damage assessment in structures using vibration ... · damage assessment in structures using...

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



5.1.4 Modal strain energy change ....................................................................... 115

5.2 Damage assessment in multiple-girder composite bridge ................................... 119

5.2.1 Model description....................................................................................... 119



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


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


damage index (right) on deck.

117


damage index (right) on girders.

118

Fig. 5.7 Isometric view of FE model with numbering system on

girders.

120


Fig. 5.9 Damage cases (D3-D6) on girders. 121



damage index (right) on the deck.

126


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. 5.19 Damage cases (D3-D6) on truss. 134

Fig. 5.20 Damage cases (D7-D8) on deck and truss. 135



damage index (right) on deck.

140


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.


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


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.


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.


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,


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.


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


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


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


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


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


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.


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


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.


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


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


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


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.


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.


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)


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.


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


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).


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


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.


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)


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


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


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.


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


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


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


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.


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.


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


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


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.


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∆ .


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:


∑∑

∑

==

==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.


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


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.


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


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.


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.


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 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.


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


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.


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)


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.


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)


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:


"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)


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)


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).


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)


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)


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)


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:


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)


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


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.


(a) Test specimen

(b) Cross bracings on test specimen

Fig. 3.4 Slab-on-girder bridge.


(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.


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


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.


(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.


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.


Fig. 3.11 Analytical deflection vs experimental deflection.

Fig. 3.12 Convergence of the static deflection at location ‘A’.


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


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.


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.


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


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.


(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.


(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).


(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


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


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).


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.


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.


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.


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


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


Fig. 3.27 Simply supported beam.

Fig. 3.28 Boundary condition of the beam.

Fig. 3.29 Accelerometer on the beam.


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.


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


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.


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


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.


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).


(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).


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


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.


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


2nd bending mode 5th bending mode


3rd bending mode

(c) Mode 3, f3=52.45Hz

Fig. 4.5 First five vibration modes of undamaged FE model.


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


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


∆ 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


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.


4.3 Damage assessment in plate (slab)


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


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)


(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.



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


1st bending mode 2nd bending and torsional mode


1st bending and torsional mode 3rd bending mode


2nd bending mode

(c) Mode 3, f3=3.07Hz

Fig. 4.10 First five vibration modes of undamaged plate with simply supported

condition.


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)


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


vibrating modes not influencing the corresponding natural frequencies, this feature

further demonstrates the need for multi-criteria damage assessment.


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


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.


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-


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.


Chapter 5 Application II - Bridges

5.1 Damage assessment in slab-on-girder bridge


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.


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)


Material Concrete Steel

Length 20 m 20 m

Width 4 m 8 mm

Depth 200 mm 1.5 m




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


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.


‘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.



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.


1st bending mode 3rd bending mode


2nd bending mode 2nd bending and torsional mode


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)


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%


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


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.


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


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.


(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.


(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.


5.2 Damage assessment in multiple-girder composite bridge


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.


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



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.


Table 5.4 Geometric and material properties of deck and girders

Flexural member Deck Girder



Length 30 m 30 m

Width 12.8 m 0.02 m

Depth 0.4 m 1.75 m





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.


Table 5.5 Natural frequencies from FEM for multiple-girder composite bridges


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


1st bending and torsional mode 3rd bending and torsional mode


2nd bending and torsional mode

(c) Mode 3, f3=12.31Hz





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.


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


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.


(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.


(a) D1

(d) D1

(b) D5

(e) D5

(c) D6

(f) D6


index (right) on the girders.


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.)


5.3 Damage assessment in truss bridge


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.


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


Length 18 m various

Width 4 m 100 mm

Depth 200 mm 6 mm




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


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


(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.


Deck (reduced stiffness 0.5E)

(a) D1

Deck (0.5E)

(b) D2



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.


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.



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


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)


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.



(c) Mode 3, f3=15.63Hz





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.


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. 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.


(a) D1 (e) D1

(b) D2 (f) D2

(c) D3 (g) D3

(d) D8 (h) D8


index (right) on deck.


(a) D3

(d) D3

(b) D6

(e) D6

(c) D7

(f) D7


index (right) on truss.


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 �


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.


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


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)


deck (MFC)


girder(s) (MSEC)


girder(s) (MFC)


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


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,


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.


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


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


test should be carried out with densely distributed sensors on the suspected damage

zone to obtain certain identification.


Bibliography

Abdel Wahab, M. M., and De Roeck, G. (1999). “Damage detection in bridges using

modal curvatures: application to a real damage scenario.” Journal of Sound and

Vibration, 226(2), 217-235.

Aktan, A. E., Catbas, F. N., Grimmelsman, K. A., and Tsikos, C. J. (2000). “Issues

in infrastructure health monitoring for management.” Journal of Engineering

Mechanics, ASCE, 126(7), 711–724.

Aktan, A. E., Farhey, D. N., Helmicki, A. J., Hunt, V. J., Lee, K. L., and Levi, A.

(1997). “Structural identification for condition assessment: experimental arts.” J.

Struct. Engrg., ASCE, 123(12), 1674-1684.

Alvandi, A., and Cremona, C. (2006). “Assessment of vibration-based damage

identification techniques.” Journal of Sound and Vibration, 292(1-2), 179-202.

Ang, A. H. S., and De Leon, D. (2005). “Modelling and analysis of uncertainties for

risk-informed decisions in infrastructures engineering.” Structure and Infrastructure

Engineering, 1(1), 19–31.

Bajaba, N. S., and Alnefaie, K. A. (2005). “Multiple damage detection in structures

using wavelet transforms.” Emirates Journal for Engineering Research, 10 (1), 35-

40.

Barker, R. M., and Puckett J. A. (1997). Design of highway bridges. John Wiley &

Sons, Inc., New York.

Catbas, F. N., Gul, M., and Burkett, J. (2008). “Damage assessment using flexibility

and flexibility-based curvature for structural health monitoring.” Smart Materials

and Structures, 17(1), 15-24.


Catbas, F. N., Grimmelsman, K. A., Barrish, R. A., Tsikos, C. J., and Aktan, A. E.

(1999). “Structural identification and health monitoring of a long-span bridge.”

Structural Health Monitoring 2000, Stanford University, Palo Alto, California, 417–

429.

Chang, P. C., and Liu, S. C. (2003). “Recent research in nondestructive evaluation of

civil infrastructures.” Journal of Materials in Civil Engineering, 15(3), 298-304.

Clough, R. W., and Penzien, J. (1993). Dynamics of structures. McGraw-Hill, USA.

Cornwell, P., Doebling, S. W., and Farrar, C. R. (1999). “Application of the strain

energy damage detection method to plate-like structures.” Journal of Sound and

Vibration, 224(2), 359-374.

Coronelli, D., and Gambarova, P. G. (2004). “Structural assessment of corroding RC

beams: modelling guidelines.” Journal of Structural Engineering, 130(8), 1214-

1224.

De Silva, C. W. (2007). Vibration: fundamentals and practice. CRC Press, United

States of America.

Doebling, S. W., Farrar, C. R., Prime, M. B., and Shevitz, D. W. (1996). “Damage

identification and health monitoring of structural and mechanical systems from

changes in their vibration characteristics: a literature review.” Report no. LA-13070-

MS, Los Alamos National Laboratory, Los Alamos (USA).

Ewins, D. J. (2001). “Model validation: correlation for updating.” Proc. Indian

Academy Sciences, Sadhana, 25(3), 221-246.

Farrar, C. R., Worden, K., Todd, M. D., Park, G., Nichols, J., Adams, D. E., Bement,

M. T., and Farinholt, K. (2007). “Nonlinear system identification for damage

detection.” Report no. LA-14353-MS, Los Alamos National Laboratory, Los Alamos

(USA).


Farrar, C. R., and Doebling, S. W. (1999). “Damage detection II: field applications

to large structures.” in J. M. M. Silva and N. M. M. Maia, edts., Modal Analysis and

Testing, Nato Science Series, Kluwer Academic Publishers, Dordrecht, Netherlands,

345-378.

Farrar, C. R., and Cone, K. M. (1995). “Vibration testing of the I-40 Bridge before

and after the introduction of damage.” Proceedings of 13th IMAC, Vol.I, 203-209.

Gere, J. M., and Goodno, B. J. (2008). Mechanics of materials. Cengage learning,

United States of America.

Hsieh, K. H., Halling, M. W., and Barr, P. J. (2006). “Overview of vibrational

structural health monitoring with representative case studies.” J. Bridge Engrg,

11(6), 707-715.

Hu, H., Wang, B. T., Lee, C. H., and Su, J. S. (2006). “Damage detection of surface

cracks in composite laminates using modal analysis and strain energy method.”

Composite Structures, 74(4), 399-405.

Huth, O., Maeck, J., Kilic, N., and Motavalli, M. (2005). “Damage identification

using modal data: experiences on a prestressed concrete bridge.” Journal of

Structural Engineering, 131(12), 1898-1910.

Iskhakov, I., and Ribakov, Y. (2005). “Experimental and numerical investigation of

a full-scale multistory RC building under dynamic loading.” The Structural Design

of Tall and Special Buildings, 14(4), 299-313.

Ko, J. M., and Ni, Y. Q. (2005). “Technology developments in structural health

monitoring of large-scale bridges.” Engineering Structures SEMC 2004 Structural

Health Monitoring, Damage Detection and Long-Term Performance, 27(12), 1715-

1725.


Ko, J. M. (2004). “Structural health monitoring of large scale bridges: research &

experience.” Proceeding of the Second International Conference on Structural

Engineering, Mechanics and Computation, 5-7 July 2004, Cape Town, South Africa.

Ko, J. M., Sun, Z. G., and Ni, Y. Q. (2002). “Multi-stage identification scheme for

detecting damage in cable-stayed Kap Shui Mun Bridge.” Engineering Structures,

24(7), 857-868.

Koh, B. H., and Dyke, S. J. (2007). “Structural health monitoring for flexible bridge

structures using correlation and sensitivity of modal data.” Computers and

Structures, 85(3–4), 117–130.

Lee, L. S., Karbhari, V. M., and Sikorsky, C. (2004). “Investigation of integrity and

effectiveness of RC bridge deck rehabilitation with CFRP composites.” SSRP-

2004/08, Department of Structural Engineering, University of California, San Diego.

Li, H., Deng, X., and Dai, H. (2007). “Structural damage detection using the

combination method of EMD and wavelet analysis.” Mechanical Systems and Signal

Processing, 21, 298–306.

Li, H. N., Li, D. S., and Song, G. B. (2004). “Recent applications of fiber optic

sensors to health monitoring in civil engineering.” Engineering Structures, 26(11),

1647-1657.

Li, J., Samali, B., Choi, F. C., and Dackermann, U. (2005). “Damage identification

of timber bridges using vibration based methods.” Proceedings of the 11th Asia–

Pacific Vibration Conference, Langkawi, Malaysia, 662-668.

Lu, Q., Ren, G., and Zhao, Y. (2002). “Multiple damage location with flexibility

curvature and relative frequency change for beam structures.” Journal of Sound and

Vibration, 253(5), 1101-1114.

Maia, N. M. M., and Silva, J. M. M. (1997). Theoretical and experimental modal

analysis. Research Studies Press Ltd., Great Britain.


Messina, A., Williams, J. E., and Contursi, T. (1996). “Structural damage detection

by a sensitivity and statistical-based method.” Journal of Sound and Vibration,

216(5), 791–808.

Ndambi, J. M., Vantomme, J., and Harri, K. (2002). “Damage assessment in

reinforced concrete beams using eigenfrequencies and mode shape derivatives.”

Engineering Structures, (24), 501-515.

Ni, Y. Q., Wang, B. S., and Ko, J. M. (2002). “Constructing input vectors to neural

networks for structural damage identification.” Smart Materials and Structures,

11(6), 825-833.

O’Connor, C. (1971). Design of bridge superstructures. Wiley, New York.

Ohtsu, M. (1987). “Acoustic emission characteristics in concrete and diagnostic applications.” Journal of Acoustic Emission, 6(2), 99-106.

Pandey, P. C., and Barai, S. V. (1995). “Multilayer perceptron in damage detection

of bridge structures.” Computers and Structures, 54(4), 597- 608.

Pandey, A. K., and Biswas, M. (1994). “Damage detection in structures using

changes in flexibility.” Journal of Sound and Vibration, 169(1), 3-17.

Pandey, A. K., Biswas, M., and Samman, M.M. (1991). “Damage detection from

changes in curvature mode shapes.” Journal of Sound and Vibration, 145(2), 321-

332.

Parloo, E., Guillaume, P., and Van Overmeire, M. (2003). “Damage assessment

using mode shape sensitivities.” Mechanical Systems and Signal Processing, 17(3),

499-518.

Patjawit, A., and Kanok-Nukulchai, W. (2005). “Health monitoring of highway

bridges based on a Global Flexibility Index.” Engineering Structures, 27(9), 1385-

1391.


Paz, M., and Leigh, W. (2004). Structural dynamics: theory and computation.

Springer Science + Business Media. Inc., United States of America.

Petros, P. X. (1994). Theory and design of bridges. John Wiley & Sons, Inc., New

York.

Quek, S. T., Tua, P. S., and Wang, Q. (2003). “Comparison of Hilbert-Huang,

wavelet and Fourier transforms for selected applications.” Mini-Symposium on

Hilbert-Huang Transform in Engineering Applications, October 31 - November 01,

2003, Newark, Delaware.

Ren, W. X., and Zong, Z. H. (2003). “Output-only modal parameter identification of

civil engineering structures.” Struct Eng Mech, 17(3-4), 429–444.

Roy, S., Chakraborty, S., and Sarkar, S. K. (2006). “Damage detection of coupled

bridge deck-girder system.” Finite Elements in Anal. and Design, (42), 942-949.

Rytter, A. (1993). “Vibration based inspection of civil engineering structures.”

Doctoral Dissertation, Department of Building Technology and Structural

Engineering, University of Aalborg, Aalborg, Denmark.

Saadat, S. A., Buckner, G. D., and Noori, M. N. (2007). “Structural system

identification and damage detection using the intelligent parameter varying

technique: an experimental study.” Structural Health Monitoring, 6(3), 231-243.

Shah, A. A., Ribakov, Y., and Hirose, S. (2009). “Non-destructive evaluation of

damaged concrete using non-linear ultrasonics.” Materials and Design, 30, 775-782.

Sikorsky, C. (1999). “Development of a health monitoring system for civil structures

using a level IV non-destructive damage evaluation method.” Structural Health

Monitoring 2000, 68–81.


Sohn, H., Farrar, C. R., Hemez, F. M., Shunk, D. D., Stinemates, D. W., and Nadler,

B. R. (2004). “A review of structural health monitoring literature:1996-2001.” Los

Alamos National Laboratory Report no. LA-13976-MS, Los Alamos, New Mexico.

Stubbs, N., Kim, J. T., and Farrar, C. R. (1995). “Field verification of a non-

destructive damage localization and severity algorithm.” 13th International Modal

Analysis Conference, 210-218.

Thambiratnam, D. P. (1995). “Vibration analysis of storey bridge.” Australian Civil

Engineering Transactions, CE37(2), 91-94.

Ugural, A. C. (1999). Stresses in plates and shells. McGraw-Hill, Singapore.

Ulz, M. H., and Semercigil, S. E. (2008). “Vibration control for plate-like structures

using strategic cut-outs.” Journal of Sound and Vibration, 309(1-2), 246-261.

Wong, K. Y., Lau, C. K., and Flint, A. R. (2000). “Planning and implementation of

the structural health monitoring system for cable-supported bridges in Hong Kong.”

Nondestructive Evaluation of Highways Utilities, and Pipelines IV, A.E. Aktan and

S.R. Gosselin (eds.), SPIE, (3995), 266–275.

Xia, Y., Hao, H., Zanardo, G. and Deeks, A. J., (2006). “Long term vibration

monitoring of a RC slab: temperature and humidity effect.” Engineering Structures,

28 (3), 441-452.

Xia, Y., Hao, H., Brownjohn, J. M. W., and Xia, P. Q. (2002). “Damage

identification of structures with uncertain frequency and mode shape data.”

Earthquake Engineering and Structural Dynamics, 31(5), 1053-1066.

Xu, Z. D., and Wu, Z. (2007). “Energy damage detection strategy based on

acceleration responses for long-span bridge structures.” Engineering Structures,

(29), 609-617.


Yan, A., and Golinval, J. C. (2005). “Structural damage localization by combining

flexibility and stiffness methods.” Engineering Structures, (27), 1752-1761.

Yan, Y. J., Cheng, L., Wu, Z. Y., and Yam, L. H. (2007). “Development in

vibration-based structural damage detection technique.” Mechanical Systems and

Signal Processing, 21, 2198–2211.

Yan, Y. J., Yam, L. H., Cheng, L., and Yu, L. (2006). “FEM modeling method of

damage structures for structural damage detection.” Comput Struct, 72, 193–199.

Zapico, J. L., and Gonzalez, M. P. (2006). “Vibration numerical simulation of a

method for seismic damage identification in buildings.” Engineering Structures,

(28), 255-263.

Zárate, B. A., and Caicedo, J. M. (2008). “Finite element model updating: multiple

alternatives.” Engineering Structures, 30, 3724-3730.

Zhao, J., and DeWolf, J. T. (2002). “Dynamic monitoring of steel girder highway

bridge.” Journal of Bridge Engineering, 7 (6), 350–356.

Zimmerman, D. C., and Kaouk, M. (1994). “Structural damage detection using a

minimum rank update theory.” ASME J. of Vibration and Acoustics, 116(2), 222-

231.

Zivanovic, S., Pavic, A., and Reynolds, P. (2007). “Finite element modelling and

updating of a lively footbridge: the complete process.” Journal of Sound and

Vibration, 301, 126-145.

damage assessment in structures using vibration ... · damage assessment in structures using...

Documents