEE and MSEE methods.
Chapter 8: Conclusions and Future Studies Page 161
© 2017 Khac-Duy Nguyen Page 161
8Chapter 8: Conclusions and Future Studies
8.1 SUMMARY AND CONCLUSIONS
The research in this thesis provided a new damage identification strategy
using correlation approach. Two damage identification algorithms using modal strain
energy-eigenvalue ratio (MSEE) were developed. Firstly, a method using a
simplified term of MSEE called geometric modal strain energy-eigenvalue ratio
(GMSEE) was developed. Secondly, a modification of this method was presented
considering use of the full term of MSEE. The modified method is capable for
damage identification with fewer modes compared with the original one although it
requires both material and sectional properties of the structural element. Thirdly, a
sensitivity-weighted search space was proposed to reduce the effect of measurement
noise on the damage identification for low-sensitivity elements. Performance of the
proposed damage identification algorithms and enhanced technique has been
validated with a simple numerical model and some experimental models of various
scales from small to large. The main findings of this thesis can be summarized as
follows:
1. A new damage identification algorithm using geometric modal strain
energy-eigenvalue ratio (GMSEE) has been developed (Chapter 3). The
sensitivity of GMSEE to stiffness reduction in structural elements has
been derived. Damage is identified by optimizing the correlation level
between a measured GMSEE change vector and a numerical one. The
method requires only measured modal parameters (i.e., natural frequency
and mode shape) and geometric information of the structural element.
Comparative studies of the proposed method have been made against the
conventional MSE correlation method on a 2D numerical truss bridge
Chapter 8: Conclusions and Future Studies Page 162
© 2017 Khac-Duy Nguyen Page 162
model. From the results, the proposed GMSEE correlation method could
predict damage location and damage extent with reasonable accuracy
compared to the MSE correlation method. All damaged elements for
single damage case, double damage case and multiple damage case were
successfully detected by the proposed method. Also, the GMSEE
correlation showed its robustness under noise conditions. Performance of
the proposed method was equivalent with the MSE correlation method
with regard to damage detection probability and false alarm probability.
Moreover, damage quantification results from the proposed method were
found to be less affected by noise than those from the existing method.
The GMSEE method has also been verified for two simple experimental
models, an 8-DOF spring-mass system and a 4-DOF multi-story building.
From the result of the 8-DOF spring-mass system, the GMSEE method
successfully predicted the actual damaged elements with no false
detection. Although the damage extent was not well predicted, it was due
to the uncertainty induced from the change in the friction between the
mass and the spring. For the 4-DOF multi-story building, the GMSEE
method could accurately identify the actual damage location for most of
the examined scenarios. The damage extents were successfully predicted
for some scenarios. Also, the increasing trend of damage severity was
well detected by the GMSEE method.
2. A modification of the initially developed GMSEE correlation method was
presented considering use of the full term of modal strain energy-
eigenvalue ratio (MSEE) (Chapter 4). The sensitivities of elemental
MSEE and total MSEE to structural stiffness reduction have been derived.
A MSEE change vector was constructed including both elemental MSEE
change and total MSEE change. Incorporating the total MSEE change in
Chapter 8: Conclusions and Future Studies Page 163
© 2017 Khac-Duy Nguyen Page 163
the MSEE change vector can overcome the calculation error caused by the
assumption made for the GMSEE method that is the fractional modal
strain energy is unchanged after damage. However, the modified method
requires not only geometric information of the structural element but also
material and cross sectional properties of the element.
Numerical verification of the proposed MSEE method has been made for
the 2D numerical truss bridge model. From the results, the proposed
MSEE correlation method could predict damage location and damage
extent with better accuracy compared to the GMSEE correlation method
when fewer modes were used. All damaged elements for single damage
case, double damage case and multiple damage case were successfully
detected by the MSEE method with only 4 modes. Meanwhile, the
initially developed GMSEE method showed some false detection when
only 4 modes were used. Also, the MSEE correlation showed its
robustness under noise conditions. The MSEE method showed excellent
results in noise condition when 8 modes were used. Its performance
reduced when fewer modes (i.e., 4 modes) were used but it still could
detect most of the damage with good accuracy. When modelling errors
were introduced, the MSEE method showed significant better
performance compared to the traditional MSE method. There were small
errors in the prediction of damage location and damage extent by the
MSEE method even at modelling error of 20%.
By applying the MSEE method to the experimental models (i.e., 8-DOF
spring-mass system and 4-DOF multi-story building), it also showed the
improvement in damage identification results. For the spring-mass
system, similar results were obtained by the MSEE method compared to
the GMSEE method. For the multi-story building, the MSEE method
Chapter 8: Conclusions and Future Studies Page 164
© 2017 Khac-Duy Nguyen Page 164
performed better than the GMSEE method in the first two damage states
representing damage in the first floor. The MSEE method gave higher
probability of damage for the actual damaged floor and fewer errors.
3. Improvement has been made for the correlation-based damage
identification method regarding search space of damage variables. The
technique refers sensitivity-weighted search space in which the search
space of an element is weighted with its MSEE sensitivity. The technique
was validated for the numerical 2-D truss model. Results demonstrated
that the technique could significantly reduce fault detection caused by
high noise effect in low-sensitivity elements.
4. The proposed GMSEE and MSEE correlation methods and the enhanced
technique have been validated for the QUT through-truss bridge model.
Due to sensor shortage, a roving strategy with total 6 layouts was
designed to capture the responses of most DOFs of one plane of the truss
bridge model. Three modes were successfully extracted from the acquired
acceleration signals. These modes were well verified with the numerical
mode obtained from a FE model.
Three damage states including two single and one multiple damage cases
have been simulated by loosening the bolts connecting the elements and
the gusset plates. Damage identification by the GMSEE and MSEE
correlation methods incorporating with the sensitivity-weighting
technique was performed. It was found that both methods successfully
detected the actual damaged elements in all the three damage states. Also,
both methods generated small rates of false detection which were up to
about 2% and 1% by the GMSEE and MSEE methods, respectively. The
MSEE method showed slightly clearer prediction in which damaged
elements were more distinguished from the undamaged elements.
Chapter 8: Conclusions and Future Studies Page 165
© 2017 Khac-Duy Nguyen Page 165
It is also observed that damage in a truss member can lead to slighter
damage on its adjacent members. In the case of severe damage in element
10, both methods showed false detection at element 99 which is adjacent
to the actual damaged element 10. This may be due to the change in the
orientation of element 99 or the force redistribution of this element after
element 10 was damaged. From this observation, it is recommended to
check all the elements within the region of the identified damaged
element in a truss structure.
Besides, results obtained from the experiment demonstrated the advantage
of using the sensitivity-weighted search space for modal correlation
methods. Significant false detection was reduced by adopting the
technique, especially for the GMSEE method.
5. The proposed GMSEE and MSEE correlation methods and the enhanced
technique have also been validated for a real bridge, the I-40 Bridge in
NM, USA. Only modal properties of the first two modes were used due
to their higher reliability compared to the higher order modes. Also, the
two most severe damage levels, E-3 and E-4, were used for the analysis.
Damage identification by the GMSEE and MSEE correlation methods
incorporating with the sensitivity-weighting technique was performed.
Both methods successfully detected the actual damage location for the
two damage scenarios. Also, both methods generated a false detection at
the mid span of the south girder for the damage case E-3 but this is
acceptable as the two girders were well connected by a system of concrete
deck, stringers and floor beams. It was found that the MSEE method
showed slightly better prediction in both damage cases. The damage
regions were more localized by the MSEE method rather than by the
GMSEE method.
Chapter 8: Conclusions and Future Studies Page 166
© 2017 Khac-Duy Nguyen Page 166
Results also demonstrated the important role of the sensitivity-weighted
search space in reducing false-positive detection. Performance of both
damage identification methods generally reduced when the conventional
search space was used instead of the sensitivity-weighted search space.
The actual damage location could still be detected but more false
locations were observed.
In summary, this research provided an innovative development of correlation-
based damage identification scheme including robust damage identification
algorithms and effective search space technique. The new damage identification
algorithms (i.e., GMSEE- and MSEE-based methods) incorporated with the
enhanced search space technique are capable of detecting the locations of damage
and estimating their damage extents. The methods work reliably for single and
multiple damage scenarios. Generally, the MSEE correlation method gives better
damage identification results compared to the GMSEE method. However, the MSEE
requires information of structural stiffness. Moreover, by employing the sensitivity-
weighted search space, the predictions by the both methods can significantly
improve. Therefore, in the cases where sufficient modes can be measured and
structural information is not reliable, the GMSEE correlation method incorporated
with the sensitivity-weighting technique would be preferred. In the case where only
few modes can be measured and the structural stiffness can be obtained properly, the
MSEE correlation method incorporated with the sensitivity-weighting technique
would be preferred.
Chapter 8: Conclusions and Future Studies Page 167
© 2017 Khac-Duy Nguyen Page 167
8.2 FUTURE STUDIES
Although the proposed damage identification algorithms and enhanced search
space technique have shown their promise for several illustrative numerical and
experimental examples, there are several issues which need to be addressed to make
them more applicable for practice. Recommendations for future work include the
following:
1. The present study used MDLAC as a basis for estimating the correlation
level between the measured modal vector and numerical one. It would be
beneficial to conduct a comparative study on using different correlation
functions. Then ones can choose appropriate function for their particular
problem.
2. The proposed damage identification methods and enhanced technique
have been verified for several structure types such as spring-mass system,
multi-story building, truss structure and plate-on-girder bridge. There is a
need to extent the study to other types of structures such as plates, frames,
cable-supported bridges, arch bridges.
3. The methodologies developed in this study were derived by considering
damage as stiffness reduction in structural elements. Although it is
popularly used to represent many types of damage, there is a need to
extent this study for real damage types such as crack, corrosion, material
degradation, etc. The information of damage in real forms would help to
evaluate more accurately the impact of the damage on the structure.
4. The study showed some limitations on prediction accuracy especially on
damage extent estimation. The undamaged stiffness matrix was used to
calculate the damaged MSEE sensitivity which could generate some
errors. Therefore, development of a more effective approach for damage
extent estimation is necessary. Besides, more effort should be made on
developing techniques of reducing false-positive identification.
Chapter 8: Conclusions and Future Studies Page 168
© 2017 Khac-Duy Nguyen Page 168
5. In the experimental study for the QUT through truss bridge mode, a
damage threshold of 10% was used for the discrimination of undamaged
and damaged states. The identification of the damage threshold should be
systematically studied in the future. Moreover, there are some issues in
the experimental test that should be carefully examined in the future.
They include effect of mass variation due to sensor roving, and
improvement of frequency resolution. Identification of true stiffness
constants of truss members at various bolt torques is also recommended as
future study.
6. This research only considered damage at some elements in both numerical
and experimental studies. In future, a study on the sensitivity of the
proposed methods to damage at different locations with different extents
should be carried out.
Reference Page 169
© 2017 Khac-Duy Nguyen Page 169
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Appendices Page 177
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Appendices
APPENDIX – A: EXAMPLES OF GEOMETRIC QUANTITY OF STIFFNESS MATRIX
Elem
ent ty
pe
Stiffn
ess matrix
G
eom
etric quan
tity o
f stiffness m
atrix
Flex
ural elem
ent
Ax
ial elem
ent
Flex
ural +
axial E
lemen
t
For flex
ural co
mp
onen
t:
For ax
ial com
ponen
t:
Appendices Page 178
© 2017 Khac-Duy Nguyen Page 178
APPENDIX – B: PUBLICATIONS DERIVED FROM THIS PROJECT
Journal Articles:
Nguyen, K.D., Chan, T.H.T. and Thambiratnam, D.P. (2016). Structural damage
identification based on change in geometric modal strain energy-eigenvalue
ratio. Smart Materials and Systems, 25, 14pp.
Nguyen, K.D., Chan, T.H.T., Thambiratnam, D.P. and Nguyen, A. A new modal
strain energy method for damage identification using a correlation technique
(Under preparation).
Nguyen, K.D., Chan, T.H.T., Thambiratnam, D.P. and Nguyen, A. Damage
identification for a complex truss structure using correlation approach and
sensitivity-weighted search space technique (Under preparation).
Conference papers:
Nguyen, K.D., Chan, T.H.T. and Thambiratnam, D.P. (2015). Correlation-based
damage detection using geometric modal strain energy and natural frequency.
Proceedings of the 7th International Conference on Structural Health
Monitoring of Intelligent Infrastructures (SHMII-7), Turin, Italy (July 2015)