damage assessment in reinforced concrete flexural...

DAMAGE ASSESSMENT IN

REINFORCED CONCRETE

FLEXURAL MEMBERS USING

MODAL STRAIN ENERGY BASED

METHOD

Buddhi Lankananda Wahalathantri

BSc Eng (Hons 1)

Submitted in partial fulfilment of the requirements for the degree of

Doctor of Philosophy

School of Civil Engineering and Built Environment

Science and Engineering Faculty

Queensland University of Technology

May 2012

Page | i

Dedication

To my son Bihan Seynila Wahalathantri, as you bring together all my memories.....

You will continue my passion and style for the next generation........... Always bring your word to the reality.......

It is neither easy nor too hard........

Page | iii

Keywords

Modal strain energy based method, reinforced concrete, flexural members, flexural

cracks, damage assessment, damage detection, damage localization, damaged

plasticity model, damage index method, vibration based damage identification

technique, structural health monitoring, MATLAB based program

Page | v

Abstract

Damage assessment (damage detection, localization and quantification) in structures

and appropriate retrofitting will enable the safe and efficient function of the

structures. In this context, many Vibration Based Damage Identification Techniques

(VBDIT) have emerged with potential for accurate damage assessment. VBDITs

have achieved significant research interest in recent years, mainly due to their non-

destructive nature and ability to assess inaccessible and invisible damage locations.

Damage Index (DI) methods are also vibration based, but they are not based on the

structural model. DI methods are fast and inexpensive compared to the model-based

methods and have the ability to automate the damage detection process. DI method

analyses the change in vibration response of the structure between two states so that

the damage can be identified. Extensive research has been carried out to apply the DI

method to assess damage in steel structures. Comparatively, there has been very little

research interest in the use of DI methods to assess damage in Reinforced Concrete

(RC) structures due to the complexity of simulating the predominant damage type,

the flexural crack. Flexural cracks in RC beams distribute non- linearly and

propagate along all directions. Secondary cracks extend more rapidly along the

longitudinal and transverse directions of a RC structure than propagation of existing

cracks in the depth direction due to stress distribution caused by the tensile

reinforcement. Simplified damage simulation techniques (such as reductions in the

modulus or section depth or use of rotational spring elements) that have been

extensively used with research on steel structures, cannot be applied to simulate

flexural cracks in RC elements. This highlights a big gap in knowledge and as a

consequence VBDITs have not been successfully applied to damage assessment in

Page | vi

RC structures. This research will address the above gap in knowledge and will

develop and apply a modal strain energy based DI method to assess damage in RC

flexural members.

Firstly, this research evaluated different damage simulation techniques and

recommended an appropriate technique to simulate the post cracking behaviour of

RC structures. The ABAQUS finite element package was used throughout the study

with properly validated material models. The damaged plasticity model was

recommended as the method which can correctly simulate the post cracking

behaviour of RC structures and was used in the rest of this study. Four different

forms of Modal Strain Energy based Damage Indices (MSEDIs) were proposed to

improve the damage assessment capability by minimising the numbers and

intensities of false alarms. The developed MSEDIs were then used to automate the

damage detection process by incorporating programmable algorithms. The developed

algorithms have the ability to identify common issues associated with the vibration

properties such as mode shifting and phase change. To minimise the effect of noise

on the DI calculation process, this research proposed a sequential order of curve

fitting technique. Finally, a statistical based damage assessment scheme was

proposed to enhance the reliability of the damage assessment results. The proposed

techniques were applied to locate damage in RC beams and slabs on girder bridge

model to demonstrate their accuracy and efficiency.

The outcomes of this research will make a significant contribution to the technical

knowledge of VBDIT and will enhance the accuracy of damage assessment in RC

structures. The application of the research findings to RC flexural members will

enable their safe and efficient performance.

Page | vii

Table of Contents

Dedication ................................................................................................................. i

Keywords ................................................................................................................ iii

Abstract .................................................................................................................... v

Table of Contents ................................................................................................... vii

List of Figures ........................................................................................................ xv

List of Tables........................................................................................................ xxv

List of Abbreviations.......................................................................................... xxvii

Statement of Original Authorship ....................................................................... xxxi

Acknowledgements ............................................................................................ xxxv

1. Introduction ......................................................................................................... 1-1

1.1. Background .................................................................................................... 1-1

1.2. Research Problem........................................................................................... 1-4

1.3. Aims and Objectives ...................................................................................... 1-7

1.4. Research Scope .............................................................................................. 1-8

1.5. Significance and Innovation ........................................................................... 1-8

1.6. Thesis Outline .............................................................................................. 1-10

2. Literature Review.............................................................................................. 2-13

2.1. Background .................................................................................................. 2-13

2.1.1. Importance of efficient transport networks ........................................... 2-13

Page | viii

2.1.2. Bridge maintenance cost in Australia .................................................... 2-14

2.1.3. Bridge failures and deterioration ........................................................... 2-16

2.1.4. Bridge management systems ................................................................. 2-17

2.2. Structural Health Monitoring (SHM) ........................................................... 2-17

2.2.1. Definition of SHM ................................................................................ 2-17

2.2.2. Applications of SHM (on bridges) ........................................................ 2-18

2.2.3. Steps of a SHM system ......................................................................... 2-19

2.2.4. Research areas related to the SHM process .......................................... 2-23

2.3. Vibration Based Damage Identification Techniques (VBDITs) .................. 2-24

2.3.1. Hypothesis of VBDIT ........................................................................... 2-24

2.3.2. Procedure of developing VBDITs ......................................................... 2-24

2.3.3. Classifications of VBDITs .................................................................... 2-29

2.3.4. Damage Index (DI) methods ................................................................. 2-33

2.3.5. Practical issues ...................................................................................... 2-39

2.4. Chapter Summary ......................................................................................... 2-40

3. Methodology and Selection of a Flexural Crack Simulation Technique ...... 3-43

3.1. Methodology ................................................................................................ 3-43

3.2. Concrete Crack Models Available in ABAQUS .......................................... 3-45

3.2.1. ABAQUS damaged plasticity model .................................................... 3-46

3.3. Numerical Models for Stress-Strain Curves ................................................. 3-50

3.3.1. Compressive stress-strain curve ............................................................ 3-50

Page | ix

3.3.2. Tensile stress-strain curve ..................................................................... 3-52

3.4. Validation of the Damaged Plasticity Model ............................................... 3-53

3.4.1. Case study 1 - Four point bending test .................................................. 3-54

3.4.2. Case Study 2 - Three point bending test ............................................... 3-63

3.4.3. Case study 3: RC slab ........................................................................... 3-67

3.5. Evaluation of Simplified Damage Simulation Techniques .......................... 3-69

3.5.1. Basics of simplified damage simulation techniques ............................. 3-69

3.5.2. FE simulation ........................................................................................ 3-70

3.6. New Displacement based Damage Severity Index ...................................... 3-73

3.7. Chapter Summary......................................................................................... 3-74

4. Theory and Equations of Damage Indices ...................................................... 4-77

4.1. Principles of Vibration ................................................................................. 4-77

4.2. Basic Parameters .......................................................................................... 4-78

4.2.1. Change in frequency ............................................................................. 4-78

4.2.2. Modal Assurance Criteria (MAC) ......................................................... 4-78

4.2.3. Modal Sensitivity Value (MSV) ........................................................... 4-79

4.3. Mode Shape Based Damage Indices ............................................................ 4-79

4.3.1. Change in mode shapes ......................................................................... 4-79

4.3.2. Change in higher order derivatives of mode shapes ............................. 4-80

4.4. Modal Flexibility Based Damage Indices .................................................... 4-81

4.5. Modal Strain Energy Based Damage Indices............................................... 4-81

Page | x

4.5.1. Modal strain energy ............................................................................... 4-82

4.5.2. Equations of SEDIs ............................................................................... 4-82

4.6. Improved SEDI ............................................................................................ 4-84

4.7. Statistical Based Damage Assessment Scheme ........................................... 4-85

4.7.1. Average Probability Distribution Index (APDI) ................................... 4-85

4.7.2. Generalized Damage Localization Index (GDLI) ................................. 4-86

4.8. Chapter Summary ......................................................................................... 4-87

5. Damage detection of cracked RC beams without noise ................................. 5-89

5.1. Details of the RC Beam Setup ...................................................................... 5-89

5.1.1. Single damage cases .............................................................................. 5-90

5.1.2. Multiple damage cases .......................................................................... 5-93

5.2. Evaluation of Damage Indices on Localizing Single Damage ..................... 5-97

5.2.1. Evaluation flexibility based DIs ............................................................ 5-97

5.2.2. Evaluation of damage indices based on mode shapes and derivatives ........

....................................................................................................................... 5-106

5.2.3. Evaluation of strain energy based damage indices .............................. 5-125

5.3. Localizing multiple Damages ..................................................................... 5-133

5.3.1. Evaluation of DIs on individual mode basis ....................................... 5-133

5.3.2. Evaluation of DIs on combined mode basis ........................................ 5-136

5.3.3. Localization of different crack patterns using β11 .............................. 5-138

5.4. Baseline Damage Elements ........................................................................ 5-150

Page | xi

5.5. Chapter Summary....................................................................................... 5-151

6. Improved Damage Detection Process ............................................................ 6-153

6.1. Definitions .................................................................................................. 6-154

6.1.1. Baseline (Undamaged) & Evaluation (Damaged) states ..................... 6-154

6.1.2. Line of sampling/measurements ......................................................... 6-154

6.1.3. Sampling point / node ......................................................................... 6-155

6.1.4. Spatial resolution ................................................................................. 6-155

6.1.5. Sampling Interval ................................................................................ 6-155

6.2. Sequential Order to Address The Forward Problem .................................. 6-155

6.2.1. Step 1: Change the spatial resolution (*f*) ......................................... 6-158

6.2.2. Step 2: Adding Noise .......................................................................... 6-166

6.2.3. Step 3: Curve fit and Normalise Mode Shapes ................................... 6-167

6.2.4. Step 4: Mode Shape Consistency Check ............................................. 6-172

6.2.5. Step 5: Phase Check ............................................................................ 6-177

6.2.6. Damage Index Calculation .................................................................. 6-178

6.3. Sequential order to address the inverse problem........................................ 6-178

6.4. Chapter Summary....................................................................................... 6-178

7. MATLAB based Software .............................................................................. 7-179

7.1. Graphical User Interfaces ........................................................................... 7-179

7.2. Step by Step Guidance ............................................................................... 7-181

7.2.1. Basic Requirements ............................................................................. 7-182

Page | xii

7.3. Enter/Browse Files ..................................................................................... 7-183

7.3.1. Read Files ............................................................................................ 7-184

7.3.2. Basic Information ................................................................................ 7-185

7.3.3. Step 01: Add Random Noise ............................................................... 7-186

7.3.4. Step 02: Select Curve Fitting Technique............................................. 7-187

7.3.5. Step 03: Activate/Deactivate Normalization ....................................... 7-190

7.3.6. Step 04: Check Consistency of Modes ................................................ 7-190

7.3.7. Step 05:Select Modes .......................................................................... 7-190

7.3.8. Check/Update ...................................................................................... 7-191

7.3.9. Calculate .............................................................................................. 7-191

7.3.10. Results ............................................................................................... 7-191

7.4. Chapter Summary ....................................................................................... 7-192

8. Applications ..................................................................................................... 8-193

8.1. Damage localization of a RC T-beam bridge using β11 ............................ 8-193

8.1.1. RC T-Beam Bridge Model .................................................................. 8-193

8.1.2. DDSIsw at serviceability limit state ..................................................... 8-202

8.1.3. Case studies ......................................................................................... 8-204

8.2. Ranking curve fitting techniques ................................................................ 8-208

8.2.1. Preliminary evaluation of nine curve fitting techniques ..................... 8-208

8.2.2. Evaluation of CFT04, CFT05, CFT07, CFT08 and CFT09 based on

probabilities of correct condition detection and false alarms ........................ 8-213

Page | xiii

8.3. Damage localization using GDLI(β11) ...................................................... 8-215

8.3.1. Improved damage localization results achieved from GDLI .............. 8-215

8.3.2. Minimum number of samples required to achieve different accuracy

levels ............................................................................................................. 8-216

8.3.3. Maximum accuracy level (ALmax) ...................................................... 8-220

8.3.4. Confirming damage states of elements ............................................... 8-223

8.4. Chapter Summary....................................................................................... 8-225

9. Conclusion & Future Work ........................................................................... 9-227

9.1. Conclusion ................................................................................................. 9-227

9.2. Future Work ............................................................................................... 9-230

List of Reference ............................................................................................... 9-231

Page | xv

List of Figures

Figure 2.1: Picture of the Sydney Harbour Bridge ("Sydney Harbour Bridge") .... 2-15

Figure 2.2: Pictures of (a): The Storey Bridge, (b): The Victoria Bridge ............... 2-16

Figure 2.3: Progressive crack patterns under three point bending test as observed by

(Maeck 2003) .......................................................................................................... 2-28

Figure 3.1: Compressive Stress-Strain Relationship ("Abaqus Analysis User Manual

– Abaqus Version 6.8. " 2008) ............................................................................... 3-48

Figure 3.2: Tension Stiffening Model ("Abaqus Analysis User Manual – Abaqus

Version 6.8. " 2008) ............................................................................................... 3-49

Figure 3.3: Compressive Stress-Strain Relationship for ABAQUS ....................... 3-52

Figure 3.4: Tension Stiffening Model (Nayal and Rasheed 2006) ......................... 3-53

Figure 3.5: Modified Tension Stiffening Model ..................................................... 3-53

Figure 3.6: Cross Section of the RC Beam ............................................................. 3-54

Figure 3.7: Experiment setup with added inertia loads ........................................... 3-55

Figure 3.8: Experimental setup of the four point bending test ................................ 3-55

Figure 3.9: Linear portion of the experimental load-displacement curve (Perera and

Huerta 2008)............................................................................................................ 3-56

Figure 3.10: Partitioned cross section to assign the smeared reinforcement layers .......

................................................................................................................................. 3-57

Page | xvi

Figure 3.11: Load-Displacement curves obtained from experimental and FE model

results ...................................................................................................................... 3-62

Figure 3.12: Experimental setup (Peeters et al. 1996) ............................................ 3-63

Figure 3.13: Cross section details of the RC beam (Peeters et al. 1996) ................ 3-63

Figure 3.14: Experiment Load vs. Displacement (Peeters et al. 1996) ................... 3-66

Figure 3.15: Finite Element Load vs. Displacement ............................................... 3-66

Figure 3.16: Crack Patterns for 3 point bending test at 8kN ................................... 3-67

Figure 3.17: Crack Patterns for 3 point bending test at 32kN ................................. 3-67

Figure 3.18: Experimental test setup (Bakhary 2009) ............................................. 3-68

Figure 3.19: Damage simulation by reducing the E-value ...................................... 3-69

Figure 3.20: Damage simulation by creating a notch/cut ........................................ 3-69

Figure 3.21: Reduction factor λ1 for a four point bending test setup ...................... 3-70

Figure 5.1: First four flexural modes of the undamaged RC beam ......................... 5-90

Figure 5.2: Loading arrangement for single damage cases ..................................... 5-90

Figure 5.3: Load arrangement to create multiple damage locations ....................... 5-94

Figure 5.4: Variation of α1 for the damage case SDM5 with first four flexural modes

................................................................................................................................. 5-99

Figure 5.5: Variation of α1 for the damage case SDQ5 with first four flexural modes

............................................................................................................................... 5-101

Figure 5.6: Variation of α2 for the damage case SDM5 with first flexural mode 5-102

Figure 5.7: Variation of α2 for the damage case SDQ5 with first flexural mode ..........

............................................................................................................................... 5-102

Page | xvii

Figure 5.8: Variation of α2 for the damage case SDM5 with third flexural mode

............................................................................................................................... 5-103

Figure 5.9: Variation of α2 for the damage case SDQ5 with second flexural mode .....

............................................................................................................................... 5-103

Figure 5.10: Variation of α3 for the damage case SDM5 with first four flexural

modes .................................................................................................................... 5-104

Figure 5.11: Variation of α4 for the damage case SDM5 with first four flexural

modes .................................................................................................................... 5-105

Figure 5.12: Variation of γ for the damage case SDM1 with first flexural mode..........

............................................................................................................................... 5-107

Figure 5.13: Variation of γ for the damage case SDM5 with first flexural mode..........

............................................................................................................................... 5-107

Figure 5.14: Variation of γ for the damage case SDQ5 with second flexural mode......

............................................................................................................................... 5-107

Figure 5.15: Variation of γ for the damage case SDQ5 with first flexural mode ..........

............................................................................................................................... 5-108

Figure 5.16: Variation of γ1, γ2, γ3 and γ4 for damage case SDM1 with first flexural

mode ...................................................................................................................... 5-110

Figure 5.17: Variation of γ1, γ2, γ3 and γ4 for damage case SDQ1 with first flexural

mode ...................................................................................................................... 5-111

Figure 5.18: Variation of γ1, γ2, γ3 and γ4 for damage case SDM1 with second

flexural mode ........................................................................................................ 5-113

Page | xviii

Figure 5.19: Variation of γ1, γ2, γ3 and γ4 for damage case SDQ1 with fourth

flexural mode ......................................................................................................... 5-114

Figure 5.20: Variation of γ2 based on first flexural mode with different damage

severities ................................................................................................................ 5-117

Figure 5.21: Variation of γ2 for damage case SDQ5 with first flexural mode ..... 5-117

Figure 5.22: Variation of γ4 for damage case SDM5 with first flexural mode ..... 5-118

Figure 5.23:Variation of γ2c(1:4) for damage cases SDM1, SDM5, SDQ1 and SDQ5

............................................................................................................................... 5-120

Figure 5.24: Curvature of first four mode shapes without normalizing ................ 5-123

Figure 5.25: Normalized curvature of first four mode shapes .............................. 5-123

Figure 5.26: Variation of γ2cn(1:4) for damage cases SDM1, SDM5, SDQ1 and

SDQ5 ..................................................................................................................... 5-125

Figure 5.27:Damage localization results using β1 for the damage case SDM1 with

third mode ............................................................................................................. 5-127

Figure 5.28: Damage localization results using β1 for the damage case SDQ1 with

second mode .......................................................................................................... 5-127

Figure 5.29: Damage localization results using β2, β3, β8 and β10 for the damage

case SDQ1 with second mode ............................................................................... 5-129


fourth mode ........................................................................................................... 5-129


fourth mode ........................................................................................................... 5-130

Page | xix


first four modes ..................................................................................................... 5-131


first four mode ....................................................................................................... 5-131

Figure 5.34: Damage localization results using β4, β6, β9 and β11 for the damage

case SDQ1 with first four mode ............................................................................ 5-132

Figure 5.35: Damage localization results using α2, γ2, γ4 and β10 for damage case

MD1 ...................................................................................................................... 5-135

Figure 5.36 : Damage localization using β10 of fourth mode for the damage case,

MD1 ...................................................................................................................... 5-136

Figure 5.37: Damage localization results using α4, γ2cn, γ4cn and β11 for damage

case MD1 .............................................................................................................. 5-138

Figure 5.38 : Damage localization results for MD2 using β11 ............................. 5-139

Figure 5.39: Damage localization results for MD7 using β11 .............................. 5-139

Figure 5.40 : Variation of β11 for damage case MD3 after each damage state (using

undamaged state as baseline) ................................................................................ 5-141

Figure 5.41: Proposed method to detect onset of cracking using different baselines ....

............................................................................................................................... 5-142

Figure 5.42: Localizing the second damage at quarter span for MD3 using proposed

baseline update method ......................................................................................... 5-143

Figure 5.43: Damage localization results of MD4 using proposed baseline update

method ................................................................................................................... 5-144

Page | xx


method ................................................................................................................... 5-144


method ................................................................................................................... 5-145


method ................................................................................................................... 5-146

Figure 5.47: Applied loads, W1, W2 and W3 vs. Stage number .......................... 5-147

Figure 5.48: Variation of DDSIs at each stage ...................................................... 5-147

Figure 5.49: Damage localization results of β11 using proposed baseline update

method ................................................................................................................... 5-149

Figure 6.1: Sequential Order to address the forward problem .............................. 6-157

Figure 6.2: Three stage procedure to change the spatial resolution ...................... 6-159

Figure 6.3: Reducing the spatial resolution ........................................................... 6-163

Figure 6.4: Increasing the spatial resolution ......................................................... 6-164

Figure 6.5: Fifth flexural mode shape with and without noise .............................. 6-164

Figure 6.6: Reduction in spatial resolution of noisy mode shape ......................... 6-165

Figure 6.7: Increasing spatial resolution of noisy mode shape ............................. 6-166

Figure 6.8: Skeleton diagram on obtaining correlated modes ............................... 6-173

Figure 6.9: 3D and 1D views of first five modes of T-Beam Bridge at undamaged

state ........................................................................................................................ 6-175

Figure 7.1: View of the Main GUI ........................................................................ 7-180

Figure 7.2: View of the Secondary GUI (Visualizing results) ............................. 7-180

Page | xxi

Figure 7.3: First part of a typical ABAQUS output file (output_file_abaqus.rpt)

............................................................................................................................... 7-182

Figure 7.4: Fist part of a typical ABAQUS Input File (input_file_abaqus.inp) ............

............................................................................................................................... 7-182

Figure 8.1: Basic layout of the selected RC T-beam bridge (Pike) ...................... 8-194

Figure 8.2: Reinforcement arrangement of the selected RC T-beam bridge (Pike) .......

............................................................................................................................... 8-194

Figure 8.3: Cross section of the simulated RC T-beam bridge model .................. 8-195

Figure 8.4: Grid layout (Isometric View) ............................................................. 8-195

Figure 8.5: Grid layout (on XY plane - top of slab).............................................. 8-196

Figure 8.6:Region with the section Sect_Con_Und .............................................. 8-197

Figure 8.7: Region with the section Sect_Con_Dam ............................................ 8-197

Figure 8.8: Smeared Layer with the section Sect_RF_Beam ............................... 8-198

Figure 8.9: Reinforcement properties assigned to the Sect_RF_Beam................. 8-198

Figure 8.10: Material orientation of the reinforcement layers .............................. 8-198

Figure 8.11: Smeared Layer with the section Sect_RF_Bot ................................. 8-199

Figure 8.12: Reinforcement properties assigned to the Sect_RF_Bot .................. 8-199

Figure 8.13: Smeared Layer with the section Sect_RF_Top ................................ 8-199

Figure 8.14: View of the meshed bridge model .................................................... 8-200

Figure 8.15: Points where displacements were recorded ...................................... 8-201

Figure 8.16: Lines where the modal displacements (mode shapes) were measured ......

............................................................................................................................... 8-201

Page | xxii

Figure 8.17: First 12 mode shapes of the simulated T-beam bridge model .......... 8-202

Figure 8.18: Damage state to reach the maximum allowable deflection (at

serviceability limit state) ....................................................................................... 8-204

Figure 8.19: Flexural crack pattern of case study 1 .............................................. 8-205

Figure 8.20: Damage localization using β11 for the case study 1 ......................... 8-205

Figure 8.21: Flexural crack patterns of case study 2 ............................................. 8-206


Figure 8.23: Flexural crack patterns of case study 3 ............................................. 8-207


Figure 8.25: Damage localization using β11 for the case study 3 (without girder-G) ...

............................................................................................................................... 8-208

Figure 8.26: Comparison of nine curve fitting techniques for the damage case SDM5

with 1% noise based on SEDI β11 ........................................................................ 8-212

Figure 8.27: Comparison of five curve fitting techniques at 1% noise ................. 8-214

Figure 8.28: Comparison of five curve fitting techniques at 2% noise ................. 8-215

Figure 8.29: Damage localization results using non-standardized probability values

for SDM5 at 5% noise ........................................................................................... 8-216

Figure 8.30: Damage localization results using GDLI(β11) for SDM5 at 5% noise .....

............................................................................................................................... 8-216

Figure 8.31: Damage localization results of GDLI(β11) for SDM1 using 16 samples

with 1% noise ........................................................................................................ 8-217

Figure 8.32: PCCD vs NS for SDM1 with 1% noise ............................................ 8-218

Page | xxiii

Figure 8.33: PCCD vs NS for SDM1 with 3% noise ............................................... 8-218

Figure 8.34: PCCD vs NS for SDM1 with 5% noise ............................................... 8-219

Figure 8.35: PDD vs NS for SDM1 with 5% noise ................................................ 8-219

Figure 8.36: Damage localization results of GDLI(β11) for SDM5 using 20 samples

with 5% noise ........................................................................................................ 8-220

Figure 8.37: PCCD vs NS for SDM5 at 1%, 3% and 5% noise ............................ 8-220

Figure 8.38: ALmax for five mid span damage cases at 1%, 3% and 5% noise levels ...

............................................................................................................................... 8-223

Figure 8.39: ALmax for five quarter span damage cases at 1%, 3% and 5% noise

levels ..................................................................................................................... 8-223

Figure 8.40: GDLI(β11) at 3% noise for SDM1 ................................................... 8-224

Figure 8.41: Cumulative probability values at 3% noise for SDM1 ..................... 8-224

Figure 8.42: Reliability Indices at 3% noise for SDM1 ........................................ 8-224

Page | xxv

List of Tables

Table 2-1: Four categories of VBDITs as proposed by (Lee, Karbhari and Sikorsky

2004) ....................................................................................................................... 2-31

Table 3-1: Material properties for concrete under compression ............................. 3-58

Table 3-2:Material properties for concrete under tension ....................................... 3-59

Table 3-3: Frequencies of mode 1 and mode2 ........................................................ 3-62

Table 3-4: Tensile Stress-Strain Values for Concrete with 51.2MPa ..................... 3-64

Table 3-5: Compressive Stress-Strain Values for Concrete with 51.2MPa ............ 3-65

Table 3-6: Comparison of present FE results with experimental results of (Bakhary

2009) ....................................................................................................................... 3-68

Table 3-7: Reduction factor, λ1 ............................................................................... 3-71

Table 3-8: Reduction factor, λ2 ............................................................................... 3-72

Table 3-9: Comparison of reduction in E-value method with the experimental results

using first two frequencies ...................................................................................... 3-72

Table 3-10: Comparison of notch type damage simulation method with the

experimental results using first two frequencies ..................................................... 3-73

Table 5-1: Ten Single Damage Cases ..................................................................... 5-91

Table 5-2: Multiple damage cases........................................................................... 5-94

Table 5-3: Peak values of α4 for different combinations of modes ...................... 5-105

Table 5-4: Maximum γ2-Mid Span ....................................................................... 5-121

Table 5-5: Maximum γ4 at mid span .................................................................... 5-121

Page | xxvi

Table 5-6:Elements with positive β11D1:i at ends of each loading steps ............... 5-150

Table 5-7: Damaged elements identified by β11 at the baseline state .................. 5-151

Table 8-1: False alarms at 1% noise ...................................................................... 8-213

Table 8-2: Reliability indices of localized damage elements ................................ 8-225

Page | xxvii

List of Abbreviations

SHM : Structural Health Monitoring

VBDIT: Vibration Based Damage Identification Techniques

DI: Damage Index

MSEDI: Modal Strain Energy based Damage Index

Symbols

i : ith

mode (or mode number)

j : jth

node (or node number)

e: jth

element (or element number) in case of strain energy based damage

indices

M : total number of measured modes

N : total number of nodes / elements in case of strain energy based

damage indices

Ne: Total number of elements

Subscripts

d : to represent the damaged state

u : to represent the undamaged state

(i): to denote that the damage index has been calculated based on ith

mode

c : to denote that damage indices have been calculated by combining

modes

c(1:m): to denote that the modes 1 to m are combined

Page | xxviii

Note: These subscripts may have been omitted upon providing a clear descriptions

upfront.

Superscripts

' : First derivative

'' : Second derivative

''': Third derivative

'''': Fourth derivative

Matrix name, intended purpose and size

ω : Contains frequency values- size 1xM

φ or Ф : Contains Mode shapes - size NxM

FRQ : Saves percentage change in frequency - a single row matrix with size

1 x M

[MAC]MuXMd : Saves MAC values - size Mu x Md; where Mu and Md

represent the total number of measured modes at undamaged and damaged states.

MSV : Saves Modal Sensitivity Values - a single row matrix with size 1 x M

γ : Saves DIs based on change in mode shapes - size N x M

γ1: Saves DIs based on change in first derivatives of mode shapes - size N

x M

γ2: Saves DIs based on change in second derivatives of mode shapes -

size N x M

γ3: Saves DIs based on change in third derivatives of mode shapes - size

N x M

Page | xxix

γ4: Saves DIs based on change in fourth derivatives of mode shapes - size

N x M

F : Saves flexibility values - size N x M

[1/ω2] : represents the diagonal matrix which contains reciprocal of natural

frequencies of modes - size MxM.

α1, α2, α3, α4 : DIs based on change in flexibility values - size N x M

β1-β11: Strain Energy based Damage Indices - size Ne x M

Page | xxxi

Statement of Original Authorship

The work contained in this thesis has not been previously submitted to meet

requirements for an award at this or any other higher education institution. To the

best of my knowledge and belief, the thesis contains no material previously

published or written by another person except where due reference is made.

Signature: _________________________

Date: _________________________

Page | xxxii

PUBLICATION LIST

The published papers based on this research are listed below.

Conference Papers

Wahalathantri, B.L., D.P. Thambiratnam, T.H.T. Chan and S. Fawzia. (2012)

"Evaluation of strain energy based damage indices for flexural crack detection of RC

beams." In Proceedings of the International Conference on Advances in

Computational Mechanics (ACOME), August 2012, Vietnam National University,

Ho Chi Minh City, Vietnam.

Wahalathantri, B.L., D.P. Thambiratnam, T.H.T. Chan and S. Fawzia. (2011) "A

material model for flexural crack simulation in reinforced concrete elements using

ABAQUS." In Proceedings of the First International Conference on Engineering,

Designing and Developing the Built Environment for Sustainable Wellbeing, April

2011, Queensland University of Technology, Brisbane, Australia, 260-264.

Wahalathantri, B.L., D.P. Thambiratnam, T.H.T. Chan and S. Fawzia. (2010) "An

improved modal strain energy method for damage assessment." Proceedings of the

Tenth International Conference on Computational Structures Technology (CST

2010), September 2010, Universidad Politécnica de Valencia, Spain.

Wahalathantri, B.L., D.P. Thambiratnam, T.H.T. Chan and S. Fawzia. (2009)

Assessment of 'change in flexibility method' in structural health monitoring systems,

Proceedings for the 3rd Smart Systems Postgraduate Student Conference, 2009,

Queensland University of Technology, Brisbane, Australia, 120-125.

http://eprints.qut.edu.au/41712/





Page | xxxiii

Journal Papers

Wahalathantri, B.L., D.P. Thambiratnam, T.H.T. Chan and S. Fawzia. (2012) "An

Improved Method to Detect Damage Using Modal Strain Energy Based Damage

Index." Advances in Structural Engineering, 15, 5, 2012.

Page | xxxv

Acknowledgements

My sincere gratitude is first expressed to Professor David Thambiratnam for

being the principal supervisor and giving me this opportunity together with his

motivation, great support and excellent guidance provided throughout this research

study. Secondly, I would like to thank my associate supervisors, A/Professor Tommy

Chan and Dr. Sabrina Fawzia for their valuable advices, useful suggestions and

professional guidance given during the entire period of time.

I must thanks all the academic and non academic staff members at QUT for

their support given in many ways specially in SEF research portfolio office and HPC

unit for their assistance and cooperation during the research and enthusiastic

responses to my numerous requests for assistance.

Special thank should go to Prof. Andy Tan, for being the examiner in my

Confirmation of Candidature Seminar (COC) and the Final Seminar. His advices,

suggestions and comments given in my COC were helped to enhance the quality of

this research study.

I would like to express my sincere gratefulness to my parents, my loving wife,

Santhushi Sondrangalla, son, Bihan Seynila Wahalathantri and my heartiest siblings

for their great understanding, encouragement and support given throughout this

carrier in many ways.

I gratefully acknowledge the financial support granted through IPRS and QUT

PRA to succeed my research work for entire period of my candidature.

I wish to gratitude my colleagues at QUT for sharing knowledge and

encouragement at friendly and fruitful atmosphere and all those who helped me in

Page | xxxvi

many ways to my success. Last but not least, I must acknowledge all of my teachers

from preschool to higher school level (at Gnanodaya Maha Vidyalay, Kalutara, Sri

Lanka and Panadura Sri Sumangala Boys School, Panadura Sri Lanka), lectures at

University of Moratuwa, Sri Lanka (specially Prof. W.P.S. Dias and Dr. K. Baskaran

for being my research supervisors) for laying the foundation in my early career.

Buddhi Lankananda Wahalathantri

School of Civil Engineering and Built Environment

Science and Engineering Faculty

Queensland University of Technology

Brisbane, Australia

May 2012

Page | 1-1

1. Introduction

1.1. Background

Bridges are subjected to changes in magnitude and pattern of loading in

addition to deterioration with age during their service life and hence are more

vulnerable to periodical damage than other structures. Some examples in the local

context are; excessive vibrations in the Storey Bridge (Brisbane, Australia) caused by

heavy traffic loads (Thambiratnam 1995), torsional vibrations in the Victoria Bridge

(Brisbane, Australia) due to modifications made to accommodate bus lanes on one

side of the bridge. It has been reported that overloading and lateral excitation caused

by heavy traffic loads accounted for 20% of bridge failures in USA (Wardhana

2003). Natural disasters, human induced activities such as terrorist attacks,

maintenance issues and imperfections in design, detailing and material are some

other causes of structural damage in bridges. Any form of structural damage can

have a detrimental influence on the serviceability and ultimate capacity of bridges

and may result in loss of lives and property in case of catastrophic failures or may

incur higher maintenance and repair costs.

Keeping bridge infrastructure safe, accessible and serviceable is therefore, vital

for both community and industry. Providing safe and efficient bridges to enhance the

community lifestyles and to promote economic prosperity is a key aspect of road

authorities in all countries. Many road authorities are therefore keen on deploying

proper bridge management systems, which incorporate Structural Health Monitoring

(SHM) schemes, to establish safer performance levels and prevent catastrophic

failures. In this context the Queensland Department of Main Roads and the Brisbane

Page | 1-2

City Council, both of which are bridge owners, have funded an Australian Research

Council project on SHM of bridges.

Damage assessment is an integral part of a SHM scheme which aims to detect,

locate and quantify the damage. Various techniques have been attempted to assess

damage in structures, including local methods such as acoustic or ultrasonic,

magnetic-field, radiograph, eddy-current and thermal field methods and global

methods such as Vibration Based Damage Identification Techniques (VBDIT). Local

methods require the vicinity of the damage to be known and readily accessible and

hence limited to detect damage on or near the surface of the structure. Global

methods such as Vibration Based Damage Identification Techniques (VBDITs) have

achieved significant research interest in recent years, mainly due to their non-

destructive nature, ability to provide uninterrupted use and ability to detect

inaccessible and invisible damage locations.

The basis of the VBDIT is that the damage alters the stiffness of a structure,

which in turn, modifies the vibration characteristics of the structure. The primary

vibration properties, which are modified due to the presence of damage, are natural

frequencies, mode shapes and damping values. This results in variation in secondary

vibration properties such as flexibility, mode shape curvatures, stiffness and strain

energy values of structural elements. Conversely, any changes in either primary or

secondary vibration properties can be used as an indication of presence of damage

which in turn creates the platform for the VBDITs. Formulation of relationships

between damage characteristics and changes in vibration properties, therefore, has

the potential to assess damage.

Different damage assessment techniques have been presented in the field of

VBDITs using either the Damage Index (DI), Artificial Neural Network (ANN), or

Page | 1-3

Correlation based methods (eg: Genetic Algorithm (GA)). Among these three

methods, the former one, DI method, uses changes in vibration properties such as

frequencies, mode shapes, damping values, or their derivatives to assess the damage.

DI method is, therefore, categorised as an output based method which has the ability

to eliminate model updating techniques during damage assessment process. In other

terms, the measured vibration properties can be directly used to assess the damage

with the DI method, whereas other methods require the model updating techniques to

be deployed in parallel with the measurements. DI method can therefore be easily

programmable and extendable to automate the damage assessment process.

Extensive investigations have been carried out to apply the DI method to assess

damage in steel structures but not for the Reinforced Concrete (RC) structures.

Further, measurement noise is another limiting fact for the DI method as with the

other two damage assessment techniques, ANN and GA based methods. This

background and gaps in knowledge motivated the present study to develop a proper

DI method that can be applied to detect and assess damage in RC structures.

The DI method is a two-stage process, which involves; 1. addressing the

forward problem and 2. addressing the inverse problem, in succession. When the

forward problem is addressed, known damage properties are induced and changes in

vibration properties are measured. The reverse happens when the inverse problem is

addressed to assess the unknown damage states of the candidate structure using the

changes in vibration characteristics. Simplified damage simulation methods, such as

reductions in the values of the Young’s Modulus or the second moment of the area

and notch type damage, used widely in VBDITs, assume a concentrated and/or

uniform distribution of the damage. The ability of the DI method to localise such

Page | 1-4

simulated damage has been verified both experimentally and numerically, especially

for structures made of steel or similar type of homogeneous materials.

However, there is an uncertainty of implementing such simplified damage

simulation techniques for Reinforced Concrete (RC) elements due to the nature of

the predominant damage type, the flexural cracks. Flexural cracks in RC beams

indicate complex behaviour as they propagate along all directions. Cracks distribute

uniformly along the width of a RC beam under flexural loading, but they do not

propagate uniformly along the longitudinal and depth directions as the severity

reduces away from the centre of the cracking zone. Secondary cracks extend more

rapidly along the longitudinal direction than propagation of existing cracks in the

depth direction due to stress distribution caused by the tensile reinforcement.

Because of this, flexural cracks in RC beams take a parabolic shape and cause

complexities in finite-element simulations. Simplified damage simulation techniques,

therefore, cannot be applied to simulate flexural cracks in RC elements.

As a result, the forward problem of the DI method has not been adequately

addressed for RC structures. This highlights a big gap in knowledge and reduces the

confidence of applying the DI method to address the inverse problem of the damage

assessment process in RC structures.

1.2. Research Problem

This research will address the above mentioned knowledge gaps and problems

and will develop and apply a modal strain energy based DI method to assess damage

in RC flexural members.

The research problems addressed in this study are listed below. The first three

research problems identified in this study are based on the knowledge gaps, while the

Page | 1-5

others two are selected to improve the accuracy and efficiency of the proposed

damage assessment technique.

The Research problems are;

1. Damage simulation technique

As highlighted above, simplified damage simulation techniques cannot be used

to simulate the post cracking behaviour of flexural cracks due to its non linear

distribution across the structure. An appropriate damage simulation techniques needs

to be proposed to treat RC structures.

2. Definition of damage severity

At present, the damage severity is defined in terms of reduction in E-value, I-

value, or flexural rigidity and assumes a uniform distribution across the damage

zone. As flexural cracks in RC structures do not distribute uniformly across the

damage zone, an alternative method is needed to define the severity of flexural

cracks.

3. Evaluation of damage indices

Different forms of Damage Indices (DIs) have been presented in the literature

based on changes in frequencies, mode shapes, mode shape derivatives, flexibility

and strain energy values of the structure. They have indicated different levels of

success on damage detection of structures made of steel. Few studies have attempted

to apply DI method to detect damage in RC structures. All the previous studies have

been conducted using simplified damage simulation techniques. This research

proposes four new DIs based on modal strain energy. It then evaluates these four new

DIs as well as existing DIs based on the damage detection ability of RC structures

and ranks them.

Page | 1-6

Evaluation and ranking of multiple DIs require a large number of damage cases

and demands huge amount of time. Existing research studies have therefore confined

to assess the applicability of a few selected damage indices.

4. Minimising false alarms

Two types of false alarms have been reported with the DI methods, namely; 1.

positive false alarms and 2. negative false alarms. Positive false alarms lead to over-

estimate the extent of damage while negative false alarms cause under-estimation.

Higher maintenance and repair cost may incur if the damage is over estimated, or it

may cause unexpected failure in case of under-estimation. Therefore, it is essential to

minimise false alarms associated with the DI method.

False alarms are caused either by the DI calculation process itself or due to

presence of measurement noise or combination of both. False alarms associated with

the DI calculation process itself can be observed at nodal points of higher-order

modes. DIs derived by obtaining the ratio of the candidate vibration property

between two states may cause abrupt changes at nodal points of higher-order modes

and lead to generate positive false alarms with higher intensities. A proper approach

is, therefore, needed to minimise the intensities of false alarms generated at nodal

points of higher order modes.

False alarms can be caused by the measurement noise which is unavoidable as

structures are subjected to different environmental conditions and loading patterns. In

case of Bridges, measurement noise is further increased due to moving traffic loads.

Although improvements are continuously being achieved in terms of reducing the

measurement noise, elimination of all measurement noise cannot be practically

Page | 1-7

achieved. Levels of accuracy that can be achieved with the candidate DI should,

therefore, be defined under different noise levels.

5. Automating the damage assessment procedure

Damage assessment procedures are complex and time consuming. Thus far,

developing an automated system to assess damage using DI method has not been a

success. An automated system requires programmable criteria to be developed to

detect damage zone and address practical issues such as mode shifting and phase

change. Programmable criteria should therefore be developed.

1.3. Aims and Objectives

The main aim of this research is to develop and apply a robust Modal Strain

Energy based Damage Index (MSEDI) method to detect and localize flexural cracks

in RC structures and to improve the damage assessment process by minimising the

number and intensity of false alarms.

The six specific research objectives which address the research problems listed

above and which enable to achieve the aim of this research are the following:

1. To evaluate and determine appropriate technique to simulate post cracking

behaviour of RC structures.

2. To develop a method to define the damage severity of cracked RC structures.

3. To identify problems associated with the existing DIs and propose improved

MSEDIs and evaluate their performance.

4. To develop programmable algorithms which have the ability to automate the

damage assessment process and address practical issues (such as mode shifting,

phase change and measurement noise)

Page | 1-8

5. To develop a probability based approach which can apply to structural health

monitoring schemes with continuous or periodical data acquisition systems and

recommend the accuracy levels that can be achieved at different noise levels.

6. Illustrate the procedure developed in this research through its application to assess

the damage in RC beams and slabs on girder bridge model.

1.4. Research Scope

The complete damage assessment process using the DI method is a broad

research area which involves three inter-related steps, namely, 1. data acquisition, 2.

data processing and 3. damage assessment. This research focuses on the third step

which is the damage assessment of structures. It addresses the first two aspects of the

damage assessment namely, damage detection and localization of flexural cracks in

RC structures, but not the third step which is the damage quantification.

Improvements to damage localization results have been achieved in this research

using a modified Modal Strain Energy based Damage Index (MSEDI), sequential

order of curve fitting technique and a standardization approach. Proposed

modifications and improvements were first formed and established using a validated

RC beam at both intact and different damage states. The developed method was

applied and verified for its damage localization ability along the longitudinal

direction of a simulated RC slab on girder bridge structure.

1.5. Significance and Innovation

This research is significant as its outcomes will contribute towards damage

assessment in Reinforced Concrete structures (bridges and buildings) and enable

their safe and efficient operation.

Page | 1-9

Until this point in time, a proper damage simulation technique has not been

used to simulate flexural cracks in RC structures for use with VBDTIs. Because of

this, the forward problem of the damage detection process in RC structures has not

been adequately addressed and the inverse problem has not been a success. This

study has proposed a damage simulation technique which can correctly simulate

flexural cracks in RC structures with sufficient accuracy. Accurate damage

simulation techniques will reduce the number of experimental studies and will allow

analysing large number of damage cases during the forward problem study. This

research will, therefore, promote extensive future research activities on damage

detection of RC structures using VBDITs. As a result, continuous improvements in

the damage detection process can be expected. From a broader aspect, this research

will lay the platform to establish safer performance levels of bridges and other

structures made of RC. From a technical context, the present study has introduced

significant and innovative concepts to improve the damage detection results of the DI

method as listed below.

1. An appropriate technique to simulate flexural cracks in RC members.

2. Development and application of new strain energy based DIs which can minimise

the numbers and intensities of false alarms generated at nodal points of higher-

order modes

3. A sequential order of the DI calculation process.

4. A sequential curve fitting technique to reduce the negative influence of

measurement noise on the DI calculation process.

5. A statistical approach to improve the reliability of damage assessment results of a

continuously monitoring structure with the presence of noise.

Page | 1-10

To save time and preserve the accuracy of the DI calculation process, a

MATLAB based programme has been developed. This programme has a Graphical

User Interface (GUI) and has the ability to address both forward and inverse problem

of the damage detection process. The first three novel concepts listed above are

already incorporated to the developed MATLAB programme as options to be

selected. Although the current version of the programme is limited to read ABAQUS

input and output files to obtain vibration properties of the candidate structure at two

different states, it can be modified to compatible file formats of other FE packages or

experimental data.

1.6. Thesis Outline

This thesis is organised using 10 chapters listed below.

Chapter 1: Introduction - This chapter introduces the background information of this

research and presents research problems that have been addressed.

Research aims and objectives, scope, significance and innovation in this

research are presented.

Chapter 2: Literature Review - This chapter highlights gaps in knowledge on

VBDITs through a comprehensive literature review.

Chapter 3: Methodology and Flexural Crack Simulation - The methodology of the

present study is briefly elaborated in the first section of this chapter.

Then, it establishes the suitability of the damaged plasticity model to

simulate post cracking behaviour of flexural cracks in RC structures.

Chapter 4: Theory and Equations of Damage Indices- Chapter 4 presents the theory

and equations of damage indices used in this study.

Page | 1-11

Chapter 5: Damage Detection of Cracked RC Beam without Noise - Damage

localization results of 25 different forms of DIs are presented to rank

them based on the number and intensity of false alarms. Results indicate

that the Modal Strain Energy based Damage Indices (MSEDI) proposed

in this study have the highest accuracy in localizing flexural cracks in RC

structures.

Chapter 6: Improved Damage Detection Process - This chapter presents the details of

the proposed damage detection process which can be programmable.

This process is named as the Multi-Step Damage Detection Process

(MSDDP).

Chapter 7: MATLAB based Programme - Overview of the MATLAB based

programme is presented with a step by step guidance. This chapter

provides the step by step guidance to use the proposed MATLAB based

program.

Chapter 8: Case Studies - Some case studies are presented to validate the

applicability of the proposed MSEDI.

Chapter 9: Conclusions and Future Work - This chapter presents main conclusions

drawn from this study and suggests future work related to this research.

Page | 1-12

Page | 2-13

2. Literature Review

This chapter reviews current knowledge and gaps on VBDITs, damage

simulation techniques, damage detection methods and their applications.

The first section highlights the need for proper bridge management systems.

The second section overviews the bridge management process and defines the

Structural Health Monitoring (SHM) process. Applications of SHM in fields such as

rotating machineries, aerospace industry and civil engineering field are discussed in

this section to identify some of the main technical challenges. Available vibration

based damage indices are discussed in section 2.2.4 of the report.

Formulas for damage indices are presented in section 2.3 of the report with

some technical details. Some of the other aspects are defined in next section,

including forward problem and inverse problem. Also, this section attempts to

identify the technical gaps in terms of multiple damage, damage types and

localization of damage in both longitudinal and transverse directions in case of plate

like structure. The time consuming step involved in damage detection techniques in

terms of calculation perspective is discussed in section 2.5. The section 2.6

summarise the available experiment and numerical results which can be used to

validate FEMs. The last section summarizes all the important facts to elaborate the

existing gaps in knowledge

2.1. Background

2.1.1. Importance of efficient transport networks

Maintaining efficient transport networks is essential to the economy of any

country. Annual budget allocation of Australia can be used as an indication to

Page | 2-14

illustrate the importance of transport networks in the local context. The Australian

Federal Government had allocated a total of $ 8.5 billion in 2009/2010 financial year

towards transport infrastructures with roads at $ 3.4 billion, rails at $ 4.6 billion and

ports at $ 389 million (Budget at a glance 2009). In 2011/2012 financial year, $36

billion investments had been proposed for roads, rails and ports (Budget at a glance

2011).

The transport network in Australia is linked by more than 33,500 bridges

across the country. Failures in bridges can have a detrimental effect to the

community and the economic prosperity of the country. Prevention of bridge failures

are therefore important as much as developing new infrastructures to meet

continuously rising traffic flows.

2.1.2. Bridge maintenance cost in Australia

Bridges need regular maintenance and repair to keep them safe, accessible and

serviceable. The sections given below illustrate some examples to indicate the extent

of bridge maintenance cost in Australia.

2.1.2.1. Bridge maintenance cost in Western Australia

The Main Roads Western Australia, which is the responsible organization for

maintaining one of the largest geographically spread road networks in the world, has

estimated cost of fixing 1032 bridges in the state road network as $250 million

according to the Western Australian Auditor General’s Report “Maintaining the State

Road Network” (Murphy 2009). This report further states that about 10% of bridges

on the 20 designated heavy haulage routes restrict heavy vehicle access, diverting

them to alternative routes. According to the facts given in the report, there are 689

Page | 2-15

concrete bridges and the majority of them are 40-50 years old and at the age where

deterioration begins.

2.1.2.2. Sydney Harbour Bridge

Figure 2.1: Picture of the Sydney Harbour Bridge (Sydney Harbour Bridge)

The Sydney Harbour Bridge (Figure 2.1), one of the most famous landmarks in

Sydney, needs approximately $ 5 million for its annual maintenance hitting the half

of the initial cost of the bridge, which was $13.5 million. The toll has been increased

by 60 times from the initial value of 5 cents to the current value of 300 cents ($3.00)

due to the increased maintenance cost over past 77 years (1932 -2009). The safer

operational condition needs to be essentially maintained as more than 150,000

vehicles per day are crossing the Sydney Harbour Bridge (Sydney Harbour Bridge).

2.1.2.2.1. Queensland

In Queensland, there were approximately 100 old and obsolete road bridges

identified in 2005. As a result, Queensland government allocated $350 million in

2005 to replace these bridges within five year period from 2006 to 2010 (Liu et al.

2008). Further, the famous landmark in Brisbane, the Story Bridge (Figure 2.2 (a)) is

experiencing excessive vibration under heavy traffic loads (Thambiratnam 1995).

Another main bridge in Brisbane, the Victoria Bridge (Figure 2.2 (b)) is vulnerable to

Page | 2-16

twisting due to functional changes as one side of the Victoria Bridge has been

allocated for bus lanes.

Figure 2.2 (a) Figure 2.2 (b)

Figure 2.2: Pictures of (a): The Storey Bridge, (b): The Victoria Bridge

2.1.3. Bridge failures and deterioration

A comprehensive report on major bridge failures around the world has been

presented by McLinn (2009). The paper lists the number of bridge failures recorded

between 1970-2009 with examples of major bridge failures. Most bridges are failed

due to natural disasters such as floods and accidents such as collisions (McLinn

2009; Wardhana 2003). Wardhana (2003) has reported that about 20% of bridge

failures are caused by overloading and lateral excitation caused by heavy trucks.

Other common sources for bridge failures are deterioration, imperfections in design,

detailing and material, maintenance issues and human induced activities such as

terrorist attacks. It has been reported that more than 40% of bridges in USA are at

age of structural deterioration (Special Structures (Bridges, Tanks)).

Page | 2-17

2.1.4. Bridge management systems

Many bridges in countries all over the world including Australia are now at age

of structural deterioration and hence susceptible to structural failures as evident from

the above section. Therefore, effective management of bridge asset through the

identification, prioritisation and planning of maintenance work is essential to a safe,

accessible and efficient road network. Bridge management systems have been

designed to manage bridges throughout the design, construction, operation and

maintenance stages. Some examples for bridge management systems are BRIDGIT

(Hawk and Small 1998) and Pontis (Thompson et al. 1998; Thomson et al. 2001).

They aim to establish safer performance levels and recommend correct maintenance

and repair works during the operational stage using Structural health monitoring

(SHM) schemes.

2.2. Structural Health Monitoring (SHM)

2.2.1. Definition of SHM

A comprehensive definition for SHM has been given by Chan et al (2011) who

defined the SHM as "the use of on-structure sensing systems to monitor the

performance of a structure and evaluate its health state". They emphasized that the

SHM should perform two main tasks, namely; 1. Structural Performance Monitoring

(SPM) and, 2. Structural Safety Evaluation (SSE). They defined SPM as the process

of "monitoring the performance of structure and its components under the designated

performance limits at serviceability limit states using on-structure instrumentation

system". SSE refers to "the evaluation of possible damage in structure or its

components and/or the assessment of its health status by analytical tools, which are

Page | 2-18

developed and calibrated in the course of SHM, basing on its designated performance

limits at ultimate limit states".

Some other definitions of SHM which are confined more or less to the damage

detection process are given below.

1. Housner et al. (1997) refers SHM to the "use of in-situ, continuous or regular

(routine) measurement and analyses of key structural and environment parameters

under operating conditions, for the purpose of warning impending abnormal states or

accidents at an early stage to avoid casualties as well as giving maintenance and

rehabilitation advice".

2. Aktan et al. (2000) defines the SHM as "the measurement of the operating

and loading environment and the critical responses of a structure to track and

evaluate the symptoms of operational incidents, anomalies and/or deterioration or

damage indicators that may affect operation, serviceability, or safety reliability".

3. Sohn (2004) refers SHM to "the process of implementing a damage

detection strategy for aerospace, civil and mechanical engineering infrastructure".

2.2.2. Applications of SHM (on bridges)

SHM systems have been applied in many bridges around the world. Few

examples are listed below.

1. Skarnsundet Bridge in Norway has been instrumented with a fully automatic

data acquisition system to monitor wind, acceleration, inclination, strain, temperature

and dynamic displacements using 37 sensors (Myrvoll, Dibiagio and Hansvold

1994).

Page | 2-19

2. New HaengJu Bridge in Korea consists with an online monitoring system. It

is instrumented with 65 sensors and a signal processor (Chang and Kim 1996).

3. Storck's Bridge in Switzerland instrumented with a fibre optical surveillance

system with 14 and 26 optical and electrical sensors , respectively.

4. WASHMS which stands for Wind And Structural Health Monitoring

System, is a SHM system installed on a six cable-supported bridges in Hong Kong. It

is claimed to be the most heavily instrumented bridge project in the world (Chan et

al. 2011).

5. Benicia Martinez Bridge in US includes instrumentations to monitor the

bridge for short term and long term behaviour

6. Pereria-Dos Quebradas Viaduct Bridge in Colombia (Thomson et al. 2001)

7. El Hormiguero Bridge in Colombia (Thomson et al. 2001)

Brownjohn (2007) has documented large number of examples to illustrate

applications of SHM on dams, bridges, offshore structures, building towers, nuclear

plants and tunnels.

2.2.3. Steps of a SHM system

Steps involved in a SHM process have been classified in different forms by

Chan et al. (2011), Aktan et al. (2000), Farrar, Doebling and Nix (2001), Farrar and

Worden (2007), Li, Li and Song (2004) and Tsamasphyros et al. (2006). Based on

the comprehensive literature review carried out in this study, four common steps of a

SHM system are identified as; 1. Instrumentation, 2. Data collection, cleansing, and

manipulation, 3. Condition assessment, and 4. Health Evaluation of the structure.

Page | 2-20

2.2.3.1. Instrumentation

Instrumentation can be defined as the deploying sensors on the candidate

structure. This process involves operational evaluation, and decision making on

instrumentation and sensor layout.

1. Operational Evaluation (Farrar, Doebling and Nix 2001; Sohn 2004)

The operational evaluation is mainly aimed to set the objectives of the

structural health monitoring system. Decisions on what to be monitored

(identification of critical structural components) and the method of monitoring are

drawn during the operational evaluation. This process includes, identifying probable

damage types, limitations on data acquisition, and economic and life safety motives

on SHM within the operational and environmental conditions of the structure.

2. Decision making on instrumentation and sensor layout (Farrar, Doebling and

Nix 2001; Chan et al. 2011)

This step involves four main tasks, namely; 1. selecting the types of sensors, 2.

forming the sensor grid (locations where the sensors are mounted on the structure), 3.

calculation of the number of sensors, and 4. identifying other instrumentations such

as data acquisition system, storage devices, transmittal hardware and software. The

interval for data acquisition (sampling interval) is also determined during this step.

The sampling interval depends on types of sensors, data acquisition system, and the

storage device. The decisions made in this step depend on the required accuracy

level, expected behaviour of critical members, and operational and environmental

conditions. Knowledge gathered from past experience is important as it is with the

results obtained from detailed analysis under simulated operational conditions.

Page | 2-21

2.2.3.2. Data collection, cleansing, and manipulation

In the next step, the instrumented sensors and data acquisition system are used

to monitor the response of the structure continuously or periodically or immediately

before and after a severe event. Information collected at this stage includes response

of the structure (output data such as strains, displacements, and acceleration), input

measurements (such as traffic load, and wind load) and environmental variations

(such as temperature, and humidity) (Murugesh 2001). The collected data should be

normalized in the next step to cater for environmental and operational variations.

Sohn et al. (2001) has summarized a procedure to normalize the measured responses

using measured inputs. When the environment and operating condition are changed,

the measured data can be compared with previously obtained response under similar

conditions (Farrar, Doebling and Nix 2001). During data cleansing process, collected

data will be selectively chosen or rejected based on the experience and knowledge

gained from persons involved with the data acquisition systems. The selected data

after cleansing process will then be manipulated and stored to facilitate the condition

detection process.

2.2.3.3. Condition assessment

The process of assessing the structure or structural components to diagnose

presence of damage can be defined as the condition detection. There are four main

steps to be performed during the condition detection process, in which last three

depends on the outcome of the first one. The first step of this process is aimed to

identify the presence of damage. Changes in structural responses are used to

differentiate between damaged and undamaged states. The next three steps will be

performed only if the presence of damage is confirmed. The second step is the

localization of damage. The method should be capable of identifying both single and

Page | 2-22

multiple damage locations as well as minimise the influence of measurement noise.

Identification of damage type and quantification will be carried out during the last

two steps.

Different condition detection methods have been attempted in the SHM

research context. Different classifications have been made for such methods. One

common classification separate damage detection approaches into one of the two

categories, namely, 1. local methods, and 2. global methods (Doebling, Farrar and

Prime 1998; Doebling et al. 1996). Local methods are based on visual observations

or localized experimental testing. Examples for local methods are; acoustic or

ultrasonic methods, magnetic field methods, radiography, eddy-current methods or

thermal field methods (Doherty 1988). Local methods require the vicinity of the

damage is known a priori and readily accessible and hence limited to detect damage

on or near the surface of the structure. On the other hand, global methods use the

global properties of the structure to assess the damage. Research interest has

continuously gained towards the global methods that examine changes in vibration

properties of the structure to assess the damage. These methods are defined as

Vibration Based Damage Identification Techniques (VBDIT) or vibration based

damage identification methods or vibration based damage detection methods in the

literature. Comprehensive literature reviews on VBDIT reflect the extensive research

works that have been carried out in the field of VBDITs. Author referred to

publications made by Doebling, Farrar and Prime (1998), Doebling et al. (1996),

Farrar and Doebling (1997) and Wang and Chan (2009).

2.2.3.4. Health evaluation

Once condition of the structure is determined, the health level of the structure

can be assessed in terms of level of damage and remaining service life. The

Page | 2-23

prognostic carried out to predict the future condition progression. Therefore, this step

often involves with reliability analysis which is mainly based on long-term failure

data such as mean time to failure, and reliability models. Also, the reliability

prediction techniques use condition monitoring data to improve the results. The

management decisions are mainly based on results from both health assessment and

prognostic. Need for repair, replacement or rebuild are the main management

decisions that need to be drawn in this step. However, management decisions are

influenced by reliability and expected costs, resource availability, timing with other

projects, losses due to outages, and safety permits for personnel. Further, it needs to

measure and compare the loading response with design assumptions for future design

purposes.

2.2.4. Research areas related to the SHM process

As highlighted in the previous section, the SHM process itself is a broader

research area. There are many research topics that have been formed based on the

SHM process. Scopes of individual researches are normally confined to the limited

areas. From authors speculation, four individual but interrelated research groups on

SHM are defined; 1. instrumentation, 2. data processing, 3. damage assessment, and

4. health assessment. Instrumentation is more related with the electronic and

mechanical engineering fields and aimed on improving the accuracy of the sensors

and data acquisition systems. During the data processing stage, acquired responses of

the structure will be converted to vibration properties such as frequencies and mode

shapes. Different software packages such as LabVIEW (LabVIEW 2010), MATLAB

(MATLAB 2011) and mathematical concepts such as Fourier transformation

techniques are widely used at the data processing stage. Researches aimed on

Page | 2-24

minimising the measurement noise can also be categorised into the data processing

group. Researches in the next group aims on developing damage detection

algorithms to detect, localize, quantify the damage. Finally, analytical tools will be

used to assess the health state of the structure to predict the remaining service life

and capacity.

The scope of this research has been confined to the damage assessment process

using VBDITs. The remaining parts of this literature review has therefore been

narrowed down to the VBDITs.

2.3. Vibration Based Damage Identification Techniques (VBDITs)

2.3.1. Hypothesis of VBDIT

Any localised damage and structural deterioration modify the physical

properties of the structure such as mass distribution, stiffness, and damping.

Modifications in physical properties are allied with change in vibration properties,

namely, natural frequencies, mode shapes, and damping values. Such modifications

will also depend on the nature, location and severity of damage and will influence

natural frequencies, damping values, and mode shapes of vibration modes in

different ways. It is therefore possible to detect structural damage by comparing and

analysing the vibration properties between the intact and damaged stage, across a few

lower modes of vibration.

2.3.2. Procedure of developing VBDITs

Development stage of VBDITs is a two stage process, 1. forward problem

study, 2. inverse problem study. During the forward problem study, known damage

parameters (i.e. damage types, locations, and severities) are induced to measure the

Page | 2-25

vibration response of the structure. At this stage, changes in vibration properties such

as frequencies, mode shapes, and damping values are measured. Induced and

measured changes are then analytically examined to build necessary algorithms to

address the inverse problem of the damage detection process.

The inverse problem is the reverse of the forward problem. At this stage,

unknown damage properties are identified using developed algorithms which are

based on measured vibration responses of the structure. The inverse problem

involves two steps; 1. validation of the developed algorithms using few case studies,

and 2. application to real structure monitoring.

Finite Element (FE) technique based studies are extensively used during the

first step to minimize cost and time involved with experimental studies. In some

cases numerical techniques have been attempted as an alternative to the FE

simulations. The accuracy of the developed VBDIT is normally verified with respect

to the large number of simulated damage cases and limited number of experimental

testing prior to deploy the method in the real structures. Accuracy of the forward

problem study should therefore be preserved to the maximum level as any

inaccuracies may result with false damage detection results.

The damage simulation technique used in the forward problem study, therefore,

should be capable of correctly simulating the behaviour of the real structure under all

potential damage types. Any incompatibilities between simulated and real behaviour

may cause positive or negative false alarms which in turn results with higher

maintenance and repair cost or damage to properties and lives. The damage

simulation technique used in the FE/numerical based studies, therefore, plays an

important role in the success or failure of the developed VBDIT.

Page | 2-26

2.3.2.1. Review of damage/crack simulation techniques

This section examines damage simulation techniques used in the field of

VBDITs.

Dimarogonas (1996) and Ostachowicz and Krawczuk (2001) have given a

comprehensive review of crack simulation techniques. Friswell (2007) has defined

three broad categories of damage simulation techniques; 1. local stiffness reduction,

2. discrete spring elements, 3. complex models in two or three dimensions.

Local stiffness reduction is the simplest among all three crack simulation

approaches. This can be achieved either reducing the Young's Modulus (E-value) or

second moment of area (I) or creating a notch type damage. Park, Kim and Stubbs

(2002) has simulated the damage in a numerical model of steel truss bridge by

reducing the stiffness of selected members. Similar approach has been used by

Alvandi and Cremona (2006) and Adewuyi, Wu and Serker (2009) on simply

supported beam made of steel. Jaishi and Ren (2006) have used reduction in elastic

modulus as the mean to reduce the flexural rigidity of numerically simulated simply

supported concrete beam. In all above studies, uniform reduction of stiffness at the

damage location has been assumed.

Shih, Thambiratnam and Chan (2009) have used flaws of 10mm x 5mm and

20mm x 5mm (length x depth) to simulate damage on the tested steel beam, while

reduction in E-value has been used for plate like structures. Notch type damage on

aluminium beams by thin saw cuts was the damage simulation technique used by

Owolabi, Swamidas and Seshadri (2003) during the experimental study aimed at

using frequencies and amplitudes to detect cracks in beams. Rectangular cuts have

been created by Li et al. (2007) on timber beams to simulate damage during their

Page | 2-27

study. Kumar, Shenoi and Cox (2009) have removed the skin at selected locations of

composite sandwich beams to induce damage.

In the second damage simulation approach, the structure is divided into two

parts at the damage location and joined by rotational spring elements. Some of the

examples of using this technique for damage simulation are evident from work

carried out by Palacz and Krawczuk (2002), Kisa and Arif Gurel (2007), Nandwana

and Maiti (1997), Patil and Maiti (2005), Chinchalkar (2001) and Khiem and Lien

(2001).

Complex damage simulation techniques were less examined due to high

computational cost. Maeck (2003) has used the discrete crack approach to simulate

damage in Reinforced Concrete (RC) structures.

This implies that existing VBDITs have not adequately examined the influence

of the damage model on the damage detection results. Simplified damage simulation

techniques are normally confined to a smaller region or assume a uniform reduction

in stiffness across the whole region. Such methods do not account for the non-

linearity in material properties. However, simplified damage simulation techniques

have been widely accepted in the research context of VBDITs towards structures

made of steel or homogeneous materials.

The predominant damage type in RC structures, the flexural cracking, behaves

in a significantly different manner. For an example, flexural cracks in a RC beam

distribute in a parabolic shape along the longitudinal and depth directions. Figure 2.3

illustrates the progressive crack patterns observed by Maeck (2003) under three point

bending test. This figure clearly illustrates that flexural cracks extend more rapidly

along the longitudinal direction than propagation of existing cracks in the depth

Page | 2-28

direction. This is because of the stress distribution caused by the tensile

reinforcement. Because of this nature, simplified damage simulation techniques

cannot be used to simulate the post-cracking behaviour of RC structures. Previous

studies have not addressed this issue except Maeck (2003) and Wang (2010).

Maeck (2003) has attempted to use the discrete crack simulation approach. The

discrete crack simulation requires additional calculations for a new set of elements.

Two of the major drawbacks of this method are; 1. some parameters such as distance

between cracks are case dependent and 2. determination of other parameters such as

spring stiffness in steel-concrete interface, and vertical and horizontal stiffness in the

crack plane are difficult (Maeck 2003).

Figure 2.3: Progressive crack patterns under three point bending test as observed by (Maeck

2003)

Wang (2010) has highlighted the adaptability of smeared cracking technique to

simulate load induced flexural cracks. However, Wang (2010) has used the notch

type damage and reduction in E-value during his/her study. Hence, no attempt has

been given to evaluate the damage detection ability of flexural cracks using smeared

cracking technique.

Irrespective of the above differential behaviour, few studies on RC structures

have been conducted using simplified damage simulation techniques (Perera and

Page | 2-29

Huerta 2008; Zhou, Wegner and Sparling 2007). However, there are few attempts

that have been made to evaluate the applicability of VBDITs on RC structures using

experimental testings (Perera and Huerta 2008; Maeck 2003; Zhou, Wegner and

Sparling 2007; Ndambi, Vantomme and Harri 2002; Maeck et al. 2000; Ismail,

Abdul Razak and Abdul Rahman 2006; Curadelli et al. 2008).

2.3.3. Classifications of VBDITs

There are many VBDITs that have been presented in the literature. These

methods have been classified using different criteria in the literature (Rytter, 1993;

Lee, Karbhari and Sikorsky 2004; Bakhary 2009; Worden and Dulieu-Barton 2004;

Wang 2010; Farrar and Doebling 1997; Saleshi et al. 2011; Huang, Gardoni and

Hurlebaus 2012)

Rytter (1993) has classified VBDITs into four levels based on the extent of

damage detection. These four levels are;

1. Level 1: (Damage detection) - Identification of occurrence of damage,

2. Level 2: (Damage localization) - Identification of the occurrence, and

determination of the damage location

3. Level 3: (Damage quantification) - Identification of the occurrence, determination

of the damage location and quantification,

4. Level 4: (Prediction and health assessment) Identifying and determining the

location and severity of the damage, Evaluating the impact of damage on the

structure or the estimating the remaining service life.

Level 1 of VBDIT simply uses the changes in primary vibration properties

such as natural frequencies, mode shapes, and damping properties or similar other

Page | 2-30

features such as time histories and frequency response functions (FRF) to identify the

structural changes. Damage detection algorithms are needed for levels 2 and 3 where

the vibration characteristics are used and analysed to locate and quantify the damage.

In essence, the final goal of a SHM strategy is to determine the amount of damage

that the structure has been subjected to at the time of health evaluation so that the

remaining service life and state can be predicted.

Worden and Dulieu-Barton (2004) introduced an intermediate level to the

above classification and defined five levels of damage detection process. These five

levels are the following:

1) Level 1: Damage detection

2) Level 2: Damage localization

3) Level 3: Classification - This step aims at identifying the damage type

4) Level 4: Damage quantification

5) Level 5: Prediction and health assessment

Lee, Karbhari and Sikorsky (2004) classified VBDITs into four broad

categories based on the features used in the damage detection process. These four

categories and corresponding features used in the damage detection process are

tabulated in Table 2-1.

Bakhary (2009) had classified VBDITs into three categories, namely, 1. direct

methods, 2. model updating methods, and 3. Artificial Neural Network (ANN)

methods.

Page | 2-31

Another broad classification has been presented by Wang (2010) who

classified VBDITs into three categories, namely, 1. Methods based on vibration, 2.

Methods based on Artificial Intelligence, and 3. Methods based on wavelet analysis.

Another classification method divides VBDITs into two groups; 1. Model-

based, and 2. Non model-based (Farrar and Doebling 1997; Saleshi et al. 2011;

Huang, Gardoni and Hurlebaus 2012).

Table 2-1: Four categories of VBDITs as proposed by (Lee, Karbhari and Sikorsky 2004)

Category Features used in the damage detection

process

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 FRFs

Page | 2-32

The model-based methods use analytical models during the damage detection

process. They need detailed computer models (either FE or numerical models) to be

employed during the damage detection process. There are two main categories within

the model-based methods, 1. Correlation based approach, and 2. Artificial Neural

Network (ANN) based approach. In the former approach, the simulated computer

model will be continuously updated using the real measurements so that the changes

in structural properties at the current state can be obtained relative to a previous state.

In ANN based approach, a neural network will be trained to relate the structural

changes to the vibration response of the structure. This will be carried out a priori.

The model-based approach is more expensive and time consuming, but more suitable

for complex structures (Abdel Wahab and De Roeck 1999). Some other drawbacks

that are associated with the model-based approaches are; difficulty in accounting for

measurement errors, and inability to guarantee the accuracy of the computer

developed computer model of the structure (due to modelling errors, assumptions,

and simplifications) (Friswell and Penny 1997; Whalen 2008). Also, such methods

require individual computer models to be developed for each of the structure under

evaluation as each civil infrastructure is unique.

On the other hand, the non model-based damage detection methods are

relatively straightforward, fast and inexpensive. Non model based damage detection

methods are also named as the Damage Index (DI) methods (Saleshi et al. 2011) or

the response-based methods (Abdel Wahab and De Roeck 1999; Whalen 2008).

Simply, the DI methods analyse the changes in response of the structure between two

states so that the damage can be identified. These methods have the ability to avoid

modelling errors and can accommodate for measurement noise. Furthermore, DI

Page | 2-33

methods have the potential to extend for automated damage detection schemes

(Whalen 2008).

The model based damage assessment techniques are difficult to deploy on large

civil structures due to high level of modelling uncertainties. For an example,

reinforced concrete structures are not a homogeneous material and hence contradict

with the modelling assumptions. Further, they are subjected to continuous variations

in material properties due to creep and shrinkage. These effects are difficult to

incorporate during the numerical simulation. The damage index method is therefore

selected as the most suitable method for the present study.

2.3.4. Damage Index (DI) methods

The DI method uses changes in vibration properties between two states to

assess the damage in the structure. Different forms of Damage Indices (DIs) have

been attempted in the literature. These DIs can broadly categorise into two groups; 1.

DIs based on primary vibration properties (frequencies, mode shapes, and damping

factor), and 2. DIs based on secondary vibration properties such as mode shape

derivatives, flexibility values, and strain energy values.

2.3.4.1. Frequency based methods

The history of the DI method runs towards late 1970s. The literature has

documented that Cawley and Adams (1979) pioneered the research on using dynamic

parameters for damage detection. These authors have described a method to detect

damage using changes in frequencies. Since then, large number of studies has been

conducted to detect damage using changes in frequencies. Salawu (1997a) has

presented a comprehensive review on the use of frequency changes for damage

detection. In few occasions, frequency changes have been successfully applied to

Page | 2-34

detect and localize both single and multiple damage cases (Chinchalkar 2001;

Owolabi, Swamidas and Seshadri 2003; Patil and Maiti 2005). Nandwana and Maiti

(1997) have successfully applied the frequency changes to locate and quantify a

single damage in a cantilever beam. Doebling et al. (1996) have noticed that

frequency changes have more robust with the random error sources than the other

modal parameters. However, the ability of damage detection using frequency

changes are limited to simple structural forms. Also, they have indicated low

sensitivity to the damage severity. In most cases, frequency changes have been used

as an indication of the presence of damage because of the global nature of this

property (Salawu 1997a; Doebling, Farrar and Prime 1998; Farrar and Doebling

1997; Bakhary 2009).

2.3.4.2. Damping values based methods

Use of damping factors towards damage detection has not achieved much

research interest in the literature. Among the few studies carried out on damping

values based damage detection schemes, many have indicated that the damping

factor is not sensitive to the damage (Farrar and Jauregui 1998; Salawu and Williams

1995; Casas and Aparicio 1994). However, there are couple of successful

applications have been reported by Razak and Choi (2001) and Curadelli et al.

(2008).

2.3.4.3. Mode shape based methods

As frequencies and damping factors are more global in nature, the damage

localization attempt has been extensively attempted with the mode shape based DIs.

The simplest format of the DI is based on the absolute change in mode shape values

between two states (Whalen 2008; Elshafey, Marzouk and Haddara 2011).

Page | 2-35

Two of the other common approaches using mode shapes are; 1. Modal

Assurance Criterion (MAC), and 2. Coordinate Modal Assurance Criterion

(COMAC). MAC has been originated in early 1980s as a quality assurance criteria

but not as a damage localization index (Allemang 2003; Allemang and Brown 1982).

This method is a measure of consistency check between two modal vectors. Lower

MAC values near zero indicate that the modal vectors are not consistent. This can be

due to many reasons such as, due to system nonlinearity, measurement noise, errors

in modal parameter estimation, and modal vectors are from linearly unrelated mode

shape vectors. MAC value near unity indicates that the two modal vectors are

consistent. However, it should note that MAC can indicates a value near unity due to

incomplete measurements of modal vectors, or as a result of a forced excitation other

than the desired input, or coherent noise (Allemang 2003). Some other similar

criterions have been derived in the literature, such as, the modal scale factor,

weighted modal analysis criterion, partial modal analysis criterion, modal assurance

criterion square root, etc... A comprehensive review on these methods has been

presented by Allemang (2003).

The other widely used mode shape based criterion, COMAC, compares two

sets of modes at each degree-of-freedom or node. However, consistency of modes

need to be established a priori using MAC or similar criterion.

Examples of studies and applications of MAC and COMAC are provided by

Yuen (1985), Rizos, Aspragathos and Dimarogonas (1990), Salawu and Williams

(1995) and Salawu (1995).

Pandey (1991), Abdel Wahab and De Roeck (1999), Dutta and Talukdar

(2004) and Whalen (2008) have highlighted that the DIs based on curvature mode

shape changes are more sensitive than mode shape changes. The correlation between

Page | 2-36

the local loss of stiffness and change in mode shape curvature was proven by Pandey

(1991) with assumption that the structural damage only affects the stiffness matrix

and not the mass matrix. Further studies on this topic has been carried out by Abdel

Wahab and De Roeck (1999) for simply supported and continuous pre-stressed

concrete bridges to detect the damage using simulated damage at different locations.

They used the curvature damage factor (CDF) as a damage indicator which combines

the difference in curvature mode shapes across all modes.

Both mode shape derivatives and eigenvectors obtained from the modal tests

were used by Ismail, Abdul Razak and Abdul Rahman (2006) to determine the

location of damage of reinforced concrete beams having either a single crack or

honeycombs. However, the technique had difficulties of detecting the damage closer

to support similar to most of other damage detection methods.

Dutta and Talukdar (2004) studied the change in mode shape curvature in more

detail to detect and localize multiple damages in simply supported and continuous

bridge decks using the first five modes. The authors noticed the higher peaks in

modal curvature changes at damage location along the beam both in longitudinal and

transverse directions.

Whalen (2008) has evaluated DIs those based on changes in first four

derivatives of mode shapes and highlighted that the higher order derivatives are more

sensitive to damage.

Curvature mode shapes based DIs have shown low probabilities of correct

damage localization with high noise levels and with complex and simultaneous

damage cases (Alvandi and Cremona 2006).

Page | 2-37

2.3.4.4. Flexibility based methods

Pandey and Biswas (1994) presented a technique based on changes in modal

flexibility of the structure to study the effect of damage with different boundary

conditions. Different changes in patterns of the modal flexibility matrix were noticed

with change in boundary conditions. Patjawit and Kanok-Nukulchai (2005)

introduced a Global Flexibility Index (GFI) to identify the global health

deteriorations, where ageing of a highway bridge was reflected by the gradual

increase of GFI.

Change in flexibility curvature based methods combine some aspects from the

mode shape curvature method and the change in flexibility method (Zhang 1995).

The basic concept of this approach is that a localized loss of stiffness will produce a

curvature increase at the same location and this is almost same as the mode shape

curvature method.

2.3.4.5. Modal strain energy based methods

Stubbs, Kim and Topole (1992) pioneered the research on using changes in

modal strain energy values to detect damage and proposed the first Modal Strain

Energy based Damage Index (MSEDI) for 1D elements. Stubbs (1995) confirmed the

applicability of the MSEDI, by detecting the damage in a steel bridge. Later,

Cornwell, Doebling and Farrar (1999) extended the above MSEDI for 2D structural

elements such as plates. Since 1992, different formulae for MSEDIs have been

presented by Stubbs, Kim and Topole (1992), Cornwell, Doebling and Farrar (1999),

Park, Kim and Stubbs (2002), Li et al. (2007) and Shih, Thambiratnam and Chan

(2009). Some other damage detection studies based on MSEDI have been presented

by Petro (1997), Osegueda (1997 ), Carrasco (1997 ), Cornwell (1998), Yoo and Kim

Page | 2-38

(2000), Pereyra (2000) , and Alvandi and Cremona (2006). Li (2010) has presented a

comprehensive list of researches on modal strain energy based methods.

2.3.4.6. Comparison of DIs

It is scientifically true to state that the frequencies and damping values are

global properties of a structure. Because of this, damage detection ability of DIs

based on frequency or damping changes are limited to damage detection stage (Level

1). This fact has been widely accepted in the literature and highlighted in the section

2.3.4.1. Damping factors achieve the least motivation in the damage detection

process.

Whalen (2008) has performed a detailed study to evaluate the changes in mode

shapes and first four derivatives of mode shape based DIs. This study has clearly

demonstrated that the DIs based on changes in higher order derivatives of mode

shapes are more sensitive to damage. Similar conclusions have been presented by

Pandey (1991), Abdel Wahab and De Roeck (1999) and Dutta and Talukdar (2004)

using curvature of mode shapes.

The sources in the literature to make conclusions to rank DIs based on changes

in mode shape derivatives, flexibility values, and strain energy values are inadequate.

Alvandi and Cremona (2006) have presented a detailed evaluation to evaluate four

DIs, namely, 1. mode shape curvature method, 2. flexibility method, 3. flexibility

curvature method, and 4. modal strain energy method. The results of this study

indicated that modal strain energy based method has the highest stability in damage

detection even with presence of noise compared to the other three counterpart

techniques. They highlighted that the MSEDI cannot be used to quantify the damage

whereas the other three techniques has the quantification ability. Further, they have

used a variable threshold or datum level (i.e. elements with higher MSEDI values

Page | 2-39

than the datum level value indicate the presence of damage) during the damage

localization process using MSEDI. Authors were unable to define a fixed datum

level. This highlights that the damage localization results of MSEDI is a subjective

matter. This emphasizes the need for a comprehensive study to evaluate DIs.

2.3.5. Practical issues

Accuracy of the damage assessment results using DI method has negatively

influenced by many of the practical issues such as presence of measurement noise,

calculation errors, mode shifting, phase change, etc. Some of these issues have been

individually addressed in the literature. For example, use of MAC value has been

widely accepted as a mean to establish the consistency of modes. This enables to

identify any mode shifting.

In most cases, mode shape curvatures are computed based on the displacement

mode shapes using computational techniques such as the "central difference" formula

(Saleh et al. 2004; Pandey 1991). Chance (1994 ) indicated that some false alarms

are associated with the computational techniques when using the displacement mode

shapes. Such false alarms will amplify in case of presence of noise. Hence, Chance

(1994 ) proposed an alternative method to obtain mode shape curvatures using direct

strain measurements. On the other hand, attempts have been made to use smoothing

techniques such as the weighted residual penalty based technique to improve the

calculation process involving displacement mode shapes (Maeck 1999 ). The method

proposed by Maeck (1999 ) has drawbacks as the selection of the penalty factor is a

trial-and-error process, which indicated problems of convergence if higher penalty

factors are used. Wang (2010) has used the cubic spline interpolation technique to

Page | 2-40

compute the curvature values, but recommended further studies to minimise the

difference between reconstructed and real mode shape values.

Existing DI methods have not proposed a robust sequential steps to be followed

during the calculation process. Also, existing DIs do not have a robust and

programmable criteria to separate the damaged zones and the undamaged zones.

Because of this, the hypothetical level of automated damage detection has not yet

been achieved.

2.4. Chapter Summary

SHM system is a key component in modern bridge management systems to

ensure safer performance levels. The SHM process is a very broad research area

which can be categorised into four main discipline areas, 1. instrumentation, 2. data

processing, 3. condition detection, and 4. health assessment. The research in this

thesis is confined to the condition detection process. In this context, the VBDIT has

been achieved great research interest in recent years. Literature presents large

number of studies on damage detection of steel structures using VBDITs, but not for

RC structures. This opens a wide research area due to huge gaps in knowledge as

highlighted below. From the literature review, the following knowledge gaps have

been identified and these form the basis of the proposed research.

1. Literature pronounced that the forward problem of the damage detection process

highly relies on the FE/numerical simulation studies. Simplified damage

simulation techniques such as reduction in E-value, I-value, introducing spring

elements or creating notches have been extensively used in the literature,

particularly towards steel structures. These simplified damage simulation

Page | 2-41

techniques do not account for material non-linearity and limited to simulate the

confined damage zones.

The predominant damage type in RC structures, the flexural cracks, distribute non-

uniformly and extend more rapidly along the longitudinal direction than

propagation of existing cracks in the depth direction. Simplified damage

simulation techniques cannot be applied to simulate such a complex behaviour.

This highlights the necessity to establish a proper damage simulation technique to

represent the post-cracking behaviour of RC structures.

2. Commonly used damage severity definitions are based on the reduction in E-value

or I-value or the size of the induced notch. Non-uniform nature of the flexural

cracking zone, makes it difficult to adopt such a damage severity definition. This

highlights the necessity of a proper definition for the damage severity.

3. VBDITs fit into two main groups, model-based, and non model-based approaches.

Non-model based approaches (or DI method) do not need a detailed computer

model of the structure and hence is an output based method. Further, DI method

has the ability to automate the damage detection process. Because of this, the DI

method was selected for the present study.

Owing to the differential behaviour of cracked RC structures, the necessity has been

arisen to evaluate the existing DI methods. A comprehensive study should

therefore be performed to rank the individual DIs based on the damage detection

ability of flexural cracks.

Page | 2-42

4. DI method has the ability to automate the damage detection process. But, there are

many technical challenges ahead due to practical problems involved with the

measurements such as presence of noise, mode shifting and phase change. A

proper sequence and programmable algorithms should therefore be derived to

automate the damage detection process.

To address the above listed research gaps, the present research was designed to

develop a robust and programmable strain energy based DI method to assess the

damage in flexural RC members.

Page | 3-43

3. Methodology and Selection of a Flexural Crack

Simulation Technique

The first section (i.e. section 3.1) of this chapter briefly elaborates the

methodology followed in the present study. The section 3.2 presents the details of the

ABAQUS damaged plasticity model. The next section (section 3.3) presents two

numerical techniques used to derive the stress-strain curves of concrete for

compressive and tensile behaviours. Necessary modifications for those two

numerical techniques are proposed so that they are compatible with the ABAQUS

damaged plasticity model. The section 3.4 presents FE simulation and validation of

the damaged plasticity model. Section 3.5 highlights the problems associated with

the two simplified damaged simulation techniques, 1. reduction in Youngs' Modulus

(E-value), and 2. notch type damage. The section 3.5 highlights the problems

associated with the traditional approach of damage severity definition (i.e.

percentage reduction of E-value or section depth), and defines a new damage severity

index.

3.1. Methodology

The present research methodology was designed to achieve the six research

objectives listed in the section 1.3. Firstly, a comprehensive evaluation of damage

simulation techniques was carried out to select an accurate method to simulate the

post cracking behaviour of RC structures. Three damage simulation techniques,

namely, 1. reduction in E-value, 2. notch type damage, and 3. the damaged plasticity

model, were evaluated at this stage. Two numerical techniques were employed to

obtain complete stress-strain curves of concrete under compression and tension.

Page | 3-44

Experimental results were used to examine the accuracy of the each damage

simulation techniques. Research findings indicated that the damaged plasticity model

has the ability to represent the post cracking behaviour of RC structures in accurate

manner, but not the other two damage simulation techniques. The damaged plasticity

model was therefore used to simulate different flexural crack patterns as deemed in

this research. The ABAQUS finite element package (Abaqus v. 6.8 [Software] ,

2008) was used in all the simulation works.

Flexural cracks in RC structures distribute non-linearly and hence difficult to

express in terms of percentage reduction in E-value or depth of the section. An

alternative method was therefore proposed to define the damage severity based on

area under the displacement curves and to achieve the objective 2.

A comprehensive evaluation of different damage indices was performed as

highlighted in the third objective. This stage was designed to identify the problems

associated with the DI calculation process and the DI itself. The effect of

measurement noise was therefore not examined. Different load induced flexural

crack patterns were simulated on a RC beam model using the damaged plasticity

model and the ABAQUS finite element package (Abaqus v. 6.8, 2008). Problems of

existing DIs were identified in terms of numbers and intensities of false alarms. Four

Modal Strain Energy based Damage Indices (MSEDI) were developed to minimise

the numbers and intensities of false alarms. These four MSEDIs were denoted by β8-

β11. 21 existing and 4 newly proposed DIs were then ranked and β11 was selected as

the best DI. β11 was therefore used in the rest of the present study. The selected

MSEDI, β11, was then applied to localize flexural cracks on a simulated 15m long

RC T-beam bridge. Results confirmed the applicability of β11 to localize damage in

large civil structures.

Page | 3-45

A MATLAB based programme was developed incorporating a Graphical User

Interface (GUI) in parallel to the DI evaluation process. Basics and details of this

programme are presented in the Chapter 6 and 7. This programme was incorporated

with necessary algorithms to address many practical issues such as mode shifting and

phase change. All the algorithms defined in this programme were developed to have

the ability to extend for an automated damage assessment process. Also, special

features were introduced to address the forward problem in research studies.

The effect of noise on damage localization results was studied at the last stage

of this research with aim to improve the reliability of damage localization results. As

a result, several corrective measures were introduced to the DI calculation process.

Firstly, the influence of curve fitting techniques on DI calculation process was

studied. A sequential order of three MATLAB based curve fitting techniques (csaps,

fourier, spaps) were proposed as a mean to minimise the influence of measurement

noise. Further enhancements to the damage localization results were proposed using

a statistical based approach which standardizes the damage localization results

obtained from different number of samples measured at different times either

continuously or periodically. This standardization process was named as the

Generalized Damage Localization Index (GDLI) method. A detailed study was then

carried out to recommend the accuracy levels that can be achieved at different noise

levels and number of samples.

3.2. Concrete Crack Models Available in ABAQUS

ABAQUS Finite Element (FE) package has three crack models that can be

used to simulate damage in reinforced concrete elements. They are; (1) Smeared

crack concrete model, (2) Brittle crack concrete model, and (3) Concrete damaged

Page | 3-46

plasticity model. The concrete damaged plasticity model was selected for this study

over the other two concrete damage models due to its superiority in damage

simulation due to reasons listed below. (Abaqus Analysis User Manual – Abaqus

Version 6.8. 2008).

1. The concrete damaged plasticity model allows defining the complete inelastic

behaviour of concrete under tension and compression including the damage

parameters.

2. The damaged plasticity model is the only model which supports with both

analysis methods, ABAQUS/Standard and ABAQUS/Explicit and allows

transferring the results between each other.

3. Because of the above fact, the damaged plasticity model allows the analysis of RC

structures under any loading conditions including both static and dynamic

loading.

3.2.1. ABAQUS damaged plasticity model

The ABAQUS damaged plasticity model requires some basic parameters and

complete stress-strain relationships (including linear and non-linear regions) to be

defined for both compressive and tensile regions. These details are briefly presented

in sections 3.2.1.1- 3.2.1.3. The ABAQUS Help Documentation (Abaqus Analysis

User Manual – Abaqus Version 6.8. 2008) has more information and definitions.

3.2.1.1. Basic parameters

The basic parameters required for the ABAQUS damaged plasticity model are;

1. Density,

2. Young's Modulus (E0),

3. Poisson's Ratio,

Page | 3-47

4. Dilation angle,

5. Eccentricity,

6. σb0/σc0 (i.e. ratio of initial equi-biaxial compressive yield stress to initial

uniaxial compressive yield stress),

7. K (i.e. ratio of the second stress invariant on the tensile meridian to that on

the compressive meridian at initial yield for any given value of the pressure invariant

such that the maximum principal stress is negative ), and

8. Viscosity parameter.

Experimentally obtained densities and Young's Modulus values were assigned

in this study. Default parameters were used for all the other parameters. They are;

Poisson's ration = 0.2, Dilation angle = 36, Eccentricity = 0.01, fb0/fc0 = 1.16, K=

0.6667, and Viscosity parameter =0. ("Abaqus Analysis User Manual – Abaqus

Version 6.8. " 2008).

3.2.1.2. Compressive stress-strain relationship

The concrete damaged plasticity model in the ABAQUS FE package needs a

complete stress-strain curve under compression to be defined (similar to Figure 3.1)

as explained in this section.

The ABAQUS FE package requires three parameters to be defined in a tabular

format, namely, 1. compressive stress (σc) (i.e. Yield Stress in the data entry column

in the ABAQUS FE package), 2. inelastic strain ( ), and 3. compressive damage

parameter (dc).

The compressive stress (σc) vs. total compressive strain (εc) curve can be

obtained from either material testing or proper numerical techniques. A numerical

Page | 3-48

technique (presented in section 3.3) was employed in the present study as an

alternative to the material tests.

The inelastic strain is derived using Equation 3-1 in terms of total

compressive strain εc and the elastic strain corresponding to the undamaged

material.

Equation 3-1

where is calculated from Equation 3-2 as below.

Equation 3-2

The concrete damage parameter, dc, should be defined if unloading steps are

involved during the simulation. In such situations, corrective measures should be

taken to ensure that the plastic strain values (

) calculated using Equation 3-3 are

neither negative nor decreasing with increased stresses.

Equation 3-3

Figure 3.1: Compressive Stress-Strain Relationship (Abaqus Analysis User Manual – Abaqus

Version 6.8. 2008)

Page | 3-49

3.2.1.3. Tensile stress-strain relationship

The post-cracking behaviour of RC structures is governed by the stress-strain

relationship of concrete under tension. A typical stress-strain curve under tension

(including non-linear region) is illustrated in Figure 3.2 which includes the

parameters used in the ABAQUS FE package.

The complete stress-strain curve under tension should be defined in a tabular

format similar to the compressive curve.

Figure 3.2: Tension Stiffening Model (Abaqus Analysis User Manual – Abaqus Version 6.8. 2008)

User needs to enter the cracking strain values (ε tck) against the tensile stress, σt,

in a tabular format to define the complete stress-strain curve. The cracking strain, ε tck,

can be calculated from Equation 3-4 if the total strain, εt, tensile stress, σt, and the

Young's modulus, E0 are known.

ε tck

Equation 3-4

where, the Elastic strain corresponding to the undamaged material, , is

calculated using Equation 3-5.

Equation 3-5

Page | 3-50

The tensile damage parameter, dt, needs to be defined when the unloading steps

are involved in the analysis process. ABAQUS FE package checks the accuracy of

the entered data before it starts the analysis process and generates an error message if

negative and/or decreasing tensile plastic strain values, ε tpl

, are recorded. This error

message can be avoided by calculating the tensile plastic strain values using

Equation 3-6.

ε tpl

ε tck

dt

(1 dt)

σt

E0

Equation 3-6

3.3. Numerical Models for Stress-Strain Curves

It is evident that stress-strain curves under compression and tension are

necessities to define the concrete damaged plasticity model in the ABAQUS FE

package. This requires either material testing or proper numerical models. In this

study, two numerical techniques were used to obtain stress-strain curves of concrete

under compression and tension. Details of these two numerical techniques and

necessary modifications are presented in this section so that they are compatible with

the requirements of the ABAQUS damaged plasticity model.

3.3.1. Compressive stress-strain curve

The numerical method proposed and experimentally validated by Hsu and Hsu

(1994) (this will be referred as 'the Hsu and Hsu model' here after) was used to obtain

complete stress-strain relationships for all simulated case studies in this study. The

Hsu and Hsu model has the potential to develop the stress-strain relationship under

uni-axial compression up to a stress of 0.3 in the descending portion. This model

requires only the maximum compressive strength of concrete, σcu. The method can

be applicable to both normal and high strength concrete with minor changes. This

Page | 3-51

section presents necessary formulae to develop stress-strain curves for normal

strength concrete with σcu less than 62MPa. Modifications can be made for high

strength concrete and details are presented in the paper of Hsu and Hsu (1994).

Figure 3.3 illustrates the complete numerical model used to obtain the

compressive stress-strain relationship. σcu, and ε0 stand for the ultimate compressive

stress and the corresponding strain at this stress level. The yield point has been

defined at 0.5 σc assuming a linear stress-strain variation up to 50% of the ultimate

compressive strength ( ) in the ascending portion. The Hsu and Hsu model is used

to derive the stress-strain values in the non-linear region between the yield point (at

0.5 ) and the 0.3 in the descending portion using Equation 3-7.

Equation 3-7

where; the parameter β which depends on the shape of the stress-strain diagram

is derived from Equation 3-8 and the strain ε0 at peak stress, is calculated using

Equation 3-9.

Equation 3-8

Equation 3-9

The initial tangential modulus, E0 is given by;

Equation 3-10

The εd, strain at the 0.3σcu in the descending portion, is iteratively calculated

using Equation 3-7 when .

Page | 3-52

Note: in the above equations, , and E0 are in kip/in2 (Conversion factor: 1MPa

= 0.145037743 kip/in2).

Figure 3.3: Compressive Stress-Strain Relationship for ABAQUS

3.3.2. Tensile stress-strain curve

The tensile stiffening model proposed and experimentally validated by Nayal

and Rasheed (2006) was used to derive the stress-strain curve required for the

damaged plasticity model under tension. This model has the ability to simulate both

primary and secondary cracking stages. The model proposed by Nayal and Rasheed

(2006) is shown in Figure 3.4. This model indicates a sudden stress drop from σt0 to

0.8σt0 at the critical tensile strain, εcr. It was identified in this research that this

sudden drop leads to run time errors in the ABAQUS FE package. A gradual

reduction in stress was therefore introduced to overcome the above run time error.

The stress was gradually reduced from σt0 to 0.77σt0 between strain values εcr and

1.25εcr. The modified tensile stiffening model is shown in Figure 3.5.

Page | 3-53

Figure 3.4: Tension Stiffening Model (Nayal and Rasheed 2006)

Figure 3.5: Modified Tension Stiffening Model

3.4. Validation of the Damaged Plasticity Model

This section presents the validation of the damaged plasticity model using three

case studies. Two of them are based on results obtained from RC beam tests, and the

other is from a RC slab test presented in the literature.

Page | 3-54

3.4.1. Case study 1 - Four point bending test

The experiment setup of Perera and Huerta (2008) was selected as the first case

study for validating the present finite element simulation work. Perera and Huerta

(2008) have published the details of experimental tests on RC beam which include

material properties, reinforcement arrangement, load vs. displacement curve, and

frequency measurements at different damage states under four point bending test.

The frequency measurements due to added inertia loads were also presented in the

paper. The missing information on reinforcement cover was obtained from direct

contacts made to the corresponding author, Perera (Perera and Huerta 2008). The

section 3.4.1.1 and 3.4.1.2 present details of the experimental setup and FE model

validation , respectively.

3.4.1.1. Details of the experimental setup

The tested RC beam was 4.54m in length with cross section dimensions of

0.32m and 0.22m in depth and width, respectively. Figure 3.6 illustrates the cross

section details of the RC beam including reinforcement arrangement. The elastic

modulus, density and yield strength of reinforcement bars were taken as 210GPa,

7850kg/m3 and 510MPa , respectively. The experimentally measured compressive

strength of concrete was 32MPa.

Figure 3.6: Cross Section of the RC Beam

Page | 3-55

At the undamaged state, three tests were performed to obtain the frequency

values of first two flexural modes. The first test was performed on the intact RC

beam under self weight. The second and third tests were performed with added

inertia loads of M=100kg and 160kg as shown in Figure 3.7.

Then the beam was subjected to four point bending test as shown in the Figure

3.8. Perera and Huerta (2008) continuously measured the load-displacement

throughout the loading process up to maximum load value, W=52kN. The first two

frequencies were extracted at the end of each load step, W=8kN, 20kN, and 52kN. At

each stage, the beam was unloaded before they performed the dynamic testing on the

RC beam.

Figure 3.7: Experiment setup with added inertia loads

Figure 3.8: Experimental setup of the four point bending test

3.4.1.1.1. Obtaining E0 at undamaged state

The Young's modulus of concrete, E0, was derived using the experimental

load-displacement curve. From the basic principles, the mid span deflection, y, of a

uniform beam under four point bending loads can be derived as shown in Equation 3-

11.

Page | 3-56

Equation 3-11

where; a = distance between support and the adjacent loading point (a=1.32m

in Figure 3.8), L=Length of the beam (L=4.54m in Figure 3.8), W=load, I=second

moment of area, and E0,equi = equivalent Young's modulus of RC section.

The linear portion of the experimental load-displacement curve was used to

compute the E0,equi. Initial part of the experimental load-displacement curve which

includes the linear behaviour is illustrated in Figure 3.9. The load-displacement curve

indicates a linear relationship up to W=20kN where the yielding starts. The

equivalent Young's modulus of the RC section, E0,equi, was calculated as 41.25GPa

based on W and y values at point A (W=18kN, y=2mm) which lies within the linear

portion. The Young's modulus of concrete at undamaged state, E0, was then

computed using the equivalent area of the RC section and a value of 38.9GPa was

obtained.

Figure 3.9: Linear portion of the experimental load-displacement curve (Perera and Huerta

2008)

3.4.1.2. Validation of the FE model

Section 3.4.1.2.1 presents details of the finite element simulation. The next

sub-section, 3.4.1.2.2 compares the FE results with the experiment results.

Page | 3-57

3.4.1.2.1. FE simulation

The above experiment test setup was then simulated in the ABAQUS FE

package using a three dimensional model.

Coordinate system

The RC beam was simulated in a X-Y-Z coordinate system, in which X, Y, and

Z axes were parallel to the longitudinal, width, and depth directions of the beam.

Geometric properties

A solid section was used to simulate the concrete region. The smeared

reinforcement layer technique was used to simulate the tensile, compressive and

shear reinforcement layers. The beam was therefore partitioned at centres of the

tensile, compressive, and shear reinforcement layers as shown in Figure 3.10. Layers

1, 3, 4, and 6 were used to represent the shear links. Layer 2 and 5 were assigned

with compressive and tensile reinforcement properties , respectively. All

reinforcement layers were embedded into the concrete solid section with the default

ABAQUS parameters.

Figure 3.10: Partitioned cross section to assign the smeared reinforcement layers

Material properties

Two types of material properties were assigned to the FE model; 1. Steel, and

2. Concrete. The elastic-plastic relationship was assumed for the steel. The elastic

Page | 3-58

modulus, density and yield strength of the steel material type were taken as 210GPa,

7850kg/m3 and 510MPa , respectively. These properties were assigned to each of the

six smeared reinforcement layer under material definition.

The concrete material type was assigned with the complete stress-strain curves

under compression and tension. The stress-strain curve under compression was

derived from the numerical technique presented in the 3.3.1. The concrete damage

parameter, dc, was taken as the ratio between (the inelastic strain) and the εd

(strain at the 0.3σcu in the descending portion). Table 3-1 tabulates the σc, , and dc

values of concrete.

Table 3-1: Material properties for concrete under compression

σc (N/m2)

(m/m) dc

1.60E+07 0.00E+00 0.00E+00

2.47E+07 3.04E-04 4.43E-03

2.97E+07 7.07E-04 1.03E-02

3.16E+07 1.19E-03 1.73E-02

3.20E+07 1.70E-03 2.48E-02

2.38E+07 8.54E-03 1.24E-01

1.90E+07 1.53E-02 2.22E-01

1.62E+07 2.20E-02 3.20E-01

1.44E+07 2.86E-02 4.17E-01

1.31E+07 3.53E-02 5.13E-01

1.21E+07 4.20E-02 6.10E-01

Page | 3-59

The tensile stress-strain curve obtained from the numerical technique

(presented in the section 3.3.2) was further calibrated based on the experimental

results. The calibrated tensile properties, σt, ε tck, and dt, are tabulated in Table 3-2.

Table 3-2:Material properties for concrete under tension

σt (N/m2) ε t

ck (m/m) dt

0.00E+00 0.00E+00 0.00E+00

5.88E+06 0.00E+00 0.00E+00

5.15E+06 6.58E-05 5.00E-02

4.56E+06 2.21E-04 1.75E-01

4.54E+06 2.67E-04 2.00E-01

4.53E+06 3.06E-04 2.40E-01

4.50E+06 4.62E-04 3.75E-01

4.41E+06 8.12E-04 5.50E-01

4.35E+06 1.43E-03 7.00E-01

4.29E+06 2.98E-03 9.00E-01

Element types

Two element types were assigned for the FE model, namely;

1. C3D8 - Three dimensional eight noded, fully integrated linear element type:

This element type was assigned for the solid concrete section, and

2. SFM3D4 - Four noded quadrilateral surface element type: This element type

was assigned for all six smeared reinforcement layers (Layers 1-6).

Page | 3-60

Boundary conditions and loading

The beam was supported using ideal pinned and roller supports during the

loading and unloading stages. The self-weight of the RC beam was calculated and

applied on top surface as a sustained pressure load. Point loads (W) were then

applied in a sequential manner and similar to the experimental setup. Static

deflection values at mid span were continuously recorded over the loading process

up to the maximum W of 52kN. The boundary conditions were slightly modified

during the frequency extraction step at W=0kN, 8kN, 20kN, and 52kN as explained

below.

The experimental test setup was not restrained using ideal support conditions,

rather it had been rested on two steel I-beams. Perera and Huerta (2008) reported this

non-ideal boundary conditions and thus introduced spring elements during the

frequency extraction steps of their FE simulations. Similar approach was used in the

present FE simulation to extract natural frequencies and mode shapes. Six parallel

spring elements were used at each of the support locations to restrain the vertical

movement and to provide a certain level of degrees of freedom. The concrete density

and the spring constant were initially assumed as variables in the FEM validation

process. Frequencies of first two flexural modes (experimentally measured) were

used to calibrate the spring constant and the density of concrete. After several trial

attempts, the spring constant and the density of concrete were found to be 1.025 x

107N/m and 3200kg/m

3, respectively. It should be noted that six parallel spring

elements were introduced at each of the support locations (resultant spring constant

at a support = 6.15 x 107N/m).

Page | 3-61

At ends of each loading steps (at W=0kN, 8kN, 20kN and 52kN), the W was

set to zero (unloaded). Then the vertical restraints were removed and spring supports

were introduced before extracting the frequencies and mode shapes of the structure.

Mesh details

The FE model wasn't meshed along the Y-axis, as no lateral forces were

applied. However, beam was partitioned into five parts along the Y-axis to facilitate

the smeared reinforcement layers as shown in the Figure 3.6. The approximate

element sizes along the X-axis and Z-axis were 90mm and 20mm.

3.4.1.2.2. Comparison of FE results with experimental results

This section compares the FE model results with the published experimental

results by Perera and Huerta (2008).

Firstly, the load-displacement curves were compared. The load-displacement

curve obtained from the present FE model was superposed on the experimental curve

of Perera and Huerta (2008) as shown in Figure 3.11. The FE model results agree

well with the experimental results. This confirms that the damaged plasticity model

can simulate the post-cracking behaviour of RC structures with acceptable level of

accuracy.

Secondly, the frequencies of FE mode results were compared with the

experimental results under 6 structural/loading arrangements. Table 4 compares the

frequency values between the present ABAQUS FE model and the experimental

results of Perera and Huerta (2008), under six different structural arrangements: 1.

Undamaged beam, 2. Undamaged beam with two 100kg masses at 1.32m away from

ends of the beam, 3. Undamaged beam with two 160kg masses at 1.32m away from

ends of the beam, 4. Beam after load step of 8kN, 5. Beam after load step of 20kN

Page | 3-62

(stage at damage initiation), and 6. Beam after load step of 52kN (at damaged state).

The maximum percentage difference recorded in Table 4 is 2.56% and establishes

the validation of the present ABAQUS finite element model under different

structural arrangements including damage states.

Figure 3.11: Load-Displacement curves obtained from experimental and FE model results

Table 3-3: Frequencies of mode 1 and mode2

Structural

Arrangement

Frequency –Mode 01 Frequency –Mode 02

Experiment FEM (Change %) Experiment FEM (Change %)

1 25.32 24.715 (2.39%) 74.76 73.725 (1.38%)

2 22.03 22.208 (0.81%) 68.35 66.642 (2.50%)

3 20.50 21.024 (2.56%) 64.14 63.123 (1.59%)

4 25.32 24.715 (2.39%) 74.70 73.723 (1.31%)

5 24.47 24.708 (0.97%) 74.21 73.717 (0.66%)

6 22.94 22.493 (1.95%) 71.10 71.318 (0.31%)

Page | 3-63

3.4.2. Case Study 2 - Three point bending test

Experiment test results published by Peeters et al. (1996) and Maeck (2003)

were selected for the second case study. Experimentally load-displacement curve and

crack patterns were used during the validation process.

3.4.2.1. Experiment test setup

Peeters et al. (1996) and Maeck (2003) have presented the experiment results

of a RC beam tested under three point bending. The 6m long RC beam was

supported at 1.2m distance from either of the edges and loaded at centre. The

experiment load vs. deflection curve at different loading and unloading steps was

presented by Peeters et al. (1996). Crack patterns at different loading steps of the

same experimental setup were presented by Peeters et al. (1996) and Maeck (2003).

The experimental setup and the cross section details of the RC beam are shown

in Figure 3.12 and Figure 3.13 , respectively. The maximum compressive strength of

concrete was recorded as 51.2 MPa (Peeters et al. 1996).

Figure 3.12: Experimental setup (Peeters et al. 1996)

Figure 3.13: Cross section details of the RC beam (Peeters et al. 1996)

Page | 3-64

3.4.2.2. FE simulation

The above experimental setup was simulated in the ABAQUS FE package

following the similar procedure of the case study 1. Reinforcements were assigned as

smeared layers and embedded into the solid concrete section assuming the full bond

between reinforcement and concrete. C3D8 and SFM3D4 element types were

assigned to the concrete section and reinforcement layers. Steel material type was

assigned to the reinforcement layers. However, the E-value was changed to 218GPa,

the experimentally measured value.

Complete stress-strain curves were assigned to the concrete sections. Stress-

strain curves of concrete under compression and tension were obtained from two

numerical techniques presented in sections 3.3.1 and 3.3.2 , respectively. The tensile

properties, σt, ε tck, and dt, are tabulated in Table 3-4.

Table 3-5 tabulates σc, , and dc values of concrete for compression.

The beam was meshed with approximate element size with 50mm x 50mm x

12.5mm in longitudinal, transverse and thickness direction respectively based on the

mesh convergence study carried out.

Table 3-4: Tensile Stress-Strain Values for Concrete with 51.2MPa

Stress

(σt)N/m2

Cracking Strain

(ε tck)

Damage

(dt)

2.36E+06 0.00E+00 0.00E+00

1.89E+06 4.07E-05 3.85E-01

9.45E+05 2.93E-04 9.00E-01

2.13E+05 8.07E-04 9.91E-01

Page | 3-65

Table 3-5: Compressive Stress-Strain Values for Concrete with 51.2MPa

Stress (σc

N/m2

Inelastic Strain

(ε cin

Damage

(dc)

2.56E+07 0.00E+00 0.00E+00

3.64E+07 1.00E-04 1.05E-02

4.49E+07 2.81E-04 2.95E-02

4.97E+07 5.87E-04 6.16E-02

5.12E+07 1.01E-03 1.06E-01

4.90E+07 1.76E-03 1.85E-01

4.43E+07 2.60E-03 2.73E-01

3.89E+07 3.46E-03 3.63E-01

3.37E+07 4.31E-03 4.53E-01

2.92E+07 5.14E-03 5.40E-01

2.54E+07 5.95E-03 6.25E-01

2.22E+07 6.74E-03 7.07E-01

1.95E+07 7.51E-03 7.88E-01

3.4.2.3. Comparison of FE and Experimental results

This section compares the present FE model results with the experimental

result of Peeters et al. (1996) and Maeck (2003). Crack patterns and static load-

displacement curve were used during the comparison of results.

Page | 3-66

The load vs. displacement curves (up to 32kN load) of experimental and FE

simulation are shown in Figure 3.14 and Figure 3.15 , respectively. Although it can

observe some differences in two figures with respect to the behaviour during the

unloading steps, FE model has correctly simulated the load-displacement behaviour

during the loading steps.

Figure 3.14: Experiment Load vs. Displacement (Peeters et al. 1996)

Figure 3.15: Finite Element Load vs. Displacement

The experimentally obtained crack patterns (Maeck 2003) and FE crack

patterns at two loading steps (at 8kN, and 24kN) are compared in Figure 3.16 and

Figure 3.17. Results indicate that the present FE results agree well with the

experiment results.

0

10

20

30

40

0 10 20 30

Po

int

Lo

ad

a

t C

en

tre (

kN

)

Displacement (mm)

Page | 3-67

Figure 3.16: Crack Patterns for 3 point bending test at 8kN

Figure 3.17: Crack Patterns for 3 point bending test at 32kN

3.4.3. Case study 3: RC slab

Experiment test results (RC slab) published by Bakhary (2009) were selected

for the second case study, which is shown in Figure 3.18. The author has presented

the variation of first two frequencies at 10 loading steps (by varying P1 and P2).

The slab was 6400mm x 800mm x 100mm in dimensions and designed to carry

a uniform load of 5kN/m2. It was reinforced with 11N6 bars at 75mm intervals with

bottom steel continues throughout the slab. Top reinforcement continues over the

length of about 900mm either side of the central support. The reinforcement cover

was 20mm throughout the slab. Ultimate strength and density of concrete were

recorded as 32MPa, and 2550kg/m3, respectively.

Experiment Crack Pattern at 32kN

FEM Crack Pattern at 32kN

Experiment Crack Pattern at 8kN

FEM Crack Pattern at 8kN

Page | 3-68

Figure 3.18: Experimental test setup (Bakhary 2009)

The same setup was simulated in the present FE simulation. The experimental

and present FE results are compared in for the first five load steps and given

Table 3-6: Comparison of present FE results with experimental results of (Bakhary 2009)

Loading

Step

P1

(kN)

P2

(kN)

Mode 1 Frequency (Hz) Mode 2 Frequency (Hz)

Experiment FE Experiment FE

Undamaged 0 0 17.810 17.845 24.458 25.272

1 6 0 17.874 17.843 25.167 25.260

2 12 0 17.512 17.701 24.167 24.870

3 18 0 16.893 17.012 24.057 24.015

04 18 3 16.714 16.705 23.854 23.420

5 18 6 16.471 16.143 23.598 22.700

Page | 3-69

Remarks 3-1:

Results presented in the section 3.4 confirm that the damaged plasticity model

in FE package has the ability to represent the post-cracking behaviour of RC

structures.

3.5. Evaluation of Simplified Damage Simulation Techniques

Two simplified damage simulation techniques were evaluated in this section,

namely, 1. Reduction in E-value, and 2. Notch type damage. The four point bending

test of Perera and Huerta (2008) was selected for the comparison.

3.5.1. Basics of simplified damage simulation techniques

Figure 3.19 illustrates the reduction in E-value method for a simply supported

beam with a uniform cross section. This method modifies the E-value at the damage

location using a reduction factor λ1 (0 ≤ λ1 ≤ 1).

The second simplified method reduces the depth of the section at the damage

location by creating a notch as shown in Figure 3.20. λ2 is the reduction factor of the

depth of the section.

Figure 3.19: Damage simulation by reducing the E-value

Figure 3.20: Damage simulation by creating a notch/cut

0 < λ2 ≤ 1

0 < λ1 ≤ 1

E λ1E E

λ2d E E E d

Page | 3-70

3.5.2. FE simulation

This section provides details of the FE simulation strategies carried out to

simulate the experimental setup of Perera and Huerta (2008). In FE simulation, the

reinforcement layers were not modelled. But, the equivalent Young’s modulus of

concrete was assigned to the material properties (E0,equi = 41.25GPa).

3.5.2.1. Reduction in E-value

Expression for the reduction factor, λ1, was first derived referring to the Figure

3.21. A uniform reduction in E-value was assumed for the region between the two

load points of the four point bending test setup. The reduction factor, λ1, was derived

from the first principles and shown in Equation 3-12. The symbols y and I represent

the mid span deflection and the second moment of area of the beam.

Figure 3.21: Reduction factor λ1 for a four point bending test setup

Equation 3-12

Perera and Huerta (2008) had measured and recorded the frequencies of first

two modes at W=0kN, 8kN, 20kN, and 52kN. But, they have stated that the beam

was undamaged at W=0 and 8kN. Because of this, only three models were created in

the ABAQUS FE package with the reduction factors listed in Table 3-7. The

equivalent Young's modulus was assigned to the undamaged regions (two regions

W W

a 2b

L

a

E λ1E E

Page | 3-71

between supports and the adjacent load points). The reduced E-value listed in the last

column of the Table 3-7 was assigned to the damaged zone within the two load

points.

Table 3-7: Reduction factor, λ1

Model ID Load (W)

(kN)

Experiment Deflection at

Centre (mm) : y

λ1 λ1E (GPa)

FEM1 0 Not recorded 1.00 41.25

FEM2 20 2.22 0.81 33.41

FEM3 52 13.50 0.35 14.44

3.5.2.2. Notch type damage

The reduction factor, λ2, was first derived from the first principles and shown

in Equation 3-13.

Equation 3-13

The reduced section depths (λ2d values) were then calculated for the two

damaged states of the Perera and Huerta (2008) at W=20kN, and 52kN. In FE

simulation, the notch type damage was easily simulated by reducing the depth of the

section as required. However, this technique also needs 3 discrete models to

represent the undamaged and two damaged states. The FE model at the undamaged

Page | 3-72

state was exactly similar to the FEM1 as λ2 is equal to 0 at this state. Computed λ2

values are tabulated in Table 3-8.

Table 3-8: Reduction factor, λ2

Model ID Load (W) (kN) λ2 λ2d (mm)

FEM4 ≡ FEM1 0 1.00 320

FEM5 20 0.932 298

FEM6 52 0.705 226

3.5.2.3. Comparison of results

Table 3-9 and Table 3-10 compare the first two frequencies obtained from two

simplified damage simulation techniques with the experimental values. As both

tables indicate, the percentage difference between simplified damage simulation

techniques and the experimental values have been increased with the damage

severity. This implies that the simplified damage simulation techniques do not

correctly represent the behaviour of cracked RC structures.

Table 3-9: Comparison of reduction in E-value method with the experimental results using first

two frequencies

Model ID Frequency –Mode 01 Frequency –Mode 02

Experiment FEM (Change %) Experiment FEM(Change %)

FEM1 25.32 25.00 (1.26%) 74.76 74.66 (0.13%)

FEM2 24.47 23.21 (5.15%) 74.21 72.97 (1.67%)

FEM3 22.94 16.58 (27.72%) 71.10 62.73 (11.77%)

Page | 3-73

Table 3-10: Comparison of notch type damage simulation method with the experimental results

using first two frequencies

Model ID Frequency –Mode 01 Frequency –Mode 02

Experiment FEM (Change %) Experiment FEM

FEM4 25.32 25.00 (1.26%) 74.76 74.66 (0.13%)

FEM5 24.47 23.30 (4.78%) 74.21 73.10 (1.50%)

FEM6 22.94 19.17 (16.43%) 71.10 69.155 (2.74%)

3.6. New Displacement based Damage Severity Index

The above section highlights that two simplified damage simulation techniques

have failed to accurately represent the post-cracking behaviour of RC structures. In

VBDITs, the damage severity has been commonly defined in terms of reduction in

E-value or reduction in section depth. The above two facts are contradictory to each

other, and hence need some amendment for the damage severity definition.

The parabolic shape of flexural cracks also highlights the need for a better

definition for the damage severity index. The severity of the flexural cracks reduces

in a non-linear manner with the distance away from the centre of the cracked zone.

Uniform reductions in E or depth are therefore not applicable.

This study has proposed a new damage severity index to be employed for the

VBDITs. This new damage severity index was denoted by DDSI which stands for

the Displacement based Damage Severity Index.

Page | 3-74

DDSI defines the damage severity of the flexural cracks using the static

displacement values of the structure at undamaged and damaged states under a same

load condition. Equation 3-14 represents the DDSI which is defined using

displacement of the undamaged and damaged states under self weight. Subscript 'sw'

is used to indicate that the DDSI has been calculated using displacement values

obtained under self weight. However, different sustained loading arrangements can

be used depending on the situation. It should be noted that these sustained loads are

serviceability loads but not cracking loads (i.e. loads that may cause crack

propagation).

Equation 3-14

Asw,u and Asw,d represent the area of the displacement curve under self weight at

undamaged and damaged states.


This chapter has evaluated three damage simulation techniques, namely, 1. The

damaged plasticity model, 2. Reduction in E-value based method, and 3. Notch type

damage. First two sections, sections 3.2, and 3.3, of the chapter have presented

details of the damaged plasticity model and two numerical techniques to derive the

stress-strain curves of concrete under compression and tension. The results in the

section 3.4 confirmed that the damaged plasticity model can simulate the post-

cracking behaviour or RC structures.

Higher percentage change in frequency values were observed with the two

simplified damage simulation techniques, the reduction in E-value, and notch type

Page | 3-75

damage. Based on these results, the damaged plasticity model was selected to

simulate different flexural crack patterns as required for the study.

Need for a better definition for the damage severity of flexural cracks has been

highlighted in the section 3.6. A new damage severity index, which was named as

DDSI, was therefore proposed.

The outcomes of this chapter highlight that the first two objectives of this

research have been achieved. In brief, this chapter has recommended the damaged

plasticity model for the future studies. The DI evaluations carried out in the

subsequent chapters of this research were therefore based on simulated flexural

cracks using the damaged plasticity model. The damage severity was defined using

the proposed damage severity index, DDSI.

Page | 4-77

4. Theory and Equations of Damage Indices

This chapter presents theories and formulas related to the vibration based

damage indices used in this study. The first section, section 4.1, presents the

formulas for, 1. Percentage change in frequency, 2. Modal Assurance Criteria

(MAC), and 3. Modal Sensitivity Value (MSV). Next three sections, section 4.2, 4.3,

and 4.4, present formulas of mode shape (including derivatives), flexibility values,

and modal strain energy values based DIs. Symbols α, β, and γ are used to represent

flexibility, modal strain energy, and mode shape based DIs.

This study proposes four new SEDIs to minimise the number and intensity of

false alarms. Formulas for these four SEDIs, β8-β11, are given in the section 4.5.

Next section succinctly explain the calculation steps of proposed Generalized

Damage Localization Index (GDLI) which standardize the damage localization

results obtained from different samples. Next, Displacement based Damage Severity

Index (DDSI) is defined in the section 4.7.

Notations and symbols used in this chapter are listed under abbreviations.

4.1. Principles of Vibration

The equation for the dynamic response of a structure is given by:

Equation 4-1

where, [M], [C], and [K] are the mass, damping and stiffness matrix of the

structure, respectively, {P} is the force vector and {U}, and are the

displacement, velocity and acceleration vectors , respectively.

Page | 4-78

For free vibration analysis, in the absence of damping, the above equation

reduces to:

Equation 4-2

This equation can be used to determine the natural frequencies and mode

shapes of the structure both in its healthy and damaged states. The DIs used in

damage assessment are based on the natural frequencies (, mode shapes (and

their derivatives.

In this study, the linear perturbation technique was used to extract frequencies

and mode shapes of the undamaged and damaged states using the ABAQUS FE

package.

4.2. Basic Parameters

4.2.1. Change in frequency

Equation 4-3 indicates the percentage change of ith

mode frequency, FRQ(1,i),

between undamaged and damaged states. ωd(1,i) and ωu(1,i) are the frequencies of ith

mode at damaged and undamaged states , respectively.

FR 1,i 1 d( i)

u( i) 100 %

Equation 4-3

4.2.2. Modal Assurance Criteria (MAC)

MAC value is used to compare the correlation of the ith

mode to kth

mode

obtained at two different states. Higher MAC values indicate higher correlation. If

mu and md number of modes have been measured at undamaged and damaged states,

dimensions of MAC matrix becomes mu X md. Entry for the ith

row and kth

column of

Page | 4-79

the [MAC]muXmd matrix is calculated using Equation 4-4. N represents the number of

measurement points.

MAC i,k

Equation 4-4

4.2.3. Modal Sensitivity Value (MSV)

The MSV value can be used to measure the sensitivity level of individual

modes to the damage (Salawu 1997b). MSV value for the ith

mode, MSV, between

undamaged and damaged states can be expressed as in Equation 4-5.

Equation 4-5

4.3. Mode Shape Based Damage Indices

Necessary equations for DIs those based on change in mode shapes and higher

order derivatives are presented in this section. Mode shape based DIs include those

derived using changes in mode shapes (γ), and changes in first four derivatives of

mode shapes (γ1-γ4).

4.3.1. Change in mode shapes

Equation 4-6 calculated the change in mode shape values for ith

mode on

individual mode basis. The DI derived by combining 'M' number of modes are shown

in Equation 4-7.

Page | 4-80

,i ,i ,i Equation 4-6

,1 ,i

Equation 4-7

4.3.2. Change in higher order derivatives of mode shapes

Similar approach is used to calculate DIs derived using first four derivatives of

mode shapes. Equations 4-8 - 4-11 gives individual mode based DIs, whereas

Equations 4-12 - 4-15 are derived by combining 'M' number of modes.

,i ,i

,i Equation 4-8

,i ,i

,i Equation 4-9

,i ,i

,i Equation 4-10

,i ,i

,i Equation 4-11

,1 ,i

Equation 4-12

,1 ,i

Equation 4-13

,1 ,i

Equation 4-14

,1 ,i

Equation 4-15

Page | 4-81

4.4. Modal Flexibility Based Damage Indices

Modal flexibility matrix [F] can be obtained using Equation 4-16 (Shih 2009).

[1/ω2] represents the diagonal matrix which contains reciprocal of natural

frequencies of modes.

Equation 4-16

Four different forms of DIs those based on changes in modal flexibility values

are presented in Equation 4-17 - 4-20.

Equation 4-17

Equation 4-18

Equation 4-19

Equation 4-20

4.5. Modal Strain Energy Based Damage Indices

Eleven different forms of Modal Strain Energy based Damage Indices

(MSEDIs) are evaluated in this study. Five of them are based on individual modes

and the rests are derived by combining multiple modes. Expressions for these

MSEDIs are presented in this section along with some preliminary information.

Page | 4-82

4.5.1. Modal strain energy

Modal strain energy associated with jth

element of a beam, U(j,i), for the ith

mode is given by Equation 4-21 (Alvandi and Cremona 2006; Shih 2009), where EI

flexural rigidity of the beam, φi = ith

mode shape, Xj = x-distance at node j, Xj+1 =

x-distance at node +1 and φ''i = curvature of the i

th mode shape.

Equation 4-21

The total modal strain energy of a beam for the ith

mode, Ui, can be calculated

by adding elementary modal strain energy values or using Equation 4-22, where L is

the beam length.

Equation 4-22

Fractional strain energy of the jth

element for the ith

mode, FSEji, is defined in

the Equation 4-23.

Equation 4-23

4.5.2. Equations of SEDIs

Three different forms of MSEDIs based on individual modes and denoted by

β1, β2 and β3 are proposed by Cornwell, Doebling and Farrar (1999), Stubbs, Kim

and Topole (1992) and Park, Kim and Stubbs (2002). β1, β2, and β3, are defined in

Equations 4-24, 4-25, and 4-26, respectively.

Page | 4-83

Equation 4-24

Equation 4-25

Equation 4-26

β4, β5, β6, and β7 denote multiple mode based MSEDIs proposed by Cornwell,

Doebling and Farrar (1999), Alvandi and Cremona (2006), Park, Kim and Stubbs

(2002) and Shih (2009) , respectively. Equations of these four DIs are given in

Equations 4-27 - 4-30.

Equation 4-27

Equation 4-28

Equation 4-29

Equation 4-30

Page | 4-84

4.6. Improved SEDI

Four different forms of modifications to the MSEDI were studied at the

preliminary stage of this research. This lead to derive four MSEDIs which are

labelled as β8, β9, β10 and β11. This section presents formulas for these four

MSEDIs. β8 and β9 are based on individual modes and introduce mode shape and

curvature of mode shapes based modification functions (MF1 and MF2),

respectively. These modification functions, MF1 and MF2, have the ability to reduce

or eliminate the false alarms generated at or near the nodal points by assigning zero

weights at these points and very low values at locations next to these nodal points.

Equations of two modifications functions, MF1 and MF2, are given in Equations 4-

31 and 4-32, respectively.

The value of the modification function MF1 at the centre of the jth

element of

the ith

mode is obtained using Equation 4-31, in which φ( ,i) and φ(i)max are the mode

shape values at the centre of the jth

element and maximum mode shape value of the ith

mode. In Equation 4-32, mode shape values have been substituted by mode shape

curvature values, φ''( ,i), and φ

''(i)max. Equations of β8 and β9 are given in Equation 4-

33 and 4-34 , respectively.

Equation 4-31

Equation 4-32

Equation 4-33

Equation 4-34

Page | 4-85

β10 and β11 are the average of β8 and β9 across measured M number of

modes. In the traditional approach, equal weight is assigned to all the modes.

However, this study has introduced the MSV value (defined in section 4.2.3) to

assign different weights based on the sensitivity of each mode to the damage.

Equation 4-35 and 4-36 gives the formulas for β10 and β11 , respectively.

Equation 4-35

Equation 4-36

4.7. Statistical Based Damage Assessment Scheme

This section provides the details of the proposed statistical based damage

assessment scheme which combines three new indices. The first one is named as

Average Probability Distribution Index (APDI) and the second one is named as the

Generalized Damage Localization Index (GDLI). Finally, a Reliability Index (RI) is

derived. The procedure is explained using the MSEDI, β11.

4.7.1. Average Probability Distribution Index (APDI)

APDI is used to calculate the average probability values of positive β11. Three

steps are involved in this process.

4.7.1.1. Step 1

In the first step, positive β11 values of the kth

sample are saved into a new

matrix [β11+]NxK as given in Equation 4-37. N and K represents the total number of

elements and samples , respectively. (k = 1, 2, 3, ....., K)

Page | 4-86

Equation 4-37

4.7.1.2. Step 2

The second step assigns value of 1 for the [β11k+]Nx1.

Equation 4-38

4.7.1.3. Step 3

In the last step, the average values of P+(j,k) are being calculated across all the

samples acquired at the time of calculation.

Equation 4-39

4.7.2. Generalized Damage Localization Index (GDLI)

GDLI standardize the positive β11 values across all measured samples using

two steps.

4.7.2.1. Step 1

Positive β11 values are multiplied by the average probability of damage

indication value calculated using Equation 4-39.

Equation 4-40

4.7.2.2. Step 2

In this step, the GDLIβ11 is calculated using Equation 4-41.

Page | 4-87

Equation 4-41

where;

=Mean value of the as given in Equation 4-42,

and = Standard deviation of

as given in Equation 4-43.

Equation 4-42

Equation 4-43

4.7.2.3. Reliability Index (RI)

RI is proposed to define the reliability of the damage localization result of the

candidate structural portion or element under investigation. The derivation of the

proposed RI, first involves standardizing GDLIβ11 values, so that the highest value

becomes unity. Standardized GDLIβ11 is multiplied by the given in

Equation 4-39. This will be further explained in section 8.3.4 under applications.


This chapter presents equations of 25 DIs including new four SEDIs developed

in this research. Step by step guidance has been given to the proposed probability

based damage detection process which incorporates the proposed concept of

Generalized Damage Localization Index method.

The next chapter evaluates 25 DIs (γ, γc, γ1 - γ4, γ1c - 4c, α1 - α4, β1-β11)

based on the damage localization ability of flexural cracks in RC beam.

Page | 5-89

5. Damage detection of cracked RC beams without noise

This chapter ranks 25 DIs based on their localization ability of flexural cracks

in RC beams without noise. Total of 21 ( 10 single and 11 multiple) damage cases

are examined during this ranking process. Detail of the RC beam setup and 21

damage cases are presented in the section 5.1. The section 0 highlights that some DIs

fail to localize single flexural cracks leading to eliminate them in the future analysis.

The remaining DIs are further evaluated based on damage localization ability of

multiple cracks in the section 5.3.

5.1. Details of the RC Beam Setup

The RC beam setup presented in the section 3.4.1 is used for the result

presented in this chapter. However, boundary conditions are changed so that left and

right ends are supported by pinned and roller supporters , respectively (i.e. a simply

supported beam setup is used). The beam is first loaded with a sustained self weight

using a uniformly distributed pressure load. Frequencies and mode shapes obtained

at this stage are saved as vibration properties of the undamaged state of the RC beam.

Mode shapes and frequencies of first four flexural modes of the undamaged beam are

shown in Figure 5.1 (a)-(d). The beam is then loaded with a concentrated load, W, so

that different damage cases are obtainable by changing the location and magnitude of

W. At each stage of the damage, frequencies and mode shapes are recorded once W

is unloaded. Multiple point loads are used to induce multiple damage locations.

Details of all simulated damage cases are presented in sections 5.1.1 and 0 for single

and multiple damage cases.

Page | 5-90

Figure 5.1(a): Mode 1-Frequency of 24.716 Hz Figure 5.1 (b): Mode 2-Frequency of 90.026 Hz

Figure 5.1 (c): Mode 3-Frequency of 221.98 Hz Figure 5.1 (d): Mode 4-Frequency of 352.54 Hz

Figure 5.1: First four flexural modes of the undamaged RC beam

5.1.1. Single damage cases

Figure 5.2 shows the loading arrangement used to create 10 single damage

cases. During DI ranking process, two damage locations are selected: 1. mid span

(XW=2.270m) and 2. quarter span (XW = 1.135m). The concentrated load value, W, is

gradually increased in five steps for each of the damage locations to vary the

severity. Table 5-1 tabulates load value (W), damage severity (DDSI), and observed

crack pattern of 10 single damage cases. It should note that the beam is divided into

30 equal size elements during the damage index calculation process. At initial stage

of this study, damage states of these 30 elements are identified by visual comparison

made with the ABAQUS crack patterns. Identified damaged elements are shown in

bold red colour numbers.

Figure 5.2: Loading arrangement for single damage cases

4.54 m XW

W

Page | 5-91

Table 5-1: Ten Single Damage Cases

Dam

age

Cas

e

XW

, W

, D

DS

I

Damage Elements identified from ABAQUS Simulation

(Bold red colour element numbers indicate damaged elements) S

DM

1

2.2

70

m,

30

kN

, 3

7%

SD

M2

2.2

70m

, 36kN

, 64%

SD

M3

2.2

70

m,

42kN

, 77%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-92

SD

M4

2.2

70

m, 4

8k

N, 8

4%

SD

M5

2.2

70m

, 6

0k

N,

92%

SD

Q1

1.1

35m

, 42kN

, 40%

SD

Q2

1.1

35

m,

48

kN

, 5

7%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-93

5.1.2. Multiple damage cases

Figure 5.3 demonstrates the loading arrangement used to create multiple

flexural cracks. The load, W1 is initially applied at a distance XW1 from the left end

to induce the first crack. Once W1 is unloaded, the second load, W2 is introduced at

a distance of XW2. The W2 is then removed so that beam is subjected only to the

sustained self-weight during vibration property extraction. Similar loading and

SD

Q3

1.1

35

m, 5

4k

N, 7

0%

SD

Q4

1.1

35m

, 6

0k

N,

78%

SD

Q5

1.1

35m

, 75kN

, 89%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-94

unloading procedure is repeated for another step to simulate cases with three

damages using a concentrated load W3 which is applied at a distance of Xw3 from

the left end. Eleven multiple damage cases are simulated by varying XW1, W1, XW2,

W2, Xw3, and W3 as necessary. Details of these damage cases are presented in Table

5-2.

Figure 5.3: Load arrangement to create multiple damage locations

Table 5-2: Multiple damage cases

Dam

age

Cas

e

XW

1,

W1,

DD

SI

XW

2,

W2,

DD

SI Damage Elements identified from ABAQUS Simulation

(Bold red colour numbers indicate damaged elements)

MD

1

1.1

35

m, 4

8k

N, 5

7%

3

.78

m,

72

kN

, 76

%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

XW1

W1

XW2

W2

Page | 5-95

MD

2

2.2

70

m, 4

2k

N, 7

7%

1.1

35

m, 5

4k

N, 8

3%

MD

3-

2.2

70

m, 60

kN

, 92

%

1.1

35

m, 42

kN

, 92

%

MD

4

1.1

35

m, 60

kN

, 78

%

2.2

70

m, 36

kN

, 81

%

MD

5

1.1

35

m, 54

kN

, 70

%

2.2

70

m, 42

kN

, 83

%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-96

MD

6

2.2

70

m, 30

kN

, 37

%

1.1

35

m, 75

kN

, 89

%

MD

7

2.2

70

m, 3

0k

N, 3

7%

1.1

35

m, 4

2k

N, 5

2%

MD

8

2.2

70

m, 3

0k

N,

1.1

35

m, 5

4k

N, 7

2%

MD

9

2.2

70

m, 3

6k

N,

1.1

35

m, 5

4k

N, 7

8%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-97

MD

10

2.2

70

m, 4

2k

N,

1.1

35

m, 6

0k

N, 8

5%

MD

11

2.2

70

m, 4

2k

N,

1

.13

5m

, 6

0k

N,

W3=

60k

N,

XW

3=

3.7

8m

,

DD

SI=

86

%

5.2. Evaluation of Damage Indices on Localizing Single Damage

5.2.1. Evaluation flexibility based DIs

This section examines damage localization ability of four flexibility based DIs

(α1-α4) for single damage cases. α1 and α2 are calculated on the individual mode

basis whereas α3 and α4 are calculated by combining higher modes.

5.2.1.1. Flexibility based damage index, α1

Figure 5.4(a)-(d) illustrate damage localization results of α1 for the mid span

damage case, SDM5 (refer Table 5-1), with use of first four flexural modes on the

individual mode basis. Figure 5.4(a) clearly identifies the centre of the damage zone

if the maximum α1 value is used as the criteria to localize damage zones. Although,

the first flexural mode indicates a promising damage localization result, higher order

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-98

modes do not behave in such a manner. Remaining three figures (Figure 5.4 (b)-(d))

those correspond to second, third and fourth flexural modes indicate some local

peaks at locations other than the centre of the damage zone. Such local peaks are

increased in the number when the order of the mode is increased. In other terms,

higher the order of modes, higher the tendency to have positive false alarms in the

damage localization process using α1.

Figure 5.5 (a)-(d) which plot variation of α1 across first four flexural modes for

the damage case SDQ5, also confirm the fact that number of false alarms are

increased with the mode number. For this damage case, it is observed that α1 fails to

localize centre of the damage zone with sufficient accuracy even with the

fundamental flexural mode. The peak α1 value in Figure 5.5 (a) is recorded closer to

element 13, which lies away from the centre of the cracked zone, but leans towards

the mid span of the beam. This implies that peak α1 value does not always

correspond to the centre of the damage zone.

Figure 5.4(a): Damage Index: α1 Mode No: 1 Damage Case: SDM5

Peak Locations: Element 16

0.00E+00

1.00E-03

2.00E-03

3.00E-03

α1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-99

Figure 5.4(b): Damage Index: α1, Mode No: 2, Damage Case: SDM5

Peak Locations : Elements 11 & 21

Figure 5.4(c): Damage Index: α1, Mode No: 3, Damage Case: SDM5

Peak locations: Element 16. Two local peaks at elements 7 & 25

Figure 5.4 (d): Damage Index: α1, Mode No: 4, Damage Case: SDM5

Peak locations: Element 18. Local peaks at elements7, 12 & 25


-5.00E-06

0.00E+00

5.00E-06

1.00E-05

1.50E-05

α1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

-1.00E-05

0.00E+00

1.00E-05

2.00E-05

3.00E-05

4.00E-05

α1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

-2.00E-05

0.00E+00

2.00E-05

4.00E-05

6.00E-05

8.00E-05

1.00E-04

α1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-100

Figure 5.5(a): Damage Index: α1, Mode No: 1, Damage Case: SDQ5

Peak locations: Element 13

Figure 5.5 (b): Damage Index: α1, Mode No: 2, Damage Case: SDQ5

Peak locations: Elements 8 & 21 (Highest value at element 21)

Figure 5.5 (c): Damage Index: α1, Mode No: 3, Damage Case: SDQ5

Peak locations: Elements 8 & 15. One local peak at elements 23

-1.50E-05

-1.00E-05

-5.00E-06

0.00E+00

5.00E-06

1.00E-05

1.50E-05

α1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

-5.00E-05

0.00E+00

5.00E-05

1.00E-04

1.50E-04

α1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

2.50E-03

α1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-101

Figure 5.5 (d): Damage Index: α1, Mode No: 4, Damage Case: SDQ5

Peak locations: Element 11. Local peaks at elements 5, 18 & 25

Figure 5.5: Variation of α1 for the damage case SDQ5 with first four flexural modes

Remarks 5-1:

Peak value of α1 does not coincide with the centre of the damage zone when

damage is at a location other than the mid span of the beam.

Significant numbers of positive false alarms are recorded with α1 of higher

order modes.

Due to above reasons, it is difficult to define a robust criterion for α1 to

localize the damage with sufficient accuracy. The damage index α1 is

therefore, not recommended to for damage localization purposes.


From Figure 5.6, it can observe that the peak value of α2 of the first mode

coincides with the centre of the damage zone. A similar result is observed for the

quarter span damage case, SDQ5, as shown in Figure 5.7. The peak value of α2 in

Figure 5.7 is recorded at element 8, which agrees with the centre of the damage zone.

This is a significant improvement compared to the α1.

-1.00E-05

-5.00E-06

0.00E+00

5.00E-06

1.00E-05

1.50E-05

α1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-102

Figure 5.6: Variation of α2 for the damage case SDM5 with first flexural mode

Figure 5.7: Variation of α2 for the damage case SDQ5 with first flexural mode

Although maximum α2 correctly localizes the centre of the damage zone, it is

difficult to define a criterion to identify the entire damage zone. In addition, α2

indicates positive false alarms at nodal points of higher modes. Two typical examples

are shown in Figure 5.8 and Figure 5.9 for the damage cases SDM5 and SDQ5 using

third and second flexural modes, respectively. In both examples, positive false

alarms can be observed at the nodal points of corresponding modes.

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

α2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01 α

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-103

Figure 5.8: Variation of α2 for the damage case SDM5 with third flexural mode

Figure 5.9: Variation of α2 for the damage case SDQ5 with second flexural mode

Remarks 5-2:

The maximum value of α2 can be used to localize the centre of the damage

zone with a sufficient accuracy. However, α2 does not localize the entire

damage zone in addition to the associated false alarms at nodal points of

higher modes.


α3 is derived by adding α2 values across measured multiple modes. α3 is

therefore identical to the performance of α2, on individual mode basis. When

multiple modes are used, false alarms associated with higher modes are intensified in

-1.00E+00

-5.00E-01

0.00E+00

5.00E-01

1.00E+00

1.50E+00

α2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

-1.00E+00

-5.00E-01

0.00E+00

5.00E-01

1.00E+00

α2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-104

numbers as they combined together. An example is illustrated in Figure 5.10 using

damage case, SDM5, in which α3 is derived by combining first four flexural modes.

Abrupt changes can observe all over the beam which leads to generate significant

numbers of positive false alarms. The maximum α3 also lies in the undamaged zone,

which leads to a totally unacceptable result.



Figure 5.11 plots variation of α4, which is derived using first four flexural

modes for the damage case SDM5. Figure 5.11 indicates that α4 has correctly

localized the centre of damage but failed to define boundaries of the cracked zone.

Table 5-3 tabulates the peak values of α4 for damage cases, SDM5 and SDQ5, under

different numbers of mode combinations. Peak values do not change significantly

with addition of higher modes implying that higher modes do not have significant

contribution to localization result using α4 for single damage cases. This DI is further

investigated with multiple damage cases.

-5.00E-01

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

α3

(1-4

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-105


Table 5-3: Peak values of α4 for different combinations of modes

Mode Combinations Peak α4 value for SDM5

(around mid span)

Peak α4 value for SDQ5

(around quarter span)

1 0.161910 0.168512

1-2 0.161608 0.160424

1-3 0.161934 0.159484

1-4 0.161924 0.159603

Remarks 5-3:

All four flexibility based DIs fail to define the boundaries of flexural cracking

but have some success in localizing the centre of the damage zone for α2 and

α4.

α1 is not recommended to localize flexural cracks as it does not possess a

robust criterion to detect at least the centre of the damage zone. The other

eliminated flexibility DI, α3, intensifies the number of false alarms when

different numbers of modes are combined.

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

α4

(1-4

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-106

α4 produces the best damage localization ability compared to other

counterpart flexibility based DIs. α2 can be ranked to the second place, but

damage localization result is acceptable only for the fundamental flexural

mode. α4 is therefore selected for further investigation.

5.2.2. Evaluation of damage indices based on mode shapes and derivatives

5.2.2.1. γ - based on change in mode shapes

Damage localization result of γ for SDM1 and SDM5 are illustrated in Figure

5.12 and Figure 5.13 , respectively. Peak values of these two figures do not coincide

with the mid span of the beam or the damage location due to the reason explained in

the section that follows.

When displacement based normalization is carried out, the maximum modal

displacement value of the fundamental mode (at mid span) is set to unity. Because of

this, the mode shape difference at centres of damage zones in SDM1 and SDM5

becomes zero and hence DI, γ, fails to localize centres of damage zones in Figure

5.12 and Figure 5.13. Another similar example is illustrated in Figure 5.14 for the

damage case SDQ5 with use of second flexural mode. In this case, the centre of the

damage zone lies at quarter span at where the modal displacement is highest for the

second flexural mode.

However, if the centre of the damage zone does not coincide with a point

where the modal displacement becomes highest, γ has the ability to localize the

damage. A typical example is illustrated in Figure 5.15 which plots γ derived from

the fundamental mode for the damage case SDQ5. Although the peak γ value

coincides with the centre of the damage zone in SDQ5, another local peak is

observed at the undamaged quarter span towards the right end of the beam. All the

Page | 5-107

above facts implies that γ does not have a robust criterion to localize the damage in

accurate manner.

Figure 5.12: Variation of γ for the damage case SDM1 with first flexural mode

Figure 5.13: Variation of γ for the damage case SDM5 with first flexural mode

Figure 5.14: Variation of γ for the damage case SDQ5 with second flexural mode

0.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

γ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

5.00E-03

1.00E-02

1.50E-02

2.00E-02

γ(1

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

γ(1

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-108

Figure 5.15: Variation of γ for the damage case SDQ5 with first flexural mode

5.2.2.2. Damage indices based on higher order derivatives of mode shapes (γ1 ,

γ2, γ3 & γ4)

Results of this section indicate that sensitivity to damage increases with the

order of derivatives. This implies that γ4 localizes damage more accurately than the

other three DIs. Accuracy of damage localization result reduces in the order of γ3,

γ2, and γ1 for the remaining three DIs.

Two typical examples are illustrated in Figure 5.16 (a)-(d) and Figure 5.17 (a)-

(d) for damage cases, SDM1 and SDQ1 , respectively to prove the above fact. In

both cases, the fundamental mode is used in the DI calculation process. γ1 has

widely spread variations in Figure 5.16 (a) and Figure 5.17 (a), which leads to

failures in damage localization process. Such Variations for γ2, γ3, and γ4 tend to

concentrate on the centre of the damage, among which γ4 produces the best result. It

should note that the peak values in Figure 5.16 (a)-(d) have increased in the order of

γ1, γ2, γ3, and γ4. Similar variation is observed in Figure 5.17 (a)-(d) for the damage

case SDQ1.

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03 γ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-109

Figure 5.16(a): Damage Index: γ1, Mode No: 1, Damage Case: SDM1

Figure 5.16(b): Damage Index: γ2, Mode No: 1, Damage Case: SDM1

Figure 5.16(c): Damage Index: γ3, Mode No: 1, Damage Case: SDM1

0.00E+00

5.00E-03

1.00E-02

1.50E-02

2.00E-02

γ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

5.00E-03

γ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

2.00E-04

4.00E-04

6.00E-04

8.00E-04

1.00E-03

γ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-110

Figure 5.16(d): Damage Index: γ4, Mode No: 1, Damage Case: SDM1

Figure 5.16: Variation of γ1, γ2, γ3 and γ4 for damage case SDM1 with first flexural mode

Figure 5.17(a): Damage Index: γ1, Mode No: 1, Damage Case: SDQ1

Figure 5.17(b): Damage Index: γ2, Mode No: 1, Damage Case: SDQ1

0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

γ2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

γ1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

-5.00E-02

0.00E+00

5.00E-02

1.00E-01

1.50E-01

γ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-111

Figure 5.17(c): Damage Index: γ3, Mode No: 1, Damage Case: SDQ1

Figure 5.17(d): Damage Index: γ4, Mode No: 1, Damage Case: SDQ1

Figure 5.17: Variation of γ1, γ2, γ3 and γ4 for damage case SDQ1 with first flexural mode

Next, performance of γ1, γ2, γ3, and γ4 of higher-order modes are presented

using two examples; 1. SDM1 with use of second mode in Figures 8.18 (a)-(d), and

2. SDQ1 with use of fourth mode in Figures 8.19 (a)-(d). The former example,

illustrates an improvement in localization results for all four DIs. However, the

accuracy level has significantly reduced in the latter case, particularly for γ1, γ2 , and

γ3. Only exception is γ4, which still localizes the damage zone at quarter span in

Figures 8.19 (d). This implies; that numbers of positive false alarms have increased

with the mode number, particularly for the DIs derived using low order derivatives of

mode shapes.

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

2.50E-01

γ4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

5.00E-03

1.00E-02

1.50E-02

2.00E-02

2.50E-02

3.00E-02

γ3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-112



Figure 5.18(c): Damage Index: γ3,Mode No: 2, Damage Case: SDM1

-1.00E-02

0.00E+00

1.00E-02

2.00E-02

3.00E-02

γ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

γ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

5.00E-04

1.00E-03

1.50E-03

γ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-113


Figure 5.18: Variation of γ1, γ2, γ3 and γ4 for damage case SDM1 with second flexural mode

Figure 5.19(a): Damage Index: γ1, Mode No: 4, Damage Case: SDQ1

Figure 5.19(b): Damage Index: γ2, Mode No: 4, Damage Case: SDQ1

0.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

6.00E-02

γ2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

5.00E-03

1.00E-02

1.50E-02

γ1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

-5.00E-02

0.00E+00

5.00E-02

1.00E-01

1.50E-01

γ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-114

Figure 5.19(c): Damage Index: γ3, Mode No: 4, Damage Case: SDQ1

Figure 5.19(d): Damage Index: γ4, Mode No: 4, Damage Case: SDQ1

Figure 5.19: Variation of γ1, γ2, γ3 and γ4 for damage case SDQ1 with fourth flexural mode

Remarks 5-4

γ4 indicates the best stability in damage localization result compared to the

other four mode shape based DIs, γ, γ1, γ2, and γ3. As γ2 is a widely used DI

in previous studies, both γ2 and γ4 are selected for further evaluation.

5.2.2.3. Performance of γ2 and γ4 under different damage severities

Damage localization results of γ2 for damage cases, SDM1-SDM5, are

illustrated in Figure 5.20 (a)-(e). Peak values in all five damage cases are recorded at

the mid span of the beam. However, the number of positive false alarms are

increased at higher damage severities as evident from Figure 5.20 (d) and (e). This

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

γ4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

2.50E-01

3.00E-01

γ3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-115

observation depicts that flexural cracks behave differently from other damage types.

Previous studies have shown that damage localization result improves at higher

severities (Whalen 2008). The reason for the controversial observation can be

explained with respect to the distribution of flexural cracks as below.

Once a flexural crack occurs, secondary cracks are formed along the

longitudinal axis very quickly due to the stress distribution caused by tensile

reinforcement layer. Propagation rate of existing cracks along the depth direction is

comparatively slow as the reinforcement layer prevents further increments of tensile

stress in adjacent concrete layers. Flexural cracks, therefore, become parabolic in

shape that is elongated along the longitudinal axis than the depth of the beam.

Because of this, the influential zone at higher damage severities becomes significant

and intensifies false alarms of γ2.

A similar result is shown in Figure 5.21 for SDQ5 that has a severe damage at

quarter span. Although peak value of γ2 lies at quarter span in Figure 5.21,

weakening of damage localization result due to higher severity can easily be

visualised by comparing with the low damage severity case SDQ1 shown in Figure

5.17(b).

γ4 also endures the same problem of having false alarms at higher severities as

illustrated in Figure 5.22 for the severe damage case SDM5. γ4 has indicated two

positive false alarms at quarter span locations at higher severity whereas it has not

had any false alarms at low damage severity as shown in Figure 5.16 (d). However, it

should be noted that γ4 has slightly improved the damage localization results

compared to that of γ2 shown in Figure 5.20(e). This proves the fact that DIs based

on higher order derivatives produce more robust damage localization results than the

ones with the low order derivatives.

Page | 5-116



Figure 5.20(c): Damage Index: γ2, Mode No: 1, Damage Case: SDM3

0.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

γ2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

5.00E-03

1.00E-02

1.50E-02

2.00E-02

2.50E-02

3.00E-02

γ2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

5.00E-03

γ2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-117


Figure 5.20(e): Damage Index: γ2, Mode No: 1, Damage Case: SDM5

Figure 5.20: Variation of γ2 based on first flexural mode with different damage severities

Figure 5.21: Variation of γ2 for damage case SDQ5 with first flexural mode

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

γ2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

γ4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

6.00E-02

γ2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-118

Figure 5.22: Variation of γ4 for damage case SDM5 with first flexural mode

Remarks 5-5:

Distribution of flexural cracks along the longitudinal direction of the beam

has a negative influence on damage localization result as false alarms are

increased at higher damage severities. This is a remarkable but contrary

observation compared to damage localization results of other damage types or

simulation techniques presented in the literature.

The above fact confirms the necessity to have correct damage simulation

techniques when addressing the forward problem.

5.2.2.4. Damage index, γ2c(1:4)

This section evaluates the curvature based DI, γ2c(1:m), that combines first 'm'

number of flexural modes. Results indicate that the traditional approach of

combining modes (averaging γ2 values across different modes) intensified the

number of false alarms. This is due to the higher contribution factors associated with

higher order modes when averaging γ2 values across different modes. Normalization

of curvature of mode shapes is therefore proposed as a corrective measure to improve

the overall damage localization result. The normalized curvature based damage index

is denoted by γ2cn(1:m), that is derived by combining 'm' number of modes. Some

-2.00E-01

0.00E+00

2.00E-01

4.00E-01

6.00E-01

8.00E-01

1.00E+00 γ4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-119

examples and more explanations are given in below paragraphs to confirm above

mentioned facts.

Firstly, presence of false alarms is elaborated using damage cases SDM1,

SDM5, SDQ1, and SDQ5. Localization results of γ2c(1:4) for above four damage

cases are presented in Figure 5.23 (a)-(d). It should note that little success is achieved

in damage localization result at low severities such as SDM1 and SDQ1. However,

accuracy has been reduced in SDM5 and SDQ5 at higher severities as false alarms at

adjacent areas are intensified. Reason for such results is linked with the nature of

flexural cracks explained in section 5.2.2.3 and the calculation method of γ2c(1:m).

The latter reason is further elaborated in the next paragraph.

Figure 5.23(a): Damage Index: γ2c(1:4), Modes: 1:4, Damage Case: SDM1

Figure 5.23(b): Damage Index: γ2c(1:4), Modes: 1:4, Damage Case: SDM5

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

γ2

c(1

:4)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

5.00E-03

1.00E-02

1.50E-02

γ2c(

1:4

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-120

Figure 5.23(c): Damage Index: γ2c(1:4), Modes: 1:4, Damage Case: SDQ1

Figure 5.23(d): Damage Index: γ2c(1:4), Modes: 1:4, Damage Case: SDQ5

Figure 5.23:Variation of γ2c(1:4) for damage cases SDM1, SDM5, SDQ1 and SDQ5

Table 5-4 and

Table 5-5 show that order of maximum γ2 values at damage location changes

significantly across four modes for all ten damage cases. Maximum γ2 values of the

third and fourth modes are much higher than the first and second modes and thus

γ2c(1:4) is predominantly governed by higher modes. Because of this, false alarms

associated with higher modes become significant in γ2c(1:4).

As Figure 5.24 illustrates, curvature values of higher order modes are much

higher than the low order modes. Because of this, contribution factors are differed

between modes. Averaging γ2 values does not apply any correction for differences in

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

γ2

c(1

:4)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

5.00E-03

1.00E-02

1.50E-02

2.00E-02

2.50E-02

3.00E-02 γ2

c(1

:4)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-121

contribution factors. This study proposes to normalize the curvature values as

explained below so that the damage localization results can be improved.

Table 5-4: Maximum γ2-Mid Span

Maximum γ2 at mid span

Mode 1 Mode 2 Mode 3 Mode 4

SDM1 3.98E-03 3.33E-03 3.27E-02 3.11E-02

SDM2 2.48E-02 2.19E-02 2.09E-01 2.02E-01

SDM3 4.47-02 6.59E-02 3.69E-01 4.39E-01

SDM4 5.38E-02 1.15E-01 4.55E-01 6.39E-01

SDM5 5.78E-02 2.01E-01 5.78E-01 9.36E-01

Table 5-5: Maximum γ4 at mid span

Maximum γ4 at mid span


SDM1 1.23E-01 1.24E-01 1.27E+00 1.19E+00

SDM2 7.78E-01 8.56E-01 8.50E+00 7.96E+00

SDM3 9.16E-01 1.97E+00 1.16E+01 1.76E+01

SDM4 8.37E-01 2.00E+00 1.24E+01 1.95E+01

SDM5 7.73E-01 2.32E+00 1.36E+01 2.18E+01

Page | 5-122

Equations 5-1 and 5-2 are used to normalize curvatures of mode shapes at

undamaged and damaged states , respectively. It should note that, a common

denominator (the absolute maximum curvature value of the intact structure,

AbsMaxh(j)) has been used in both equations. Figure 5.25 clearly indicates that

curvature values are in the same order across all modes. Difference in normalised

curvatures between undamaged and damaged states, γ2n, are then calculated on

individual mode basis as in Equation 5-3. Finally, average of γ2n across 'm' number

of modes are calculated to obtain the DI, γ2cn(1:m) using Equation 5-4.

Equation 5-1

Equation 5-2

Equation 5-3

Equation 5-4

Where;

NMSCh(i,j) = Normalized curvature value of ith

node for jth

mode at the

undamaged state,

AbsMaxh(j) = Absolute maximum curvature value of jth

mode,

MSCh(i,j) = Curvature of jth

mode at ith

node at undamaged state,

NMSCd(i,j) = Normalized curvature value of ith

node for jth

mode at the

damaged state, and

MSCd(i,j) = Curvature of jth

mode at ith

node at damaged state.

Damage localization results of γ2cn(1:4) for SDM1, SDM5, SDQ1, and SDQ5

are plotted in Figure 5.26 (a)-(d). Red colour dotted lines indicate mean values of

Page | 5-123

γ2cn(1:4) or the datum level used during the damage localization process. Regions

above the datum level are the signs for presence of damage. These figures (Figure

5.26 (a)-(d)) indicate significant improvement in damage localization results

compared to the γ2c(1:4) shown in Figure 5.23 (a)-(d).

Figure 5.24: Curvature of first four mode shapes without normalizing

Figure 5.25: Normalized curvature of first four mode shapes

-5.E+00

-4.E+00

-3.E+00

-2.E+00

-1.E+00

0.E+00

1.E+00

2.E+00

3.E+00

4.E+00

5.E+00

-3.44 -2.305 -1.17 -0.035 1.1

Cu

rva

ture

of

Mo

de

Sh

ap

e

Longitudinal Distance

MS-D: Mode10 Mode 1 Mode 2 Mode 3

-1.50E+00

-1.00E+00

-5.00E-01

0.00E+00

5.00E-01

1.00E+00

1.50E+00

-3.44 -2.305 -1.17 -0.035 1.1

No

rma

lize

d c

urv

atu

re

Longitudinal distance


Page | 5-124

Figure 5.26(a): Damage Index: γ2cstd(1:4), Modes: 1:4, Damage Case: SDM1

Figure 5.26(b): Damage Index: γ2cstd(1:4), Modes: 1:4, Damage Case: SDM5

Figure 5.26(c): Damage Index: γ2cstd(1:4), Modes: 1:4, Damage Case: SDQ1

0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

1.20E-02

γ2

cn(1

:4)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

γ2

cn(1

:4)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

5.00E-03

γ2

cn(1

:4)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-125

Figure 5.26(d): Damage Index: γ2cstd(1:4), Modes: 1:4, Damage Case: SDQ5

Figure 5.26: Variation of γ2cn(1:4) for damage cases SDM1, SDM5, SDQ1 and SDQ5

Remarks 5-6:

The DI, γ2cn, has improved the damage localization results because of the

proposed normalization process. Mean value of γ2cn is recommended as the

datum level so that the damage zone is localized by the region above this

level.

5.2.3. Evaluation of strain energy based damage indices

This section evaluates 11 different forms of SEDIs in total. Five of them are

based on individual modes whereas rests are based on combined modes. These 11

SEDIs are differed from each other by the method of derivation. Although, there are

minor changes in derivation method, damage localization results are significantly

differed between some of SEDIs. Noise is not added to mode shape values, so that

problems arisen from calculation method can be distinguished.

Section 5.2.3.1 examines five individual mode based SEDIs to rank them based

on damage detection abilities. Remaining six SEDIs that combines multiple modes,

are evaluated in section 5.2.3.2. Three SEDIs are eradicated from further evaluation

0.00E+00

2.00E-02

4.00E-02

6.00E-02

8.00E-02

1.00E-01

1.20E-01

γ2

cn(1

:4)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-126

due to the higher number of false alarms compared with other counterparts. Section

5.2.1.3.1 highlights problems associated with these three SEDIs, namely, β1, β5, and

β7. Among these three, former one (β1) is based on individual modes whereas latter

two are based on combined modes.

5.2.3.1. Ranking individual mode based SEDIs

This section presents evaluation results of β1, β2, β3, β8, and β10 with

selective examples to elaborate their merits and demerits. Results indicate that β1

creates more false alarms at nodal points of modes other than the fundamental

modes. Among remaining four SEDIs, β8 and β10 are marginally outperformed β2

and β3 based on the intensity of false alarms. Damage localization results of all five

SEDIs deteriorate when the order of the mode is increased.

Figure 5.27 plots the localization results obtained from β1 for SDM1 based on

the 3rd

mode with a datum level equals to 1. As shown in the figure, β1 has indicated

three peak locations; one at mid point and another two at third points of the beam.

The former location correctly points out to the centre of the damage zone. However,

the latter two locations which are circled by dotted red colour lines, are positive false

indications. Another illustrative example is shown in Figure 5.28 for the damage

case, SDQ1 using 2nd

mode. In this case, false alarms are observed at the centre of

the beam. Similar results are observed for other single damage cases. β1 is therefore,

not recommended for further analysis.

Page | 5-127

Figure 5.27:Damage localization results using β1 for the damage case SDM1 with third mode

Figure 5.28: Damage localization results using β1 for the damage case SDQ1 with second mode

Improved damage localization results of the other four SEDIs are illustrated in

Figure 5.29(a)-(d) for the damage case SDQ1 using second mode. However, all four

plots have minor false fluctuations around elements 16-19. The ratio between peak

false fluctuation to the highest DI value above the datum level of 1 is about 8% for

β2 and β3, whereas this ratio above the datum level of 0 is halved to 4% by β8 and

β10. β8 and β10 are therefore outperformed the other three SEDIs.

1.00E+00

1.01E+00

1.02E+00

1.03E+00

1.04E+00

β1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1.00E+00

1.01E+00

1.01E+00

1.02E+00

1.02E+00

β1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-128

Figure 5.29(a): Damage Index: β2, Mode: 2, Damage Case: SDQ1

Figure 5.29(b): Damage Index: β3, Mode: 2, Damage Case: SDQ1

Figure 5.29(c): Damage Index: β8, Mode: 2, Damage Case: SDQ1

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

β8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1.0000E+00

1.0002E+00

1.0004E+00

1.0006E+00

1.0008E+00

1.0010E+00

β3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1.0000E+00

1.0005E+00

1.0010E+00

1.0015E+00

1.0020E+00

β2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-129

Figure 5.29(d): Damage Index: β10, Mode: 2, Damage Case: SDQ1

Figure 5.29: Damage localization results using β2, β3, β8, and β10 for the damage case SDQ1

with second mode

However, the intensity of these false alarms becomes significant for fourth

mode or above. A typical example is illustrated in Figure 5.30, using results of β10

for the damage case SDQ1. Similar problem is observed with the other three SEDIs,

β2, β3, and β8.

Intensity of such positive false alarms reduces when the severity of damage

increases. But, another problem arises as negative false alarms are observed at higher

damage severity. A typical example is illustrated in Figure 5.31 using damage

localization results of β10 for the damage case SDQ5. The figure does not indicate

positive β10 values for elements 13-16 and leads for negative false alarms.

Figure 5.30: Damage localization results using β10 for the damage case SDQ1 with fourth mode

0.00E+00

2.00E-04

4.00E-04

6.00E-04

β1

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

β1

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-130

Figure 5.31: Damage localization results using β10 for the damage case SDQ5 with fourth mode

Remarks 5-7:

Among five SEDIs that are calculated on individual mode basis, β8 and β10

are produced slightly improved damage localization results over β2 and β3.

The remaining SEDI, β1, is not recommended for further investigation due to

higher number of false alarms.

When higher order modes are used on individual basis, SEDIs indicate

positive false alarms at low damage severity and negative false alarms at

higher damage severity.

5.2.3.2. Ranking combined mode based SEDIs

Among six SEDIs those derived by combining multiple modes, β5 and β7

indicate false alarms particularly at low damage severities. Damage localization

results of β5 and β7 for the damage case, SDQ1 are illustrated in Figure 5.32 and

Figure 5.33 , respectively. In both cases, the first four flexural modes are combined.

Regions those circled by red colour dotted lines in both figures highlight the positive

false alarms of β5 and β7 for this damage case.

Damage localization results of the remaining four SEDIs, β4, β6, β9, and β11,

for the above damage case, SDQ1, are illustrated in Figure 5.34 (a)-(d). All of these

0.00E+00

5.00E-03

1.00E-02

1.50E-02 β

10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-131

DIs have successfully localized the quarter span as the damage region with zero false

alarms. Similar level of accuracy is recorded for all of the ten single damage cases

confirming the robustness of β4, β6, β9, and β11.

Figure 5.32: Damage localization results using β5 for the damage case SDQ1 with first four

modes

Figure 5.33: Damage localization results using β7 for the damage case SDQ1 with first four

mode

Figure 5.34(a): Damage Index: β4, Modes: 1:4, Damage Case: SDQ1

1.00E+00

1.01E+00

1.02E+00

1.03E+00

β4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1.0000E+00

1.0002E+00

1.0004E+00

1.0006E+00

1.0008E+00

β7

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

4.00E+00

4.05E+00

4.10E+00

4.15E+00

β5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-132

Figure 5.34(b): Damage Index: β6, Modes: 1:4, Damage Case: SDQ1

Figure 8.34(c): Damage Index: β9, Modes: 1:4, Damage Case: SDQ1

Figure 5.34(d): Damage Index: β11, Modes: 1:4, Damage Case: SDQ1

Figure 5.34: Damage localization results using β4, β6, β9, and β11 for the damage case SDQ1

with first four mode

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

β1

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

β9

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

1.0000E+00

1.0010E+00

1.0020E+00

1.0030E+00

1.0040E+00 β

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-133

Remarks 5-8:

Among six SEDIs that combine multiple modes, β5 and β7 are not

recommended for further evaluation. Remaining four SEDIs, namely, β4, β6,

β9, and β11, have successfully identified the correct damage locations

without any false alarms. β4, β6, β9, and β11 are therefore recommended to

localize the flexural cracks.

If multiple modes are not measured, the best SEDIs are β8 and β10 as

highlighted in the section 5.2.3.1. β9 and β11 become identical to β8 and β10

when individual modes are used. Because of this, β9 and β11 can be

recommended as the best selections for the damage localization process.

5.3. Localizing multiple Damages

5.3.1. Evaluation of DIs on individual mode basis

Section 5.2 has recommended five individual mode base DIs, namely, α2, γ2,

γ4, β8, and β10, for further investigation. Evaluation of β8 is intentionally avoided as

significant difference is not recorded between β10 and β8 in terms of damage

localization abilities. This section, therefore, presents damage localization results of

remaining four DIs (α2, γ2, γ4, and β10) for multiple damage cases with varied

locations, severities and order of damage occurrence.

Figure 5.35 plots damage localization results of four DIs for the damage case

MD1 with respect to the first flexural mode. Among the four DIs, α2 does not have

the ability to define a clear criterion to separate two damage locations. For γ2 and γ4,

the mean values shown in red colour dotted lines in Figure 5.35 (b) and (c) are used

as the datum level for localizing damage zones. γ2 has a positive false alarm between

Page | 5-134

elements 14 and 17. On the other hand, γ4 has correctly identified centres of two

damage regions as indicated by the peak values near elements 8-9 and 24-25.

However, the regions above the mean datum of γ4 have been lean towards the edges

of the beam. Because of this, positive and negative false alarms that lean towards

supports and mid span of the beam respectively are observed at both damage

locations. Figure 5.35 shows that β10 has correctly localized the centres of both

damage zones. Although β10 has negative false alarms at edges of the damage zones,

the overall damage localization ability has been improved compared to the other

three counterpart techniques. Based on these observations, the four DIs are ranked as

β10, γ4, γ2, and α2 in the order of reduction in damage localization ability.

Figure 5.35(a): Damage Index: α2, Mode: 1, Damage Case: MD1

Figure 5.35(b): Damage Index: γ2, Mode: 1, Damage Case: MD1

0.00E+00

2.00E-02

4.00E-02

6.00E-02

γ2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

2.00E-02

3.00E-02

4.00E-02

5.00E-02

α2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-135

Figure 5.35(c): Damage Index: γ4, Mode: 1, Damage Case: MD1

Figure 5.35(d): Damage Index: β10, Mode: 1, Damage Case: MD1

Figure 5.35: Damage localization results using α2, γ2, γ4, and β10 for damage case MD1

When higher order modes are used on individual basis, false alarms are

observed with all four DIs. A typical example is illustrated in Figure 5.36 when the

fourth flexural mode is used during damage localization process of MD1 using β10.

Positive and negative false alarms have been circled by red colour and blue colour

dotted lines , respectively.

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

2.50E-03

β1

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

-1.00E-01

0.00E+00

1.00E-01

2.00E-01

3.00E-01

4.00E-01

γ4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-136

Figure 5.36 : Damage localization using β10 of fourth mode for the damage case, MD1

5.3.2. Evaluation of DIs on combined mode basis

This section evaluates four DIs, namely, α4, γ2cn, γ4cn, and β11, those

combine multiple modes. Result of β9 is not presented as it had similar damage

localization ability of β11. The damage localization results of α4, γ2cn, γ4cn, and

β11 for MD1 are shown in Figure 5.37 (a)-(d).

The modal flexibility based DI, α4, shows two local peaks near the centres of

two damage zones and thus has a slight improvement over α2 in Figure 5.37 (a).

However, α4 lacks from a criterion which can be programmable and therefore, limits

for manual diagnosis systems. Among the two mode shape derivative based DIs,

γ4cn in Figure 5.37 (c) indicates accurate localization result of both damage locations

than γ2cn in Figure 5.37 (b). This reconfirms the fact that higher order derivatives

based DIs are more accurate than the low order derivatives based DIs. For both γ2cn

and γ4cn, the mean value is used as the datum level. However, γ4cn values above the

datum level have been lean towards the edges of the beam at both damaged locations

and thus creates positive and negative false alarms.

Positive values of β11 in Figure 5.37 (d) are recorded at the centres of two

damage locations. Although there are some negative false alarms at edges of both

0.00E+00

1.00E-03

2.00E-03

3.00E-03 β

10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-137

cracked zones, β11 still has the ability to correctly localize centres or most severe

damage zones of both damaged locations. β9 also indicates a similar level of

accuracy.

Figure 5.37(a): Damage Index: α4, Modes: 1:4, Damage Case: MD1

Figure 5.37(b): Damage Index: γ2cn, Modes: 1:4, Damage Case: MD1

Figure 5.37(c): Damage Index: γ4cn, Modes: 1:4, Damage Case: MD1

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

2.50E-01

γ4

cn

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

γ2cn

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

α4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-138

Figure 5.37(d): Damage Index: β11, Modes: 1:4, Damage Case: MD1

Figure 5.37: Damage localization results using α4, γ2cn, γ4cn, and β11 for damage case MD1

Remarks 5-9 :

SEDIs have highest accuracy and robustness in localizing single and multiple

damages compared to the flexibility and mode shape based DIs. β8 or β10 is

recommended to use on the individual mode basis. However, if multiple

modes are measured, either β9 or β11 should be used to localize the damage.

Results presented in subsequent sections are limited to β11.

5.3.3. Localization of different crack patterns using β11

To check the robustness of β11, damage localization abilities of different crack

patterns should be examined by varying damage locations, severities and the order of

occurrence. Figure 5.37 (d) confirms that β11 can detect two medium-level

damages, which are apart from each other. Damage localization ability of β11 is

further investigated in this section under different crack patterns.

5.3.3.1. Overlapping cracks

The damage case, MD2, is an example for an overlapping crack. Elements 7-20

are identified as damaged elements with aid of the crack pattern shown in Table 5-2.

Crack patterns and DDSI values confirm that both damages have medium and

0.00E+00

5.00E-03

1.00E-02

1.50E-02

2.00E-02

β1

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-139

approximately equal severities. Figure 5.38 shows positive values for elements 7-18,

which indicates that those elements are at the damaged state. Although there are two

negative false alarms at elements 19 and 20, β11 has the ability to localize the severe

damage zone with sufficient accuracy. This confirms that β11 can localize

overlapping flexural cracks that are approximately equal in severity.

Figure 5.38 : Damage localization results for MD2 using β11

5.3.3.2. Two low severity damages

MD7 has two low severity damages at mid and quarter spans. Elements 7-17

are identified as damaged from ABAQUS crack pattern shown in Table 5-2. Positive

β11 values can be observed over the same elements in Figure 5.39. This result

confirms that β11 has the ability to localize multiple damages with low severities.

Figure 5.39: Damage localization results for MD7 using β11

0.000E+00

5.000E-09

1.000E-08

1.500E-08

2.000E-08

β1

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

1.00E-05

2.00E-05

3.00E-05

4.00E-05

β1

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 5-140

5.3.3.3. Order of sequential damage occurrence

When flexural cracks are forming at different stages, damage localization

results of β11 may sub ect to change. Possible sequential occurrences can be broadly

categorized into three groups for two damage cases as below.

1. A higher severity damage followed by a low severity damage at another

location (MD3 and MD 4)

2. Two damages with approximately equal severities at two locations (MD2

and MD5)

3. A low severity damage followed by a higher severity damage at another

location (MD6)

5.3.3.3.1. Detecting onset of a new crack

The first group represents initiation of a second crack at a different location of

a cracked RC beam. Damage cases, MD3 and MD4 in Table 5-2 can be categorized

into this group. MD3 represents a crack initiation at quarter span of a RC beam

which has a severe mid span damage. Figure 5.40 (a) and (b) illustrate the damage

localization results of β11 before and after the initiation of second crack at quarter

span. β11 only detects the most severe damage zone at mid span and fails to detect

crack initiation at quarter span as shown in Figure 5.40 (b). In both Figure 5.40 (a)

and (b), the undamaged state has been used as the fixed baseline. This illustrates that

the traditional approach of using a fixed baseline throughout the damage localization

process does not detect the crack initiation at another location at a secondary stage.

This problem can be overcome using the baseline update method presented in the

next section.

Page | 5-141

(a): β11 after first damage at mid span (b): β11 after first damage at quarter span

Figure 5.40 : Variation of β11 for damage case MD3 after each damage state (using undamaged

state as baseline)

5.3.3.3.2. Improved method to detect formation and/or propagation of cracks

The proposed method which has been illustrated in Figure 5.41, continuously

updates the baseline once formation or propagation of cracks is detected. At the first

instance, undamaged or initial state is employed as the baseline (BL1). At each of the

monitoring stages, β11 values are calculated for all elements to evaluate the

condition of the structure. Regions with positive β11 values are identified as

damaged zones. However, process may need several data sets or repetition in close

by time intervals to confirm the damage state in real application, mainly due to

presence of noise. Chapter 9 presents a probability based approach, which can be

used to confirm the damage state. Once the damage state is confirmed, the baseline is

updated to the current state of the structure (D1). This second baseline is labelled as

BL2 for clarity of explanation. In all other cases, no updating will be carried out and

the process will be repeated using BL1 until confirmed damage is detected. Positive

β11 values at the first damage state are recorded as β11D1 for future reference. BL2 is

used in the next stage which is aimed to monitor the formation of new cracks and

propagation of existing cracks. Once, formation or propagation of cracks is detected

using β11, the tertiary baseline, BL3, is introduced. Positive β11 values are recorded

0.0E+00

1.0E-04

2.0E-04

3.0E-04

4.0E-04

5.0E-04

1 4 7 10 13 16 19 22 25 28

β1

1

Element Numbers

0.00E+00

1.00E-04

2.00E-04

3.00E-04

4.00E-04

5.00E-04

1 4 7 10 13 16 19 22 25 28

β1

1

Element Numbers

Page | 5-142

as β11D2 which are then combined with β11D1 to derive β11D1:2. β11D1:2 is used to

localize all damage elements of the structure between the initial and current

monitoring stages.

Figure 5.41: Proposed method to detect onset of cracking using different baselines

Damage localization results for MD3 using updated baseline approach is

illustrated in Figure 5.42. The state of the structure after the mid span damage is

taken as the second baseline, BL2. This example confirms the improved results

obtained from β11D1. Next two sections (i.e. sections 5.3.3.3.3 and 5.3.3.3.4) further

Continue the process at each stage of monitoring

Step 5 : Introduce third baseline (BL3= Damaged State 2, D2)

Step 4 : Examine for formation of new cracks or propagation of new cracks

Step 3 : Introuduce second baseline (BL2 =Damaged State 1, D1)

Step 2 : Determine the state of the structure

Step 1 : First baseline (BL1= Undamaged /

Initial State

Calculate & plot β11 values

Identify positive β11

Undamaged : Go to Step 1

Damaged (D1)

Calculate & plot β11 values using

BL2


No propagation or formation of new

cracks:

Go to step 3

New cracks /cracks

propagated (D2)

Calculate β11 values using BL3


Repeat the process similar to Step 4

& 5 (with relevant updating)

Record positive β11 values at D2

(β11D2)

Calculate β11D1:2 (β11D1:2 = β11D1 +

β11D2 )

Record positive β11 values at D1

(β11D1)

Page | 5-143

examine the ability of β11D1 on localizing a second crack with different severities.

Use of β11D1:2 is illustrated in section 5.3.3.3.5.

Figure 5.42: Localizing the second damage at quarter span for MD3 using proposed baseline

update method

Damage localization result for MD4 is illustrated in Figure 5.43 (a)-(c). MD4

has the first damage at quarter span with a higher severity and second damage at mid

span with low severity in the order of occurrence. Figure 5.43 (b) illustrates that

traditional approach of fixed baseline approach using undamaged state could only

detect the quarter span and misses the onset of mid span cracks. Whereas, the

proposed method indicates a promising result as Figure 5.43 (c) correctly localizes

onset of second damage at mid span when the first cracked state is used as the

secondary baseline.

5.3.3.3.3. Localizing approximately equal size damages

Damage localization results for damage cases MD7 and MD5 are shown in

Figure 5.44 and Figure 5.45 , respectively. In MD7, two damages with low severities

are occurred first at mid span and then at quarter span. MD5 represents a case with

medium severe damages at quarter and mid spans in the order of occurrence. For

these two damage cases, traditional and modified approaches have indicated similar

level of accuracy in the damage localization process.

0

2E-11

4E-11

6E-11

8E-11

1E-10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

β1

1

Element Numbers

Page | 5-144

(a): β11 after the first crack

using undamaged state as

the baseline

(b): β11 after the second

crack using undamaged

state as the baseline

(c): β11 after the second

crack using secondary

baseline

Figure 5.43: Damage localization results of MD4 using proposed baseline update method


using undamaged state as the

baseline


crack using undamaged state

as the baseline



baseline


0.E+00

1.E-05

2.E-05

3.E-05

1

6

11

16

21

26

Element Numbers

0.0E+00

1.2E-05

2.4E-05

3.6E-05

1

6

11

16

21

26

Element Numbers

0.E+00

2.E-07

4.E-07

6.E-07

1

5

9

13

17

21

25

29

Element Numbers

0

1E-09

2E-09

3E-09

4E-09

5E-09

6E-09

7E-09

1

5

9

13

17

21

25

29

Element Number

0

5E-09

1E-08

1.5E-08

2E-08

1

6

11

16

21

26

Element Numbers

0

2E-09

4E-09

6E-09

8E-09

1E-08

1.2E-08 1

6

11

16

21

26

Element Numbers

Page | 5-145



the baseline


crack using undamaged

state as the baseline



baseline


5.3.3.3.4. Localizing a case where the second damage severity is higher than the

first damage

Figure 5.46 shows damage localization results for the damage case MD6 in

which the second crack at quarter span is higher than the initial mid span crack. Mid

span damage is identified by Figure 5.46 (a) which plots the variation of β11 after the

first damage but before the occurrence of second damage. Once the higher damage

occurs at quarter span, β11 has failed to localize the smaller damage at mid span

using either approaches of fixed or updated baselines as illustrated in Figure 5.46 (b)

and (c). This highlights the importance of continuous monitoring as first crack may

not detect if a larger crack appears at another location at a secondary stage. The

proposed continuous monitoring scheme that uses DI, β11D1:n, is elaborated in the

next section.

0.E+00

1.E-06

2.E-06

3.E-06

4.E-06

5.E-06

6.E-06

1

4

7

10

13

16

19

22

25

28

Element Numbers

0.0E+00

5.0E-06

1.0E-05

1.5E-05

2.0E-05

2.5E-05

3.0E-05

3.5E-05

1

5

9

13

17

21

25

29

Element Numbers

0.0E+00

5.0E-06

1.0E-05

1.5E-05

2.0E-05

2.5E-05

1

4

7

10

1

3

16

1

9

22

2

5

28

Element Numbers

Page | 5-146



the baseline


crack using undamaged state

as the baseline



baseline


5.3.3.3.5. Use of β11 for a damage case with continuously propagating cracks

This section presents damage localization results of β11D1:n for a beam with

cracks propagating at six different stages. A step wise loading pattern shown in

Figure 5.47 is used during FEM simulation. W1, W2, and W3 are the loads applied at

L/2, L/4 and 5L/6. Corresponding DDSI variation at each loading steps is plotted in

Figure 5.48.

The first crack is initiated at the mid span due to a concentrated load, W1 =

30kN (SMD1 in Table 5-1). This is followed by initiating the second crack at quarter

span in the next loading step at W2=42kN (MD7 in Table 5-2). The load, W2 is then

increased up to 54kN so that second crack is propagated more than the first one

(MD8 in Table 5-2). W1 is increased up to 36kN at the fourth stage, so that mid span

crack is further propagated (MD9 in Table 5-2). Fifth stage represents a case where

two cracks are propagating simultaneously (MD10 in Table 5-2). The sixth stage

demonstrates initiation of a new crack at a third location due to the concentrated load

W3=60kN (MD11 in Table 5-2).

0

1E-09

2E-09

3E-09

4E-09

5E-09

6E-09

7E-09

1

4

7

10

13

16

19

22

25

28

Element Number

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

2.5E-04

1

4

7

10

13

16

19

22

25

28

Element Numbers

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

2.5E-04

1

4

7

10

1

3

16

1

9

22

2

5

28

Element Numbers

Page | 5-147

Figure 5.47: Applied loads, W1, W2, and W3 vs. Stage number

Figure 5.48: Variation of DDSIs at each stage

Figure 5.49 shows the variation of β11Di (where; i=1-6) and β11D1:i for each

loading steps. β11Di shown in Figure 5.49(a1)-(a6) are used to identify the

propagation of existing cracks and/or formation of new cracks between two

successive evaluation states. β11D1:i in Figure 5.49(b1)-(b6) are used to identify all

damage locations that the structure has been subjected from the initial state. Figure

5.49(b3) and (b6) indicate two examples in which figures have been failed to

visualise damage locations at elements 14-19 and 21-27, respectively.

Table 5-6 lists all the elements with positive β11D1:i values for each damage

stage. As this table shows elements 14-19 and 21-27 are recorded with positive

values for damaged stage 3 and 6, respectively. Positive β11D1:i values are

recommended to use as the criterion for localizing damage elements as some damage

locations may not be clearly visible in graphical illustration.

0

20

40

60

80

100

0 1 2 3 4 5 6

DD

SI

(%)

Stage

DDSI(i,u) DDSI(i,i-1)

0

10

20

30

40

50

60

70

0 1 2 3 4 5 6

Lo

ad

(k

N)

Stage

W1 at L/2 W2 at L/4 W3 at 5L/6

Page | 5-148

Figure 5.49(a1): β11D1 after stage 1 loading -baseline :

undamaged state Figure 5.49(b1): β11D1:1 after stage 1 loading

Figure 5.49(a2): β11D2 after stage 2 loading -baseline : stage

1 Figure 5.49(b2): β11D1:2 after stage 2 loading



0

1E-09

2E-09

3E-09

4E-09

5E-09

6E-09

7E-09

1

4

7

10

13

16

19

22

25

28

Element Number

0

1E-09

2E-09

3E-09

4E-09

5E-09

6E-09

7E-09

1

4

7

10

13

16

19

22

25

28

Element Number

0

2E-09

4E-09

6E-09

8E-09

1E-08

1.2E-08

1

4

7

10

13

16

19

22

25

28

β1

1D

2

Element Numbers

0

2E-09

4E-09

6E-09

8E-09

1E-08

1.2E-08

1

4

7

10

13

16

19

22

25

28

β1

1D

1:2

Element Number

0.0E+00

1.0E-06

2.0E-06

3.0E-06

4.0E-06

5.0E-06

1

4

7

10

13

16

19

22

25

28

β1

1D

3

Element Numbers

0.E+00

5.E-07

1.E-06

2.E-06

2.E-06

3.E-06

3.E-06

4.E-06

4.E-06

5.E-06

1

4

7

10

13

16

19

22

25

28

β1

1D

1:3

Element Number

Page | 5-149







Figure 5.49: Damage localization results of β11 using proposed baseline update method

0.E+00

2.E-07

4.E-07

6.E-07

8.E-07

1.E-06

1

4

7

10

13

16

19

22

25

28

β1

1D

4

Element Numbers

0.E+00

1.E-06

2.E-06

3.E-06

4.E-06

5.E-06

1

4

7

10

13

16

19

22

25

28

β1

1D

1:4

Element Number

0.E+00

1.E-06

2.E-06

3.E-06

4.E-06

5.E-06

6.E-06

1

4

7

10

13

16

19

22

25

28

β1

1D

2

Element Number

0.E+00

1.E-06

2.E-06

3.E-06

4.E-06

5.E-06

6.E-06

7.E-06

1

4

7

10

13

16

19

22

25

28

β1

1D

1:5

Element Number

0

2E-09

4E-09

6E-09

8E-09

1E-08

1.2E-08

1.4E-08

1

4

7

10

13

16

19

22

25

28

β1

1D

3

Element Number

0.E+00

1.E-06

2.E-06

3.E-06

4.E-06

5.E-06

6.E-06

7.E-06

1

4

7

10

13

16

19

22

25

28

β1

1D

1:6

Element Number

Page | 5-150

Table 5-6:Elements with positive β11D1:i at ends of each loading steps

Stage 1 2 3 4 5 6

Damaged

Elements

12-19 7-19 6-19 6-19 6-19 6-19,

21-27

Remarks 5-10:

The traditional approach that uses fixed baseline, does not localize

propagation of existing cracks and formation of new cracks of a structure that

has been subjected to a severe damage in previous state.

The proposed method which continuously update the baseline, has the ability

to mitigate the above drawback and localize all damage locations. DI, β11Di is

recommended to determine the formation and/or propagation of cracks with

respect to the previous monitoring stage. The DI, β11D1:i, is recommended to

identify all damage locations that the structure has been subjected from its

initial state.

5.4. Baseline Damage Elements

The term, baseline damage elements, is defined as the damaged elements

detected at zero noise level. The baseline damaged elements are used to evaluate the

damage localization results at different noise levels. For single damage cases,

baseline damage elements are identified using β11. For multiple damage cases, the

damage index β11D1:i is used to identify the baseline damage elements. Table 5-7

lists all baseline elements for all single and first ten multiple damage cases presented

in Table 5-1 and Table 5-2 , respectively.

Page | 5-151

Table 5-7: Damaged elements identified by β11 at the baseline state

Single Damage Cases Multiple Damage Cases

Damage case Elements with positive

β11

Damage case Elements with positive β11D1:i

SDM1 12-19 MD1 6-12, 19-27

SDM2 12-19 MD2 6-19

SDM3 12-19 MD3 5-9,12-20

SDM4 12-19 MD4 6-20,22-23

SDM5 12-20 MD5 6-20

SDQ1 6-13 MD6 4-19

SDQ2 6-12 MD7 7-19

SDQ3 6-14 MD8 6-19

SDQ4 6-14 MD9 6-19

SDQ5 6-15 MD10 6-19, 21-27


This chapter presents the comprehensive evaluation study that was carried out

to evaluate 25 DIs based on the damage localization ability of flexural cracks in RC

structures. Effect of noise was not examined as the aim was to identify problems

associated with each of the DI itself. Results indicated that the SEDIs have better

Page | 5-152

damage localization results than the mode shape and flexibility based DIs. Among 11

different forms of SEDI, the four SEDIs (β8-β11) proposed in this study, indicated

the highest robustness. Among all, β11 was recommended for future studies.

Page | 6-153

6. Improved Damage Detection Process

The fourth objective of this research was aimed at developing a robust damage

detection process based on the DI method. From the authors' speculation, a robust

damage detection process should not only incorporate a robust DI, but also be

capable of addressing practical issues that may be encountered in the DI calculation

process. In addition, a robust damage detection process in the modern context should

have the ability to automate the damage detection process. Some of the practical

issues associated with measurements are; presence of noise, reduced spatial

resolution, phase change, and mode shifting. A robust damage detection process

should, therefore, be capable of addressing the above problems. The present study

has proposed a multistep damage detection process (MSDDP) with the ability to

address the aforementioned practical issues. Special interest was given to incorporate

features that can be programmable. The outcome was presented as a MATLAB based

program, including a Graphical User Interface (GUI).

This chapter presents details of the proposed MSDDP, which improves the

damage detection ability and provides the base for an automated damage detection

scheme. It follows six steps in a sequential order, during the DI calculation process.

The MSDDP was initially developed to address the forward problem which was

extensively needed for the present study. This lead to introduce additional features

such as steps for introducing artificial noises and changing the spatial resolution.

However, the thought process was always aligned with the features required to

address the inverse problem. So that all the additional features required for the

forward problem were included as optional features in the developed MATLAB

based program.

Page | 6-154

The content of this chapter has been organised as below.

The section 6.1 defines some general terms used in this chapter so that the

reader can easily grasp the information. The section 6.2 provides detailed

descriptions and information on the proposed sequential order of the MSDDP

including the features required to address the forward problem. The next section

highlights the steps those needs to address the inverse problem of the damage

detection process.

6.1. Definitions

6.1.1. Baseline (Undamaged) & Evaluation (Damaged) states

The damage is always a relative measure between two states of the structure,

the baseline state and the evaluation state. The objective of any damage detection

scheme aims on detecting damage at the evaluation state relative to the baseline state.

The undamaged level is normally selected as the baseline, so that all the damage

properties can be identified. However, the undamaged state is hard to be defined for

old structures. In such situations, either the current state or the best possible

estimated healthy level is used as the baseline state.

In subsequent sections, the baseline state is called as the undamaged state, whereas

the evaluation state is called as the damaged state.

6.1.2. Line of sampling/measurements

The scope of the present study was limited to one directional measurements. So

that it was assumed that all the sampling points are aligned along a straight line on

the structure. This can be taken as a situation where the sensors are being placed

Page | 6-155

along a straight line in case of real measurements. In FE based analysis this can be

easily achievable by obtaining the output values along a straight line.

The axis parallel to the line of sampling was defined as the X-axis.

6.1.3. Sampling point / node

The sampling point of a real structure is a sensor location at where the modal

responses are being measured. In FEM based studies, the sampling point is an output

request point. The sampling point is also called as a node.

6.1.4. Spatial resolution

The number of sampling points per unit length is defined as the spatial

resolution. (Sazonov and Klinkhachorn 2005)

6.1.5. Sampling Interval

The sampling interval means the distance between two measurement points. If

measurements are taken with a uniform spacing, the sampling interval is equal to the

reciprocal of the spatial resolution.

6.2. Sequential Order to Address The Forward Problem

Research and development process of any VBDITs starts with addressing the

forward problem of the damage detection process at the first instance. This stage

involves an extensive number of numerical or FE based simulations. Numerical or

FE based simulations do not generate practical issues such as measurement noise.

Higher number of sampling points (higher sampling resolutions) are intuitively being

obtained during the numerical/FE based studies. However, real measurements

produce contradictory results due to inherent measurement noise and a limited

Page | 6-156

number of sampling points. The MSDDP process, therefore, has started with

introducing two steps; 1. Step one to change the spatial resolution, and 2. Step two to

introduce artificial noise to the measurements. These two steps are predominantly

designed to address the forward problem of the damage detection process.

Uneven sampling interval is another practical issue that may be encountered

with the measurements. In real structures, the uneven sampling intervals may be

generated due to intervals between sensors. In FE based simulations, this is caused

by uneven mesh sizes. Evenly distributed sampling points simplify the DI calculation

process and facilitates the programming codes. This highlights the necessity of a step

to generate sampling points on a uniformly distributed grid.

The other two practical issues as highlighted in the first paragraph of this

chapter, mode shifting, and phase change, are addressed next. Both these issues are

commonly encountered in the real measurements as well as in the numerical/FE

simulations. DI calculation will be performed at the end of the process.

In summary, there are six main steps in the proposed MSDDP which are

aligned in a sequential order as presented in Figure 6.1. Some intermediate

calculation steps such as mode shape normalization, MSV calculation, etc. are also

incorporated into some of the main steps. Detailed discussion on each of these six

steps are presented from section 6.2.1 to section 6.2.6.

Page | 6-157

Figure 6.1: Sequential Order to address the forward problem

The notation *f* was used to highlight the optional steps those designed to

address the forward problem of the damage detection.

It should note that all the programming codes were developed within the MATLAB

program.

Note:

The first three steps of the MSDDP handles vibration properties at undamaged

and damaged states separately, but in the same manner. In other terms, same

• Inputs: Frequencies & Mode Shapes

• At undamaged and damaged states Inputs

• Change the spatial resolution

• To generate mode shape values at uniform spacing Step 1*f*

• Add Noise* (*Optional)

• Allows to examine the performance with presence of noise

Step 2*f*

• Curve fit and normalise* (*Optional) mode shapes

• Changes the spatial resolution to obtain evenly distibuted sampling points

Step 3

• Mode Shape Consistency Check

• MAC + change in frequencies, MSV calculation

Step 4

• Phase Check

• Correct any phase changes betwen two evaluation states

Step 5

• Selection of sensitivity of modes to damage &

• Damage Index Calculation Step 6

Page | 6-158

procedure and programming codes have been used during the first three steps. From

fourth steps onward, the MSDDP needs to combine or compare the vibration

properties at undamaged and damaged states. The first three steps of the MSDDP has

been explained without using superscripts 'u' and 'd' those used to denote undamaged

and damaged states. Superscripts 'u' and 'd' have been introduced during the

explanations made from the step 4 and onwards.

6.2.1. Step 1: Change the spatial resolution (*f*)

6.2.1.1. Background

Spatial resolution is one of the governing factor which controls the accuracy of

the damage detection process. Previous studies have indicated that higher spatial

resolution (over sampling) improves the accuracy of the damage detection results of

DIs that are based on change in mode shapes (Sazonov and Klinkhachorn 2005; Yoo

and Kim 2000). However, Sazonov and Klinkhachorn (2005) indicated that the

above hypothesis (i.e. higher the sampling resolution, higher the accuracy) is not

valid for DIs based on change in mode shape curvatures or strain energy values.

They have indicated that both under sampling and over sampling can cause problems

in the damage detection process due to presence of measurement noise. In large civil

structures, the number of mountable sensors (such as accelerometers and transducers)

are often limited due to higher cost. Under sampling is therefore, a more prominent

problem that may be encountered in the practical circumstances than the over

sampling. Truncation errors have a significant influence on the damage detection

results in case of under sampling. However, it is creditable to note that researchers

tend to use higher spatial resolutions in FE based studies or data-acquisition systems

utilizing Laser Doppler Vibrometers (LDVs) (Sazonov and Klinkhachorn 2005). In

Page | 6-159

such situations, over sampling needs to be addressed with attention given to the

measurement noise.

The first step of the sequential order was therefore designed to change the

spatial resolution of inputs. This facilitates to study the effect of sampling resolution

on damage detection results during the forward problem study in an efficient manner.

The section 6.2.1.2 describes the procedure of the first step of the MSDDP and

presents necessary MATLAB codes. The next two sections evaluate the application

of this method to provide recommendations as well as limitations.

6.2.1.2. Procedure to change the spatial resolution

The procedure shown in Figure 6.2 was designed to change the spatial

resolution at the first step of the MSDDP.

Figure 6.2: Three stage procedure to change the spatial resolution

6.2.1.2.1. Read & Save Inputs

The MSDDP starts with reading and converting the input values into a unique

format, (mode shapes, frequencies, and coordinates of the sampling points). X-

coordinates and mode shape values are saved in two matrices, [MS0u] and [MS0

d] ,

where superscripts u and d stand for undamaged and damaged states. As stated

earlier, superscripts are not shown in following explanations up to step 4. However,

it should be noted that two separate matrices are formed at the same time

corresponding to the undamaged and damaged states. For an example, [MS0] has

Read & Save Inputs

Generate spline interpolation curve

Change the spatial resolution

Page | 6-160

been used during the explanation made in this section instead of repeating the same

procedure for [MS0u] and [MS0

d].

During this step x-coordinates and mode shape values are saved in a same

matrix to reduce the number of variables. This matrix is denoted by [MS0] and

defined below.

N) (M+1) [

Where; X is a column matrix ([X](N)x(1)) containing x-coordinates of the

sampling points, and φ is the mode shape matrix ([φ](N)x(M)). N and M denotes the

number of sampling points and number of modes , respectively.

Frequency values are saved into another matrix, which is denoted by [F](1)x(M).

6.2.1.2.2. Generate spline interpolation curves

Next, the third order piecewise polynomial interpolation technique is used to

generate spline curves for each of the mode. Third order piecewise polynomial

interpolation function (cubic spline technique in MATLAB) is selected to avoid the

Runge's phenomenon (i.e. problem of oscillation at the edges of the interval when

using the higher order polynomial interpolation technique). The minimum number of

sample points required for the cubic spline technique is four whereas higher order

polynomial interpolation techniques requires more sample points. Therefore the

cubic spline technique can be used with low spatial resolutions compared to others.

The MATLAB function 'csapi' is therefore used in the MSDDP. The full MATLAB

command to generate the cubic spline curve, 'splineci', for mode 'i' is given below.

here; MS0(:,1) = All (row) entries in the first column of [MS0](N)x(M+1) (i.e.

the X-coordinates of sampling points), MS0(:,i+1) = All (row) entries in the (i+1)th

column (i.e. mode shape values of ith

mode).

Page | 6-161

6.2.1.2.3. Change the spatial resolution

Finally, the spatial resolution is changed based on the user preference. A

provision has been made in the developed program so that the user can change the

required number of elements at the beginning of the DI calculation process (This will

be discussed in the next chapter).

First, a new X-coordinate matrix, [X1](nmax)x(1), is formed using;

[ : int : x ]'

Where;

x0 = MS0(1,1) = X-coordinate of the first sampling point (i.e. the lowest X-

coordinate),

xe = MS0(N,1) = X-coordinate of the last sampling point (i.e. the highest X-

coordinate),

int = Distance between two adjacent sample points = L / (nmax -1),

L = Length between first and last sampling points = xe - x0,

nmax = total number of sampling points as defined by the user.

The MATLAB command 'fnval' is then used interpolate the cubic spline curve

'splineci' to obtain modal displacement values at all X-coordinates defined in [X1] for

the ith

mode. Full MATLAB command for this operation is given below. The same

code will be repeated for all the modes so that the mode shape values are saved into a

new matrix, [MS1](nmax)x(M).

MS (:,i) fnval( , 1)

Page | 6-162

6.2.1.3. Application for cases with no measurement noise

This section provides two examples to prove that the first step of the MSDDP

process can apply to either reduce or increase the spatial resolution for cases without

measurement noise.

The fifth mode shape obtained from the simulated RC beam presented in the

section 3.3.1 is used for both examples.

6.2.1.3.1. Example 1: Reducing the spatial resolution

Firstly, the mode shape values of the fifth flexural mode were obtained on a

finer grid with 55 sample points (the original sampling points). The mode shapes

obtained from the original sampling points are shown in Figure 6.3 with the series

name 'SP55'.

Then the mode shape values on 11 sample points were calculated using the

proposed procedure in the above section (reduced sampling points) and superposed

on the Figure 6.3 with the series name ' SPLINEC_SP55_11SP'.

As Figure 6.3 illustrates, the reduced sample points have been coincided with

the spline curve (shown in black colour line) generated using original 55 sample

points. This observation confirms that three stage procedures shown in Figure 6.2

can successfully apply to reduce the spatial resolution.

Page | 6-163

Figure 6.3: Reducing the spatial resolution

6.2.1.3.2. Example 2: Increasing the sample resolution

Next, performance of the above procedure to increase the spatial resolution was

examined and illustrated in Figure 6.4. In this example, 12 sample points were first

obtained from the FE simulation outputs. These 12 sample points are labelled as

SP12 in the Figure 6.4. Then the spline curve 'splinec5' was generated using the Code

(6-2). The spline curve, splinec5, was then used to obtain 51 sample points which are

shown with the series name SPLINEC_SP12_51SP. For the comparison purpose,

SP55 series also replotted in the same figure.

Figure 6.4 clearly depicts that the sample points of SPLINEC_SP12_51SP are

concurred with the SP55. This confirms that the proposed method can apply to

increase the spatial resolution as well.

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-4 -3 -2 -1 0 1 2

SP55 SPLINEC_SP55_11SP

Page | 6-164

Figure 6.4: Increasing the spatial resolution

6.2.1.4. Application to noisy data

Measurement noise always presents with real or experimentally obtained

modal displacements. The validity of the method should therefore be tested for noisy

mode shapes. Fifth flexural mode shape was therefore polluted with 5% random

noise. Figure 6.5 illustrates the polluted (SP55N) and non-polluted (SP55) mode

shapes of fifth flexural mode each consisting 55 node points.

Figure 6.5: Fifth flexural mode shape with and without noise

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5

SP55 SP12 SPLINEC_SP12_51SP

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5

SP55N SP55

Page | 6-165

6.2.1.4.1. Reducing the spatial resolution of noisy measurements

Cubic spline curve which passes through noisy sample points (SP55N) was

used to reduce the number of sample points to 11. Figure 6.6 depicts that these 11

points (SPLINEC_SP55N_11SP) have coincided with the spline curve, SP55N. This

confirms that the proposed method has the ability to reduce the spatial resolution

with noisy data.

Figure 6.6: Reduction in spatial resolution of noisy mode shape

6.2.1.4.2. Increasing the spatial resolution with noisy measurements

Twelve noisy sample points marked in red colour dots (SP12N) were used to

generate the spline curve, splinec5, which was then used to increase the number of

samples to 51 (SPLINEC_SP12N_51SP). As Figure 6.7 indicates,

SPLINEC_SP12N_51SP has not coincided with SP55N.

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5

SP55N SPLINEC_SP55N_11SP

Page | 6-166

Figure 6.7: Increasing spatial resolution of noisy mode shape

Remarks 6-1: Application and limitation for using cubic spline interpolation

function

The first step of the MSDDP can correctly apply to either reduce or increase

the spatial resolution for the mode shapes without any noise, but cannot be used to

increase the sample resolution with noisy measurements.

6.2.2. Step 2: Adding Noise

Mode shapes obtained from Numerical/FE simulations do not include

measurement noise. The second step of MSDDP was therefore designed to add

artificial noise to the mode shape values. This enables to study the performance of

DIs under different noise levels. Addition of randomly distributed noise has been

commonly used in previous studies (Alvandi and Cremona 2006). MATLAB

command 'rand' was therefore selected for the present study. The full command

given below was used to vary the noise level between (1-NL) and (1+NL), where,

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5

SP55N SP12N SPLINEC_SP12N_51SP

Page | 6-167

'NL' is the percentage of noise. Default value for the variable 'NL' was set to zero

with provisions to change this value in the GUI of the developed program.

2 NL rand(nmax,1) + (1 NL)

Polluted mode shape value, NMS(j,i) at jth

node of the ith

mode was obtained

from;

MS( ,i) MS1( ,i) noise( ,1)

[NMS](nmax)X(M) is the new mode shape matrix with 'nmax' number of equally

distributed sample points and 'M' number of modes.

6.2.3. Step 3: Curve fit and Normalise Mode Shapes

This step allows to change the number of nodes or elements at where the

damage indices are being calculated. New X-coordinate matrix, '[X2](Ne)x(1)' was first

defined as below, where 'Ne' is the new number of sample points.

[ : Int : x ]'; where,

x0

xe

Length between x0 and xe x0 xe

sample points.

Page | 6-168

Three non-smoothing, three smoothing and three sequential curve fitting

techniques are embedded into the software, so that the most appropriate curve fitting

technique can be selected. Sequential curve fitting techniques are used to minimise

the effect of noise on damage detection results. One of the above nine technique is

used to generate the interpolation curve, 'curvefi', of the ith

mode modal displacement

values of mode 'i', NMS(:, i) at all X-coordinate points, [X1](nmmax)x(1). The curve

'curvefi' is then used to obtain modal displacement values at new sample points

defined by [X2](Ne)x(1).

6.2.3.1. Non-smoothing curve fitting techniques

Details of three non-smoothing curve fitting techniques embedded into the

software are as below.

1. Sixth order piecewise polynomial interpolation function (sixth order spline

curve)

MATLAB command 'spaps' is used to generate sixth order spline curve

'curvefi', which is then interpolated using 'fnval' command at all X-coordinate

points defined in [XNe]Nex1. New modal displacement values at all sample

points for the mode 'i' are saved into the ith

column of [MS]NexM matrix (i.e.

MS(:,i)).

spapi(6, , MS1(:,i))

MS(:,i) fnval( , )

2. Third order piecewise polynomial interpolation function (cubic spline curve)

Similar MATLAB codes are used, except the function is changed to 'csapi'

instead 'spapi' when generating the curve 'curvefi' as below.

csapi(6, , MS1(:,i))

MS(:,i) fnval( , )

Page | 6-169

3. Fourier series based curve fitting technique

MATLAB command 'fit' is used to generate the curve 'curvefi' using Fourier

series based curve fitting technique. The order of the Fourier series is

determined by the number of sample points, 'Ne', generated at the first step.

curvef(m)

fit , MS1 :,i ,'fourier8') ; if Ne 17








MATLAB command 'feval' is then used to interpolate modal displacement

values at all new sample points.

MS(:,i) feval( , )

6.2.3.2. Smoothing curve fitting techniques

Three smoothing techniques supported in the software are;

1. Cubic smoothing spline

The MATLAB function 'csaps' is provided in the software with the

smoothing parameter equals to 0.99 as below.

csaps( )

MS(:,i) fnval( , )

2. Smoothing spline

The MATLAB function 'spaps' is provided in the software with a pre-set

tolerance value of 1.00E-4 as below.

spaps( )

MS(:,i) fnval( , )

Page | 6-170

3. Smooth response data

The moving average filter based smoothing technique is provided using

MATLAB function 'smooth'. The method for the 'smooth' function is selected

as 'rloess' with span value equals to 10%. This method cannot directly be used

to obtain modal displacement values at new coordinate points. Therefore, the

sixth order spline curve is used after the 'smooth' function. Smoothed modal

displacement values are first saved into a variable called, 'MSs'. MSs values

are then used to generate the sixth order spline curve, which is finally used to

obtain modal displacement values at new coordinate points, [X2].

MSs smooth( )

spapi(6, , MSs)

MS(:,i) fnval( , )

6.2.3.3. Sequential order of curve fitting technique

There are three sequential curve fitting techniques programmed in the software.

The first sequential curve fitting technique uses cubic smoothing spline curve

followed by the Fourier series based curve fitting technique. The cubic smoothing

spline curve first generates smoothed modal displacement values at all new sample

points. In the next step, Fourier series based curve fitting technique is used. The

order of the Fourier series should determine based on new number of sample points,

Ne, and as mentioned in the section 6.1.3.1. Full MATLAB commands for a typical

case when Ne > 17 is as below.

curvef1 csaps( )

MSm fnval(curvef1 )

fit( ) ; when Ne 17

Page | 6-171

MS(:,i) feval( , )

The second sequential curve fitting technique uses 'smooth' function followed

by 'spaps' function as below.

MSs smooth( )

spaps( )

MS(:,i) fnval( , )

The third sequential curve fitting technique adds an additional step to smooth

modal displacements further using the 'spaps' command. The full command for this

step therefore is as below.

curvef1 csaps( )

MSm1 fnval(curvef1, )

curvef2 fit( ) ; when Ne 17

MSm2 feval(curvef2, )

spaps( )

MS(:,i) fnval( , )

6.2.3.4. Normalising Mode Shapes

Mode shape normalization is another optional step which is programmed in the

software. Displacement based normalised mode shape of mode 'm' is obtained by

dividing the modal displacement values of mode 'm' by the absolute maximum value.

The MATLAB coding to obtain the absolute maximum value and displacement

based normalised mode shape for mode 'i', NMS(:,i), are as below.

absmax(i) max(abs(MS(:,i))

NMS(:,i) MS(:,i) / absmax(i)

Page | 6-172

6.2.4. Step 4: Mode Shape Consistency Check

Mode shape correlation is checked in the fourth step which identifies correlated

undamaged modes to the damaged modes. An intermediate step to the traditional

approach of correlation of modes using Modal Assurance Criteria (MAC) is

introduced so that the MAC can be successfully used in the developed software. The

complete procedure, including details of this intermediate step is presented in the

section 6.1.4.1. The section 6.1.4.2. proposes another approach which combines the

frequency ratio to the MAC based approach.

6.2.4.1. Procedure of the improved mode shape correlation method

Firstly, MAC values for all available modes are calculated and saved into the

matrix, [MAC]UmxDm; where, Um and Dm denote the number of modes obtained at

undamaged and damaged states , respectively. An additional step is introduced to the

MAC based procedure to distinguish flexural and torsion modes when measurements

are taken along a single direction (1-D) as in this study. In this additional step,

maximum value of each row and the corresponding column number are saved

respectively into the first row and the second row of another matrix, [MaxMAC]2 x

Um. In other terms, [MaxMAT]2 x Um saves highest MAC values and corresponding

undamaged and damaged mode numbers. For an example, if the first and second row

values of ith

column of [MaxMAC]2xUm matrix are 'MACij' and 'j', then the highest

MAC value for ith

undamaged and jth

damaged mode shapes is given by MACij.

Modes with highest MAC values are then evaluated based on the MAC limit. As

stated in chapter 2, higher the MAC value, higher the correlation of two modes. The

default MAC limit used in the software is set to 0.975 with provisions for changes.

Modes with higher MAC values than the MAC limit are taken as the correlated

Page | 6-173

modes. Mode shapes and frequency values of these correlated modes at undamaged

and damaged states are then saved into new four matrices which are listed below.

1. [MSH] matrix contains correlated mode shapes at undamaged state,

2. [MSD] matrix contains correlated mode shapes at damaged state,

3. [FH] matrix contains correlated frequencies at undamaged state, and

4. [FD] matrix contains correlated frequencies at damaged state.

Figure 6.8 shows the complete procedure in a skeleton format.

Figure 6.8: Skeleton diagram on obtaining correlated modes

Below example illustrates the importance of the checking the maximum MAC

value. Modal displacement values along the first beam (B1) of the T-Beam bridge

(Chapter 8, Section 8.1) are used in this illustration. 3D views of first five modes are

illustrated in Figures 6.9 (a) - (e). Figures 6.9 (f)-(j) illustrate the 1D views of mode

shapes plotted using modal displacement values along the first beam.

Page | 6-174

Figure 6.9 (a): 3D view of Mode 1 Figure 6.9 (f): 1D view of Mode 1

Figure 6.9 (b): 3D view of Mode 2 Figure 6.9 (g): 1D view of Mode 2

Figure 6.9 (c): 3D view of Mode 3 Figure 6.9 (h): 1D view of Mode 3

Figure 6.9 (d): 3D view of Mode 4 Figure 6.9 (i): 1D view of Mode 4

0

0.2

0.4

0.6

0.8

1

0 3 6 9 12 15

No

rm

ali

zed

Mo

da

l

Dis

pla

cem

en

t

Distance (m)

0

0.5

1

0 3 6 9 12 15

No

rm

ali

zed

Mo

da

l

Dis

pla

cem

en

t

Distance (m)

0

0.5

1

0 3 6 9 12 15

No

rm

ali

zed

Mo

da

l

Dis

pla

cem

en

t

Distance (m)

-0.4

-0.3

-0.2

-0.1

0

0 3 6 9 12 15

No

rm

ali

zed

Mo

da

l

Dis

pla

cem

en

t

Distance (m)

Page | 6-175

Figure 6.9 (e): 3D view of Mode 5 Figure 6.9 (j): 1D view of Mode 5

Figure 6.9: 3D and 1D views of first five modes of T-Beam Bridge at undamaged state

[MAC]5x5 matrix for above mode shapes is as below. To avoid the repetition

of values and to enhance the clarity, only the upper triangular matrix of [MAC]5x5 is

shown.

MAC 5x5

0 999995 0 998117 0 992576 0.931456 0 994613

0 999999 0 997878 0.933055 0 997935

1 000000 0.941344 0 999012

1 000000 0.952004

0 999996

Bold values in [MAC]5x5 matrix are higher than the default MAC limit, 0.975.

This will lead to generate multiple correlated modes if MAC criterion alone is used

and hence will create problems in programming. [MaxMAC]2x5 matrix for the above

[MAC]5x5 matrix is as below. All entries in the first row of [MaxMAC]2x5 matrix are

higher than the MAC limit, 0.975. For this case, jth

damaged mode is correlated with

the ith

undamaged mode, where, i=column number (1-5) and j= ith column entry in

the second row of the [MaxMAC]2x5 matrix (i.e. j=MaxMAT(2,i)).

MaxMAC 2x5 0.999995 0.999999 1.000000 1.000000 0.999996

1 2 3 4 5

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 No

rm

ali

zed

Mo

da

Dis

pla

cem

en

t

Distance (m)

Page | 6-176

6.2.4.2. Combined MAC and Frequency ratio to check the correlation of modes

Ratio of the frequency values of damaged state to the undamaged state can be

combined with the MAC based correlation procedure presented above. The

frequency ratio of the jth

damaged mode to ith

undamaged mode, FR(i,j) is given by;

FR(i, )

Fd(1, )Fdh 1, ;if

Fd(1, )Fdh 1, ≤1

0 ;if Fd(1, )

Fdh 1, 1

FR is the matrix with Um and Dm number of rows and columns to save all

frequency ratios. The [FR]UmxDm matrix is formed parallel with forming the

[MAC]UmxDm matrix. A new matrix, [MACFR]UmxDm is then formed in which ith

row,

jth

column entry, MACFR(i,j), is given by, MAC(i,j)xFR(i,j). The [MaxMAC]2xUm

matrix is then replaced with a new matrix called [MaxMACFR]2xUm, which saves

maximum row entries and locations of [MaxMACFR]UmxDm. Finally, correlated

modes are identified using the MAC limit.

Matrices [MACFR]5x5 (upper triangular matrix) and [MaxMACFR]2x5 for the

above example are as below which demonstrates improvements achieved during the

correlation. Although this method produces promising results for all the case studies

examined in this study, the method may need further validation for different case

studies. Therefore, the software allows this method to be selected as an alternative to

the procedure presented in the section 6.1.4.1.

MACFR 5x5

0.988468 0 0 0 0

0.994292 0 0 0

0.995782 0 0

0.998692 0

0.997117

MaxMACFR 2x5 0.988468 0.994292 0.995782 0.998692 0.997117

1 2 3 4 5

Page | 6-177

Remarks 6-2: Application and limitation of the improved MAC based

correlation methods

Proposed first improvement introduces an intermediate step to produce a

logical arguments that can be programmed. As MAC based correlation method is

widely accepted, this method is selected as the default option for the software. The

method presented in the section 6.1.4.2. enhances the correlation results further.

However, author recommends to check this method before adopting to a new damage

detection scheme. Provisions are provided in the software to select this method as an

alternative to the method presented in the section 6.1.4.1.

6.2.5. Step 5: Phase Check

Mode shapes obtained at two stages may subject to 180° change which is

checked and corrected in the fifth step. The logical procedure to check the phase

change of correlated ith

modes between undamaged and damaged states is as follow.

First, the absolute maximum modal displacement value of the ith

undamaged mode is

obtained with the node number, Na. ith

modal displacement values at node Na are then

obtained at both undamaged (MShmax) and damaged (MSdmax) states as follow.

MShmax MSH(Na,i)

MSdmax MSD(Na,i)

MShmax is then multiplied by MSdmax to obtain the answer, MultVal, which is

then used to identify phase changes. If MultVal is negative, ith

damaged mode shape

has a 180° phase change compare to the ith

undamaged mode shape. Damaged mode

shapes with phase changes are then corrected by multiplying modal displacement

values by -1.

Page | 6-178

MultVal MShmax MSdmax

MSD(:,i) ( 1) MSD(:,i) ; if MultVal 0

MSD(:,i) ; if MultVal 0

6.2.6. Damage Index Calculation

Twenty damage indices are programmed to calculate in the sixth step in the

following order.

1. DIs based on changes in mode shapes

2. DIs based on changes in higher order derivatives of mode shapes

3. DIs based on change in flexibility values

4. DIs based on changes in strain energy values

6.3. Sequential order to address the inverse problem

Inverse problem study also follows the above sequential order except the step

2. During inverse problem study, no artificial noise is added, so that step 2 in Figure

6.1 is skipped.


This chapter presents the improved damage detection scheme proposed in this

research. The proposed six sequential steps have been elaborated with the developed

MATLAB codes. These MATLAB coding are incorporated to the developed

MATLAB based software presented in the next chapter.

Page | 7-179

7. MATLAB based Software

This chapter presents the MATLAB based programme developed to achieve

one objective of this research. The MATLAB based programme utilizes two

Graphical User Interfaces (GUIs) to interact with the user and provides many user

friendly features and step by step guidance to calculate 25 damage indices. The

programme capable to analyse both the forward and inverse problem of the damage

detection process and provides many user preference options as presented in

following subsections. Some of the important features that have been inbuilt of the

programme are; ability to change the number of data points, modes, noise levels, and

curve fitting technique. These features help to address the forward problem of the

damage detection process in an efficient manner.

To enhance the user friendly features, the programme consists of several

checks to identify erroneous inputs such as numeric checks, file availability etc... .

Also some of the user interacting tools are deactivated automatically as necessary.

7.1. Graphical User Interfaces

The programme uses two MATLAB based graphical user interfaces shown in

Figure 7.1 and Figure 7.2. Figure 7.1 illustrates the main GUI which is the main part

of the programme, which includes all the necessary MATLAB codes to calculate the

25 damage indices. The GUI shown in Figure 7.2 is used to visualise variations of

DIs which will be a valuable tool for the inverse problem study.

Page | 7-180

Figure 7.1: View of the Main GUI

Figure 7.2: View of the Secondary GUI (Visualizing results)

The main GUI provides three main functions; 1. Select user inputs, 2. Select

some of the vital options during damage index calculation process, and 3. Save

output files. To provide sufficient capabilities to achieve these three main features,

the GUI consists of push buttons, edit boxes, check boxes, radio buttons, list boxes,

Page | 7-181

drop down boxes and plots as necessary. Static text boxes are used to indicate

necessary texts to guide the user whereas panel boxes are used to group some user

interacting tools to enhance the efficiency. However, the programme is lack from a

help menu. But, this chapter provides all the necessary information to interact with

the programme. All MATLAB '.m' files are embedded with extensive number of

comments which will allow for further possible enhancements proposed in this

chapter.

Section 7.2 provides step by step guidance to calculate damage indices using

ABAQUS output files.

7.2. Step by Step Guidance

This section provides the guidance to calculate programmed 25 damage indices

using ABAQUS output and input files. In this context, ABAQUS output files means

the report files (.rpt) which contains mode shape data extracted from a frequency

extraction step. First part of a typical ABAQUS output file is shown in Figure 7.3

and a first part of a typical ABAQUS input file is shown in Figure 7.4.

Figure 7.3 and 7.4 shows with line numbers to clarify some of the discussions made

in section 7.2.1.

Page | 7-182

Figure 7.3: First part of a typical ABAQUS output file (output_file_abaqus.rpt)

Figure 7.4: Fist part of a typical ABAQUS Input File (input_file_abaqus.inp)

7.2.1. Basic Requirements

To initiate damage index calculation, it is required to have two input and two

output files ready for the same structure corresponding to undamaged/baseline state

Page | 7-183

and damaged/evaluation state. Default input and output file names for undamaged

state are "inp_undamaged.inp" and "output_undamaged.rpt" whereas for that of

damaged state are "inp_damaged.inp" and "output_damaged.rpt".

The input file and output file need to meet some of the requirements to avoid

termination of programme as listed below.

1. The current programme is designed for a input file which contains 17 header

lines before the nodal co-ordinate section. User must therefore ensure that this

requirement is satisfied (deletion or insertion of some header lines may necessary),

unless the changes made within the MATLAB coding.

2. The output file (.rpt) should contain 8 header lines before the line

corresponding to the mode number. The same line (9th

line) should contain the

corresponding frequency value. This will be followed by another 11 lines to have

total of 20 lines before node labels and mode shape values are appeared. When

ABAQUS report file is generating, user should uncheck both column total and

maximum and minimum check boxes. Also, user must ensure that only one direction

at a time is recorded along a single longitudinal axis. Alternatively, user may wish to

change the MATLAB codes.

7.3. Enter/Browse Files

If files are in the parent directory of the MATLAB programme, enter input and

output file names for undamaged and damaged states in correct edit boxes. If files are

not in the parent directory of the MATLAB programme, enter file names with their

file paths. Alternatively, four Browse buttons may be used to select four files.

1. Enter input file name for undamaged state in the edit box with name

"inp_undamaged.inp".

Page | 7-184

2. Enter output file name for undamaged state in the edit box with name

"output_undamaged.rpt".

3. Enter input file name for damaged state in the edit box with name

"inp_damaged.inp".

4. Enter output file name for damaged state in the edit box with name

"output_damaged.rpt".

7.3.1. Read Files

Once all the above listed four files are selected, click "Read Files" push button

to start reading files.

Existence of entered file names will be checked before performing any further

calculation. However, user should ensure to enter correct files as this step does not

check the format of files. If wrong file names are entered, either wrong results will be

resulted or programme will be terminated with error messages returned to the

MATLAB workspace.

If correct files are selected, this operation will generate two matrices which

contain all the vibration properties with variable names 'undvibdata' and

'damavibdata' for undamaged and damaged states , respectively. Both matrices

follow the matrix format given below.

The first two columns of the first and second rows consist with zero values.

Node numbers are listed in the first row from third column onwards. Values in

similar column numbers in the second row includes the frequency values. The first

column from third row onwards save the node number, whereas second column from

third row save the X-co-ordinate value. The rest of the entries consist of mode shape

values for the relevant node and mode numbers.

Page | 7-185

The read file operation further identifies the minimum and maximum co-

ordinate values and will report these values to the panel named "Distance Regions".

The programme allows user to save 'undvibdata' and 'damvibdata' matrices as

text files. For this user may enter two file names in two edit boxes with default

names "Und_Structure_Name.txt" and "Dam_Structure_Name.txt". These files will

be saved in the parent directory of the programme unless, the user defines a new

location. Alternatively, user may use two push buttons names as "Save As" to

provide new file name and directory.

7.3.2. Basic Information

User needs to adjust the default basic information in the entry fields in the

panel named "Basic Information". The values to be entered for each field are as

follows.

X0 : The minimum X co-ordinate of the region to be analysed. This value is

automatically updated to the minimum X co-ordinate during the "Read File"

operation. However, provision is made to allow for further changes. A numerical

value must enter. Necessary error messages will generate if otherwise.

Xe : The maximum X co-ordinate of the region to be analysed. This value is

automatically updated to the maximum X co-ordinate during the "Read File"

operation. However, provision is made to allow for further changes. A numerical

value must enter. Necessary error messages will generate if otherwise.

nx : The number of elements examined during the damage index calculation

process. Each element will have the same size along the line of data acquired.

Maximum recommended value is 100. If more than 100 elements are needed, user

may update the Excel file, 'resultsolve.xlsx'. This file can be found in the parent

Page | 7-186

directory of the programme. Input should be a positive integer, otherwise, an error

message will generate.

nf : The number of degrees of freedom. Changes to this value is not allowed at

this stage, as the current version is limited to analyse structures along only one

direction at a time.

nxmax : The maximum number of elements to be used for the curve fitting

techniques. Default value is 100. This allows to simulate to study the performance of

damage indices with change in number of data acquisition points when addressing

the forward problem of the damage detection. Input should be a positive integer,

otherwise, an error message will generate.

nmmax: Maximum number of modes used for the damage index calculation.

Default maximum is 10. If more than 10 modes to be used during damage index

calculation process, user may update the Excel file, 'resultsolve.xlsx'. This file can be

found in the parent directory of the programme. However, visualization of mode

shapes is allowed only up to 10 modes. Input should be a positive integer, otherwise,

an error message will generate.

MAC : Modal Assurance Criterion ratio. Default is 0.975. Same ratio will be

used during the mode shape consistency check using both the MAC and frequency

ratio. Input should be a ratio between 0 and 1, otherwise, an error message will

generate. Also, warning message will generate if low MAC ratio is entered to warn

the user on possible inconsistent mode pairing.

7.3.3. Step 01: Add Random Noise

When addressing the forward problem, noise is an important factor to be

examined. To allow for this, the programme has provided the capability to add

Page | 7-187

random noise to the mode shape data. User can evaluate damage detection ability of

damage indices at different noise levels presence in either the undamaged, damaged

or both mode shapes.

The panel named "Step 01: Add Random Noise" allows user to enter random

noise level to be added for the mode shape data. The noise level should enter as the

noise to signal ratio. For an example 5% noise should enter as 0.05. Input should be a

value between 0 and 1, otherwise an error message will generate. If the noise to

signal ratio is greater than 0.10, a warning message will generate.

To add noise to the undamaged mode shape, tick "Undamaged Mode Shapes"

check box in the "Step 01: Add Random Noise" panel and enter the noise to signal

ration in the edit box "Und_Noise (5%=0.05)". Default value is 0 which will not add

any noise to the signal.

Similar procedure may be repeated to add noise to the damaged mode shapes

using "Damaged Mode Shapes" check box and "Dam_Noise (5%=0.05)" edit box.

To change the random noise pattern, user may use the "New Noise" button.

7.3.4. Step 02: Select Curve Fitting Technique

The programme allows user to select one of the curve fitting technique listed

below under two categories.

Category 01: Direct curve fitting techniques

There are three direct curve fitting techniques available in the programme as

listed below. The dropdown box for this is visible in default. For more information

about these curve fitting technique, refer MATLAB help (MATLAB® R2010b Help

Page | 7-188

Browser, 2010. MATLAB® R2010b [Software], The MathWorks, Inc. 3 Apple Hill

Drive Natick, MA 01760-2098 2010).

1. CF01_spapi : uses spline interpolation technique to obtain the mode shape

values at each node.

2. CF02_csapi: uses cubic spline interpolation technique to obtain the mode

shape values at each node.

3. CF03_fourier: uses Fourier series based curve fitting technique to obtain the

mode shape values at each node. The order is determined by the programme based on

the number of elements, nxmax.

Category 02: Sequential curve fitting techniques

Six other curve fitting techniques presented in Chapter 6 are programmed in

the current version. These six curve fitting techniques include, three smoothing

techniques and three sequential curve fitting techniques listed below. To select one of

below six curve fitting technique, user need to check the "Use Sequential Curve

Fitting" check box. This operation will deactivate selection of one of the above three

curve fitting techniques.

1. FF01_csaps: uses cubic smoothing spline technique to obtain the mode


2. FF02_spaps: uses 6th order smoothing spline technique to obtain the mode


3. FF03_smooth: uses smooth function to obtain the mode shape values at each

node.

Page | 7-189

4. FF04_FF01+fourier: This sequential curve fitting technique uses the cubic

smoothing spline technique followed by the Fourier series based technique. The

order of the Fourier series will automatically be determined by the programme. The

highest order, order 8 will be first attempted.

5. FF05_smooth+spaps: This sequential curve fitting technique uses smoothing

function followed by the smoothing spline function to obtain the mode shape values

at each node.

6. FF06_FF04+spaps: The fourth curve fitting technique (FF04_FF01+fourier)

will be followed by the smoothing spline technique. Among all the curve fitting

techniques, this research highlights the enhanced damage detection ability of this

sequential curve fitting techniques (Chapter 8).

For the future development, some of these curve fitting techniques may be

combined with filter techniques to minimise the influence of noise on damage index

calculation.

The programme allowed to use random function based noise removal

technique. An estimated noise to signal ration should be provided by the user in the

edit boxes either "Und_Filter (5%=0.05)" or "Dam_Filter (5%=0.05) or both.

However, this random distribution based noise removal technique may need several

trial and error attempts and will not have significant use for this operation. In future

development of the programme, these allocations may be replaced with values

required for filter function.

Page | 7-190

7.3.5. Step 03: Activate/Deactivate Normalization

User may check the "Displacement based Normalization" check box to activate

the displacement mode shape normalization. The current version of the programme,

does not allow for mass normalization.

7.3.6. Step 04: Check Consistency of Modes

One of the consistency check methods presented in chapter 6 can be selected

using the drop down box in this panel. Default consistency check method is based on

MAC. However, user may select the second method, MAC+FREQUENCY ratio, if

appropriate. In both cases, the MAC ratio provided under basic information will be

used to check the consistency of modes. During this operation, modes below the

MAC value will be discarded.

User also, have another option to activate the MSV based method to check the

consistency of modes. If this is selected, modes with MSV less than zero, will be

assigned with a MSV value of zero. Otherwise, negative MSV values may use when

calculating damage indices by combining modes.

7.3.7. Step 05:Select Modes

This panel allows user to activate MSV factor when combining modes. For

this, user need to tick the check box "Use MSV factor to Combine Modes".

Also, user may select all or few of the modes out of the first 10 modes to be

used during damage index calculation process. This section further allows to

visualise the mode shapes using "View" push buttons. The mode shape for both

undamaged and damaged states with their frequency values will be displayed in the

plot right to the panel "Step 05:Select Modes".

Page | 7-191

7.3.8. Check/Update

After following the above steps, "Check/Update" button will be used to

perform initial checks and to update all the information and to generate the curve

fitted data. This operation should be performed to activate the "Calculate" button

after each change. However, selection of modes can be performed even after this

operation. During this operation, the programme will interact with Excel. User

should not therefore open or close any Excel file, as the current version does not

account for the error message that may arise due to such operation.

7.3.9. Calculate

"Calculate" button should use to start damage index calculation process. All the

20 damage indices will be calculated. Selection of preferred damage indices are not

allowed at this stage. All 20 damage indices will be saved in the Excel file

"resultsolver.xlsx" in the parent directory of the programme. User should not

therefore open or close any Excel file, as the current version does not account for the

error message that may arise due to such operation.

Once all the damage indices are calculated, "Save Output File" button will be

activated to allow user to save the "resultsolver.xlsx" file in a different name.

7.3.10. Results

Pushbutton "Results" can be used to visualize previous result files at any time.

This will open the second GUI for this operation. This GUI allows to visualise six

plots simultaneously. Six damage indices therefore can be visualize for the

comparison or during damage identification purposes. Each plot consists with three

drop down boxes, among those two drop down boxes are used to select damage

Page | 7-192

index type, and the remaining drop down box to select the mode or mode

combination.

By default, the output file "resultsover.xlsx" will be used for plots. However,

provision is given to change the output file to a desired file using "Change Default

File01" button in the left panel. The left side panel also allows user to read

previously save three more output files and then to superpose those in above plots.

User need to check the corresponding check boxes in the order and "Browse" button

to browse the file. The "Update All Plots" button will then be used to update all plots

after reading all four or lesser number of output files.

The "Back" button will be used to return to the main GUI.


This chapter provides a quick guide to the developed MATLAB programme.

This can be used as a step by step guidance or help for the program for another user.

More information have been given in the MATLAB coding files. All the analysis

performed in this research was conducted with aid of this program.

Page | 8-193

8. Applications

The first section (section 8.1) of this chapter presents case studies to confirm

the applicability of developed MSEDI, β11, on large civil structures. Details of the

simulated 15m long RC T-beam bridge model is presented in the sub-section 8.1.1.

The sub-section 8.1.2 defines the DDSI value corresponding to the serviceability

limit state as defined in the Australian Bridge Design code, AS5100. Sub-section

8.1.3 presents damage localization results obtained from three case studies with

different crack patterns.

The next section of this chapter, section 8.2, evaluates the effect of nine curve

fitting techniques on damage localization results of β11 to recommend the best curve

fitting technique which has the highest ability to minimise the effect of noise. The

section 8.3 presents the application of the proposed probability based approach to

confirm the damage elements.

8.1. Damage localization of a RC T-beam bridge using β11

8.1.1. RC T-Beam Bridge Model

8.1.1.1. Geometric properties

Figure 8.1 and Figure 8.2 show the basic layout and the reinforcement

arrangement of the selected RC T-beam bridge model. Figure 8.3 illustrates the cross

section details with slightly modified dimensions as simulated in the present study

(Note: All dimensions shown in Figure 8.3 are in metre). The parapet walls, side

walkways, and asphalt layer were not simulated in the FE model. The length of the

bridge was taken as 15m instead of the 15.24m (50'). It was assumed that the girders

and slabs have been monolithically casted.

Page | 8-194

Figure 8.1: Basic layout of the selected RC T-beam bridge (Pike)

Figure 8.2: Reinforcement arrangement of the selected RC T-beam bridge (Pike)

Page | 8-195

Figure 8.3: Cross section of the simulated RC T-beam bridge model

The cross section of the simulated bridge model lies on the YZ plane in which

Y and Z axes are parallel to the width and depth directions as shown in Figure 8.3.

X-axis lies parallel to the longitudinal direction of the bridge. Figure 8.4 and Figure

8.5 illustrate the partitioned bridge model so that material properties and different

loading arrangements can be applied to the structure. Grid lines parallel to the X-axis

are denoted by A-G. Grid lines A, C, E, and G correspond with the mid points of

girders while others correspond with the mid points of slab spans. Grid lines 1-9 are

parallel to the Y-axis.

The bridge model was partitioned at points 0.1m away in either side of the grid

lines in X and Y directions except at 1st, 2

nd, 8

th, and 9

th grid lines at where the

partitioned were coincided with those four grid lines. Partitioned made along the Z-

direction were at Z=0.075m, 0.875m, 1.0m, 1.025m, and 1.125m.

Figure 8.4: Grid layout (Isometric View)

Page | 8-196

Figure 8.5: Grid layout (on XY plane - top of slab)

8.1.1.2. Material properties

Three material models assigned for this FE model are as follows.

1. STEEL - Similar to the steel properties defined in section 3.4.1.2.1.

2. CON_DAM - Similar to the concrete properties defined in section 3.4.1.2.1,

which includes inelastic and damaged properties of concrete.

3. CON_UND - Similar to the concrete properties defined in section 3.4.1.2.1,

but without inelastic and damaged properties.

8.1.1.3. Sectional properties

The simulated FE model were separated into five sections for material property

assignment. Sections with names Sect_Con_Und and Sect_Con_Dam were defined

to assign for the solid concrete areas. Sect_RF_Beam, Sect_RF_Bot, and

Sect_RF_Top were defined to assign reinforcement properties of the smeared layers

in the FE model.

0.325m

1.575m

2.825m

4.075m

5.325m

6.575m

7.825m

0.6

00m

2.5

00m

6.0

00m

7.5

00m

10.0

00m

12.0

00m

14.4

00m

15.0

00m

Page | 8-197

1. Sect_Con_Und: This was a solid homogeneous section and assigned to the

ends of the bridge model between grid lines 1 & 2 and 8 & 9 as shown in Figure 8.6

(highlighted by red colour). The material model assigned to this section property was

CON_UND so that the modelling errors caused by the stress concentration could be

eliminated. As a result, the damage near the support were not examined during the

case studies.

Figure 8.6:Region with the section Sect_Con_Und

2. Sect_Con_Dam: The material model, CON_DAM, was assigned to this solid

homogeneous section. This section property was assigned to the region between grid

lines 2 and 8 as shown in Figure 8.7.

Figure 8.7: Region with the section Sect_Con_Dam

3. Sect_RF_Beam: This surface section was assigned to the smeared

reinforcement layer shown in Figure 8.8 at the bottom of the girders. Reinforcement

area, spacing, and orientation angle that were assigned to this section are shown in

Figure 8.9.

Page | 8-198

It should note that the local 1-axis of the three smeared reinforcement layers

have been defined parallel to the global X-axis as shown in Figure 8.10. This sets the

orientation angles of the longitudinal and transverse reinforcement layers to 0° and

90° , respectively.

Figure 8.8: Smeared Layer with the section Sect_RF_Beam

Figure 8.9: Reinforcement properties assigned to the Sect_RF_Beam

Figure 8.10: Material orientation of the reinforcement layers

4. Sect_RF_Bot: This surface section was assigned to the smeared

reinforcement layer shown in Figure 8.11 at the bottom of the slab. Reinforcement

properties assigned to this section are shown in Figure 8.12.

Page | 8-199

Figure 8.11: Smeared Layer with the section Sect_RF_Bot

Figure 8.12: Reinforcement properties assigned to the Sect_RF_Bot

5. Sect_RF_Top: Properties of this section were similar to the Sect_Con_Bot

(Figure 8.12). This section was assigned to the top reinforcement layer of the slab as

shown in Figure 8.13.

Figure 8.13: Smeared Layer with the section Sect_RF_Top

8.1.1.4. Boundary conditions

Pinned and roller supports were assigned to the bottom of girders along the

grid line 1 and 9 , respectively. The self weight was applied as a sustained pressure

load (magnitude = 10kN/m2) over the top surface of the slab.

Page | 8-200

8.1.1.5. Mesh properties

C3D8 element type was assigned to two solid sections (Sect_Con_Dam and

Sect_Con_Und). Other three sections were assigned with surface element SFM3D4.

The model was meshed with approximate element sizes of 200mm x 200mm

along the X & Y axis. Smaller element sizes were used along the Z axis to facilitate

smooth crack propagation and to avoid early termination of the analysis caused by

stress singularities. The element depths were approximately 50mm in the girders and

25mm in the slab as shown in Figure 8.14.

Figure 8.14: View of the meshed bridge model

8.1.1.6. Data acquisition

Three types of data were acquired from the ABAQUS FE simulation. They are: 1. Z

or vertical component of static displacement (denoted by Uz), 2. Frequencies

(denoted by f) and 3. Z or vertical component of modal displacement values (denoted

by z). These data were first acquired at the undamaged state with self weight of the

bridge model. Pressure loads over a concentrated area of 200mm x 200mm were then

applied to simulate different crack patterns on the bridge model (locations of loads

Page | 8-201

are discussed in sections 8.1.2 and 8.1.3). These concentrated pressure loads were

unloaded before obtaining Uz, f, and z. Uz values were obtained at node points A3-

A7, C3-C7, E3-E7, G3-G7 as shown in Figure 8.15 (red colour dots). Vertical

components of the modal displacements values, z, were initially obtained at all the

node points along the four lines shown in Figure 8.16 (shown in red colour lines on

girders). However, the developed MATLAB based programme was used to reduce

the number of sampling points to 8 per each girder and 32 in total. This can therefore

be taken as an artificially simulated case with 32 sensors in total to measure the

response of the structure under free vibration.

DDSIsw at the serviceability limit state was calculated in the section 8.1.2. All of the

case studies shown in section 8.1.3 were designed to be within the serviceability limit

state.

Figure 8.15: Points where displacements were recorded

Figure 8.16: Lines where the modal displacements (mode shapes) were measured

Page | 8-202

8.1.1.7. Selection of flexural modes to DI calculation

The first 12 vibration modes are shown in Figure 8.17. From these 12 mode

shapes, first and second flexural modes were identified as mode 1 and mode 6. These

two modes were used during the DI calculation process.




Figure 8.17: First 12 mode shapes of the simulated T-beam bridge model

8.1.2. DDSIsw at serviceability limit state

The Australian Bridge Design Standard, AS5100.2 (Australian Standard

AS5100. 2-2004, Bridge design—Part 2: design loads 2004), defines the maximum

Page | 8-203

allowable deflection as l/600 of the span under live load plus the dynamic load

allowance. It recommends to use the live load of M1600 moving traffic load, without

the uniformly distributed load.

For this bridge model, the deflection at the serviceability limit state was

calculated as 25mm. The axle load of M1600 was selected as the live load applied on

the bridge as recommended in AS5100.2 (Australian Standard AS5100. 2-2004,

Bridge design—Part 2: design loads 2004). The number of lanes according to the

AS5100.2 (Australian Standard AS5100. 2-2004, Bridge design—Part 2: design

loads 2004) were calculated as 2, and located at the centre of the bridge model to

create the worst circumstance or the maximum deflection.

DDSIsw =72% was recorded at the damaged state shown in Figure 8.18 (by

applying a concentrated pressure load of 6x107 N/m

2 over 0.2 m x 0.2m area at the

grid point C5). The deflection at this damaged state with M1600 axle load was

recorded as 22mm. This value was less than the maximum allowable deflection of

25mm at the serviceable limit state.

The maximum stress in reinforcement at this damage state with ultimate limit

state design factors and load combinations (factored self weight and M1600) was

recorded as 126MPa and hence reinforcement bars have not yielded.

Based on this, it can conclude that this bridge structure will not exceed the

serviceable and ultimate limit states at or below DDSIsw of 72%.

Page | 8-204

Figure 8.18: Damage state to reach the maximum allowable deflection (at serviceability limit

state)

8.1.3. Case studies

As stated earlier, all case studies presented in this section were designed to be

within the serviceable limit state. The maximum DDSIsw was therefore limited to

72%. The MSESDI, β11, was calculated at equally distributed 30 elements using the

developed MATLAB based programme.

8.1.3.1. Case study 1

The first case study has a smaller damage severity at the mid span of the

girder-A as shown in Figure 8.19. This damage was simulated by applying a

concentrated pressure load of 1x107 N/m

2 over 200mm x 200mm area at the grid

point A5. The percentage increment of the maximum deflection was as low as 2.7%

which indicated that this damage has a negligible influence on the serviceable level

of the structure. The calculated DDSIsw value for this damage case was 7%.

Damage localization results obtained from the Figure 8.20 confirms that the

MSEDI, β11, has the ability to correctly localize this low severe damage case.

Page | 8-205

Figure 8.19: Flexural crack pattern of case study 1

Figure 8.20: Damage localization using β11 for the case study 1


The second case study has damage on the girder-A and girder-C with 33% and

16% severities respectively as shown in Figure 8.21. This damage pattern was

simulated by applying a pressure load of 2x107 N/m

2 over 200mm x 200mm area at

the grid point A5.

As Figure 8.22 illustrates, β11 had correctly localized mid span damage on

girder-A and girder-C. β11 values correspond to the girder-A are higher than the

girder-C. This indicate that the magnitude of β11 can be related with the damage

severity in future research studies.

0.00E+00

5.00E-08

1.00E-07

1.50E-07

2.00E-07

2.50E-07

3.00E-07

3.50E-07

4.00E-07

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

β1

1

Element Number

Girder A Girder C Girder E Girder G

Page | 8-206

Figure 8.21: Flexural crack patterns of case study 2



The next case study has damage on girder-C and girder-G. Concentrated

pressure loads were applied in a sequential order, first at the grid point C3 with

magnitude of 3x107 N/m

2 and then at grid point G7 with magnitude of 2x10

7 N/m

2.

From the crack pattern shown in Figure 8.23, three damage zones were identified,

namely, 1. girder-G near G7, 2. girder-E near mid span but leaned towards the right

end, and 3. girder-C near C3. Calculated DDSIsw values for girder-G, E, and C were

3.2%, 2.5%, and 2.3%.

0.00E+00

2.00E-07

4.00E-07

6.00E-07

8.00E-07

1.00E-06

1.20E-06

1.40E-06

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

β1

1

Element Number

Girder A Girder C Girder E Girder G

Page | 8-207

Similar conclusions can easily be made using the damage localization results of

β11 as shown in Figure 8.24. Positive β11 values can be observed on girder-C,

girder-E, and girder-G with comparatively higher values on girder G. Figure 8.25

clearly illustrates the identified damage zones on girder-C and girder-E.

Figure 8.23: Flexural crack patterns of case study 3


0.00E+00

5.00E-08

1.00E-07

1.50E-07

2.00E-07

2.50E-07

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

β1

1

Element Number

GIRDER A GIRDER C GIRDER E GIRDER G

Page | 8-208

Figure 8.25: Damage localization using β11 for the case study 3 (without girder-G)

Remarks 8-1:

Above results confirm that the SEDI, β11 has the ability to locate flexural cracks on

other types of reinforced concrete structures. Also, one may notice that the β11 value

have been increased with the damage severity. This implies that β11 has the potential

to quantify the damage. This can be addressed in future research studies, as the

present study was limited to damage detection and localization.

8.2. Ranking curve fitting techniques

8.2.1. Preliminary evaluation of nine curve fitting techniques

The ranking process is started with a preliminary evaluation which is aimed to

shortlist some of the curve fitting techniques for further analysis. The damage

severity and the noise level are selected to have the minimum negative influence on

damage localization results. It should note that negative influence of noise on

damage localization results degrade with the increase in damage severity and

reduction in noise level. In other terms, most favourable results can be expected at

higher damage severities and low noise levels. SDM5 with presence of 1% noise is

therefore used during the preliminary study. Based on results obtained at the

0.00E+00

1.00E-09

2.00E-09

3.00E-09

4.00E-09

5.00E-09

6.00E-09

7.00E-09

8.00E-09

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

β1

1

Element Number

GIRDER A GIRDER C GIRDER E

Page | 8-209

preliminary study, four curve fitting techniques, namely, CFT01, CFT02, CFT03 and

CFT06, are directly eliminated from further analysis.

Figure 8.26 (a)-(i) illustrate damage localization results for SDM5 at 1% noise for

each of the nine curve fitting techniques. Baseline damage elements at 0% noise

levels for each of the curve fitting techniques are calculated separately. These

baseline elements are shown in bold red colour numbers, so that false alarms can be

visualized clearly in each figure itself. Damage localization results of CFT01,

CFT02, CFT03 and CFT06 are unacceptable as Figure 8.26 (a), (b), (c), and (f) have

a considerable number of negative and positive false alarms with an irregular

distribution across the longitudinal direction of the beam. These four curve fitting

techniques are therefore eliminated from further analysis.

Table 8-1 lists element numbers with positive β11 values at 0% and 1% noise levels

for the remaining five curve fitting techniques, CFT04, CFT05, CFT07, CFT08, and

CFT09. The last column of Table 8-1 shows number of positive and negative false

alarms recorded at 1% noise relative to the 0% noise. The highest number of false

alarms, seven, is associated with CFT07, whereas the least number of false alarms,

zero, is recorded with CFT09. These five curve fitting techniques are further

evaluated using probabilities of correct condition detection and false alarms.

Page | 8-210

Figure 8.26(a): Damage Index: β11(1:4), Damage Case: SDM5, CFT01, Noise: 1%

Figure 8.26(b): Damage Index: β11(1:4), Damage Case: SDM5, CFT02, Noise: 1%

Figure 8.26 (c): Damage Index: β11(1:4), Damage Case: SDM5, CFT03, Noise: 1%

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

β1

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

β1

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03 β

11

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 8-211

Figure 8.26 (d): Damage Index: β11(1:4), Damage Case: SDM5, CFT04, Noise: 1%

Figure 8.26 (e): Damage Index: β11(1:4), Damage Case: SDM5, CFT05, Noise: 1%

Figure 8.26 (f): Damage Index: β11(1:4), Damage Case: SDM5, CFT06, Noise: 1%

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

β1

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

1.00E-04

2.00E-04

3.00E-04

4.00E-04

5.00E-04

6.00E-04

β1

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

5.00E-04

1.00E-03

1.50E-03

β1

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 8-212

Figure 8.26 (g): Damage Index: β11(1:4), Damage Case: SDM5, CFT07, Noise: 1%

Figure 8.26 (h): Damage Index: β11(1:4), Damage Case: SDM5, CFT08, Noise: 1%

Figure 8.26 (i): Damage Index: β11(1:4), Damage Case: SDM5, CFT09, Noise: 1%

Figure 8.26: Comparison of nine curve fitting techniques for the damage case SDM5 with 1%

noise based on SEDI β11

0.00E+00

1.00E-04

2.00E-04

3.00E-04

4.00E-04

5.00E-04

β1

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

2.00E-04

4.00E-04

6.00E-04

8.00E-04

1.00E-03

β1

1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

0.00E+00

2.00E-04

4.00E-04

6.00E-04

8.00E-04

1.00E-03

1.20E-03 β

11

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 8-213

Table 8-1: False alarms at 1% noise

Curve fitting

technique

Localized damage elements with

β11>0

Number of false alarms

Without Noise With 1% Noise Positive Negative

CFT04 12-19 13-16,19-21 2 3

CFT05 12-19 1,12-20,30 3 0

CFT07 12-19 1, 13-16, 19-21, 30 4 3

CFT08 10-20 8-21 3 0

CFT09 12-20 12-20 0 0

8.2.2. Evaluation of CFT04, CFT05, CFT07, CFT08, and CFT09 based on

probabilities of correct condition detection and false alarms

Ranking of CFT04, CFT05, CFT07, CFT08, and CFT09 is carried out using

probabilities of correct condition detection and false alarms at two noise levels, 1%,

and 2%. Probability of correct condition detection, PCCD, is the average of Pdd

(Probability of correctly determining a damage element) and Puu (Probability of

correctly determining an undamaged element). Average of Pdu (Probability of

negative false alarms) and Pud (Probability of positive false alarms) gives the

probability of false alarms, PFA. 10 randomly generated noise patterns are used at

each noise level and each curve fitting technique with aid of the MATLAB based

program.

Page | 8-214

Figure 8.27 (a) illustrates Puu, Pdd, and PCCD values at 1% noise for all five curve

fitting techniques. Probabilities of false alarms, Pud, Pdu, and PFA, are plotted in

Figure 8.27 (b). CFT05, CFT08, and CFT09 have detected all damage elements

without any negative false alarms (i.e. Pdu = 0%). Among these three curve fitting

techniques, CFT05 and CFT08 have approximately 15% chance of generating

positive false alarms. Highest probability of false alarms, 16%, is associated with

CFT04 and followed by CFT07 (Pud = 13%).

Figure 8.27 (a): Puu, Pdd, and PCCD Figure 8.27 (b): Pud, Pdu, and PFA

Figure 8.27: Comparison of five curve fitting techniques at 1% noise

At 2% noise, CFT05 has indicated unacceptable damage localization results as

indicated by PCCD, which equals to 48%. CFT05 cannot be recommended to use in

the damage index calculation process at moderate or higher noise levels. Remaining

four curve fitting techniques are ranked as CFT09, CFT08, CFT04, and CFT07 in the

descending order of accuracy. The highest accuracy level is indicated by CFT09 with

PCCD=98% followed by CFT08 with PCCD=85%. PCCD values at 3% noise for CFT09,

CFT08, CFT04, and CFT07 are recorded as 95%, 73%, 71%, and 69% , respectively.

Results of this section, therefore, confirm that CFT09 can minimize the influence of

noise on damage localization results.

0

20

40

60

80

100

Puu Pdd PCCD

Pro

ba

bil

ity

(%

)

CFT04

CFT05

CFT07

CFT08

CFT09 0

5

10

15

20

25

Pud

(%)

Pdu

(%)

PFA

(%)

Pro

ba

bil

ity

(%

) CFT04

CFT05

CFT07

CFT08

CFT09

Page | 8-215

Figure 8.28 (a): Puu, Pdd, and PCCD Figure 8.28 (b): Pud, Pdu, and PFA

Figure 8.28: Comparison of five curve fitting techniques at 2% noise

Remarks 8-2:

CFT09 indicates the highest accuracy level in the damage localization process with

presence of noise. CFT09 is therefore recommended to use during the damage index

calculation process.

8.3. Damage localization using GDLI(β11)

Presence of noise makes it difficult to confirm the damage state straightforward

using a single sample. The proposed approach which uses the GDLI to confirm the

damage state is illustrated in this section. Section 8.3.1 illustrates improved damage

localization ability associated with the GDLI using damage case SDM5 at 5% noise

level. The next section, section 8.3.2, examines the minimum number of samples

required to confirm the damage state for different damage cases and noise levels. A

qualitative assessment of GDLI is carried out so that recommendations can be made

on limitations and/or further enhancements.

8.3.1. Improved damage localization results achieved from GDLI

Figure 8.29 plots the variation of probabilities of positive β11 values, Pd(β11), of 30

elements for the damage case SDM5 using 10 samples at 5% random noise.

Elements 8-11 and 21-23 are positive false alarms compared to the baseline damage

0

20

40

60

80

100

Puu Pdd PCCD

Pro

ba

bil

ity

(%

) CFT04

CFT05

CFT07

CFT08

CFT09 0

10 20 30 40 50 60 70

Pud Pdu PFA

Pro

ba

bil

ity

(%

) CFT04

CFT05

CFT07

CFT08

CFT09

Page | 8-216

localization results of the SDM5. GDLI(β11) has improved the damage localization

results by reducing the number of false alarms to one (i.e. one negative false alarm at

element 12) as shown in Figure 8.30.

Figure 8.29: Damage localization results using non-standardized probability values for SDM5 at

5% noise

Figure 8.30: Damage localization results using GDLI(β11) for SDM5 at 5% noise

8.3.2. Minimum number of samples required to achieve different

accuracy levels

This section examines the number of samples required to achieve 60% or higher

accuracy levels (PCCD 60%) at three different noise levels, 1%, 3%, and 5%. The

notation NSAL is used to denote the minimum number of samples required to achieve

0.00

0.20

0.40

0.60

0.80

1.00

Pd(β

11

)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers

Page | 8-217

AL% accuracy level. The probability based damage index, GDLI(β11) is used

throughout this study.

8.3.2.1. NSAL for SDM1 at 1%, 3%, and 5% noise levels

Figure 8.31 illustrates the damage localization results obtained from GDLI(β11)

using 20 samples at 1% noise level. It indicates two positive false alarms at elements

20 and 21 and one negative false alarm at element 12. The corresponding PCCD value

is calculated as 89% (i.e. average of Puu = 91% and Pdd = 88%). Similar procedure is

adopted to obtain PCCD values for all number of samples.

PCCD vs. number of samples, NS, is plotted in Figure 8.32 which indicates the

minimum accuracy level of 70% at NS=2 (NS70=2). When the number of samples

becomes 3 or more (NS 3), the PCCD value turns out to be always greater than

77.5%. The minimum number of samples required to achieve 77.5% accuracy level,

NS77.5, is therefore taken as 3. Similarly, NS80, NS85, and NS87.5 are obtained as 4, 6,

and 8 , respectively. This implies that accuracy level can be improved by increasing

the number of samples.

Figure 8.31: Damage localization results of GDLI(β11) for SDM1 using 16 samples with 1%

noise

0

1

2

3

4

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

GD

LI(

β1

1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers (Red colour - Damaged elements)

Page | 8-218

Figure 8.32: PCCD vs NS for SDM1 with 1% noise

Accuracy of damage localization result is reduced at higher noise levels as expected.

The minimum accuracy level has been reduced to 52.5% with NS52.5=2 at 3% noise

as shown in Figure 8.33. Minimum PCCD value observed after 3rd

sample is recorded

as 65%. NS65, therefore, equals to 3. Similarly, NS67.5 and NS72.5 are taken as 11 and

16 , respectively.

As shown in Figure 9.9, maximum PCCD value at 5% noise has reduced to 35%. PDD

vs. NS is plotted in Figure 8.35 which indicates zero or low PDD values across all

samples. This implies that GDLI(β11) fails to localize damage elements with low

severities at higher noise levels.


60

70

80

90

100

0 2 4 6 8 10 12 14 16 18 20

PC

CD (

%)

Number of Samples (NS)

50

60

70

80

0 2 4 6 8 10 12 14 16 18 20

PC

CD

(%

)


Page | 8-219


Figure 8.35: PDD vs NS for SDM1 with 5% noise

8.3.2.2. NSP for SDM5 at 1%, 3%, and 5% noise levels

Figure 8.36 illustrates damage localization results for SDM5 at 5% noise using 20

samples. As shown in Figure 8.36, GDLI(β11) indicates negative false alarms at

elements 12 and 20. For all other samples, negative false alarms are observed at

either of elements 12, 20 or both. For all other elements, accurate condition detection

is observed for all samples. Because of this, the accuracy level is fluctuated between

89% and 94% as shown in Figure 8.37. The minimum accuracy level, PCCD=87.5,

does not change across all 20 samples. A single accuracy level, NS87.5 =1, is

therefore defined for SDM5 at 5% noise level. Similar observation is made at 3%

noise with NS87.5 =1. Higher PCCD value of 92.5% is observed for all samples at 1%

noise. Comparison of results in previous and this sections for SDM1 and SDM5

indicates that negative influence of noise degrades when the damage severity is

increased. Greater accuracy levels can therefore be obtained at higher damage

severities using lesser number of samples.

30

40

50

0 2 4 6 8 10 12 14 16 18 20

PC

CD (

%)

Number of samples (NS)

0

10

20

30

40

0 2 4 6 8 10 12 14 16 18 20

PD

D (

%)


Page | 8-220

Figure 8.36: Damage localization results of GDLI(β11) for SDM5 using 20 samples with 5%

noise

Figure 8.37: PCCD vs NS for SDM5 at 1%, 3%, and 5% noise

8.3.3. Maximum accuracy level (ALmax)

The maximum accuracy level (ALmax) is defined as the minimum PCCD value

recorded after half the number of sample points. The half the number of sample

points is selected as the limit so that the reliability of results can be guaranteed. The

first half of the sample points serves the purpose of stabilization of results. For this

case, half the number of samples is equal to 10. Cases with ALmax less than 60% are

not interested as they indicate more false alarms or failures in damage localization

process.

80

90

100

0 2 4 6 8 10 12 14 16 18 20

PC

CD (

%)


Noise = 1% Noise = 3% Noise = 5%

0.00

0.50

1.00

1.50

2.00

2.50

3.00

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

GD

LI(

β1

1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Element Numbers (Red colour - Damaged elements)

Page | 8-221

8.3.3.1. ALmax for SDM1

At 1% noise level, the minimum PCCD value after 10th sample point is recorded as

87.5% (rounded off to the nearest lower 2.5 value) at 16th sample (Figure 8.32).

Therefore, ALmax at 1% noise level for the damage case SDM1 is taken as 87.5%.

ALmax values at 3% and 5% become 67.5% and 35% as obtained from Figure 8.33

and Figure 8.34 , respectively. Damage localization results at 5% noise, cannot be

accepted as ALmax is less than 60%.

Similar procedure is used to obtain ALmax values for other damage cases at 1%, 3%,

and 5% noise levels as presented in the next section.

8.3.3.2. Variation of ALmax at different damage severities and noise levels

Figure 8.38 illustrates ALmax against damage severities of five mid span damage

cases, SDM1-SDM5, at 1%, 3%, and 5% noise levels. Accuracy of damage

localization results have been gradually improved with the damage severity. Highest

accuracy can be observed at the lowest noise level as expected. Similar observations

can be made for five quarter span damage cases, SDQ1-SDQ5, as shown in Figure

8.39. Detailed analysis of these two figures is carried to perform a qualitative

assessment of GDLI(β11) on damage localization ability at different noise levels.

Accuracy level, ALmax for all 10 single damage cases are higher than 75% at 1%

noise level. This implies that GDLI(β11) can be successfully used to localize low

damage severities with a data acquisition system that can provide mode shape values

with 1% or lesser noise.

Figure 8.38 illustrates ALmax against damage severities of five mid span damage

cases, SDM1-SDM5, at 1%, 3%, and 5% noise levels. Accuracy levels greater than

80% (ALmax > 80%) can be observed for all five damage cases at 1% noise. At 3%

Page | 8-222

noise, ALmax for SDM2-SDM5 are greater than 80%, while SDM1 indicates

moderate accuracy level of 67.5%. Damage localization results of SDM1 cannot be

accepted at 5% noise, as ALmax becomes less than 60%, but results have been

gradually increased to moderate accuracy levels (60% < ALmax < 80%) for SDM2

and SDM3, and to higher accuracy levels (ALmax > 80%) for SDM4 and SDM5.

Figure 8.39 illustrates ALmax values for the five quarter span damage cases, SDQ1-

SDQ5. ALmax value at 5% noise levels for SDQ1 have been recorded as less than

60% at 5% noise. Accuracy has been gradually increased to moderate and higher

accuracy levels at reduced noise levels of 3% and 1% , respectively. For SDM2 and

SDM3, moderate accuracy levels can be observed at 5% noise, while SDM4 and

SDM5 indicate further improvements in damage localization results. Probability of

correct condition detection for SDM2-SDM5 are always higher than 80% at 3% and

1% noise.

These two figures illustrate that, GDLI(β11) has accuracy of 60% or more on

localizing damage zones with DDSI greater than 60% even at higher noise levels of

5%. GDLI(β11) can produce moderate accuracy levels at low damage severities (i.e.

DDSI < 50%) at 3% noise level. Highest accuracy levels (ALmax > 80%) can be

achieved at 1% noise level for all damage severities. If the damage severity is greater

than 80%, effect of noise (up to 5%) is negligible and 80% or more accuracy levels

can be attainable. GDLI(β11) therefore, can localize higher damage severities (i.e.

DDSI > 80%) correctly. However, GDLI(β11) needs mode shapes with less than 3%

noise to localize damage cases with DDSI less than 60%.

Page | 8-223

Figure 8.38: ALmax for five mid span damage cases at 1%, 3%, and 5% noise levels

Figure 8.39: ALmax for five quarter span damage cases at 1%, 3%, and 5% noise levels

8.3.4. Confirming damage states of elements

When addressing the inverse problem, damage should be defined as a reliable index

using a probability based approach. This procedure has been explained in this

section. The damage case SDM1 at 3% noise was used as the illustrative example.

Figure 9.15 indicates the GDLI(β11) values recorded at 30 elements. Positive

GDLI(β11) values have been recorded at elements 9,10, 14-21, and 25. Figure 9.16

illustrates the cumulative probability of damage detection of each of the element as

recorded after the 20th

sample.

Then the GDLI(β11) values are normalized in such a manner that the highest

GDLI(β11) becomes unity. In other terms, all GDLI(β11) values are divided by the

maximum GDLI(β11). The normalized values of GDLI(β11) are then multiplied by

the cumulative probabilities of damage detection to obtain the reliability indices of

damage localization results of each of the element. Figure 9.17 illustrates the

60

80

100

30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00 AL

max (%

) Damage Severity :DDSI (%)

Noise=1% Noise=3% Noise=5%

60

80

100

30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00

AL

max (

%)

Damage Severity: DDSI (%)

Noise=1% Noise=3% Noise=5%

Page | 8-224

reliability index of 30 elements for the SDM1 at 3% noise. Table 8-2 lists the

reliability index of the localized damage elements. These indices can be used to

express the reliability of the damage localization results. Elements with higher

reliability indices indicate the presence of damage. Others are more associate with

false alarms, and hence can be neglected.

Figure 8.40: GDLI(β11) at 3% noise for SDM1

Figure 8.41: Cumulative probability values at 3% noise for SDM1

Figure 8.42: Reliability Indices at 3% noise for SDM1

0.00

1.00

2.00

3.00

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

GD

LI(

β1

1)

Element Number

0.00

0.20

0.40

0.60

0.80

1.00

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

Cu

mu

lati

ve p

rob

abili

ty

of

dam

age

ind

icat

ion

Element Number

0.000

0.200

0.400

0.600

0.800

1.000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Rel

iab

lity

In

dex

Element Number

Page | 8-225

Table 8-2: Reliability indices of localized damage elements

Element Number Reliability Index of Damage

Localization Results

9 0.005

10 0.013

14 0.058

15 0.256

16 0.326

17 0.591

18 0.493

19 0.471

20 0.800

21 0.220

25 0.022


This chapter was intended to verify the applicability of the proposed new techniques.

The section 8.1, illustrates that β11 can successfully applied to localize damage in

large civil structures. The next section has comprehensively evaluated the nine curve

fitting techniques that have been programmed in the MATLAB based program, to

Page | 8-226

give recommendations. The sequential curve fitting technique, CFT09 has indicated

the best accuracy compared to the other eight counterpart techniques.

The last section illustrates the improvements that have been achieved through the

proposed probability based damage localization process which utilise the GDLI.

Page | 9-227

9. Conclusion & Future Work

9.1. Conclusion

This research developed an improved Modal Strain Energy based Damage

Index (MSEDI) method to assess flexural cracks in Reinforced Concrete (RC)

structures. The study was performed using properly validated Finite Element (FE)

models of RC structures. The conceptual design was formed based on analysis results

obtained from simulated flexural crack patterns on a simply supported RC beam. The

outcome was presented as a MATLAB based program, which can address both

forward and inverse problems of the damage detection process. The programming

codes that have been developed in this thesis lay the platform for an automated

damage detection process. The designed damage detection process was named as the

Multi-Step Damage Detection Process (MSDDP) which uses a modified Modal

Strain Energy based Damage Index.

The main findings of this research are listed and discussed below

1. The damaged plasticity model studied and used in this research can correctly

simulate post-cracking behaviour of flexural RC structures. Problems associated

with two of the simplified and common damage simulation techniques, viz; reduction

in E-value and notch type damage, were highlighted. This study therefore provides

the correct guidance on simulating flexural cracks in RC structures within a FE

platform.

The damaged plasticity model was employed throughout the present study to

simulate the post-cracking behaviour of RC structures in an accurate manner.

2. A new damage severity index was proposed to define the extent of flexural

cracks in RC structures. Commonly used damage severity indices in the field of

Page | 9-228

VBDITs assume a uniform reduction in E-value, I-value or flexural rigidity and thus

cannot be used to define the extent of non-uniformly distributed flexural cracks. The

proposed damage severity index can be directly linked with the serviceable limits of

the structure.

3. The comprehensive evaluation performed in this study has recommended

that the MSEDI has higher potential of localizing flexural cracks than the mode

shapes and flexibility based DIs. An improved MSEDI was proposed to minimise the

number and intensities of false alarms. Detailed evaluation of MSEDIs recommends

the use of the proposed β9 to localize flexural cracks if a single mode has been

measured. In all other cases, where multiple modes have been measured, the

proposed MSEDI, β11 should be used.

4. This research has proposed a baseline update method to be applied during

the damage localization process. This enables to detect formation of new cracks and

propagation of existing cracks on a cracked structure. It is important to note that RC

structures always have cracks and hence difficult to obtain vibration properties at

intact state. The proposed baseline update method has the ability to avoid the

necessity of undamaged state during the damage localization process and hence

useful in damage localization in RC structures.

5. A Multi-Step Damage Detection Process (MSDDP) was developed to

address many of the practical issues associated with the DI calculation process. The

developed MSDDP process consists of a six steps to address the forward problem of

the damage detection to facilitates, changing the spatial resolution, adding random

noise, identifying consistent modes, correct phase changes, minimising influence of

noise in DI calculation process, and finally calculating the DIs. During inverse

problem study, first two steps can be avoided and. Required criterions were

Page | 9-229

developed with the aim to develop a MATLAB program so that the MSDDP can

leads for an automated damage detection scheme.

6. A sequential order of curve fitting technique was proposed to minimise the

effect of noise on curvature calculation. A detailed study was conducted to compare

and recommend this technique over eight other curve fitting techniques.

7. The research recommends using the MSV during the combination process of

multiple modes. The traditional approach to combine modes does not account for the

sensitivity of individual modes to the damage.

8. All the above features were incorporated into a MATLAB based program

which has a Graphical User Interface (GUI). Chapter 7 of this thesis was dedicated as

a Help/Manual for this program, so that future research can get advantage from this

research. All the 25 damage indices evaluated in this study are incorporated into this

program to allow future evaluations if necessary. In addition, this program calculates

MAC, COMAC, and MSV. Results can be saved into Excel file and/or visualise

within the program itself. The proposed improvements are incorporated as optional

features, so that it can easily switch to the traditional approach if necessary.

However, it should be noted that this thesis has presented enough evidence to

confirm the importance of all proposed improvements.

9. Finally, a Generalized Damage Localization Index (GDLI) was proposed to

improve the accuracy of the damage detection process using the proposed β11. This

approach combines the damage index β11 with the probabilities of damage detection

during the damage detection process. This is a statistical approach to qualitatively

assess the accuracy of the damage detection results. The approach can be applied to a

Page | 9-230

health monitoring scheme which continuously or periodically measure the vibration

properties to assess the condition of the structure.

9.2. Future Work

The scope of the present work was limited to damage localization of flexural

cracks in RC beams or similar type of structures. Because of this limitation,

following future studies are recommended.

1. Extend this research to damage quantification of RC structures.

The outcome of this research will provide the platform for this type of study.

This may require simulating large number of flexural cracks progressively, so that

the necessary algorithms can be designed for damage quantification. The damaged

plasticity model will be an aid for such studies to simulate load induced cracking.

2. Evaluate and extend the proposed method damage in decks of plate like

structures

The proposed study has achieved a great success in localizing damage along

the longitudinal direction of the structure. However, this technique may have some

drawbacks when attempting to localize damage in both longitudinal and transverse

direction, as the DIs used in the present study are calculated using one dimensional

measurements. Future research can therefore aim on using two dimensional

measurements and appropriate DIs to detect damage in plate like structures.

3. Extend the MATLAB based program for 2D measurements

The developed MATLAB program can handle only one dimensional

measurements. Future research can therefore extend this program to 2D

measurements.

Page | 9-231

List of Reference

Abaqus Analysis User Manual – Abaqus Version 6.8. 2008. Accessed 05/11/2010.

http://bee-pg-031941:2080/v6.8/books/usb/default.htm

Abaqus v. 6.8. 2008. Dassault Systèmes Simulia Corp. Software: Windows XP OS.

Abdel Wahab, M.M. and G. De Roeck. 1999. "Damage detection in bridges using

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

and Vibration 226 (2): 217-235.

Adewuyi, A.P., Z. Wu and N.H.M.K. Serker. 2009. "Assessment of vibration-based

damage identification methods using displacement and distributed strain

measurements." Structural Health Monitoring 8 (6): 443-461.

Aktan, A.E., F.N. Catbas, K.A. Grimmelsman and CJ Tsikos. 2000. "Issues in

infrastructure health monitoring for management." Journal of engineering

mechanics 126 (7): 711-724.

Allemang, R.J. 2003. "The modal assurance criterion–twenty years of use and

abuse." Journal of Sound and Vibration 37 (8): 14-23.

Allemang, R.J. and D.L. Brown. 1982. "A correlation coefficient for modal vector

analysis." Proceedings of the 1st International Modal Analysis Conference &

Exhibit, Union Coll, Schenectady, NY, USA, 1982, 110-116.

Alvandi, A. and C. Cremona. 2006. "Assessment of vibration-based damage

identification techniques." Journal of Sound and Vibration 292 (1): 179-202.

Bakhary, N. 2009. "Structural condition monitoring and damage identification with

artificial neural network." PhD diss., University of Western Australia.

Brownjohn, J.M.W. 2007. "Structural health monitoring of civil infrastructure."

Philosophical Transactions of the Royal Society A: Mathematical, Physical

and Engineering Sciences 365 (1851): 589-622.

Budget at a glance. 2009. Accessed 28/04/2012. http://www.budget.gov.au/2009-

10/content/at_a_glance/html/at_a_glance.htm.

Page | 9-232

Budget at a glance. 2011. Accessed 28/04/2012. http://www.budget.gov.au/2011-

12/content/at_a_glance/html/at_a_glance.htm.

Carrasco, C., R. Osegueda, C. Ferregut and M. Grygier. 1997 "Damage localization

in a space truss model using modal strain energy." Proceedings of 15th

International Modal Analysis Conference, Orlando, FL, USA, 1997, 1786-

1792.

Casas, J.R. and A.C. Aparicio. 1994. "Structural Damage Identification from

Dynamic Test Data." Journal of Structural Engineering 120: 2437.

Cawley, P. and R.D. Adams. 1979. "The location of defects in structures from

measurements of natural frequencies." The Journal of Strain Analysis for

Engineering Design 14 (2): 49.

Chan, T.H.T., K.Y. Wong, Z.X. Li and Y.Q. Ni. 2011. "Structural health monitoring

for long span bridges: Hong Kong experience & continuing onto Australia."

In Strctural Helath Monitoring in Australia, edited by Tommy .H.T. Chan

and David P. Thambiratnam, 1-32. Nova Publishers, United States of

America.

Chance, J., G.R. Tomlinson and K. Worden. 1994 "Simplified approach to the

numerical and experimental modelling of the dynamics of a cracked beam."

Proceedings of 12th

International Modal Analysis Conference, Honolulu,

Hawaii, USA, 1994, 778-785.

Chang, S. and S. Kim. 1996. "On-line structural monitoring of a cable-stayed

bridge." Proceedings of SPIE 2719, Smart Structures and Materials 1996:

Smart Systems for Bridges, Structures, and Highways, San Diego, CA,

February 1996, 150-158.

Chinchalkar, S. 2001. "Determination of crack location in beams using natural

frequencies." Journal of Sound and Vibration 247 (3): 417-429.

Cornwell, P., S.W. Doebling and C.R. Farrar. 1999. "Application of the strain energy

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

Vibration 224 (2): 359-374.

Cornwell, P., M. Kam, B. Carlson, B. Hoerst, S. Doebling and C. Farrar. 1998.

"Comparative study of vibration-based damage ID algorithms." Proceedings

of 16th

International Modal Analysis Conference (IMAC-XVI), Santa

Barbara, CA, 1998, 1710-1716.

Page | 9-233

Curadelli, R.O., J.D. Riera, D. Ambrosini and M.G. Amani. 2008. "Damage

detection by means of structural damping identification." Engineering

structures 30 (12): 3497-3504.

Dimarogonas, A.D. 1996. "Vibration of cracked structures: a state of the art review."

Engineering Fracture Mechanics 55 (5): 831-857.

Doebling, S.W., C.R. Farrar and M.B. Prime. 1998. "A summary review of vibration-

based damage identification methods." Shock and Vibration Digest 30 (2):

91-105.

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

identification and health monitoring of structural and mechanical systems

from changes in their vibration characteristics: a literature review." Los

Alamos National Laboratory Report LA-13070-MS, May 1996, United

States.

Doherty, JE. 1988. "Nondestructive Evaluation." In Handbook on Experimental

Mechanics, edited by A.S. Kobayashi, Society for Experimental Mechanics,

inc.

Dutta, A. and S. Talukdar. 2004. "Damage detection in bridges using accurate modal

parameters." Finite Elements in Analysis and Design 40 (3): 287-304.

Elshafey, A.A., H. Marzouk and M.R. Haddara. 2011. "Experimental damage

identification using modified mode shape difference." Journal of Marine

Science and Application 10 (2): 150-155.

Farrar, C.R. and S.W. Doebling. 1997. "An overview of modal-based damage

identification methods." Proceedings of the DAMAS Conference, Sheffield,

UK, 1997, .

Farrar, C.R., S.W. Doebling and D.A. Nix. 2001. "Vibration–based structural

damage identification." Philosophical Transactions of the Royal Society of

London. Series A: Mathematical, Physical and Engineering Sciences 359

(1778): 131-149. Accessed February 25, 2009. doi: 10.1098/rsta.2000.0717

Farrar, C.R. and D.A. Jauregui. 1998. "Comparative study of damage identification

algorithms applied to a bridge: I. Experiment." Smart Materials and

Structures 7: 704-719. Accessed April 21, 2009. doi:10.1088/0964-

1726/7/5/013

Page | 9-234

Farrar, C.R. and K. Worden. 2007. "An introduction to structural health monitoring."

Philosophical Transactions of the Royal Society A: Mathematical, Physical

and Engineering Sciences 365 (1851): 303-315.

Friswell, M.I. 2007. "Damage identification using inverse methods." Philosophical

Transactions of the Royal Society A: Mathematical, Physical and

Engineering Sciences 365 (1851): 393-410. Accessed March 08, 2009. doi:

10.1098/rsta.2006.1930

Friswell, M.I. and J.E.T. Penny. 1997. "Is damage location using vibration

measurements practical." Proceedings of EUROMECH 365 International

Workshop: DAMAS 97, Structural Damage Assessment using Advanced

Signal Processing Procedures, 351-362. University of Sheffield, UK.

Hawk, H. and E.P. Small. 1998. "The BRIDGIT bridge management system."

Structural Engineering International 8 (4): 309-314.

Housner, G. W., L. A. Bergman, T. K. Caughey, A. G. Chassiakos, R. O. Claus, S. F.

Masri, R. E. Skelton, T. T. Soong, B. F. Spencer and J. T. P. Yao. 1997.

"Structural control: Past, present, and future." Journal of engineering

mechanics 123 (9): 897-971.

Hsu, L.S. and C.T.T. Hsu. 1994. "Stress-strain behavior of steel-fiber high-strength

concrete under compression." ACI structural journal 91 (4): 448-457.

Huang, Q., P. Gardoni and S. Hurlebaus. 2012. "A probabilistic damage detection

approach using vibration-based nondestructive testing." Structural Safety 38:

11-21.

Ismail, Z., H. Abdul Razak and A.G. Abdul Rahman. 2006. "Determination of

damage location in RC beams using mode shape derivatives." Engineering

structures 28 (11): 1566-1573.

Jaishi, B. and W.X. Ren. 2006. "Damage detection by finite element model updating

using modal flexibility residual." Journal of Sound and Vibration 290 (1):

369-387.

Khiem, N.T. and T.V. Lien. 2001. "A simplified method for natural frequency

analysis of a multiple cracked beam." Journal of Sound and Vibration 245

(4): 737-751.

Page | 9-235

Kisa, M. and M. Arif Gurel. 2007. "Free vibration analysis of uniform and stepped

cracked beams with circular cross sections." International journal of

engineering science 45 (2-8): 364-380.

Kumar, M., R.A. Shenoi and S.J. Cox. 2009. "Experimental validation of modal

strain energies based damage identification method for a composite sandwich

beam." Composites Science and Technology 69 (10): 1635-1643.

LabVIEW. 2010. LabVIEW: National Instruments Corp. Software: Windows XP OS.

Lee, L.S., V.M. Karbhari and C. Sikorsky. 2004. "Investigation of integrity and

effectiveness of RC bridge deck rehabilitation with CFRP composites."

Report no. SSRP-2004/08, Department of Structural Engineering, University

of California, San Diego, 2004.

Li, H.N., D.S. Li and G.B. Song. 2004. "Recent applications of fiber optic sensors to

health monitoring in civil engineering." Engineering structures 26 (11):

1647-1657.

Li, J., FC Choi, B. Samali and K. Crews. 2007. "Damage localisation and severity

evaluation of a beam-like timber structure based on modal strain energy and

flexibility approaches." Journal of Building Appraisal 2 (4): 323-334.

Li, Y. Y. 2010. "Hypersensitivity of strain-based indicators for structural damage

identification: A review." Mechanical Systems and Signal Processing 24 (3):

653-664.

Liu, D.K., G. Dissanayake, P.B. Manamperi, P.A. Brooks, G. Fang, G. Paul, S.

Webb, N. Kirchner, P. Chotiprayanakul and N.M. Kwok. 2008. "A robotic

system for steel bridge maintenance: research challenges and system design."

Proceedings of the Australian Conference on Robotics and Automation

(ACRA 08), December 2008, Canberra, Australia.

Maeck, J. 2003. "Damage Assessment of Civil Engineering Structures by Vibration

Monitoring." PhD diss. Department of Civil Engineering, K. U. Leuven,

Belgium.

Maeck, J., M. Abdel Wahab, B. Peeters, G. De Roeck, J. De Visscher, W.P. De

Wilde, J.M. Ndambi and J. Vantomme. 2000. "Damage identification in

reinforced concrete structures by dynamic stiffness determination."

Engineering structures 22 (10): 1339-1349.

Page | 9-236

Maeck, J., M. Abdel Wahab and G. De Roec. 1999 "Damage localization in

reinforced concrete beams by dynamic stiffness determination." Proceedings

of 17th

International Modal Analysis Conference, Orlando, FL, 1289–1295.

MATLAB® R2010b. 2010. The MathWorks, Inc. 3. Software: Windows XP OS.

MATLAB® R2010b Help Browser, 2010. The MathWorks, Inc. 3 Apple Hill Drive

Natick, MA 01760-2098.

McLinn, J. 2009. "Major Bridge Collapse in the US, and Around the World." IEEE

Reliability Society Annual Technology Report, September 2010, 449-482.

IEEE Reliability Society.

Murphy, C. 2009. "Western Australian Auditor Genral's Report - Maintaining the

state road network." Accessed 27/03/2012. www.audit.wa.gov.au.

Murugesh, G. 2001. "Health monitoring of the new Benicia Martinez bridge."

Proceedings of SPIE: Smart Structures and Materials and Nondestructive

Evaluation for Health Monitoring and Diagnostics, 2001, 4337, 256-267.

Myrvoll, F., E. Dibiagio and C. Hansvold. 1994. "Instrumentation for monitoring the

Skarnsundet cable-stayed bridge." Strait crossings 94: 207-215.

Nandwana, B.P. and S.K. Maiti. 1997. "Detection of the location and size of a crack

in stepped cantilever beams based on measurements of natural frequencies."

Journal of Sound and Vibration 203 (3): 435-446.

Nayal, R. and H.A. Rasheed. 2006. "Tension stiffening model for concrete beams

reinforced with steel and FRP bars." Journal of materials in civil engineering

18 (6): 831-841.

Ndambi, J.M., J. Vantomme and K. Harri. 2002. "Damage assessment in reinforced

concrete beams using eigenfrequencies and mode shape derivatives."

Engineering structures 24 (4): 501-515.

Osegueda, R.A., Carrasco, C.J., and Meza, A. . 1997 "A modal strain energy

distribution method to localize and quantify damage." Proceedings of 15th

International Modal Analysis Conference, Orlando, FL, 1298-1304.

Ostachowicz, W.M. and M. Krawczuk. 2001. "On modelling of structural stiffness

loss due to damage." Key Engineering Materials 204-205: 185-200.

Page | 9-237

Owolabi, G.M., A.S.J. Swamidas and R. Seshadri. 2003. "Crack detection in beams

using changes in frequencies and amplitudes of frequency response

functions." Journal of Sound and Vibration 265 (1): 1-22.

Palacz, M. and M. Krawczuk. 2002. "Vibration parameters for damage detection in

structures." Journal of Sound and Vibration 249 (5): 999-1010.

Pandey, A.K., M. Biswas, M., and Samman M.M. . 1991. "Damage detection in

bridges from changes in curvature mode shapes." Journal of Sound and

Vibration 145 (2): 312–332.

Pandey, A.K. and M. Biswas. 1994. "Damage detection in structures using changes

in flexibility." Journal of Sound and Vibration 169 (1): 3-17.

Park, S., Y.B. Kim and N. Stubbs. 2002. "Nondestructive damage detection in large

structures via vibration monitoring." Electronic Journal of Structural

Engineering 2: 59-75.

Patil, D.P. and S.K. Maiti. 2005. "Experimental verification of a method of detection

of multiple cracks in beams based on frequency measurements." Journal of

Sound and Vibration 281 (1): 439-451.

Patjawit, A. and W. Kanok-Nukulchai. 2005. "Health monitoring of highway bridges

based on a Global Flexibility Index." Engineering structures 27 (9): 1385-

1391.

Peeters, B., M. Abdel Wahab, G. De Roeck, J. De Visscher, W.P. De Wilde, M.

Ndambi and J. Vantomme. 1996. "Evaluation of structural damage by

dynamic system identification." 21th Int. seminar on Modal Analysis (ISMA

21), 1349-1362: Katholieke Universiteit Leuven, Belgium.

Perera, R. and C. Huerta. 2008. "Identification of damage in RC beams using indexes

based on local modal stiffness." Construction and Building Materials 22 (8):

1656-1667.

Pereyra, L., R. Osegueda, C. Carrasco and C. Ferregut. 2000. "Detection of damage

in a stiffened plate from fusion of modal strain energy differences."

Proceedings of 18th

International Modal Analysis Conference, San Antonio,

TX, 2000, 1556-1562.

Page | 9-238

Petro, S.H., S. En, S. GangaRao and S. Venkatappa. 1997. "Damage detection using

vibration measurement." Proceedings of 15th

International Modal Analysis

Conference, Orlando, FL, 1997, 113-119.

Pike, John. "FM 3-34.343 Appendix F." Accessed 7/11/2011.

http://www.globalsecurity.org/military/library/policy/army/fm/3-34-343/.

Razak, H.A. and F.C. Choi. 2001. "The effect of corrosion on the natural frequency

and modal damping of reinforced concrete beams." Engineering structures 23

(9): 1126-1133.

Rizos, P.F., N. Aspragathos and A.D. Dimarogonas. 1990. "Identification of crack

location and magnitude in a cantilever beam from the vibration modes."

Journal of Sound and Vibration 138 (3): 381-388.

Rytter, A. , 1993. "Vibration based inspection of civil engineering structures." PhD

diss., Department of Building Technology and Structural Engineering,

University of Aalborg.

Salawu, O.S. and C. Williams. 1995. "Bridge assessment using forced-vibration

testing." Journal of Structural Engineering 121 (2): 161-173.

Salawu, O.S. 1995. "Non-destructive assessment of structures using the integrity

index method applied to a concrete highway bridge." Insight 37 (11): 875-

878.

Salawu, O.S. 1997a. "Detection of structural damage through changes in frequency:

a review." Engineering structures 19 (9): 718-723.

Salawu, O.S. 1997b. "An integrity index method for structural assessment of

engineering structures using modal testing." Insight 39 (1): 33-37.

Saleh, F., B. Supriyadi, B. Suhendro and D. Tran. 2004. "Damage Detection In Non-

Prismatic Reinforced Concrete Beams Using Curvature Mode Shapes."

Structural Integrity and Fracture International Conference: SIF2004,

Brisbane, Australia, September 2004, 331-337.

Saleshi, M., R. Ziaei, M. Ghayour and M.A. Vaziry. 2011. "A non model-based

damage detection technique using dynamically measured flexibility matrix."

Iranian Journal of Science and Technology Transaction B-Engineering 35

(M1): 1-13.

Page | 9-239

Sazonov, E. and P. Klinkhachorn. 2005. "Optimal spatial sampling interval for

damage detection by curvature or strain energy mode shapes." Journal of

Sound and Vibration 285 (4): 783-801.

Shih, H.W. 2009. "Damage assessment in structures using vibration characteristics."

PhD diss., School of Urban Development, Queensland University of

Technology.

Shih, H.W., D.P. Thambiratnam and T.H.T. Chan. 2009. "Vibration based structural

damage detection in flexural members using multi-criteria approach." Journal

of Sound and Vibration 323 (3-5): 645-661.

Sohn, H., C. Farrar, N. Hunter and K. Worden. 2001. "Applying the LANL statistical

pattern recognition paradigm for structural health monitoring to data from a

surface-effect fast patrol boat." Los Alamos National Laboratory Report, Los

Alamos, New Mexico.

Sohn, H. 2004. "A review of structural health monitoring literature: 1996-2001." Los

Alamos National Laboratory Report, Los Alamos, New Mexico.

Special Structures (Bridges, Tanks). Accessed 28/04/2012.

http://www.exponent.com/special_structures_bridges_tanks/.

Standards Association of Australia. 2004. Australian standards for Bridge Design -

Part 2: Design loads. AS 5100.2-2004. Accessed January 10, 2012.

http://www.saiglobal.com.

Stubbs, N., J.T. Kim and K. Topole. 1992. "An efficient and robust algorithm for

damage localization in offshore platforms." Proceedings ASCE 10th

Structures Congress, 543-546.

Stubbs, N., J.T. Kim and C.R. Farrar. 1995. "Field verification of a non-destructive

damage localization and severity estimation algorithm." Proceedings of 13th

International Modal Analysis Conference, 210-218.

Sydney Harbour Bridge. Accessed 28/04/2012.

http://www.sydney.com.au/bridge.html.

Sydney Harbour Bridge. Accessed 28/04/2012. http://sydney-harbour-

bridge.bos.nsw.edu.au/images-sepea/bridge5.jpg.

Page | 9-240

Thambiratnam, D. 1995. "Vibration analysis of storey bridge." Transactions of the

Institution of Engineers, Australia. Civil engineering 37 (2): 91-97.

Thompson, P.D., E.P. Small, M. Johnson and A.R. Marshall. 1998. "The Pontis

bridge management system." Structural Engineering International 8 (4): 303-

308.

Thomson, P., J.M. Casas, J.M. Arbelaez and J. Caicedo. 2001. "Real-time health

monitoring of civil infrastructure systems in Colombia." Health monitoring

and management of civil infrastructure systems, The International Society for

Optical Engineering, Bellingham (WA), 2001, 113–121.

Tsamasphyros, G.J., E.A. Koulalis, G.N. Kanderakis, N.K. Furnarakis and V.Z.

Astreinidis. 2006. "Structural Health Monitoring of a Steel Railway Bridge

using Optical Fibre Bragg Grating Sensors and Numerical Simulation."

Proceedings of the third European Workshop: Structural Health Monitoring,

2006, edited by Alfredo Guemes, 341.

Wang, L. and T.H.T. Chan. 2009. "Review of vibration-based damage detection and

condition assessment of bridge structures using structural health monitoring."

The Second Infrastructure Theme Postgraduate Conference: Rethinking

Sustainable Development, Planning, Engineering, Design and Managing

Urban Infrastructure, March 2009, Queensland University of Technology,

Brisbane.

Wang, Y. 2010. "A non-destructive damage detection method for reinforced concrete

structures based on modal strain energy." PhD diss., University of

Technology, Sydney.

Wardhana, K. 2003. "Analysis of recent bridge failures in the United States." Journal

of Performance of Constructed Facilities 17 (3): 144-150.

Whalen, T.M. 2008. "The behavior of higher order mode shape derivatives in

damaged, beam-like structures." Journal of Sound and Vibration 309 (3-5):

426-464.

Worden, K. and J.M. Dulieu-Barton. 2004. "An overview of intelligent fault

detection in systems and structures." Structural Health Monitoring 3 (1): 85-

98.

Page | 9-241

Yoo, S.H. and B.S. Kim. 2000. "Characterization of a crack in a plate using strain

mode shapes." Proceedings of the 18th

International Modal Analysis

Conference, San Antonio, TX, 2000, 1790-1795.

Yuen, M.M.F. 1985. "A numerical study of the eigenparameters of a damaged

cantilever." Journal of Sound and Vibration 103 (3): 301-310.

Zhang, Z., and Aktan, A. E. 1995. "The damage indices for constructed facilities."

Proceedings of the 13th

International Modal Analysis Conference, 1995,

1520–1529.

Zhou, Z., L.D. Wegner and B.F. Sparling. 2007. "Vibration-based detection of small-

scale damage on a bridge deck." Journal of Structural Engineering 133 (9):

1257-1267.

damage assessment in reinforced concrete flexural...

Documents