Contenido Risk and Reliability in Geotechnical Engineering

Establishes Geotechnical Reliability as Fundamentally Distinct from Structural Reliability

Reliability-based design is relatively well established in structural design. Its use is less mature in geotechnical design, but there is a steady progression towards reliability-based design as seen in the inclusion of a new Annex D on "Reliability of Geotechnical Structures" in the third edition of ISO 2394. Reliability-based design can be viewed as a simplified form of risk-based design where different consequences of failure are implicitly covered by the adoption of different target reliability indices. Explicit risk management methodologies are required for large geotechnical systems where soil and loading conditions are too varied to be conveniently slotted into a few reliability classes (typically three) and an associated simple discrete tier of target reliability indices.

Provides Realistic Practical Guidance

Risk and Reliability in Geotechnical Engineering makes these reliability and risk methodologies more accessible to practitioners and researchers by presenting soil statistics which are necessary inputs, by explaining how calculations can be carried out using simple tools, and by presenting illustrative or actual examples showcasing the benefits and limitations of these methodologies.

With contributions from a broad international group of authors, this text:

    Presents probabilistic models suited for soil parameters
    Provides easy-to-use Excel-based methods for reliability analysis
    Connects reliability analysis to design codes (including LRFD and Eurocode 7)
    Maximizes value of information using Bayesian updating
    Contains efficient reliability analysis methods

Accessible To a Wide Audience

Risk and Reliability in Geotechnical Engineering presents all the "need-to-know" information for a non-specialist to calculate and interpret the reliability index and risk of geotechnical structures in a realistic and robust way. It suits engineers, researchers, and students who are interested in the practical outcomes of reliability and risk analyses without going into the intricacies of the underlying mathematical theories.

Part I Properties

1 Constructing multivariate distributions for soil parameters
Jianye Ching and KoK-Kwang Phoon

1.1 Introduction
1.2 Normal random variable
    1.2.1 Random data
    1.2.2 Normal random variable
          1.2.2.1 Probability density function
          1.2.2.2 Cumulative distribution function
    1.2.3 Estimation of normal parameters
          1.2.3.1 Method of moments
          1.2.3.2 Percentile method
          1.2.3.3 Maximum likelihood method
          1.2.3.4 Normal probability plot
          1.2.3.5 Statistical uncertainties in theμ and σ estimators
    1.2.4 Simulation of a normal random variable
          1.2.4.1 Simulating standard uniform random variable U
          1.2.4.2 Simulating standard normal random variable X
          1.2.4.3 Simulating normal random variable Y

1.3 Bivariate normal vector
    1.3.1 Bivariate data
    1.3.2 Bivariate normal distribution
          1.3.2.1 Bivariate standard normal
          1.3.2.2 Correlation coefficient
    1.3.3 Estimation of δ 12
          1.3.3.1 Method of moments
          1.3.3.2 Maximum likelihood method
          1.3.3.3 Rank correlation method
          1.3.3.4 Statistical uncertainties in the δ12 estimate
          1.3.3.5 Goodness-of-fit test (the line test)
    1.3.4 Simulation of bivariate standard normal random variables
1.4 Multivariate normal vector
    1.4.1 Multivariate data
    1.4.2 Multivariate normal distribution
    1.4.3 Estimation of correlation matrix C
          1.4.3.1 Positive definiteness of the correlation matrix C
          1.4.3.2 Goodness-of-fit test
    1.4.4 Simulation of multivariate standard normal random vector X
    1.4.5 Conditional normal and updating

1.5 Non-normal random variable
    1.5.1 Non-normal data
    1.5.2 Non-normal distribution
          1.5.2.1 Lognormal and shifted lognormal distributions
          1.5.2.2 Johnson system of distributions
    1.5.3 Selection and parameter estimation for the Johnson distribution
          1.5.3.1 Probability plot and the goodness-of-fit test (K–S test)
    1.5.4 Simulation of the Johnson random variable
    1.5.5 Some practical observations
          1.5.5.1 Choice of z
          1.5.5.2 Parameter estimation under prescribed lower and/or upper bound
1.6 Multivariate non-normal random vector
    1.6.1 Multivariate non-normal data
    1.6.2 CDF transform approach
    1.6.3 Estimation of the marginal distribution of Y
    1.6.4 Estimation of the correlation matrix C
    1.6.5 Simulation
    1.6.6 Some practical observations
1.7 Real example
    1.7.1 Clay/10/7490 database
    1.7.2 Construction of multivariate distribution
          1.7.2.1 Fit a Johnson distribution to each component (Yi)
          1.7.2.2 Convert Yi into standard normal Xi
          1.7.2.3 Compute the correlation matrix for(X1,X2, ..., X10)
          1.7.2.4 Problem of nonpositive definiteness
    1.7.3 Conditioning: Bayesian analysis
1.8 Future challenges
List of symbols
References

2 Modeling and simulation of bivariate distribution of shear strength parameters using copulas
dian-Qing Li and Xiao-Song Tang

2.1 Introduction
2.2 Copula theory
    2.2.1 Definition of copulas
    2.2.2 Dependence measures
          2.2.2.1 Pearson’s rho
          2.2.2.2 Kendall’s tau
    2.2.3 Four selected copulas

2.3 Modeling bivariate distribution of shear strength parameters
    2.3.1 Measured data of cohesion and friction angle
    2.3.2 Identification of best-fit marginal distributions
    2.3.3 Identification of best-fit copula
2.4 Simulating bivariate distribution of shear strength parameters
    2.4.1 Algorithms for simulating bivariate distribution
          2.4.1.1 Gaussian copula
          2.4.1.2 Plackett copula
          2.4.1.3 Frank and No.16 copulas
    2.4.2 Simulation of copulas and bivariate distribution

2.5 Impact of copula selection on retaining wall reliability
    2.5.1 Retaining wall example
    2.5.2 Probability of failure using direct integration
    2.5.3 Nominal factor of safety for retaining wall stability
    2.5.4 Reliability results produced by different copulas
          2.5.4.1 Effect of geometrical parameters on probability of failure
          2.5.4.2 Effect of COV of shear strength parameters on probability of failure
          2.5.4.3 Effect of correlation between cohesion and friction angle on probability of failure
    2.5.5 Discussions
2.6 Summary and conclusions
Acknowledgments
Appendix 2A: MATLAB® codes
List of symbols
References

Part II Methods evaluating reliability in geotechnical engineering
J. MichaeL duncan and Matthewd. Sleep

3.1 Purpose of reliability analysis
3.2 Probability of failure and risk
3.3 Language of statistics and probability
    3.3.1 Variables
    3.3.2 Correlated and uncorrelated variables
    3.3.3 Standard deviation
    3.3.4 Coefficient of variation
    3.3.5 Histograms and relative frequency diagrams
    3.3.6 Probability and probability theory
    3.3.7 Probability density function
    3.3.8 Normal and lognormal distributions
    3.3.9 Lognormal distribution
    3.3.10 Cumulative density function
    3.3.11 Probability of failure
    3.3.12 Reliability
    3.3.13 Reliability index
    3.3.14 Probability of failure on the CDF curve
    3.3.15 Reliability index for normally distributed factor of safety
    3.3.16 Reliability index for a lognormally distributed factor of safety
    3.3.17 Effect of standard deviation on estimated value of probability of failure
3.4 Probability of failure and factor of safety
    3.4.1 What is “failure?”
    3.4.2 Assumed distribution of the factor of safety
3.5 Methods of estimating standard deviations
    3.5.1 Computation from data
    3.5.2 Published values
    3.5.3 The “three-sigma rule”
    3.5.4 The “N-sigma rule”
    3.5.6 Graphical N-sigma rule
3.6 Computing probability of failure
    3.6.1 Deterministic analyses
    3.6.2 Factor of safety against sliding on top of the silty sand layer
    3.6.3 Factor of safety against sliding on the clay foundation
    3.6.4 Factor of safety against bearing capacity failure
3.7 Monte Carlo analysis using @Risk™
    3.7.1 Accuracy of calculations
3.8 Hasofer Lind method
    3.8.1 Summary of the Hasofer Lind method
3.9 Taylor Series method with assumed normal distribution of the factor of safety
3.10 Taylor Series method with a lognormal distribution of the factor of safety
    3.10.1 Summary of the Taylor Series method
3.11 PEM with a normal distribution for the factor of safety
3.12 PEM with a lognormal distribution for the factor of safety
    3.12.1 Summary of the PEM
3.13 Comments on the methods
    3.13.1 Significance of the variables
    3.13.2 Accuracy
3.14 Summary
References

4 Maximum likelihood principle and its application in soil liquefaction assessment  
  Charng Sein Juang, Sara Khoshnevisan, and Jie Zhang

4.1 Introduction
4.2 Principle of maximum likelihood
    4.2.1 Independent observations
    4.2.2 Correlated observations
    4.2.3 Censored observations
    4.2.4 Ranking of competing models
    4.2.5 Limitations of the maximum likelihood method
4.3 Liquefaction probability based on generalized linear regression
    4.3.1 Predicting liquefaction probability based on generalized linear models
    4.3.2 Calibration database
    4.3.3 Evaluation of sampling bias
    4.3.4 Calibration of liquefaction models
    4.3.5 Ranking of liquefaction models
4.4 Converting a deterministic liquefaction model into a probabilistic model
    4.4.1 Probabilistic model
    4.4.2 Calibration and ranking of PL–Fs relationships
4.5 Estimation of liquefaction-induced settlement
    4.5.1 Probabilistic model for predicting liquefaction-induced settlement
    4.5.2 Calibration database
    4.5.3 Maximum likelihood estimation of statistics of model bias factor
4.6 Summary and conclusions
Acknowledgments
Appendix 4A: Model of Robertson and Wride (1998) and Robertson (2009)
Appendix 4B: Notation
References

5 Bayesian analysis for learning and updating geotechnical parameters and models with measurements
Daniel Straub and iason Papaioannou

5.1 Introduction
5.2 Bayesian analysis
5.3 Geotechnical reliability based on measurements: Step-by-step procedure for Bayesian analysis
    5.3.1 Initial probabilistic model: Prior distribution
          5.3.1.1 Modeling spatially variable parameters
    5.3.2 Computing the reliability and risk based on the prior model
    5.3.3 Describing observations and data: The likelihood
          5.3.3.1 Measurement xi of a parameter X
          5.3.3.2 Samples of a spatially variable parameter
          5.3.3.3 Measurement of site performance parameters
    5.3.4 Updating the model
          5.3.4.1 Conjugate priors
          5.3.4.2 Numerical integration to determine the proportionality constant
          5.3.4.3 Advanced sampling methods
          5.3.4.4 Multinormal approximation of the posterior
          5.3.4.5 Direct updating of the reliability
          5.3.4.6 Predictive distributions
    5.3.5 Updating reliability and risk estimates
    5.3.6 Communicating the results

5.4 Advanced algorithms for efficient and effective Bayesian updating of geotechnical models
    5.4.1 Markov chain Monte Carlo
    5.4.2 Sequential Monte Carlo
    5.4.3 Bayesian updating with structural reliability methods
5.5 Application: Foundation of transmission towers under tensile loading
    5.5.1 Prior probabilistic model
    5.5.2 Reliability analysis based on the prior model
    5.5.3 Updating with CPT test outcomes
    5.5.4 Updating with survived loading conditions
5.6 Application: Finite-element-based updating of soil parameters and reliability
    5.6.1 Prior probabilistic model
    5.6.2 Updating the soil parameters with deformation measurements
    5.6.3 Updating the reliability with deformation measurements
5.7 Concluding remarks
Acknowledgment
References

6 Polynomial chaos expansions and stochastic finite-element methods
  Bruno Sudret

6.1 Introduction
6.2 Uncertainty propagation framework
    6.2.1 Introduction
    6.2.2 Monte Carlo simulation
6.3 Polynomial chaos expansions
    6.3.1 Mathematical setting
    6.3.2 Construction of the basis
          6.3.2.1 Univariate orthonormal polynomials
          6.3.2.2 Multivariate polynomials
    6.3.3 Practical implementation  
          6.3.3.1 Isoprobabilistic transform
          6.3.3.2 Truncation scheme
          6.3.3.3 Application example
    6.3.4 Computation of the coefficients
          6.3.4.1 Introduction
          6.3.4.2 Projection
          6.3.4.3 Least-square minimization
    6.3.5 Validation
          6.3.5.1 Error estimators
          6.3.5.2 Leave-one-out cross-validation
    6.3.6 Curse of dimensionality
    6.3.7 Adaptive algorithms
6.4 Post-processing for engineering applications
    6.4.1 Moment analysis
    6.4.2 Distribution analysis and confidence intervals
    6.4.3 Reliability analysis
    6.4.4 Sensitivity analysis
          6.4.4.1 Sobol decomposition
          6.4.4.2 Sobol indices
          6.4.4.3 Sobol indices from PC expansions
6.5 Application examples
    6.5.1 Load-carrying capacity of a strip footing
          6.5.1.1 Independent input variables
          6.5.1.2 Correlated input variables
    6.5.2 Settlement of a foundation on an elastic two-layer soil mass
    6.5.3 Settlement of a foundation on soil mass with spatially varying Young’s modulus
    6.5.4 Conclusions
6.6 Summary and outlook
Acknowledgments
Appendix 6A: Hermite polynomials
List of symbols
References

7 Practical reliability analysis and design by Monte Carlo Simulation in spreadsheet
  Yuw ang and ZiJun Cao

7.1 Introduction
7.2 Subset Simulation
    7.2.1 Algorithm
    7.2.2 Simulation procedures
7.3 Expanded RBD with Subset Simulation
    7.3.1 Expanded RBD approach
    7.3.2 Desired sample number in direct MCS
    7.3.3 Integration of expanded RBD approach with Subset Simulation
7.4 Probabilistic failure analysis using Subset Simulation
    7.4.1 Hypothesis testing
    7.4.2 Bayesian analysis
    7.4.3 Integration of probabilistic failure analysis with Subset Simulation
7.5 Spreadsheet implementation of MCS-based reliability analysis and design
    7.5.1 Deterministic modeling
    7.5.2 Uncertainty modeling
    7.5.3 Uncertainty propagation
7.6 Illustrative example I: Drilled shaft design
    7.6.1 Deterministic model worksheet
    7.6.2 Uncertainty model worksheet
    7.6.3 Subset Simulation and RBD Add-In
    7.6.4 Determination of feasible designs
    7.6.5 Results comparison
    7.6.6 Effects of the driving variable
7.7 Illustrative example II: James Bay Dike design scenario
    7.7.1 Subset Simulation results
    7.7.2 Hypothesis test results
    7.7.3 Bayesian analysis results
7.8 Summary and concluding remarks

Acknowledgment
List of symbols
References

Part III Design

8 Lrfd calibration of simple limit state functions in geotechnical soil-structure design
Richard J. Bathur St

8.1 Introduction
8.2 Preliminaries
8.3 Bias value distributions
8.4 Calculation of β,ΥQ,and φ
    8.4.1 Generation of bias values
    8.4.2 Selection of load factor
    8.4.3 Selection of target reliability index
    8.4.4 Calculation of φ
          8.4.4.1 MC simulation
          8.4.4.2 Closed-form solutions
8.5 Example
          8.5.1 General
          8.5.2 Load data
          8.5.3 Pullout (resistance) data
          8.5.4 Calibration
                8.5.4.1 Resistance factor using MC simulation
                8.5.4.2 Resistance factor using closed-form solution

8.6 Additional considerations
8.7 Conclusions
References

9 Reliability-based design: Practical procedures, geotechnical examples,and insights
Bak-Kong Low

9.1 Introduction
    9.1.1 Three spreadsheet FORM procedures and intuitive dispersion ellipsoid perspective
9.2 Example of reliability-based shallow foundation design
     9.2.1 RBD compared with EC7 or LRFD design, and complementary roles of RBD to EC7 and LRFD design
9.3 SORM analysis on the foundation of FORM results for a rock slope
    9.3.1 Constrained optimizational FORM spreadsheet approach with respect to the u vector
    9.3.2 Positive reliability index only if the mean-value point is in the safe domain
9.4 Probabilistic analyses of a slope failure in San Francisco Bay mud
9.5 Reliability analysis of a Norwegian slope accounting for spatial autocorrelation
9.6 System FORM reliability analysis of a soil slope with two equally likely failure modes
9.7 Multicriteria RBD of a laterally loaded pile in spatially autocorrelated clay
    9.7.1 Illustrative example of multicriteria RBD of a laterally loaded pile
9.8 FORM design of an anchored sheet pile wall
9.9 Reliability analysis of roof wedges and rockbolt forces in tunnels
9.10 Probabilistic settlement analysis of a Hong Kong trial embankment on soft clay
    9.10.1 LSS and performance functions g(x) pertaining to magnitude and rate of soft clay settlement
    9.10.2 Distinguishing positive and negative reliability indices
    9.10.3 Reliability analysis for different limiting state surfaces
    9.10.4 Obtaining probability of failure (Pf) and CDF from β indices
    9.10.5 Obtaining PDF curves from β index
9.11 Coupling of stand-alone deterministic program and spreadsheet-automated reliability procedures via response surface or similar methods
9.12 Summary and conclusions
References

10 Managing risk and achieving reliable geotechnical designs using eurocode 7
 Trevor l.l.orr

10.1 Introduction
10.2 Geotechnical complexity and risk
     10.2.1 Factors affecting complexity
     10.2.2 Levels of risk and Geotechnical Categories
     10.2.3 Risks due to adverse water pressures
     10.2.4 Geotechnical investigations and geotechnical risks
10.3 Reliability requirements in designs to Eurocode 7
     10.3.1 Basic requirement
     10.3.2 Measures to achieve reliable designs
     10.3.3 Design assumptions for reliable designs
10.4 Verification of designs to Eurocode 7
     10.4.1 Limit state design method
     10.4.2 Verification by use of calculations
            10.4.2.1 Design equations and their components
            10.4.2.2 Design geometrical data
            10.4.2.3 Design actions
            10.4.2.4 Design geotechnical parameters
            10.4.2.5 Design effects of actions and design resistances
     10.4.3 Characteristic parameter values
            10.4.3.1 Definition and selection of characteristic values
            10.4.3.2 Aleatory variability and epistemic uncertainty
            10.4.3.3 Selection of aleatory characteristic parameter values
            10.4.3.4 Example 10.1: Selection of characteristic parameter values
            10.4.3.5 Characteristic pile compressive resistances  
     10.4.4 Partial factors, safety levels and reliability 419
            10.4.4.1 Types of ultimate limit state and recommended partial factor values
            10.4.4.2 Example 10.2: Determination of the design soil resistance on walls against uplift
            10.4.4.3 Example 10.3: Design of a basement against uplift

10.5 Reliability levels
     10.5.1 Partial factors, uncertainty, calibration, and target reliability
     10.5.2 Partial factors in spread and pile foundations designs
     10.5.3 Reliability differentiation
10.6 Conclusions
Acknowledgments
References

Part IV risk and decision

11 Practical risk assessment for embankments, dams, and slopes
Luis aLtarejos-García, Fracisco Silva-Tulla,ignacio Escuder-Bueno, Adrián Moas-Torres

11.1 Introduction
11.2 Estimation of conditional probability as a function of safety factor
     11.2.1 FS versus p(f) charts for slope instability and soil transport
     11.2.2 Example of risk assessment for an earth dam based on the empirical FS versus p(f) charts
            11.2.2.1 Estimation of failure probabilities versus peak pool elevation: Example from engineering practice
            11.2.2.2 Estimation of peak pool elevation annual exceedance probabilities
            11.2.2.3 Estimation of potential loss of life versus peak pool elevation at time of failure
            11.2.2.4 Comparison of results with risk evaluation guidelines
11.3 Role of fragility curves to evaluate the uncertainty in probability estimates
     11.3.1 Concept of uncertainty
     11.3.2 Concept of fragility curves
     11.3.3 Role of fragility curves in risk analysis
11.4 Mathematical roots and numerical estimation of fragility curves
     11.4.1 Introduction
     11.4.2 Conditional probability of failure versus FS
     11.4.3 Building fragility curves
     11.4.4 Example of fragility analysis for stability failure mode of an earth dam
11.5 From fragility curves to annualized probability of failure commonly used in risk analysis
11.6 Summary of main points
Acknowledgments
List of main symbols and acronyms
References

12 evolution of geotechnical risk analysis in north american practice
Gregory B. Baecher and John T. Christian

12.1 Introduction
12.2 Beginnings
12.3 Geotechnical reliability (1971–1996)
     12.3.1 Probabilistic veneer on deterministic models
     12.3.2 Variability of soil-engineering properties
     12.3.3 Slope stability analysis
     12.3.4 Lumped versus distributed parameter models
     12.3.5 Aleatory versus epistemic uncertainty
12.4 Mining engineering (1969–1980)
12.5 Offshore reliability (1974–1990)
12.6 Environmental remediation (1980–1995)
12.7 Dam safety (1986–ongoing)
12.8 Systems risk assessment (2005–ongoing)
     12.8.1 New Orleans
     12.8.2 California delta
     12.8.3 Risk registers
12.9 Emerging approaches: System simulation, stress testing, and scenario appraisals
     12.9.1 Systems simulation methods
     12.9.2 Stress testing and scenario analysis
     12.9.3 Dynamic risk analysis and management
12.10 Ten unresolved questions
12.11 Concluding thoughts
Acknowledgments
References

13 assessing the value of information to design site investigation and construction quality assurance programs
Robert B.giLberT and Mahdiha Bibi

13.1 Introduction
13.2 Value of information framework
     13.2.1 Decision analysis
     13.2.2 Illustrative example: Remediation of contaminated lagoon

13.3 Insights from Bayes’ theorem
     13.3.1 Prior probabilities
     13.3.2 Likelihood functions
     13.3.3 Illustrative example: Design of pile foundation
13.4 Implementation of value of information assessment
     13.4.1 Analytical methods
     13.4.2 Illustrative example: Design quality control program for compacted fill
     13.4.3 Numerical methods
     13.4.4 Illustrative example: Pile foundation load tests
13.5 Case-history applications
     13.5.1 Site investigation for foundation design
     13.5.2 Remedial investigation for a contaminated site
     13.5.3 Exploration program for resources
     13.5.4 QA/QC testing
13.6 Summary
Acknowledgments
References

14 verification of geotechnical reliability using load tests and integrity tests
 Limin Zhang

14.1 Introduction
14.2 Within-site variability of pile capacity
14.3 Updating pile capacity with proof load tests
     14.3.1 Proof load tests that pass
     14.3.2 Proof load tests that do not pass
     14.3.3 Proof load tests conducted to failure
     14.3.4 Multiple types of tests
14.4 Updating pile capacity with integrity tests
     14.4.1 Reliability updating based on integrity tests
     14.4.2 Updating occurrence probability of toe debris
     14.4.3 Updating mean thickness of toe debris
     14.4.4 Cases of test outcome
14.5 Reliability of piles verified by proof load tests
     14.5.1 Calculation of reliability index
     14.5.2 Example: Design based on SPT and verified by proof load tests
     14.5.3 Accuracy effect of design methods
14.6 Reliability of piles verified by integrity tests
     14.6.1 Worked example
     14.6.2 Survey of toe debris
     14.6.3 Updating the priors based on interface coring tests
     14.6.4 Updating reliability of piles based on interface coring tests
14.7 Summary
Acknowledgment
List of symbols
References

Part V Spatial variability

15 application of the subset simulation approach to spatially varying soils
Ashra Fahm ed and abduL-hamid Soubra

15.1 Introduction
15.2 Karhunen–Loève expansion methodology for the discretization of a random field
15.3 Brief overview of the subset simulation approach
15.4 Method of computation of the failure probability by the SSapproach in the case of a spatially varying soil property
15.5 Example applications
     15.5.1 Example 1: Generation of a random field by K–L expansion
     15.5.2 Example 2: Computation of the failure probability by SS approach in the case of random variables
     15.5.3 Example 3: Computation of the failure probability by an SS approach in the case of random fields
15.6 Conclusion
     Appendix 15A: Modified M–H algorithm List of symbols
     References
     Index