Kriging in Slope Reliability Analysis

Lei-Lei Liu is an associate professor in the Department of Geological Engineering at Central South University, China. He is the co-author of Analysis, Design, and Construction of Foundations, also published by CRC Press. Jing-Ze Li is a research associate at Central South University, China. His PhD research was undertaken jointly with Central South University, China and Université Grenoble Alpes, France. Lei Huang is an associate professor at Sanming University, China. He received his PhD in The Hong Kong Polytechnic University.

1 Introduction 1

1.1 Background 1

1.1.1 Uncertainties in slope engineering 1

1.1.2 Reliability analysis of slopes 3

1.1.3 Reliability-based design of slopes 4

1.1.4 Kriging in slope reliability analysis 5

1.2 Layout of the book 6

References 8

2 Overview of geostatistics and spatial sampling 11

2.1 Background of geostatistics 11

2.2 Review of geostatistics 11

2.3 Variogram and variogram modeling 13

2.3.1 Introduction of variogram 13

2.3.2 Modeling of variogram 14

2.4 Applications of geostatistics 17

2.5 Spatial sampling 19

References 21

3 Basic theory of Kriging 23

3.1 Introduction 23

3.2 Ordinary Kriging theory 24

3.3 Other types of Kriging 26

3.3.1 Simple Kriging 26

3.3.2 Universal Kriging 27

3.3.3 Co-Kriging 27

3.3.4 Disjunctive Kriging 28

3.3.5 Bayesian Kriging 29

3.4 Determination of model parameter 29

References 31

4 Application of Kriging in slope reliability analysis 33

4.1 Introduction 33

4.2 Reliability analysis of slopes 33

4.2.1 Slope stability analysis 33

4.2.2 Slope reliability analysis 35

4.2.3 Slope reliability considering parameter uncertainty 39

4.3 Kriging-based surrogate model 40

4.4 Kriging-based conditional random field modeling 41

References 43

5 Genetic algorithm-optimized Taylor Kriging surrogate model for system reliability analysis of soil slopes 47

5.1 Introduction 47

5.2 Kriging methodology 49

5.2.1 Classical Kriging theory 49

5.2.2 Theory of TK 50

5.3 GATK surrogate model 51

5.3.1 Genetic algorithm 51

5.3.2 GATK model 52

5.3.3 Analytical validation of GATK−example #1 53

5.3.4 Analytical validation of GATK−example #2 57

5.4 System reliability analysis using GATK surrogate model 59

5.5 Illustrative examples 59

5.5.1 A homogeneous c–ϕ slope 60

5.5.2 A heterogeneous two-layered soil slope 64

5.6 Discussions 70

5.7 Conclusions 73

References 73

6 Adaptively selected-autocorrelation structure-based Kriging metamodel for slope reliability analysis 76

6.1 Introduction 76

6.2 The proposed GAWMK method 78

6.3 Implementation procedure of the proposed method for slope reliability analysis 80

6.4 Validation of the proposed method and the modified DACE toolbox 83

6.4.1 A one-dimensional cubic function 83

6.4.2 A three-dimensional data fitting problem 88

6.5 Applications to slope reliability analysis 90

6.5.1 Example 1: a homogeneous c–ϕ slope 90

6.5.2 Example 2: a two-layered cohesive soil slope 96

6.5.3 Example 3: a three-layered cohesive soil slope 98

6.5.4 Example 4: a three-layered c–ϕ slope 101

6.6 Summary and conclusions 102

References 104

7 System reliability analysis of soil slopes using an advanced Kriging metamodel and quasi Monte Carlo simulation 108

7.1 Introduction 108

7.2 Probabilistic analysis of soil slope stability using QMCS 111

7.3 Advanced Kriging metamodel 112

7.3.1 Genetic algorithm optimized Kriging 112

7.3.2 Construction of the advanced Kriging method 113

7.4 AKQMCS for system reliability analysis of soil slopes 116

7.5 Illustrative examples 119

7.5.1 Example #1: a two-layered cohesive slope 119

7.5.2 Example #2: a three-layered c–ϕ slope 124

7.5.3 Example #3: a single-layered sand slope 129

7.6 Summary and conclusions 132

References 134

8 Efficient slope reliability analysis and risk assessment based on multiple Kriging surrogate models 138

8.1 Introduction 138

8.2 The proposed MK method for slope reliability analysis and risk assessment 140

8.2.1 General idea of MK method 140

8.2.2 Slope reliability analysis based on the proposed MK method 142

8.2.3 Slope risk assessment based on the proposed MK method 144

8.3 Implementation procedure of the proposed MK method 145

8.4 Illustrative examples 147

8.4.1 Example 1: a two-layered cohesive soil slope 148

8.4.2 Example 2: Congress Street cut slope 153

8.5 Discussions 158

8.6 Conclusions 160

References 161

9 A new active learning Kriging surrogate model for structural system reliability analysis with multiple failure modes 165

9.1 Introduction 165

9.2 The proposed ALK-SD method for system reliability analysis 167

9.2.1 Basic idea of ALK-SD 167

9.2.2 Identification of significant domain 169

9.2.3 Determination of ATSs 173

9.2.4 System reliability analysis based on ALK-SD 174

9.2.5 Implementation procedure 175

9.3 Numerical examples 177

9.3.1 Example 1: a series system with four branches 177

9.3.2 Example 2: a parallel system with three failure modes 181

9.3.3 Example 3: a series system with three failure modes 182

9.3.4 Example 4: a parallel system with disconnected failure regions 186

9.3.5 Example 5: a mass gravity retaining wall with five random variables 187

9.4 Discussion 192

9.4.1 The determination of Φ(δ) 192

9.4.2 Comparison with other U-function series methods 194

9.4.3 Comparison of the computational efficiency and robustness 195

9.4.4 The locations of the ATSs 197

9.5 Conclusion 199

References 202

10 New Kriging methods for efficient system slope reliability analysis considering soil spatial variability 205

10.1 Introduction 205

10.2 Review of MK-based slope reliability analyses 208

10.3 The proposed new Kriging methods 208

10.3.1 Basic idea 208

10.3.2 RALK method 209

10.3.3 MK-RSS-SIR method 218

10.3.4 MK-RSS method 218

10.4 Example 1: a three-layered cohesive slope 218

10.4.1 Results of RALK method 220

10.4.2 Results of MK-RSS-SIR method 231

10.4.3 Results of MK-RSS method 234

10.5 Example 2: a four-layered slope with a soft band 236

10.5.1 Results of RALK method 239

10.5.2 Results of MK-RSS-SIR method 241

10.5.3 Results of MK-RSS method 243

10.6 Discussion 244

10.6.1 Comparison of the computational accuracy 244

10.6.2 Comparison of the computational efficiency 245

10.6.3 Slope types applicable to three methods 246

10.7 Summary and conclusions 247

References 249

11 Conditional random field reliability analysis of a cohesion-frictional slope 255

11.1 Introduction 255

11.2 Simulation of unconditional random field 257

11.3 Simulation of conditional random field 260

11.4 Probabilistic analysis of a slope based on SS 262

11.5 Implementation procedure of conditional probabilistic analysis 264

11.6 Illustrative example 267

11.6.1 Basic model 267

11.6.2 Reliability results based on unconditional random fields 268

11.6.3 Reliability results based on conditional random fields 270

11.7 Summary and conclusions 280

References 283

12 Reliability analysis and risk assessment of pile-reinforced slopes considering spatial soil variability and site investigation 286

12.1 Introduction 286

12.2 Simulation of soil spatial variability based on random field theory 288

12.2.1 Conditional random field 288

12.2.2 Conditional stationary random field based on investigation boreholes 289

12.3 Probabilistic analysis of pile-reinforced slope 291

12.3.1 Stability analysis of pile-reinforced slopes 291

12.3.2 RFDM for slope reliability analysis and risk assessment 294

12.4 Implementation procedure for the proposed framework 295

12.5 Illustrative example 297

12.5.1 Influence of investigation scheme on soil uncertainty 301

12.5.2 Influence of investigation scheme on probabilistic characteristics of slope safety 305

12.5.3 Influence of investigation scheme on slope failure probability and quantitative risk 311

12.5.4 Influence of investigation scheme on pile structural responses 313

12.6 Summary and conclusions 316

References 317

13 Summary and concluding remarks 321

Index 323