Title Page

Abstract

Contents

Chapter 1. Introduction 24

1.1. Background 24

1.1.1. Historical review of structural reliability 24

1.1.2. Current context of bridges deterioration 26

1.2. Research objectives 28

1.3. Thesis organization 29

Chapter 2. Corrosion mechanisms for deteriorated bridges 33

2.1. Introduction 33

2.2. Corrosion mechanism of a steel box girder bridge 37

2.3. Corrosion mechanism of shear reinforcement of a PC box girder bridge 38

2.4. Pitting corrosion in post-tensioned tendon 40

2.4.1. Pitting corrosion mechanisms 41

2.4.2. Sources of Chloride 42

2.4.2.1. Grouting process 42

2.4.2.2. Material properties of grout 43

2.4.2.3. Corrosion of anchorage 43

2.4.2.4. Duct problems 44

2.5. Corrosion scenario 45

2.6. Corrosion initiation 46

2.7. Corrosion propagation-the stress corrosion cracking 49

2.8. Conclusions 50

Chapter 3. Failures analysis of PC box girder bridges under corrosion attack 51

3.1. Introduction 51

3.2. Structural configuration 53

3.2.1. Geometrical properties 53

3.2.2. Load models 56

3.3. Shear strength degradation modeling 59

3.3.1. Corrosion of shear reinforcements 59

3.3.2. Limit state function 60

3.4. Flexural strength degradation modeling 61

3.4.1. Ductile model 61

3.4.2. Ductile-brittle model 65

3.4.3. Brittle model 68

3.4.3.1. Corrosion initiation 68

3.4.3.2. Corrosion propagation 68

3.4.3.3. Computational procedure 73

3.5. Analysis results and discussions 74

3.5.1. Shear failure 74

3.5.2. Moment failure: ductile model 74

3.5.3. Moment failure: ductile-brittle model 77

3.5.4. Moment failure: brittle model 79

3.6. Stochastic data for RBDO analysis 86

3.7. Equivalence between ductile and brittle model 91

3.8. Conclusions 94

Chapter 4. Efficient approaches for reliability analysis and reliability-based design optimization of structures 96

4.1. Introduction 96

4.2. First order approximation for the reliability analysis 99

4.2.1. Background of reliability-based design theory 99

4.2.2. First and second order approximation for limit state functions 100

4.3. Probabilistic optimal approaches 103

4.3.1. Problem definitions 103

4.3.2. Probabilistic optimal approaches for RBDO problems 104

4.4. Numerical examples 111

4.4.1. RC girder design 111

4.4.1.1. Example 1: Design for minimizing the initial cost (initial and failure costs) with equality reliability constraints 114

4.4.1.2. Example 2: design for minimizing the total cost (initial and failure costs) 117

4.4.1.3. Example 3: Design for minimizing the total cost (initial and failure costs) with time-variant probabilistic inequality constraints 119

4.4.1.4. Results of the second order approximation 120

4.4.2. Example 4: PC girder design 125

4.5. Conclusions 133

Chapter 5. Time variant reliability-based design optimization of highway bridges 135

5.1. Introduction 135

5.2. RBDO of PC box girder bridges under corrosion attack 138

5.2.1. Problem definition 138

5.2.2. Load models 139

5.2.3. RBDO formulation of a PC box girder bridge 141

5.2.3.1. Sectional configurations 142

5.2.3.2. Objective function 144

5.2.3.3. Probabilistic constraints 145

5.2.3.4. Deterministic constraints 153

5.2.3.5. Analysis results and discussion 154

5.3. RBDO of the steel box girder bridge under corrosion attack 163

5.3.1. Problem definition 163

5.3.2. RBDO formulation of steel box girder bridge 163

5.3.3. Analysis results and discussion 167

5.4. Conclusions 173

Chapter 6. Effective target reliability indices for highway bridges 175

6.1. Introduction 175

6.2. Background of the target reliability index for highway bridges in bridge design specification codes 176

6.2.1. Target reliability index using in AASHTO and OHBDC codes 177

6.2.2. Target reliability index using in Euro code 180

6.2.3. Target reliability index using in JCSS and other codes 182

6.3. Effective target reliability index for highway bridges 183

6.4. Application of the predefined reliability index for the practical design 185

6.5. Conclusions 186

Chapter 7. Summary, conclusions and recommendations 187

7.1. Summary and conclusions 187

7.2. Recommendation for future research 195

Bibliography 197

Table 3-1. Geometry properties of cross section of PC box girder bridge at mid-span 54

Table 3-2. Geometry properties of cross section of PC box girder bridge at support 55

Table 3-3. Random variables 67

Table 3-4. Summary of ductile, ductile-brittle and brittle models 87

Table 3-5. Recommendation for the estimation of tendon area loss due to the pitting corrosion for RBDO of PC bridges 91

Table 3-6. Equivalence between ductile and brittle model 94

Table 4-1. Definition of the design variables for the RC girder 113

Table 4-2. Normal random parameters for the RC girder bridge 114

Table 4-3. Results for the minimum initial cost at component level of reliability indices 115

Table 4-4. Results for the minimum initial cost at system level of reliability indices 116

Table 4-5. Results for the minimum initial cost and failure cost with time-invariant and time-variant reliability constraints 121

Table 4-6. Results of the first, second approximation and existing method (EX1) 122

Table 4-7. Results of the first, second approximation and existing method (EX2) 123

Table 4-8. Results of the first, second approximation and existing method (EX3) 124

Table 4-9. Definition of design variables for the PC beam 127

Table 4-10. Normal random parameters for the PC I-beam 127

Table 4-11. Results for the minimum initial cost at different span lengths 132

Table 5-1. Random parameters for the PC box girder bridge 141

Table 5-2. Design variables definition of the PC box girder bridge 144

Table 5-3. Results for design variables of the PC box girder versus target reliability indices 156

Table 5-4. Results for actual reliability indices versus target reliability indices 157

Table 5-5. Results for design variable versus span lengths 159

Table 5-6. Results for actual reliability indices versus span lengths 162

Table 5-7. Random parameters for a steel box girder bridge 168

Table 5-8: Results of optimum design variables and actual reliability index 169

Table 5-9. Optimum total costs versus reliability indices 170

Table 6-1. Definition of consequences classes 181

Table 6-2. Recommended minimum values for reliability index for ultimate strength using in Eurocode 0 (2007) 182

Table 6-3. Target reliability index for one year reference period according to JCSS Probabilistic Model Code 183

Table 6-4. Target reliability index for one year reference period in some codes 183

Figure 2-1. The collapse of the Ynys-y-Gwas Bridge in UK in 1985 34

Figure 2-2. The collapse of the Silver Bridge in USA in 1967 35

Figure 2-3. The collapse of the Mianus River Bridge USA in 1983 35

Figure 2-4. Mechanism of thin oxide formation 38

Figure 2-5. Corrosion of shear reinforcements under environmental agents 40

Figure 2-6. Electrochemical process of tendon corrosion 42

Figure 2-7. Incomplete grouting process leading the corrosion 44

Figure 2-8. Corrosion of anchorages 45

Figure 2-9. The duct corrosion and bleed water void 45

Figure 2-10. Typical grout void inside the duct 46

Figure 2-11. Hydrogen generation inside the pit 49

Figure 3-1. Elevation of Back-Jun Bridge in Korea 56

Figure 3-2. Critical section at midspan 56

Figure 3-3. Critical section at support (diaphragm) 57

Figure 3-4. Moment diagram due to dead load 58

Figure 3-5. Moment diagram due to live load 58

Figure 3-6. Shear diagram due to live load 59

Figure 3-7. Tendon detail for maximum positive moment section at the external mid-span 63

Figure 3-8. Calculation of residual area of a pitted bar 63

Figure 3-9. The discretization of girder into elements 69

Figure 3-10. The stress redistribution between wires in the tendon 71

Figure 3-11. Computational procedure for failure analysis of pitting corrosion 72

Figure 3-12. Distribution of shear reinforcement area 75

Figure 3-13. Deterioration of reliability index for shear capacity 75

Figure 3-14. Distribution of tendon area 76

Figure 3-15. Deterioration of reliability index for moment capacity 76

Figure 3-16. Deterioration of reliability index for fracture resistance 78

Figure 3-17. Reliability index for tendon fracture with values of diameter 78

Figure 3-18. Reliability index for tendon fracture with different values of pitting factor 80

Figure 3-19. Deterioration of reliability index after tendons failure 80

Figure 3-20. Failure process of PC girder with 1000 samples 82

Figure 3-21. Failure process of PC girder with 5000 samples 83

Figure 3-22. Failure process of PC girder with 10000 samples 83

Figure 3-23. Histogram of the time to failure of the critical element from the program started until the time of failure 84

Figure 3-24. Histogram of the reliability index of the critical element during the failure process 84

Figure 3-25. Histogram of tendon area loss of critical element from the program started until the time of failure 85

Figure 3-26. Distribution of reliability index of each element over the length of girder (horizontal axis: element number, vertical axis: reliability index) 86

Figure 3-27. Failure process of the PC box girder with β＝4 88

Figure 3-28. Failure process of the PC box girder with β＝4.5 88

Figure 3-29. Failure process of the PC box girder with β＝5 89

Figure 3-30. Histogram of tendon area loss with the reliability index level β＝4 89

Figure 3-31. Histogram of tendon area loss with the reliability index level β＝4.5 90

Figure 3-32. Histogram of tendon area loss with the reliability index level β＝5 90

Figure 3-33. The loss of tendon area with 30% corroded tendon in 45 year according to ductile model (wire diameter D＝12.7 mm) 93

Figure 3-34. Failure process of girder according to the brittle and ductile model 93

Figure 4-1. Transverse cross-section of the bridge 112

Figure 4-2. Geometry of the cross-section of the RC girder 112

Figure 4-3. Shear reinforcement at intervals of the RC girder 113

Figure 4-4. Optimal design variables: X₂ - X5 vs. target reliability indices(이미지참조) 117

Figure 4-5. Optimal design variables: X₁ and X₂ vs. target reliability indices 118

Figure 4-6. Example 1: comparison of the two results with respect to design variables 119

Figure 4-7. Example 2: comparison of the two results with respect to design variables 120

Figure 4-8. Example 3: comparison of the two results with respect to design variables 122

Figure 4-9. Comparison of results (example 1) 123

Figure 4-10. Comparison of results (example 2) 124

Figure 4-11. Comparison of results (example 3) 125

Figure 4-12. Loading condition and tendon profile for the simply-supported PC I-beam 126

Figure 4-13. Geometry of the cross-section and notation for the design variables 126

Figure 4-14. Design variables vs. span length 133

Figure 5-1. The torsion moment due to truck load 140

Figure 5-2. Geometry design variables 143

Figure 5-3. The tendon envelope 144

Figure 5-4. Calculation of residual area of a pitted bar 152

Figure 5-5. The tendon area versus target reliability indices 155

Figure 5-6. Shear and torsion reinforcements versus target reliability indices 156

Figure 5-7. Actual reliability indices versus target reliability indices 158

Figure 5-8. Optimal total costs versus target reliability indices 160

Figure 5-9. Total Costs versus span lengths with different levels of target reliability indices 160

Figure 5-10. Total Costs versus span lengths and target reliability indices 161

Figure 5-11. Actual reliability indices versus span lengths 161

Figure 5-12a. Typical cross section and design variables of steel box girder bridge 164

Figure 5-12b. Corrosion of steel box girder bridge 164

Figure 5-13. Graph of design variables vs. span length 170

Figure 5-14. Graph of total coats versus span length and target reliability indices 171

Figure 5-15. Graph of optimal cost with different level of target reliability indices 172

Figure 5-16. Combination effects of span length and target reliability indices 173

Figure 6-1. Load effect, resistance and safety margin probability distribution functions 177

Figure 6-2. Target reliability index for moment for non-LRFD and calibrated LRFD AASHTO code 178

Figure 6-3. Target reliability index for shear for non-LRFD and calibrated LRFD AASHTO code 178

Figure 6-4. Target reliability index for moment for non-LRFD and calibrated LRFD OHBDC code 179

Figure 6-5. Target reliability index for moment for non-LRFD and calibrated LRFD OHBDC code 179

Figure 6-6. Target reliability index using in Euro, Spain and AASHTO LRDF codes 180

Figure 6-7. Combination effects of span lengths and target reliability indices for PC bridges 184

Figure 6-8. Combination effects of span length and target reliability indices for steel bridges 185