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Damage assessment in structures using
vibration characteristics
by
Shih Hoi Wai
A thesis submitted to the School of Urban Development
Queensland University of Technology
in partial fulfilment of the requirements for the degree of
Doctor of Philosophy
2009
Damage assessment in structures using vibration characteristics i
ABSTRACT
Changes in load characteristics, deterioration with age, environmental influences and
random actions may cause local or global damage in structures, especially in bridges,
which are designed for long life spans. Continuous health monitoring of structures
will enable the early identification of distress and allow appropriate retrofitting in
order to avoid failure or collapse of the structures. In recent times, structural health
monitoring (SHM) has attracted much attention in both research and development.
Local and global methods of damage assessment using the monitored information are
an integral part of SHM techniques. In the local case, the assessment of the state of a
structure is done either by direct visual inspection or using experimental techniques
such as acoustic emission, ultrasonic, magnetic particle inspection, radiography and
eddy current. A characteristic of all these techniques is that their application requires a
prior localization of the damaged zones. The limitations of the local methodologies
can be overcome by using vibration-based methods, which give a global damage
assessment. The vibration-based damage detection methods use measured changes in
dynamic characteristics to evaluate changes in physical properties that may indicate
structural damage or degradation. The basic idea is that modal parameters (notably
frequencies, mode shapes, and modal damping) are functions of the physical
properties of the structure (mass, damping, and stiffness). Changes in the physical
properties will therefore cause changes in the modal properties. Any reduction in
structural stiffness and increase in damping in the structure may indicate structural
damage.
This research uses the variations in vibration parameters to develop a multi-criteria
method for damage assessment. It incorporates the changes in natural frequencies,
modal flexibility and modal strain energy to locate damage in the main load bearing
elements in bridge structures such as beams, slabs and trusses and simple bridges
involving these elements. Dynamic computer simulation techniques are used to
develop and apply the multi-criteria procedure under different damage scenarios. The
effectiveness of the procedure is demonstrated through numerical examples. Results
show that the proposed method incorporating modal flexibility and modal strain
energy changes is competent in damage assessment in the structures treated herein.
Damage assessment in structures using vibration characteristics ii
Damage assessment in structures using vibration characteristics iii
KEYWORDS
Structural health monitoring, damage assessment, free vibration characteristics,
beam, plate, truss, slab-on-girder bridge, finite element method, natural frequencies,
mode shape, modal flexibility and modal strain energy.
Damage assessment in structures using vibration characteristics iv
Damage assessment in structures using vibration characteristics v
ACKNOWLEDGEMENTS
This thesis is the result of three and half years of work whereby I have been
accompanied and supported by many people. It is a pleasant aspect that I have now
the opportunity to express my gratitude for all of them.
I would like to express my deep and sincere gratitude to my principal supervisor,
Professor David Thambiratnam for his supervision, guidance and advice as well as
providing me continuous encouragement and momentum throughout this work. His
professional and energetic support helps me in writing this thesis; a task which I
could not have achieved alone. In particular, I am grateful for enlightening me the
first glance of research in the area of structural dynamics. Moreover, his wide
knowledge, logical way of thinking, detailed and constructive comments have been
of great value for me. I would like to extend my appreciation to my associate
supervisor, Assoc. Professor Tommy Chan for his essential assistance in my work. I
would like to thank him who kept an eye on the progress of my work and was always
available when I needed his advice. His valuable comment, advice and crucial
contribution to the thesis, especially in related to structural health monitoring of
bridge structures, are gratefully acknowledged. Thank you for both supervisors for
patiently correcting and editing my manuscript during the PhD period. I have learned
a lot of skills from them in numerous ways, such as information retrieval technique,
data analysis, lateral thinking, presentation and writing technical paper skills etc. In
addition, I am grateful for having the training opportunity to be the class tutor in
structural engineering units of Bachelor Degree of Civil Engineering during my
research.
Alternatively, I would like to acknowledge the financial support of the Research
Capacity Building Doctoral Scholarships and travel grants from the Faculty of Built
Environment and Engineering (BEE) at Queensland University of Technology
(QUT). I am also grateful for the School of Urban Development for providing me an
excellent work environment, library and computer facilities during the past years.
During this work I have collaborated with many laboratory technicians for whom I
have great regard, and I wish to extend my warmest thanks to all those who have
Damage assessment in structures using vibration characteristics vi
helped me in my experimental testing: they are Arthur Powell, Brian Pelin, Terry
Beach, Anthony Tofoni, Melissa Johnston, Jonathan James and Lincoln Hudson. I
would also like to thank fellow Jehangir Madhani from School of Engineering
Systems for providing advice to me about experimental instruments in dynamic
testing. In addition, I would like to thank librarian Peter Fell for delivering excellent
presentation in the workshops, which is about advanced information retrieval skills
(AIRS). I also want to thank Donald Lam from the department of information
technology support for dealing with software license of finite element analysis
program. I would also like to thank all fellows in my office and friends for their
support and sharing the research time. Last but not least, I am very grateful to my
parents and sisters for their love, encouragement, patience and support during the
PhD period.
Damage assessment in structures using vibration characteristics vii
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 higher education institution. To the best of
my knowledge and belief, this thesis contains no material previously published or
written by another person except where due reference is made.
Signature: Shih Hoi Wai
Date: 6 Sep 2009
Damage assessment in structures using vibration characteristics viii
PUBLICATION LIST
The published papers based on the work presented in this thesis are listed as follows:
Conference Paper
1. H.W. Shih, D.P. Thambiratnam, and T.H.T. Chan (2009). “Damage assessment
in multiple-girder composite bridge using vibration characteristics.” Proceedings
BEE Postgraduate Infrastructure Theme Conference, Gardens Point Campus,
Queensland University of Technology, 11-21.
2. H.W. Shih, T.H.T. Chan, and D.P. Thambiratnam (2008). “Structural damage
localization in slab-on-girder bridges using vibration characteristics.” The Third
World Congress on Engineering Asset Management and Intelligent Maintenance
System Conference (WCEAM-IMS), Beijing, China, 1408-1418.
3. H.W. Shih, D.P. Thambiratnam, and M. Humphreys (2007). “Damage
assessment in beams using vibration characteristics.” The Third International
Conference on Structural Engineering, Mechanics and Computation (SEMC),
Cape Town, South Africa, 709-710.
4. H.W. Shih, D.P. Thambiratnam, and M. Humphreys (2007). “Review of
structural health monitoring & damage assessment of bridge structures.”
Proceedings BEE Postgraduate Infrastructure Theme Conference, Gardens Point
Campus, Queensland University of Technology, 51-64.
5. T.H.T. Chan, H.W. Shih, and D.P. Thambiratnam (2009). “Case studies on
vibration based damage identification: multi-criteria approach.” International
Conference on Reliability Maintainability and Safety (ICRMS), Chengdu, China,
1242-1247.
Damage assessment in structures using vibration characteristics ix
Journal Paper
1. H.W. Shih, D.P. Thambiratnam and T.H.T. Chan (2009). “Vibration based
structural damage detection in flexural members using multi-criteria approach.”
Journal of Sound & Vibration, 323, 645–661.
2. H.W. Shih, D.P. Thambiratnam and T.H.T. Chan (2009). “Damage assessment in
slab-on-girder bridges using vibration characteristics.” Computers & Structures.
(Submitted)
Damage assessment in structures using vibration characteristics x
TABLE OF CONTENT
Abstract .......................................................................................................................... i
Keywords ..................................................................................................................... iii
Acknowledgements ........................................................................................................v
Statement of original authorship ................................................................................. vii
Publication list ........................................................................................................... viii
Table of content .............................................................................................................x
List of figures ...............................................................................................................xv
List of tables ............................................................................................................. xviii
Notations ......................................................................................................................xx
Abbreviations ........................................................................................................... xxiii
Chapter 1 Introduction
1.1 Background ..............................................................................................................1
1.2 Aim, objectives and scope .......................................................................................2
1.3 This research ............................................................................................................3
1.3.1 Hypothesis.......................................................................................................3
1.3.2 Research problem............................................................................................3
1.3.3 Significance and innovation of this research ..................................................5
1.4 Research method ......................................................................................................5
1.5 Thesis content ..........................................................................................................5
Chapter 2 Literature review
2.1 Introduction ..............................................................................................................7
2.2 Structural form of bridges ........................................................................................8
2.2.1 General bridge configurations .........................................................................8
2.2.2 Slab-on-girder bridge ......................................................................................9
2.2.3 Truss bridge ..................................................................................................10
2.3 Structural health monitoring ..................................................................................11
2.3.1 Advantage of structural health monitoring ...................................................14
Damage assessment in structures using vibration characteristics xi
2.4 Structural damage in bridges ................................................................................. 15
2.5 Forward and inverse problem................................................................................ 16
2.6 Classification of damage detection techniques ..................................................... 17
2.6.1 Vibration-based damage detection methods ................................................ 19
2.6.1.1 Frequency change ............................................................................ 19
2.6.1.2 Modal flexibility............................................................................... 20
2.6.1.3 Modal strain energy.......................................................................... 22
2.6.1.4 Flexibility curvature ......................................................................... 23
2.6.1.5 Mode shape curvature ...................................................................... 23
2.6.1.6 Uniform load surface curvature ....................................................... 25
2.6.1.7 Stiffness change ............................................................................... 25
2.6.2 Multiple criteria methods ............................................................................. 27
2.6.2.1 Flexibility and stiffness .................................................................... 27
2.6.2.2 Flexibility and strain energy............................................................. 27
2.6.2.3 Flexibility difference and modal curvature difference ..................... 28
2.6.2.4 Energy difference and energy curvature difference ......................... 28
2.6.3 Advanced damage detection methods .......................................................... 29
2.6.3.1 Wavelet analysis............................................................................... 29
2.6.3.2 Artificial neural network .................................................................. 30
2.6.3.3 Acoustic emission monitoring ......................................................... 31
2.7 Structural identification ......................................................................................... 32
2.7.1 Modal identification ..................................................................................... 32
2.7.2 Fundamentals of modal testing .................................................................... 33
2.7.3 Types of excitation ....................................................................................... 35
2.7.3.1 Forced vibration excitation .............................................................. 35
2.7.3.2 Ambient vibration excitation............................................................ 36
2.7.3.3 Free vibration ................................................................................... 36
2.7.4 Model updating ............................................................................................ 37
2.7.4.1 Manual tuning model updating ........................................................ 37
2.7.4.2 Automatic model updating ............................................................... 38
2.7.4.3 Comparison of modal properties ...................................................... 39
2.7.5 Uncertainties and limitations ....................................................................... 40
2.7.5.1 Uncertainties .................................................................................... 41
Damage assessment in structures using vibration characteristics xii
2.7.5.2 Effect of environmental and operational variations ..........................41
2.7.5.3 Low sensitivity to damage ................................................................42
2.7.5.4 Nonlinear analysis .............................................................................43
2.8 Case study 1 – Wind and structural health monitoring system ..............................43
2.9 Case study 2 – Damage identification of timber bridge ........................................44
2.10 Summary of literature review ..............................................................................45
2.10.1 Natural frequency and mode shape ..........................................................45
2.10.2 Modal flexibility method and modal strain energy method .....................46
2.10.3 Environmental effect and operational variations .....................................46
2.10.4 Multiple criteria approach ........................................................................47
Chapter 3 Theory and validation of finite element models
3.1 Introduction ............................................................................................................48
3.2 Basic dynamic equations........................................................................................49
3.3 Frequency response function .................................................................................51
3.4 Damping ratio ........................................................................................................52
3.4.1 Logarithmic decrement .................................................................................52
3.4.2 Bandwidth method ........................................................................................53
3.5 Rayleigh’s method .................................................................................................53
3.6 Modal flexibility matrix .........................................................................................55
3.7 Elastic strain energy ..............................................................................................56
3.7.1 Modal strain energy – Beam .........................................................................57
3.7.2 Modal strain energy – Plate ..........................................................................59
3.7.3 Modal strain energy – Truss .........................................................................61
3.8 Validation of finite element models .......................................................................62
3.9 Static test ................................................................................................................63
3.9.1 Description of the model ...............................................................................63
3.9.2 Instrument setup ............................................................................................66
3.9.3 Test methodology..........................................................................................66
3.9.4 Experimental results and discussions............................................................69
3.10 Free vibration test I: Slab-on-girder bridge ........................................................71
3.10.1 Description of the model .............................................................................71
Damage assessment in structures using vibration characteristics xiii
3.10.2 Instrument setup ......................................................................................... 71
3.10.3 Test methodology ....................................................................................... 73
3.10.4 Experimental results and discussions ......................................................... 74
3.10.5 Model updating .......................................................................................... 82
3.11 Free vibration test II: Simply supported beam .................................................... 85
3.12 Summary ............................................................................................................. 88
Chapter 4 Application I – Load bearing elements of structures
4.1 Introduction ........................................................................................................... 90
4.2 Damage assessment in beam ................................................................................. 91
4.2.1 Model description......................................................................................... 91
4.2.2 Frequency change......................................................................................... 94
4.2.3 Modal flexibility change .............................................................................. 94
4.2.4 Modal strain energy change ......................................................................... 95
4.3 Damage assessment in plate (slab) ........................................................................ 98
4.3.1 Model description......................................................................................... 98
4.3.2 Frequency change....................................................................................... 101
4.3.3 Modal flexibility change ............................................................................ 103
4.3.4 Modal strain energy change ....................................................................... 104
4.4 Summary ............................................................................................................. 106
Chapter 5 Application II – Bridges
5.1 Damage assessment in slab-on-girder bridge ...................................................... 108
5.1.1 Model description....................................................................................... 108
5.1.2 Frequency change....................................................................................... 112
5.1.3 Modal flexibility change ............................................................................ 114
5.1.4 Modal strain energy change ....................................................................... 115
5.2 Damage assessment in multiple-girder composite bridge ................................... 119
5.2.1 Model description....................................................................................... 119
5.2.2 Frequency change....................................................................................... 122
5.2.3 Modal flexibility change ............................................................................ 124
Damage assessment in structures using vibration characteristics xiv
5.2.4 Modal strain energy change ........................................................................124
5.3 Damage assessment in truss bridge ......................................................................129
5.3.1 Model description .......................................................................................129
5.3.2 Frequency change .......................................................................................136
5.3.3 Modal flexibility change .............................................................................138
5.3.4 Modal strain energy change ........................................................................138
5.4 Summary ..............................................................................................................142
5.4.1 Flowchart for multi-criteria approach .........................................................145
Chapter 6 Conclusions
6.1 Summary ..............................................................................................................146
6.2 Contributions to scientific knowledge .................................................................148
6.3 Recommendations for further research ...............................................................149
Bibliography .........................................................................................................152
Damage assessment in structures using vibration characteristics xv
LIST OF FIGURES
Fig. 2.1 Common truss types used in bridges. 11
Fig. 3.1 Dynamic equilibrium of a single degree-of-freedom system. 49
Fig. 3.2 Free-vibration response of an underdamped system. 52
Fig. 3.3 Typical frequency response curve. 53
Fig. 3.4(a) Slab-on-girder bridge (Test specimen). 64
Fig. 3.4(b) Slab-on-girder bridge (Cross bracings on test specimen). 64
Fig. 3.4(c) Slab-on-girder bridge (Boundary condition). 65
Fig. 3.5 FE model of slab-on-girder bridge. 65
Fig. 3.6 Layout of load frame. 67
Fig. 3.7(a) Hydraulic jack system (Vertical jack). 67
Fig. 3.7(b) Hydraulic jack system (Hydraulic pump). 67
Fig. 3.8(a) Field measurement equipment used in static test (Load cell). 68
Fig. 3.8(b) Field measurement equipment used in static test (LVDT). 68
Fig. 3.8(c) Field measurement equipment used in static test (Data
acquisition system).
68
Fig. 3.9 Loading position and LVDT layout on the deck. 68
Fig. 3.10 Plot of load vs deflection for the static test. 69
Fig. 3.11 Analytical deflection vs experimental deflection. 70
Fig. 3.12 Convergence of the static deflection at location ‘A’. 70
Fig. 3.13 Instrument setup in dynamic test. 72
Fig. 3.14 Schematic of the dynamic measurement system. 72
Fig. 3.15 Piezoelectric accelerometer. 73
Fig. 3.16 Measurement grid and accelerometer locations. 74
Fig. 3.17 Typical acceleration time history. 75
Fig. 3.18 Typical power spectrum density plot. 75
Fig. 3.19 Experimentally obtained vibration modes of undamaged deck. 76
Fig. 3.20 First five vibration modes of undamaged slab-on-girder bridge
(FEM).
77
Fig. 3.21 First five vibration modes of undamaged deck (FEM). 78
Fig. 3.22 Plot of analytical vs experimental natural frequencies. 80
Fig. 3.23 Modal flexibility change on girders based on experimental data in damage scenario.
81
Damage assessment in structures using vibration characteristics xvi
Fig. 3.24 Modal strain energy based damage index on girders based on
experimental data in damage scenario.
81
Fig. 3.25(a) Finite element model updating (Initial FEM). 83
Fig. 3.25(b) Finite element model updating (Test specimen). 83
Fig. 3.25(c) Finite element model updating (Updated FEM). 83
Fig. 3.26 First five vibration modes of undamaged slab-on-girder bridge
after model updating.
84
Fig. 3.27 Simply supported beam. 86
Fig. 3.28 Boundary condition of the beam. 86
Fig. 3.29 Accelerometer on the beam. 86
Fig. 3.30 Flaw at mid-span of the beam. 87
Fig. 3.31 Flaw size (10mmx5mmx40mm) in the FE model. 87
Fig. 3.32(a) Measured natural frequency of the undamaged beam
(Mode 1).
87
Fig. 3.32(b) Measured natural frequency of the undamaged beam
(Mode 2).
87
Fig. 3.33(a) Measured natural frequency of the damaged beam (Mode 1). 87
Fig. 3.33(b) Measured natural frequency of the damaged beam (Mode 2). 87
Fig. 4.1 Damage case (D) for single-span beam (2.8m span length). 92
Fig. 4.2 Damage case (D) for 2-span beam (2.8m span length). 93
Fig. 4.3 Damage case (D) for 3-span beam (2.8m span length). 93
Fig. 4.4 Flaw size ‘A’ simulated in FEM. 94
Fig. 4.5 First five vibration modes of undamaged FE model. 96
Fig. 4.6 Modal flexibility change (left) and Modal strain energy based
damage index (right) on beam.
97
Fig. 4.7 Damage case (D) for plate with all edges clamped. 99
Fig. 4.8 Damage case (D) for simply supported plate. 100
Fig. 4.9 Damage case (D) for 2-span plate. 100
Fig. 4.10 First five vibration modes of undamaged plate with simply
supported condition.
102
Fig. 4.11 Modal flexibility change (left) and Modal strain energy based
damage index (right) on plate.
105
Fig. 5.1 Isometric view of FE model. 109
Damage assessment in structures using vibration characteristics xvii
Fig. 5.2 Damage cases (D1-D3) on deck. 110
Fig. 5.3 Damage cases (D4-D7) on girders. 111
Fig. 5.4 First five vibration modes of FE model. 113
Fig. 5.5 Modal flexibility change (left) and Modal strain energy based
damage index (right) on deck.
117
Fig. 5.6 Modal flexibility change (left) and Modal strain energy based
damage index (right) on girders.
118
Fig. 5.7 Isometric view of FE model with numbering system on
girders.
120
Fig. 5.8 Damage cases (D1-D2) on deck. 120
Fig. 5.9 Damage cases (D3-D6) on girders. 121
Fig. 5.10 First five vibration modes of FE model. 123
Fig. 5.11 Modal flexibility change (left) and Modal strain energy based
damage index (right) on the deck.
126
Fig. 5.12 Modal flexibility change (left) and Modal strain energy based
damage index (right) on the girders.
127
Fig. 5.13 Relationship between modal strain energy based damage index
and structural state of girders.
128
Fig. 5.14 Isometric view of truss model. 130
Fig. 5.15 The classification of truss members. 131
Fig. 5.16 Numbering system for truss nodes. 132
Fig. 5.17 Numbering system for truss members. 132
Fig. 5.18 Damage cases (D1-D2) on deck. 133
Fig. 5.19 Damage cases (D3-D6) on truss. 134
Fig. 5.20 Damage cases (D7-D8) on deck and truss. 135
Fig. 5.21 First five vibration modes of FE model. 137
Fig. 5.22 Modal flexibility change (left) and Modal strain energy based
damage index (right) on deck.
140
Fig. 5.23 Modal flexibility change (left) and Modal strain energy based
damage index (right) on truss.
141
Fig. 5.24 Flowchart of damage detection in proposed structures.
145
Damage assessment in structures using vibration characteristics xviii
LIST OF TABLES
Table 2.1 Common structural material used in slab-on-girder bridges 10
Table 2.2 Functions of sensors in SHM system 13
Table 2.3 Summary of damage detection categories and methods 18
Table 3.1 Geometric and material properties for the test specimen 66
Table 3.2 Correlation between experimental and initial FE model 79
Table 3.3 MAC using experimental & analytical data in undamaged cases 80
Table 3.4 Estimated damping ratio by half-power method 80
Table 3.5 Correlation between experimental & manually tuned FE model 83
Table 3.6 Geometric and material properties of the beam 85
Table 3.7 Validation of the FEM for the simply supported beam 88
Table 4.1 Geometric and material properties of beam 92
Table 4.2 Dimension of flaws in beam 92
Table 4.3 Natural frequencies of undamaged beam from FEM 95
Table 4.4 Natural frequencies of damaged beam from FEM 96
Table 4.5 Geometric and material properties of plate 98
Table 4.6 Validation of FEM for plate with clamped boundaries 99
Table 4.7 Natural frequencies from FEM for undamaged plate 101
Table 4.8 Natural frequencies from FEM for damaged plate 103
Table 4.9 Performance of damage detection algorithms for beam and plate 104
Table 5.1 Geometric and material properties for the slab-on-girder bridge 109
Table 5.2 Natural frequencies from FEM for slab-on-girder bridges 112
Table 5.3 The relationship between fundamental frequency change ratio
and damage severity with certain locations on deck and girders
114
Table 5.4 Geometric and material properties of deck and girders 122
Table 5.5 Natural frequencies from FEM for multiple-girder composite
bridges
123
Table 5.6 Geometric and material properties of deck and truss 130
Table 5.7 Numbering systems for truss members 130
Table 5.8 Truss damage configurations 131
Table 5.9 Natural frequencies from FEM for truss bridges 136
Damage assessment in structures using vibration characteristics xix
Table 5.10 Performance of damage detection algorithms for slab-on-girder
bridge
142
Table 5.11 Performance of damage detection algorithms for multiple-girder
composite bridge
142
Table 5.12 Performance of damage detection algorithms for truss bridge 143
Damage assessment in structures using vibration characteristics xx
NOTATIONS
The following symbols are used in this thesis:
A Cross-sectional area
a Amplitude of motion
C Contribution coefficient, correlation value
[C] Damping matrix
c Viscous damping coefficient
D Bending stiffness of the plate
d Damaged structural state
E Modulus of elasticity
EI Flexural rigidity of the cross section
F Flexibility
][ F Global flexibility matrix
{F} Force vector
)(ωF Vector of discrete Fourier transforms of external forces
f Frequencies; time dependent excitation force
)(ωH FRF matrix
h Plate thickness, healthy state
I Second moment of a plane area, identity matrix
gI Gross section moment of inertia
i i-th mode
j j-th element
K Modal stiffness
][ K Global stiffness matrix
k Elemental stiffness
L Length, distance
l Length of element
M Bending moment at a section
][ M Global mass matrix
N Total number of modes
Damage assessment in structures using vibration characteristics xxi
m Mass
n, nn Number of degree of freedom
P Load
T Period
t Time of motion
U Strain energy
u Displacement
u& Velocity
u&& Acceleration
1u , 2u Two successive peak amplitudes in the free vibration
V Volume of the body
v Poisson’s ratio, mode shape vector
vv ′′′, 22, dxvddxdv
W Work
EW Virtual external work
IW Virtual internal work
w Transverse displacement of the plate
)(ωX Vector of discrete Fourier transforms of displacement responses
x Distance measured along the length of the structure
x,y,z Rectangular axes (origin at point O)
y Vertical deflection
Z Normalized damage index
α Phase angle
β Damage index
γ Shear strain
∆ Change
δ Logarithmic decrement, displacement
ε Normal strain
θ Angle of rotation
λ Eigenvalue, material parameters to be calibrated
µ Mean
ξ Damping ratio
Damage assessment in structures using vibration characteristics xxii
σ Normal stress, standard deviation
τ Shear stress
φ Mass normalized modal vectors
ω Angular frequency
2ω Eigenvalue
* Damaged structural state
{} Vector
[] Matrix
T[] Transpose of matrix
1[]− Inverse of matrix
Damage assessment in structures using vibration characteristics xxiii
ABBREVIATIONS
AE Acoustic emission
ANN Artificial neural network
CA Condition assessment
CDF Curvature damage factor
CIP Concrete impregnated with polystyrene
COMAC Coordinate modal assurance criterion
D Damage case
Denom Denominator
DLAC Damage location assurance criterion
DOF Degree of freedom
DQM Differential quadrature method
EMA Experimental modal analysis
FE Finite element
FEA Finite element analysis
FEM Finite element model
FFT Fast Fourier transform
FRF Frequency response function
GA Genetic algorithm
GFI Global flexibility index
GIS Geographic information system
GPS Global positioning system
HHT Hilbert–Huang transform
IMF Intrinsic mode functions
KE Kinetic energy
L Left girder
LVDT Linear voltage displacement transducer
MAC Modal assurance criterion
MC Modal curvature
MFC Modal flexibility change
MSC Multi-span continuous
MSEC Modal strain energy change
Damage assessment in structures using vibration characteristics xxiv
MSSS Multi-span simply supported
NDE Non-destructive evaluation
NM Number of vibration modes
Num Numerator
PSD Power spectral density
R Right girder
RC Reinforced concrete
SDDI Sensitivity damage detection index
SE Strain energy
SHM Structural health monitoring
SS Simply supported
VB Vibration based
WASHMS Wind and structural health monitoring system
2D Two dimensional
Damage assessment in structures using vibration characteristics 1
Chapter 1 Introduction
1.1 Background
Bridges are an important and integral part of modern transportation systems and play
a vital role in the lives of a community. They are normally designed to have long life
spans. Changes in load characteristics, deterioration with age, environmental
influences and random actions may cause local or global damage to structures.
Bridge failure or poor performance will disrupt the transportation system and may
also result in loss of lives and property. It is therefore very important to ensure that
bridges perform safely and efficiently at all times by monitoring their structural
integrity and undertaking appropriate remedial measures. Many of the bridges in
Queensland Australia were built several decades ago and are now decaying due to
aging, deterioration, fatigue, lack of repair and in some cases, because they were not
designed for the current demand. Today, these structures are subjected to heavier and
faster moving loads, compared to their original design loads. As an example,
significant levels of vibrations have been monitored in some spans of the Storey
Bridge in Brisbane, which was built over 60 years ago (Thambiratnam 1995).
Another example is the Victoria Bridge in Brisbane, which has been subjected to
modified use to accommodate bus lanes on one half of the bridge width. This can
lead to torsional vibration which may not have been considered before. These
changes in loading patterns, together with normal deterioration with age, can bring
about localised failure and if this goes undetected the failure can extend and cause
partial or even total collapse of the structure. At the time of writing this thesis, there
is interest expressed by the Brisbane City Council and the Queensland Department of
Main Roads (the owners of the bridges) to monitor some of the bridges in order to
evaluate their performance and carry out appropriate retrofitting. This cannot be
achieved without established methods of damage evaluation and failure prediction on
bridges, which forms the basis of this PhD study.
Damage assessment in structures using vibration characteristics 2
1.2 Aim, objectives and scope
The main aim of this research is to develop a multi-criteria procedure for damage
assessment of structures.
In addition, the specific research objectives are as follows:
• Incorporate the variations in natural frequencies, modal flexibility and modal
strain energy between the healthy and damaged structures into this procedure.
• Treat the main load bearing elements of structures, viz beam and plate, and
complete bridge structures for damage assessment under different damage scenarios.
• Demonstrate the feasibility and capability of the proposed procedure through
numerical examples.
A typical structural health monitoring (SHM) system includes three major
components: a sensor system, a data processing system and a health evaluation
system (Li et al. 2004). As SHM is such a broad scope of the field, this research will
focus on the third component of the SHM system, which is the health evaluation
system. Usually there are four different levels of damage evaluation in a structure
(Rytter 1993): damage detection (Level 1), damage localization (Level 2), damage
quantification (Level 3), and predication of the acceptable load level and of the
remaining service life of the damaged structure (Level 4). The emphasis of this study
will be on Level 1 and Level 2, using non-destructive vibration-based damage
detection methods (changes in frequency, modal flexibility and modal strain energy)
to detect and locate damages on the proposed structures. These are (i) beams, (ii)
slabs (or plates), (iii) slab-on-girder bridges and (iv) truss bridges. Beams and slabs
are selected for investigation as they are common load bearing elements of bridge
structures. In addition, slab-on-girder bridges and truss bridges are investigated as
they are coupled by beam, slab and truss elements and also they are the most
common types of bridge structures. The feasibility and capability of the proposed
multi-criteria approach will be demonstrated through numerical examples which
cover axial, flexural elements and the complete bridge structures as well. In general,
damage can be defined as changes introduced into a structure that adversely affects
Damage assessment in structures using vibration characteristics 3
its current or future performance (Doebling et al. 1996). The simulated damage used
in this study has been adopted from established work in the literature, such as
Cornwell et al. (1999), and will be limited to changes to the material and/or
geometric properties of structural components which affect the performance of the
entire structure.
1.3 This research
1.3.1 Hypothesis
It is possible to quantify damage in a structure by observing the variations in its
dynamics properties. As the dynamic characteristics of a structure, namely natural
frequencies and mode shapes, are known to be functions of its stiffness and mass
distributions, variations in modal frequencies and mode shapes can be an effective
indication of structural deterioration. Deterioration of a structure results in a
reduction of its stiffness which causes changes in its dynamics characteristics. Thus,
the damage state of a structure can be inferred from the changes in its vibration
characteristics (Doebling et al. 1996).
1.3.2 Research problem
This research treats the problem of damage evaluation in (older) bridges in order to
ensure their integrity and safety. In recent times, structural health monitoring (SHM)
has attracted much attention in both research and development. SHM encompasses
both local and global methods of damage identification (Zapico and Gonzalez 2006).
In the local case, the assessment of the state of a structure is performed either by
direct visual inspection or using experimental techniques such as ultrasonic,
magnetic particle inspection, radiography and eddy current. A characteristic of all
these techniques is that their applications require a prior localization of the damaged
zones. The limitations of the local methodologies can be overcome by using
vibration-based methods, which give a global damage assessment.
Damage assessment in structures using vibration characteristics 4
A number of vibration-based methodologies have been found in the recent literature
to identify, locate and estimate the severity of damage in structures using numerical
simulations. The most common vibration-based damage detection techniques include
changes to mode shapes, modal curvatures, flexibility curvatures, strain energy
curvatures, modal strain energy, flexibility and stiffness matrices. The other
vibration-based techniques include numerical model updating and neural network
based methods. The amount of literature in non-destructive vibration methods is
quite large for treating single damage scenarios, however is limited for multiple
damage scenarios. Most existing methods are based on a single criterion and most
authors demonstrate these methods mainly in beam-like or plate-like elements. Also
existing methods, which depend only on changes in frequencies and mode shapes,
are limited in scope and may not be useful in several realistic situations. It is noted
that changes in natural frequencies alone may not provide enough information for
integrity monitoring (Farrar and Cone 1995). It is common to have more than one
damage case giving a similar frequency-change characteristic ensemble. In the case
of symmetric structures, the changes in natural frequency due to damage at two
symmetric locations are exactly the same. Alternatively, no changes in the mode
shapes can be detected if the mode has a node point at the location of damage. It is
noted that those methods utilizing mode shapes are the most developed in terms of
displaying the ability to identify, locate and estimate the severity of damage.
Therefore the modal flexibility and modal strain energy methods are chosen for
damage localization in this thesis as their corresponding algorithms can be applied to
beams, plates, trusses and their coupled structures. A multi-criteria based damage
identification procedure, incorporating these vibration parameters, is developed and
applied to chosen structures. As mentioned earlier, there is a need to assess the
performance of Queensland bridges, many of which have been built several decades
ago with some showing signs of distress. There is thus a need for a comprehensive
and reliable non-destructive method for damage assessment of these structures,
which is the aim of the proposed research.
Damage assessment in structures using vibration characteristics 5
1.3.3 Significance and innovation of this research
This research is significant as it will contribute towards the safe and efficient
operation of our bridge structures, which form an integral and important part of the
national infrastructure. The research findings can be applied to predict distress in a
bridge so that appropriate retrofitting can be carried out to prevent bridge failure.
The proposed multi-criteria approach is novel and will provide damage assessment
more accurate than hitherto possible. This research is also innovative as it will be
able to treat several damage scenarios, which include single and multi-damages in
bridge girders, decks and trusses.
1.4 Research method
Fast computers and sophisticated finite element techniques have enabled the
possibility of analysing hitherto intractable problems in structural engineering while
simplifying the analyses of other problems. This research study uses dynamic
computer simulation techniques to develop and apply a procedure using non-
destructive vibration based methods for damage identification in the chosen
structures - beam, slab (plate), slab-on-girder bridges and truss bridges. Limited
experimental testing is carried out to establish the hypothesis and validate the
computer model.
1.5 Thesis content
This thesis is organised into 6 chapters as follows:
Chapter 1 introduces the background of this thesis and depicts the research problem
in the area of structural health monitoring and damage detection in bridge structures.
It also defines the scope and gives the objectives and motivation of the research. The
methodology and research plan are also outlined.
Chapter 2 presents an overview of the structural health monitoring. It also reviews
the structural form of bridges, damage detection methods, structural identification,
Damage assessment in structures using vibration characteristics 6
modal testing, model updating, with emphasis placed on the non-destructive
vibration-based damage identification methods. Two case studies are given: (i) wind
and structural health monitoring system and (ii) damage identification of timber
bridges. Finally, a summary of the literature review findings related to the selection
of damage detection methodologies and their limitations is presented.
Chapter 3 discusses the basic structural dynamic theories including governing
equations for structural systems and damping. The (homogeneous) linear differential
equations relating the effects of the mass, stiffness and damping lead to the
determination of natural frequencies, mode shapes and damping ratios of the
idealized structural systems. For damage identification purpose, the modal flexibility
matrix and modal strain energy based damage index, which are based on these modal
parameters, are then derived for beams, plates and trusses. This chapter also
describes the static and free vibration testing of the slab-on-girder bridge model.
These were carried out to capture the physical behaviour as well as to validate the
computer models. The instrumentation, test methodology and experimental results in
each test are described and the results are compared with those from the computer
model.
Chapter 4 and the next chapter illustrate the application of the proposed method
through numerical examples. This chapter treats the main load baring elements in
slab-on-girder bridges, viz beams and slabs (plates), under different damage
scenarios. Results are presented and discussed.
Chapter 5 illustrates the applications in two types of bridges (i) slab-on-girder
bridge and (ii) truss bridge, under different damage scenarios. Results are discussed
and a flowchart for the proposed multi-criteria approach is also presented.
Chapter 6 presents the main identification findings of this research and its
contributions to scientific knowledge. Recommendations for further research are also
provided herein.
Damage assessment in structures using vibration characteristics 7
Chapter 2 Literature Review
2.1 Introduction
This chapter presents a literature review confining to the areas of structural health
monitoring, damage detection algorithms and modal testing. The review begins by
describing general structural forms of bridges to be investigated in the numerical
studies. Next, the basic components of the structural health monitoring system (SHM)
that appeared in the recent technical literature are defined, and the advantages of
implementing SHM are presented. A global non-destructive monitoring procedure
which consists of four levels of damage identification (existence, localization, extent
and prognostic) is discussed. Following this, definitions and essential concepts of
forward and inverse problems are introduced and compared with each other. After
this, classification for health monitoring and damage identification methods
according to required measured data and analysis techniques are presented. The
categories include global and local techniques and also model-based and non-model-
based approaches. Among all damage detection approaches, the review focuses on the
global vibration-based damage detection methods including changes in frequency
approach, modal flexibility approach and modal strain energy approach, while some
advanced damage detection techniques (e.g. wavelet analysis, artificial neural
network and acoustic emission monitoring) are also included.
In addition, this chapter also provides an overview of the vibration testing for modal
analysis. The modal testing, including modal identification, model validation and
updating, enables extraction and analysis of the dynamic properties for all the
required modes of the structure. Following this, constraints associated with
uncertainties, environmental and operational variations which limit the successful use
of dynamic testing and damage identification algorithms, are summarised. This
chapter concludes by illustrating two examples of the application of the structural
Damage assessment in structures using vibration characteristics 8
health monitoring system that have been implemented on bridges recently. Two
selected case studies include (i) a wind and structural health monitoring system
deployed in three long-span cable-supported bridges in Hong Kong and (ii) a damage
index method applied on timber bridges for damage assessment. Finally, the main
findings of the vibration-based damage detection methodologies and their limitations
are summarised. By considering the advantages and disadvantages of these damage
detection methods, a multi-criteria approach which incorporates (i) changes of
frequency, (ii) changes of modal flexibility and (iii) changes of modal strain energy
based damage index, is proposed. These damage detection methodologies are
selected as they are applicable to the structures treated within the scope of this thesis
and also they can be adapted to a wide range of other structures. The feasibility and
capability of this multi-criteria approach will be demonstrated through numerical
examples presented in Chapters 4 and 5.
2.2 Structural form of bridges
Two types of bridge structures are reviewed: (i) slab-on-girder bridge and (ii) truss
bridge. Through understanding the general bridge configurations and structural
behaviour of these bridges, detailed structural system of the bridges are modelled in
Chapters 4 and 5 for damage identification purpose.
2.2.1 General bridge configurations
A bridge deck is a system that has a floor resting on top of suitable carrying members,
such as beams or girders, so that overhead bracing is not required. The number of
beams, girders, box girders, trusses, is at least two and possibly three or more.
Classification according to the structural layout of the principal components, beam,
girder, and truss bridges can be the simple-span, multi-span continuous, or cantilever
type (Barker and Puckett 1997; Petros 1994). The multi-span simply supported
(MSSS) and multi-span continuous (MSC) bridges could be different in their lengths
and weights of typical spans, presence and size of gaps, and bearing configuration
and the primary difference between these two types of bridges is that the latter will
develop hogging moment at the supports. The simply supported bridge has alternating
Damage assessment in structures using vibration characteristics 9
fixed and expansion bearings supporting each deck, while the continuous bridge has
expansion bearing at each abutment and fixed bearings between the continuous deck
and pier. Multiple-span bridges are required where single-span design becomes too
long for an economical solution. Simple spans require less engineering effort; they do
not require field splices, making possible faster erection; differential support
settlements do not have to be considered in the beam design; and expansion devices
accommodate single-span movement only. On the other hand, continuous spans allow
a reduction of materials or longer spans and fewer piers for the same steel section;
they result in less deflection and live load vibration effects; and they enhance
improvements in appearance through variation in span length and beam depth. A
continuous bridge, whether concrete or steel, usually implies a beam system of a
variable second moment of area. For short spans, the continuous steel beam is a wide-
flange rolled section with welded cover plates in the region of maximum negative
moments or with heavier sections between field splices.
2.2.2 Slab-on-girder bridge
Slab-on-girder bridges are one of the common types of structural forms for bridge
superstructure. Typical structural material used for slab and girders are listed in Table
2.1 (Barker and Puckett 1997; Petros 1994). Slab-on-girder bridges are frequently
used for road and railway traffic. The girder span follows the direction of traffic and
is used as a primary load bearing structure. The slab is normally connected to the
girders, which increases the rigidity of the girders and provides a plane surface for
live traffic. Transverse components or diaphragms are provided to enhance the
transverse loading distribution. The spans of girder bridges seldom exceed 150m,
with a majority of them less than 50m. Girders are not as efficient as trusses in
resisting loads over long spans (Petros 1994). However, for short and medium spans
the difference in material weight is small and girder bridges are competitive. In
addition, the girder bridges have greater stiffness and are less subject to vibrations
which was important to the railroads. I-beam bridges utilize I rolled sections as the
main support system of the superstructure. An I-beam floor system consists of the
roadway and the supporting rolled beams. Floor systems are usually provided with a
concrete slab, with reinforcement perpendicular to traffic. Structural components like
Damage assessment in structures using vibration characteristics 10
I-sections are major in flexure behaviour that carry transverse loads perpendicular to
their longitudinal axis primarily in a combination of bending and shear. The
resistance of I-sections in flexure is largely dependent on the degree of stability
provided, either locally or in an overall manner. Axial loads are usually small in most
bridge girder applications and are often neglected. If axial loads are significant, then
the cross section would be considered as a beam column. If the transverse load is
eccentric to the shear centre of the cross section, then combined bending and torsion
would be considered.
Table 2.1 Common structural material used in slab-on-girder bridges
Slab material Girder material
Concrete Steel
Steel Steel
CIP concrete CIP concrete
CIP concrete Concrete
Concrete Concrete
Wood Wood
Note: CIP means concrete impregnated with polystyrene.
2.2.3 Truss bridge
A truss bridge consists of two main planar trusses tied together with cross girders and
lateral bracing to form a three-dimensional truss that can resist a general system of
loads. Some of the most commonly used trusses (e.g. Howe, Pratt and Warren truss)
suitable for both road and rail bridges are shown in Fig. 2.1. Members of the truss
girder bridges can be classified as chord members and web members. Generally, the
chord members resist overall bending moment in the form of direct tension and
compression and web members carry the shear force in the form of direct tension or
compression. Lateral bracing in truss bridges is provided for transmitting the
longitudinal live loads and lateral loads to the bearings and also to prevent the
compression chords from buckling. This is done by providing stringer bracing,
braking girders and chord lateral bracing. Concrete deck of highway truss bridges
also acts as lateral bracing support system. A truss bridge has two major structural
Damage assessment in structures using vibration characteristics 11
advantages: (i) the primary member forces are in axial action; (ii) the open web
system permits the use of greater overall depth than for an equivalent solid web
girder. Both these factors lead to economy in materials and a reduced dead weight.
The increased depth also leads to reduced deflections, and hence forms a rigid
structure. The relative light weight of a truss bridge has an erection advantage. It may
be assembled member by member using lifting equipment of small capacity.
Alternatively, the number of field connections maybe reduced by fabrication and
erecting the trusses bay by bay, rather than one member at a time. Two types of
connections for tension members are typically used: bolted and welded. In case of
truss bridges that are continuous over many supports, the depth of the truss is usually
larger at the supports and smaller at mid-span. Due to their efficiency, truss bridges
are built over wide range of spans. The conventional truss bridges are most likely to
be economical for medium spans in the range 150-500m (O’Connor 1971).
(a) Howe truss
(b) Pratt truss
(c) Warren truss
Fig. 2.1 Common truss types used in bridges.
2.3 Structural health monitoring
A number of definitions have been given in terms of structural health monitoring
(SHM). SHM is defined by Aktan et al. (2000) as tracking the responses of a structure
Damage assessment in structures using vibration characteristics 12
along with inputs, if possible, over a sufficiently long duration to determine
anomalies, to detect deterioration and to identify damage for decision making. For
more specific, measurement of the operating and loading environment and the critical
responses of a structure are able to evaluate the symptoms of operational incidents,
anomalies and/or deterioration or damage indicators that may affect operation,
serviceability, safety reliability.
Li et al. (2004) stated that a typical structural health monitoring system includes three
major components: a sensor system, a data processing system (including data
acquisition, transmission and storage) and a health evaluation system (including
diagnostic algorithms and information management). The first component, the sensor
system, is usually attached to older bridges or embedded into newer bridges. It
measures the structural response including stress, strain, displacement and
acceleration, and the environmental parameters of temperature and wind speed.
Examples of the sensors used to measure structural response are accelerometers,
strain meters and linear voltage displacement transducers (LVDT), while the
environmental variations can be measured with thermistor and anemometers. The
combination of sensors ensures that all of the variables can be measured and the
structural responses can be distinguished from the environmental variables. The
function of some types of sensors used in planning and implementation of the
structural health monitoring system for cable-supported bridges in Hong Kong are
listed in Table 2.2 (Wong et al. 2000). The data processing system which includes
wireless, global positioning system (GPS) or geographic information system (GIS)
based data acquisition, transmission methods and data archival and management
architectures are used to transfer raw data from these sensors to either the on-site lab
or off-site desktop computer. The final step is to interpret correctly the data from
various types of sensors to reach critical decisions regarding the load capacity and
system reliability, i.e. the health status of the structure. Prognostic and diagnostic
algorithms based on modal analysis, pattern recognition and time series analysis are
commonly used to detect the presence, location, magnitude, and the extent of
structural faults.
Damage assessment in structures using vibration characteristics 13
Table 2.2 Functions of sensors in SHM system
Sensors Functions of sensors
Anemometers
To measure wind speeds in 3 orthogonal directions
To derive wind incidences, wind turbulent intensities & wind
spectra
Accelerometers
(fixed)
To derive frequencies, mode shapes & modal damping
To derive bridges response spectra for comparison with
loading spectra such as wind, seismic & traffic
To monitor wind-induced vibrations on bridges
To monitor accelerations of bridge for user comfort
To monitor dynamic motion of decks and towers
Level sensing
stations
To monitor vertical deflection & rotation of decks
To derive static deck load-deformation relationship
To derive deck displacement influence lines
Displacement
transducers
To measure deck longitudinal & transverse motions
To verify the adequacy of predicted motions
To monitor any deck motions due to creep of concrete and/or
relaxation of steel
Strain gauges
To measure peak stresses and stress-cycles for fatigue life
estimation
To monitor the performance of bearings
To derive forces influence lines for load assessment
Temperature
sensors
To measure temperatures in steel sections, suspension cables,
concrete sections & asphalt pavement section
To measure ambient (air) temperature for references
To derive appropriate values of differential temperatures for
steel bridge design
Dynamic weigh-
in-motion sensors
To measure traffic flow and loads
To derive parameters for traffic load design & assessment
To derive stress levels for fatigue life estimation
Damage assessment in structures using vibration characteristics 14
Alternatively, Sohn et al. (2004) described SHM as a statistical pattern recognition
process to implement a damage detection strategy for civil and mechanical
engineering infrastructure and it is composed of four portions: (i) operational
evaluation, (ii) data acquisition, fusion and cleansing, (iii) feature extraction, and (iv)
statistical model development. In the first portion, operational evaluation deals with
the types of defined damage, conditions in the operational environment and also the
limitations on acquiring data. In the second portion, data acquisition deals with the
selection the types, number and places of the sensors and defining the other hardware,
data normalization. This portion also discusses about data fusion which relate to the
integration of different sets of data from different types of sensors and also cleansing
regarding choosing the data to accept or reject. In the third portion, feature extraction
deals with the identification of the metrics, which help to differentiate the damaged
and undamaged structure. This portion also discusses about the data compression. In
the last portion, statistical model development gives the information about the
damage state of the structure by analysing the identified features.
2.3.1 Advantage of structural health monitoring
There are several advantages of implementing structural health monitoring systems
on infrastructures compared to the traditional local damage detection methodologies.
Conventional non-destructive tests include penetrant, magnetic particle, eddy current,
ultrasonic and radiographic testing. They have several limitations when testing large
structures. First, they have limited depth of penetration. Second, their application
requires a prior localization of the damaged zones of the test structures. Lastly, there
is no way to easily determine the health of the structure at the boundaries and joints.
Advantages of SHM include obtaining data for structural identification, identifying
global and local structural characteristics, effective maintenance and operation
(Catbas et al. 1999). The continuous monitoring data and the findings can also be
used for improving future designs and diagnosing pre- and post-hazard conditions.
Sikorsky (1999) pointed out that a proficient structural health monitoring system is
capable of determining and evaluating the serviceability of the structure, the
reliability and the remaining functionality of the structure in terms of durability.
Damage assessment in structures using vibration characteristics 15
Ko and Ni (2005) presented that structural health monitoring systems are generally
envisaged to: (i) validate design assumptions and parameters with the potential
benefit of improving design specifications and guidelines for future similar structures;
(ii) detect anomalies in loading and response, and possible damage/deterioration at an
early stage to ensure structural and operational safety; (iii) provide real-time
information for safety assessment immediately after disasters and extreme events; (iv)
provide evidence and instruction for planning and prioritizing bridge inspection,
rehabilitation, maintenance and repair; (v) monitor repairs and reconstruction with the
view of evaluating the effectiveness of maintenance, retrofit and repair works; and
(vi) obtain massive amounts of in-situ data for leading edge research in bridge
engineering, such as wind-and earthquake-resistant design, new structural types and
smart material applications.
2.4 Structural damage in bridges
Damage is not meaningful without a comparison between two different states of the
system, one of which is assumed to represent the initial and often undamaged state
(Sohn et al. 2004). For damaged state, structural damage includes reduction of the
structural bearing capacity during their service period. This reduction is usually
caused by degradation and deterioration of structural components and connections.
Alternatively, Li et al. (2007) presented structural damage as weakening of the
structure that negatively affects its performance and can be defined as any deviation
in structure’s original geometric or material properties that may cause undesirable
stresses, displacements or vibration on the structure. These weakening and deviation
may be due to cracks, loose bolts, broken welds, corrosion, fatigue, aging etc. which
may be caused by rapid changes due to impact loads, strong earthquakes, hurricanes,
and blast. In all cases damage can severely affect safety and serviceability of the
structure (Roy et al. 2006).
The physical local damages in reinforced concrete (RC) structures include
microcracking and crushing of concrete, yielding of the reinforcement bars and bond
deterioration at the steel concrete interfaces (Coronelli and Gambarova 2004). The
overall effect of local damage at various locations is the stiffness and strength
Damage assessment in structures using vibration characteristics 16
deterioration of the whole structure. RC structures can be modelled by non-linear
mechanical theories, while local damage at a cross-section of the structure can
adequately be measured by the degradation of bending stiffness and moment
capacity of the cross-section. A global damage indicator can be defined as a
functional of such continuously distributed local damage which characterizes the
overall damage state and serviceability of the structure. The simulated damage used
in this research has been adopted from established work in the literature, and will be
limited to changes to the material and/or geometric properties of structural
components. Such a damage is expected to affect the performance of the entire
structure.
2.5 Forward and inverse problem
Forward problem is defined as modelling and analysing a nonexisting structure for
new design. The differential equations of models, mass M, damping D, stiffness K of
the system along with the appropriate initial conditions and forcing functions are
assumed to be known, and the theory developed consists of calculating and
characterizing the response of the system to known inputs (Maia and Silva 1997). The
forward problem which usually falls into the category of Level 1 damage
identification consists of pattern recognition. The model is used to generate a
database of feature vectors that correspond to many damage scenarios. The database
serves to train a classifier or a regressor. They can output a damage diagnosis in a
discrete or continuous form, respectively, from the measured features during the
monitoring phase after training. Inverse problem is defined as modelling a
constructed structure based on experimental measurement, visual inspections, and
other information. While forward problems have a unique solution for linear systems,
inverse problems do not. The inverse problem, on the other hand, determines the
matrices M, D and K from knowledge of the measurements of the responses
(position, velocity, or acceleration). Model updating is used to update and match the
analytically derived values of M, D and K with measured modal data. The modal
testing problem which determines the dynamic characteristics of structures is a
subclass of inverse problems. It is carried out to recover natural frequencies, mode
shapes, and damping ratios from response measurements. Level 2 or level 3 damage
Damage assessment in structures using vibration characteristics 17
identification methods, which consist of calculated damage parameters, e.g. crack
length and location, are deduced from the changes in the mechanical properties of the
elements of the model.
2.6 Classification of damage detection techniques
Saadat et al. (2007) presented that health monitoring and damage detection techniques
can be classified according to either their detection capability (global techniques
merely infer the existence of damage, while local techniques assist in locating it) or
based on the extent of prior knowledge required (model-based techniques utilized
explicit mathematical descriptions of the system dynamics, while non-model-based or
feature techniques based rely on signal processing of measured responses). Global
methods attempt to simultaneously assess the condition of the whole structure
whereas local methods focus non-destructive evaluation tools on specific structural
components. Global monitoring techniques can be used either intermittently or
continuously to gauge the health of the structure, and may be used to guide the
assessment of suspect areas and thus to obtain efficient use of the inspection time.
Local non-destructive evaluation methods may be used to detect, locate and
characterize defects more precisely. All such available information may be combined
and analysed by experts to assess the structural state of the structure. Model-based
approaches typically rely on parametric system identification using linear, time
variant methods. The model based methods are basically a model updating procedure,
in which the physical parameters of the mathematical model are calibrated or updated
using vibration measurements from the structural response. A fundamental difficulty,
however, lies in the fact that the physical parameters obtained from the automatic
updating procedure may be unrelated to the actual damage scenarios, though they can
be consistent with the measured modal data. Non-model-based or feature based
approaches include modal analysis, dynamic flexibility measurements, matrix update
methods and wavelet transform techniques. These methods typically seek to identify
damage from changes in structural vibration characteristics. These approaches detect
structural changes by using some damage features, without the need of a detailed
model of the structure. Features for damage identification have used to be based on
natural frequencies, mode shapes, mode shape derivatives, stiffness matrix and
Damage assessment in structures using vibration characteristics 18
flexibility matrix etc. These features are extracted from measured responses and with
certain degree of sensitivity to structural changes.
Lee et al. (2004) summarised the features in damage detection algorithms utilizing
vibration properties as listed in Table 2.3.
Table 2.3 Summary of damage detection categories and methods
Category Methodology
Modal parameters
Natural frequencies Frequency changes
Residual force optimization
Mode shapes
Mode shape changes
Modal strain energy
Mode shape derivatives
Matrix methods Stiffness-based
Optimization techniques
Model updating
Flexibility-based Dynamically measured flexibility
Machine learning
Genetic algorithm Stiffness parameter optimization
Minimization of the objective function
Artificial neural
network
Back propagation network training
Time delay neural network
Neural network systems identification with
neural network damage detection
Other techniques
Time history analysis
Evaluation of frequency response functions
(FRF)
Yan et al. (2007) considered that the development of vibration-based structural
damage detection can be divided into traditional and modern types. The traditional-
type refers to detection method for structural damage only utilising dynamic
characteristics of structures, such as natural frequency, modal damping, modal strain
energy or modal shapes, etc. These methods generally require experimental modal
analysis or transfer function measurement. The modern-type vibration-based
structural damage detection, also called intelligent damage diagnosis, is a type of
method which uses online measured structural vibration responses to detect damage.
Damage assessment in structures using vibration characteristics 19
These methods mainly take modern signal-processing technique and artificial
intelligence as analysis tools. The measured structural dynamic responses may
indicate the change of structural dynamic parameters at the structural damaged status.
Among the modern-type methods for structural damage detection, the representative
ones include wavelet analysis, genetic algorithm (GA) and artificial neural network
(ANN) etc. These methods are often implemented based on a few measured data and
a large number of simulation data from structural vibration responses. The vibration-
based damage detection methods, multi-criteria methods and advanced damage
detection methods are briefly described as follows.
2.6.1 Vibration-based damage detection methods
2.6.1.1 Frequency change
Koh and Dyke (2007) utilized the concept of linear correlation to detect the location
of damage in simple structural systems. In linear correlation, the angle between the
two parameter vectors is calculated to estimate correlation value. The parameter
vectors used for evaluating correlation coefficients consist of the ratios of the first n
modal frequency changes due to damage to the undamaged modal frequencies.
Correlation value is expressed as follows:
j
jT
jCδωωδωω
∆∆
= (2.1)
where ( )
h
dh
ωωωω −=∆ (2.2)
Here, hω and dω denote the natural frequency vectors of the healthy and damaged
structure, respectively. Likewise the corresponding hypothesis vector, predicted from
an analytic modal denotes jδω . The subscript j indicates the hypothesized location of
damage (j=1,2,…r). The level of correlation between the measured and predicted
(hypothesis) modal frequencies is used to provide a simple statistical tool for locating
damage.
Damage assessment in structures using vibration characteristics 20
Messina et al. (1996) proposed a similar correlation concept based on the modal
assurance criterion (MAC) to develop the damage location assurance criterion
(DLAC). The DLAC method measures the correlation of a vector of experimental
natural frequency change ratios instead of mode shapes.
))((
2
jTjj
T
jT
jDLACδωδωωω
δωω∆∆∆
= (2.3)
Similar to Eq. (2.2), DLAC compares two frequency change vectors, one based on
measurements obtained from the test structure, the other from the jth hypothesis of an
analytical model of the structure. Eqs. (2.2) and (2.3) can only be used to detect single
damage occurrences. Methods using eigenfrequencies have a number of limitations.
They cannot distinguish damage at symmetrical locations in a symmetric structure.
The effectiveness of a correlation-based technique to locate damage depends on
sufficient and accurate set of the system’s modal parameters in numerical and
experiment model. Obtaining an accurate set of analytical parameters which
correspond well to the real structure as-built is not always available in practice.
Meanwhile, it is common that only a limited number of natural frequencies can be
realized experimentally which introduces errors into identification and uniquely
localization of damage. In this proposed research, the multi-criteria approach will be
demonstrated to show its feasibility and capability to treat several damage scenarios,
which include single and multi-damages in bridge girders, deck and trusses.
2.6.1.2 Modal flexibility
The modal flexibility matrix includes the influence of both the mode shapes and the
natural frequencies. It is defined as the accumulation of the contributions from all
available mode shapes and corresponding natural frequencies (Huth et al. 2005). The
modal flexibility matrix associated with the referenced degrees of freedom can be
established as follows:
TF ]][/1][[][ 2 φωφ= (2.4)
Damage assessment in structures using vibration characteristics 21
where ][F is the modal flexibility matrix; ][φ is the mass normalized modal vectors;
and ]/1[ 2ω is a diagonal matrix containing the reciprocal of the square of (circular)
natural frequencies in ascending order. The modal contribution to the flexibility
matrix decreases as the frequency increases, i.e., the flexibility matrix converges
rapidly with increasing values of frequency. From only a few of the lower frequency
modes, therefore, a good estimate of the flexibility can be made. The modal
flexibility change MFC or the change in flexibility matrix ][F∆ due to structural
deterioration is given by
MFC = ][][][ hd FFF −=∆ (2.5)
where index ‘h ’ and ‘d ’refer to the healthy and damaged state respectively.
Theoretically, structural deterioration reduces stiffness and increases flexibility.
Increase in structural flexibility can therefore serve as a good indicator of the degree
of structural deterioration.
Pandey and Biswas (1994) presented the flexibility matrix for detecting the presence
and location of structural damage. All predictions of the state of damage were made
from the full experimental data from modal testing. The authors treated a simply
supported beam, a cantilever beam and a free-free beam to gain an insight into how
the flexibility matrix is affected by the presence of damage. It was shown that the
flexibility change pattern is different for different support conditions.
Patjawit and Kanok-Nukulchai (2005) introduced a global flexibility index (GFI) to
identify global health deteriorations of highway bridges. The index is the spectral
norm of the modal flexibility matrix obtained in association with selected sensitive
reference points to the deformation of the bridge structure. The modal flexibility
matrix is evaluated from the dynamic responses at these reference points under forced
vibration. Aging of a bridge over a period of time will be reflected by the gradual
increase of GFI. The change in the GFI has been shown to be sufficiently sensitive to
the global weakening of the structure and its increase in magnitude is a good
indication for structural deterioration.
Damage assessment in structures using vibration characteristics 22
2.6.1.3 Modal strain energy
Cornwell et al. (1999) applied the strain energy damage detection method to plate-
like structures. The method only requires the mode shapes of the structure before and
after damage and the modes do not need to be mass normalized making it very
advantageous when using ambient excitation. The algorithm was found to be effective
in locating areas with stiffness reductions as low as 10% using relatively few modes.
The algorithm was also demonstrated successfully using experimental data.
Hu et al. (2006) applied strain energy method and modal analysis to the damage
detection of a surface crack in composite laminated plates. Both experimental modal
analysis (EMA) and finite element analysis (FEA) were performed to obtain the
mode shapes of the laminated plates. The mode shapes were then used to calculate
strain energy using differential quadrature method (DQM). The authors indicated
that only a few grid points in the test plate are required for DQM to provide an
accurate and rapid approach to obtain strain energy. Consequently, a damage index
was established to locate the surface crack using the fractional strain energy of
laminated plates before and after damage. Experimental results showed that surface
crack locations in various composite laminates were successfully identified by the
damage indices.
Li et al. (2007) evaluated the performance of the modal strain energy based damage
identification algorithm for detecting damage in a timber structure. It was shown that
the method was capable of detecting single damage in timber but will experience
some problems when faced with multiple damage detection. A modified algorithm
was proposed by the authors to overcome the problems associated with reliable
detection of multiple damages in terms of damage location and severity.
Alvandi and Cremona (2006) studied the performance of both flexibility method and
strain energy method on a simply supported beam. Measured modal parameters which
use only few mode shapes and modal frequencies of the structure obtained by random
force excitation were used. The authors assessed the performance of these techniques
by introducing different noise levels to the response signals of a simulated beam
which was excited by a random force. They concluded that both methods are capable
Damage assessment in structures using vibration characteristics 23
of detecting and localising damaged elements but in the case of complex and
simultaneous damages, the flexibility method is less efficient. Moreover, the strain
energy method demonstrates stability in the presence of noisy signals.
2.6.1.4 Flexibility curvature
Lu et al. (2002) proposed a flexibility curvature method to locate multiple damages in
continuous structure. The flexibility curvature vector will possess a smooth curve
shape in undamaged cases. Therefore the local peaks on the curve can be used to
indicate abnormal flexibility/stiffness changes at that position, i.e., peaks normally
means damage has occurred at the corresponding positions.
Flexibility curvature can be approximated as follows:
21,1,1,1)( )2(
l
FFFF iiiiiic
i ∆+−
= ++−− 1,...,2 −= nni (2.6)
where iiF , and )(ciF are the i-th diagonal element of the flexibility matrix and the i-th
item of the flexibility curvature vector, respectively, and l∆ is the length of the
elements, nn is the number of nodes
2.6.1.5 Mode shape curvature
Pandey et al. (1991) assumed that structural damage only affects the structure’s
stiffness matrix and its mass distribution. The pre and post-damage mode shapes are
first extracted from an experimental analysis. Curvature of the mode shapes for the
beam in its undamaged and damaged conditions are estimated numerically from the
displacement mode shapes with a central difference approximation or other means of
differentiation. Given the before- and after-damage mode shapes, the beam cross-
section at location x is subjected to a bending moment M(x).
Damage assessment in structures using vibration characteristics 24
The curvature at location x along the length of the beam is
)/()()( EIxMxv =′′ (2.7)
where M(x) is the bending moment at a section, E is the modulus of elasticity, and I is
the moment of inertia of the section. From Eq. (2.7), it is evident that the curvature is
inversely proportional to the flexural stiffness (EI). Thus, a reduction of stiffness
associated with damage will, in turn, lead to an increase in curvature. Differences in
the pre and post-damage curvature mode shapes will, in theory, be large in the
damaged region. For multiple modes, the absolute values of change in curvature
associated with each mode can be summed up in a damage parameter for a particular
location.
Abdel Wahab and De Roeck (1999) presented mode shape curvature method for
damage detection in bridges. A central difference approximation which used to derive
the curvature mode shapes from the displacement mode shapes is obtained by the
following equation.
211'' )2(
h
vvv iii −+ +−=ν (2.8)
where h is the distance between two successive measured locations and v is the
displacement mode shape. A damage indicator called “curvature damage factor” is
introduced as
∑=
−=N
ndioi vv
NCDF
1
''''1 (2.9)
where N is the total number of modes to be considered, ''oν is the curvature mode
shape of the intact structure and ''dν is that of the damaged structure. The difference
in curvature mode shape for all modes can be summarized in one number for each
measured point. By plotting the difference in modal curvature (MC) between the
intact and the damaged case, a peak appears at the damaged element indicating the
Damage assessment in structures using vibration characteristics 25
presence of a fault. Therefore modal curvatures are sensitive to damage and can be
used to localize it.
2.6.1.6 Uniform load surface curvature
Farrar and Doebling (1999) presented the change in curvature of the uniform load
surface to determine the location of damage. The coefficients of the i-th column of
the flexibility matrix represent the deflected shape assumed by the structure with a
unit load applied at the i-th degree of freedom. The sum of all columns of the
flexibility matrix represents the deformed shape assumed by the structure if a unit
load is applied at each degree of freedom, and this shape is referred to as the uniform
load surface. In terms of the curvature of the uniform load surface, the curvature
change at location is evaluated as follows:
"*""iii FFF −=∆ (2.10)
where *"iF and "
iF are the damaged and undamaged curvature of the uniform load
surface at i-th degree of freedom respectively, "iF∆ represents the absolute curvature
changes. The curvature of the uniform load surface can be obtained with a central
difference operator.
2.6.1.7 Stiffness change
Since damage reduces the stiffness and increases the flexibility of structures,
Zimmerman and Kaouk (1994) have developed a damage detection method based on
changes in the stiffness matrix that is derived from measured model data. The
eignenvalue problem of an undamaged, undamped structure is
}0{}]){[][( =+ ii KM φλ (2.11)
where iλ is the square of the i-th modal frequency, iφ is the i-th unit-mass normalized
mode.
Damage assessment in structures using vibration characteristics 26
The eigenvalue problem of the damaged structure is formulated by first replacing the
pre-damaged eigenvectors and eigenvalues with a set of post-damaged modal
parameters and second, subtracting the perturbations in the mass and stiffness
matrices caused by damage from the original matrices. Letting dM∆ and
dK∆ represent the perturbations to the original mass and stiffness matrices, the
eigenvalue equation becomes
}0{}]){[][( ** =∆−+∆− iddi KKMM φλ (2.12)
where the asterisk * signifies properties of the damaged structure.
Two forms of a damage vector }{iD , for the i-th mode are then obtained by separating
the terms containing the original matrices from those containing the perturbation
matrices. Hence,
}]){[][(}]){[][(}{ ****iddiiii KMKMD φλφλ ∆+∆=+= (2.13)
To simplify the investigation, damage is considered to alter only the stiffness of the
structure (i.e. ][ dM∆ = [0]), therefore the damage vector reduces to
}]{[}{ *idi KD φ∆= (2.14)
To obtain the i-th damage vector }{ iD , Eq. (2.15) is subtracted from Eq. (2.16) to
obtain ][ dK∆ as shown in Eq. (2.17), and this matrix is multiplied by the ith damaged
mode shape vector }{ *iφ .
Tii
n
iiK **
1
2** ][ φφω∑=
≈ (2.15)
Tii
n
iiK φφω∑
=
≈1
2][ (2.16)
][][][ *KKKd −=∆ (2.17)
Damage assessment in structures using vibration characteristics 27
where iω is the i-th modal frequency, iφ is the i-th unit-mass normalized mode, n is
the number of measured modes.
2.6.2 Multiple criteria methods
2.6.2.1 Flexibility and stiffness
Yan and Golinval (2005) adopted a combined analysis on the measured flexibility
and stiffness constructed from modal parameters for damage localization in a
cantilever beam and a simulated three-span bridge. Damage localization was realized
by observing the difference in the diagonal entries of measured matrices between the
reference state and the damaged state. Since the flexibility matrix was easy to
construct, and diagonal changes of the stiffness matrix were directly related to
damage locations, a combined consideration of the two matrices provided more
reliable information on damage location. It was found that the approach required a
sufficient number of well distributed sensors and that if damage was too small, it
could be masked by numerical errors. Overall, the results showed that a combined
consideration of the two matrices provided more reliable information on damage
location.
2.6.2.2 Flexibility and strain energy
Ndambi et al. (2002) applied the flexibility method and strain energy method to
reinforced concrete (RC) beam to examine their capability for detecting and
identifying location of damage. RC beams of 6 m length were subjected to
progressing cracking introduced in different steps. The damaged sections were
located in symmetrical or asymmetrical positions along the beam. Modal analysis
was carried out to observe the changes in dynamic characteristics. It appeared from
the experimental results that eigenfrequencies decreased with the crack damage
accumulation, thus the damage severity could be followed using this approach.
Alternatively, eigenfrequencies were not influenced by the crack damage locations.
They also found that the strain energy method appears to be more precise than the
flexibility method. The change in flexibility matrices had difficulties in detecting the
Damage assessment in structures using vibration characteristics 28
crack (damage) in RC beams due to the fact that the cracks spread over a certain
distance on both sides of the loaded section. In this case, it became very difficult to
identify the damaged zones.
2.6.2.3 Flexibility difference and modal curvature difference
Ko et al. (2002) proposed a multi-stage scheme for detection of the occurrence,
location and extent of the structural damage for the cable-stayed Kap Shui Mun
Bridge by using measured modal data from an on-line instrumentation system. They
performed the damage-identification simulation based on a precise three-dimensional
finite element model of the bridge. In the first stage, a novel detection technique
based on auto-associative neural networks was proposed for damage alarming. In the
second stage, the modal curvature index and the modal flexibility index were used
for localization of damaged deck segment or section. In the third stage, a multi-layer
perceptron neural network with back-propagation training algorithm was used to
identify specific damaged members within the segment and the damage extent. They
showed that the multi-stage scheme provided reliable damage monitoring of the Kap
Shui Mun Bridge in operation. The method needed only a series of measured
frequencies of the structure in intact and damaged states, and was inherently tolerant
of measurement error and uncertainties in ambient conditions. For damage occurring
at the bearing and supporting systems, the modal flexibility index was better than the
modal curvature index in locating the damage. When damage occurred at the deck
sections close to the bridge towers, the modal curvature index presented more correct
damage indication than the modal flexibility index. Therefore, the modal flexibility
index in conjunction with the modal curvature index was used for detection of the
damaged segment.
2.6.2.4 Energy difference and energy curvature difference
Xu and Wu (2007) proposed an energy index based on acceleration response and
power spectral density (PSD) function, for detecting the two damaged elements in
the girder of a long-span cable-stayed bridge. Numerical analysis was performed by
using the proposed strategy and the traditional energy difference and energy
curvature strategy. They showed that the proposed strategy had a good damage
Damage assessment in structures using vibration characteristics 29
quantification ability and anti-noise pollution ability. In addition the strategy was
able to estimate slight, moderate and severe damage which corresponds to 10%, 30%
and 70% reduction of stiffness in the two elements in the girder. It was found that
there were some difficulties in carrying out mode shape curvature strategy, due to
perturbation of measured frequencies and mode shapes, a limited number and
omission of measured mode shapes, and deviation of the mode shapes’ solution from
the measured data, especially for long-span cable-stayed bridges. As the change
ratios of the energy curvature difference of the damaged bridge to the initial energy
of the undamaged bridge were higher than the change ratios of the energy difference
of the damaged bridge in all damage scenarios, they concluded that the energy
curvature difference was more competent for damage detection under three damage
scenarios.
2.6.3 Advanced damage detection methods
2.6.3.1 Wavelet analysis
Wavelet analysis is a theory based on the idea that any signal can be broken down
into a series of local basis functions called “wavelets” (Bajaba and Alnefaie 2005).
The wavelet analysis has been applied to the space domain of a structure rather than
a time series. This method uses wavelet transforms to detect damages by sensing
local perturbations at damage sites. Damages are obtained from the continuous
wavelet, transform coefficients. The magnitudes of the wavelet coefficients at the
damage location show the damage severity. The wavelet transform is a recent
solution to overcome the shortcomings of the Fourier transform. Wavelet analysis
represents a windowing technique with variable-sized regions. It allows the use of
longer time intervals where precise low-frequency information is needed and
shortened time intervals where high-frequency information is required. This is a
distinct advantage over the short time Fourier analysis where the precision of the
time and frequency information is limited by the window size used.
Damage assessment in structures using vibration characteristics 30
2.6.3.2 Artificial neural network
An artificial neural network (ANN) is a software simulation or a hardware
implementation of a structure derived from studying the physiology of groups of
nerve cells or neurons (Pandey and Barai 1995). Based on the understanding of
neurons, a computational model is developed. Structural damage detection based on
ANN is usually constructed by three layers: an input layer, a hidden layer and an
output layer. The ANN includes the following steps: (i) determining the network
structure; (ii) selecting the network parameters; (iii) normalising the learning
samples; (iv) giving initial weight value and (v) detecting structural damage. In order
to train the constructed ANN, the known feature information (ANN input) and the
corresponding status (ANN output) of structural damage were taken as train samples.
These damage information as train sample can be obtained by experiments or
numerical simulations for a structure to be detected. When the ANN has been well
trained, the experimentally measured real structural damage feature index can be
input into the trained ANN, and the output of the trained ANN will be able to give the
location and severity of the structural damage.
ANN is one of the effective computation tools in pattern recognition and
classification, data interpretation, function approximation for structural damage
detection. ANN can exhibit considerable tolerance of noisy, partially incomplete and
partially faulty data, which are particularly useful for damage detection application of
large civil engineering structures where the in-situ measured data are expected to be
incomplete and noise-corrupted. The advantages of using ANN also include the self-
organisation and learning capabilities which eliminate the need for explicitly
extracting the cause and effect relationship between the system responses and the
damage patterns. By interchanging the input and output roles during the network
training, a functional mapping for the inverse relation is directly established which
can be used for diagnostic purpose. For general neural network based damage
detection approaches, the damage location and severity are simultaneously identified
with a one-stage scheme. In order to accurately identify damage extent, the one-stage
scheme requires the network to be trained with different damage levels at each
possible damage location. Therefore the range of training samples is expected to
cover the largest damage magnitude which may occur. When dealing with large-scale
Damage assessment in structures using vibration characteristics 31
structures with a lot of possible damage locations, the network requires a tremendous
amount of sampling data and exceedingly long learning process. This may
significantly jeopardize the training efficiency and accuracy of the neural network (Ni
et al. 2002).
2.6.3.3 Acoustic emission monitoring
Acoustic emission (AE) testing is used as a type of nondestructive testing technology
for local damage detection (Ohtsu 1987). AE is based on the principle that ultrasonic
acoustic signals are emitted as materials are stressed. Imperfections such as the
initiation and growth of fatigue cracks, failure of bonds, area of corrosion, and
loosening of bolts, emit mechanical waves as the structure is stressed. As a result,
different frequencies are produced for different size and type of defects. Acoustic
emissions can be monitored and detected by transducers with a typical frequency
between 10 Hz and 2MHz. Transducers are attached to the testing material to detect
these waves. These AE burst can be used both to locate flaws and to evaluate their
rate of growth as a function of the applied stress (Chang and Liu 2003). This
approach is a cost effective and sensitive technique for detecting and locating
potential problem areas. This process can be followed by other non-destructive test
techniques to quantify problems in the areas identified by acoustic emission testing.
AE methods have the advantage that they detect and locate all of the activated flaws
in one test. However, a structure can have acoustic signals other than flaws including
friction, cavitation and impact which complicate AE tests. Non-linear ultrasonic
testing can also be used for detecting early damage in materials. For example, Shah et
al. (2009) carried out such nondestructive evaluation of cubic concrete specimens.
Based on the literature review on vibration-based damage detection techniques, it is
observed that each of the existing damage localisation algorithms, by itself, is not
effective in locating all multiple damages and evaluating the severity of damages. It is
possible to develop a multi-criteria approach to localize multiple damages in the
proposed structural systems and cross check the results. The present research will
utilise this multi-criteria approach, which incorporates (i) changes of frequency, (ii)
changes of modal flexibility and (iii) modal strain energy based damage index to
Damage assessment in structures using vibration characteristics 32
detect and locate damage in the main load bearing elements of structures and
complete bridges.
2.7 Structural identification
Structural identification applications for large constructed systems generally require
the integration of: (i) structural conceptualization; (ii) analytical (geometric-FE)
modelling; (iii) designing and executing various experiments; (iv) data processing
and identifying modal and other characteristics; (v) model calibration and validation;
(vi) simulation and interpretation and (vii) decisions and heuristics (Aktan et al.
1997). Structural identification can be done both under static and dynamic conditions.
When a structure undergoes various degrees of damage, certain characteristics have
been found to undergo changes. In order to identify those changes, a sequence of tests
may be conducted and the resulting data such as load, displacements, strains,
acceleration, etc. can be measured. From such data, mechanical properties, such as
stiffness/strength, and dynamic characteristics, such as natural frequency and
damping can be estimated. Modelling a structure or its components can be based on
either continuum or discrete approaches. A continuum representation is useful for a
study of complex phenomena such as wave-propagation, heat transfer, etc. In the
discrete approach, a structure maybe characterized within the numerical, modal or
geometric spaces. The numerical space models the structure in terms of its mass,
damping and stiffness with appropriate assumptions on the size, form, and coupling
of the state matrices. The geometric space denotes microscopic finite element,
element-level, macro, or mixed models. Linearized and deterministic geometric
models are generally used for condition assessment of constructed facilities. The
parameters that need to be investigated when using for damage detection includes the
effect of boundary conditions, the frequency range of the excitation, the effect of
damage magnitude and location of detection (Maia and Silva 1997).
2.7.1 Modal identification
In order to obtain a dynamically realistic numerical model of a structure, it is
necessary to: (i) identify the dynamic characteristics of the real structure (e.g. natural
Damage assessment in structures using vibration characteristics 33
frequencies, mode shapes and damping ratios) and (ii) develop a numerical model
that can emulate that dynamic behaviour. The procedure is called modal identification
of system identification (Maia and Silva 1997). The dynamic properties of a system
with N degree of freedoms (DOFs) maybe described by three different types of
complete models: the spatial model, the modal model and the response model. In the
first case, the system dynamic characteristics are contained in the spatial distribution
of its mass, stiffness and damping properties, described by the NxN mass, stiffness,
and damping matrices, respectively. The spatial model given by mass [M], stiffness
[K] and damping [C] matrices leads to an eigenproblem which, have been solved,
yields the modal model constituted by the modal properties (N natural frequencies, N
modal damping values and N mode shape vectors) contained in matrices [ω2], [C] and
[Ф]. Furthermore, the modal model yields the response model (e.g. frequency
response function FRF). In the forward problem, it is assumed that the systems are
described by complete models, i.e., that all their mass, stiffness and damping
properties are known, or that all the eigenvalues and all the elements in the
eigenvectors are known, or that all the elements in the FRF matrix are known. If the
system is too complex and therefore cannot be modelled analytically, experimental
analysis will be carried out where the starting point is the measurement of the system
FRF. There were many techniques that allow derivation of modal characteristics of a
given system from the experimentally obtained response model. The fundamentals of
modal testing are briefly described as follows.
2.7.2 Fundamentals of modal testing
Vibration testing for experimental modal analysis is commonly known as modal
testing. Modal analysis is an important tool in the analysis, diagnosis, design, and
control of vibration. Modal testing includes instrumentation, signal processing,
parameter estimation, vibration analysis (De Silva 2007). An experimental vibration
system generally consists of three main measurement mechanisms: (i) the excitation
mechanism; (ii) the sensing mechanism; and (iii) the data acquisition and processing
mechanism.
Damage assessment in structures using vibration characteristics 34
The excitation mechanism is constituted by a system which provides the input motion
to the structure under analysis, generally under the form of a driving force applied at a
given coordinate. A popular excitation device is the impulse or impact hammer,
which consists of a hammer with a force transducer attached to its head. This device
does not need a signal generator and a power amplifier. The hammer, by itself, is the
excitation mechanism and is used to impact the structure and thus excite a broad
range of frequencies.
The sensing mechanism is basically constituted by sensing devices known as
transducers. There is a large variety of such devices and the most commonly used in
experimental modal analysis are the piezoelectric transducers either for measuring
force excitation (force transducers) or for measuring acceleration response
(accelerometers). The transducers generate electric signals that are proportional to the
physical parameter of measurement target. Most of the time, the electric signals
generated by the transducers are not amenable to direct measurement and processing.
This problem, usually related to the signals being very weak and to electric
impedance mismatch, is solved by the conditioning amplifiers. These conditioning
amplifiers which may be charge amplifiers or voltage amplifiers, match and often
amplify signal in terms of both magnitude and phase over the frequency range of
interest.
The data acquisition and processing mechanism measure the signals developed by the
sensing mechanism and ascertain the magnitudes and phases of the excitation forces
and responses. Analysers are used to extract and derive the modal characteristics (e.g.
natural frequencies, damping ratios and mode shapes) of structures. The most
common analysers are based on the Fast Fourier Transform (FFT) algorithm and
provide direct measurement of the FRFs. They are known as spectrum analysers or
FFT analysers. There are two main subsets of analysis procedures developed, time
domain and frequency domain methods. Time domain methods produce modal
characteristics directly from the response records in the time domain. Frequency
domain methods accomplish the same tasks by converting the response signals into
the frequency domain.
Damage assessment in structures using vibration characteristics 35
The Hilbert–Huang transform (HHT) is a method to decompose a signal into so-
called intrinsic mode functions (IMF), and obtain instantaneous frequency data. It is
designed to work well for data that are nonstationary and nonlinear (Quek et al.
2003). In contrast to other common transforms like the Fourier transform, the HHT is
more like an algorithm (an empirical approach) that can be applied to a data set,
rather than a theoretical tool. Almost all the case studies reveal that the HHT gives
results much sharper than any of the traditional analysis methods in time-frequency-
energy representation. Additionally, it reveals true physical meanings in many of the
data examined. The HHT will not be adopted in this research as it is basically for non
linear vibration.
2.7.3 Types of excitation
Modal identification can be performed based on different types of test, including: (i)
forced vibration; (ii) ambient vibration test and (iii) free vibration (Ren and Zong
2003). These three types of excitation are briefly discussed as follows:
2.7.3.1 Forced vibration excitation
The methods commonly used for forced vibration excitation include shakers and
impact hammers. Linear variable mass shakers can be used for both vertical and
horizontal excitation and can be used for various types of excitation. They can
generate and maintain a steady state sinusoidal forcing, or other wave forms which
may include combinations of steady state or transient waves. Impact hammers are
used in impact testing utilizes to excite the structure. The weight of the impact
hammers can be adjusted to produce different force levels to the structure. These
hammers can be hand held, suspended by chains, or dropped. The advantage of using
impact hammers are that they are fast to use and the test can be repeated numerous
times. There are several advantages of using forced vibration for structural vibration
monitoring. It can be designed the type, location, amplitude, frequency content,
duration of the forcing and also the time of the day that the forcing is applied
(Iskhakov and Ribakov 2005). By utilizing a known forcing function, many of the
uncertainties in the data collection and processing can be avoided. Although forced
Damage assessment in structures using vibration characteristics 36
excitations such as heavy shakers and drop weights and correlated input-output
measurements are available in some cases, testing method, structural complexity,
and/or achievable data quantity restrict these approaches to implement in practical
applications. The input or excitation level of the real large structure in its operational
condition is not easy to be quantified, which make controlled force excitation largely
impractical to use. Also, the use of known excitation methods requires the temporary
closing of the structure while tests are performed, significantly increasing the cost
related with each test (Zárate and Caicedo 2008).
2.7.3.2 Ambient vibration excitation
Structural vibration monitoring is often carried out utilizing ambient vibrations for the
excitation of the structure. A variety of ambient excitation sources include wind,
seismic activity, traffic, waves or tidal fluctuations etc. The advantage of using
ambient excitation includes low cost, little to no disruption to traffic, long term
excitation, and in some cases the frequency content is appropriate for the structure
(Hsieh et al. 2006). However, the disadvantage of using ambient excitation includes
the variability in amplitude, duration, direction, frequency content, and difficulty in
accurately measuring excitation. Once the excitation source is collected, the data
analysis is critical as noise in the data makes determination of the vibration
characteristics more difficult. As a result, reliance must be placed on the measurement
of vibrations induced by ambient excitation sources. This involves using measured
excitation data with certain assumptions including stationary, white, and
unidirectional or pre-determined multidirectional.
2.7.3.3 Free vibration
Free vibration occurs in a flexible system when a body moves away from its original
or rest position. No external force acts on the system after release the structure to
vibrate. Free vibration can be induced by impacting. In real structures, energy is lost
as a result of friction or heat generation, resulting in the free vibration decay. Free
vibration testing is a kind of output-only data dynamic test method. This type of
dynamic testing has an advantage of being inexpensive since no equipment is needed
to excite the structure (Ren and Zong 2003). Therefore, this technique will be applied
Damage assessment in structures using vibration characteristics 37
in the experimental testings for modal identification of slab-on-girder bridge in this
research.
2.7.4 Model updating
Most of the existing model-based damage identification methods require an accurate
FE model to represent the intact structure. But in practice, the uncertainties existing in
the FE model along with errors in the measured vibration data limit the successful use
of these models (Xia et al. 2002). The FE model updating procedures are therefore
applied to minimise the differences between the numerical and experimental modal
properties. Most model updating techniques are based on the minimization of
structural parameters to minimize an error function between the measured and
numerical responses. In general, there are two groups of updating methods: direct
methods and iterative (or parametric) methods (Zivanovic et al. 2007). The former is
based on updating of stiffness and mass matrices directly, in a way that often has no
physical meaning. The latter, on the other hand, concentrates on the direct updating of
physical parameters which indirectly updates the stiffness and mass matrices.
Iterative methods are slower than their direct counterparts. However, their main
advantage is that changes in the updated model can be interpreted physically. Also
iterative methods can be implemented easily using existing FE codes. The updating
process has four key phases: initial FE modelling, modal testing, manual model
tuning and automatic updating. Manual tuning model updating and automatic
updating are briefly discussed as follows:
2.7.4.1 Manual tuning model updating
The manual tuning involves manual changes of the model geometry and modelling
parameters by trial and error, guided by engineering judgement (Zivanovic et al.
2007). The aim of this is to bring the numerical model closer to the experimental one.
Small number of key parameters, e.g. boundary conditions and non-structural
elements are adjusted to improve the initial structural idealisation in this process. It is
very important to determine a suitable initial value of a selected parameter to provide
a reasonable starting point.
Damage assessment in structures using vibration characteristics 38
For obtaining a reasonable approximation of uncertain parameter of boundary
condition in this proposed research, manual tuning technique will be used in the
model updating process to adjust the structural parameters of the finite element model
such that the error between the identified critical experimental parameters and the
numerical model is minimised. Manual tuning is carried out by engineering
judgement to improve the simulation of the boundary condition.
2.7.4.2 Automatic model updating
The aim of automatic updating is to improve further the correlation between the
numerical and experimental modal properties by taking into account a larger number
of uncertain parameters (Zivanovic et al. 2007). The iterative methods used in
automatic model updating are mainly sensitivity based. This requires the sensitivity
matrix to be calculated in iteration. The sensitivity matrix of ordernm× is as follows:
][][j
iij P
RSS
δδ== (2.18)
Sij is the sensitivity of the target response Ri (i=1,2,…m) to a certain change in
parameter Pj ( j=1,2….,n). m and n are the number of target responses and parameters
respectively. Operator δ presents the variation of the variable.
Elements of the sensitivity matrix can be calculated numerically using the forward
finite difference approach.
jjj
jijji
PPP
PRPPRS
−∆+−∆+
=)(
)()( (2.19)
where )( ji PR is the value of the i-th response at the current state of the parameter jP ,
while )( jji PPR ∆+ is the value of the same i-th response when the parameter jP is
increased by value jP∆ .
Damage assessment in structures using vibration characteristics 39
The target responses mainly considered are natural frequencies, mode shapes and
frequencies response functions (FRFs), or some combination of these. As natural
frequencies are normally measured quite accurately, they are almost always selected.
Once relevant (measured) target responses and structural parameters for updating
have been selected, the sensitivity matrix can be calculated. Since in the iterative
model updating process the updating parameters change at every step, the sensitivity
matrix has to be recalculated in iteration. The iterative process is required because the
relationship between target responses and parameters that is mainly nonlinear is
approximated by the linear term. The updating, which targets larger number of
measured responses at a time, is preferable because it puts more constraints to the
optimisation process.
The targeted experimental response vector eR can be approximated via vectors 0R ,
uP and 0P and using the linear term in a Taylor’s expansion series:
)( 00 PPSRR ue −+≈ (2.20)
where 0P and 0R denote the starting parameter and target response vectors
respectively, uP represents the vector of updated parameters in the current iteration.
An updating process which produces good correlation between experimental and
analytical responses can be regarded as successful only if finally obtained parameters
are physically viable. Success updating will give more confidence in the results.
2.7.4.3 Comparisons of modal properties
The success of the updating process is usually judged by comparing two sets of
modal parameters using modal assurance criterion (MAC) and the coordinate modal
assurance criterion (COMAC). For damage identification purposes, differences
between mode shapes of the structure in a reference condition and a damaged
condition can be assessed (Parloo et al. 2003). MAC and COMAC express the
correlation between two measured mode shapes obtained from two sets of tests are
defined as follows:
Damage assessment in structures using vibration characteristics 40
∑∑
∑
==
==oo
o
N
i
dik
N
i
uij
N
i
dik
uij
kjMAC
1
2
1
2
2
1
))((
)(
),(φφ
φφ (2.21)
∑∑
∑
==
==mm
m
N
j
dij
N
j
uij
N
j
dij
uij
iCOMAC
1
2
1
2
1
2
))((
)(
)(
φφ
φφ (2.22)
where uijφ and d
ijφ represent the normal mode shape j (real-valued) evaluated in i-th
degree of freedom (DOF) for the undamaged and damaged conditions of the test
structure respectively, mN and oN represent number of modes and DOFs,
respectively.
MAC provides a measure of the least squares deviation or “scatter” of the points
from the straight line correlation (Ewins 2001). The MAC values vary from 0 to 1,
with 0 for no correlation and 1 for full correlation. Therefore, the derivation of these
factors from 1 could be interpreted as a damage indicator in structures. The diagonal
values of the MAC matrix indicate which modes are most affected by the damage.
The COMAC factors are generally used to identify where the mode shapes of a
structure from two sets of measurement do not correlate. In this research, MAC
will be used to quantify the correlation between measured mode shapes and
analytical mode shapes in free vibration testing.
2.7.5 Uncertainties and limitations
Developing a numerical model of a civil structure that has sufficiently reliable
dynamic properties requires a wide range of skills and expertise in the areas of FE
modelling, modal testing, FE model correlation, tuning and updating with the regard
to experimental modal properties. Therefore, reducing the mathematical modelling
errors to an acceptable level is important in order to reliably apply the vibration-based
structural condition monitoring methods to civil engineering structures.
Damage assessment in structures using vibration characteristics 41
2.7.5.1 Uncertainties
Basically, uncertainty can be categorized into two groups, aleatory and epistemic
(Ang and De Leon 2005). Aleatory uncertainty represents the inherent randomness of
natural occurrences governed by probabilistic models. The inherent variability’s of
random environment forces and mechanical property of structural materials constitute
aleatory uncertainty. On the other hand, epistemic uncertainty is the uncertainty due
to lack of complete knowledge, such as insufficient data, inaccuracies in
measurement, inadequate models, etc. The idealization of loading, modelling of
structure and the structural responses to environmental loading contribute to the
additional uncertainty of the epistemic type. Increased knowledge and accuracy
information decrease the epistemic uncertainty, making the predictions more reliable.
In practice, the uncertainty existing in the model along with errors in the measured
vibration data limit the successful use of these models. The inevitably uncertainties
come from the inaccurate stiffness parameter (owing to material and geometrical
variations) of the FE model, modelling of boundary conditions, and effects of non-
structural elements, noises of measured frequencies and mode shapes during modal
testing and environment effect (temperature). The influence of uncertainties can cause
the damage not to be detected or exaggerates the damage: both are adverse false
identifications. Therefore it is very important to analyse the influences of FE
modelling error, environmental and operational variations on the damage
identification results.
2.7.5.2 Effect of environmental and operational variations
Civil structures are subjected to varying environmental and operational conditions
such as traffic, temperature, wind, humidity. These environmental effects can cause
changes in modal parameters which may mask the change caused by structural
damage. If the effect of uncertainties on structural vibration properties is larger than
or comparable to the effect of structural damage on its vibration properties, the
structural damage cannot be reliably identified. For reliability performance of damage
detection algorithms, it is important to distinguish modal parameter changes caused
by structural damage from changes due to environmental variations. Therefore
Damage assessment in structures using vibration characteristics 42
modelling of the effect of environment is made by including vibration transducers,
such as temperature sensors or anemometer in long-term structural health monitoring
systems. By using the measurement data covering a full cycle of in-service
environmental conditions, various statistical/regression/learning methods can be
conducted for modelling the effect of environment on modal property of structures
(Xia et al. 2006).
2.7.5.3 Low sensitivity to damage
Generally, the size of structural damage can be approximately divided into three
levels: (1) micro-damage, i.e., damage size is smaller than 0.1% of structural size; (2)
small-damage, i.e., damage size is about 1% of structural size; (3) macro-damage, i.e.,
damage size is greater than 10% of structural size (Yan et al. 2006). Vibration-based
detection method is generally not successful for structural micro-damage and it
should be detected using instruments with high precision, such as acoustic emission
transducer. The reason is that vibration characteristic are global properties of the
structure, and although they are affected by local damage, they may not be very
sensitive to such damage. As a result, the change in global properties may be difficult
to identify unless the damage is very severe or the measurements are very accurate
and made with extra care.
Alternatively, a real structure normally possesses a large number of degree of
freedom (DOF) and hence a large number of frequency and mode shapes. However,
the higher frequencies and mode shapes can rarely be measured with sufficient
accuracy. Vibration-based damage detection therefore depends on the measurements
of a limited number of the lower vibration frequencies, or such frequencies and the
associated mode shapes. In general, the measurement of mode shapes is more difficult
than the measurement of frequencies. Measurement errors as well as mode truncation
and incomplete mode shapes introduce errors in damage prediction and may make
such a prediction unreliable. In order to produce reliable and accurate results, a
relatively large number of sensors are required to produce the fine coordinates of the
mode shapes.
Damage assessment in structures using vibration characteristics 43
2.7.5.4 Nonlinear analysis
The structural dynamics background theory and the modal parameter estimation
theory are based on two major assumptions: (i) the system is linear; (ii) the system is
stationary. Many existing non-destructive damage detection methods assume linear
damage behaviour for damage. These methods cannot deal with situations where the
damage introduces nonlinearity in the structure. Nonlinearity can result from the
presence of cracks and loose connections that slip under load. Closing and opening of
the cracks alters the stiffness of the structure, introducing nonlinearity in its
behaviour. Most structures are nonlinear to some degree. Damping of real structures
related to the general problem of nonlinearity and the actual physical mechanisms of
damping in structures are many and complex. Although the dynamic behaviour of
damping elements are always expressed in very complicated and nonlinear
expressions, it is found that the effects of the nonlinearity are barely perceptible
against the other measurement uncertainties in most vibration tests (Farrar et al.
2007).
2.8 Case study 1 - Wind and structural health monitoring
system
In Hong Kong, a sophisticated long-term monitoring system, called Wind And
Structural Health Monitoring System (WASHMS), has been devised by the Hong
Kong SAR Government Highways Department to monitor structural performance
and evaluate health and safety conditions of three long-span cable-supported bridges:
(i) Tsing Ma Bridge, (ii) Kap Shui Mun bridge and (iii) Ting Kau Bridge (Ko et al.
2002). The WASHMS consists of over 800 sensors permanently installed on the
bridges, including accelerometers, strain gauges, displacement transducers,
anemometers, temperature sensors, level sensors, weigh-in-motion sensors, and
global positioning systems. Since GIS technology provides an efficient computerized
database management system for capture, storage retrieval, analysis, and display of
temporal-spatial data, it has been adopted as a platform for developing a visualized
monitoring and management system in the Hong Kong Polytechnic University (Ko
Damage assessment in structures using vibration characteristics 44
2004). An initial measurement of an undamaged structure is acted as the baseline for
future comparisons of measured response. This model is used for modal analysis and
classification, damage-scenario simulation and generating training samples. A multi-
stage identification strategy is proposed for vibration-based damage detection. Using
a hierarchical strategy, different identification algorithms and different sensor
deployment schemes can be designed in view of the objectives of the different
stages. In a total of four stages, the second stage of the proposed strategy is to locate
the deck segment or section that contains damaged members. The modal curvature
index is used in conjunction with the modal flexibility index for detection of the
damaged segment. In this research, the modal flexibility change and modal strain
energy change is complementary to the frequency change for damage identification
and localization of a wide range of structures. This approach is reliable and
innovative.
2.9 Case study 2 - Damage identification of timber bridge
Li et al. (2005) conducted a study to investigate the capabilities and limitation of
using the damage index method for locating inflicted damage, especially multiple
damages, in timber bridges. The proposed damage index method is developed by
Stubbs et al. (1995) based on change in modal strain energy as an indicator of
localized damage or stiffness loss in a structure. The FE model is constructed based
on a laboratory timber bridge configuration which is built for experimental
investigations. Common damages found in timber bridges are simulated in the FE
model by an opening on the web. The strain-energy based damage index method is
used to compare the normalized mode shape vector for each girder from each of the
damage cases versus the corresponding normalized undamaged mode shape vector.
The confidence in detecting the damage using the damage index method increases
with increasing severity of damage. The effectiveness of damage localization method
is closely related to the number of elements of the structure or components. The
damage index method is demonstrated successfully in single damage localization, but
it encounters problem during the identification of multiple damage cases. As there are
limitations of using the damage index method during identification of multiple
damages found in the literature, a multi-criteria approach is proposed in this research.
Damage assessment in structures using vibration characteristics 45
The feasibility and capability of multi-criteria approach for treating multi-damage
localisation will be demonstrated through numerical examples in Chapters 4 and 5. It
provides evidence that damage assessment is more accurate than hitherto possible
through the use of the multi-criteria approach.
2.10 Summary of literature review
Important literature findings related to damage detection methodologies are
summarised as follows:
2.10.1 Natural frequency and mode shape
Early damage detection methods use natural frequency changes as the damage
indicator. This is because the modal frequencies can be measured easily and
accurately. However, frequencies alone may not be sufficient for a reasonably
accurate assessment of the location and severity of the damage. This is because
frequencies are not spatially specific and are not very sensitive to damage, such that
its application is limited to simple structures. Mode shapes can be described as a
vibration form in which the structure oscillates with the corresponding natural
frequencies. The mode shapes have the advantage of being spatially specific, however
the measurement is more complex and relatively not very accurate for large-sized
structures. Moreover, one particular difficulty in using mode shape data is that the
number of measurement locations is usually much less than the size of the analytical
model. In analysis, it needs to either expand the measured data to full degrees of
freedom of the finite element model, or reduce the FE model to the measured degree
of freedom. Both approaches generate extra errors and make damage detection more
difficult. It is often very difficult or impracticable to measure the response along all of
the DOF necessary for the complete definition of a given mode shape. In order to
capture the incomplete mode shapes, a dense array of sensors would be needed.
Damage assessment in structures using vibration characteristics 46
2.10.2 Modal flexibility method and modal strain energy method
A number of methodologies have been found in the recent literature to identify, locate
and estimate the severity of damage in structures using numerical analysis. It is
noticed that those methods utilizing mode shapes are the most developed in terms of
displaying the ability to identify the location of damage and estimate the severity of
damage. The advantage of using the modal flexibility method is that the flexibility
matrix is most sensitive to changes in the lower-frequency modes of the structures
due to the inverse relationship to the square of the natural frequencies. Therefore, a
good estimation of the modal flexibility can be made with the inclusion of the first
few natural frequencies and their associated mode shapes. The advantage of using the
modal strain energy method is that only measured mode shapes are required in the
damage identification, without knowledge of the complete stiffness and mass
matrices of the structure. Only the mode shapes of the first few modes and their
corresponding derivatives are required in this proposed algorithm to accurately locate
damage. Therefore, the modal flexibility and modal strain energy methods are chosen
in this study as their corresponding algorithms can be applied to beams, plates and
integrated structures.
2.10.3 Environmental effect and operation variations
Theoretically, the damage can be identified by examining the changes in the stiffness
property of the updated model. However in practice, the environmental effects and
operation variations limit the successful use of vibration-based methods. For
example, global vibration characteristics are often affected by thermal effects caused
by temperature variation and changes in boundary conditions. Whenever the
structural system is constrained or indeterminate, thermal effects introduce axial
stresses in the structural elements. The presence of such axial stresses changes the
stiffness of the structure and may alter its vibration characteristics. The boundary
conditions in a structure can have a significant effect on its stiffness, and if these
boundary conditions, such as at bridge bearings, are prone to change with the age of
the structure, they may lead to a change in the vibration characteristics even when
there is no damage in the structure. Therefore a valid structural health monitoring and
Damage assessment in structures using vibration characteristics 47
damage detection technique is necessary to distinguish modal parameter changes
caused by structural damage from changes due to environmental variations. One of
the possible ways is that statistical analysis can be implemented to decrease the
number of false positives and false negatives.
2.10.4 Multiple criteria approach
Structural health monitoring requires clearly defined performance criteria, a set of
corresponding condition indicators, and also global and local damage deterioration
indices, which would help diagnose reasons for changes in structural condition.
Condition indicators such as the mechanical characteristics (flexibility, frequencies
etc.) of structures change continuously due to the interactions between environment,
foundation, and operational conditions. Condition indicators also change following
important events in the life cycle such as a significant accident or a major retrofit.
Condition indicators can also be affected by accumulated deterioration, although
deterioration mechanisms typically take a long time (years) before their impact
becomes measurable. The numbers of the natural causes and uncertainties make
damage detection approach necessary to rely on a multitude of local, regional and
global indices describing changes in the global and local mechanical characteristics
and intrinsic properties of the materials. It is unrealistic to expect that damage can be
reliably detected or tracked by using a single damage index as evidenced in the
numerical examples (Chapters 4 and 5). The best approach is to use a suite of
validated features depending on the structure, damage type, and expected
performance from the structure (Catbas et al. 2008). Therefore, this thesis proposes a
novel multi-criteria approach which can be applied to several structures under multi-
damages scenarios.
Damage assessment in structures using vibration characteristics 48
Chapter 3 Theory and Validation of Finite Element Models
3.1 Introduction
This chapter begins by presenting basic structural dynamic theories, which include
governing equations for structural systems and damping. Dynamic equilibrium
equation is introduced for single and multiple degree-of-freedom systems. The linear
differential equations relate the effects of the mass, stiffness and damping in a way
that leads to determination of natural frequencies, mode shapes and damping ratios
of the idealized system. Differential equations of motion also lead to the dynamic
response of the system when excitation is applied. As the input and output relation of
the linear system can be written in the frequency domain, the general frequency
response transfer function is used for modal analysis. Next, two common methods
(i.e. logarithmic decrement and bandwidth method) are introduced for determining
the damping ratio in a system experimentally (Clough and Penzien 1993).
Following this, an alternative method of considering the dynamic equilibrium of the
system, Rayleigh’s method, is introduced (Paz and Leigh 2004; Clough and Penzien
1993). Rayleigh’s method determines the natural frequency of a vibrating system
based on the principle of conservation of energy. It is applied by equating the
maximum potential energy with the maximum kinetic energy of the system. As the
flexibility and strain energy are fundamental concepts in applied mechanics and are
widely used for determining the response of structures to both static and dynamic
loads, their simplest static forms are presented following the expression of modal
forms. These damage detection algorithms derived herein form the basis for damage
identification in numerical examples including load bearing elements of bridge, viz
beam, slab and truss, and also the simple bridge involving these elements in Chapters
4 and 5. Next, the chapter shows an experimental study consisting of a static and a
Damage assessment in structures using vibration characteristics 49
dynamic test on a slab-on-girder bridge model to validate the finite element model.
The model description, instrument setup, testing procedure, and data analysis for the
two tests are briefly discussed.
3.2 Basic dynamic equations
The structure shown in Fig. 3.1 is called a lumped parameter system of a single
degree-of-freedom because its physical properties are “lumped” into the mass,
spring, and damper elements (Clough and Penzien 1993).
Fig. 3.1 Dynamic equilibrium of a single degree-of-freedom system.
When the elastic structure is excited by a force or displacement motion, the forced
linear vibration of the structure can be described by a homogeneous dynamic
equilibrium equation given as follows:
)(tfkuucum =++ &&& (3.1)
where m and k are the mass and spring constant of the oscillator respectively and c is
the viscous damping coefficient. )(tf is the time dependent excitation force applied
to the system. u&& , u& and u are the corresponding response of acceleration, velocity
and displacement, respectively.
Damage assessment in structures using vibration characteristics 50
Eq. (3.1) is a statement of Newton’s second law of motion; a force balance among
three types of internal forces in any structure made of elastic materials. These
internal forces are the inertial (mass), dissipative (damping), and restoring (stiffness)
forces. Some forms of damping (e.g. viscous) are always present in all real
structures. The free vibration without damping of the linear multiple degree-of-
freedom system requires that the force vector {F} and damping matrix [C] equal zero
in Eq. (3.1). The general form of this equation is given as follows:
{ } { } 0][][ =+ uKuM && (3.2)
The solution of Eq. (3.2) is in the form as
)sin( αω −= tau ii ni ,...,2,1= (3.3)
or in vector notation
{ } { } )sin( αω −= tau (3.4)
where ia is the amplitude of motion of the i-th coordinate and n is the number of
degrees of freedom, t is the time of motion, ω is the circular frequency and α is the
phase angle.
After substituting Eq. (3.4) into Eq. (3.2), and rearranging the terms, it forms Eq.
(3.5), which is an important mathematical problem known as the Eigen problem.
{ } }0{][][ 2 =− aMK ω (3.5)
This eigenproblem is then used to find the nontrivial solution which yields
0][][ 2 =− MK ω (3.6)
Damage assessment in structures using vibration characteristics 51
By using Eq. (3.6), the circular frequency (ω ), natural frequency (f), and the period
of motion (T) are then determined as follows:
m
k=ω (3.7)
fπω 2= (3.8)
Tf /1= (3.9)
ωπ2=T (3.10)
For each of these values of 2ω satisfying the characteristic Eq. (3.6), they are used to
solve Eq. (3.5) for1a , 2a ,... na in terms of an arbitrary constant.
3.3 Frequency response function
The frequency response function (FRF) for a linear single degree-of-freedom system
is usually established as the relationship between the Fourier transform of the input
signal )(ωF and the output signal )(ωX . For example, when the impulse force and
the resulting acceleration response of the vibration system are measured, the
resulting data are used to generate the FRF for the system (Maia and Silva 1997).
The general relationship can be given as follows:
)()()( ωωω FHX = or )(
)()(
ωωω
F
XH = (3.11, 3.12)
where )(ωH is the FRF matrix, )(ωX is the vector of discrete Fourier transforms of
displacement responses, and )(ωF is the vector of discrete Fourier transforms of
external forces. The FRF of a system is a complex-value function of the real-valued
independent variable ω and therefore has real and imaginary components.
Damage assessment in structures using vibration characteristics 52
3.4 Damping ratio
3.4.1 Logarithmic decrement
In order to determine the damping coefficient of a system experimentally, a free
vibration is carried out on the structure to obtain a record of its oscillatory motion,
such as the one shown in Fig. 3.2, and measure the rate of decay of the amplitude of
motion.
Fig. 3.2 Free-vibration response of an underdamped system.
The decay may be conveniently expressed by the logarithmic decrement δ which is
defined as the natural logarithm of the ratio of any two successive peak amplitudes
for the displacement or acceleration in the free vibration as shown in Eqs. (3.13) and
(3.14) respectively.
2
1lnu
u=δ or 2
1lnu
u&&
&&=δ (3.13, 3.14)
where u and u&& are the corresponding response of displacement and acceleration
respectively, subscript 1 and 2 denote two consecutive peaks.
After determining the amplitudes of two successive peaks of the system in free
vibration experimentally, the damping ratio ξ can be calculated as follows:
21
2
ξπξδ−
= (3.15)
Damage assessment in structures using vibration characteristics 53
3.4.2 Bandwidth method
The bandwidth method, also known as the half-power method, is an alternative way
to estimate the damping ratio (Clough and Penzien 1993). A typical frequency
amplitude curve obtained experimentally for a moderately damped structure is
shown in Fig. 3.3.
Fig. 3.3 Typical frequency response curve.
The shape of the curve is controlled by the amount of damping presented in the
system; in particular, the bandwidth, that is the difference between two frequencies
corresponding to the same response amplitude, is related to the damping in the
system. In the evaluation of damping it is convenient to measure the bandwidth at
21 of the peak amplitude. The frequencies corresponding to this bandwidth, 1f
and 2f are referred to as half-power points. The damping ratio ξ is then calculated
as follows:
12
12
ff
ff
+−=ξ (3.16)
3.5 Rayleigh’s method
Rayleigh’s method is used to find the approximate value of the fundamental natural
frequency of a discrete system. Rayleigh’s principle can be stated as follows:
Damage assessment in structures using vibration characteristics 54
"The frequency of vibration of a conservative system vibrating about an equilibrium
position has a stationary value in the neighbourhood of a natural mode. This
stationary value, in fact, is a minimum in the neighbourhood of the fundamental
natural mode. This method, in which the natural frequency is obtained by equating
maximum kinetic energy with maximum potential energy, is known as Rayleigh’s
method." (Paz and Leigh 2004)
By considering the principle of conservation of energy, if no external forces are
acting on the system and there is no dissipation of energy due to damping, maximum
strain energy )( maxSE equals maximum kinetic energy )( maxKE .
maxmax KESE = (3.17)
Strain Energy in spring (SE) = 2
21
ku (3.18)
Kinetic Energy of body (KE) = 2
21
um& (3.19)
tu ωφ sin= (3.20)
where m and k are the mass and spring constant of the oscillator respectively, u& and
u are the response of velocity and displacement respectively, φ denotes the vector of
amplitudes (mode shape), ω represents the natural frequency of vibration, and t is
the time of motion.
Substituting Eqs. (3.18) and (3.19) into Eq. (3.17), the fundamental natural
frequency of a discrete system is given as follows:
φφωφφ MK TT 2
21
21 = (3.21)
φφφφω
M
KT
T
=2 (3.22)
Damage assessment in structures using vibration characteristics 55
3.6 Modal flexibility matrix
The modal flexibility includes the influence of both the modes and natural
frequencies. It is defined as the accumulation of the contributions from all available
mode shapes and corresponding natural frequencies. It is found that modal flexibility
is more sensitive to damage than the mode shapes and natural frequencies alone
(Zhao and DeWolf 2002). Damage in a structure results in stiffness reduction and the
flexibility increment in the corresponding elements near the damages. Increase in
structural flexibility can therefore serve as a good indicator of the degree of
structural deterioration.
The modal flexibility matrix is derived as follows:
0}]{[}]{[ =+ uKuM && (3.23)
where tu ωφ cos}{}{ = (3.24)
Substituting Eq. (3.24) into Eq. (3.23), it becomes
0]][][[]][[ 2 =− φωφ MK (3.25)
Pre-multiplying Eq. (3.25) by the transpose of the modal vector T][φ
0]][[]][[]][[][ 2 =− φφωφφ MK TT (3.26)
For normalized eigenvectors, the orthogonality condition is given by
][]][[][ IMT =φφ , (3.27)
Substituting Eq. (3.27) into Eq. (3.26), it becomes
0]][[]][[][ 2 =− IKT ωφφ (3.28)
]][1
[][][][2
11 IK T
ωφφ =−−− (3.29)
Damage assessment in structures using vibration characteristics 56
TK ]][1
][[][2
1 φω
φ=− (3.30)
TF ]][1
][[][2
φω
φ= (3.31)
where ][][ 1 FK =− (3.32)
With mode shapes normalized to unit mass, the flexibility matrix can be obtained
approximately by using only a few of the lower modes.
The change of modal flexibility matrix is given as follow:
][][][ hd FFF −=∆ (3.33)
where ][F is the modal flexibility matrix, ][K is the stiffness matrix, ][M is the
mass matrix, ][φ is the mass normalized modal vectors, ]/1[2ω is a diagonal matrix
containing the reciprocal of the square of natural frequencies in ascending order.
Index ‘h ’ and ‘d ’ refer to the healthy and damaged state respectively (Huth et al.
2005, Paz and Leigh 2004).
3.7 Elastic strain energy
The modal strain energy based damage index method uses the change in modal strain
of the undamaged and damaged structure to detect and locate damage in a structure.
The strain energy U stored in an elastic body, for a general state of stress, is
expressed by
dxdydzU yzyzxzxzxyxy
V
zzyyxx )(2
1 γτγτγτεσεσεσ +++++= ∫∫∫ (3.34)
where σ and ε are the stress components at a point in a body, V is the volume of the
three dimensional body in a coordinate system (x, y and z-axis).
Damage assessment in structures using vibration characteristics 57
For completeness the derivation of the damage indicator will be shown for beam
plate and truss elements. Further details can be found in Ugural (1999).
3.7.1 Modal strain energy – Beam
The strain energy stored in a beam is given as follows:
dxdx
ydEIU ∫
=
2
2
2
2 (3.35)
The change of strain energy is
hd UUU −=∆ (3.36)
where x is the distance measured along the length of the beam, y is the vertical
deflection, EI is the flexural rigidity of the cross section and 22 dxyd is the
curvature of the deformed beam. Index ‘h ’ and ‘d ’ refer to the healthy and
damaged state respectively (Stubbs et al. 1995).
By using principle of virtual work:
Virtual external work = Virtual internal work
IE WW = (3.37) 2
iiE KW δ= (assume 1=iδ ) (3.38)
∫=L
I dxMW0
)( θ (3.39)
where )(2
2
xdx
d
dx
d
EI
M φφθ ′′=== (3.40)
The thi modal stiffness iK of the beam is given by
∫ ′′′′=L
iii dxxxxkK0
])()][()([ φφ where EIxk =)( (3.41)
∫ ′′=L
ii dxxxkK0
2)]()[( φ (3.42)
The contribution of the thj member of the thi modal stiffness, ijC , is given by
∫ ′′=j ijij dxxkC 2)]([φ (3.43)
Damage assessment in structures using vibration characteristics 58
where jk is the stiffness of the thj member.
The fraction of the modal stiffness (element sensitivity) for the thi mode that is
concentrated in thj member is given by
iijij KCF /= (3.44)
For the damaged structure,
*** / iijij KCF = (3.45)
where scalars *ijC and *
iK are given by
∫ ′′=j ijij dxxkC 2*** )]([ φ (3.46)
∫ ′′=L
ii dxxkK0
2*** )]([φ (3.47)
For any modei , the term ijF and *ijF have the following properties:
11
*
1
==∑∑==
NE
jij
NE
jij FF ; and 1<<ijF , 1* <<ijF (3.48, 3.49)
where NE is the number of elements in a member.
Therefore, an expression which connects the behaviour of the damaged and
undamaged structures is developed from the approximation.
*11 ijij FF +≅+ (3.50)
Substituting Eqs. (3.44) and (3.45) into Eq. (3.50)
*
*
11i
ij
i
ij
K
C
K
C+=+ (3.51)
*
**
)(
)(1
iiij
iiij
KKC
KKC
++
= (3.52)
Utilizing the expressions for ijC and *ijC and the mean value theorem of calculus, Eq.
(3.52) is transformed to
∫∫∫
∫∫∫
′′
′′+′′
′′
′′+′′
=L
i
L
ij
j ij
L
i
L
ij
j ij
dxxxkdxxkk
dxxk
dxxxkdxxkk
dxxk
0
2**
0
22
0
2
0
2***
2**
)]([)()]([1
)]([
)]([)()]([1
)]([
1
φφφ
φφφ
)
)
(3.53)
Damage assessment in structures using vibration characteristics 59
By approximating )()( * xkxk)) ≅ (3.54)
∫∫∫
∫∫∫′′
′′+′′
′′
′′+′′
==L
i
L
ij i
L
i
L
ij i
j
jji
dxxdxxdxx
dxxdxxdxx
k
k
0
2*
0
22
0
2
0
2*2*
*
)]([)]([)]([
)]([)]([)]([
φφφ
φφφβ (3.55)
or ∑ ∑∑ ∑
′′′′+′′′′′′+′′
==])(][)()[(
])(][)()[(2*22
22*2*
*jijiji
jijiji
j
jji k
k
φφφφφφ
β (3.56)
To account for all available modes, a single indicator for each location is given by
∑
∑
=
== NM
iij
NM
iij
j
Denom
Num
1
1β (3.57)
where NM is the number of modes, Num and Denom are the numerator and
denominator respectively.
The normalized damage index jZ is obtained:
j
jjjZ
β
β
σµβ −
= (3.58)
where jβµ is the mean of jβ values for all j elements and jβσ is the standard
deviation of jβ for all j elements. A judgement-based threshold value is selected and
used to determine which of the j elements are possibly damaged. This is based on
what level of confidence is required for localisation of damage within the structure.
3.7.2 Modal strain energy – Plate
The strain energy U for a plate of size ba× is given as follows:
dxdyyx
wv
y
w
x
w
y
w
x
wDU
b a
∫ ∫
∂∂∂−+
∂∂
∂∂+
∂∂+
∂∂=
0 0
22
2
2
2
22
2
22
2
2
))(1(2))((2)()(2
ν (3.59)
Damage assessment in structures using vibration characteristics 60
where )1(12 23 vEhD −= is the bending stiffness of the plate, v is the Poisson’s ratio,
h is the plate thickness, w is the transverse displacement of the plate, 22 xw ∂∂ and
22 yw ∂∂ are the bending curvatures, and yxw ∂∂∂22 is the twisting curvature of the
plate (Cornwell et al. 1999). For a particular mode shape ),(yxiφ of the undamaged
structure, the strain energy Ui associated with that mode shapes is
dxdyyx
vyxyx
DU
b aiiiii
i ∫ ∫
∂∂∂−+
∂∂
∂∂+
∂∂+
∂∂=
0 0
22
2
2
2
22
2
22
2
2
))(1(2))((2)()(2
φφφνφφ (3.60)
where 22 xi ∂∂ φ and 22 yi ∂∂ φ are the mode shape curvatures, yxi ∂∂∂ φ22 is the
twisting mode shape curvature for the i -th mode of the plate. If the plate is
subdivided into xN subdivisions in the x direction and yN subdivisions in y the
direction, then the energy Uijk associated with sub-region jk for the i -th mode is
given by
dxdyyx
vyxyx
DU
k
k
j
j
b
b
a
a
iiiiijkijk ∫ ∫
+ +
∂∂∂−+
∂∂
∂∂+
∂∂+
∂∂= 1 1 2
2
2
2
2
22
2
22
2
2
))(1(2))((2)()(2
φφφνφφ (3.61)
and
∑∑= =
=y x
N
k
N
jijki UU
1 1
(3.62)
The fractional energy at location jk is defined as:
i
ijkijk U
UF = and 1
1 1
=∑∑= =
y xN
k
N
jijkF (3.63, 3.64)
Similar expressions can be written using the modes of the damaged structure *iφ ,
where the superscript * indicates damaged state. A ratio of parameters can be
determined that is indicative of the change of stiffness in the structure as follows:
Damage assessment in structures using vibration characteristics 61
ijk
ijk
jk
jk
f
f
D
D *
*= (3.65)
where
dxdyyx
vyxyx
dxdyyx
vyxyx
fb a
iiiii
b
b
a
a
iiiii
ijk
k
k
j
j
∫ ∫
∫ ∫
∂∂∂−+
∂∂
∂∂+
∂∂+
∂∂
∂∂∂−+
∂∂
∂∂+
∂∂+
∂∂
=
+ +
0 0
22
2
2
2
22
2
22
2
2
22
2
2
2
22
2
22
2
2
))(1(2))((2)()(
))(1(2))((2)()(1 1
φφφνφφ
φφφνφφ
(3.66)
and an analogous term *ijkf can be defined using the damaged mode shapes. In order
to account for all measured modes, the following formulation for the damage index,
or MSEC, for sub-region jk is used:
∑
∑
=
== m
iijk
m
iijk
jk
f
f
1
1
*
β (3.67)
3.7.3 Modal strain energy – Truss
The strain energy U stored in a bar, which equals the work done W by the load, is
given as follows:
2δP
U = (3.68)
The relationship between the load P and the elongation δ for a bar of linearly elastic
material is
EA
PL=δ (3.69)
Damage assessment in structures using vibration characteristics 62
whereE is the modulus of elasticity, A is the cross-sectional area and L is the length
of the prismatic bar (Gere and Goodno 2008).
The strain energy of a linearly elastic bar can be expressed in alternative form as
follows:
EA
LPU
2
2
= (3.70)
The modal strain energy U stored in the bar is as follows:
}]{[}{2
1 φφ KU T= (3.71)
where LEAK /][ = (3.72)
The change of modal strain energy
hd UUU −=∆ (3.73)
where ][K is the modal stiffness matrix, }{φ is the mass normalized modal vectors,
and indices ‘h ’ and ‘d ’ refer to the healthy and damaged states respectively (Paz
and Leigh 2004).
3.8 Validation of finite elements models
The experimental studies consisting of a static and free vibration test on a slab-on-
girder bridge model are carried out. As a precursor to the free vibration test, the
static test is performed to verify the static behaviour of the bridge and to check the
stiffness properties and boundary conditions used in the analytical study. The free
vibration test is then performed to (i) obtain natural frequencies of the structure; (ii)
obtain mode shapes and damping information for the structure; (iii) correlate the FE
Damage assessment in structures using vibration characteristics 63
model of the structure with measured results from the experimental model and (iv)
obtain a validated computer model of the structure that can then be used to assess the
effects of a range of damage simulation to that structure. The free vibration test
provides a check on differences of primary modal parameters (e.g. natural frequency
and the corresponding mode shape) between the experimental results and the FEA
results. To minimise the differences between the analytical and experimental results,
the initial FE models of the bridge are calibrated at local and global levels with static
and free vibration test data. As a result of this calibration, all modal parameters,
modelling assumptions, connections, and support conditions of the finite element
model are updated to represent the experimental model as accurately as possible. The
model description, instrument setup, testing procedure, and experimental data
analysis for the two tests are briefly discussed.
3.9 Static test
3.9.1 Description of the model
The experimental model is a slab-on-girder bridge comprised of a continuous deck
supported by two steel girders as shown in Figs. 3.4. A single-span simply supported
bridge with a span length and deck width of 1.8m and 1.2m, respectively, is
designed. The spacing of the girders is 800mm centre to centre. Steel diagonal
bracings are installed at the two ends (over end bearings) of the model. Although
most decks in real bridges are made of reinforced concrete, a steel plate of thickness
3mm is used in this study instead after careful consideration of the laboratory
condition, the project duration and the availability of materials. All steel structural
components including deck, girders and diagonal bracings are connected by welding.
Testing a full scale structure is recommended in future research.
Damage assessment in structures using vibration characteristics 64
(a) Test specimen
(b) Cross bracings on test specimen
Fig. 3.4 Slab-on-girder bridge.
Damage assessment in structures using vibration characteristics 65
(c) Boundary condition
Fig. 3.4 Slab-on-girder bridge.
The FE software SAP2000 was used to model the bridge. The general modelling
scheme for bridge is depicted in Fig. 3.5. The details of the bridge are listed in Table
3.1. Both bridge deck and girders are modelled as shell elements. The deck and each
girder are divided into 216 and 108 elements respectively. Steel diagonal bracings at
the two exterior support lines are modelled as truss elements. Shell elements are
widely used to idealize the bridge deck since behaviour of this structural component
is governed by flexure and in this case a mesh of shell elements is computationally
more efficient when compared to one of solid elements. It is assumed that there is a
complete connection between the girders and slab. Twin-girders having the same
span are simply supported at their ends.
Fig. 3.5 FE model of slab-on-girder bridge.
Damage assessment in structures using vibration characteristics 66
Table 3.1 Geometric and material properties for the test specimen
Flexural member Deck (2D) Girder (2D)
Element type Shell Shell
Material Steel Steel
Length 1800 mm 1800 mm
Width 1200 mm 3 mm
Depth 3 mm 300 mm
Poisson's ratio 0.3 0.3
Mass density 7800 kg/m3 7800 kg/m3
Modulus of elasticity 200 GPa 200 GPa
3.9.2 Instrument setup
The point loading is applied on the deck in the static test. The layout of the load
frame is shown in Fig. 3.6. The test rig of load frame consisted of a hydraulic jack
system is shown in Fig. 3.7. A calibrated hydraulic jack with a capacity of 3 tons is
used for exciting static point load at mid-span of the test specimen. The load applied
by the jack is transmitted through a load-cell with a sensitivity of 5.96 kN/volt. This
kind of loading system is the most common type of loading arrangement and is
favoured for laboratory experiments. The details of the field measurement
equipments are shown in Fig. 3.8. Two displacement measuring instruments, called
linear voltage displacement transducer (LVDT), each with sensitivity of 10.27
mm/volt are used for measuring the vertical static displacement at two points along
the centreline of the deck. The locations of applied load and sensors are shown in
Fig. 3.9. The LVDT and load cell are connected to a data acquisition system and the
data is recorded and stored automatically in a computer during loading.
3.9.3 Test methodology
For the static test, the point load is applied gradually to a maximum value of 6kN.
This maximum value is selected based on the capacity of the load cells and the
prediction of linearity behaviour of the steel. The two LVDTs located underneath the
deck at the designated locations are used to measure the static deflections. The data
Damage assessment in structures using vibration characteristics 67
acquisition system and a computer are used to capture and record the measured data.
All instruments are carefully calibrated before installation to minimize problems due
to instrument errors. In addition, two repeated tests are carried out to reduce the
measurement errors and ensure the consistency and reliability of the experiment.
Fig. 3.6 Layout of load frame.
(a) Vertical jack (b) Hydraulic pump
Fig. 3.7 Hydraulic jack system.
Damage assessment in structures using vibration characteristics 68
(a) Load cell (b) LVDT
(c) Data acquisition system
Fig. 3.8 Field measurement equipment used in static test.
Fig. 3.9 Loading position and LVDT layout on the deck.
Damage assessment in structures using vibration characteristics 69
3.9.4 Experimental results and discussions
The static test investigates the behaviour of the bridge model under the planned static
load. The deflection data at location ‘A’ in two repeated tests are plotted in Fig. 3.10.
The plot of load (kN) versus deflection (mm) indicates that the test specimen
behaves linearly up to the load of 6kN. Similar conclusions are drawn for the
deflection data at location ‘B’. An alternative plot, pairing the experimental and
analytical deflection at location ‘A’, is shown in Fig. 3.11. The plot shows a very
close correlation between experimental and analytical deflection and only small
deflection errors occur. From the test results, it can be concluded that the data
obtained from the experiment and FE model are in good agreement, which is within
an acceptable error. The small discrepancies between the experimental results and
the computational results are attributed to measurement error from instruments,
simulated supported conditions of the FE model, and assumptions made for FE
modelling. Since the accuracy of the FE model can be further improved by
increasing the mesh density, a convergence study is therefore conducted and the
result is shown in Fig. 3.12. It is found that the proposed FE model consisting 216
elements provides a reasonably good result (11.8mm) compared with a higher mesh
density (12mm). Since the test results compared well with the finite element results,
it is concluded that the computer model generated for the bridge is verified by the
static test.
Fig. 3.10 Plot of load vs deflection for the static test.
Damage assessment in structures using vibration characteristics 70
Fig. 3.11 Analytical deflection vs experimental deflection.
Fig. 3.12 Convergence of the static deflection at location ‘A’.
Damage assessment in structures using vibration characteristics 71
3.10 Free vibration test I: Slab-on-girder bridge
Upon completion of the static test, modal testing is performed on the bridge model
using dynamic testing equipment including an impact hammer, accelerometers, and a
data acquisition system. All these instruments are carefully inspected and calibrated
to ensure that they work effectively as intended. Details of the dynamic test are
discussed as follows. The major objective of the dynamic test is to obtain dynamic
properties, such as natural frequencies, mode shapes and damping for validation of
the finite element models. The dynamic test provides a check on differences of
primary modal parameters (e.g. natural frequency and the corresponding mode
shape) between the experimental result and calculated result from FEA. To minimise
the differences between the analytical and experimental modes, the initial FE models
of the bridge are calibrated with static and dynamic test data. As a result of this
calibration, all modal parameters, modelling assumptions, connections, and support
conditions of the finite element model are updated to represent the experimental
model as accurately as possible.
3.10.1 Description of the model
The slab-on-girder bridge model is tested before and after damage. The testing
procedure consists of measuring the dynamic properties in both the undamaged and
damaged models. Damage is induced by physically removing one of the boundary
supports (un-seating on bearings).
3.10.2 Instrument setup
The instruments used in the modal testing are shown in Fig. 3.13 and the hardware
components are presented schematically in Fig. 3.14. The experimental vibration
system consists of three main components; (i) impact hammer (ii) accelerometer (iii)
charge amplifier and data acquisition system. The impact hammer is used to provide
a source of excitation to the test specimen. The accelerometer is used to convert the
mechanical motion of the structure into an electrical signal. The charge amplifier is
used to match the characteristics of the transducer to the input electronics of the
Damage assessment in structures using vibration characteristics 72
digital data acquisition system. Software called "OriginPro8" is then used to execute
signal processing and modal analysis.
Fig. 3.13 Instrument setup in dynamic test.
Fig. 3.14 Schematic of the dynamic measurement system.
Damage assessment in structures using vibration characteristics 73
3.10.3 Test methodology
Free vibration is conducted on the test specimen to obtain its dynamic characteristics
including natural frequencies, mode shapes and damping ratios. The impact is
initially applied at the mid-span of the test specimen by using a hammer. During free
vibration, the dynamic responses of the bridge are measured through 2 uni-axial
piezoelectric accelerometers with nominal sensitivity of 10.33mV/g as shown in Fig.
3.15. As there are only two channels available for each test, a series of dynamic tests
with different locations of accelerometers is carried out in order to extract the modal
parameters. The layout of the sensors on the test specimen is depicted in Fig. 3.16.
The vertically mounted accelerometer at Grid B2 is used primarily for reference
purposes. A data acquisition system is used to store the record data and transfer
measured data to the PC for data post-processing. The sampling frequency is 500Hz
and the average sampling length is 1900 samples per channel. The modal parameters
for the undamaged and damaged state of the test specimen are extracted from the
measured acceleration using a commercial data analysis and graphing software
‘OrignPro8’. The modal identification method, namely Fourier Spectral Analysis
Method, which uses response-only measurements, is applied to the measured free
vibration data. The results of identified natural frequencies, mode shapes and
damping ratios are presented.
Fig. 3.15 Piezoelectric accelerometer.
Damage assessment in structures using vibration characteristics 74
Fig. 3.16 Measurement grid and accelerometer locations.
3.10.4 Experimental results and discussions
The natural frequencies and vibration mode shapes are determined based on the
Fourier Amplitude Spectra (FAS) of the multi-channel response data. Each output
channel of the bridge is subjected to a Fast Fourier Transform (FFT). As frequency
and damping are global properties, they do not vary across the structure, and can be
estimated from any frequency response measurement taken from the structure except
from measurements at any point where the mode shape has zero amplitude. The
natural frequencies of the bridge are identified as the frequencies corresponding to
FAS peaks present in the channels. The mode shape associated with an identified
natural frequency is obtained as the ratio of the magnitudes of the FAS peaks at the
various channels to the magnitude of the FAS peak at a reference channel.
Figs. 3.17 and 3.18 show a typical acceleration response and power spectrum density
respectively. After the data post-processing in MATLAB, the captured experimental
mode shapes are shown Fig. 3.19. The first five vibration modes of finite element
models for the slab-on-girder bridge and the slab alone are plotted in Figs. 3.20 and
Damage assessment in structures using vibration characteristics 75
3.21 respectively. It is evident that the experimental and FEA results compare well.
Moreover, from the FEA results, it appears that the vibration response of the bridge
is governed by vertical bending modes, coupled with torsional modes, in the
frequency range of 11 - 26 Hz. The fundamental mode is the vertical bending mode
of the deck with lateral vibration of the girder and corresponds to a natural frequency
of 11.06 Hz. It can be seen that all modes involved both slab and girder vibrations
and most include coupled vertical bending and torsional modes of slab and girders.
Fig. 3.17 Typical acceleration time history.
Fig. 3.18 Typical power spectrum density plot.
Damage assessment in structures using vibration characteristics 76
(a) Mode 1, f1=10.74Hz
(b) Mode 3, f3=18.57Hz
(c) Mode 4, f4=24.56Hz
Fig. 3.19 Experimentally obtained vibration modes of undamaged deck.
Damage assessment in structures using vibration characteristics 77
(a) Mode 1, f1=11.06Hz
(d) Mode 4, f4=23.67Hz
(b) Mode 2, f2=17.58Hz
(e) Mode 5, f5=25.54Hz
(c) Mode 3, f3=18.01Hz
Fig. 3.20 First five vibration modes of undamaged slab-on-girder bridge (FEM).
Damage assessment in structures using vibration characteristics 78
(a) Mode 1, f1=11.06Hz
(d) Mode 4, f4=23.67Hz
(b) Mode 2, f2=17.58Hz
(e) Mode 5, f5=25.54Hz
(c) Mode 3, f3=18.01Hz
Fig. 3.21 First five vibration modes of undamaged deck (FEM).
From the free vibration testing, three vibration modes (modes 1, 3 and 4) are
captured instead of five modes. This is due to the fact that the computed frequency
from the finite element model in modes 2 and 3 are close to each other, with a value
of 17.58 Hz and 18.01Hz respectively; therefore the measuring equipments and data
analyser are not sensitive enough to distinguish these two modes. In addition, the
instruments are not able to capture the high mode, which is mode 5. Comparisons
between the natural frequencies obtained by the free vibration measurement and
finite element analysis are listed in Table 3.2. It is observed that the differences
between the measured and computed natural frequencies of the coupled bending and
torsional modes before updating reached a maximum value of 7.5%, indicating the
need of updating the preliminary finite element model. As the measured frequency is
Damage assessment in structures using vibration characteristics 79
higher than the corresponding analytical frequency in modes 1, 3 and 4, it indicates
that the actual structure is stiffer than the initial finite element model. Overall, it is
concluded that the measured and computed natural frequencies are in good
agreement as evidenced by the plot of the analytical frequency against experimental
frequency as shown in Fig. 3.22.
To quantify the correlation between measured mode shapes and analytical mode
shapes in free vibration, modal assurance criterion (MAC) is calculated by using Eq.
(2.21). MAC values vary from 0 to 1, with 0 for no correlation and 1 for full
correlation. MAC values in undamaged and damaged cases are listed in Table 3.3,
with those in damaged cases are listed within brackets. It is found that the identified
experimental mode shapes correlate well with the corresponding analytical mode
shapes. As there is good agreement between two sets of data, this provides further
confidence that each mode identified from the experimental data is a unique pattern
of motion for the test specimen.
The bandwidth method (half-power) is used to determine the damping ratio of the
test specimen experimentally. The damping ratio is calculated by using Eq. 3.16. The
modal damping ratios of modes 1, 3 and 4 are listed in Table 3.4. Overall, it is found
that the damping ratio in damaged state is slightly increased comparing with the
corresponding one in undamaged state.
Table 3.2 Correlation between experimental and initial FE model
Mode Undamaged Damaged
Modal
testing
(Hz)
Initial
FEM
(Hz)
Frequency
difference
(%)
Modal
testing
(Hz)
Initial
FEM
(Hz)
Frequency
difference
(%)
1 10.74 11.06 2.89 10.22 10.54 3.04
3 18.57 18.01 3.11 15.82 16.84 6.06
4 24.56 23.67 3.76 18.07 19.53 7.48
Damage assessment in structures using vibration characteristics 80
Fig. 3.22 Plot of analytical vs experimental natural frequencies.
Table 3.3 MAC using experimental and analytical data in undamaged cases
(Damaged cases are listed within brackets)
Analytical data
Mode 1 Mode 3 Mode 4
Experimental
data
Mode 1 0.92 (0.94) 0.18 (0.20) 0.21 (0.20)
Mode 3 0.12 (0.12) 0.86 (0.82) 0.16 (0.13)
Mode 4 0.15 (0.17) 0.24 (0.21) 0.91 (0.90)
Table 3.4 Estimated damping ratio by half-power method
Mode no. Undamaged state
(%)
Damaged state
(%)
1 0.26 0.28
3 0.23 0.24
4 1.24 1.25
After modal analysis, the measured natural frequencies and associated mode shapes
obtained from the free vibration testing are used to calculate the modal flexibility
change and modal strain energy based damage index as shown in Figs. 3.23 and 3.24
respectively. It is found that there is a distinct peak at the end-support of the right
girder, which conforms well with the damage scenario (un-seating of bearings).
Damage assessment in structures using vibration characteristics 81
Therefore, it is concluded that the MFC and MSEC are both competent o localize
damages in slab-on-girder bridge, and this provides further confidence on damage
identification of structures using multi-criteria approach.
Fig. 3.23 Modal flexibility change on girders based on experimental data in damage scenario.
Fig. 3.24 Modal strain energy based damage index on girders based on experimental data in damage scenario.
Damage assessment in structures using vibration characteristics 82
3.10.5 Model updating
In this study, manual tuning technique is used in the mode updating process to adjust
the structural parameters of the finite element model such that the error between the
identified critical experimental parameters and the numerical model is minimised.
Initially, the finite element model of the test specimen is generated in which the
corresponding parameters in the model (Young’s modulus, member geometry,
connectivity, conditions at supports) are chosen as best as possible for an initial
analysis. With the measured vibration data of the test structure, the initial FE model
is updated to match the measured vibration properties as closely as possible. As the
static test shows a good correlation between the experimental deflection and
analytical deflection, only those parameters that will most directly affect the dynamic
responses are considered in the manual tuning. For the initial FE model as shown in
Fig. 3.25(a), shell elements are used to model the deck and girders, while truss
elements are used for diagonal bracings, which do not fully simulate the realistic
connection of the test structure as shown in Fig. 3.25(b). Therefore, four rectangular
shell elements as shown in Fig. 3.25(c) are added to the initial FE model to simulate
the gusset plates which connect the diagonal bracings with the deck and girders. The
objective of this updating is to improve simulation of the boundary conditions,
continuity conditions and structural geometry. The first five vibration modes of
updated finite element models are plotted in Fig. 3.26. Comparisons of measured and
analytical natural frequencies are made to assess the effectiveness of this model
updating. Comparing the results between Table 3.2 (initial FE model) and Table 3.5
(tuned FE model), it is found that the frequency difference between experimental and
analytical data is improved after the updating process. It can be concluded that
manual tuning leads to a good correlation between the experimental and analytical
models, and the resulting FE model can be used in further analyses with significant
confidence.
Damage assessment in structures using vibration characteristics 83
Table 3.5 Correlation between experimental and manually tuned FE models
Mode Undamaged Damaged
Modal
testing
(Hz)
Tuned
FEM
(Hz)
Frequency
difference
(%)
Modal
testing
(Hz)
Tuned
FEM
(Hz)
Frequency
difference
(%)
1 10.74 10.96 2.05 10.22 10.60 3.72
3 18.57 18.05 2.80 15.82 16.66 5.31
4 24.56 23.66 3.66 18.07 19.24 6.47
(a) Initial FEM
(b) Test specimen
(c) Updated FEM (Additional 4 shell elements)
Fig. 3.25 Finite element model updating.
Damage assessment in structures using vibration characteristics 84
1st bending and torsional mode 4th bending and torsional mode
(a) Mode 1, f1=11.28Hz (d) Mode 4, f4=25.37Hz
2nd bending and torsional mode 5th bending and torsional mode
(b) Mode 2, f2=17.88Hz (e) Mode 5, f5=26.97Hz
3rd bending and torsional mode
(c) Mode 3, f3=18.91Hz
Fig. 3.26 First five vibration modes of undamaged slab-on-girder bridge after model
updating
Damage assessment in structures using vibration characteristics 85
3.11 Free vibration test II: Simply supported beam
In another dynamic test, free vibration is conducted on a steel beam to obtain the
natural frequency of the first few modes and to validate the FE model. The simply
supported steel beam is shown in Figs. 3.27 and 3.28. The geometry and material
properties of the test beam are listed in Table 3.6. The undamaged beam is first
excited by an impact hamper and the dynamic responses are measured by an
accelerometer fixed at the mid-span of the beam as shown in Fig. 3.29. A software
known as "SignalCalc ACE Dynamic Signal Analyser" (which was available) is used
to extract the pre-damage modal parameters. The test beam is then cut at mid-span
with the flaw size (10mmx5mmx40mm) as shown in Fig. 3.30 and the testing is
repeated to extract the post-damage modal parameters. The corresponding flaw size,
which is simulated in the FE model, is shown in Fig. 3.31. As a low frequency range
of accelerometer is used in the measurement of dynamic responses, only the two
lowest natural frequencies are captured from the experiments. Figs. 3.32 and 3.33
show the measured frequency for undamaged and damaged beam respectively. There
is no change of frequency for mode 2 in FEM because the damage elements are
located at the nodes of vibration modes. The obtained experimental results are
compared with those from FE analysis to validate the FEM. It is noted that the
experimental and FE results show good agreement as listed in Table 3.7.
Table 3.6 Geometric and material properties of the beam
Element type 2D Geometry type Plane stress
Material Isotropic
Width 40mm
Depth 20mm
Span 2.8m
Boundary condition Simply supported
Poisson's ratio 0.3
Mass density 7850kg/m3
Modulus of elasticity 200GPa
Damage assessment in structures using vibration characteristics 86
Fig. 3.27 Simply supported beam.
Fig. 3.28 Boundary condition of the beam.
Fig. 3.29 Accelerometer on the beam.
Damage assessment in structures using vibration characteristics 87
Fig. 3.30 Flaw at mid-span of the beam.
Fig. 3.31 Flaw size (10mmx5mmx40mm) in the FE model.
(a) Mode 1, f1=5.94Hz (b) Mode 2, f2=24.38Hz
Fig. 3.32 Measured natural frequency of the undamaged beam.
(a) Mode 1, f1=5.63Hz (b) Mode 2, f2=23.13Hz
Fig. 3.33 Measured natural frequency of the damaged beam.
Damage assessment in structures using vibration characteristics 88
Table 3.7 Validation of the FEM for the simply supported beam
State Frequency
mode
Experiment
(Hz)
FEM
(Hz)
Difference
(%)
Undamaged ƒ1 5.94 5.84 1.7
ƒ2 24.38 23.33 4.3
Damaged at
mid-span
ƒ1* 5.63 5.65 0.4
ƒ2* 23.13 23.33 0.9
3.12 Summary
The free vibration of an elastic structure (as a single or multiple degree-of-freedom
system) is described by homogeneous dynamic equilibrium equation. This equation,
which relates the effects of the mass, stiffness and damping, leads to the calculation
of natural frequencies, mode shapes and damping factors of the structure through
Eigen value analysis. These obtained modal parameters, which will be utilized in
multi-criteria approach (changes in frequencies, modal flexibility and modal strain
energy), form the basis for damage assessment in structures in Chapters 4 and 5. The
theories underlying these two parameters of modal flexibility and model strain
energy are developed and presented for the beam, plate and truss elements. In
addition, two experiments: (i) static test and (ii) free vibration test are carried out on
the test specimen to calibrate and validate the finite element model of the slab-on-
girder bridge. The results of the static test conducted on the test structure show that
the structure behaved linearly as predicted by the finite element model. Good
correlation between experimental deflection and analytical deflection is obtained.
The free vibration test is conducted on the model bridge structure to evaluate its
vibration characteristics. Three vibration modes of natural frequencies and
corresponding mode shapes of slab-on-girder bridge are successfully identified using
the Fast Fourier Transform. Good agreement is obtained between the experimental
and analytical modal properties by assessing the error of natural frequencies and also
the modal assurance criterion. The bandwidth method is used to estimate the
damping ratio experimentally to provide an indication of the level of damping
present in the free vibration. To further improve the accuracy of the finite element
model which represents the test structure, manual tuning is carried out. This model
Damage assessment in structures using vibration characteristics 89
updating process involves changing the connectivity of the initial finite element
model to simulate the boundary conditions of the real structure. The validated finite
element model then provides further confidence for vibration-based damage
detection applications, which are treated in later chapters.
Damage assessment in structures using vibration characteristics 90
Chapter 4 Application I - Load Bearing Elements of Structures
4.1 Introduction
This chapter and next chapter present a non-destructive multi-criteria approach
which incorporates (i) changes of frequency, (ii) changes of modal flexibility and
(iii) modal strain energy based damage index to detect and locate damage in some of
the main load bearing elements of structures. The validity of the proposed approach
is demonstrated using numerical models calibrated by experimental data. Two load
bearing elements of bridges, viz beams and slab (plate) are treated in this chapter.
The next chapter will deal with two types of bridges (i) slab-on-girder bridge and (ii)
truss bridge. The geometric and material properties, boundary condition and element
types used in the computer models of the beam and plate (slab) structures are
described in detail. Different damage scenarios (e.g. changes of damage location and
severity) are introduced into the numerical models to assess the performance of the
damage identification techniques. Free vibration analyses of the beam and plate
structures both in their healthy (intact) states and under the selected damage
scenarios are carried out. First, natural frequencies obtained from the undamaged and
damaged state of models are used as a reference and for damage alarming in the
proposed multi-criteria technique. This approach for level 1 in structural health
monitoring (SHM) or alarming is based on the fact that natural frequencies are
sensitive indicators to structural integrity, and therefore damage existence causes
changes of frequency. The changes of modal flexibility and modal strain energy
based damage index, are then evaluated using the first five modes of modal
parameters (natural frequencies and mode shapes) and used to locate damages.
Distinguishing peaks over a threshold level in the plots indicate the location of
damage. The findings of the proposed multiple criteria approach in the numerical
Damage assessment in structures using vibration characteristics 91
examples are summarised and a general flowchart with the sequences of the
vibrated-based damage detection approach is provided.
4.2 Damage assessment in beam
4.2.1 Model description
Finite element models (FEM) of the undamaged and damaged simply supported
beams, tested previously, are generated using the FE software SAP2000. Plane
elements are used for the FE modelling. The details of the beam are given in Table
4.1. The flexural rigidity EI is assumed constant over the beam span and damping
effect is not taken into account. Modal analysis is performed to obtain the natural
frequencies and the associated mode shapes of the beam. The free vibration test on
simply supported beam is carried out to validate the FE model. The obtained
experimental results are compared with those from FE analysis, and it is found that
they are in good agreement as evident in Chapter 3. This provides confidence in the
FE modelling and analysis of other beam models. Further FE modelling and
analysis are carried out on 2-span and 3-span continuous beam structures to extract
the modal parameters. All continuous beams have the same span length of 2.8m and
are simply supported at their ends, similar to the validated single span beam model.
To simulate damage, the selected plane elements are removed from the bottom of the
beams in the FE models. Nine such damage cases are investigated with two different
sizes of flaws as listed in Table 4.2, in which size B flaw represents larger damage
than size ‘A’ flaw. Parametric studies are carried out to investigate the feasibility of
the multi-criteria damage detection approach on changes of parameters, such as
damage severity and locations. Fig. 4.1 shows the first three damage scenarios in a
single-span beam. In damage cases D1 and D2, a single damaged element is
simulated on the beam at the mid-span with different damage severity to observe the
changes of frequency, changes of modal flexibility and modal strain energy based
damage index corresponding to the damage severity. The severity of damage
increases in damage case D3 compared with D1 and D2, as two damaged elements
are simulated on the beam, one located at the mid-span and the other at quarter-span.
Damage assessment in structures using vibration characteristics 92
The other six damage scenarios in the 2-span and 3-span beams with different
damage severity and locations are shown in Figs. 4.2 and 4.3 respectively. Fig. 4.4
shows the details of flaw size ‘A’ simulated in the FE model.
Table 4.1 Geometric and material properties of beam
Flexural member Beam (2D)
Element type Plane stress
Material Steel
Length 2.8 m
Width 40 mm
Depth 20 mm
Poisson's ratio 0.3
Mass density 7850 kg/m3
Modulus of elasticity 200 GPa
Table 4.2 Dimension of flaws in beam
Size Length (mm) Depth (mm) Width (mm)
A 10 5 40
B 20 5 40
(a) D1
(b) D2
(c) D3
Fig. 4.1 Damage case (D) for single-span beam (2.8m span length).
Damage assessment in structures using vibration characteristics 93
(a) D4
(b) D5
(c) D6
Fig. 4.2 Damage case (D) for 2-span beam (2.8m span length).
(a) D7
(b) D8
(c) D9
Fig. 4.3 Damage case (D) for 3-span beam (2.8m span length).
Damage assessment in structures using vibration characteristics 94
Fig. 4.4 Flaw size ‘A’ simulated in FEM.
4.2.2 Frequency change
The natural frequencies of the first five modes of the beam before and after damage
in nine damage scenarios obtained from the results of the FEA are shown in Tables
4.3 and 4.4 respectively. Percentage changes in the natural frequencies between the
undamaged and damaged states are listed within brackets. The first five vibration
modes of undamaged FE model are plotted in Fig. 4.5. It is observed that the
presence of damage in beams causes a decrease in the natural frequencies in all
damage cases, with very few exceptions. If the damage cases D1 and D2 for the
single-span beam are considered, change (i.e. decrease) in the frequency ∆ f is
evident for the 1st, 3rd and 5th modes and there is no change for the 2nd and 4th
modes. This is because the damage elements are located at the nodes of these anti-
symmetric modes of vibration and hence have no influence on the corresponding
natural frequencies. By observing the changes in natural frequency of the first five
modes, it is possible to achieve the Level 1 of identification that damage might be
present in the beam-like structure.
4.2.3 Modal flexibility change
The first five natural frequencies and associated mode shapes obtained from the
results of the FE analysis are used to calculate the MFC by using Eqs. (3.31) and
(3.33). The plots of MFC as a percentage along the beam for some representative
cases are shown in Figs. 4.6(a)-(d). In all cases, the peak values correctly indicate the
location of damage in the beams. Figs. 4.6(a) and (b) show the results for single
damage cases and it is evident that the peak for the more severe damage case, D2, is
higher than that for case D1. In Fig. 4.6(c) there are two un-equal peaks
corresponding to the 2 different damages in this beam and once again it is seen that
Damage assessment in structures using vibration characteristics 95
greater damage in the beam attracts a greater peak in the MFC. Finally, Fig. 4.6(d)
clearly shows that this damage case with triple damages has three distinct peaks in
the MFC. The other damage cases showed analogous results and further confirm the
feasibility of MFC in locating damage in a beam structure under a variety of damage
scenarios. From the above observations, it is evident that the damage detection
algorithm for MFC is able to locate the damage in the beam structure correctly in all
damage cases and also give an indication of its severity. This confirms that the
modal flexibility method is sufficiently sensitive to the damages in the beams.
4.2.4 Modal strain energy change
The first five mode shapes and their corresponding mode shape curvatures obtained
from the results of the FE analysis are used to calculate the modal strain energy
based damage index on beams by using Eqs. (3.56) and (3.57). The plot of damage
indices along the beam for damage cases D1, D2, D5 and D6 are shown in Figs.
4.6(e)-(h). The spikes with magnitudes greater than 1 indicate the location of
damaged elements. Comparison of Figs. 4.6(e) and (f) shows that the peak in the
damage index increases with the severity of damage. The peaks in Figs. 4.6(g) and
(h) clearly indicate the multiple damages in the beam. From the results of all cases, it
is evident that the damage index on the modal strain energy method is able to
correctly locate the damage in beams in all damage cases.
Table 4.3 Natural frequencies of undamaged beam from FEM
Member
type
Boundary
condition
Mode 1
ƒ1 (Hz)
Mode 2
ƒ2 (Hz)
Mode 3
ƒ3 (Hz)
Mode 4
ƒ4 (Hz)
Mode 5
ƒ5 (Hz)
Beam
SS (1-span) 5.84 23.33 52.45 93.10 145.16
SS (2-span) 5.84 9.12 23.33 29.52 52.45
SS (3-span) 5.84 7.48 10.92 23.33 26.58
Note: SS means simply supported.
Damage assessment in structures using vibration characteristics 96
Table 4.4 Natural frequencies of damaged beam from FEM
(Percentage changes wrt the undamaged conditions are listed within brackets)
Damage
case
Mode 1
ƒ1 (Hz)
Mode 2
ƒ2 (Hz)
Mode 3
ƒ3 (Hz)
Mode 4
ƒ4 (Hz)
Mode 5
ƒ5 (Hz)
D1 5.80 (0.68) 23.33 (0.00) 52.08 (0.71) 93.10 (0.00) 144.13 (0.71)
D2 5.77 (1.20) 23.33 (0.00) 51.90 (1.05) 93.10 (0.00) 143.65 (1.04)
D3 5.74 (1.71) 23.08 (1.07) 51.62 (1.58) 93.10 (0.00) 142.95 (1.52)
D4 5.82 (0.34) 9.10 (0.22) 23.33 (0.00) 29.51 (0.03) 52.26 (0.36)
D5 5.78 (1.03) 9.07 (0.55) 23.33 (0.00) 29.49 (0.10) 51.99 (0.88)
D6 5.77 (1.20) 9.05 (0.77) 23.25 (0.34) 29.39 (0.44) 51.90 (1.05)
D7 5.80 (0.68) 7.43 (0.67) 10.90 (0.18) 23.33 (0.00) 26.57 (0.04)
D8 5.79 (0.86) 7.45 (0.40) 10.86 (0.55) 23.28 (0.21) 26.48 (0.38)
D9 5.77 (1.20) 7.42 (0.80) 10.86 (0.55) 23.16 (0.73) 26.42 (0.60)
Note: Changes of natural frequencies result in decrease in all damage cases.
1st bending mode 4th bending mode
(a) Mode 1, f1=5.84Hz (d) Mode 4, f4=93.10Hz
2nd bending mode 5th bending mode
(b) Mode 2, f2=23.33Hz (e) Mode 5, f5=145.16Hz
3rd bending mode
(c) Mode 3, f3=52.45Hz
Fig. 4.5 First five vibration modes of undamaged FE model.
Damage assessment in structures using vibration characteristics 97
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8
Position along beam (m)
∆ F
(%
)
0.99
1
1.01
1.02
1.03
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
Position along beam (m)
β
(a) Plot of MFC (%) in D1 (e) Plot of damage index in D1
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8
Position along beam (m)
∆ F
(%
)
0.99
1
1.01
1.02
1.03
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
Position along beam (m)
β
(b) Plot of MFC (%) in D2 (f) Plot of damage index in D2
(c) Plot of MFC (%) in D5 (g) Plot of damage index in D5
(d) Plot of MFC (%) in D6 (h) Plot of damage index in D6
Fig. 4.6 Modal flexibility change (left) and Modal strain energy based damage index (right)
on beam.
Damage assessment in structures using vibration characteristics 98
4.3 Damage assessment in plate (slab)
4.3.1 Model description
A rectangular steel plate with the size of 2.5m in length, 1m in width and 2mm in
thickness is chosen for the numerical analysis. The material properties of the steel
plate are listed in Table 4.5. The steel plate is divided into 250 plate elements. FE
techniques are used to carry out modal analysis of the structure. Initially, a FE model
of the steel plate clamped along all four edges is analysed and the first three natural
frequencies and the associated mode shapes of the plate obtained from the modal
analysis are compared with those provided by Ulz and Semercigil (2008). The two
sets of results are in good agreement as seen in Table 4.6, providing adequate
confidence in the present FE modelling and analysis of plate structures. Additional
FE models of single and two span plate structures are developed and their modal
analysis is carried out before and after damage. Damage is simulated by reducing the
elastic modulus (E) to 80% and 50% in selected elements as shown in Figs. 4.7-4.9.
An assumption is made that the mass of the plate does not change appreciably as a
result of the damage. No structural damping is used in the FE analysis. Nine damage
cases are investigated in this study. Among these cases, three different boundaries
conditions in addition to different damage severity of selected elements are
simulated to investigate the feasibility and capability of the multi-criteria damage
detection approach. Fig. 4.7 shows three damage scenarios for the plate with all
edges clamped. Figs. 4.8 and 4.9 show the other six damage scenarios for simply
supported plates with single and 2 spans respectively.
Table 4.5 Geometric and material properties of plate
Flexural member Plate
Material Steel
Length 2.5 m
Width 1 m
Depth 2 mm
Poisson's ratio 0.3
Mass density 7800 kg/m3
Modulus of elasticity 210 GPa
Damage assessment in structures using vibration characteristics 99
Table 4.6 Validation of FEM for plate with clamped boundaries
Structural
state
Frequency
ƒi
From Reference
(Ulz and Semercigil, 2008)
(Hz)
From
SAP2000
(Hz)
Difference
(%)
Undamaged
ƒ1 11.81 11.78 0.3
ƒ2 13.89 13.77 0.8
ƒ3 17.68 17.44 1.4
(a) D1 (0.8E)
(b) D2 (0.5E)
(c) D3 (0.8E)
Fig. 4.7 Damage case (D) for plate with all edges clamped.
(a) D4 (0.8E)
Damage assessment in structures using vibration characteristics 100
(b) D5 (0.5E)
(c) D6 (0.5E)
(d) D7 (0.8E)
(e) D8 (0.8E)
Fig. 4.8 Damage case (D) for simply supported plate.
(a) D9 (0.8E)
Fig. 4.9 Damage case (D) for 2-span plate.
Damage assessment in structures using vibration characteristics 101
4.3.2 Frequency change
The natural frequencies of the first five modes of the plates before and after damage
obtained from the results of the FE analysis in nine damage scenarios are shown in
Tables 4.7 and 4.8 respectively. Percentage changes in the natural frequencies
between the undamaged and damaged conditions are listed within brackets. The first
five vibration modes of undamaged FE model are plotted in Fig. 4.10. It can be
observed that in general the presence of damage in the plate causes a small decrease
in the natural frequencies in all damage cases, with very few exceptions. If the
damage cases D1 and D2 for the plate are considered, change (i.e. decrease) in the
frequency ∆f is evident for the 1st, 3rd and 5th modes, while there is no change for
the 2nd and 4th modes. This is because the damage elements are located at the nodes
of these anti-symmetric modes of vibration and hence have no influence on the
corresponding natural frequencies. It may be concluded that by observing the
changes in the natural frequencies, it is more possible to achieve Level 1 of
identification of macro-damage in plate, rather than of micro-damage or small
damage. The detection of small damage can be supplemented by advanced
techniques such as acoustic emission monitoring.
Table 4.7 Natural frequencies from FEM for undamaged plate
Member
Type
Boundary condition
(no. of span)
Mode 1
ƒ1 (Hz)
Mode 2
ƒ2 (Hz)
Mode 3
ƒ3 (Hz)
Mode 4
ƒ4 (Hz)
Mode 5
ƒ5 (Hz)
Plate
Edges clamped
(1-span)
11.78 13.77 17.44 22.91 30.15
Simply supported
(1-span)
0.76 2.66 3.07 5.95 6.95
Simply supported
(2-span)
3.04 4.90 5.88 7.36 12.19
Damage assessment in structures using vibration characteristics 102
1st bending mode 2nd bending and torsional mode
(a) Mode 1, f1=0.76Hz (d) Mode 4, f4=5.95Hz
1st bending and torsional mode 3rd bending mode
(b) Mode 2, f2=2.66Hz (e) Mode 5, f5=6.95Hz
2nd bending mode
(c) Mode 3, f3=3.07Hz
Fig. 4.10 First five vibration modes of undamaged plate with simply supported
condition.
Damage assessment in structures using vibration characteristics 103
Table 4.8 Natural frequencies from FEM for damaged plate
(Percentage changes wrt to the undamaged conditions are listed within brackets)
Damage
case
Mode 1
ƒ1 (Hz)
Mode 2
ƒ2 (Hz)
Mode 3
ƒ3 (Hz)
Mode 4
ƒ4 (Hz)
Mode 5
ƒ5 (Hz)
D1 11.77 (0.14) 13.77 (0.01) 17.42 (0.12) 22.91 (0.00) 30.11 (0.15)
D2 11.74 (0.39) 13.77 (0.01) 17.38 (0.38) 22.91 (0.02) 30.02 (0.43)
D3 11.77 (0.14) 13.77 (0.02) 17.41 (0.15) 22.90 (0.04) 30.10 (0.18)
D4 0.76 (0.08) 2.66 (0.00) 3.07 (0.00) 5.94 (0.06) 6.95 (0.08)
D5 0.76 (0.24) 2.66 (0.00) 3.07 (0.00) 5.94 (0.17) 6.94 (0.23)
D6 0.76 (0.13) 2.66 (0.04) 3.06 (0.08) 5.94 (0.09) 6.95 (0.11)
D7 0.76 (0.11) 2.66 (0.15) 3.07 (0.07) 5.93 (0.19) 6.94 (0.18)
D8 0.76 (0.13) 2.66 (0.04) 3.06 (0.08) 5.94 (0.09) 6.95 (0.11)
D9 3.03 (0.16) 4.90 (0.12) 5.87 (0.10) 7.36 (0.10) 12.18 (0.08)
4.3.3 Modal flexibility change
Plots of MFC for damage cases D1, D2 and D9 are shown in Figs. 4.11(a), (b) and
(d) respectively. To optimise the damage detection results for the plate structure, the
plot of MFC for damage case D8, as shown in Fig. 4.11(c), is expressed as a
percentage with respect to the undamaged modal flexibility matrix. The peak values
indicate the location of damage in the plate. Comparison of Figs. 4.11(a) and (b)
pertaining to single damage detection in a plate, shows that as the severity of the
single damage at mid-span increases, the corresponding MFC also increases, as
demonstrated by the higher peak. For the case of multiple damage detection in
damage case D9, the modal flexibility method is able to correctly locate the damage.
The other damage cases showed analogous results and further confirm the feasibility
of MFC in locating damage in a beam structure under a variety of damage scenarios.
However for multiple damage case D8, the results in Fig. 4.11(c) do not clearly
indicate the damage locations and the damage indicator has partly missed the
damage at the mid-span of the plate. Overall, the results show that the modal
flexibility method is able to correctly locate the damage in most multiple damage
cases, except in cases D7 and D8 where the damage indicator seems to have partly
missed the damage at the mid-span of the plate. Similar to damage at nodes of
Damage assessment in structures using vibration characteristics 104
vibrating modes not influencing the corresponding natural frequencies, this feature
further demonstrates the need for multi-criteria damage assessment.
4.3.4 Modal strain energy change
The modal strain energy based damage index for plate is calculated by using Eqs.
(3.66) and (3.67). The plot of damage indices for damage cases D1, D2, D8 and D9
are shown in Figs. 4.11(e)-(h) respectively. The spikes with the magnitudes greater
than 1 indicate the location of damaged elements. The peak in Fig. 4.11(f)
corresponding to a more severe damage case is higher than the peak in Fig. 4.11(e).
Multi-peaks in Figs. 4.11(g) and (h) indicate the locations of multiple damages
correctly in the plate. Overall, the results show that the strain energy method is
capable of detecting multiple damages in plates for all damage cases.
From the extensive numerical analyses, the performance of proposed damage
detection methods for beam and slab(plate) are summarised in Table 4.9.
Table 4.9 Performance of damage detection algorithms for beam and plate
Damage case Beam Plate
MFC MSEC MFC MSEC
D1 � � � �
D2 � � � �
D3 � � � �
D4 � � � �
D5 � � � �
D6 � � � �
D7 � � �* �
D8 � � �* �
D9 � � � �
Note: �= accurate damage localization
* = partially successful damage localization
X = damage indication failure
Damage assessment in structures using vibration characteristics 105
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
2.1
2.3
2.5
1.0
0.50
2
4
6
8
∆ F
X
Y
x10-9
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
2.1
2.3
2.5
1.0
0.51
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
β
X
Y
(a) Plot of MFC in D1 (e) Plot of damage index in D1
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
2.1
2.3
2.5
1.0
0.50
2
4
6
8
10
12
14
16
18
20
∆ F
X
Y
x10-9
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
2.1
2.3
2.5
1.0
0.51
1.5
2
2.5
3
3.5
4
4.5
β
X
Y
(b) Plot of MFC in D2 (f) Plot of damage index in D2
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
2.1
2.3
2.5
1.0
0.50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
∆ F
(%
)
X
Y
x10-6
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
2.1
2.3
2.5
1.0
0.51
1.05
1.1
1.15
1.2
1.25
1.3
β
X
Y
(c) Plot of MFC (%) in D8 (g) Plot of damage index in D8
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
2.1
2.3
2.5
1.0
0.50
10
20
30
40
50
60
∆ F
X
Y
x10-9
0.1
0.3
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
2.1
2.3
2.5
1.0
0.51
1.05
1.1
1.15
1.2
1.25
β
X
Y
(d) Plot of MFC in D9 (h) Plot of damage index in D9 Fig. 4.11 Modal flexibility change (left) and Modal strain energy based damage index (right) on plate.
Damage assessment in structures using vibration characteristics 106
4.4 Summary
The main findings can be summarized as follows:
From the extensive numerical examples, it is observed that the change of frequency
is less than 1% in most damaged cases. Based on the results of frequency sensitivity
studies in all damage cases, it is concluded that change of frequency method (Level
1) is suitable to detect macro-damage in proposed structures rather than micro-
damage or small damage. For reference purpose, the size of structural damage can be
approximately divided into three levels: (1) micro-damage, i.e., damage size is
smaller than 0.1% of structural size; (2) small-damage, i.e., damage size is about 1%
of structural size; (3) macro-damage, i.e., damage size is greater than 10% of
structural size. To develop a comprehensive and reliable structural health monitoring
system with both global and local assessment capability, the detection of the local
small damage can be supplemented by the advanced techniques, e.g. acoustic
emission monitoring.
In beam example, two types of damage severity with damage size of (10mm x 5mm
x 40mm) and (20mm x 5mm x 40mm) are simulated at selected damage location(s).
The single and continuous span (2-span and 3-span) of beam with simply supported
condition are investigated. The result shows that both modal flexibility change
method (MFC) and modal strain energy method (MSEC) are effective to detect
single and multiple damages. It is found that the two methods are sensitivities to the
damages in both single span and multiple span of beam. Alternatively, in the case of
single damage on single-span beam, it is evidenced that the two methods are able to
quantify the damage severity (extent) based on the magnitude of damage index. It is
concluded that Level-3 damage quantification is achieved only in cases of single
damage. The value of the peak in the plot is usually an indication of damage severity
in cases of multiple damages, but this needs further research.
In plate example, two types of damage severity with flexural stiffness reduction of
80% and 50% are simulated at selected damage location(s). The single-span plate
with two boundary conditions (all edges clamped and simply supported), and also 2-
Damage assessment in structures using vibration characteristics 107
span plate with simply supported are investigated. The results show that MSEC is
capable of detecting single and multiple damages in all damage cases, while MFC is
capable of detecting all cases of single damage but shows less effective in cases of
multiple damages as evidenced in example. It is concluded that MSEC is more
competent than MFC to localize damages in plate. Again, the value of the peak in the
plot is usually an indication of damage severity in cases of multiple damages, but
this needs further research. The next chapter will treat the damage assessment in
complete bridge structures – (i) slab on girder bridge and (ii) truss bridge.
Damage assessment in structures using vibration characteristics 108
Chapter 5 Application II - Bridges
5.1 Damage assessment in slab-on-girder bridge
5.1.1 Model description
The superstructure used as the basis for the damage assessment is a zero-skew, single
span slab-on-girder bridge with 4m wide deck consisting of two steel plate girders
spanning 20m. The spacing between twin-girders is 3m and the thickness of concrete
deck is 200mm. To provide the lateral restraint required for the development of
transverse bending stiffness of the slab and the stability for twin-girders, steel
diagonal bracings are installed between girders at spacing of 1m. The general
modelling scheme for bridge is depicted in Fig. 5.1. The geometric and material
properties for the bridge are listed in Table 5.1.
Both bridge deck and girders are modelled as shell elements. The deck and each
girder are divided into 160 and 80 elements respectively. Cross bracings between
girders are modelled as truss elements. Shell elements are widely used to idealize the
bridge deck since behaviour of this structural component is governed by flexure and
in this case a mesh of shell elements is computationally more efficient when
compared to one of solid elements. It is assumed that there is a complete connection
between the girders and slab. Twin-girders having the same span are simply
supported at their ends and rotations about all 3 axes are allowed in order to simulate
the desired boundary conditions.
Damage assessment in structures using vibration characteristics 109
Fig. 5.1 Isometric view of FE model.
Table 5.1 Geometric and material properties for the slab-on-girder bridge
Flexural member Deck (2D) Girder (2D)
Element type Shell Shell
Material Concrete Steel
Length 20 m 20 m
Width 4 m 8 mm
Depth 200 mm 1.5 m
Poisson's ratio 0.2 0.3
Mass density 2400 kg/m3 7800 kg/m3
Modulus of elasticity 24 GPa 200 GPa
A total of 7 damage cases are investigated for the damage identification of this
bridge. The first three damage cases involve deck damage only and the last four
cases girder(s) damage only as shown in Figs. 5.2 and 5.3, respectively. Damage on
deck is simulated by reducing the elastic modulus (E) of selected elements, while
damage on the girder is simulated by removing selected element with the size of 500
mm x 375 mm from the bottom of the girder. The corresponding reduction in
stiffness for the selected deck and girder damage elements are 0.5E and 0.6Ig
respectively, in which Ig is the gross second moment of area. In damage cases D1 and
D2, a single damaged element is simulated on the deck at the mid-span and quarter-
span respectively. In damage case D3, two damaged elements are simulated on the
deck, one located at the mid-span and the other at quarter-span. In damage cases D4
and D5, one damage element is simulated on the right ‘R’ girder at the quarter-span
Damage assessment in structures using vibration characteristics 110
and mid-span respectively. In damage case D6, two damage elements are simulated
both on the ‘R’ girder, one at the mid-span and the other at quarter-span. In damage
case D7, a total of two damage elements are simulated on the girders, one on the ‘R’
girder at the three quarter-span and the other on the left ‘L’ girder at the mid-span.
(a) D1 (0.5E)
(b) D2 (0.5E)
(c) D3 (0.5E)
Fig. 5.2 Damage cases (D1-D3) on deck.
Damage assessment in structures using vibration characteristics 111
‘R’ girder
‘L’ girder
(a) D4
‘R’ girder
‘L’ girder
(b) D5
‘R’ girder
‘L’ girder
(c) D6
‘R’ girder
‘L’ girder
(d) D7
Fig. 5.3 Damage cases (D4-D7) on girders.
Damage assessment in structures using vibration characteristics 112
5.1.2 Frequency change
The natural frequencies of the first five modes of slab-on-girder bridge before and
after damage in seven damage scenarios obtained from the results of the FE analysis
are listed in Table 5.2. The mode shapes corresponding to the first five vibration
modes of intact bridge are illustrated in Fig. 5.4. It appears that the dynamic
behaviour of this bridge is governed by vertical bending modes, coupled with
torsional modes, in the frequency range of 0.7 – 4.3 Hz. The fundamental mode of
the bridge is the vertical bending mode with natural frequency of 0.74 Hz. It can be
seen that modes 1, 2 and 4 are vertical bending modes, while modes 3 and 5 are
coupled vertical bending and torsional modes.
Table 5.2 Natural frequencies from FEM for slab-on-girder bridges
(Percentage changes w.r.t. to the undamaged conditions are listed within brackets)
Situation Mode 1
ƒ1 (Hz)
Mode 2
ƒ2 (Hz)
Mode 3
ƒ3 (Hz)
Mode 4
ƒ4 (Hz)
Mode 5
ƒ5 (Hz)
Original 0.7413 2.2735 2.4692 3.6471 4.2961
Deck
damage
D1 0.7409 2.2730 2.4691 3.6464 4.2956
(0.05) (0.02) (0.00) (0.02) (0.01)
D2 0.7410 2.2733 2.4682 3.6451 4.2937
(0.03) (0.01) (0.04) (0.05) (0.06)
D3 0.7409 2.2729 2.4681 3.6463 4.2942
(0.05) (0.03) (0.04) (0.02) (0.04)
Girder(s)
damage
D4 0.7298 2.2425 2.4594 3.6295 4.2877
(1.55) (1.36) (0.40) (0.48) (0.20)
D5 0.7244 2.2690 2.4672 3.6434 4.2880
(2.27) (0.20) (0.08) (0.10) (0.19)
D6 0.7138 2.2372 2.4573 3.6264 4.2798
(3.70) (1.60) (0.48) (0.57) (0.38)
D7 0.7178 2.2443 2.4534 3.5846 4.2720
(3.16) (1.28) (0.64) (1.71) (0.56)
Note: Changes of natural frequencies result in decrease in all damage cases.
Damage assessment in structures using vibration characteristics 113
1st bending mode 3rd bending mode
(a) Mode 1, f1=0.7413Hz (d) Mode 4, f4=3.6471Hz
2nd bending mode 2nd bending and torsional mode
(b) Mode 2, f2=2.2733Hz (e) Mode 5, f5=4.2961Hz
1st bending and torsional mode
(c) Mode 3, f3=2.4692Hz
Fig. 5.4 First five vibration modes of FE model.
In order to relate the location and severity of damage with damage-induced
frequency change levels, frequency change ratios for all the damage cases are
calculated. The frequency change ratio for the i-th mode caused by damage is
defined as
%100*
×−=∆i
iii f
fff (5.1)
Damage assessment in structures using vibration characteristics 114
where if and *if are computed natural frequencies for the i-th mode of the intact
structure and the damaged structure respectively. The frequency change ratios for the
first 5 modes, using Eq. (5.1) are listed within brackets in Table 5.2. Based on the
observation in all damage cases, the frequency change ratios corresponding to the
locations and severity of damage are summarised in Table 5.3. It is noted that high
damage severity pertaining to multiple damage in the deck causes high frequency
change ratio as expected. Also the damage occurring at the mid-span of the deck
causes higher frequency change ratio in the first mode than that at quarter-span.
Similar conclusions apply for the girders as well. It will be seen later that the damage
severity is also indicated to a certain extent by the maximum value of the MFC or
MSEC based damage index β. For example, the maximum value of MFC in the
bridge deck under damage case D2 is more than that under damage case D1.
Table 5.3 The relationship between fundamental frequency change ratio and damage
severity with certain locations on deck and girders
Damage case 1 damage at
quarter-span
(D2/D4)
1 damage at
mid-span
(D1/D5)
2 damages (1 at quarter span
or edge and 1 at mid-span)
(D3/D6/D7)
Deck damage
(D1-D3)
0.03% 0.05% 0.05%
Girder(s) damage
(D4-D7)
1.55% 2.27% 3.70%, 3.16%
5.1.3 Modal flexibility change
After frequency analysis, which indicated the occurrence of damage, the first five
natural frequencies and associated mode shapes obtained from the Eigen value
analysis are used to calculate the modal flexibility change (MFC) by using Eqs.
(3.31) and (3.33). Plots of MFC in deck for damage cases D1-D3 are shown in Figs.
5.5(a)-(c). The peak values of the plots indicate the damage locations on deck. In
Figs. 5.5(a) and (b), there are distinct peaks at the mid-span and quarter-span
respectively, which conform well with the damage cases D1 and D2 respectively. In
Fig. 5.5(c) there should be two peaks in the graph corresponding to the 2 damaged
Damage assessment in structures using vibration characteristics 115
elements on the deck in damage case D3, but it seems that this plot has missed one
of the peaks, which is the damage at mid-span. Plots of MFC on deck for damage
case D7, which pertain to girder damage (only) are shown in Fig. 5.5(d). As expected
that there are no distinguishing peak(s) in the plots of MFC, as these are `randomly
distributed across the intact deck and most importantly these MFC plots have much
smaller magnitudes compared to those at damage locations. MFC in the deck for the
other girder damage cases (D4, D5 and D6) are not shown as they too do not convey
definite any information and have smaller peaks.
The plots of MFC along the twin-girders for damage case D1 are shown in Fig.
5.6(a). It can be seen that the curves corresponding to the girders have very small
values (of ∆F) across a range of 0 to 6x10-9 m/N. From an enlarged view of these
plots, it is evident that their peaks are mostly in phase and that the shapes of the
curves were probably half-sine vibration modes (initially) for the intact beam
elements alone. The distortions of the curve from the half-sine wave shape is due to
the interference caused by the damaged elements on deck and also the connection
effects of beam and deck system. The plots of MFC along the girders for damage
cases D5-D7 are shown in Figs. 5.6(b)-(d). It can be seen that the curves for the
damaged girders have higher amplitudes compared to those for the undamaged
girders and most importantly the peaks (or maxima) in the curves for the damaged
girders correspond to the damage locations. The intact deck does not seem to
intervene much on the modal vectors of the damaged girders and as a consequence
comparatively smooth sine wave curves are obtained along the girders.
5.1.4 Modal strain energy change
The first five mode shapes obtained from the FE analysis are used to calculate the
MSEC (β). Plots of MSEC on deck for damage cases D1-D3 are shown in Figs.
5.5(e)-(g). The peak values of the plots indicate the location of damage on the deck.
It is found that the MSEC method is able to detect and localize damage zones on
deck precisely in all cases of deck damage. Fig. 5.5(h) shows the variation of MSEC
in the deck when there is damage only in the girders. The peak values in this Figure
Damage assessment in structures using vibration characteristics 116
are smaller than those in Figs. 5.5(e)-(g) and they do not offer any meaningful
interpretation as they vary randomly across the deck.
The plots of MSEC (β) along the twin-girders for damage cases (D1, D5-D7) are
shown in Figs. 5.6(e)-(h). It can be seen that the curves for the undamaged girders in
Fig. 5.6(e), corresponding to deck damage case D1 oscillate over a range 0.998 -
1.002 about the base line value of 1 and that the peaks in these curves have smaller
values than those corresponding to the girder damage cases (D5-D7). The latter
curves in Figs. 5.6(f)-(h) for the damaged girders oscillate in a comparatively larger
range of 0.985 – 1.015 about the base line value of 1. It is clearly evident that these
Figures, corresponding to girder damage cases (D5-D7) have distinct peaks at the
damage locations. This feature confirms that the MSEC method can accurately
identify and locate damage in bridge girders.
The compatibility of the MFC and MSEC methods is clearly evident from Figs. 5.5
and 5.6. Figs. 5.5(a) – (c) (from MFC) co-relate very well with Figs. 5.5(e) – (g)
(from MSEC) and establish the damage locations on the bridge deck. Similarly, Figs.
5.6(b) – (d) (from MFC) and Figs. 5.6(f) – (h) (from MSEC) co-relate well to
establish damage locations in the bridge girders.
Damage assessment in structures using vibration characteristics 117
(a) D1 (e) D1
(b) D2 (f) D2
(c) D3 (g) D3
(d) D7 (h) D7
Fig. 5.5 Modal flexibility change (left) and Modal strain energy based damage index (right) on deck.
Damage assessment in structures using vibration characteristics 118
(a) D1 (e) D1
(b) D5 (f) D5
(c) D6 (g) D6
(d) D7 (h) D7
Fig. 5.6 Modal flexibility change (left) and Modal strain energy based damage index (right) on girders.
Damage assessment in structures using vibration characteristics 119
5.2 Damage assessment in multiple-girder composite bridge
5.2.1 Model description
A multiple-girder composite bridge as shown in Fig. 5.7 is treated in this study. The
superstructure used as the basis for the investigation is a zero-skew, single span with
12.8m wide concrete deck spanning 30m. The deck is supported by four welded-steel
plate girders which are I-section assembled of flange and web plates. The details of
the multiple-girder composite bridge are given in Table 5.4. Cross bracing at spacing
of 2m is provided between girders. Both bridge deck and girders are modelled as shell
elements while steel diagonal bracings are modelled as truss elements. The deck and
each girder are divided into 480 and 240 elements respectively. Four girders having
the same span are simply supported at their ends and rotations about all 3 axes are
allowed in order to simulate the desired boundary conditions.
A total of six damage cases are investigated for the damage identification on the
bridge. The first two damage cases involve deck damage only and the last four cases
involve girders damage only, as shown in Figs. 5.8 and 5.9 respectively. Damage on
the deck and girders are simulated either by reducing the elastic modulus (E) of
selected elements (0.5E) or removing the selected elements. In damage cases D1 and
D2, a single and three damaged elements are simulated on the deck respectively, as
shown in Figs. 5.8(a) and (b). In damage case D3, a selected element with the size of
1000mm x 400mm x 20mm is removed from the bottom flange of the girder (G1) to
simulate the damage as shown in Fig. 5.9(a). In damage cases D4-D6, damaged
elements are simulated on the web of girders at different locations as shown in Figs.
5.9(b)-(d). It is assumed that linear behaviour of the bridge occurs in all damage
cases. The nonlinear effects associated with the crack are not studied in this
investigation.
Damage assessment in structures using vibration characteristics 120
Fig. 5.7 Isometric view of FE model with numbering system on girders.
Deck (reduced stiffness 0.5E)
(a) D1
Deck (0.5E)
(b) D2
Fig. 5.8 Damage cases (D1-D2) on deck.
Damage assessment in structures using vibration characteristics 121
Damage on bottom flange of G1 at mid-span
(a) D3
Web of G2 (0.5E)
(b) D4
Web of G2 (0.5E)
Web of G4 (0.5E)
(c) D5
Web of G1 (0.5E)
Web of G2 (0.5E)
Web of G3 (0.5E)
(d) D6
Fig. 5.9 Damage cases (D3-D6) on girders.
Damage assessment in structures using vibration characteristics 122
Table 5.4 Geometric and material properties of deck and girders
Flexural member Deck Girder
Element type Shell Shell
Material Concrete Steel
Length 30 m 30 m
Width 12.8 m 0.02 m
Depth 0.4 m 1.75 m
Poisson's ratio 0.2 0.3
Mass density 2400 kg/m3 7800 kg/m3
Modulus of elasticity 24 GPa 200 GPa
5.2.2 Frequency change
Natural frequencies of the first five modes of the multiple-girder composite bridge
before and after damage in six scenarios obtained from the FE analysis results are
shown in Table 5.5. Percentage changes in the natural frequencies between the
undamaged and damaged conditions are listed within brackets. It is observed that the
presence of damage in the multiple-girder composite bridge causes a decrease in the
natural frequencies in all damage cases, with very few exceptions. There is no
change of frequency for some modes (e.g. 2nd mode in damage case D4) because the
damage elements are located at the nodes of vibration modes and hence have no
influence on the corresponding natural frequencies. The five undamaged vibration
mode shapes are illustrated in Fig. 5.10. It appears that the dynamic behaviour of the
bridge is governed by vertical bending modes, coupled with torsional modes, in the
frequency range of 3 - 15 Hz. The fundamental mode is the vertical bending mode of
the deck and girders and corresponds to a natural frequency of 3.75 Hz. The second
vertical bending mode appears in mode 4 with a natural frequency of 12.55 Hz. In
mode 2, 3 and 5, it can be seen that they all involve coupled bending and torsional
vibration of the slab and girders.
Damage assessment in structures using vibration characteristics 123
Table 5.5 Natural frequencies from FEM for multiple-girder composite bridges
(Percentage changes wrt to the undamaged conditions are listed within brackets)
Situation Mode 1 ƒ1 (Hz)
Mode 2 ƒ2 (Hz)
Mode 3 ƒ3 (Hz)
Mode 4 ƒ4 (Hz)
Mode 5 ƒ5 (Hz)
Original 3.75 5.02 12.31 12.55 14.48
Deck damage
D1 3.73
(-0.42) 5.01
(-0.03) 12.28 (-0.24)
12.54 (-0.06)
14.48 (0.01)
D2 3.72
(-0.73) 5.00
(-0.24) 12.22 (-0.72)
12.49 (-0.47)
14.40 (-0.59)
Girder(s) damage
D3 3.72
(-0.68) 5.00
(-0.40) 12.30 (-0.11)
12.55 (-0.01)
14.48 (0.00)
D4 3.75
(-0.01) 5.02
(0.00) 12.31 (0.00)
12.55 (-0.02)
14.48 (-0.01)
D5 3.74
(-0.07) 5.02
(-0.03) 12.31 (-0.01)
12.54 (-0.10)
14.47 (-0.05)
D6 3.74
(-0.09) 5.01
(-0.04) 12.31 (-0.02)
12.54 (-0.07)
14.47 (-0.06)
1st bending mode 2nd bending mode
(a) Mode 1, f1=3.75Hz (d) Mode 4, f4=12.55Hz
1st bending and torsional mode 3rd bending and torsional mode
(b) Mode 2, f2=5.02Hz (e) Mode 5, f5=14.48Hz
2nd bending and torsional mode
(c) Mode 3, f3=12.31Hz
Fig. 5.10 First five vibration modes of FE model.
Damage assessment in structures using vibration characteristics 124
5.2.3 Modal flexibility change
The first five natural frequencies and associated mode shapes obtained from the
eigenvalue analysis are used to calculate the MFC. Plots of MFC in deck for damage
cases D1 and D2 are shown in Figs. 5.11(a) and (b). The peak values of the plots
indicate the damage locations on the deck. In Fig. 5.11(a), there is a peak at the mid-
span, which conforms well with the damage case D1. In Fig. 5.11(b), it is noted that
the peaks in the plots do not match well with the corresponding damage in multiple
locations. Therefore, it is concluded that MFC is only able to detect single deck
damage, and it fails to detect multiple deck damages on the multiple-girder
composite bridge. Plots of MFC on the deck for damage case D3, which pertains to
girder damage only are shown in Fig. 5.11(c). As expected, there are no
distinguishing peaks in the plots of MFC, as the plots are randomly distributed
across the intact deck. MFC in the deck for the other girder damage cases (D4, D5
and D6) are not shown as they draw the same conclusion as in damage case D3.
The plots of MFC along the four-girders for damage cases D1, D5 and D6 are shown
in Fig. 5.12. It is found that the plots do not provide any information for localization
of damage, which means that MFC is not feasible for application on the multiple-
girder composite bridge.
5.2.4 Modal strain energy change
The first five mode shapes obtained from the eigenvalue analysis are used to
calculate the MSEC (β). Plots of MSEC on the deck for damage cases D1 and D2 are
shown in Figs. 5.11(d) and (e). The peak values of the plots indicate the location of
damage on the deck. In Fig. 5.11(d), there is a distinct peak at the mid-span, which
conforms well with the damage case D1. In Fig. 5.11(e) there are three un-equal
peaks which correspond to three damaged elements on the deck in damage case D2.
It is concluded that the MSEC method is able to detect and localize damage zones on
the deck precisely in all deck damage cases. Fig. 5.11(f) shows the MSEC in the
deck when there is damage only in the girders. As expected, there are no
distinguishing peaks in the plots of MSEC, as the plots are randomly distributed
Damage assessment in structures using vibration characteristics 125
across the intact deck. It is also noted that the MSEC value in this figure, shown in
an enlarged scale, are much smaller than those in Figs. 5.11(d) and (e).
The plots of MSEC (β) along four girders for damage cases D1, D5 and D6 are
shown in Figs. 5.12(d)-(f). It can be seen that the plots for the undamaged girders in
Fig. 5.12(d), corresponding to deck damage case D1, oscillate over a range 0.995 -
1.005 about the base line value of 1, and that the peaks in these curves have smaller
values than those corresponding to the girder damage cases D5 and D6. The latter
curves in Figs. 5.12(e)-(f) for the damaged girders oscillate in a comparatively larger
range of 0.99 – 1.015 about the base line value of 1. It is clearly evident that these
figures, corresponding to girder damage cases D5 and D6, have distinct peaks (β
over 1.005) at the damage locations. In damage case D5, there should be a total of
three distinct peaks (β over 1.005) in the plots corresponding to girder damage. It is
noted that the MSEC curve corresponding to this case obtains two peaks only, and
one peak is missed.
A total of 24 MSEC curves corresponding to damaged and undamaged girders for all
damage cases are plotted in Fig. 5.13 for comparison of amplitude. It is observed that
the damaged girders have higher maximum amplitudes compared to the undamaged
girders at corresponding damage locations. Due to this fact, a damage limit on the
change of modal strain energy is defined at 1.005 in order to localize all damages in
the girders. Damage limit is established to discriminate structural health status. This
can be done by using statistical hypothesis based on statistical confidence bounds on
the normalized values of the damage index adopted. From the observation in the
figure, there should be seven peaks in the graph to be plotted. It is found that one has
been missed in a girder (G4) in damage case D5. Overall, the results show that the
modal strain energy method is competent to locate the damaged elements in both
bridge deck and girders.
Damage assessment in structures using vibration characteristics 126
(a) D1
(d) D1
(b) D2
(e) D2
(c) D3
(f) D3
Fig. 5.11 Modal flexibility change (left) and Modal strain energy based damage
index (right) on the deck.
Damage assessment in structures using vibration characteristics 127
(a) D1
(d) D1
(b) D5
(e) D5
(c) D6
(f) D6
Fig. 5.12 Modal flexibility change (left) and Modal strain energy based damage
index (right) on the girders.
Damage assessment in structures using vibration characteristics 128
Fig. 5.13 Relationship between modal strain energy based damage index and
structural state of girders. (The legend is the same as in Fig. 5.12.)
Damage assessment in structures using vibration characteristics 129
5.3 Damage assessment in truss bridge
5.3.1 Model description
A truss bridge model is treated in this study and the general modelling scheme for
bridge is depicted in Fig. 5.14. The superstructure used as the basis for the
investigation is a zero-skew, single span truss bridge with 4m wide deck consisting
two steel truss girders spanning 18m. The concrete slab thickness is 200mm and the
spacing between twin-girders is 3m. Details of geometry and material properties for
the bridge are listed in Table 5.6. The model is a nine-panel truss bridge with a truss
depth of 3m, and a width of 2m for each bay. The classification of truss members is
shown in Fig. 5.15, while the numbering system for truss nodes and members are
shown in Fig. 5.16 and Fig. 5.17, respectively. Steel diagonal bracings are installed
at the two ends (over end bearings) and bottom cross bracing are provided between
the two truss panels to prevent lateral buckling failure of compression truss
members. For finite element modelling, bridge deck is modelled as shell elements
while truss panels and steel bracing are modelled as truss elements. The deck and
each truss girder are divided into 288 and 102 elements respectively. It is assumed
that there is a complete connection between the girders and slab. Twin girders having
the same span are simply supported at their ends and rotations about all 3 axes are
allowed in order to simulate the desired boundary condition.
A total of 8 damage cases are investigated for the damage identification of this
bridge. The first two damage cases D1 and D2 are simulated for deck damage, the
next four damage cases D3-D6 for truss damage, and the last two damage cases D7
and D8 for deck and truss damage simultaneously. Damage on deck is simulated by
reducing the elastic modulus (E) of selected elements with size of 500mm x 500mm
while damage on the truss is simulated by reducing the cross-section area (A) of the
selected one-third of total length of truss members. The corresponding reduced
stiffness for the selected deck and truss damage elements are 0.5E and 0.5Ag
respectively. All damage scenarios of deck and truss are shown in Figs. 5.18-5.20.
The numbering systems for truss members are listed in Table 5.7, while the truss
damage configurations are listed in Table 5.8.
Damage assessment in structures using vibration characteristics 130
In damage cases D1 and D2, a single damage and three damages are simulated on the
deck respectively. In damage cases D3, D4 and D8, damage are simulated in right
truss panel only, while in D5-D7, damage are simulated on right and left truss panel.
Fig. 5.14 Isometric view of truss model.
Table 5.6 Geometric and material properties of deck and truss
Structural member Deck Truss
Element type Shell Truss
Material Concrete Steel
Length 18 m various
Width 4 m 100 mm
Depth 200 mm 6 mm
Poisson's ratio 0.2 0.3
Mass density 2400 kg/m3 7800 kg/m3
Modulus of elasticity 24 GPa 200 GPa
Table 5.7 Numbering systems for truss members
Truss type Element no.
Left truss panel Right truss panel
Bottom chord member 103-129 1-27
Top chord member 130-147 28-45
Vertical member 148-177 46-75
Diagonal member 178-204 76-102
Damage assessment in structures using vibration characteristics 131
Fig. 5.15 The classification of truss members.
Table 5.8 Truss damage configurations
Damage
case
Left truss panel Right truss panel
Element no. Node no. Element no. Node no.
D3 - - 14 14-15
D4 - - 14,50,98 14-15,50-51,82-83
D5 110,182,185 93-94,155-156,157-158 14,50,98 14-15,50-51,82-83
D6 188,200 159-160,167-168 23,47,74 23-24,48-49,66-67
D7 173 149-150 5,92 5-6,78-79
D8 - - 14 14-15
Damage assessment in structures using vibration characteristics 132
(a) Right truss panel
(b) Left truss panel
Fig. 5.16 Numbering system for truss nodes.
(a) Right truss panel
(b) Left truss panel
Fig. 5.17 Numbering system for truss members.
Damage assessment in structures using vibration characteristics 133
Deck (reduced stiffness 0.5E)
(a) D1
Deck (0.5E)
(b) D2
Fig. 5.18 Damage cases (D1-D2) on deck.
Damage assessment in structures using vibration characteristics 134
Right girder (0.5A)
(a) D3
Right girder (0.5A)
(b) D4
Right girder (0.5A)
Left girder (0.5A)
(c) D5
Right girder (0.5A)
Left girder (0.5A)
(d) D6
Fig. 5.19 Damage cases (D3-D6) on truss.
Damage assessment in structures using vibration characteristics 135
Deck (0.5E)
Right girder (0.5A)
Left girder(0.5A)
(a) D7
Deck (0.5E)
Right girder (0.5A)
(b) D8
Fig. 5.20 Damage cases (D7-D8) on deck and truss.
Damage assessment in structures using vibration characteristics 136
5.3.2 Frequency change
The first five natural frequency and mode shapes are extracted from the eigenvalue
analysis. No structural damping is used in the free vibration analysis. The natural
frequencies of the first five modes of truss bridge before and after damage in eight
damage cases are shown in Table 5.9. It is found that frequency decreases in all
damage cases D1-D8 including deck damage, truss damage and combined damages.
It is also found that damage on deck (stiffness reduction of 0.5E) causes
comparatively small change of frequency than truss damage (stiffness reduction of
0.5A). Meanwhile, by comparing the rate of frequency change between D3 (girder
damage) and D8 (combine deck and girder damage), it draws the same conclusion
that deck damage cause small change of frequency for the truss model. Therefore, it
is concluded that the girder damage is dominant for the frequency change in all
modes in this model. In addition, by observing the rate of frequency change among
D3, D4 and D5 (truss damage), it is noteworthy that the higher level of damage
severity on truss panel(s) leads to greater change of frequency in all vibration modes
(Mode 1 - Mode 5).
Table 5.9 Natural frequencies from FEM for truss bridges
(Percentage changes wrt to the undamaged conditions are listed within brackets)
Damage case Mode 1 ƒ1 (Hz)
Mode 2 ƒ2 (Hz)
Mode 3 ƒ3 (Hz)
Mode 4 ƒ4 (Hz)
Mode 5 ƒ5 (Hz)
Intact 7.90 12.05 15.63 17.95 22.55
Deck damage D1
7.87 (0.49)
12.00 (0.42)
15.59 (0.24)
17.93 (0.10)
22.52 (0.12)
D2 7.84
(0.81) 11.95 (0.85)
15.53 (0.63)
17.92 (0.16)
22.37 (0.79)
Girder(s) damage
D3 7.82
(1.12) 12.05 (0.03)
15.63 (0.01)
17.94 (0.04)
22.54 (0.05)
D4 7.80
(1.30) 12.05 (0.05)
15.47 (0.98)
17.93 (0.06)
22.34 (0.93)
D5 7.69
(2.70) 12.04 (0.12)
15.41 (1.36)
17.80 (0.81)
22.18 (1.66)
D6 7.79
(1.42) 11.88 (1.44)
15.34 (1.82)
17.71 (1.32)
22.23 (1.42)
Deck & girder(s) damage
D7 7.85
(0.74) 11.97 (0.72)
15.44 (1.22)
17.85 (0.55)
22.19 (1.59)
D8 7.78
(1.60) 12.00 (0.46)
15.59 (0.23)
17.92 (0.14)
22.51 (0.17)
Damage assessment in structures using vibration characteristics 137
The first five vibration mode shapes in undamaged case are illustrated in Fig. 5.21. It
appears that the dynamic behaviour of the bridge model is governed by vertical
bending modes of deck, coupled with local vibration mode of truss members, in the
frequency range of 7.9-23Hz. The fundamental mode is the vertical bending mode of
the deck along with local vibration mode of the truss and it corresponds to a natural
frequency of 7.9 Hz. As all modes are dominant by the deck vibration mode, it is
concluded that deck structure is much flexible (or weaker) compared to the truss
structure.
(a) Mode 1, f1=7.9Hz (d) Mode 4, f4=17.95Hz
(b) Mode 2, f2=12.05Hz (e) Mode 5, f5=22.55Hz
(c) Mode 3, f3=15.63Hz
Fig. 5.21 First five vibration modes of FE model.
Damage assessment in structures using vibration characteristics 138
5.3.3 Modal flexibility change
The first five natural frequencies and associated mode shapes obtained from the
Eigen value analysis are used to calculate the MFC. Plots of MFC in deck for
damage cases D1, D2, D3 and D8 are shown in Figs. 5.22(a)-(d). The peak values of
the plots indicate the damage locations on deck. In Fig. 5.22(a), there is a distinct
peak at the mid-span of deck, which conforms well with the damage cases D1. In
Fig. 5.22(b) there are three un-equal peaks which correspond to the 3 damaged
elements on the deck in damage case D2. Plots of MFC on deck for damage case D3,
which pertain to girder damage (only) are shown in Fig. 5.22(c). As expected that
there are no distinguishing peak(s) in the plots of MFC, as these are randomly
distributed across the intact deck. Plots of MFC on deck for damage case D8, which
pertain to both deck and girder damage are shown in Fig. 5.22(d). As the damage
indicators are randomly distribution across the deck (with no distinct peak), it is
found that MFC method is unable to detect damaged elements in this damage case
D8. MFC in the deck for the other girder damage cases (D4, D5 and D6) and
combined damage case D7 are not shown as similar conclusion with D3 and D8 will
be drawn respectively.
The plots of MFC along the truss for damage cases D3, D6 and D7 are shown in Fig.
5.23. In Fig. 5.23(a), the distinct peak of the plot indicates correctly the damage
locations on truss elements of bottom chord member. For damage cases D5 and D7
as shown in Figs. 5.23(c) and (d), MFC shows great ability on localizing all damaged
element accurately. Overall, it is concluded that MFC provides feasible, reliable and
effective results on localization of truss damage.
5.3.4 Modal strain energy change
The first five mode shapes obtained from the eigenvalue FE analysis are used to
calculate the MSEC. Plots of MSEC on deck for damage cases D1, D2, D3 and D8
are shown in Figs. 5.22(e)-(h). The peak values of the plots indicate the location of
damage on the deck. Similar to the MFC method, it is found that the MSEC method
is able to detect and localize damage zones on deck precisely in all cases of deck
Damage assessment in structures using vibration characteristics 139
damage. In Fig. 5.22(g), random distribution of all damage indicators (β<1.0) of the
plot implies that no damage occurs on the deck. For damage case D8 (deck and
girder combined damage) as shown in Fig. 5.22(h), MSEC is able to detect the
damage element precisely. MSEC in deck for the other girder damage cases D4, D5
and D6 and combined damage case D7 are not shown as they are drawn the same
conclusion with D3 and D8 respectively.
The plots of MSEC (β) along the truss for damage cases D3, D6 and D7 are shown
in Figs. 5.23(d)-(f). Similar to the MFC method, the MSEC method is able to detect
and locate multiple damages on truss panel(s) precisely in all truss damage cases.
The same conclusion is drawn for other damage cases.
Damage assessment in structures using vibration characteristics 140
(a) D1 (e) D1
(b) D2 (f) D2
(c) D3 (g) D3
(d) D8 (h) D8
Fig. 5.22 Modal flexibility change (left) and Modal strain energy based damage
index (right) on deck.
Damage assessment in structures using vibration characteristics 141
(a) D3
(d) D3
(b) D6
(e) D6
(c) D7
(f) D7
Fig. 5.23 Modal flexibility change (left) and Modal strain energy based damage
index (right) on truss.
Damage assessment in structures using vibration characteristics 142
5.4 Summary
To investigate the feasibility of the two damage detection methods (i) modal
flexibility method and (ii) modal strain energy method on proposed structures,
parametric studies and sensitivities studies are carried out for assessment on
numerical examples. The parameters considered include type of structure, span
length, support condition, damage element, damage location, damage severity. From
the extensive numerical analyses, the performance of proposed damage detection
methods for slab-on-girder bridges and truss bridge, based on the corresponding
damage index in each method, are summarised in Tables 5.10-5.12.
Table 5.10 Performance of damage detection algorithms for slab-on-girder bridge
Damage case Deck Girder
MFC MSEC MFC MSEC
Deck
damage
D1 � � � �
D2 � � � �
D3 � � � �
Girder(s)
damage
D4 � � � �
D5 � � � �
D6 � � � �*
D7 � � � �
Table 5.11 Performance of damage detection algorithms for multiple-girder
composite bridge
Damage case Deck Girder
MFC MSEC MFC MSEC
Deck
damage
D1 � � x �
D2 x � x �
Girder(s)
damage
D3 � � x �
D4 � � x �
D5 � � x �*
D6 � � x �
Damage assessment in structures using vibration characteristics 143
Table 5.12 Performance of damage detection algorithms for truss bridge
Damage case Deck Truss
MFC MSEC MFC MSEC
Deck damage D1 � � � �
D2 � � � �
Girder(s)
damage
D3 � � � �
D4 � � � �
D5 � � � �
D6 � � � �
Deck &
girder(s)
damage
D7 x � � �
D8 x � � �
The main identification result findings can be summarized as follows:
In slab-on-girder bridge example, two types of damage severity with flexural
stiffness reduction of 50% on plate and 60% on girder are simulated at selected
damage location(s). The bridge is single-span with simply supported condition. It is
found that MSEC and MFC are both reasonably well in detecting deck and girder
damage in all damage cases. It is concluded that two proposed methods show
promise as detecting damage in slab-on-girder bridge consisting two steel plate
girders.
In multiple-girder composite bridge example, three types of damage severity
including flexural stiffness reduction of 50% on plate, reduction of 50% on web of
girder and also removing of element with size of 1000mm x 400mm x 20mm from
the bottom flange of girder are investigated. The bridge is single-span with simply
supported condition. It is found that the MSEC shows promise as detecting deck and
girder damage. On the contrary, MFC is failure to detect damage on either deck or
girder. Comparing between two methods, it is concluded that MSEC is suitable for
application on multiple-girder composite bridge, while MFC is not.
Damage assessment in structures using vibration characteristics 144
In truss bridge example, two types of damage severity including flexural stiffness
reduction of 50% on plate and also axial stiffness reduction of 50% on truss are
investigated at selected damage location(s). A total of eight damage cases are
investigated in the study. Two combined damage cases, in which deck and girder are
damaged simultaneously, are studied along with two deck damage cases and four
girder damage cases. It is found that MSEC is capable to detect damages in all
damage cases, while MFC is capable to detect damage in most cases, except the
combined damage cases.
A sensitivity study of the damage detection technique vis-a-vis the positioning and
size of the flaws on the slab-on-girder bridge is carried out. The sensitivity analysis
results include the relationship between fundamental frequency change ratio and
damage severity with certain locations on deck and girders. The frequency change
ratios corresponding to the locations and severity of damage are discussed in section
5.1.2. It is found that high damage severity pertaining to multiple damages in the
deck causes high frequency change ratio in the first mode. Also the damage
occurring at the mid-span of the deck causes higher frequency change ratio in the
first mode than that at quarter-span. Similar conclusion applies to the girders as well.
Moreover, sensitive damage detection index (%) for two proposed methods (modal
flexibility method and modal strain energy method) on deck and girders are also
studied. According to the sensitive damage detection index (%), it is found that
MSEC is highly sensitively to detect damage on deck than using MFC, while both
methods show similar results for detecting damage on girders.
It is concluded that applying modal flexibility and modal strain energy methods to
the proposed structures provides sensitive, reliable and accurate results for multiple
damage localization.
From the analysis result, it can be concluded that the two methods (changes of modal
flexibility and modal strain energy based damage index) show varying levels of
success when applied to proposed structures with different damage cases. Overall,
the strain energy method presents the best stability and capability regarding the
damage detection on the deck and girders among the five proposed structures. The
change of modal flexibility method is also capable to detect and localise damaged
Damage assessment in structures using vibration characteristics 145
elements in most structures but in the case of multiple girder composite bridges with
simultaneous damages, this technique shows less effectiveness.
5.4.1 Flowchart for multiple criteria approach
A flowchart for the proposed multiple criteria damage detection system on proposed
structures is shown in Fig. 5.24. Firstly, an initial FEM is generated using FE
software. Then the experimental dynamic testing is carried out to capture the primary
modal parameters. Calibration techniques based on sensitivity analysis could be used
for FE model updating. After that, the primary modal parameters including natural
frequencies and mode shapes are obtained from the validated baseline model using
eigenvalue analysis. Finally Level-1 damage alarming is achieved by observing the
change of natural frequencies of proposed structures. Level-2 damage localization is
achieved accurately by using two complementary damage identification methods, (i)
modal flexibility method and (ii) modal strain energy method, which utilize the
primary modal parameters produced from FE models.
Fig. 5.24 Flowchart of damage detection in proposed structures.
Damage detection on proposed structures
Experimental dynamic testing
Initial FE modelling
Calibration, model updating
Modal analysis of baseline FE model
Level-1 Damage alarming (change of fundamental frequencies)
Frequency change (Deck or girder damage)
Frequency no change (Intact structural state)
Level-2 Damage localization of
deck (MSEC)
Level-2 Damage localization of
deck (MFC)
Level-2 Damage localization of
girder(s) (MSEC)
Level-2 Damage localization of
girder(s) (MFC)
Damage assessment in structures using vibration characteristics 146
Chapter 6 Conclusions
6.1 Summary
This thesis uses dynamic computer simulation techniques to develop and apply a
multi-criteria based non-destructive damage detection methodology for five types of
structures: beam, plate, slab-on-girder bridge, multiple-girder composite bridge and
truss bridge. The proposed procedure involves two vibration based damage detection
parameters (i) changes in modal flexibility matrix and (ii) changes in modal strain
energy based damage index, in addition to changes in natural frequencies, all of
which are evaluated from the results of free vibration analysis of the undamaged and
damaged finite element models of the structure. Experimental testings have been
used to validate the proposed FE models of the structure. The experimental data
obtained from static and dynamic tests have been compared with those predicted
from updated finite element models. A good agreement between numerical and
experimental results is observed. Moreover the results from the experimental testing
of the slab on girder bridge model confirmed the feasibility of the proposed damage
detection strategy using the selected vibration parameters. This provides confidence
for using the computer models for further investigation towards establishing the
proposed multi-criteria damage detection method.
Based on the results from the extensive dynamic computer simulations, the
following conclusions are drawn:
• The proposed multi-criteria approach is feasible for damage assessment in the
chosen structures.
As a starting point, changes in natural frequencies can be used to detect the presence
of damage, since this can be done from a single point measurement. This approach
Damage assessment in structures using vibration characteristics 147
could provide an inexpensive structural assessment technique as frequency
measurement is easily acquired. The presence of damage could be triggered by a
change in natural frequency and then detected by using high sensitive transducers.
The modal flexibility and modal strain energy methods can then be used to locate the
damage in the proposed structures. The most attractive feature of the two methods is
that they can be implemented using the first few vibration modes. The changes in the
modal flexibility matrix and modal strain energy between the undamaged and
damaged structure provide a basis for locating the damage.
• Damages in beams, slabs, slab-on-girder bridges and truss bridges can be
correctly located by the proposed procedure.
Results from several numerical results have confirmed the feasibility of the
proposed approach. A comparison between results of the modal flexibility and modal
strain energy methods in the several numerical examples reveal that MSEC is more
sensitive for damage localization than MFC.
It is also evident that both methods (MFC and MSEC) are able to quantify the extent
of a single damage cases based on the magnitude of the damage index - peaks in the
plots of MFC and MSEC diagrams.. Moreover, damage at mid-span causes a larger
spike compared to damage elsewhere. It is concluded that Level-3 damage
quantification is achieved at least for cases of single damage through the proposed
technique. Further research is needed for Level -3 damage quantification in the cases
of multiple damage.
• The proposed multi-criteria approach has the capability to treat multi-damage
localisation. The illustrated numerical examples evidenced that both MFC and
MSEC can locate single and multiple damage locations accurately in most of the
cases, except in the case of the multiple-girder composite girder in which MFC
fails to detect damage in girders.
Overall, it can be concluded that the multi-criteria vibration based approach provides
reasonably reliable and accurate tools for damage identification of multiple damages,
Damage assessment in structures using vibration characteristics 148
and severity estimation of single damage on chosen structures. As there are some
discrepancies in both damage assessment methods (MFC and MSEC), the
combination of MFC & MSEC together with the natural frequencies provides the
optimum chances of accurate damage assessment as demonstrated through the
examples. Due to the major advances in the fields of structural dynamics and
experimental modal analysis, the multiple criteria approach shows promise in being
used for detecting and locating damage in structures.
6.2 Contributions to scientific knowledge
The major contributions of this research study are that a multiple criteria based non-
destructive damage detection methodology is developed and applied on four types of
structures: beams, plates, slab-on-girder bridges and truss bridges. A two-stage
identification strategy is proposed for vibration-based damage detection: (i) Level 1
uses the change of frequency method to identify the damage and (ii) Level 2
incorporates the modal flexibility method and modal strain energy method to localize
single and multiple damages. This research has established possibility of treating
different damage scenarios including deck damage, girder damage and combined
damage. This research has also shown that the two damage localization methods are
able to localize single and multiple damage locations accurately in most cases. It has
been shown that in general the MSEC parameters are more competent and robust for
damage localization than MFC in the selected case studies. In addition, some
sensitivity analyses for the beam and plate structures have shown the possibility of
achieving Level 3 damage assessment (i.e. damage quantification) in the case of single
damage cases. For these cases, it has been demonstrated that the magnitude of the spikes
(or peaks) in the MFC and MSEC diagrams, are proportional to the damage intensity.
Moreover, damage at mid-span causes a larger spike compared to damage elsewhere.
However, the situation with multiple damage cases is more complex and needs further
investigation. Finally, a flow chart of the multi- criteria based damage detection
approach and a table which summarizes the performance of damage detection algorithms
for five types of proposed structures, have been developed. The research findings will
contribute towards the safe and efficient performance of structures.
Damage assessment in structures using vibration characteristics 149
6.3 Recommendations for further research
Although the proposed damage detection methodology has shown promise in the
illustrated numerical examples, there are several issues which need to be improved to
make it applicable in practice. In continuation of this study, future research efforts
can be focused on the following issues:
Damages cause changes in structural parameters, namely the matrices of mass,
damping, and stiffness (flexibility) of structures which form the basis of damage
detection methods. As structural characteristics would change according to the
environmental conditions such as temperature, it is very critical to monitor
environmental conditions and structural responses in order to interpret the data in
terms of damage indices. One of the possible ways to separate damage from
environmentally induced conditions is that the structure and the environmental
effects are monitored continuously such that seasonal and yearly environmental
cycles are captured. In further research, a statistical approach (e.g. regression study)
can be studied for elimination of environmental effects from the data.
It was also found that there is a limited amount of literature related to the non-
destructive damage detection method detailing the real situation of concrete damage
or steel damage. Concrete damage includes pounding, spalling effects and
continuous cracking, while steel damage includes loose bolts, broken welds,
corrosion and fatigue. In all cases damage can severely affect the safety and
serviceability of the structure. To evaluate the performance of damage detection
techniques in practical applications, a strategy which involves simulating realistic
damage and using non-linear analysis on complex and large structures could be
developed in the future work.
From the illustrated numerical examples, it is noted that the two damage localization
methods (MFC and MSEC) are effective for locating areas with stiffness reductions
ranging from 20%-50% by using the first five mode shapes of the finite element
models before and after damage. At present, there is no guideline for determining the
numbers of vibration mode shapes to be involved in the damage detection
Damage assessment in structures using vibration characteristics 150
algorithms. It is suggested to establish for selecting criteria the number of modes for
the application of nondestructive vibration-based damage detection methods in
structures in the future.
Two chosen bridge structures: (i) slab-on-girder bridges and (ii) truss bridge, are
investigated for the feasibility of non-destructive damage detection algorithms in this
thesis. The geometry and boundary conditions of proposed structures are mainly
designed in single-span with simply supported, while structural members act in
flexure and axial actions. Further research can be done in other structural forms such
as cable-stayed or suspension bridges with multiple spans to evaluate the
performance, robustness and sensitivity of the damage detection methods.
In the literature related to structural health monitoring, simple damage cases, in
which damages are simulated at the mid-span or quarter-span of flexural members,
are typically adopted by authors. Further research work may lie in damage detection
at the boundary supports of the structures (e.g. bearings and substructures). It is also
suggested that damping effects and various types of damage including complex and
simultaneous damage on bearings and superstructures are considered in the studies
of damage detection methods.
A system of classification for damage-identification methods defines four levels of
damage identification (Rytter 1993), which is presented in Chapter 2. It is found that
Level 3 (quantification of the severity of the damage) and Level 4 (prediction of the
remaining service life of the structure) are not fully addressed in the literature. Since
a robust damage detection methodology is able to detect the locations and extent of
small to large levels of damage accurately, and also to evaluate the impact of damage
on the structures reliably, it is suggested that global condition assessment is
employed in conjunction with local monitoring techniques to evaluate the reliability
of each portion of the structure. Effort can be focused in the field of fracture
mechanics, fatigue life analysis, or structural design assessment (Level 4) in further
research, to provide reliable information for decision making. In experimental
testing, it is suggested that sensors should be well-distributed, and when a rough
estimate of damage presence and location is made from some sensors, an additional
Damage assessment in structures using vibration characteristics 151
test should be carried out with densely distributed sensors on the suspected damage
zone to obtain certain identification.
Damage assessment in structures using vibration characteristics 152
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