Copyright. Shadi Sam Najjar

Transcription

1 Copyright by Shadi Sam Najjar 2005

2 The Dissertation Committee for Shadi Sam Najjar Certifies that this is the approved version of the following dissertation: The Importance of Lower-Bound Capacities in Geotechnical Reliability Assessments Committee: Robert B. Gilbert, Supervisor Roy E. Olson Jorge G. Zornberg John L. Tassoulas Eric B. Becker

3 The Importance of Lower-Bound Capacities in Geotechnical Reliability Assessments by Shadi Sam Najjar, B.E., M.E. Dissertation Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy The University of Texas at Austin May 2005

4 Dedication This dissertation is dedicated to my family Sam, Jackline, Rula, and Fadi Najjar and to my faithful friend Asli Kurtulus

5 Acknowledgements I would like to thank my advisor, Dr. Robert Gilbert, for his valuable guidance and support throughout the phases of this research. I feel privileged to have been mentored by Dr. Gilbert for the last four years, and I will forever be grateful for his role in making me what I am today. I am also sincerely grateful to Professor Roy Olson for serving on my doctoral committee and for taking valuable time out of his schedule to read my dissertation and provide valuable feedback on various technical issues. I would also like to thank the other members of my committee for reviewing the dissertation. Throughout the last four years, I have been surrounded by friends and classmates who provided me with emotional and technical support and gave me the strength to continue. I would like to thank them all and wish them the best of luck in their endeavors. I want to specifically thank my best friend, Miss Asli Kurtulus, for standing by me in stressful times and for her true friendship and unconditional love. Finally, I want to thank my family for believing in me at all times. Your prayers and unlimited love have guided me in this journey to the shore of safety. I thank god every day for having you as my father, mother, sister, and brother. v

6 The Importance of Lower-Bound Capacities in Geotechnical Reliability Assessments Publication No. Shadi Sam Najjar, Ph.D. The University of Texas at Austin, 2005 Supervisor: Robert B. Gilbert The ability to manage risk in geotechnical engineering relies on a realistic assessment of the probability of failure for designs. Most reliability analyses focus on the mean and variance and an assumed, mathematically convenient distribution to model the left-hand tail of the distribution for capacity. However, the reliability of a geotechnical engineering system is governed by a physical constraint on the smallest available capacity. This lower-bound capacity is usually neglected in conventional reliability analyses. In this study, databases of load tests conducted on offshore and onshore deep foundations are analyzed to provide evidence for the existence of a lower-bound capacity that can be calculated using site-specific soil properties and information about the geometry of the foundation. Next, realistic probability distributions that can accommodate a lower-bound capacity are proposed and used to relate reliability to the lower-bound capacity. Multiple Load and Resistance Factor Design (LRFD) design- vi

7 checking formats that include information on the lower-bound capacity in addition to the conventional design information are then introduced. Finally, practical approaches are presented for updating information about lower-bound capacities using installation data, proof-load data, and historical performance of foundations under load. Databases with deep foundations show clear evidence of the existence of a lowerbound capacity that typically ranges from 0.4 to 0.8 of the predicted capacity in both cohesive and cohesionless soils. Results from reliability analyses indicate that the presence of a lower-bound capacity can have a significant effect on increasing the reliability of a deep foundation. The effect of the lower-bound capacity increases as the coefficient of variation for the capacity increases and as the target reliability index increases. This result indicates that reliability-based design codes need to incorporate information about lower-bound capacities. Incorporation of a lower-bound capacity into design is expected to provide a more realistic quantification of reliability for decisionmaking purposes and therefore a more rational basis for design. vii

8 Table of Contents List of Tables...xv List of Figures... xxi Chapter 1. Introduction Background Objective and Approach of Research Organization of Dissertation...5 Chapter 2. Lower-Bound Axial Capacity for Driven Piles Introduction Axial Capacity of Driven Piles in Cohesive Soils Lower Bound Capacities of Driven Piles in Cohesive Soils Lower Bound Capacity Based on Undrained Remolded Shear Strength Definition and Measurement of Undrained Remolded Shear Strength Correlation Between Undrained Remolded Strength and Liquidity Index Evidence of a Lower Bound Axial Capacity Lower Bound Capacity Based on Residual Interface Shear Resistance Lower Bound Horizontal Effective Stress at Failure Lower Bound Interface Friction Angle Evidence of a Lower Bound Axial Capacity Axial Capacity of Driven Piles in Cohesionless Soils Lower Bound Capacities for Driven Piles in Cohesionless Soils Summary...36 Chapter 3. Predicted and Lower-Bound Axial Capacities of Suction Caissons in Normally Consolidated Clays Introduction...37 viii

9 3.2 Database of Pullout Tests of Suction Caissons in NC Clays Uplift Capacity of Suction Caissons in Normally Consolidated Clay Calculation of Side Resistance Calculation of the Net Reverse End Bearing Analysis of Load Test Results Evidence of a Lower-Bound Axial Capacity Summary...51 Chapter 4. Models for Lower-Bound Capacity Introduction Conventional Probability Distributions for Capacity Normal Distribution Lognormal Distribution Bounded Probability Distributions for Capacity Uniform Distribution Triangular Distribution Shifted Exponential Distribution Shifted Lognormal Distribution Beta Distribution Truncated Probability Distributions for Capacity Mixed Probability Distributions for Capacity Comparison Between Probability Distributions for Capacity General Distributions Based on Hermite Polynomials Hermite Polynomials Procedure for Fitting Hermite Polynomials to Probability Distributions Fitting Hermite Polynomials to Different Probability Distributions Bounded Distributions Truncated Distributions Mixed Distribution ix

10 Remarks on Fitting Hermite Polynomials to Different Probability Distributions Summary Chapter 5. Modeling Uncertainty in the Capacity using Databases of Load Tests Introduction Procedure for Modeling Distributions for Capacity using Databases Statistical Evidence of the Existance of Lower-Bound Capacities Likelihood Functions and Parameters for Different Probability Distributions for Capacity Bounded, Truncated, and Mixed Distributions Beta Distribution Truncated Lognormal Distribution Mixed Lognormal Distribution Mixed Seventh Order Hermite Polynomials Modeling Uncertainty in the Capacity for Driven Piles Piles in Cohesive Soils Piles in Siliceous, Cohesionless Soils Modeling Uncertainty in the Capacity of Suction Caissons in Normally Consolidated Clays Summary Chapter 6. Reliability Assessments in the Presence of a Lower-Bound Capacity Introduction General Assessment of the Probability of Failure First Order Second Moment Reliability Method (FOSM) First and Second Order Reliability Methods (FORM and SORM) Transformation of g(r,s) Approximation of g(u R,u S ) Evaluation of the Probability of Failure Curvature vs. Point-Fitting SORM x

11 Point-Fitting SORM (Zhao and Ono 1999b) Example of a Reliability-Based Analysis Using SORM Numerical Integration Assessment of Reliability for Distributions with a Lower Bound First Order Second Moment Reliability Method (FOSM) First and Second Order Reliability Methods (FORM and SORM) First Order Reliability Method Second Order Reliability Method Summary Chapter 7. Effect of Lower-Bound Capacity on the Reliability Introduction Effect of Lower-Bound Capacity Effect of the Probability Distribution for Capacity Simple Bilinear Model for Assessing Reliability Summary Chapter 8. Uncertainty in the Lower-Bound Capacity Introduction Modeling Uncertainty in Lower-Bound Capacities Effect of Uncertainty in Lower-Bound Capacities on the Reliability Effect of the Degree of Uncertainty in the Lower-Bound Capacity Effect of the Relative Uncertainty in the Load and the Capacity Effect of the Median Factor of Safety Effect of the Probability Distribution of the Lower-Bound Capacity Summary Chapter 9. Incorporating Lower-Bound Capacities in LRFD Codes for Deep Foundations Introduction xi

12 9.2 Incorporating Lower-Bound Capacities into LRFD Adjusted Resistance Factor for Lower-Bound Capacity Added Design Checking Equation for Lower-Bound Capacity Illustrative Examples of Lower-Bound Capacity in LRFD Design of a Suction Caisson Foundation for an Offshore Facility Design of Driven Piles for a Foundation of a Bridge Summary Chapter 10. Approaches for Updating Lower-Bound Capacities Introduction Updating Lower-Bound Capacities of Individual Foundations Static Proof Load Tests Dynamic Tests Historical Performance of Foundations Under Load Uncertainties in Updated Lower-Bound Capacities Examples of Updating the Reliability of Foundations Updating with Static Proof Load Tests Updating with Dynamic Load Tests Updating the Reliability of Foundations at a Site with Load Tests Bayesian Updating on the Lower-Bound Capacity Bayesian Updating Example Discussion on the Design of Proof-Load Test Programs Summary Chapter 11. Conclusions Evidence of a Lower-Bound Capacity for Deep Foundations Conclusions Recommendation for Future Work Realistic Probability Distributions for Deep Foundations Conclusions Recommendation for Future Work Effect of the Lower-Bound capacity on the Reliability xii

13 11.4 Incorporating Lower-Bound Capacities in Design of Foundations Incorporating Lower-Bound Capacities into LRFD Updating Lower-Bound Capacities Recommendation for Future Work Appendix A Database of Load Tests on Piles in Cohesive Soils A.1 Introduction A.2 Normally Consolidated to Slightly Overconsolidated Clays A.2.1 Pile # 1: Seed and Reese (1955) A.2.2 Piles # 2 and 3: McCammon and Golder (1970) A.2.3 Piles # 4 to 7: Mansur and Focht (1956) A.2.4 Piles # 8: Hutchinson and Jensen (1968) A.2.5 Piles # 9, 10, and 11: Kirby and Roussel (1979) A.2.6 Piles # 12, 13, 14, 15, 16, 17, 18, 19: Cox, McClelland, and Verner (1979) A.2.7 Pile # 20: Conoco Tension Pile Study: Bogard and Matlock (1998) and Audibert and Hamilton (1998) A.2.8 Piles # 21 and 22: Stermac et al. (1969) A.2.9 Pile # 23: Pelletier and Doyle (1982) A.2.10 Pile # 24: Darragh and Bell (1969) A.2.11 Pile # 25 (Reference Unavailable) A.2.12 Piles # 26, 27, and 28 (Reference Confidential) A.2.13 Pile # 29 (Reference Confidential) A.2.14 Piles # 30 and 31 (Reference Confidential) A.2.15 Pile # 32: Texas Coast (Reference Unavailable) A.2.16 Pile # 33 and 34 (Reference Confidential) A.3 Overconsolidated Clays A.3.1 Pile # 35 and 36 (Reference Confidential) A.3.2 Pile # 37: Togrol (1973) A.3.3 Piles # 38 to 45: O'Neill et al. (1981) A.4 Summary xiii

14 Appendix B Database of Load Tests on Piles in Cohesionless Soils B.1 Introduction B.2 Database of Pile Load Tests B.2.1 Pile # 1 to 9: Arkansas River B.2.2 Pile # 10 to 18: Old River, LA B.2.3 Piles # 19 and 20: Beech, R. (Ontario) B.2.4 Piles # 21 to 25: Ogeechee, River B.2.5 Piles # 26 to 28: Helena, Ark B.2.6 Piles # 29 to 32: Kansas City B.2.7 Pile # 33: British Columbia B.2.8 Pile # 34: Florida B.2.9 Pile # 35: Japan B.2.10 Pile # 36: Michigan B.3 Summary Appendix C Database of Pullout Tests on Suction Caissons in NC Clay C.1 Introduction C.2 Database of Pullout Tests C.2.1 Tests # 1 to 7: Luke et al. (2003) C.2.2 Tests # 8 to 10: Clukey and Morrison (1993) C.2.3 Tests # 11 to 13: House and Randolph (2001) C.2.4 Tests # 14 to 18: Randolph and House (2002) C.2.5 Tests # 19 to 22: Clukey et al. (2003) C.2.6 Tests # 23 to 25: Cho et al. (2003) C.3 Summary Appendix D Matlab Code for SORM Algorithm D.1 Main Code D.2 Surfaces References Vita xiv

15 List of Tables Table 2.1. Table 2.2. Database of load tests for steel pipe piles in cohesive soils...11 Summary of measured, predicted, and lower-bound capacities for piles in normally consolidated to slightly overconsolidated clays...19 Table 2.3. Summary of measured, predicted, and lower-bound capacities for piles in overconsolidated clays Table 2.4. Summary of measured, predicted, and lower-bound capacities for piles in clay...27 Table 2.5. API method for predicting the capacity of piles in cohesionless soils...29 Table 2.6. Table 2.7. Relative density based on standard penetration blow counts...30 Database of load tests for steel pipe piles in siliceous, cohesionless soils...30 Table 2.8. Parameters for predicting the lower-bound capacity of piles in siliceous, cohesionless soils...34 Table 2.9. Summary of measured, predicted, and lower-bound capacities for piles in cohesionless, siliceous soils...35 Table 3.1. Table 3.2. Database of pullout tests on suction caissons (general description).39 Database of pullout tests on suction caissons (soil properties and measured loads)...40 Table 3.3. Biases and uncertainties in prediction models for suction caissons (α=1, N=9)...49 Table 3.4. Calibration of α and N (corrected undrained shear strength and excluding tests with setup = 0)...49 xv

16 Table 3.5. Summary of measured, predicted, and lower-bound capacities for suction caissons in normally consolidated clays...52 Table 4.1. Table 4.2. Variations of the beta distribution...69 Difference between the means of truncated and non-truncated lognormal distributions (in percent, %)...79 Table 4.3. Difference between the means of truncated and non-truncated normal distributions (in percent, %)...79 Table 4.4. Table 4.5. Table 4.6. Coefficients of variation of truncated lognormal distributions...80 Coefficients of variation of truncated normal distributions...80 Difference between the means of the mixed lognormal and conventional lognormal distributions (in percent, %)...85 Table 4.7. Table 5.1. Coefficients of variation of mixed lognormal distributions...85 Summary of measured, predicted, and lower-bound capacities for driven piles in cohesive soils Table 5.2. Parameters of bounded, truncated, and mixed distributions (piles in clays) Table 5.3. Table 5.4. Parameters of mixed Hermite Polynomials (piles in clays) Summary of measured, predicted, and lower-bound capacities for driven piles in cohesionless soils Table 5.5. Parameters of bounded, truncated, and mixed distributions (piles in sands) Table 5.6. Parameters of mixed Hermite Polynomials (piles in sands) Table 5.7. Summary of measured, predicted, and lower-bound capacities for suction caissons in normally consolidated clays xvi

17 Table 5.8. Parameters of bounded, truncated, and mixed distributions (suction caissons) Table 5.9. Parameters of truncated and mixed Hermite Polynomials (suction caissons) Table 6.1. Iterative results for the SORM example Table Accuracy of analysis using GRLWEAP (Rausche et al., 2004) Table Accuracy of analysis using CAPWAP (Likins and Rausch, 2004) 224 Table A.1. Summary of soil properties at the site near San Francisco bay Table A.2. Table A.3. Results from load tests conducted at different setup times Soil profile for Pile # 2 (length = 100 feet) Table A.4. Soil Profile for Pile # 3 (length = 158 feet) Table A.5. Measured capacities and properties of the piles Table A.6. Soil profile for Pile # Table A.7. Soil profile for Pile # Table A.8. Soil profile for Piles # 6 and Table A.9. Soil profile for Pile # Table A.10. Soil profile at the Hamilton Base site (Piles # 9, 10, and 11) Table A.11. Properties of the piles and soils at the Empire site Table A.12. Summary of the results of the load tests at Empire Table A.13. Soil profile for Piles # 12 and Table A.14. Soil profile for Piles # 14 and Table A.15. Soil profile for Piles # 16 and Table A.16. Soil profile for Piles # 18 and Table A.17. Soil profile for Pile # xvii

18 Table A.18. Results of load tests conducted at different setup times at the West Delta site Table A.19. Soil profile for Piles # 21 and Table A.20. Soil profile for Piles # Table A.21. Soil profile for Piles # Table A.22. Soil profile for Pile 25, Atchafalaya Table A.23. Soil profile for Piles 26 and 27, Venice Table A.24. Soil profile for Pile 28, Venice Table A.25. Soil profile for Pile 29, Harvey, Louisiana Table A.26. Soil profile for Pile 30, Eugine Island Table A.27. Soil profile for Pile 31, Eugine Island Table A.28. Soil profile for Pile 32, Aquatic Park Table A.29. Soil profile for Pile 33, Eugine Island Table A.30. Soil profile for Pile 34, Eugine Island Table A.31. Soil profile for Pile 35, Port Arther, Texas Table A.32. Soil profile for Pile 36, Port Arther, Texas Table A.33. Soil profile for Pile # 37, Izmir, Turkey Table A.34. Soil profile for Piles 38-45, Houston, Texas Table A.35. Summary of measured, predicted, and lower-bound capacities Table B.1. Summary of properties of the soil at the sites of Piles # 1 and Table B.2. Summary of properties of the soil at the sites of Piles # 3 and Table B.3. Summary of properties of the soil at the sites of Piles # 5 to Table B.4. Summary of properties of the soil at the sites of Piles # 10 to Table B.5. Summary of properties of the soil at the sites of Pile # xviii

19 Table B.6. Summary of properties of the soil at the sites of Piles # 15 and Table B.7. Summary of properties of the soil at the sites of Pile # Table B.8. Summary of properties of the soil at the sites of Pile # Table B.9. Summary of properties of the soil at the sites of Pile # Table B.10. Summary of properties of the soil at the sites of Pile # Table B.11. Summary of properties of the soil at the sites of Pile # Table B.12. Summary of properties of the soil at the sites of Pile # Table B.13. Summary of properties of the soil at the sites of Pile # Table B.14. Summary of properties of the soil at the sites of Piles # 24 and Table B.15. Summary of properties of the soil at the sites of Pile # Table B.16. Summary of properties of the soil at the sites of Pile # Table B.17. Summary of properties of the soil at the sites of Pile # Table B.18. Summary of properties of the soil at the sites of Piles # 29 and Table B.19. Summary of properties of the soil at the sites of Piles # 31 and Table B.20. Summary of properties of the soil at the sites of Piles # Table B.21. Summary of properties of the soil at the sites of Piles # Table B.22. Summary of properties of the soil at the sites of Piles # Table B.23. Summary of properties of the soil at the sites of Piles # Table B.24. Summary of measured, predicted, and lower-bound capacities Table C.1. Test program and soil properties for the University of Texas at Austin study xix

20 Table C.2. Results of the tests (N = 7.6 (Vented) and N = 9.0 (Sealed)) Table C.3. Results of the centrifuge tests (assume α = 1.0) Table C.4. Test Program for the House and Randolph (2001) study Table C.5. Table C.6. Table C.7. Test program for the Randolph and House (2002) study Test program for the Clukey et al. (2003) study Test program for the Cho et al. (2003) study Table C.8. Summary of measured, predicted, and lower-bound capacities xx

21 List of Figures Figure 1.1. Reliability analysis using conventional and bounded probability distributions for capacity...2 Figure 2.1. Measured versus calculated capacities for axially loaded, driven steel pipe piles in cohesive soils (uncorrected for setup or method of shear strength measurement)...10 Figure 2.2. Measured versus calculated capacities for axially loaded, driven steel pipe piles in cohesive soils (corrected for setup and method of undrained strength measurement)...13 Figure 2.3. Correlation between the undrained remolded strength and the liquidity index...17 Figure 2.4. Models for calculating predicted and lower-bound capacities of axially loaded, driven steel pipe piles in normally consolidated to slightly overconsolidated clays...18 Figure 2.5. Measured versus lower-bound capacities for axially loaded, driven steel pipe piles in normally consolidated to slightly overconsolidated clay (uncorrected measured capacities)...18 Figure 2.6. Comparison between the at-rest earth pressure coefficient, k 0, and the coefficient k, which defines the fully-equalized horizontal effective stress acting on the walls of instrumented piles in overconsolidated clays...22 Figure 2.7. Correlation between the residual interface friction angle with the plasticity index (from Ramsey et al., 1998)...23 xxi

22 Figure 2.8. Models for calculating predicted (mean) and lower-bound capacities of axially loaded, driven steel pipe piles in overconsolidated clays...24 Figure 2.9. Measured versus lower-bound capacities for axially loaded, driven steel pipe piles in overconsolidated clay (uncorrected measured capacities)...24 Figure Measured versus lower-bound capacities for axially loaded, driven steel pipe piles in all clays (uncorrected measured capacities)...26 Figure Measured versus calculated capacities for axially loaded, driven steel pipe piles in siliceous cohesionless soils...31 Figure Models for calculating predicted (mean) and lower-bound capacities of axially loaded, driven steel pipe piles in cohesionless, siliceous soils...33 Figure Measured versus calculated lower-bound capacities for axially loaded, driven steel pipe piles in siliceous cohesionless soils...34 Figure 3.1. Foundations for anchoring offshore facilities...39 Figure 3.2. Comparison between measured and predicted capacities for 25 suction caissons (uncorrected shear strength, α = 1.0, N = 9)...45 Figure 3.3. Comparison between measured and predicted capacities for 25 suction caissons (corrected shear strength; DSS, α = 1.0, N = 9)...47 Figure 3.4. Evidence of a lower-bound capacity for 25 suction caissons...51 Figure 4.1. Normal probability distribution (r median = µ R = 1000, δ R = 0.3, 0.4, 0.5)...55 Figure 4.2. Lognormal probability distribution (r median = 1000, δ R = 0.3, 0.4, 0.5)...58 xxii

23 Figure 4.3. Uniform probability density function...60 Figure 4.4. Uniform probability distribution (r median = 1000, δ R = 0.3,0.4,0.5)...61 Figure 4.5. Triangular probability density function...62 Figure 4.6. Triangular distribution (µ R = 1000, δ R = 0.3,0.4,0.5)...64 Figure 4.7. Shifted exponential distribution (µ R = 1000; δ R = 0.3,0.4,0.5)...65 Figure 4.8. Shifted lognormal distribution ( r > = 1000; δ R = 0.3,0.4,0.5, median r r LB = 400)...67 Figure 4.9. Beta distribution (µ R = 1000; δ R = 0.3,0.4,0.5, r LB = 400, r UB = 4000)...70 Figure Beta distribution (µ R = 1000; δ R = 0.3,0.4,0.5, r LB = 600, r UB = 4000)...71 Figure Truncated lognormal distribution (r median = 1000, δ R = 0.3,0.4,0.5, r LB = 400)...74 Figure Truncated lognormal distribution (r median = 1000, δ R = 0.3,0.4,0.5, r LB = 600)...75 Figure Truncated normal distribution (r median = 1000, δ R = 0.3,0.4,0.5, r LB = 400)...76 Figure Truncated normal distribution (r median = 1000, δ R = 0.3,0.4,0.5, r LB = 600)...77 Figure Mixed distribution for modeling capacity...81 Figure Mixed lognormal distribution (r median = 1000, δ R = 0.3,0.4,0.5, r LB = 400)...83 Figure Mixed lognormal distribution (r median = 1000, δ R = 0.3,0.4,0.5, r LB = 600)...84 r LB xxiii

24 Figure Comparison between different types of probability distribution (r median = 1000, δ R = 0.5, r LB = 400)...87 Figure Comparison between different types of probability distribution (r median or r mean = 1000, δ R = 0.5, r LB = 600)...88 Figure Hermite polynomials, order zero through three...90 Figure Hermite polynomials, order four through seven...91 Figure Hermite Polynomial approximations for tails of uniform distributions (µ R = 1000, r LB / µ R = 0.4 and 0.6)...96 Figure Hermite Polynomial approximations for tails of triangular distributions (µ R = 1000, r LB / µ R = 0.4 and 0.6)...98 Figure Hermite Polynomial approximations shifted exponential distributions...99 Figure Hermite Polynomial approximations for beta distributions (δ R = 0.3) Figure Hermite Polynomial approximations for beta distributions (δ R = 0.5) Figure Hermite Polynomial approximations for truncated normal distributions (δ R = 0.3) Figure Hermite Polynomial approximations for truncated normal distributions (δ R = 0.5) Figure Hermite Polynomial approximations for truncated lognormal distributions (δ R = 0.3) Figure Hermite Polynomial approximations for truncated lognormal distributions (δ R = 0.5) xxiv

25 Figure Mixed distribution functions (r median = 1000, δ R = 0.4, r LB = 400 and 600) Figure Hermite Polynomial approximations for mixed distributions (r median = 1000, δ R = 0.4) Figure 5.1. Comparison between different probability distributions for piles in cohesive soil (r predicted = 200 kips, r LB = 100 kips) Figure 5.2. Comparison between different probability distributions for piles in cohesive soil (r predicted = 200 kips, r LB = 120 kips) Figure 5.3. Comparison between different probability distributions for piles in cohesionless soils (r predicted = 200 kips, r LB = 80 kips) Figure 5.4. Comparison between different probability distributions for piles in cohesionless soils (r predicted = 200 kips, r LB = 100 kips) Figure 5.5. Comparison between different probability distributions for suction caissons in normally consolidated clays (r predicted = 200 kips, r LB = 100 kips) Figure 5.6. Comparison between different probability distributions for suction caissons in normally consolidated clays (r predicted = 200 kips, r LB = 120 kips) Figure 6.1. Performance function in the original and reduced space for the case of 2 random variables (FOSM application) Figure 6.2. Performance functions in the original and reduced space (FORM and SORM) Figure 6.3. Approximations for the performance function g(u R,u S ) by first and second order surfaces g (u R,u S ) using FORM and SORM Figure 6.4. Application of point-fitting SORM (Zhao and Ono 1999b) xxv

26 Figure 6.5. Accuracy of FOSM (truncated distributions, δ S = 0.2, δ R = 0.4) Figure 6.6. Accuracy of FOSM method (mixed lognormal distribution, δ S = 0.2, δ R = 0.4) Figure 6.7. Accuracy of FOSM method (uniform and beta distributions, δ S = 0.2) Figure 6.8. Accuracy of FORM (truncated normal distributions, δ R = 0.4, δ S = 0.2) Figure 6.9. Accuracy of FORM (truncated lognormal distributions, δ R = 0.4, δ S = 0.2) Figure Accuracy of FORM (uniform distribution, δ S = 0.2) Figure Accuracy of FORM (beta distribution, δ R = 0.4, δ S = 0.2) Figure Accuracy of FORM (mixed lognormal distribution, δ R = 0.4, δ S = 0.2) Figure Variation of the principal curvature with the lower-bound capacity for bounded capacity distributions (FS Median = 3.0, δ S = 0.2) Figure Accuracy of FORM as a function of the principal curvatures of the performance function (FS Median = 3.0, δ S = 0.2) Figure Accuracy of FORM as a function of the principal curvatures of the performance function and the reliability index (δ S = 0.2) Figure Accuracy of FORM (truncated distributions, δ R = 0.4, δ S = 0.4) Figure Magnitude of the principal curvatures of the performance function for truncated distributions (FS Median = 4.0, δ S = 0.4, δ R = 0.4) Figure Sensitivity of the maximum principal curvature and accuracy of FORM to the uncertainty in the load and capacity (truncated lognormal distribution) xxvi

27 Figure Accuracy of SORM (truncated normal and lognormal distributions, δ R =0.4, δ S =0.2) Figure Accuracy of SORM (uniform and beta distributions, δ S = 0.2) Figure Accuracy of SORM (mixed lognormal distribution, δ S = 0.2) Figure Accuracy of SORM (all distributions, δ S = 0.2) Figure 7.1. Reliability analysis using a mixed lognormal distribution for capacity Figure 7.2. Effect of lower-bound capacity on the reliability (δ R = 0.4 and δ S = 0.2) Figure 7.3. Required factors of safety for different lower-bound values (δ R = 0.4, and δ S = 0.2) Figure 7.4. Effect of median factor of safety and lower-bound capacity on reliability (δ R = 0.4, and δ S = 0.2) Figure 7.5. Effect of uncertainty in the load on the increase in the reliability (δ R = 0.4) Figure 7.6. Effect of uncertainty in the capacity on the increase in the reliability (FS median = 2.5, δ S = 0.2) Figure 7.7. Reliability calculations using different distributions for capacity (δ S = 0.2, δ R = 0.4) Figure 7.8. Reliability using different distributions for capacity (δ S = 0.1, δ R = 0.5) Figure 7.9. Reliability using different distributions for capacity (δ S = 0.4, δ R = 0.4) Figure Proposed bilinear model for evaluating the reliability Figure Reliability approximations with the proposed bilinear model xxvii

28 Figure 8.1. Effect of uncertainty in the lower-bound capacity on the reliability (FS median = 3.0, δ R = 0.4, and δ S = 0.2) Figure 8.2. Effect of uncertainty in the lower-bound capacity on the reliability (FS median = 3.0, δ R = 0.5, and δ S = 0.1) Figure 8.3. Distribution for capacity with uncertain lower bound (δ R = 0.4, r LB,median / r median = 0.4) Figure 8.4. Probability distribution for the lower-bound capacity (r LB,median / r median = 0.4) Figure 8.5. Distribution for capacity with uncertain lower bound (δ R = 0.4, r LB,median / r median = 0.6) Figure 8.6. Probability distribution for the lower-bound capacity (r LB,median / r median = 0.6) Figure 8.7. Effect of uncertainty in the lower-bound capacity on the reliability (r LB,median / r median = 0.6, FS median = 3.0, δ R = 0.5, and δ S = 0.1) Figure 8.8. Effect of median factor of safety on the degree in which uncertainty in the lower-bound capacity affects the reliability (δ R = 0.4, δ S = 0.2) Figure 8.9. Effect of the probability distribution of the lower-bound capacity on the reliability (FS median = 3.0, δ R = 0.4, and δ S = 0.2) Figure Distribution for capacity with uncertain lower bounds (δ LB = 0.3, δ R = 0.4, δ S = 0.2) Figure 9.1. Variation of the increase in the nominal resistance factor with the lower- bound capacity (c.o.v. Load = δ S = 0.15) Figure 9.2. Variation of the lower-bound resistance factor to account for a lower-bound capacity (c.o.v. Load = δ S = 0.15) xxviii

29 Figure 9.3. Variation of the nominal resistance factor and lower-bound resistance factor with the lower-bound capacity (δ S = 0.15, δ R = 0.4) Figure 9.4. Soil profile for case study example (from Goble 1996) Figure Estimates of proof loads for piles using End of Driving (EOD) and Beginning of Restrike (BOR) data Figure Effect of proof loads on the reliability (δ R = 0.4, δ S = 0.15) Figure Effect of uncertainty in the proof load (FS median = 2, δ R = 0.4, δ S = 0.15) Figure Effect of uncertainty in proof loads on the design of foundations (δ R = 0.4, δ S = 0.15) Figure Prior and updated distributions for the lower-bound capacity (r proof /r median =0.6; n=30) Figure A.1. Undisturbed and remolded shear strength profiles of Bay Mud Figure A.2. Undisturbed and remolded shear strength profiles at British Columbia Figure A.3. Variation of the total stress, pore pressure, and effective stress with depth Figure A.4. Undisturbed shear strength profile at Morganza Figure A.5. Variation of undisturbed and remolded shear strengths with depth for the site in Iran Figure A.6. Variation of undisturbed and remolded shear strengths with depth for the Hamilton Base site Figure A.7. Variation of undisturbed shear strengths with depth for the Empire site xxix

30 Figure A.8. Undisturbed and remolded shear strength profiles at the West Delta site Figure A.9. Profiles of total stress, pore pressure, and effective stress at the West Delta site (Jardine and Saldivar, 1999) Figure A.10. Undisturbed undrained shear strength profile at the Toronto site Figure A.11. Undisturbed and remolded shear strength profiles of Bay Mud Figure A.12. Variation of the undisturbed undrained shear strength with depth at the Donaldsonville site Figure A.13. Undisturbed undrained shear strength profile for site in Houston..276 Figure C.1. Properties of the soil in the test tank Figure C.2. Undrained strength profile for the soil in the Clukey and Morrison (1993) study Figure C.3. Undrained strength profiles for the soil in House and Randolph (2001) study Figure C.4. Undrained strength profiles for the soil in Randolph and House (2002) study xxx

31 Chapter 1. Introduction 1.1 BACKGROUND The ability to manage risk in geotechnical engineering relies on a realistic assessment of the probability of failure for designs. A realistic assessment is important in comparing absolute risks against what is considered to be acceptable, and it is important in performing cost-benefit analyses to manage risk. However, there is a general belief that calculated probabilities of failure are not realistic and are primarily useful as nominal or relative values for comparison purposes (Allen 1975; Simpson et al. 1981; Wu et al. 1989; Bolton 1993; Rodriguez et al. 1998). For example, in the last 30 to 40 years, observations of failure rates for offshore piles indicated that the frequencies of foundation failures are significantly lower than calculated probabilities of failure (Horsnell and Toolan 1996; Aggarwal et al. 1996(a), 1996(b); and Bea et al. 1999). The reason that calculated probabilities of failure are not realistic is related to how the tails of the probability distributions for the load and the capacity are modeled (Figure 1.1). Most analyses focus on the mean and variance and an assumed, mathematically convenient distribution (for example, normal or lognormal) to model the distribution tails. For example, in all of the reliability analyses for deep foundations, the foundation capacity is modeled using a lognormal distribution and the coefficient of variation (c.o.v.) for the capacity is relatively large, ranging from 0.3 to 1.0 (Tang 1988 and 1990; Hamilton and Murff 1992; API 1993; Horsnell and Toolan 1996; Bea et al. 1999; McVay 2000, 2002, and 2003; Kulhawy and Phoon 2002; Phoon et al. 2003; AASHTO 2004). A lognormal distribution, with a lower tail that extends to zero, does not capture the realistic possibility that there is a physical lower bound for the capacity of geotechnical systems. 1

32 Probability Density Function Lower Bound Load Capacity Load and Capacity (kn) Figure 1.1. Reliability analysis using conventional and bounded probability distributions for capacity. The existence of lower-bound capacities for geotechnical engineering systems is supported by the principals of critical state soil mechanics. These principals provide a basis for predicting the behavior of highly disturbed soils and for estimating lower-bound values for shear strength (Wroth and Wood 1978). When sheared, normally consolidated to slightly overconsolidated clays and loose sands typically tend to compress (become denser), whereas heavily overconsolidated clays and dense sands tend to dilate (become looser) as they approach failure. Ultimately, these soils will reach states at which the shear stress is a constant and there are no more volumetric strains (Atkinson 1993). These states, which are referred to as critical states, define a unique relationship between the shear stress, the normal stress, and the void ratio (Atkinson 1993). The critical state represents the final state in which it is possible to continue to shear the soil without any change in the imposed stresses and volume of the soil (Britto and Gunn 1987). 2

33 Generally, the shear strength of soils at critical state conditions can be used to estimate the minimum available shear strength (Wroth and Wood, 1978). Lower-bound estimates of the undrained shearing strength of cohesive soils can be represented by the undrained strength of the remolded soil tested at the same water content. The drained shearing strength of cohesive soils also has a lower bound that can be represented by the residual drained shearing strength. For cohesionless soils, a lower-bound estimate of the drained shearing strength can be expressed in terms of the drained critical state friction angle of the soil. Critical state or residual values of shear strength provide a basis for estimating lower-bound capacities for geotechnical engineering systems. The existence of a lower-bound capacity could help to explain why observed failure rates for offshore piles are significantly smaller than calculated probabilities of failure using conventional models. Incorporation of a lower-bound capacity into design for geotechnical structures is expected to provide a more realistic quantification of reliability for decision-making purposes and therefore a more rational basis for design. Existing literature on lower-bound capacities include the work done by Bolton (1981, 1986), Simpson et al. (1981); Fjeld (1977); and Rodriguez et al. (1998). Bolton (1981, 1986) argued that designs should be checked for the occurrence of a limit state when all the parameters are assigned their worst obtainable values and a conservative method of calculation is used. Simpson et al. (1981) presented a design method that makes use of a so called worst credible value in order to incorporate information about lower-bound capacities in design. Fjeld (1977) recognized the importance of modeling the left-hand tail of the capacity distribution to comply with physical limits in the problem under consideration. He suggested the use of truncated normal distributions in modeling these physical limits. Rodriguez et al. (1998) used truncated normal and lognormal distributions to analyze the reliability of a mooring system for an offshore 3

34 structure, and their results showed a consistent decrease in the probability of failure when information about lower-bound capacities was included. 1.2 OBJECTIVE AND APPROACH OF RESEARCH The objective of this research is to investigate the effect of having a lower-bound capacity on the reliability of geotechnical engineering systems and to provide a methodology for incorporating information about lower-bound capacities in design. These goals will be achieved through the following tasks: 1. Investigate the possibility of a lower-bound capacity using databases of load test for driven piles and suction caissons. 2. Propose realistic distributions for capacity that can accommodate a lower-bound capacity, and use these distributions to provide a realistic model of the uncertainty in the capacity of driven piles and suction caissons under axial loading. 3. Investigate the effectiveness of numerical and semi-analytical methods in assessing the probability of failure for problems where the distribution for capacity has a lower bound. 4. Study the effect of having a lower-bound capacity on the reliability of geotechnical engineering systems, and investigate the effect of uncertainty in the lower-bound capacity on the magnitude of increase in the reliability. 5. Propose an LRFD design-checking format that includes information on the lowerbound capacity in design. 6. Propose practical approaches for updating information about lower-bound capacities using installation data, proof-load data, and historical performance of foundations under load. 4

35 1.3 ORGANIZATION OF DISSERTATION In Chapter 2, results from two well-known load test databases of driven piles in cohesive and cohesionless soils are analyzed to provide evidence of the existence of a lower-bound pile capacity that can be computed using site-specific soil properties and information on pile geometry. In Chapter 3, a database of load tests conducted in uplift on suction caissons in normally consolidated clays is assembled and used to analyze biases and uncertainties in current models for predicting the capacity of suction caissons and to provide evidence of the existence of a lower-bound axial capacity. In Chapters 4 and 5, realistic probability distributions that can accommodate a lower-bound capacity are introduced and used to model the uncertainty in the capacity of piles and suction caissons. In Chapter 6, practical tools are proposed for assessing the probability of failure in the presence of a lower-bound capacity. The effect of the lower-bound capacity on the reliability of geotechnical engineering systems is investigated in Chapter 7, while the effect of uncertainty in the lower-bound capacity on the reliability is included and studied in Chapter 8. Alternative LRFD checking formats that incorporate information on lowerbound capacities of deep foundations are proposed in Chapter 9, and their use is validated using practical illustrative design examples. Finally, practical approaches for updating information on lower-bound capacities are proposed in Chapter 10. Conclusions and contributions of this research are presented in Chapter 11. 5

36 Chapter 2. Lower-Bound Axial Capacity for Driven Piles 2.1 INTRODUCTION Predictions of the axial capacity of driven piles in cohesive and cohesionless soils show significant scatter when compared to capacities that are measured in load tests (Olson and Dennis 1982, Tang 1988, Briaud and Tucker 1988, Pelletier et al. 1993, Lacasse and Nadim 1996, Kolk and van der Velde 1996, and McVay et al. 2000, 2002, and 2003). The scatter results primarily from the inability of current design methods, which are mostly empirical, to model the complicated sequence of events that follows pile driving. Uncertainties in measurements of soil properties in addition to uncertainties in measured capacities of piles provide additional reasons for the observed scatter. The objective of most design methods is to provide an unbiased prediction of pile capacity and to reduce uncertainty in the prediction of capacity by reducing scatter between measured and predicted capacities. In this research, a different approach is proposed for decreasing the uncertainty in predictions of capacity. In this approach, it is hypothesized that there is a physical lower bound to variations in the capacity of a pile foundation, and that this lower-bound capacity is greater than zero. The hypothesis is based on the principals of critical state soil mechanics, which imply that the shear strength of soil, even when substantially disturbed, is greater than zero. An analysis is presented to investigate the possibility of a lower-bound capacity for piles using two well-known databases with results of axial load tests (Olson and Dennis 1982). The first database contains results of steel pipe piles driven into cohesive soils, while the second contains closed-ended, steel pipe piles driven into cohesionless soils. These databases were used to develop design methods and resistance factors for offshore piles (API 1993). 6

37 2.2 AXIAL CAPACITY OF DRIVEN PILES IN COHESIVE SOILS The axial capacity of piles in compression is comprised of two components: (1) a component from side friction, Q s, and (2) a component from end bearing, Q p. For piles that are loaded in tension, the component of end bearing is assumed to be negligible. The weight of the pile acts as a resisting or loading component for piles in tension and compression respectively, but is typically neglected from calculations of capacity. In cohesive soils, the end-bearing component of the capacity of piles is typically calculated assuming that the soil is undrained during loading (Olson 2004). Pelletier et al. (1993) state that there is agreement that the net end bearing capacity can be calculated as: Q = N s A (2.1) P c u Tip where s u is the design undrained shearing strength, N c is an end bearing factor, and A Tip is the area of the tip of the pile. The end bearing factor, N c, is typically set to 9 although higher values have been back calculated from case histories with instrumented piles (Aschenbrenner 1984). For piles that are tipped in cohesive soils, the net end-bearing component constitutes, on the average, about 7% of the total predicted capacity (Olson 2004). As a result, uncertainty in the value of N c has a relatively small effect on the uncertainty in the total predicted capacity. Uncertainty in the capacity of piles in cohesive soils results primarily from uncertainty in predicting the magnitude of the side friction that is mobilized during an event of loading. The mobilized side friction is affected by changes in the state of effective stress and properties of the soil that occur during driving, subsequent consolidation, and loading to failure (Bond and Jardine 1995). The process of pile driving causes severe distortions and deformations in the soil adjacent to a pile, and the strains 7

38 that are created can be large enough to bring the soil close to the critical or residual state. This behavior is typically accompanied by changes in pore water pressure, effective stresses, and shear strength of the soil adjacent to the pile. After installation, the excess pore water pressures generally dissipate and the soil around the pile reconsolidates. The process of reconsolidation is typically accompanied by changes in effective stress and in the undrained strength of the soil around the pile. In the period of time between the end of driving and loading of the pile, the soil around the skin of the pile can be subjected to other time dependant changes in shear strength as a result cementation and aging. At the onset of loading, the values of shear strength that dictate the magnitude of the mobilized side friction are those of clay that was subjected to severe shearing and remolding, allowed to consolidate, and exposed to possible aging effects. There are a number of empirical methods for predicting the side friction for piles in cohesive soils. Most methods assume undrained conditions and attempt to express the side friction as a function of the undrained shearing strength of the clay. In these methods, the effects of installation, subsequent consolidation, and loading to failure are reflected in what is referred to as an adhesion factor, α. The mobilized side friction is then calculated as a function of α and the undrained shearing strength as: N ( αs u ASide ) Q = (2.2) s L= 1 L where L is the number of layers in the soil profile and A Side is the area of the pile on which side shear is mobilized in each layer. The factor α is typically correlated empirically with parameters such as the undrained shearing strength of the soil, s u, the overconsolidation ratio OCR, or the strength ratio, ψ, which is defined as the ratio of the undrained shearing strength to the effective overburden pressure. A major source of 8

39 uncertainty in design methods that express the side friction as a function of the undrained shearing strength is the uncertainty in the measurement of the undrained shearing strength. Values of undrained shearing strength are rate dependant, sensitive to disturbance during sampling, and vary with different testing techniques. Other empirical methods are also available for predicting the capacity of piles in cohesive soils. A summary of different methods is provided by Pelletier et al. (1993). Predicted capacities for piles in cohesive soils show significant scatter when compared to capacities that are measured in load tests. As an example, a plot of the measured versus predicted capacities for 45 piles that comprise a subset of a larger database that was compiled by Olson and Dennis (1982) is shown on Figure 2.1. Olson and Dennis (1982) compiled a database of 1004 load tests on piles that included data from the literature in addition to unpublished data from government agencies, oil companies, and consulting firms. Each test in the database was allocated a data quality factor, (DQF) which ranged from 1 for the poorest tests to 5 for the best tests. In this research, a subset of 45 pile load tests with DQFs that are greater than or equal to 3 and that were conducted on steel pipe piles in cohesive soils are analyzed. Descriptions for each load test are presented in Table 2.1 and in Appendix A. Predicted capacities on Figure 2.1 are calculated using the API method, which is a variation of the α-method (API, 1993). In this method, α is defined as: -0.5 α = 0.5ψ for ψ 1.0 (2.2) α = 0.5ψ for ψ 1.0 (2.3) where ψ is the ratio of the undrained shearing strength to the effective overburden pressure ( s ' u σ v ) at the depth where side friction is calculated. The method incorporates 9

40 the use of a design profile for undrained shearing strength that is typically developed from unconsolidated-undrained triaxial laboratory tests on Shelby tube samples obtained by the pushed method of sampling. Undrained strengths for the cases analyzed in the database were measured using unconsolidated-undrained, unconfined compression, minivane, torvane, or field vane tests. The quality of the samples ranged from high quality (3 diameter pushed thin-walled samplers) to low quality (2.25 diameter driven wireline samplers). Available information on testing and sampling techniques for each test in the database is presented in Appendix A. The predicted capacities shown on Figure 2.1 are calculated using uncorrected undrained shearing strength values that are reported in the original case studies. 600 Measured Capacity (kips) API Capacity (kips) Figure 2.1. Measured versus calculated capacities for axially loaded, driven steel pipe piles in cohesive soils (uncorrected for setup or method of shear strength measurement). 10

41 Table 2.1. Database of load tests for steel pipe piles in cohesive soils. Pile Load Test Diameter Length Site Location Comp. DQF Setup Time # Number* (in) (ft) or Tens. * (days) San Francisco C British Columbia C British Columbia C Morganza, LA C Morganza, LA C Morganza, LA C Morganza, LA C Iran C California C California C California C Empire, LA C Empire, LA T Empire, LA C Empire, LA T Empire, LA C Empire, LA T Empire, LA C Empire, LA C new West Delta T Toronto, Canada C Toronto, Canada T Long Beach, Cal T Louisiana C Atchafalaya C 3? Venice, LA C Venice, LA T Venice, LA C Harvey, LA C Eugine Island C Eugine Island C Texas Coast C Eugine Island C 3? Eugine Island C 4? Port Arthur, TX C Port Arthur, TX C Izmir, Turkey C Houston, TX T Houston, TX T Houston, TX T Houston, TX T Houston, TX T Houston, TX T Houston, TX C Houston, TX C * Olson and Dennis 1982 Overconsolidated Normally Consolidated to Slightly Overconsolidated 11

42 Measured capacities on Figure 2.1 correspond to those obtained using Davisson s method to reduce the load test data (Davisson 1970) unless the peak capacities are mobilized at lower displacements. In that case, the peak measured capacity is used to indicate the capacity of the pile. To account for uncertainty in the measured capacity due to test and interpretation errors (McVay 2002; Tang 1988), bounds corresponding to ±10 percent about the reported value are shown for each data point on Figure 2.1. The ratio of measured to predicted capacity for the data has a sample mean value of 0.96 (the design method is relatively unbiased), a coefficient of variation (c.o.v.) of 0.23, and a distribution that follows a lognormal distribution reasonably well (Tang 1988). This c.o.v. value of 0.23 is relatively small compared to many pile-load databases, where the c.o.v. values range from 0.3 to 1.0 (Olson and Dennis 1982, Tang 1988, Briaud and Tucker 1988, Pelletier et al. 1993, Lacasse and Nadim 1996, Kolk and van der Velde 1996, and McVay et al. 2000, 2002, and 2003); consequently, this database represents a case where there is relatively little variability in measured versus predicted capacities. To account for biases in the measured undrained shearing strength as a result of the use of different sampling and testing techniques, a correction factor F c was introduced to calculate an equivalent standard undrained shearing strength. The correction factor is defined as the ratio of the shear strength determined using unconsolidated-undrained triaxial compression tests on samples of high quality to the shearing strength measured using some other technique (Dennis and Olson 1983). Reasonable estimates for F c are 1.1 for unconfined compression tests on samples of high quality, 0.8 for field vane tests, and 1.3 for mini-vane tests on wire-line samples (Doyle et al. 1971; Koutsoftas and Fisher 1976; Young et al. 1983, Quiros et al. 1983; Dennis and Olson 1983; and Tang 1988). In addition to biases in the predicted capacity, uncertainty in the measured capacity can result due to inadequate setup time and errors in testing and interpretation 12

43 techniques for analyzing results of load tests. An attempt is made to correct the measured capacities to account for inadequate setup time using the empirical method proposed by Bogard and Matlock (1990). The method assumes that the increase in capacity after installation can be computed as a function of the diameter of the pile, the ratio of the thickness of the pile wall to the pile diameter, the setup time, and the condition of the pile during driving (plugged or unplugged). Corrected measured and predicted capacities are calculated for relevant piles in the database. A plot of the corrected measured versus predicted capacities is shown on Figure 2.2. The predicted capacities are calculated with the API method using corrected undrained strength values while the measured capacities are corrected for inadequate setup. The corrected ratio of measured to predicted capacity has a sample mean value of 0.97 (compared to 0.96 for the uncorrected case) and a c.o.v. value of 0.24 (compared to 0.23 for the uncorrected case). 600 Measured Capacity (kips) API Capacity (kips) Figure 2.2. Measured versus calculated capacities for driven steel pipe piles in cohesive soils (corrected for setup and method of undrained strength measurement). 13

44 2.3 LOWER-BOUND CAPACITIES FOR DRIVEN PILES IN COHESIVE SOILS The scatter in plots that show measured versus predicted capacities for piles in cohesive soils (Figures 2.1 and 2.2) is common and cannot be easily reduced. Empirical methods lack the ability to provide a sound theoretical basis for predicting capacity, and uncertainties in shear strength measurements resulting from spatial variability, sample disturbance, method of shear strength measurement and rate of loading will add to the total uncertainty in the predicted capacity. The scatter is amplified by uncertainties in measured capacities as a result of inadequate setup time, rate of loading, and errors in interpretation and testing techniques. Because of the factors stated above, it is difficult to reduce the uncertainty in predictions of capacity using empirical design methods. However, uncertainties in capacities of driven piles in cohesive soils can be reduced significantly by introducing a physical lower-bound capacity that can be calculated based on the principals of critical state soils mechanics using critical state or residual shear strength values. For piles in cohesive soils, the residual shear strength for undrained loading conditions can be approximated by the undrained remolded shear strength of the soil. The remolded shear strength resembles the undrained strength of clays after being subjected to very large strains at a constant void ratio. The residual shear strength can also be estimated from the drained residual interface friction angle between soil and steel provided that the minimum horizontal effective stress that is acting on the skin of the pile at failure can be estimated. These two approaches for estimating the smallest possible shear resistance can be used to calculate lower-bound estimates of the capacity for piles in clays. 14

45 2.3.1 Lower-Bound Capacity Based on Undrained Remolded Shearing Strength An estimate of the lower-bound axial capacity for piles in normally consolidated to slightly overconsolidated clays can be obtained by assuming that the shearing strength of the soil around the skin and under the tip of the pile reduces to the fully remolded undrained shearing strength. For this case, the lower-bound capacity can be calculated using the API method and replacing the undisturbed undrained shearing strength with the remolded shearing strength for these normally consolidated to slightly overconsolidated clays. For cases involving very stiff highly overconsolidated clays, remolding the soil requires a large amount of energy, and reliable measurements of the undrained remolded shearing strength are generally difficult to obtain. In these soils, the lower-bound capacity can be estimated based on the procedure proposed in Section Definition and Measurement of the Undrained Remolded Shearing Strength Most clays lose a proportion of their strength when remolded (Skempton and Northey 1952). Terzaghi (1944) defined the sensitivity of clay as the ratio of the undisturbed strength to the remolded strength measured at the same water content. The remolded strength is typically measured in the laboratory using unconfined compression tests or unconsolidated-undrained triaxial tests on soil samples that have been remolded at constant water content. For very sensitive clays, the preparation of remolded soil samples in the laboratory can be problematic and measurements can be performed using laboratory or field vane tests (Mitchell and Houston 1969). Skempton and Northey (1952) state that sensitivities between 2 and 4 are very common among normally consolidated clays. For normally consolidated clays in the Gulf of Mexico, Dutt et al. (1995) state that the remolded shearing strength varies between 20 to 50 percent of the undisturbed strength, indicating sensitivities between 2 and 5. The causes of clay 15

46 sensitivity have been studied extensively and reported elsewhere (Skempton and Northey 1952, Rosenqvist 1953; Bjerrum 1954; and Mitchell and Houston 1969) Correlation between Undrained Remolded Strength and Liquidity Index Measurements of the remolded strength of clays with different sensitivities indicate a correlation between the remolded strength and the liquidity index of the clay. The remolded strength tends to decrease as the liquidity index of the soil increases (Skempton and Northey 1952; Bjerrum 1954; Wroth and Wood 1979; Terzaghi et al. 1996). Based on critical soil mechanics concepts and on the work reported by Yousef et al. (1965) and Skempton and Northey (1952), Wroth and Wood (1978) attempted to relate the remolded strength, s ur, to the liquidity index, LI, with the following expression: ( 4.6LI) s ur = 3550e psf (2.4) To test the applicability of the proposed correlation to normally consolidated clays offshore in the Gulf of Mexico, measurements of remolded shear strength and their corresponding liquidity indices were digitized from plots presented in Dutt et al (1995) for the Auger TLP, Ewing Bank 873, Pompano Platform, and the Bullwinkle Platform sites. The reported strengths are plotted against the liquidity indices on Figure 2.3. The correlation presented by Wroth and Wood (1978) is also plotted on Figure 2.3 for comparison. Results indicate that the proposed correlation can be used to calculate an unbiased estimate of the undrained remolded shearing strength based on information on the liquidity index of the clays. The correlation will be used in this chapter to estimate the lower-bound capacity of piles for case studies where measurements of the remolded undrained shear strength are not available. 16

47 Undrained Remolded Shear Strength (psf) Mini-Vane, Dutt et al. (1995) Wroth and Wood (1978) Liquidity Index. LI Figure 2.3. Correlation between the undrained remolded strength and the liquidity index Evidence of a Lower-Bound Axial Capacity A lower-bound capacity is calculated using the model shown on Figure 2.4 for each pile in normally consolidated to slightly overconsolidated clays. Measurements of the undrained remolded shearing strength are available for 23 out of 34 tests (Piles # 1 to 23). For 11 out of 34 tests (Piles # 24 to 34), measurements of the remolded strength are not available and estimates based on the correlation proposed by Wroth and Wood (1978) (Equation 2.4) are used to predict the lower-bound capacity. A plot showing the measured capacities versus the calculated lower-bound capacities for 34 piles in normally consolidated to slightly overconsolidated clays is shown on Figure 2.5. No effort was done to correct the measured capacities to account for inadequate setup time. 17

48 API Axial Pile Capacity Skin Friction = αs u 0.5 su α= 0.5 σ ' v s u = undisturbed undrained shear strength σ ' = effective overburden pressure v Lower-Bound Axial Pile Capacity Skin Friction = αs ur α = 1.0 s ur = remolded undrained shear strength End Bearing = 9s u End Bearing = 9s ur Figure 2.4. Models for calculating predicted and lower-bound capacities of axially loaded, driven steel pipe piles in normally consolidated to slightly overconsolidated clays. Measured Capacity (kips) Measured Remolded Shear Strength Estimated Remolded Shear Strength (Correlation, Eq. 2.4) Lower-Bound Capacity (kips) Figure 2.5. Measured versus lower-bound capacities for axially loaded, driven steel pipe piles in normally consolidated to slightly overconsolidated clay (uncorrected measured capacities). 18

49 The data on Figure 2.5 support the hypothesis of a lower-bound capacity because none of the data points fall below the calculated lower bound. Each point in the database has a different calculated lower-bound capacity relative to the predicted capacity since the calculated lower bound depends on the properties of the soil and the geometry of the pile (Figure 2.4). For this database, the ratio of the lower-bound capacity to the predicted capacity ranges from about 0.3 to 1.0 and has an average of A summary of measured, predicted and lower-bound capacities is presented in Table 2.2. Table 2.2. Summary of measured, predicted, and lower-bound capacities for piles in normally consolidated to slightly overconsolidated clays. Pile Diameter Length Measured Capacity API Capacity Lower-Bound Capacity Ratio of Lower Bound to # (in) (ft) kips, (Davisson) kips (Unorrected) kips (Remolded) Predicted Capacity Normally Consolidated to Slightly Overconsolidated

50 2.3.2 Lower-Bound Capacity Based on Residual Interface Shear Resistance An effective stress approach can also be utilized to calculate an estimate of the lower-bound capacity for piles in cohesive soils. In this approach, the predicted lowerbound side resistance is calculated as: Side, LB ' h,lb ' ( δ ) Q = σ tan (2.5) LB where Q Side,LB is the lower-bound side resistance, ' σ h,lb is a lower-bound estimate of the effective horizontal stress that is acting on the walls of the pile at failure and δ ' LB is a lower-bound estimate of the interface friction angle between the pile and the soil. ' Practical approaches for estimating Q Side,LB and σ h,lb are presented in the next sections primarily for piles in overconsolidated clays. The applicability of the proposed approach to piles in normally consolidated to slightly overconsolidated clays is then explored Lower-Bound Horizontal Effective Stress at Failure Results from research studies on instrumented piles in overconsolidated clays are analyzed to study the variation of the horizontal effective stress during installation, subsequent equalization, and loading to failure (Coop and Wroth 1989; Lehane and Jardine 1992; and Bond and Jardine 1991, 1995). Results indicate that during installation, the total horizontal stresses acting on the walls of the pile increase significantly while excess negative pore pressures are typically generated as the soil dilates strongly due to shear. Bond and Jardine (1995) report that the installation of piles in heavily overconsolidated London clay increases the horizontal total stress to approximately 3 to 6 times the at-rest value (K o value). After equalization, the horizontal total stress does not change significantly while the negative pore water pressures dissipate to hydrostatic 20

51 conditions. The net effect is that the radial effective stress after full equalization is about six times the in-situ radial effective stress (K o conditions) at deep portions of the pile and decreases with increasing distance from the pile tip. At the butt of the pile, the horizontal effective stress approaches the in-situ horizontal effective stress. This observed behavior from one case study (Bond and Jardine 1995) indicates that the magnitude of the horizontal effective stress after full dissipation of pore pressures but prior to loading of the pile is expected to be bounded on the low side by the horizontal effective stress that corresponds to the at-rest condition. To test the above hypothesis, results from studies performed on instrumented driven and pushed piles were analyzed (Bond et al. 1993). For each case, the ratio of the horizontal effective stress at full equalization to the initial vertical effective stress (K c ) is plotted on Figure 2.6 as a function of the overconsolidation ratio (OCR). Typical trends of the variation of the atrest coefficient (K o ) with the OCR are then added on Figure 2.6 to allow for comparison with the data provided in Bond et al. (1993). The trends of the variation of the at-rest coefficient (K o ) with the OCR are presented in Mayne and Kalhawy (1982). Results on Figure 2.6 provide strong evidence of the existence of a lower-bound fully equalized horizontal effective stress that can be obtained by using K o conditions. As the pile is loaded to failure, the horizontal effective stress is expected to either increase or decrease depending on whether the soil dilates or contracts. Bond et al. (1993) defined a pile loading factor (f L ) that represents the ratio of the horizontal effective stress at failure to the stress after full equalization. They report values for f L that range from 0.9 to 1.3 for tests on pushed piles in heavily overconsolidated London clay, 0.8 to 1.0 for tests on pushed piles in the stiff Cowden Glacial till, and about 0.85 in the normally consolidated Bothkennar clay. In this research, 0.8 is assumed to be a lower-bound estimate for the ratio f L. 21

52 10 Bond et al. (1993) Kc or Ko 1 Mayne and Kalhawy K o for Φ' = 20 o Mayne and Kalhawy K o for Φ' = 30 o OCR Figure 2.6. Comparison between the at-rest earth pressure coefficient, K 0, and the coefficient K c Lower-Bound Interface Friction Angle The drained residual interface friction angle is used to provide a lower-bound estimate for shear strength at the pile-soil interface. The residual strength of steel-soil interfaces depends on the plasticity and mineralogy of the soil, the roughness of the interface, and on the magnitude of the effective stress acting on the interface (Tika- Vassilikos 1991; Lehane and Jardine 1992; Tsubakihara et al. 1993a,b; and Lemos and Vaughan 2000). Ramsey et al. (1998) compiled a database of available case studies in which residual friction angles for clay-steel interfaces were measured. Residual friction angles from the database are plotted on Figure 2.7 as a function of the plasticity index of the soil. An exponential trend line provides a reasonable fit to the measured data points (Figure 2.7) and is used in this research in estimating the residual interface friction angle for calculating the lower-bound capacity for piles in clays. 22

53 Residual Interface Friction Angle, δlb δ LB = e PI Plastcity Index (%) Figure 2.7. Correlation between the drained, residual friction angle between soil and steel and the plasticity index (from Ramsey et al., 1998) Evidence of a Lower-Bound Axial Capacity An estimate of the lower-bound axial capacity for piles driven in overconsolidated clays can thus be obtained using the model presented on Figure 2.8. The lower-bound side resistance is calculated using lower-bound estimates of the horizontal effective stress at failure and the interface friction angle between soil and steel. A plot showing measured capacities versus calculated lower-bound capacities is shown on Figure 2.9. This graph supports the hypothesis of a lower-bound capacity because none of the data points fall below the calculated lower bound. For this database, the ratio of the lower-bound capacity to the predicted capacity ranges from 0.4 to 0.85 and has an average of A summary of measured, predicted and lower-bound capacities for each test in overconsolidated clays is presented in Table

54 API Axial Pile Capacity Skin Friction = αs u s α = 0.5 u ' σ v 0.25 s u = undisturbed undrained shear strength σ ' v = effective overburden pressure End Bearing = 9s u Lower-Bound Axial Pile Capacity Skin Friction = K LB σ v tan(δ LB ) K LB = 0.8K o K o = (1-sinΦ )OCR (sinφ ) δ LB = Residual Interface Friction Angle (Correlation with Plasticity Index) End Bearing = 9s u Figure 2.8. Models for calculating predicted (mean) and lower-bound capacities of axially loaded, driven steel pipe piles in overconsolidated clays. 300 Measured Capacity (kips) Lower-Bound Capacity (kips) Figure 2.9. Measured versus lower-bound capacities for axially loaded, driven steel pipe piles in overconsolidated clay (uncorrected measured capacities). 24

55 Table 2.3. Summary of measured, predicted, and lower-bound capacities for piles in overconsolidated clays. Pile Diameter Length Measured Capacity API Capacity Lower-Bound Capacity Ratio of Lower Bound to # (in) (ft) kips, (Davisson) kips (Unorrected) kips (Remolded) Predicted Capacity Overconsolidated In theory, the proposed approach for estimating the lower-bound capacity for piles in overconsolidated clay is expected to work for all piles in clay, irrespective of the degree of overconsolidation of the soil. In this approach, it is assumed that the lowerbound side friction for piles in clays is governed by the residual interface friction angle between soil and steel and by horizontal effective stresses that correspond to the at-rest condition. The applicability of the proposed approach to all piles in the database (including piles in normally consolidated to slightly overconsolidated clays) is tested on Figure None of the measured data points on Figure 2.10 fall below the calculated lowerbound capacity, indicating that the proposed approach for predicting the lower-bound capacity is a sound approach. Each point in the database has a different calculated lowerbound capacity relative to the predicted capacity since the calculated lower bound depends on the properties of the soil and the geometry of the pile (Figure 2.10). For this approach, the ratio of the lower-bound capacity to the predicted capacity ranges from 0.35 to 0.9 and has an average of A summary of measured, predicted and lowerbound capacities for all tests in the database are presented in Table

56 600 Measured Capacity (kips) Lower-Bound Capacity (kips) Figure Measured versus lower-bound capacities for axially loaded, driven steel pipe piles in all clays (uncorrected measured capacities). 2.4 AXIAL CAPACITY OF DRIVEN PILES IN COHESIONLESS SOILS For piles in cohesionless soils, the end-bearing component of the capacity represents a significant portion of the capacity in compression, and averages about 26% of the total capacity of the pile (Olson 2004). The net end-bearing capacity for piles in compression is typically expressed by Equation 2.6 as (Pelletier et al. 1993): Q = σ N A (2.6) Tip ' v q Tip where N q is a bearing capacity factor, σ v is the vertical effective overburden pressure, and A Tip is the area of the tip of the pile. The side friction is calculated as: Side ' v () δ ASide Q = Kσ tan (2.7) 26

57 Table 2.4. Summary of measured, predicted, and lower-bound capacities for piles in clay. Pile Diameter Length Measured Capacity API Capacity Lower-Bound Capacity Ratio of Lower Bound to # (in) (ft) kips, (Davisson) kips (Unorrected) kips (Remolded) Predicted Capacity Normally Consolidated to Slightly Overconsolidated Overconsolidated

58 where K is an earth pressure coefficient, σ v is the vertical effective overburden pressure, δ is the soil-pile friction angle, and A side is the area of the pile on which side shear is mobilized. The problem of predicting the axial capacity of piles in cohesionless soils is more difficult to solve compared to that of predicting the capacity of piles in cohesive soils (Pelletier et al and Olson 2004). Representative values for the parameters that are required for predicting the end bearing capacity and side friction as expressed in Equations 2.6 and 2.7 are difficult to obtain. It is impractical to measure the undisturbed shearing strength of cohesionless soils in the laboratory because of disturbance that can occur during sampling and testing. In addition, the process of pile installation changes the state of stress in the soil around the pile and can result in changes in the shape, gradation, and density of the soil particles that are in contact or adjacent to the walls of the pile. Changes in the capacity of piles in cohesionless soils with time after installation have also been reported (Tavenas and Audy 1972, Chow et al. 1996, Chow et al 1998) and add to the uncertainty in predicting capacity. In addition, some experimental results from model tests on piles in cohesionless soils indicate that the end bearing capacity and side friction tend to reach maximum values at relatively shallow depths (Olson 2004). Reasons for the presence of these limits are still not clear, but many current design methods specify upper bounds or limiting values for the end bearing pressure and the side friction. Because of these reasons, most available methods for predicting the capacity of piles in cohesionless soils tend to be empirical in nature. In these methods, the parameters N q, δ, and K, in addition to values of limiting end bearing pressure and side friction, are typically correlated to the relative density and type of soil and to in-situ measurements of strength, which are typically obtained using standard penetration tests (SPT), cone penetration tests (CPT), or pressure meter tests. For example, the parameters δ and N q, in 28

59 addition to values of the limiting end bearing and skin friction, are calculated in the API method (API 1993) using one of five categories based on the type and relative density of the supporting soil (Table 2.5). The relative density is typically correlated to the SPT blow count as shown in Table 2.6. In the API method, the earth pressure coefficient, K, is specified as 1.0 for closed-ended piles and 0.8 for open-ended piles. Because of the empirical nature of current design methods for piles in cohesionless soils, predicted capacities show significant scatter when compared to capacities that are measured in load tests. As an example, a plot of the measured versus predicted capacities for 36 piles that comprise a subset of a database that was compiled by Olson and Dennis (1982) is shown on Figure The dataset is comprised of 36 tests with DQFs that are greater than or equal to two and that are conducted on closed-ended steel pipe piles in siliceous, cohesionless soils. A description of each load test is presented in Table 2.7 and in Appendix B. Table 2.5. API method for predicting the capacity of piles in cohesionless soils. Density Very Loose, VL Loose, L Medium, M Soil Description Sand Sand-Silt Silt Soil-Pile Friction Angle, δ (Deg) 15 Limiting Skin, Friction Values ksf, (kpa) 1.0 (47.8) End Bearing Factor, Nq Limiting Unit End Bearing ksf (Mpa) 8 40 (1.9) Loose, L Medium, M Dense, D Sand Sand-Silt Silt (67.0) (2.9) Medium, M Dense, D Sand Sand-Silt (81.3) (4.8) Dense, D Very Dense, VD Sand Sand-Silt (95.7) (9.6) Dense, D Very Dense, VD Gravel Sand (114.8) (12.0) 29

60 Table 2.6. Relative density based on standard penetration tests (Terzaghi et al. 1996). No of Blows, N 60 Relative Density 0-4 Very Loose 4-10 Loose Medium Dense Over 50 Very Dense Table 2.7. Database of load tests for steel pipe piles in siliceous cohesionless soils. Pile Load Test Diameter Length Site Location Compression Data Quality Plate Measured Capacity # Number* (in) (ft) or Tension Factor* kips, (Davisson) Arkansas River C Arkansas River T Arkansas River C Arkansas River T Arkansas River C Arkansas River T Arkansas River C Arkansas River C Arkansas River T Old River, LA C Old River, LA T Old River, LA C Old River, LA T Old River, LA C Old River, LA C Old River, LA C Old River, LA T Old River, LA T Beech, R. Ont. C 2 Oversized Beech, R. Ont. C 2 Oversized Ogeechee R. C Ogeechee R. C Ogeechee R. C Ogeechee R. C Ogeechee R. T Helena Ark. C 2 Oversized Helena Ark. C 2 Oversized Helena Ark. C 2 Oversized Kansas City C Kansas City C Kansas City C Kansas City C British Colombia C 3 Oversized Florida C Japan C Michigan C * Olson and Dennis

61 1000 Measured Capacity (kips) Predicted API Capacity (kips) Figure Measured versus calculated capacities for axially loaded, driven steel pipe piles in siliceous, cohesionless soils. Predicted capacities on Figure 2.11 are calculated using the API method for piles in cohesionless soils using corrected SPT values. As with cohesive soils, measured capacities correspond to those obtained using Davisson s (1970) method to reduce the load test data. To account for uncertainty in the measured capacity due to test and interpretation errors (McVay 2002; Tang 1988), bounds corresponding to ±10 percent about the reported value are shown for each data point on Figure The ratio of measured capacity to predicted capacity has a sample mean value of 1.02 (the design method is relatively unbiased), a c.o.v. value of 0.50, and a distribution that follows a lognormal distribution reasonably well (Tang 1988). This c.o.v. of 0.50 is large compared to many pile-load databases (Olson and Dennis 1982, Tang 1988, Briaud and Tucker 1988, Pelletier et al. 1993, Lacasse and Nadim 1996, Kolk and van der Velde 1996, and McVay et al. 2000, 2002, and 2003), meaning that this database represents a test case where there is large variability in measured versus predicted capacities. 31

62 2.5 LOWER-BOUND CAPACITIES FOR DRIVEN PILES IN COHESIONLESS SOILS As with the case for cohesive soils, the scatter in plots that show measured versus predicted capacities for piles in cohesionless soils (Figure 2.11) is common, irrespective of the method used to predict the capacity. Uncertainties in actual values of the endbearing factor N q, the soil-steel interface friction angle, δ, and the earth pressure coefficient, K, translate into uncertainties in the predicted capacity. The scatter is amplified by uncertainties in measured capacities as a result of differences in setup times, rate of loading, and errors in interpretation and testing techniques for each load test. Because of the factors stated above, it is difficulty to reduce the uncertainty in predictions of capacity for piles in cohesionless soils. However, a lower bound to variations in predictions of capacity can be introduced in the form of a physical lowerbound capacity that can be calculated with the help of the principals of critical state soil mechanics. The existence of a lower-bound capacity reduces the uncertainty or variability in the capacity, particularly in the left-hand tail of the probability distribution for capacity. The lower-bound capacity represents the minimum, possible capacity for the pile foundation and can be calculated by assuming that the interface shear strength between the soil and the pile reduces to the critical state shear strength. It is also assumed that increases in the effective stress around the pile as a result of pile driving are nonexistent, and that the horizontal effective stresses that act on the pile walls can be represented by free field conditions (at-rest conditions). A practical and simple model for estimating the lower-bound capacity of piles in cohesionless soils can be obtained by modifying the API method such that: the lateral coefficient of earth pressure is replaced with the at-rest value and the soil-pile friction angle and end-bearing capacity factors are replaced with the values for one-category less in density (e.g., the values for a Dense Sand are replaced with those for a Medium 32

63 Sand ). The one-category reduction in density is a simple approach for accounting for possible reductions in the interface friction angle and in the density at the tip of the pile as a result of factors such as spatial variability, pile driving and loading to failure. The reduced interface friction angle is used in this simple model to represent the critical state interface friction angle, which can be difficult to measure. For piles with oversized end plates, a reduction in the side friction is expected and the lower-bound capacity is calculated by assuming that the side friction is equal to zero. The proposed model is presented in Figure 2.12 and the corresponding parameters are presented in Table 2.8. Lower-bound capacities for each pile in the database are calculated using the proposed model (Figure 2.12). A plot showing measured capacities versus calculated lower-bound capacities is presented on Figure As with cohesive soils, none of the data points on Figure 2.13 fall below the calculated lower-bound capacity, providing support to the hypothesis of a lower-bound pile capacity. The ratio of the lower-bound capacity to the predicted capacity for these data ranges from about 0.2 to 0.75 with an average of 0.5. A summary of measured, predicted and lower-bound capacities for each test in the database is presented in Table 2.9. API Axial Pile Capacity Skin Friction = k API σ' v tanδ API k API = 1.0 for closed ended piles σ' v = effective overburden pressure δ API = soil-pile interface friction angle (1 of 5 categories based on type of soil and density from blow counts) End Bearing = σ' v N q,api N q,api = bearing capacity coefficient (1 of 5 categories based on type of soil and density from blow counts) Lower-Bound Axial Pile Capacity Skin Friction = k LB σ' v tanδ LB k LB 1 sin φ' API normal plates = 0 over-sized plates φ' API = deduced from N q, API δ LB = δ API downgraded one category End Bearing = σ' v N q,lb N q, LB = N q, API downgraded one category Figure Models for calculating predicted (mean) and lower-bound capacities of axially loaded, driven steel pipe piles in cohesionless, siliceous soils. 33

64 Table 2.8. Parameters for predicting the lower-bound capacity of piles in siliceous, cohesionless soils. Density Very Loose, VL Loose, L Medium, M Loose, L Medium, M Dense, D Medium, M Dense, D Dense, D Very Dense, VD Dense, D Very Dense, VD Soil Description Sand Sand-Silt Silt Sand Sand-Silt Silt Sand Sand-Silt Sand Sand-Silt Gravel Sand Soil-Pile Friction Angle, δ (Deg) Limiting Skin, Friction Values ksf, (kpa) End Bearing Factor, Nq Limiting Unit End Bearing ksf (Mpa) (38.2) 6 30 (1.4) (47.8) 8 40 (1.9) (67.0) (2.9) (81.3) (4.8) (95.7) (9.6) 1000 Measured Capacity (kips) Predicted Lower Bound Capacity (kips) Figure Measured versus calculated lower-bound capacities for axially loaded, driven steel pipe piles in siliceous cohesionless soils. 34

65 Table 2.9. Summary of measured, predicted, and lower-bound capacities for piles in cohesionless, siliceous soils. Pile Load Test Compression Plate Measured Capacity Predicted Capacity Lower-Bound Capacity Ratio of Lower Bound # Number* or Tension kips, (Davisson) kips, (API) kips to Median Capacity 1 89 C T C T C T C C T C T C T C C C T T C Oversized C Oversized C C C C T C Oversized C Oversized C Oversized C C C C C Oversized C C C

66 2.6 SUMMARY These two pile-load test databases for driven piles in cohesive and cohesionless soils both provide strong evidence for the existence of a lower-bound capacity. The lower-bound capacity can be estimated using simple and practical approaches that are based on the principals of critical state soil mechanics using site-specific soil properties and information about pile geometries; the lower-bound capacity is a physical variable, not a statistical parameter. Many designers already calculate lower-bound capacities in practice for design and installation checks even though they are not formally required in design codes. The ratio of the lower-bound capacity to the predicted capacity is generally greater than 0.5 and can approach 1.0. While these results correspond only to two databases for driven piles that are used primarily for offshore pile design, the same principle is expected to apply to any pile-load database that is used to develop pile design methods and resistance factors. 36

67 Chapter 3. Predicted and Lower-Bound Axial Capacities of Suction Caissons in Normally Consolidated Clays 3.1 INTRODUCTION Design methods and criteria for suction caissons have generally been adapted from those for driven pipe piles (Andersen et al. 1999). However, the accuracy of these design methods has never been thoroughly tested due to the lack of published databases of pullout tests on suction caissons. Databases are needed for evaluating biases and uncertainties in current design methods for predicting capacity. In this chapter, a database that is comprised of published laboratory model tests, centrifuge tests, and full scale field tests conducted in uplift on suction caissons in normally consolidated clays is assembled and used to achieve the following objectives: (1) evaluate biases and uncertainties that are inherent in available models for predicting the uplift capacity of suction caissons in normally consolidated clays, and (2) provide evidence of the existence of a physical lower-bound capacity that can be calculated using information on suction caisson geometry and site-specific soil properties. 3.2 DATABASE OF PULLOUT TESTS ON SUCTION CAISSONS IN NC CLAYS As the oil and gas industry moves to deeper water depths, suction caissons emerge as alternative foundations to piles for anchoring facilities to the sea floor. All facilities in deepwater in the Gulf of Mexico that are anchored in less than 4000 ft of water have used driven piles while all but one facility in greater than 4000 ft have used or plan to use suction caissons (Clukey and Phillips 2002). Suction caissons are closed-top steel tubes that are lowered to the seafloor, allowed to penetrate the soil under their own weight, and 37

68 then pushed to their final penetration using suction (Luke et al. 2003). The suction pressure required to overcome the resistance of the soil during penetration is produced by pumping water out of the interior of the caisson. After installation, the top cap of the caisson can be either sealed or vented. Clukey et al. (2003) state that the diameters of suction caissons that are used to anchor large deepwater facilities are typically 4 to 7 meters (13 to 25 ft). In the Gulf of Mexico, the length of caissons that are embedded in normally consolidated clay are typically 5 to 7 times the suction caisson diameter (L/D = 5 to 7). A comparison between piles and suction caissons is shown on Figure 3.1. There is currently a lack of published databases of pullout tests on suction caissons in normally consolidated clays. Databases are needed in calibrating models for predicting the capacity of suction caissons in pullout. However, individual efforts have been made to study the behavior of suction caissons under different test conditions. For suction caissons embedded in normally consolidated clays, these efforts include laboratory scale model tests (Luke et al. 2003), centrifuge tests (Clukey and Morrison 1993; House and Randolph 2001; Randolph and House 2002; Clukey et al. 2003; and Clukey and Phillips 2002), and full scale field tests (Cho et al. 2003). There is a need to assemble the results of these tests in a database that contains information about the geometry of the caissons, method of installation, properties of the soil, measured capacities in uplift, rate of loading, and soil setup. In this study, an effort is made to assemble a database containing available published tests performed in uplift on suction caissons in normally consolidated clay. The database is described in Table 3.1 and Table 3.2. General information about the type of soil, geometry of the caisson, loading conditions, etc. are presented in Table 3.1 while specific information regarding properties of the soil and measured capacities are presented in Table 3.2. Details regarding each test are presented in Appendix C. 38

69 (a) Typical Offshore Piles Figure 3.1. Foundations for anchoring offshore facilities. (b) Typical Suction Caissons Table 3.1. Database of pullout tests on suction caissons (general description). # Caisson ID Soil Type Diam L/D Accel. Top Cap Loading Rate of Loading Setup feet g's mm/sec 1 Luke et al 2003 Kaolinite g Vented Monotonic 5 to hrs 2 Luke et al 2003 Kaolinite g Vented Monotonic 5 to hrs 3 Luke et al 2003 Kaolinite g Vented Monotonic 5 to hrs 4 Luke et al 2003 Kaolinite g Closed Monotonic 5 to hrs 5 Luke et al 2003 Kaolinite g Closed Monotonic 5 to hrs 6 Luke et al 2003 Kaolinite g Closed Monotonic 5 to hrs 7 Luke et al 2003 Kaolinite g Closed Monotonic 5 to hrs 8 Clukey 1993 Kaolinite 49.9* g Closed Monotonic 2.6 to 4.1 days* 24 hrs 9 Clukey 1993 Kaolinite 49.9* g Closed Monotonic 14 days* 24 hrs 10 Clukey 1993 Kaolinite 49.9* g Closed Monotonic 2.6 to 4.1 days* 24 hrs 11 House 2002 Kaolinite 11.81* g Vented Monotonic House 2002 Kaolinite 11.81* g Closed Monotonic House 2002 Kaolinite 11.81* g Closed Sustained 150 days* 0 14 Randolph 2001 Kaolinite 11.81* g Vented Sustained 0.1 (10 days*) 0 15 Randolph 2001 Kaolinite 11.81* g Vented Sustained 6 months* 1 year* 16 Randolph 2001 Kaolinite 11.81* g Closed Monotonic days* 17 Randolph 2001 Kaolinite 11.81* g Closed Cyclic - 1 year* 18 Randolph 2001 Kaolinite 11.81* g Closed Sustained Long term* 1 year* 19 Clukey 2002 Kaolinite 17.4* g Closed Monotonic 8 sec 20 hrs 20 Clukey 2002 Kaolinite 17.4* g Closed Monotonic 8 sec 20 hrs 21 Clukey 2002 Kaolinite 17.4* g Closed Monotonic 8 sec 20 hrs 22 Clukey 2002 Kaolinite 17.4* g Closed Monotonic 8 sec 20 hrs 23 Cho et al 2003 CH-Clay g Closed Monotonic 5 min 3 days 24 Cho et al 2003 CH-Clay g Closed Monotonic 5 min 3 days 25 Cho et al 2003 CH-Clay g Closed Monotonic 5 min 3 days * Prototype Scale 39

70 Table 3.2. Database of pullout tests on suction caissons (soil properties and measured loads). # Caisson ID Method of Shear Strength Density Average S u Tip S u Sensitivity Measured Capacity in Uplift Measurement pcf psf psf kips 1 Luke et al T-Bar Luke et al T-Bar Luke et al T-Bar Luke et al T-Bar Luke et al T-Bar Luke et al T-Bar Luke et al T-Bar Clukey 1993 CPT to Vane* Clukey 1993 CPT to Vane* Clukey 1993 CPT to Vane* House 2002 T-Bar House 2002 T-Bar House 2002 T-Bar Randolph 2001 T-Bar Randolph 2001 T-Bar Randolph 2001 T-Bar Randolph 2001 T-Bar Randolph 2001 T-Bar Clukey 2002 Piezo. to DSS** Clukey 2002 Piezo. to DSS** Clukey 2002 Piezo. to DSS** Clukey 2002 Piezo. to DSS** Cho et al 2003 UU -Triaxial *** Cho et al 2003 UU- Triaxial *** Cho et al 2003 UU- Triaxial *** 82 CPT calibrated to Vane, ** Piezocone calibrated to Direct Simple Shear, *** Assumed value The database is comprised of seven lab-scale model tests, fifteen centrifuge tests, and three full scale field tests. Diameters range from 4 inches (model tests) to about 50 feet (prototype scale for centrifuge tests), while aspect ratios range from 2 to 10. In all the lab and centrifuge tests, kaolinite is used to model the soil profile. The majority of load tests in the database are conducted under rapid monotonic loading conditions to simulate undrained uplift under environmental loading conditions in the field. However, different 40

71 loading rates are used in different studies thus introducing a source of uncertainty in the measured loads. Another source of uncertainty in the database is the different periods of time that were allowed for the suction caissons to setup prior to undrained testing. Load tests that are conducted prior to full equalization of excess pore water pressures that result from the installation process can underestimate the ultimate capacity of the caisson. A last major source of uncertainty in the database lies in the use of different methods to measure the undrained shearing strength of the soil. A variety of direct simple shear, unconsolidated-undrained triaxial, cone penetration, vane shear, and T-bar tests are used to measure the undrained shearing strength in the different case studies analyzed. 3.3 UPLIFT CAPACITY OF SUCTION CAISSONS IN NORMALLY CONSOLIDATED CLAY Uplift tests on suction caissons can be performed with a sealed or vented top cap. For tests were the cap is sealed, the soil resistance contributing to the uplift axial capacity can be determined from the external skin friction and the reverse end bearing acting at the tip of the caisson. The reverse end bearing is mobilized due to a reduction in the inside pressure under the caisson head and the pore pressure in the soil (Clukey and Morrison 1993). In sealed caissons, the interior soil plug is typically pulled out of the deposit with the caisson. For uplift tests on suction caissons with a vented top cap, water can flow in the caisson during pullout, and the plug of soil inside the caisson is typically left behind in the soil deposit (Luke et al. 2003). In this case, the soil resistance contributing to the uplift capacity is comprised of skin friction acting on the internal and external walls of the caisson and a relatively small contribution of net reverse end bearing acting on the annulus of the wall of the caisson. 41

72 3.3.1 Calculation of Side Resistance Under rapid uplift loading, the side resistance is typically calculated using a variation of the alpha method as a function of the undrained shearing strength of the soil. For normally consolidated clays, an alpha value of 1.0 is typically used in the design of offshore piles. For suction caissons, concerns about the effect of suction installation, soil setup, and the presence of the padeye have led to a tendency for designers to reduce alpha to values that are less than 1.0 (α = 0.6 to 0.8). Randolph and House (2002) indicate that after installation, the external side resistance is expected to increase from the soil s remolded strength to a fully equalized strength that corresponds to alpha values of about 0.5 to 0.7. The fact that alpha does not reach a value of 1.0 is attributed by the authors to the high ratio of diameter to wall thickness of suction caissons. Andersen and Jostad (1999) and Clukey (2001) state that an expected reduction in the external skin friction for the full set-up condition can be attributed to a reduction in soil stresses on the portion of the caisson that is installed with suction. The rationale behind such a concern is the observation that the soil typically moves into the caisson (rather than outside the caisson) as a result of installation by suction. Another expected cause of the reduction in the external adhesion is related to the presence of the padeye. The padeye can cover up to 5 to 10% of the caisson perimeter and tends to cut a groove as the caisson penetrates into the soil (Clukey and Philips 2002, and Hwang et al. 2003). There are concerns that the soil above the padeye will not reestablish contact with the caisson wall and will lead to a reduction in the skin friction. Another factor that can affect the mobilized side friction for suction caissons is the rate of loading. The extreme loads that govern the design for a driven pile are typically applied over a short time during a storm (minutes to hours), while the extreme loads that govern the design for a suction caisson could be applied over a longer time due 42

73 to a sustained current in deepwater (days to weeks). This slower rate of loading may reduce the available external skin friction for a suction caisson (Gilbert and Murff 2001). Another source of uncertainty in the determination of side resistance is the determination of the undrained shear strength of the clay. Design methods for driven piles are based on shear strength profiles developed from unconsolidated-undrained (UU) triaxial shear tests on undisturbed samples from the field (API 1993). Different testing and sampling techniques typically result in different values for the undrained shear strength (Doyle et al. 1971; Koutsoftas and Fisher 1976; Young et al. 1983, Quiros et al. 1983; Dennis and Olson 1983; and Tang 1988). For example, Quiros et al. (1983) present and analyze data comparing different undrained shear strength measurements. Methods other than the standard UU method generally introduce biases on the order of plus or minus 10 percent with coefficients of variation on the order of 20 percent (Gilbert and Murff 2001). In addition, Gilbert and Murff (2001) state that the relative magnitude of variability in the undrained shear strength tends to decrease with depth from the mudline. Typical suction caissons are installed in the upper 50 to 100 feet of a soil profile, while typical offshore piles are driven 200 to 300 feet deep. A major consequence of this difference is that there will be more uncertainty in the design shear strength profiles for suction caissons compared to driven piles. Therefore, uncertainties in undrained shear strength will have a greater effect on the estimated skin friction for a suction caisson than for a driven pile Calculation of the Net Reverse End Bearing The net reverse end bearing is typically calculated by multiplying the undrained shearing strength at the tip of the caisson by an end bearing factor N, which for driven 43

74 piles have been typically set to 9. For suction caissons, some studies indicate an end bearing factor that can be greater than 9 (Clukey and Morrison 1993, House and Randolph 2001, Randolph and House 2002 and Clukey et al. 2002, Luke et al. 2003). A major difference between piles and suction caissons lies in the relative contribution of the net reverse end bearing to the total capacity. For driven piles, the contribution of the net end bearing is typically less than 10%. For caissons with geometries that are typical of those used in the Gulf of Mexico, the net reverse end bearing can account for about 40% to 60% of the total axial capacity (Clukey and Phillips 2002). The reverse end bearing is affected by the rate of loading, permeability of the clay, undrained shearing strength of the clay, and aspect ratio of the caisson (Huang et al. 2003). The net reverse end bearing at the tip of a suction caisson can only develop under undrained conditions since typical soils have no effective stress tensile strength (Gilbert and Murff 2001). Consequently, the duration of loading and the rate of drainage at the tip of the caisson are variables that can affect the ability to mobilize reverse end bearing for a suction caisson. These factors add uncertainty to the contribution of the end bearing to the total capacity of a suction caisson. Another source of uncertainty in the reverse end bearing capacity is the variability in the undrained shearing strength. Gilbert and Murff (2001) state that there is more variability in the reverse end bearing than in the total skin friction as a result of variability in the undrained shearing strength. The variability is greater for end bearing than skin friction because the vertical extent of soil involved in the failure mechanism is smaller and there is less averaging in the values of shear strength along the shearing planes. 44

75 3.4 ANALYSIS OF LOAD TEST RESULTS In an initial analysis, predicted capacities for the 25 tests in the database are calculated using an alpha of 1 and an end bearing factor of 9. Values of undrained strength that are reported in the original references are used in this initial analysis with no attempts to correct for the method of shear strength measurement. For pullout tests that are conducted immediately after installation (Tests 11 to 14), the undrained shearing strength of the remolded clay is used in calculating the predicted uplift capacity. Ratios of measured to predicted capacities for the 25 tests in the database are plotted on Figure 3.2. Results on Figure 3.2 indicate an average ratio of measured to predicted capacity of 0.99 (unbiased model) and a coefficient of variation in the ratio of measured to predicted capacity of Measured Capacity / Predicted Capacity Luke et al 2003 Clukey and Morrison 1993 Clukey et al 2003 Randolph and House 2002 Cho et al 2003 House and Randolph Measured Capacity (kips) Figure 3.2. Comparison between measured and predicted capacities for 25 suction caissons (uncorrected shear strength, α = 1, N = 9). 45

76 In the majority of tests in the database, the undrained shear strength is measured using T-bar penetration tests (using T-bar factors of 10.5) and the values are expected to correlate well with shear strength values measured using the direct simple shear test (Watson et al. 2000). To provide a consistent analysis of the data, a correction factor F c is introduced to calculate an equivalent standard undrained shear strength. Since most of the cases in the database utilize the T-bar test, and since results from T-bar tests are expected to correlate well to results from direct simple shear tests, the correction factor is defined as the ratio of the shear strength determined using direct simple shear tests to the shearing strength measured using some other technique. Generally, undrained shear strength values that are measured in direct simple shear tests are about 70% to 80% of shear strength values obtained from triaxial compression tests and vane shear tests (Watson et al. 2000). As a result, an F c value of 0.75 is used to calculate equivalent undrained shear strengths for the tests reported by Clukey and Morrison (1993) and Cho et al. (2003). In addition, values of undrained strength for the tests reported by Clukey et al. (2003) are increased by 20%, based on the recommendations of the authors, to account for differences in the shearing rates used in the direct simple shear tests and the centrifuge pullout tests. The centrifuge tests were conducted at a rate that was 1000 times faster than the shearing rate used in the direct simple shear tests. Additional direct simple shear tests that were conducted at higher rates of loading indicated a 7% increase in the undrained shear strength per log cycle of loading. The ratio of measured to predicted capacities is reevaluated for the 25 tests using an alpha of 1 and an N of 9 and plotted on Figure 3.3. Results indicate a small difference between the corrected and uncorrected cases with an average ratio of measured to predicted capacity of 1.03 (compared to 0.98) and a coefficient of variation in the ratio of measured to predicted capacity of 0.31 (compared to 0.28). 46

77 Measured Capacity / Predicted Capacity Luke et al 2003 Clukey and Morrison 1993 Clukey et al 2003 Randolph and House 2002 Cho et al 2003 House and Randolph Measured Capacity (kips) Figure 3.3. Comparison between measured and predicted capacities for 25 suction caissons (corrected shear strength; direct simple shear, α = 1, N = 9). This initial analysis indicates that an alpha of 1 and an end bearing factor of 9 can be used to calculate an approximately unbiased estimate of the capacity of suction caissons in normally consolidated clay. The uncertainty in the model can be represented with a coefficient of variation of 0.3 in the ratio of measured to predicted capacity. A more detailed analysis of the data was conducted by (1) disregarding tests that were conducted immediately after installation, and (2) distinguishing between tests conducted on sealed and vented caissons. The calculated values of the mean and coefficient of variation of the ratio of measured to predicted capacity for each test case are summarized in Table 3.3. Results in Table 3.3 indicate that (1) for the four vented 47

78 tests in which a seemingly adequate setup time was allowed prior to load testing, predictions of the uplift capacity using an alpha of 1 tend to overestimate the capacity by about 30%, (2) for the 17 sealed tests in which a seemingly adequate setup time was allowed prior to load testing, predictions using an alpha of 1 and an end bearing factor of 9 provide a relatively unbiased prediction of capacity, and (3) when only the 17 sealed tests are analyzed, the uncertainty in the prediction model decreases noticeably; the coefficient of variation in the ratio of the measured to predicted capacity decreases to 0.17 or 0.25 depending on whether the uncorrected or corrected undrained shear strength values are used in predicting capacity. An attempt to calibrate the parameters of the model using the available data was conducted and combinations of α and N that resulted in an unbiased model were calculated and presented in Table 3.4. Coefficients of variation in the ratio of measured to predicted capacity corresponding to these combinations of α and N were also calculated (Table 3.4). Tests where the caissons were pulled out immediately after installation were excluded from the analysis. Results indicate that different combinations of α and N can be used to predict the measured capacities in an unbiased manner. For values of α ranging from 0.7 to 1, the corresponding N factors range from 13 to 8.5 respectively, while corresponding c.o.v s range from 0.26 to It should be noted that the combination of α = 0.7 and N = 13 provides an unbiased estimate for both the vented tests and the sealed tests even when analyzed separately. 48

79 Table 3.3. Biases and uncertainties in model for suction caissons (α=1, N=9). Uncorrected Undrained Shear Strength Corrected Direct Simple Shear Strength Number of Tests Average Measured / Predicted Coefficient of Variation Measured / Predicted All Tests All Tests (excluding setup = 0) Sealed Tests (excluding setup = 0) Vented Tests (excluding setup = 0) All Tests All Tests (excluding setup = 0) Sealed Tests (excluding setup = 0) Vented Tests (excluding setup = 0) Table 3.4. Calibration of α and N (corrected undrained shear strength and excluding tests with setup = 0). Measured Capacity / Predicted Capacity α N Average Coefficient of Variation

80 3.5 EVIDENCE OF A LOWER-BOUND AXIAL CAPACITY Results from 25 axial pullout tests on suction caissons in normally consolidated clays show significant scatter in the ratio of measured to predicted capacity. Part of the scatter results from uncertainties in the values of alpha, N, and the undrained shear strength that are required in calculating estimates of the mobilized side friction and reverse end bearing for each caisson. As in the case of driven piles, the side friction and end bearing are affected by changes that occur in the properties and state of stress of the soil during installation, subsequent reconsolidation, and loading to failure. For suction caissons, the effects of installation by suction, the presence of the padeye, and the complex mechanism of failure add uncertainty to the predicted capacity. However, it is hypothesized that the uncertainty in the predicted capacity for suction caissons can be reduced by introducing a physical lower-bound capacity that can be calculated by assuming that the shear strength of the soil around the skin and under the tip of the caisson reduces to the fully remolded undrained shear strength. To explore the hypothesis of a lower-bound capacity, an analysis is presented for the axial pullout tests available in the database. In tests in which the top cap of the caisson is vented, the lowerbound side friction is calculated as the sum of the frictional resistance acting on the inner and outer walls of the caisson and the lower-bound reverse end bearing is assumed to act on the annulus of the caisson. In tests in which the top cap is sealed, the lower-bound side friction is calculated from the external skin friction and the lower-bound reverse end bearing is assumed to act on the full cross sectional area of the caisson. An end bearing factor of 9 is used in the analysis. The ratio of the predicted lower-bound capacity to the measured capacity for the 25 load tests is calculated and plotted on Figure 3.4. For all the cases studied, the calculated ratio of the predicted lower-bound capacity to the measured capacity is less 50

81 than 1.0, providing evidence for the existence of a lower-bound axial capacity. The ratio of lower-bound capacities to predicted capacities ranges from 0.35 to 1.0 and has an average value of A summary of measured, predicted and lower-bound capacities for all the tests in the database is presented in Table 3.5. Lower Bound / Measured Capacity Luke et al 2003 Clukey and Morrison 1993 Clukey et al 2002 Randolph and House 2002 Cho et al 2003 House and Randolph Measured Capacity (kips) Figure 3.4. Evidence of a lower-bound capacity for 25 suction caissons. 3.6 SUMMARY Analysis of a database containing results from 25 pullout tests conducted on suction caissons in normally consolidated clays indicates that the capacity of suction caissons in uplift can be predicted using an alpha of 1 and an end bearing factor of 9 without introducing significant bias to the predicted capacity. The coefficient of variation in the ratio of measured to predicted capacity ranges between 0.28 and 0.31 depending on 51

82 whether corrections to the method of shear strength measurements are made. Other combinations of alpha and N can also be used to predict the capacity in an unbiased manner. For values of alpha ranging from 0.7 to 1, the corresponding N factors range from 13 to 8.5 respectively, while corresponding c.o.v s range from 0.26 to Results from 25 axial pullout tests on suction caissons in normally consolidated clays provide evidence of the existence of a lower-bound capacity that can be calculated using the undrained remolded shear strength of the soil and information about the geometry of the caisson. Table 3.5. Summary of measured, predicted, and lower-bound capacities for suction caissons in normally consolidated clays. Test Measured Capacity Predicted Capacity, (kips) Lower-Bound Ratio of Lower Bound # (kips) (Uncorrected, α = 1.0, N = 9.0) Capacity (kips) to Predicted Capacity

83 Chapter 4. Models for Lower-Bound Capacity 4.1 INTRODUCTION In this chapter, practical approaches for modeling lower-bound capacities are investigated to aid in providing a mathematical framework for including lower-bound capacities in reliability assessments. Several types of probability distributions that can accommodate a lower-bound capacity are presented and discussed. These types include bounded distributions, truncated distributions, mixed distributions, and general distributions that are based on Hermite polynomial transformations. Examples of bounded distributions include the uniform, triangular, shifted exponential, shifted lognormal and beta distributions, whereas commonly used examples of truncated distributions include truncated normal and lognormal distributions. Mixed probability distributions are distributions that are characterized by a probability mass function representing values of capacity that are at the lower-bound capacity and by a continuous distribution function for values of capacity that are greater than the lower-bound capacity. Common examples of mixed distributions include mixed normal and lognormal probability distributions. Hermite Polynomials are general probability distributions that are derived from the cumulative distribution function of a standard normal distribution and can be used to represent distributions for capacity with arbitrary geometrical shapes. 4.2 CONVENTIONAL PROBABILITY DISTRIBUTIONS FOR CAPACITY Traditionally, normal and the lognormal probability distributions have been used to model the uncertainty in the capacity in conventional reliability analyses. Parameters 53

84 and mathematical forms of normal and lognormal distributions are described in the following sections. The advantages and disadvantage of each probability distribution are stated and discussed Normal Distribution The normal distribution (Gaussian distribution) has a probability density function (PDF) that is defined over a range of values that extends from a lower-bound capacity r LB equal to negative infinity to an upper-bound capacity r UB equal to positive infinity. The normal distribution is symmetrical in shape and is defined by two parameters, the mean capacity µ R and the standard deviation σ R (or coefficient of variation δ R ). The coefficient of variation is defined as the ratio of the standard deviation to the mean capacity. Because of symmetry, the mean of the normal distribution is equal to the median value (50 th percentile value). The PDF of the normal distribution is given by the following mathematical expression: f R (r) 2 1 r µ R 1 2 σ R = exp (4.1) 2πσ R Examples of normal distributions that are described by a mean capacity µ R that is equal to 1000 kips and a coefficient of variation δ R that is equal to 0.3, 0.4, and 0.5 are plotted on Figure 4.1. The probability density function (PDF) and the cumulative distribution function (CDF) of these normal distributions are shown on Figures 4.1(a) and 4.1(b) respectively. 54

85 Probability Density pdf Function, PDF (a) δ R = 0.3 δ R = 0.4 δ R = 0.5 Normal Distribution Cumulative Distribution cdf Function, CDF υr = φ -1 (FR(r)) Capacity, r 1.0 (b) Capacity, r (c) Capacity, r Figure 4.1. Normal probability distribution (r median = µ R = 1000, δ R = 0.3, 0.4, 0.5). 55

86 In addition, the relationship between the actual values of capacity (r) and the standardized or reduced values of capacity (u R ) for these distributions is presented on Figure 4.1(c). The relationship between the actual and reduced values of capacity is defined by the Rosenblatt transformation as: u R 1 = Φ (F (r)) (4.2) R where F R (r) is the cumulative distribution function of the capacity and Φ -1 () is the inverse of the cumulative standard normal function. The transformation is equivalent to finding the value of u R in standard normal space that has the same cumulative density as the variable r in the space of original variables. The reduced capacity u R represents the equivalent number of standard deviations in reduced space that separate the capacity r from the mean capacity µ R. The concept of a reduced capacity (and reduced load) is common in reliability-based analyses for evaluating the probability of failure. The normal distribution is widely used in different areas of civil engineering due primarily to the simplicity of its mathematical form and to the physical significance of the parameters describing it. However, when used to model the uncertainty in the capacity of engineering systems, the normal distribution has a major shortcoming in that the lefthand tail of the distribution can extend to values of capacity that are less than zero (see Figure 4.1). Negative values of capacity are not physically possible in engineering design. This drawback becomes more pronounced when the uncertainty in the capacity is relatively large (see Figure 4.1), which is typically the case for probability distributions describing the uncertainty in the capacity for applications in geotechnical engineering. 56

87 4.2.2 Lognormal Distribution The lognormal distribution has a probability density function (PDF) that is defined over a range of values that extends from a lower-bound capacity r LB that is equal to zero to an upper-bound capacity r UB equal to positive infinity and can be expressed as: f R (r) 2 1 ln(r) λr 1 2 ζ R = exp (4.3) 2πrζ R The lognormal distribution has positive skewness (skewed to the left) and is defined by two parameters, λ R and ζ R, which are in turn related to the median capacity (r median ) and coefficient of variation (δ R ) using the following equations: 2 λ = ln( ) and ζ ln( 1+ ) R r median = (4.4) R δ R The parameters λ R and ζ R of the lognormal distribution represent the mean and standard deviation of the natural logarithm of the capacity r. Examples of lognormal distributions that are described by a median capacity (r median ) that is equal to 1000 kips and a coefficient of variation δ R that is equal to 0.3, 0.4, and 0.5 are plotted on Figure 4.2. The lognormal distribution has been used extensively to model the uncertainty in the load and capacity in conventional reliability analyses in civil engineering in general and in geotechnical engineering in particular (Tang 1988 and 1990; Hamilton and Murff 1992; Tang and Gilbert 1993, API 1993; Horsnell and Toolan 1996; Bea et al. 1999; McVay 2000, 2002 and 2003; Kulhawy and Phoon 2002; Phoon et al. 2003; AASHTO 2004). 57

88 Probability Density pdf Function, PDF (a) δ R = 0.3 δ R = 0.4 δ R = 0.5 Lognormal Distribution Capacity, r Cumulative Distribution cdf Function, CDF 1.0 (b) Capacity, r (c) 2.0 ur = φ -1 (FR(r)) Capacity, r Figure 4.2. Lognormal probability distribution (r median = 1000, δ R = 0.3,0.4,0.5). 58

89 The main reasons for the wide-spread use of the lognormal distribution are related to the fact that it is skewed to the left and has a lower bound of zero. However, the lognormal distribution, with a lower tail that extends to zero, does not capture the realistic possibility that there is a physical minimum or lower bound for the capacity of geotechnical engineering systems. This lower-bound capacity is typically greater than zero and is not modeled properly by conventional lognormal distributions. 4.3 BOUNDED PROBABILITY DISTRIBUTIONS FOR CAPACITY Information about lower-bound capacities can be incorporated into reliability assessments using probability distributions that have finite lower-bound values that can be greater than zero. Examples of such commonly used distributions are the uniform, triangular, shifted exponential, shifted lognormal and beta distributions Uniform Distribution The uniform distribution is defined by a probability density function (PDF) that has a constant value over a range defined by a lower-bound capacity r LB and an upperbound capacity r UB (Figure 4.3) and is expressed as: f R 1 (r) = for rlb r rub (4.5) r r UB LB f R (r) = 0 elsewhere where f R (r) is the probability density function of the capacity. The mean µ R and the standard deviation σ R of a uniform distribution are given by ( r r ) 2 σ R ( r r ) 12 UB LB uniform distribution are identical. 59 µ R LB + UB = and = respectively. Because of symmetry, the mean and median of the

90 Probability Density Function, PDF pdf Uniform Distribution r LB Capacity, r r UB Figure 4.3. Uniform probability density function. Uniform distributions for capacity with a median (r median ) of 1000 and a coefficient of variation (δ R ) of 0.3, 0.4, and 0.5 are plotted on Figure 4.4. A uniform distribution is typically used to model a random variable when the information about its variability is limited to the range of values that it can take. A disadvantage of a uniform distribution is that for a given mean and standard deviation (coefficient of variation), the lower-bound and upper-bound capacities of a uniform distribution are fixed Triangular Distribution A triangular distribution is defined by a lower bound r LB, an upper bound r UB, and a vertex r V (Figure 2.5). If h is the value of the probability density function at the vertex r V, h can be calculated such that the area under the probability density function is equal to 1.0 as: h = 2 / (r UB r LB ). The triangular distribution can then be expressed as: f f R R (r) (r) h hr LB = r for LB V rv rlb rv rlb h hr UB = r + for V UB rub rv rub rv 60 r r r r (4.6) r r (4.7)

91 υr = φ -1 (FR(r)) Cumulative Distribution cdf Function, CDF Probability Density pdf Function, PDF (a) Uniform Distribution δ R = 0.3 δ R = 0.4 δ R = Capacity, r 1.0 (b) Capacity, r (c) Capacity, r Figure 4.4. Uniform probability distribution (r median = 1000, δ R = 0.3,0.4,0.5). 61

92 Probability Density Function, PDF pdf h Triangular Distribution r LB r V Capacity, r r UB Figure 4.5. Triangular probability density function. The mean µ R and the variance σ 2 R of a triangular distribution are given by Equations 4.8 and 4.9 respectively: µ R 2 σ R = 3 2 LB V ( r r ) 2( r r ) 6( r r ) 3( r r ) 2( r r ) 6( r r ) V hr 3 V LB hr V r 2 LB V LB + hr V 3 LB hrlb µ R hrlb µ RhrLB = r r V LB 2µ R hrv µ R hrv hr µ hr µ hr UB R UB R r r UB V 2µ R hrv µ R hrv LB + + µ 2 UB R + µ hr UB hr R 2 V hr 3 V V hr 3 2 V 3 V hr 3 µ 3 V hr UB 2 R µ hr 2 R r V hr V r V LB r + hr + 4 UB hr UB 4 V hr UB + 4 V V (4.8) (4.9) 62

93 Triangular probability distributions with a mean of 1000 and coefficients of variation of 0.3, 0.4, and 0.5 are shown on Figure 4.6. The parameters of the triangular distributions (r LB, r UB, and r V ) are chosen such that the means and coefficients of variation of the distributions are equal to the specified values. One disadvantage of triangular distributions is that they have been rarely used in conventional reliability analyses. However, triangular distributions are used in other areas of civil engineering and serve as a likely alternative for modeling bounded distributions for capacity Shifted Exponential Distribution A shifted exponential distribution has the same mathematical form as a conventional exponential distribution, but has a finite lower-bound value r LB. A shifted exponential probability density function (PDF) with a mean µ R and a lower-bound capacity r LB is given by: f R (r) ( r ) λ r = λe LB for r > r LB (4.10) f R (r) = 0 for r < r LB where λ 1 ( ) = µ R r LB. The standard deviation of the shifted exponential distribution is given by σ R = 1 λ. Shifted exponential distributions with a mean µ R of 1000 and with coefficients of variation of 0.3, 0.4, and 0.5 are presented on Figure 4.7. A limitation of the shifted exponential distribution is that the lower-bound capacity is fixed when the mean and the coefficient of variation are specified. For example, lower-bound capacities that are associated with the shifted exponential distributions shown on Figure 4.7 are constrained to 400, 500, and 600 kips for coefficients of variation of 0.3, 0.4, and 0.5 respectively. 63

94 Probability Density pdf Function, PDF Cumulative Distribution Function, CDF cdf (a) Triangular Distribution δ R = 0.3 δ R = 0.4 δ R = Capacity, r 1.0 (b) Capacity, r (c) 2.0 ur = φ -1 (FR(r)) Capacity, r Figure 4.6. Triangular distribution (µ R = 1000, δ R = 0.3,0.4,0.5). 64

95 ur = φ -1 (FR(r)) Cumulative Distribution Function, CDF Probability Density Function, PDF pdf cdf (a) δ R = 0.3 Shifted Exponential δ R = 0.4 δ R = Capacity, r 1.0 (b) Capacity, r 4 3 (c) Capacity, r Figure 4.7. Shifted exponential distribution (µ R = 1000; δ R = 0.3,0.4,0.5). 65

96 4.3.4 Shifted Lognormal Distribution A shifted lognormal distribution has the same mathematical form as a conventional lognormal distribution, but has a finite lower-bound value r LB. A shifted lognormal probability density function (PDF) with a median r median, a coefficient of variation δ R and a lower-bound capacity r LB is given by: f R (r) 2 1 ln(r-rlb ) λr 1 2 ζ R = exp for r > r LB 2πζ R (r - r LB ) f R (r) = 0 for r < r LB (4.11) where λ R and ζ R are defined in terms of the median capacity r median and the coefficient of variation δ R of the conventional lognormal distribution using Equation 4.4. The mean and coefficient of variation of the shifted lognormal distribution are given by: µ R r> r LB = r LB + r median 2 ( 1+ δ ) R δ R = δ r> r R (4.12) LB Shifted lognormal distributions with a median capacity ( r > ) of 1000 and median r with coefficients of variation of 0.3, 0.4, and 0.5 are presented on Figure 4.8. For illustration, all the distributions shown on Figure 4.8 are characterized by a lower-bound capacity that is equal to 400 kips (ratio of lower-bound to median capacity = 0.4). Since most reliability analyses in geotechnical engineering tend to utilize the conventional lognormal distribution to model the uncertainty in the capacity, the use of a shifted lognormal distribution to constrain the left hand tail of the distribution of capacity 66 r LB

97 Probability Density pdf Function, PDF (a) δ R = 0.5 δ R = 0.3 δ R = 0.4 Shifted Lognormal Cumulative Distribution Function, CDF cdf Capacity, r 1.0 (b) Capacity, r (c) 2.0 ur = φ -1 (FR(r)) Capacity, r Figure 4.8. Shifted lognormal distribution ( r > = 1000; δ R = 0.3,0.4,0.5, r LB = 400). median r r LB 67

98 at a certain lower-bound value seems plausible. The parameters of the shifted lognormal distribution are similar to those of the conventional lognormal distribution with the addition of one parameter, the lower-bound capacity r LB. However, results on Figure 4.8 indicate that the left-hand tail of the shifted lognormal distribution does not initiate from the value of the specified lower-bound capacity (r LB = 400), but rather from a larger value (greater than 500). This observed behavior is related to the characteristics of the mathematical form of the lognormal probability density function Beta Distribution A beta distribution with a mean µ R and a standard deviation σ R is defined over a range r LB to r UB and has the following probability density function: f R (r) ( α 1) ( β 1) LB UB = (4.12) r UB 1 r LB Γ(α + β) r r Γ(α)Γ(β) rub r LB r r UB r r LB 2 X =, Y where ( 1 X) X α 2 function defined as: ( X) α 1 β =, X X µ 68 r R LB R =, Y = and Γ() is the Gamma rub rlb rub rlb k 1 x Γ(k) = x e dx for k > 0. Depending on the values and signs of the 0 parameters α and β, the beta distribution can assume different geometrical shapes. Seven variations of the beta distribution are presented in Table 4.1. Beta distributions with a mean µ R of 1000 and coefficients of variation of 0.3, 0.4, and 0.5 are presented on Figures 4.9 and The distributions shown on Figure 4.9 represent the case where the lower-bound capacity is relatively small and is equal to 400 (ratio of lower-bound to mean capacity of 0.4) whereas the distributions on Figure 4.10 represent the case where the lower-bound capacity is relatively large and is equal to 600 σ

99 (ratio of lower-bound to mean capacity of 0.6). For all the distributions shown on Figures 4.9 and 4.10, the upper bound of the distributions is set to a fixed value of 3000 (three times the mean capacity). The beta distribution is widely used in different areas of civil engineering. The main advantages of the beta distribution are related to the fact that it allows for lower and upper bounds and for flexibility in the shape of the distribution. One main conclusion can be drawn from Figures 4.9 and 4.10 regarding the applicability of the beta distribution in modeling distributions for capacity. The shape of the left-hand tail of the beta distribution tends towards the shape of the tail of the shifted exponential distribution when (1) the coefficient of variation of the distribution increases (with all other parameters left constant) and (2) the lower-bound capacity increases relative to the mean capacity (Figures 4.9(a) and 4.10(a)). Table 4.1. Variations of the beta distribution. α β Type of Distribution Shape of Distribution < 0 < 0 Bath Tub 0 0 Uniform 1 0 Triangular 0 1 Triangular f(x) f(x) f(x) f(x) a a a a b b b b x x x x Normal f(x) a b x α > β Skewed Right f(x) a b x α < β Skewed Left 69 f(x) a b x

100 ur = φ -1 (FR(r)) Cumulative Distribution Function, CDF Probability Density Function, PDF cdf pdf (a) Beta Distribution δ R = 0.5 δ R = 0.4 δ R = Capacity, r 1.0 (b) Capacity, r (c) Capacity, r Figure 4.9. Beta distribution (µ R = 1000; δ R = 0.3,0.4,0.5, r LB = 400, r UB = 3000). 70

101 pdf Probability Density Function, PDF Cumulative Distribution cdf Function, CDF (a) δ R = 0.5 Beta Distribution δ R = 0.4 δ R = Capacity, r 1.0 (b) (c) Capacity, r 2.0 ur = φ -1 (FR(r)) Capacity, r Figure Beta distribution (µ R = 1000; δ R = 0.3,0.4,0.5, r LB = 600, r UB = 3000). 71

102 4.4 TRUNCATED PROBABILITY DISTRIBUTIONS FOR CAPACITY Most conventional reliability analyses in geotechnical engineering assume normal or lognormal distributions for modeling the uncertainty in the capacity. A simple approach for incorporating lower-bound capacities in reliability analyses is through the use of truncated or conditional probability distributions (Fjeld 1977, Rodriguez et al. 1998). Normal or lognormal distributions for capacity that are truncated at a lower-bound capacity (r LB ) can be used to fulfill this purpose. Mathematical expressions describing probability density functions of truncated normal and lognormal distributions are presented in Equations 4.13 and 4.14 respectively. f f R r> rlb R r> rlb 2 1 r µ R σ R (r r > rlb ) = exp rlb µ R 2πσ R 1 Φ σ R (4.13) 2 1 ln(r) λ R ξ R (r r > rlb ) = exp ln(rlb) λ R ) 2πξ R r 1 Φ ξ R (4.14) The first terms in Equations 4.13 and 4.14 are scaling factors needed to maintain an area of 1.0 under the probability density function of the capacity. Examples of normal and lognormal distributions that are truncated at a lower-bound capacity (r LB ) are presented in Figures 4.11 to For all distributions, the median capacity r median is specified to be 1000 and coefficients of variation of 0.3, 0.4, and 0.5 are used. The cases shown on Figures 4.11 and 4.13 represent the case where a relatively small lower-bound capacity of 400 is used (ratio of lower-bound to median capacity of 0.4), whereas the 72

103 73 cases shown on Figures 4.12 and 4.14 represent the case where a larger lower-bound capacity of 600 is used (ratio of lower-bound to median capacity = 0.6). Expressions for the mean and variance of truncated distributions are given by Equations 4.15 to 4.18 as: + = > R R LB σ µ r 2 1 R R r r R σ µ r Φ 1 1 e 2π σ µ µ 2 R R LB LB (4.15) 2 r r R 2 R R R LB σ µ r 2 1 R R LB 2 R R R LB σ µ r 2 1 R R 2 R 2 r r R LB 2 R R LB 2 R R LB LB µ σ σ µ r Φ 1 1 e σ µ r 2π σ σ µ r Φ 1 1 e 2π σ 2µ µ σ > > = (4.16) ( ) ) ξ (2λ 2 1 R 2 R R LB R R LB r r R 2 R R LB e ξ ξ λ lnr Φ 1 ξ λ lnr Φ 1 1 µ + > + = (4.17) ( ) 2 r r R ) 4ξ (4λ 2 1 R 2 R R LB R R LB 2 r r R LB 2 R R LB µ e ξ 2ξ λ lnr Φ 1 ξ λ lnr Φ 1 1 σ > + > + = (4.18) where µ R and σ R are the mean and standard deviation of the conventional non-truncated distributions respectively. Normal Log-Normal

104 Probability Density Function, PDF pdf Cumulative Distribution Function, CDF cdf (a) δ R = 0.3 Truncated Lognormal δ R = 0.4 δ R = Capacity, r 1.0 (b) Capacity, r (c) 2.0 ur = φ -1 (FR(r)) Capacity, r Figure Truncated lognormal distribution (r median = 1000, δ R = 0.3,0.4,0.5, r LB = 400). 74

105 ur = φ -1 (FR(r)) Cumulative Distribution Function, CDF Probability Density Function, PDF cdf pdf (a) δ R = 0.3 Truncated Lognormal δ R = 0.4 δ R = Capacity, r 1.0 (b) Capacity, r (c) Capacity, r Figure Truncated lognormal distribution (r median = 1000, δ R = 0.3,0.4,0.5, r LB = 600). 75

106 ur = φ -1 (FR(r)) Cumulative Distribution Function, CDF Probability Density Function, PDF cdf pdf (a) δ R = 0.3 Truncated Normal δ R = 0.4 δ R = Capacity, r 1.0 (b) Capacity, r (c) Capacity, r Figure Truncated normal distribution (r median = 1000, δ R = 0.3,0.4,0.5, r LB = 400). 76

107 ur = φ -1 (FR(r)) Cumulative Distribution Function, CDF Probability Density Function, PDF cdf pdf (a) δ R = 0.3 Truncated Normal δ R = 0.4 δ R = Capacity, r 1.0 (b) Capacity, r (c) Capacity, r Figure Truncated normal distribution (r median = 1000, δ R = 0.3,0.4,0.5, r LB = 600). 77

108 The use of truncated distributions is convenient because the parameters describing them are the same as those of the non-truncated distribution with the addition of one extra parameter, the lower-bound capacity (r LB ). However, the means and coefficients of variation of truncated normal or lognormal distributions can be quite different than the means and coefficients of variation of non-truncated distributions, especially as the lower-bound capacity increases and becomes closer to the mean or median capacity. The difference (in percent) between the means of truncated and non-truncated lognormal and normal probability distributions are calculated and presented in Tables 4.2 and 4.3 respectively. For truncated lognormal distributions of capacity, results presented in Table 4.2 indicate that the difference in percent between the means of truncated and non-truncated distributions increases as the coefficient of variation of the capacity increases and as the ratio of the lower bound to the median capacity increases. The same behavior is observed for truncated normal distributions as seen in Table 4.3. For practical ratios of lower bound to median capacity (0.4 to 0.7), the difference in the means is less than 13% for the truncated lognormal case and less than 20% for the truncated normal case. As the lower-bound capacity approaches the median, the difference in the means approaches 30% for coefficients of variation that are relatively large. Coefficients of variation of truncated distributions also differ from those of nontruncated distributions as observed in Tables 4.4 and 4.5 for truncated lognormal and normal distributions respectively. Differences in the calculated values for the coefficients of variation also become more pronounced as the coefficient of variation of the nontruncated distribution increases and as the ratio of the lower bound to the median capacity increases. The difference is relatively small for ratios of lower-bound to median capacity that are less than about 0.6, and becomes significant as the lower-bound capacity approaches the median capacity. 78

109 Table 4.2. Difference between the means of truncated and non-truncated lognormal distributions (in percent, %). Coefficient of Variation of the Capacity, δ R Ratio of Lower-Bound to Median Capacity Table 4.3. Difference between the means of truncated and non-truncated normal distributions (in percent, %). Coefficient of Variation of the Capacity, δ R Ratio of Lower-Bound to Median Capacity

110 Table 4.4. Coefficients of variation of truncated lognormal distributions. Coefficient of Variation of the Capacity, δ R Ratio of Lower-Bound to Median Capacity Table 4.5. Coefficients of variation of truncated normal distributions. Coefficient of Variation of the Capacity, δ R Ratio of Lower-Bound to Median Capacity

111 4.5 MIXED PROBABILITY DISTRIBUTIONS FOR CAPACITY Another convenient mathematical model to account for a lower-bound capacity is shown on Figure For capacities greater than the lower bound, the distribution is a conventional continuous probability density function (lognormal, beta, uniform, Hermite Polynomial, etc.). For capacities at the lower bound, there is a finite probability that corresponds to the probability of being less than or equal to the lower bound in the nontruncated distribution. In a practical application, the lower bound can be represented by a range to account for testing errors in analyzing data or uncertainty in estimating it. Expressions for the mean and variance of a mixed probability distribution with a lognormal probability density function are given in Equations 4.19 and 4.20 respectively. µ σ R r> rlb lnr ( λ ξ ) (2λ R + ξ R ) LB R R 2 = rlbp + 1 Φ e (4.19) ξ R ( ) (4λ 4ξ ) 2 lnrlb λ R + 2ξ R + R R 2 2 = rlbp + 1 Φ e (4.20) R r> rlb ξ R 2 µ R r> rlb p R (r) = p = F R (r LB ) for r = r LB f R (r) = conventional distribution for r > r LB pr(r) and fr(r) p Area = 1 - p r LB Capacity, r Figure Mixed distribution for modeling capacity. 81

112 Examples of mixed lognormal distributions that are described by a median capacity (r median ) that is equal to 1000 kips and a coefficient of variation δ R that is equal to 0.3, 0.4, and 0.5 are plotted on Figures 4.16 and The distributions shown on Figure 4.16 represent the case where a relatively small lower-bound capacity of 400 is used (ratio of lower-bound to median capacity of 0.4), whereas the distribution shown on Figure 4.17 represent the case where a larger lower-bound capacity of 600 is used (ratio of lower-bound to median capacity = 0.6). The mixed probability distribution model, with a probability mass function at the lower bound and a probability density function, is convenient because the mean and standard deviation describing it are approximately the same as for a non-truncated distribution, providing that the lower-bound capacity is not too close to the mean. Differences (in percent) between the means of mixed and non-truncated lognormal probability distributions are calculated and presented in Table 4.6. Results indicate that for all ratios of lower bound to median capacity that are less than 0.9, the difference in the means is less than 10%, which can be considered small compared to the difference in the means of truncated and non-truncated lognormal distributions (compare Table 4.6 to Table 4.2). Similar conclusions can be drawn from comparing the difference between the coefficients of variation of mixed and non-truncated lognormal distributions as shown in Table 4.7. The fact that the means and standard deviations of the mixed lognormal distribution do not differ significantly from means and standard deviations of conventional non-truncated distribution for practical values of coefficients of variation of the capacity and practical ratios of lower-bound to median capacities indicates that results from statistical analyses that are published in the literature can be incorporated directly into the mixed probability model without having to re-analyze the raw data. 82

113 Probability Density Function, PDF pdf Cumulative Distribution Function, CDF cdf (a) Mixed Lognormal δ R = 0.3 δ R = 0.4 p = p = δ R = 0.5 p = Capacity, r 1.0 (b) Capacity, r (c) 2.0 ur = φ -1 (FR(r)) Capacity, r Figure Mixed lognormal distribution (r median = 1000, δ R = 0.3,0.4,0.5, r LB = 400). 83

114 Probability Density Function, PDF pdf (a) p = 0.14 p = p = Mixed Lognormal δ R = 0.3 δ R = 0.4 δ R = 0.5 Cumulative Distribution Function, CDF cdf Capacity, r 1.0 (b) Capacity, r (c) 2.0 ur = φ -1 (FR(r)) Capacity, r Figure Mixed lognormal distribution (r median = 1000, δ R = 0.3,0.4,0.5, r LB = 600). 84

115 Table 4.6. Difference between the means of the mixed lognormal and conventional lognormal distributions (in percent, %). Coefficient of Variation of the Capacity, δ R Ratio of Lower-Bound to Median Capacity Table 2.7. Coefficients of variation of mixed lognormal distributions. Coefficient of Variation of the Capacity, δ R Ratio of Lower-Bound to Median Capacity

116 4.6 COMPARISON BETWEEN PROBABILITY DISTRIBUTIONS FOR CAPACITY The different types of probability distributions that were presented and discussed in Sections 4.3, 4.4, and 4.5 are tools for modeling the uncertainty in the capacity in a realistic manner. All these probability distributions can incorporate a lower-bound capacity in the distributional form of the capacity and can thus provide a realistic model for capacity in reliability-based calculations. A comparison between a variety of bounded, truncated and mixed probability distributions is shown on Figures 4.18 and Examples of bounded distributions on Figures 4.18 and 4.19 include uniform, shifted exponential, shifted lognormal, and beta distributions, whereas examples of truncated distributions include truncated normal and lognormal distributions. Mixed lognormal distributions are also plotted on Figures 4.18 and 4.19 for comparison. For illustration, all distributions are characterized by a lowerbound capacity of 400 (Figure 4.18) or 600 (Figure 4.19), and a median (or mean capacity for the beta and shifted exponential distribution) of For truncated normal and lognormal distributions, beta distributions, shifted lognormal distributions, and mixed lognormal distributions, a coefficient of variation of 0.5 is used in the analysis. For all other distributions, the value of the coefficient of variation of the capacity is dictated by the choice of the mean and the lower-bound capacity. The cumulative distribution functions shown on Figures 4.18 and 4.19 provide a wide variety of probabilistic models for the left-hand tail of the distribution for capacity. The choice of a particular probability distribution for modeling the uncertainty in the capacity of geotechnical engineering systems depends entirely on the dataset or database that is being modeled. 86

117 cdf Capacity, r Uniform Beta Shifted Exponential Shifted Lognormal Truncated normal Truncated Lognormal Mixed Lognormal ur = φ -1 (FR(r)) Capacity, r Figure Comparison between different types of probability distribution (r median = 1000, δ R = 0.5, r LB = 400).

118 cdf Capacity, r Uniform Beta Shifted Exponential Shifted Lognormal Truncated normal Truncated Lognormal Mixed Lognormal ur = φ -1 (FR(r)) Capacity, r Figure Comparison between different types of probability distribution (r median or r mean = 1000, δ R = 0.5, r LB = 600).

119 4.7 GENERAL DISTRIBUTIONS BASED ON HERMITE POLYNOMIALS Hermite Polynomials can be used to form a transform function which expresses the relationship between a variate with a non-normal probability distribution to that with a standard normal distribution. As a result, they can be used to model distributions for capacity with arbitrary shapes. In this section, a general description of the mathematical form and characteristics of Hermite Polynomials is presented. Then, the use of Hermite Polynomials in modeling shapes of probability distributions is investigated Hermite Polynomials Hermite Polynomials are derived from the probability density function of a standard normal distribution and are defined as follows (Journel and Huijbregts 1978): 2 u i d 2 H i (u) = e [ f U (u)] (4.21) i du where u is a variable with a standard normal distribution and i is the order of the Hermite Polynomial. Useful characteristic of Hermite Polynomials is that they can be expressed by the following recurrence relation: Hi+ 1 i i 1 (u) = H (u) ih (u) (4.22) Equations and graphs for Hermite Polynomials from the zero through seventh order are presented in Figures 4.20 and As the order of the Hermite Polynomial increases, the tails of the polynomial become more sensitive to the value of u when u is small or large. This behavior illustrates that a higher order Hermite Polynomial will be required to fit the tails of non-normal distributions. 89

120 10 H0(u) 0 H 0 (u) = u H1(u) 0 H 1 (u) = -u H2(u) u H 2 (u) = u u H3(u) H 3 (u) = -u 3 + 3u u Figure Hermite polynomials, order zero through three. 90

121 H4(u) H 4 (u) = u 4-6u 2 +3 H5(u) u H 5 (u) = -u 5 +10u 3-15u u H 6 (u) = u 6-15u 4 +45u 2-15 H6(u) u H 7 (u) = -u 7 +21u 5-105u u H7(u) u Figure Hermite polynomials, order four through seven. 91

122 The transform function that maps a variate with a non-normal probability distribution to that with a standard normal probability distribution is commonly referred to as the Gaussian transform function and can expressed mathematically as follows: Y =ϕ (U) (4.23) Y where Y is a general random variable, U is a standard normal variable, and ϕ (U) Y is the Gaussian transform function. Journel and Huijbergts (1978) showed that this Gaussian transform function for any random variable with a finite variance can be theoretically expressed as a linear combination of Hermite Polynomials: = ψi Y = ϕ Y (U) H i (U) (4.24) = i! i 0 where H i (U) denotes the Hermite Polynomials defined by Equation 4.21 and expressed iteratively by Equation 4.22, U denotes a variable with a standard normal distribution, and ψ i denote the coefficients of this transform function. Even when a polynomial of order n is used, the approximation with Hermite Polynomials provides the best fit for all possible polynomials of order n (Journel and Huijbergts 1978). The Hermite Polynomial transform function can then be expressed in terms of the mean and the variance of Y using the following substitutions: ψ 0 =µ Y σy ψ 1 = ψi n ψ i! i= 2 2 (4.25) 92

123 The transform function is then given by the following: ψ i n σ ψ Y 1 Y µ Y + H 2 1(U) + H i(u) i= 2 i! ψi n ψ 1 1+ i! i= 2 (4.26) Finally, the ratio ψ i ψ is replaced by ' i 1 ψ to simplify the notation (Wang 2002): ' σ n y ψ i Y µ Y + 2 H 1(U) + H i(u) n ' i= 2 i! ( ψi ) 1+ i! i= 2 (4.27) ψ ψ. ' 3 Therefore, the parameters that describe the transform function are µ Y, σ Y, and ψ ' 2, ' n Procedure for Fitting Hermite Polynomials to Probability Distributions Fitting a Hermite Polynomial transform function to a specified distribution for capacity requires that the parameters describing the Hermite Polynomial be determined. This can be accomplished using linear regression by minimizing the square of the error between the actual values of capacity y and their transformed values y n given by Equation Minimize: ( y y ) = dy n 2 (4.28) 93

124 In practice, Equation 4.28 is expressed as a summation from a finite lower-bound capacity to a finite upper-bound capacity. The fitting process can be summarized with the following steps: 1. Define a region of interest that represents the distribution for capacity. In this research, values of capacity that fall within the 0.1 and 99.9 percentiles (i.e., u = Φ -1 [(F Y (y)] = to +3.09) are chosen to provide a practical representation of the distribution for capacity. 2. Subdivide the region of interest into a number of standard normal variables u i that are equally spaced. In this research, the representative range is divided into 50 equally spaced intervals. 3. For each standard normal capacity u i, calculate the corresponding 1 actual value of capacity y F [ Φ(u )] i = and the transformed capacity y ni from Equation Use an optimization technique to find the parameters of the Hermite Polynomial that minimize the sum of the square error given by Equation Y i Fitting Hermite Polynomials to Different Probability Distributions The objective of this section is to determine whether Hermite Polynomial transformations can provide a reliable and flexible tool for modeling arbitrary shapes of 94

125 the left-hand tails of distributions for capacity. The main focus is on determining the order of the Hermite Polynomial that is required to properly model the left-hand tails. To accomplish these objectives, third, fifth, and seventh order Hermite Polynomials are utilized to model probability distributions that can accommodate a lower-bound capacity. These distributions include bounded, truncated, and mixed probability distributions. In a practical sense, this wide spectrum of probability distributions encompasses the different possibilities in which the probability content at the left-hand tail of the distribution for capacity can be distributed. If Hermite Polynomials of a certain order can provide a representative model for the left-hand tails of these distributions, it will be concluded that Hermite Polynomials of this order can be used with confidence to model the left-hand tails of the distribution for capacity in practical problems in geotechnical engineering Bounded Distributions The general procedure outlined in Section was used with third, fifth, and seventh order Hermite Polynomials to model distributions for capacity following uniform, triangular, shifted exponential, shifted lognormal and beta distributions. Uniform Distribution The ability of Hermite Polynomials to model uniform capacity distributions with a mean of 1000 and lower-bound capacities equal to 0.4 and 0.6 times the mean capacity is investigated on Figure Both third and fifth order Hermite Polynomials do not result in a satisfactory model for the tails of the distributions. The observed behavior is reasonable because the tail of the uniform distribution is highly non-normal (flat and has a finite lower-bound). On the other hand, the approximation provided by seventh order Hermite Polynomials results in a good fit even at very low percentile values. 95

126 UR = φ -1 (FR(r)) rlb = 400 rlb = Cummulative Distribution Function, CDF y n Uniform Distribution 3rd Order Hermite Polynomial 5th Order Hermite Polynomial 7th Order Hermite Polynomial rlb = 400 rlb = y, y n Figure Hermite Polynomial approximations for tails of uniform distributions (µ R = 1000, r LB / µ R = 0.4 and 0.6). 96

127 Triangular Distribution Two triangular distributions for capacity having a mean of 1000 and lower-bound capacities of 400 and 600 (0.4 and 0.6 of the mean capacity) are used to study the accuracy in which Hermite Polynomials can model tails of distributions for capacity that increase in a linear manner. Curves showing the fit provided by third, fifth, and seventh order Hermite Polynomials to the triangular distributions are shown on Figure For both distributions, third order Hermite Polynomials do not provide a good fit to the lefthand tails whereas both fifth and seventh order Hermite Polynomials provide a good model of the left-hand tails of the triangular distributions. Shifted Exponential Distribution Hermite Polynomial transform functions that are required to model shifted exponential distributions with a mean of 1000 and lower-bound capacities of 400 and 600 are presented on Figure Both fifth and seventh order Hermite polynomials provide a good fit to the given distributions. The fit provided by third order Hermite Polynomials is relatively poor especially at low percentile values. Beta Distribution Curves showing third, fifth, and seventh order Hermite Polynomial transform functions that are required to fit beta distributions with a mean of 1000 and lower-bound capacities of 400 and 600 are presented on Figures 4.25 and The distributions on Figure 4.25 represent the case where the coefficient of variation is relatively small (δ R = 0.3) whereas the distributions on Figure 4.26 represent the case where the coefficient of variation is relatively large (δ R = 0.5). For all the cases analyzed, the upper bound of the beta distribution was taken as 4 times the mean capacity (r UB = 4000). 97

128 UR = φ -1 (FR(r)) r LB = 400 r LB = 600 Cummulative Distribution Function, CDF y n Triangular Distribution 3rd Order Hermite Polynomial 5th Order hermite Polynomial 7th Order Hermite Polynomial r LB = 400 r LB = y, y n Figure Hermite Polynomial approximations for tails of triangular distributions (µ R = 1000, r LB / µ R = 0.4 and 0.6). 98

129 ur = φ -1 (FR(r)) r LB = 400 r LB = Cummulative Distribution Function, CDF Shifted Exponential Distribution 3rd Order Hermite Polynomial 5th Order Hermite Polynomial 7th Order Hermite Polynomial y, y n Figure Hermite Polynomial approximations shifted exponential distributions. 99

130 ur = φ -1 (FR(r)) r LB = 400 r LB = 600 Cummulative Distribution Function, CDF Beta Distribution 3rd Order Hermite Polynomial 5th Order Hermite Polynomial 7th Order Hermite Polynomial r LB = 400 r LB = y, y n Figure Hermite Polynomial approximations for beta distributions (δ R = 0.3). 100

131 0.0 ur = φ -1 (FR(r)) r LB = 400 r LB = 600 Cummulative Distribution Function, CDF r LB = 400 Beta Distribution 3rd Order Hermite Polynomial 5th Order Hermite Polynomial 7th Order Hermite Polynomial r LB = y, y n Figure Hermite Polynomial approximations for beta distributions (δ R = 0.5). Third, fifth, and seventh order Hermite Polynomial transformations provide an acceptable fit to the left-hand tail of the beta distribution with the smaller lower-bound capacity (r LB = 400) and the smaller coefficient of variation (δ R = 0.3). The fit provided by the third order polynomial becomes poor as the lower-bound capacity and coefficient of variation increase. For all the cases studied, seventh order Hermite Polynomials provide a representative model of the left-hand tails of the beta distributions. 101

132 Truncated Distributions Hermite Polynomials of different orders are used to model truncated normal and lognormal distributions having a median of 1000 and lower-bound capacities of 400 and 600. The curves on Figures 4.27 and 4.28 correspond to truncated normal distributions with coefficients of variation of 0.3 and 0.5 respectively, whereas the curves shown on Figures 4.29 and 4.30 correspond to truncated lognormal distributions with coefficients of variation of 0.3 and 0.5 respectively ur = φ -1 (FR(r)) r LB = 400 r LB = 600 Cumulative Distribution Function, CDF Truncated Normal Distribution 3rd Order Hermite Polynomial 5rd Order Hermite Polynomial 7th Order Hermite Polynomial y, y n Figure Hermite Polynomial approximations for truncated normal distributions (δ R = 0.3). 102

133 ur = φ -1 (FR(r)) rlb = 400 rlb = 600 Cumulative Distribution Function, CDF Truncated Normal Distribution 3rd Order Hermite Polynomial 5rd Order Hermite Polynomial 7th Order Hermite Polynomial y, y n Figure Hermite Polynomial approximations for truncated normal distributions (δ R = 0.5). Results on Figures 4.27 to 4.30 indicate that seventh order Hermite Polynomials provide an accurate model of the left-hand tails of all the truncated distributions that are analyzed. The approximations provided by fifth order Hermite Polynomials to the truncated distributions shift from the actual distribution in the close proximity of the truncation point. The same behavior is observed for approximations provided by third order Hermite Polynomials. 103

134 ur = φ -1 (FR(r)) r LB = 400 r LB = 600 Cumulative Distribution Function, CDF Truncated Lognormal Distribution 3rd Order Hermite Polynomial 5rd Order Hermite Polynomial 7th Order Hermite Polynomial y, y n Figure Hermite Polynomial approximations for truncated lognormal distributions (δ R = 0.3). 104

135 ur = φ -1 (FR(r)) rlb = 400 rlb = 600 Cumulative Distribution Function, CDF Truncated Lognormal Distribution 3rd Order Hermite Polynomial 5rd Order Hermite Polynomial 7th Order Hermite Polynomial y, y n Figure Hermite Polynomial approximations for truncated lognormal distributions (δ R = 0.5) Mixed Distributions Theoretically, the continuous part of a mixed distribution can be modeled using any conventional probability density function (normal, lognormal, beta, uniform etc.). A more general model for a mixed distribution can be obtained with Hermite Polynomial transformations representing capacities greater than the lower bound, with the addition of an extra parameter describing the probability mass at the lower-bound capacity r LB. 105

136 In this section, Hermite Polynomials of different orders are used to model the continuous part of mixed distributions. For illustration, two mixed lognormal distributions having a median of 1000 and a coefficient of variation of 0.4 are shown on Figure The two distributions are assumed to have lower-bound capacities of 400 and 600 with corresponding probability masses of 0.01 and p = 0.01 Mixed Mixed Lognormal (r LB = 400) pdf p = 0.13 Series2 Mixed Lognormal (r LB = 600) pdf Capacity, r Figure Mixed distribution functions (r median = 1000, δ R = 0.4, r LB = 400 and 600). 106

137 Hermite polynomials that are required to fit the continuous parts of the mixed distributions are plotted on Figure Results indicate that fifth and seventh order Hermite Polynomial transformations are adequate for modeling the lognormal distributions. On the other hand, third order Hermite Polynomials do not provide a good fit to the tail of the distribution for the case were the lower-bound capacity is large. The problem of fitting Hermite Polynomials to the continuous part of a mixed probability distribution is a subset of the more complicated problem of modeling entire continuous probability distributions. Seventh order polynomials proved to be adequate for modeling complicated cases where the left hand tail of the distribution for capacity followed highly non-normal distributions (uniform, triangular, beta with large ratio of lower-bound to mean capacity, etc.). Consequently, seventh order Hermite Polynomial transformations are expected to be adequate for modeling the continuous part of general mixed probability distributions Remarks on Fitting Hermite Polynomials to Different Probability Distributions To explore the generality of the conclusions and observations that resulted from the analysis performed in Section 4.7.3, a wide range of cases was analyzed where both the coefficient of variation and the ratio of the lower-bound to the median capacity were varied. The results of these analyses indicated that the conclusions presented in Section are valid in a general manner and are not restricted to the particular cases that are presented in Sections , , and These results prove that seventh order Hermite Polynomials can be used with confidence to model a wide variety of shapes of left-hand tails of distributions for capacity. 107

138 ur = φ -1 (FR(r)) u -1.0 r LB = r LB = 400 Mixed Lognormal Distribution 3rd Order Hermite Transform Function 5th Order Hermite Transform Function 7th Order Hermite Transform Function Cummulative Distribution Function r LB = y, yn r LB = y, y n Figure Hermite Polynomial approximations for mixed distributions (r median = 1000, δ R = 0.4). 108

139 4.8 SUMMARY In this chapter, practical probability distributions that can accommodate lowerbound capacities are presented. The mathematical form and parameters that describe each probability distribution are described and the advantages and disadvantages of each are identified. The effectiveness of Hermite Polynomial transformations in modeling different shapes of distributions for capacity is also investigated. Based on the findings presented in this chapter, the following conclusions can be drawn: 1. Bounded distributions (uniform, triangular, shifted exponential, shifted lognormal, and beta) can be used to model lower-bound capacities. One disadvantage of these distributions is that they have been rarely used in conventional reliability analyses. However, bounded distributions have been used frequently in other areas of civil engineering and serve as a likely alternative for modeling distributions for capacity. 2. Truncated distributions (normal and lognormal) provide simple and practical means for modeling lower-bound capacities. Their main advantage is that the parameters describing them are similar to those used in conventional distributions with the addition of one parameter (lower-bound capacity). 3. A mixed distribution model that is characterized by a PMF at the lower-bound capacity and a PDF for capacities greater than the lower bound is a suitable model for accommodating a lower-bound capacity and maintaining consistency with the conventional approach for modeling capacity. 4. Hermite Polynomial transformations can be used to model general capacity distributions with different tail geometries. 109

140 5. Seventh order Hermite Polynomials provide a very good fit to the left-hand tails of all distributions investigated in this chapter. Fifth order Hermite Polynomials provide an acceptable fit to the left-hand tails of truncated distributions, shifted exponential distributions, and beta distributions, but provide a relatively poor fit for the tails of the uniform and triangular distributions in the close vicinity of the lower-bound capacity. Finally, third order Hermite Polynomials provide a relatively poor fit for almost all the cases studied in this chapter. In most cases, third order polynomials provide an acceptable fit above the 5 th percentile values and for left-hand tail that are normal in shape. 110

141 Chapter 5. Modeling Uncertainty in the Capacity using Databases of Load Tests 5.1 INTRODUCTION In this chapter, databases for axial load tests are utilized to model the uncertainty in the capacity of driven pipe piles in cohesive and cohesionless soils and suction caissons in normally consolidated clays. Initially, information about predicted, measured, and lower-bound capacities are used to calibrate the parameters of bounded, truncated and mixed probability distributions for capacity. These distributions include beta, truncated lognormal, and mixed lognormal probability distributions. Mixed seventh order Hermite Polynomial distributions are then used to provide a more general and flexible model for the distribution for capacity. Seventh order Hermite Polynomials are used as a reference to which the models provided by bounded, truncated, and mixed distributions are compared. Based on the results of the comparison, the mixed lognormal distribution is recommended as a simple and practical probability distribution for capacity. 5.2 PROCEDURE FOR MODELING DISTRIBUTIONS FOR CAPACITY USING DATABASES In this section, a procedure is described for calibrating the parameters of probability distributions for capacity using information about predicted, measured, and lower-bound capacities for a limited number of load tests in a database. It is assumed in the analysis that each data point in a database has specific values for measured capacities (example, capacity from load test), predicted capacities (example, API capacity for driven piles), and lower-bound capacities (example, lower-bound models for capacity of piles as described in Chapter 2). It is also assumed for simplicity that data points in the database 111

142 constitute a set of statistically independent variables, following a probability distribution whose parameters are to be determined. Although some of the data points in the database are expected to be correlated, the assumption of statistical independence is utilized to simplify the calibration procedure. The procedure for determining the parameters of the distribution for capacity involves: (1) defining the parameters that represent the probability distribution that is being modeled, (2) defining a likelihood function that represents the likelihood of observing the measured set of data in the database, and (3) determining the values of the parameters that will maximize the likelihood of observing the measured data. The probability of observing a set of statistically independent data points in a database using a specific set of model parameters is referred to as the likelihood function and is defined as: L( v r φ v n ) = P(data parameters) = PDF(r i ) (5.1) i= 1 where φ v is a vector containing one set of parameters of the probability model for capacity, v r is a vector containing the observed or measured capacities, L( v r φ v ) is the likelihood function, n is the number of test cases in the database, PDF() is the probability density function of the capacity, and PDF(r i ) is the value of the probability density function of the capacity, evaluated at each observed or measured capacity (r i ). The calibration process involves finding the set of parameters that maximizes the value of the likelihood function expressed by Equation 5.1. To avoid working with very small numbers that may result from the use of Equation 5.1, define g( φ v ) as the natural logarithm of the likelihood function: g( φ v ) = ln(l( v r φ v n )) = ln( PDF(r )) i= i (5.2)

143 The set of parameters, φ v, that maximizes the natural logarithm of the likelihood function is then used to represent the distribution for capacity. 5.3 STATISTICAL EVIDENCE OF THE EXISTENCE OF LOWER-BOUND CAPACITIES As further support of the existence of a lower-bound capacity, a statistical hypothesis test can be formulated. If there is not a physical lower bound (that is it is zero), then one can calculate the probability that none of the data points that were measured in load tests would fall below their respective lower bounds as follows: ( ) P No Measurements < Calculated Lower Bounds if Lower Bound Doesn't Exist n i= 1 ( ) = P Measurement > Lower Bound for Test i = ( rlbi ) ln( λ Rrpredicted,i ) 2 ln( 1+ δ ) n = ln 1 - Φ (5.3) 1 R i where n is the number of data points in the database; r LBi is the calculated lower-bound capacity for data point i; r predicted,i is the calculated predicted capacity for data point i; λ R is a bias factor that relates the median capacities to the predicted capacities; δ R is the coefficient of variation of the capacity; Φ() is the standard normal function; and it is assumed that the capacity follows a conventional lognormal distribution. For all three database that were analyzed in this study (driven piles in cohesive soils, driven piles in cohesionless soils, suction caissons in normally consolidated clays), the probability of having no data points below the calculated lower bounds is less than one percent. Therefore, it is highly unlikely that the measurements that were observed correspond to a situation where there is no lower-bound capacity. 113

144 5.4 LIKELIHOOD FUNCTIONS AND PARAMETERS FOR DIFFERENT PROBABILITY DISTRIBUTIONS FOR CAPACITY In this section, likelihood functions and parameters that describe different probability distributions for capacity are presented. The probability distributions include examples from bounded, truncated, and mixed distributions, in addition to mixed Hermite Polynomial transformations Bounded, Truncated, and Mixed Distributions Beta Distribution When the uncertainty in the capacity is modeled with a beta distribution, the natural logarithm of the likelihood function is expressed as: ln r ϕ v ( L( r )) = n i= 1 ln ϕ4i 1 Γ( ϕ 1i + ϕ2i ) ri ϕ3i ( ) ϕ3i Γ( ϕ1i )Γ ϕ2i ϕ4i ϕ3i ( ϕ 1) ( ϕ 1) 1i ϕ4i ri ϕ4i ϕ3i 2i (5.4) where ϕ 1i, ϕ 2i, ϕ 3i and ϕ 4i are model parameters that can be expressed in terms of the predicted capacity, r predicted,i, the lower-bound capacity, r LBi, and the coefficient of 2 X variation, δ R, as i 1i ( i ) ( 1 X i ) ϕ 1i = 1 X X, 2 i 2i = ϕ λ prpredicted,i rlbi ϕ, X i =, Yi X i rubi rlbi δ R ( λ prpredicted,i ) Yi =, ϕ 3 = r i LBi, and ϕ 4i = rubi. In this research, the upper-bound capacity r r UBi LBi of the beta distribution is specified to be equal to six times the predicted capacity. The parameter λ p is a bias factor that was introduced to relate the mean capacity to the predicted capacity, r predicted, which is unique for each data point in the database. As a result, the parameters that need to be calibrated are the bias factor λ p and the coefficient of variation δ R. 114

145 Truncated Lognormal Distribution When the uncertainty in the capacity is modeled with a truncated lognormal distribution, the natural logarithm of the likelihood function is expressed as: ln r ϕ v ( L( r )) = n i= 1 1 ln ln( ϕ 3i) ϕ1i) 1 Φ ϕ2i 1 2πϕ r 2i i exp 1 ln(r i ) ϕ1i 2 ϕ2i 2 (5.5) where ϕ 1i, ϕ 2i, and ϕ 3i are model parameters that can be expressed in terms of the predicted capacity r predicted,i, the predicted lower-bound capacity, r LBi, and the coefficient of variation, δ R, as: 2 ϕ 1i = ln( λ prpredicted,i ), 2i = ln ( 1+ δ R ) ϕ, and ϕ 3i = rlbi (5.6) For the truncated lognormal distribution, the bias factor λ p relates the median capacity to the predicted capacity, r predicted, which is unique for each data point in the database. As in the case of the beta distribution, the parameters that need to be calibrated are the bias factor λ p and the coefficient of variation δ R Mixed Lognormal Distribution When the uncertainty in the capacity is modeled with a mixed lognormal distribution, the natural logarithm of the likelihood function is expressed as: ln r ϕ v n ( L( r )) = ln( PDF(ri ) or PMF(ri )) i= 1 (5.7) where 115

146 2 1 ln(r i ) ϕ1i 1 2 ϕ 2i PDF(r i) = exp for r i > r LBi 2πϕ2iri ln( ϕ ) 3i ϕ1i PMF(r i) = Φ for r i = r LBi ± ε (5.8) ϕ2i where ϕ 1i, ϕ 2i, and ϕ 3i are model parameters that can be expressed in terms of the predicted capacity r predicted,i, the predicted lower-bound capacity, r LBi, and the coefficient of variation, δ R, as: 2 ϕ 1i = ln( λ prpredicted,i ), 2i = ln ( 1+ δ R ) ϕ, and ϕ 3i = rlbi (5.9) The parameter ε defines a range in which the measured capacity is considered to be at the lower-bound capacity. For this research, the parameter ε is selected to be 10% of the lower-bound capacity Mixed Seventh Order Hermite Polynomials When the distribution of capacity is modeled with a mixed Hermite Polynomial, the natural logarithm of the likelihood function is expressed in terms of the reduced or normalized variable (u r ) as: ln r ϕ v n ( L( r )) = ln( PDF(ri ) or PMF(ri )) i= 1 (5.10) 1 dr PDF(r i U ri for r i > r LBi du uri where ) = ( f (u ) ( ) PMF(r ) = Φ for r i = r LBi ± ε (5.11) i u LBi 116

147 where f U (u ri ) is the standard normal probability density function evaluated at each u ri corresponding to the transformed data point r i. The parameters of the mixed Hermite Polynomial define the relationship between the actual values of capacity r i and their transformed values u ri using the following equation: r i 7 ϕ2i ϕm+ 1 = ϕ 1i + H1(u ri ) + H m (u ri ) (5.12) 7 2 ( ϕ ) m= 2 m! m m! m= 2 where ϕ = ( λ r ), = δ ( λ r ) 1i p predicted,i ϕ 2i R p predicted,i, and 3, ϕ4, ϕ5, ϕ6, ϕ7, ϕ8 ϕ correspond to the parameters ψ of the seventh order Hermite Polynomial function. ' ' ' ' ' ' 2,ψ3,ψ4,ψ5,ψ6, ψ7 5.5 MODELING UNCERTAINTY IN THE CAPACITY FOR DRIVEN PILES Piles in Cohesive Soils Values of measured, predicted, and lower-bound capacities for 45 test cases for piles in cohesive soils are summarized in Table 5.1. Lower-bound capacities for piles in normally consolidated to slightly overconsolidated clays (Piles 1 to 34) were calculated using the undrained shearing strength of the remolded soil according to the approach proposed in Section The effective stress approach that was proposed in Section was used to calculate the lower-bound capacities for piles in heavily overconsolidated clays (Piles 35 to 45). Information about measured, predicted, and lower-bound capacities from the available set of data is used to calibrate parameters of alternative probability distributions for capacity. These probability distributions include lognormal, beta, truncated lognormal, and mixed lognormal distributions, in addition to more general mixed distributions that are based on Hermite Polynomial transformations. 117

148 Table 5.1. Summary of measured, predicted, and lower-bound capacities for driven piles in cohesive soils. Pile # Measured Capacity, kips Predicted Capacity, Kips Lower-Bound Capacity, Kips

149 Parameters of the different probability distributions for capacity are evaluated by maximizing the likelihood of observing the measured capacities presented in Table 5.1. The calibration is conducted according to the procedure presented in Section 5.2 using the likelihood functions that are presented in Section 5.4. The parameters of the resulting probability distributions are presented in Tables 5.2 and 5.3. Parameters of lognormal, beta, truncated lognormal, and mixed lognormal distributions are presented in Table 5.2, whereas parameters of mixed Hermite Polynomials are presented in Table 5.3. In Tables 5.2 and 5.3, the parameters λ p and δ R are presented in place of the actual parameters, φ i, because these factors are non-dimensional and do not depend on the actual values of the predicted and lower-bound capacities. Given the parameters of the fitted probability distributions, the actual probability density functions describing the uncertainty in the capacity of two example driven piles in cohesive soils are evaluated and plotted on Figures 5.1 and 5.2. The distributions shown on Figure 5.1 correspond to a pile with a predicted capacity of 200 kips (r predicted = 200) and a lower-bound capacity of 100 kips (r LB = 100), whereas the distribution shown on Fig. 5.2 correspond to a pile having the same predicted capacity (r predicted = 200), but a larger lower-bound capacity (r LB = 120). Table 5.2. Parameters of bounded, truncated, and mixed distributions (piles in clays). Probability Distribution λ p δ R Conventional Lognormal Distribution Truncated Lognormal Distribution Mixed Lognormal Distribution Beta Distribution (r UB = 6r predicted ) Table 5.3. Parameters of mixed Hermite Polynomials (piles in clays) Probability Distribution λ p δ R ψ 2 ψ 3 ψ 4 ψ 5 ψ 6 ψ 7 Mixed Hermite

150 Probability Density Function, PDF Conventional Lognormal Beta Distribution Truncated Lognormal Mixed Lognormal Mixed Hermite Polynomial Capacity (kips) Cumulative Distribution Function, CDF Capacity (kips) Figure 5.1. Comparison between different probability distributions for piles in cohesive soil (r predicted = 200 kips, r LB = 100 kips). 120

151 Probability Density Function, PDF Conventional Lognormal Beta Distribution Truncated Lognormal Mixed Lognormal Mixed Hermite Polynomial Capacity (kips) Cumulative Distribution Function, CDF Capacity (kips) Figure 5.2. Comparison between different probability distributions for piles in cohesive soil (r predicted = 200 kips, r LB = 120 kips). 121

152 Results on Figures 5.1 and 5.2 indicate that different probability distributions result in different models for the left-hand tail of the distribution for capacity. The mixed seventh order Hermite Polynomial is expected to provide the most representative model for the uncertainty in the capacity because of the flexibility of its mathematical form. However, the important conclusion from Figures 5.1 and 5.2 is that the mixed lognormal distribution provides a very close approximation to the Hermite Polynomial distribution, and serves as a likely alternative for modeling distributions for capacity. Note that the probability distributions shown on Figures 5.1 and 5.2 correspond to two realistic cases where the ratios of the lower-bound capacity to the predicted capacity are equal to 0.5 and 0.6. Results from analyzing the database of load tests in piles in cohesive soils indicate that ratios of the lower-bound capacity to the predicted capacity range from about 0.3 to 1.0 and have an average of Piles in Siliceous, Cohesionless Soils Values of measured, predicted, and lower-bound capacities for 36 test cases of piles in siliceous, cohesionless soils are summarized in Table 5.4 and used to calibrate parameters of alternative probability distributions for capacity. Lower-bound capacities for each test case are calculated based on the approach proposed in Section 2.5. The alternative probability distributions include lognormal, beta, truncated lognormal, and mixed lognormal distributions in addition to more general mixed distributions that are based on Hermite Polynomial transformations. The parameters of the resulting probability distributions are presented in Tables 5.5 and 5.6. Parameters of lognormal, beta, truncated lognormal, and mixed lognormal distributions are presented in Table 5.5, whereas parameters of mixed Hermite Polynomials are presented in Table

153 Table 5.4. Summary of measured, predicted, and lower-bound capacities for driven piles in cohesionless soils. Pile # Measured Capacity, Kips Predicted Capacity, Kips Lower-Bound Capacity, Kips Table 5.5. Parameters of bounded, truncated, and mixed distributions (piles in sands). Probability Distribution λ p δ R Conventional Lognormal Distribution Truncated Lognormal Distribution Mixed Lognormal Distribution Beta Distribution (r UB = 6r predicted )

154 Table 5.6. Parameters of mixed Hermite Polynomials (piles in sands). Probability Distribution λ p δ R ψ 2 ψ 3 ψ 4 ψ 5 ψ 6 ψ 7 Mixed Hermite Given the parameters of the fitted probability distributions, the actual probability density functions describing the uncertainty in the capacity of two example driven piles in cohesionless soils are evaluated and plotted on Figures 5.3 and 5.4. The distributions shown on Figure 5.3 correspond to a pile with a predicted capacity of 200 kips (r predicted = 200) and a lower-bound capacity of 80 kips (r LB = 80), whereas those shown on Figure 5.4 correspond to a pile with the same predicted capacity (r predicted = 200), but a larger lower-bound capacity (r LB = 100). These probability distributions correspond to two realistic cases where the ratios of the lower-bound capacity to the predicted capacity are equal to 0.4 and 0.5. Databases of load tests of piles in cohesionless soils indicate ratios of lower-bound to predicted capacity ranging from 0.3 to 0.75, with an average of 0.5. Results on Figures 5.3 and 5.4 indicate large differences in the shapes of the lefthand tails of the fitted probability distributions for the capacity of piles in cohesionless soils. Distributions that correspond to the mixed seventh order Hermite Polynomial exhibit relatively large probability masses (percentiles between 10% and 20%) at the lower-bound capacities. These large masses are supported by the data in the database, which indicated that in six out of thirty six cases, the measured capacities were within ± 10% of the predicted lower bound. This observation is important because it indicates that the presence of the lower-bound capacity cannot be modeled by simply shifting a conventional probability distribution such that it initiates from the lower-bound capacity. Instead the data provides physical evidence for the presence of a probability mass that is concentrated in the vicinity of the lower-bound capacity. 124

155 For piles in cohesionless soils, the relatively large probability masses that are observed at the lower-bound capacity for some test cases could have resulted from different factors that might include (1) the soil having a density that is lower than expected, (2) the soil particles being crushed during the process of driving, (3) the effective stresses approaching the free field conditions (at rest conditions), and (4) other factors such as errors in performing the load test, etc. As in the case of cohesive soils, the mixed lognormal distribution provides a very close approximation to the Hermite Polynomial distribution and serves as a likely alternative for modeling distributions for capacity. None of the other probability distributions that are utilized in this study had the flexibility to reflect the relatively large probability content that exists in the vicinity of the lower-bound capacity. 5.6 MODELING UNCERTAINTY IN THE CAPACITY OF SUCTION CAISSONS IN NORMALLY CONSOLIDATED CLAYS Values of measured, predicted, and lower-bound capacities for 21 test cases for suction caissons in normally consolidated clays are summarized in Table 5.7. The test cases that are reported in Table 5.7 exclude Tests # 11 to 14, which are cases in which the suction caissons were pulled out immediately after installation. Lower-bound capacities were calculated using the undrained shearing strength of the remolded soil according to the approach proposed in Section 3.5. Information about measured, predicted, and lowerbound capacities from the available set of data is used to calibrate parameters of alternative probability distributions for capacity. These probability distributions include lognormal, beta, truncated lognormal, and mixed lognormal distributions, in addition to mixed distributions that are based on Hermite Polynomial transformations. 125

156 Probability Density Function, PDF Conventional Lognormal Beta Distribution Truncated Lognormal Mixed Lognormal Mixed Hermite Polynomial Capacity (kips) Cumulative Distribution Function, CDF Capacity (kips) Figure 5.3. Comparison between different probability distributions for piles in cohesionless soils (r predicted = 200 kips, r LB = 80 kips). 126

157 Probability Density Function, PDF Conventional Lognormal Beta Distribution Truncated Lognormal Mixed Lognormal Mixed Hermite Polynomial Capacity (kips) Cumulative Distribution Function, CDF Capacity (kips) Figure 5.4. Comparison between different probability distributions for piles in cohesionless soils (r predicted = 200 kips, r LB = 100 kips). 127

158 Table 5.7. Summary of measured, predicted, and lower-bound capacities for suction caissons in normally consolidated clays. Test # Measured Capacity Predicted Capacity, (kips) Lower-Bound Capacity (kips) (α = 1, N = 9) (kips) The parameters of the resulting probability distributions for capacity are presented in Tables 5.8 and 5.9. Parameters of lognormal, beta, truncated lognormal, and mixed lognormal distributions are presented in Table 5.8, whereas parameters of truncated and mixed Hermite Polynomials are presented in Table 5.9. Given the parameters of the fitted 128

159 probability distributions, the actual probability density functions describing the uncertainty in the capacity of two example suction caissons in normally consolidated clays are evaluated and plotted on Figures 5.5 and 5.6. As in the case for piles in cohesive soils, the distributions shown on Figure 5.5 and 5.6 correspond to suction caissons with a predicted capacity of 200 kips (r predicted = 200) and lower-bound capacities of 100 and 120 kips respectively (r LB = 100 and 120). These probability distributions correspond to two realistic cases where the ratios of the lower-bound capacity to the predicted capacity are equal to 0.5 and 0.6. When caissons that were pulled out immediately after installation are excluded from the analysis, the database of load tests on suction caissons in normally consolidated clays indicate ratios of lower-bound to predicted capacity ranging from about 0.3 to 0.8, with an average of 0.6. As in the case of piles, the probability distributions shown on Figures 5.5 and 5.6 for suction caissons in normally consolidated clays indicate a clear similarity between mixed Hermite Polynomial distributions and mixed lognormal distributions. Table 5.8 Parameters of bounded, truncated, and mixed distributions (suction caissons). Probability Distribution λ p δ R Conventional Lognormal Distribution Truncated Lognormal Distribution Mixed Lognormal Distribution Beta Distribution(r UB = 6r predicted ) Table 5.9 Parameters of truncated and mixed Hermite Polynomials (suction caissons). Probability Distribution λ p δ R ψ 2 ψ 3 ψ 4 ψ 5 ψ 6 ψ 7 Mixed Hermite

160 Probability Density Function, PDF Conventional Lognormal Beta Distribution Truncated Lognormal Mixed Lognormal Mixed Hermite Polynomial Capacity (kips) Cumulative Distribution Function, CDF Capacity (kips) Figure 5.5. Comparison between different probability distributions for suction caissons in normally consolidated clays (r predicted = 200 kips, r LB = 100 kips). 130

161 Probability Density Function, PDF Conventional Lognormal Beta Distribution Truncated Lognormal Mixed Lognormal Mixed Hermite Polynomial Cumulative Distribution Function, CDF Capacity (kips) Capacity (kips) Figure 5.6. Comparison between different probability distributions for suction caissons in normally consolidated clays (r predicted = 200 kips, r LB = 120 kips). 131

162 5.7 SUMMARY In this chapter, a procedure is described for calibrating the parameters of probability distributions for capacity using information about predicted, measured, and lower-bound capacities from a limited number of load tests in databases. Results from two pile-load test databases for driven piles in cohesive and cohesionless soils and one database of load tests on suction caissons in normally consolidated clays are used to calibrate the parameters of conventional lognormal, beta, truncated lognormal, and mixed lognormal probability distributions. The same data is then used to calibrate parameters of more general mixed seventh order Hermite Polynomial distributions, which are used as a reference to which other distributions are compared. The calibrated parameters for each type of probability distribution are then used to model the uncertainty in the capacities of several example foundations. In each example, practical values of the predicted and lower-bound capacities are chosen, and different probability distributions for capacity are plotted on the same figure for comparison. Results from all the cases analyzed indicate that the mixed lognormal distribution provides the closest approximation to the more general distribution that is based on seventh order Hermite Polynomials. As a result, it is recommended that the mixed lognormal distribution should be used as a simple and practical probability distribution for modeling the uncertainty in the capacity for driven piles in cohesive and cohesionless soils and for suction caissons in normally consolidated clays. 132

163 Chapter 6. Reliability Assessments in the Presence of a Lower-Bound Capacity 6.1 INTRODUCTION In this chapter, numerical and semi-analytical methods for calculating the probability of failure when there is a lower-bound capacity are explored. The main objective is to study the ability of semi-analytical methods to accommodate reliability problems in which the capacity distribution is bounded at the left-hand tail. The accuracy of the results obtained using the First Order Second Moment method (FOSM), First Order Reliability Method (FORM), and Second Order Reliability Method (SORM) is explored and compared to solutions obtained using numerical integration. Uniform, beta, mixed and truncated distributions are used as candidate probability distributions for modeling different mathematical shapes for the left-hand tail for capacity. 6.2 GENERAL ASSESSMENT OF THE PROBABILITY OF FAILURE In general, the probability of failure can be calculated as (Ang and Tang, 1984): = + f FR (s)fs 0 p (s)ds (6.1) where F R (s) is the cumulative distribution function of the capacity evaluated at the load s; and f S (s) is the probability density function of the load. In cases where both the load and the capacity follow independent normal or lognormal distributions, closed form solutions for the probability of failure from Equation 6.1 can be obtained and expressed by Equations 6.2 and 6.3 respectively: 133

164 p p f f ( µ µ ) ( FS 1) Φ R S mean = = Φ = Φ( β) σ R σ S FS + meanδ R + δ S (6.2) ln( FS ) Φ median = = Φ( β) 2 2 ln(1 δ R )(1 δs ) + + (6.3) where Φ() is the standard normal cumulative distribution function, µ R, µ S and σ R, σ S are the means and standard deviations of the capacity and the load respectively, FS mean and FS median are the mean and median factors of safety respectively defined as the ratio of the mean or median capacity to the mean or median load, δ R and δ S are the coefficients of variation of the capacity and load, respectively, and β is the reliability index. In most cases where the load and capacity are not normally or lognormally distributed, no exact analytical solution is available for the probability of failure and solutions based on numerical integration are required. Alternatively, semi-analytical tools that are based on second moment methods (FOSM) or on fully distributional methods (FORM and SORM) can be used to approximate the probability of failure. The First Order Second Moment reliability method (FOSM) is a simple and computationally inexpensive method for assessing reliability. However, the FOSM method incorporates only the first two moments of the load and capacity and is thus expected to have low accuracy in problems where the capacity distribution has a lower bound. The First Order Reliability Method (FORM) can reflect the effect of having a lower-bound capacity on the reliability since it includes the whole distributional form of the capacity. The approximations made in FORM make it simple and computationally less expensive than the Second Order Reliability Method (SORM) at the expense of possible inaccuracy. Finally, techniques based on numerical integration can be utilized to assess reliability for problems where the capacity distribution is bounded. However, numerical methods tend to be computationally expensive. 134

165 6.2.1 First Order Second Moment Reliability Method (FOSM) The First Order Second Moment approach for approximating the probability of failure was formalized by Cornell (1969) and Ang and Cornell (1974). In this method, the reliability is assessed entirely in terms of the first and second moments of the design variables. A failure surface is defined by a performance function g( X r ) that separates the safe region from the failure region. The cases that are addressed in this research involve only two variables, the load (S) and the capacity (R). For these cases, a simple performance function can be defined as g(r,s) = R S to describe the failure region. Failure occurs when g(r,s) < 0 (Figure 6.1). The failure surface can be plotted in the space of reduced variables (Figure 6.1) by introducing the reduced variates: R' R µ R = and σ R S' S µ S = (6.4) σ S From Figure 6.1, the shortest distance d from the origin of reduced variables to the performance function can then be used to estimate the reliability index. Using analytic geometry, the distance d is expressed by Equation 6.5 as a function of the means and standard deviations of the load and capacity (Ang and Tang, 1984). β FOSM ( µ µ ) R S = d = (6.5) 2 2 σ R + σs Comparison between Equations 6.2 and 6.5 indicates that the reliability index obtained using the FOSM reliability method leads to an exact measure of the probability of failure for normally distributed and statistically independent load and capacity. 135

166 8000 FAILURE SET Load, s Performance Function g(r,s) SAFE SET Capacity, r 6 FAILURE SET 5 Performance Function, g(r,s) d SAFE SET s' = (s - µs)/σs r' = (r - µ R )/ σ R Figure 6.1. Performance function in the original and reduced space for the case of 2 random variables (FOSM application). 136

167 6.2.2 First and Second Order Reliability Methods (FORM and SORM) First and Second Order Reliability Methods (FORM and SORM) are fully distributional analytical methods in which the probability of failure is calculated using approximations of the failure surface g(r,s). Bjerager (1990) states that the application of FORM and SORM involves three basic steps: (1) the transformation of the performance function g(r,s) into the standard normal space, (2) an approximation of the performance function in the standard normal space, and (3) a computation of the probability of failure corresponding to the approximate failure surface Transformation of g(r,s) The first step in the application of FORM and SORM involves the transformation of the performance function g(r,s) into the standard normal space g(u R, u S ). For statistically independent variables, the transformation of the vector (R,S) into the vector (u R, u S ) is obtained using the Rosenblatt transformation as: u R 1 1 = Φ (F (r)) and u = Φ (F (s)) (6.6) R S S where, F R (r) and F S (s) are the cumulative distribution functions of the load and the capacity respectively, and Φ -1 () is the inverse of the cumulative standard normal function. The transformation presented in Equation 6.6 indicates that in FORM and SORM, the whole probability distribution for capacity is used to transform the failure surface to the space of reduced variables. This implies that the effect of a lower-bound capacity can be reflected using these fully distributional methods. If variables following non-normal probability distributions are used, the performance function g(u R, u S ) is expected to be non-linear in the transformed space (Figure 6.2). 137

168 8000 FAILURE SET Load, s Performance Function g(r,s) Capacity, r SAFE SET FAILURE SET Most Probable Failure Point u* Performance Function g(u R,u S ) 3 2 US SAFE SET Figure 6.2. Performance functions in the original and reduced space (FORM and SORM). U R If Hermite Polynomial transform functions are used to model the probability distribution for capacity, the relationship between the vector (R,S) and the vector (u R, u S ) is already defined as follows (see Equation 4.27, Chapter 4), eliminating the need for the Rosenblatt transformation: 138

169 R µ R + 1+ σ R n ' ( ψi ) i= 2 i! 2 H 1 n ' ψ ( u ) + i H ( u ) R i= 2 σ S S µ S + H (6.7) 1 S i S n ' 2 ( ψ ) i= 2 i! i 1+ i! i= 2 i! n ' ψ ( u ) + i H ( u ) i R Approximation of g(u R,u S ) The second step in the application of FORM and SORM involves the approximation of the failure surface g(u R,u S ) in the standard normal space by first or second order functions g (u R,u S ). In FORM (Figure 6.3(a)), the failure surface is approximated by a hyperplane using a first order Taylor series expansion (Hasofer and Lind 1974; Rackwitz and Fiessler 1978; Ditlevsen 1981; and Hohenbichler and Rackwitz 1981). In SORM (Figure 6.3(b)), the performance function is approximated by a second order surface that can be parabolic, spherical, or hyperbolic (Fiessler et al. 1979; Breitung 1984; Der Kiureghian et al. 1987, 1991; Tvedt 1990; and Zhao and Ono 1999a, 1999b). The expansion point (u*) for the Taylor series is referred to as the most probable failure point and is defined as the point on the performance function (or limit state surface) that has the minimum distance to the origin of reduced variables (Figure 6.2). Many algorithms have been proposed in the literature for finding the most probable failure point (u*). A comparison between different algorithms is provided by Liu and Der Kiureghian (1991). For SORM, fitting a second order Taylor Series to the performance function requires additional knowledge about the principal curvatures of the performance function at the most probable failure point. Zhao and Ono (1999b) state that the principal curvatures are obtained as the eigenvalues of the rotational transformed second-order derivative matrix (Hessian matrix) of the performance function in standard normal space. 139

170 6 Performance Function g(u R,u S ) g'(u R,u S ) using FORM US (a) U R 6 5 Performance Function g(u R,u S ) g'(u R,u S ) using SORM US (b) U R Figure 6.3. Approximations for the performance function g(u R,u S ) by first and second order surfaces g (u R,u S ) using FORM and SORM Evaluation of the Probability of Failure The final step in the application of FORM and SORM involves computing the reliability that corresponds to the fitted performance function g (u R,u S ). This entails 140

171 evaluating the probability content of the failure region that lies outside the assumed failure surface g (u R,u S ) (Zhao and Ono 1999c). Using FORM, this is accomplished by computing a first-order estimate of the reliability index as the distance from the origin of reduced space to the most probable failure point u* (Hasofer and Lind 1974; Ang and Tang 1984). The probability of failure is then given by: p ( ) = Φ (6.8) f β F where β F is the first order reliability index. The accuracy of the First Order Reliability Method (FORM) is good when the performance function is nearly flat in the neighborhood of the design point (Der Kiureghian et al 1987). Accuracy problems arise when the performance function is strongly nonlinear. The Second Order Reliability Method (SORM) aims at improving the accuracy of FORM by capturing the nonlinearity in the performance function in the vicinity of the design point (Fiessler et al. 1979; Tichy 1994; and Zhao and Ono 1999a). The exact computation of the probability of failure corresponding to a second order failure surface g (u R,u S ) is relatively complicated and computationally demanding. In the early 1980 s, researchers worked on developing closed form approximations for the failure probability. Assuming a parabolic failure surface, Breitung (1984) proposed an asymptotic formula for the probability of failure, which approaches the exact failure probability as β F approaches infinity. Tvedt (1983) proposed a three-term formula that constitutes an improvement to Breitung s formula. Tvedt (1990) proposed an exact expression for the probability of failure that works for all quadratic forms of random variables, whereas Hohenbichler et al. (1988) used importance sampling to evaluate the failure probability. Koyluoglu and Nielsen (1994) and Cai and Elishakoff (1994) proposed expressions for the probability of failure based on Taylor series expansions. 141

172 Zhao and Ono (1999c) state that the accuracy of the different approximations for SORM depends on (1) the curvature radius of the fitted performance function g (u R,u S ), (2) the number of random variables n, and (3) the magnitude of the first order reliability index β F. Discrepancies between probabilities of failure calculated using SORM and those obtained using methods based on numerical techniques have been observed for cases with a relatively small curvature radius, a relatively large number of random variables, and a relatively small first order reliability index (Zhao and Ono 1999c). Zhao and Ono (1999a) proposed an empirical second order reliability index that works well for a wide range of curvature radii, first order reliability indices, and number of variables Curvature vs. Point-Fitting SORM There are two types of second order reliability approximations: curvature fitting SORM and point-fitting SORM (Zhao and Ono, 1999b). In curvature fitting SORM, the fitted performance function g (u R,u S ) is defined by matching the curvature of g (u R,u S ) to the curvature of the original performance function g(u R,u S ) at the design point. The drawback of using this method is the requirement that the performance function should be continuous and twice differentiable to allow for computation of the matrix of transformed second order derivatives needed to calculate the principal curvatures. The principal curvatures are obtained as the eigenvalues of this matrix (Hessian matrix), which is a process that can be very costly for a relatively complicated transformed performance function (Zhao and Ono 1999b). Difficulty in computing the Hessian matrix led to the development of the pointfitting SORM methods (Der Kiureghian et al 1987, 1991, Zhao and Ono 1999a,b). Der Kiureghian et al (1987, 1991) proposed an efficient algorithm to calculate the principal curvatures of a point-fitted paraboloid without having to compute the Hessian matrix. 142

173 However, this SORM method uses a point-fitted paraboloid in rotated standard normal space. The need for a rotational transformation is still present and can be complicated. To eliminate this drawback, Zhao and Ono (1999b) presented a simple point fitting SORM approximation that results in a second order reliability index that does not require the computation of the Hessian matrix and eliminates the need for rotational transformations and eigenvalue analyses. This point-fitting method will be used in this dissertation to calculate the probability of failure corresponding to the Second Order Reliability Method Point-Fitting SORM (Zhao and Ono 1999b) The general procedure for applying the point-fitting SORM method proposed by Zhao and Ono (1999b) is summarized in the following steps: 1. Select an initial central point u c (u R, u S ) in standard normal space. In this research, the initial point u c was taken as u c (-1, 1). The arbitrary choice of this initial point is based on the fact that it corresponds to a relatively small first order reliability index (β = 1.4) compared to values of reliability indices that are accepted in geotechnical engineering design. This choice of the initial point is expected to aid in achieving convergence and efficiency of the SORM algorithm. 2. Select two fitting points along each of the coordinate axes such that each fitting point is equidistant from the central point u c by a distance d. In the case of 2 variables, the 4 fitting points will have the coordinates u 1 (u R, u S +d), u 2 (u R, u S -d), u 3 (u R + d, u S ), and u 4 (u R - d, u S ). Zhao and Ono (1999) state that the value of d should be between 0.1 and 0.5 for acceptable accuracy. In this research, the distance d is taken as

174 3. Approximate the failure surface g(u R, u S ) by the second order polynomial g (u R, u S ) represented by Equation 6.9. The parameters of g (u R, u S ) are obtained by equating g(u R, u S ) with g (u R, u S ) at the 5 fitting points and by solving a system of 5 equations with 5 unknowns. The value of g(u R, u S ) at each fitting point is obtained by transforming the fitting point to the original space and by evaluating g(r,s) = R S. g' (u,u ) = a + γ u + γ u + λ u + λ u (6.9) R S o 1 R 2 S 1 2 R 2 2 S 4. For the point fitted failure surface g (u R,u S ), conduct FORM iterations and obtain the design point (most probable failure point) u*. Use u* instead of u c in Step 1 and repeat Steps 2 to 4 until convergence occurs. Zhao and Ono (1999b) state that 3 to 6 iterations are needed to achieve convergence. 5. Given the parameters of the approximated performance function, g (u R,u S ), the gradient vector at the design point is calculated and used to calculate the principal curvature K s and the radius of curvature R: = * 2 K s = λ j 1 (γ j + 2λ ju j) (6.10) 2 G' j 1 G' 1 R = (6.11) K s 2 * 2 G' = (γ + 2λ u j (6.12) j= 1 j j ) 6. The first order reliability index β F is calculated as the distance from the origin of reduced space to the design point (u*). 7. The empirical second order reliability index β S is calculated as a function of the first order reliability index β F, the radius of curvature R, and the principal curvature K s as: 144

175 2K s (1+ 2β ) 1 φ(β F ) F β = + ( ) s Φ Φ( β F ). 1 K s > 0 (6.13) RΦ β F 2.5K s 1 K + + S β s = 1+ β F K S 1 K 2 s < 0 (6.14) 2n 5R + 25(23 5β F )/R 2 40 where φ() is the standard normal probability density function. It should be noted that convergence of the algorithm proposed by Zhao and Ono (199b) is assumed to take place when the calculated values of the second order reliability index β S from two consecutive iterations indicate a difference that is less than 0.1% Example of a Reliability-Based Analysis Using SORM An example is presented to illustrate the applicability and convergence of the point-fitting SORM procedure as proposed by Zhao and Ono (1999b). In this example, the load is assumed to be lognormally distributed and is characterized by a coefficient of variation of The capacity is assumed to follow a truncated lognormal distribution with a median capacity of 1000 and a coefficient of variation of 0.4. A lower-bound capacity that is equal to 0.5 times the median capacity is assumed in the analysis. The median factor of safety used in the design is assumed to be equal to 3.0. The three fitted polynomials g (u R,u S ) corresponding to the three iterations needed for convergence in this sample problem are shown on Figure 6.4(c). The values of the most probable failure points and principal curvatures in addition to the mathematical equations that describe the point-fitted performance functions for each iteration are presented in Table 6.1. Also shown on Table 6.1 are the values of the calculated first and second order reliability indices, β F and β S. 145

176 5 First Iteration for g'(u R,u S ) 4 3 US Performance Function g(u R,u S ) First Design Point Initial Point 2 Fitting Points U R 0 5 First Iteration for g'(u R,u S ) 4 3 US Second Iteration for g'(u R,u S ) First Iteration for g'(u R,u S ) Third Iteration for g'(u R,u S ) 4 3 Second Iteration for g'(u R,u S ) Third Design Point 2 US U R Figure 6.4. Application of point-fitting SORM (Zhao and Ono 1999b)

177 Table 6.1. Iterative results for the SORM example. Iteration Central Point Fitted Performance Function, g'(u R,u S ) Most Probable β F Curvature β S # point, uc Failure Point, Ks 1 (-1, 1) g'(u R,u S ) = u R -2948u S +4072u 2 R-258u 2 S (-2.315, 2.983) (-2.315, 2.983) g'(u R,u S ) = u R -2585u S +3215u 2 R-347u 2 S (-2.293, 3.027) (-2.293, 3.027) g'(u R,u S ) = u R -2571u S +3289u 2 R-350u 2 S (-2.293,3.027) (-2.293,3.027) g'(u R,u S ) = u R -2571u S +3289u 2 R-350u 2 S (-2.293,3.027) Numerical Integration When the probability distributions describing uncertainty in the load and the capacity are known, the integral expressed by Equation 6.1 can be solved for the probability of failure using numerical integration. For the relatively simple case of two variables (load and capacity), the integral can be formed and solved using available software such as MathCAD. MathCAD12 provides four numerical tools for integrating complex integrals. These include (1) the Romberg integration method, (2) the adaptive quadrature method, (3) the infinite limit method, and (4) the singular endpoint method. In this research, the adaptive quadrature method is selected to evaluate the expression for the probability of failure numerically. The adaptive quadrature method is based on Simpson s rule and is recommended for integrating functions that change rapidly over the interval of integration (Mathcad12, 2005). Prior to the selection of the adaptive method for integration, all the available techniques for integration in MathCAD were tested on a wide range of problems. Results indicated that the Romberg, infinite limit, and singular endpoint methods did not always provide reliable estimates of the probability of failure, especially for small probabilities of failure (P f < 10-5 ). Convergence of these methods was not guarantied for all problems and the results were sensitive to changes in the specified tolerance. On the other hand, convergence was consistently achieved with the adaptive method for integration. 147

178 148 Different trials were conducted using the adaptive method to select a working value for the tolerance. Based on the results of these trials, a fixed value of was chosen for all subsequent analyses. The use of a smaller value of tolerance (10-15 ) lead to the same results as the chosen value (10-10 ) indicating that no additional accuracy was being gained using the smaller tolerance. For illustration, expressions for the probability of failure for cases where the load follows a lognormal distribution and the capacity follows truncated normal and lognormal distributions are presented in Equations 6.15 and 6.16 respectively. + = LB LB 2 R R 2 S S r s r σ µ r 2 1 R R R LB ξ λ lns 2 1 S f ds dr e 2πσ 1 σ µ r Φ 1 1 e 2πsξ 1 p (6.15) + = LB LB 2 R R 2 S S r s r ξ λ lnr 2 1 R R R LB ξ λ lns 2 1 S f ds dr e 2πrξ 1 ξ λ lnr Φ 1 1 e 2πsξ 1 p (6.16) Expressions for the probability of failure for cases where the capacity follows uniform, beta, and mixed lognormal distributions are given by Equations 6.17, 6.18, and = LB LB 2 S S r s r LB UB ξ λ lns 2 1 S f ds dr r r 1 e 2πsξ 1 p (6.17) ( ) ( ) + + = LB LB 2 S S r s r 1 β LB UB UB 1 α LB UB LB LB UB ξ λ lns 2 1 S f ds dr r r r r r r r r Γ(α)Γ(β) β) Γ(α r r 1 e 2πsξ 1 p (6.18) + = LB 2 R R 2 S S r s 0 ξ λ lnr 2 1 R ξ λ lns 2 1 S f ds dr e 2πrξ 1 e 2πsξ 1 p (6.19)

179 6.3 ASSESSMENT OF RELIABILITY FOR DISTRIBUTIONS WITH A LOWER BOUND The objective of this section is to explore the accuracy of reliability assessments provided by semi-analytical tools for problems where the capacity distribution is bounded. Probabilities of failure calculated using numerical integration are used as a reference to which results from FOSM, FORM, and SORM are compared. For all the cases studied, the load is assumed to follow a lognormal distribution as is the convention (e.g., AASHTO 2004; McVay et al. 2000, 2002 and 2003; and Scott et al. 2003). Bounded capacity distributions are represented using truncated normal and lognormal distributions, mixed lognormal distributions, and uniform and beta distributions. The parameters describing the probability distributions of the load and the capacity are varied systematically in the analysis to cover cases that represent practical problems in geotechnical engineering. These parameters include the relative uncertainty in the load and the capacity, the median factor of safety, and the ratio of the lower-bound capacity to the mean or median capacity First Order Second Moment Reliability Method (FOSM) The First Order Second Moment method is used to predict the probability of failure for cases where the capacity follows truncated normal and lognormal distributions. Initially, the load is assumed to be lognormally distributed with a typical coefficient of variation δ S of 0.2 (e.g., AASHTO 2004; McVay et al and 2003 and Paikowsky 2003). The capacity is assumed to have a coefficient of variation δ R of 0.4. Curves showing the variation of the ratio of the probability of failure calculated using FOSM to the probability of failure calculated using numerical integration are shown on Figures 6.5(a) and 6.5(b) for the truncated normal and lognormal cases respectively. Both the median factor of safety FS median and the ratio of the lower-bound 149

180 to the median capacity are varied in the analysis. The means and standard deviations of the truncated distributions are adjusted using Equations 6.12 and 6.13 to account for the presence of the lower-bound capacity, and the adjusted values are used in Equation 6.5 to calculate the First Order Second Moment reliability index (β FOSM ) (a) Truncated Normal Pf (FOSM) / Pf (Numerical) FS Median Pf (FOSM) / Pf (Numerical) Ratio of Lower-Bound to Median Capacity (b) Truncated Lognormal FS Median Ratio of Lower-Bound to Median Capacity Figure 6.5. Accuracy of FOSM (Truncated distributions, δ S = 0.2, δ R = 0.4). 150

181 Results presented on Figure 6.5 indicate that the First Order Second Moment reliability method has low accuracy. In the case where the capacity follows a truncated normal distribution, the ratio of P f(fosm) to P f(numerical) ranges from 1 to 4 for ratios of lower-bound to median capacity that are relatively small (between 0 and 0.3). For ratios of lower-bound to median capacity that are more representative of practical geotechnical engineering problems (greater than 0.3), FOSM overestimates the probability of failure by several orders of magnitude. The ratio of P f(fosm) to P f(numerical) increases as the ratio of the lower-bound to the median capacity increases and as the median factor of safety increases. In the case where the capacity follows a truncated lognormal distribution, the ratio of P f(fosm) to P f(numerical) increases compared to the truncated lognormal case, particularly at relatively small ratios of lower-bound to median capacity. These results are expected since the reliability assessments using the FOSM method are based on information regarding the mean and standard deviation of the capacity, which even if adjusted to account for the lower-bound capacity, do not seem to properly model the left-hand tail of the probability distribution for capacity. Similar conclusions can be drawn from results on Figure 6.6, which represent the case where the capacity is assumed to follow a mixed lognormal distribution. The First Order Second Moment reliability method is also used to estimate the probability of failure for cases where the capacity follows uniform and beta distributions. Curves showing the ratio of the probability of failure calculated using FOSM to the probability of failure calculated using numerical integration are plotted on Figures 6.7(a) and 6.7(b). Results on Figure 6.7 are in line with the results obtained for truncated distributions indicating that the First Order Second Moment reliability method fails at capturing the effect of the lower-bound capacity on the reliability. The method gives results that differ from the exact probabilities of failure by several orders of magnitude. 151

182 Pf (FOSM) / Pf (Numerical) Pf (FOSM) / Pf (Numerical) FS Median Ratio of Lower-Bound to Median Capacity Figure 6.6. Accuracy of FOSM method (mixed lognormal distribution, δ S = 0.2, δ R = 0.4) FSMedian FSMean Ratio of Lower-Bound to Mean Capacity Ratio of Lower-Bound to Mean Capacity (a) Uniform Distribution (b) Beta Distribution Figure 6.7. Accuracy of FOSM method (uniform and beta distributions, δ S = 0.2). 152

183 Based on results obtained for the cases discussed above, it can be concluded that the probabilities of failure calculated using the FOSM reliability method tend to overestimate the probability of failure for bounded distributions for capacity. As a result, the method cannot be used with confidence in practical reliability problems that incorporate a lower-bound capacity in the distributional form of the capacity First and Second Order Reliability Methods (FORM and SORM) The First and Second Order Reliability Methods (FORM and SORM) are used to calculate the probability of failure for cases where the capacity distribution is bounded. Estimates of the probability of failure are obtained using the First Order Reliability Method and for the more complicated Second Order Reliability Method and compared to solutions that are based on numerical integration. To achieve a sense of the magnitude of variation in the ratio of P f(form) and P f(sorm) to P f(numerical) for practical geotechnical engineering problems, a large number of cases is analyzed where the coefficients of variation of the load and capacity are varied in a systematic manner. In these problems, the load and the capacity are assumed to follow truncated normal and lognormal distributions, mixed lognormal distributions, and uniform and beta distributions, whereas the load is assumed to follow a lognormal distribution First Order Reliability Method The First Order Reliability Method (FORM) is used to calculate the probability of failure for cases where the capacity follows truncated normal and lognormal distributions. Initially, the load is assumed to be lognormally distributed with a coefficient of variation δ S of 0.2. The capacity is assumed to have a coefficient of variation δ R of 0.4. The accuracy of the First Order Reliability Method is reflected by the curves plotted on Figures 6.8 and 6.9 for truncated normal and lognormal distributions for capacity 153

184 respectively. The ratio of the probability of failure calculated using FORM to the probability of failure calculated using numerical integration varies between 1 and 2 and increases as the factor of safety FS median increases and as the ratio of the lower-bound to the median capacity increases. Similar results are obtained for distributions of capacity following uniform and beta distributions as seen in Figures 6.10 and Truncated Normal Distribution Reliability Index FS Median Numerical Integration Exact FORM Ratio of Lower-Bound to Median Capacity Pf (FORM) / Pf (Numerical) FS Median Ratio of Lower-Bound to Median Capacity Figure 6.8. Accuracy of FORM (truncated normal distributions, δ R = 0.4, δ S = 0.2). 154

185 5 Truncated Lognormal Distribution Reliability Index FS Median Numerical Integration FORM Ratio of Lower-Bound to Median Capacity 2 Pf (FORM) / Pf (Numerical) FS Median Ratio of Lower-Bound to Median Capacity Figure 6.9. Accuracy of FORM (truncated lognormal distributions, δ R = 0.4, δ S = 0.2). 155

186 6 5 Uniform Distribution Reliability Index FSMedian Numerical Integration Exact FORM Ratio of Lower-Bound to Mean Capacity Pf (FORM) / Pf (SORM) FS Median Ratio of Lower-Bound to Mean Capacity Figure Accuracy of FORM (uniform distribution, δ S = 0.2). 156

187 6 Beta Distribution Reliability Index FS M ean Numerical Integration Exact FORM Ratio of Lower-Bound to Mean Capacity 2 Pf (FORM) / Pf (Numerical) FS Mean Ratio of Lower-Bound to Mean Capacity Figure Accuracy of FORM (beta Distribution, δ R = 0.4, δ S = 0.2). 157

188 The First Order Reliability Method (FORM) is also used to calculate the probability of failure for cases where the capacity follows a mixed lognormal distribution with a coefficient of variation of 0.4. The load is assumed to be lognormally distributed with a coefficient of variation δ S of 0.2. The variation of the ratio of the probability of failure calculated using FORM to the probability of failure calculated using numerical integration is plotted on Figure Results indicate that the ratio varies between 1.0 and 1.3 and is much smaller than that obtained for truncated, uniform, and beta distributions as shown in Figures 6.8 to This behavior can be explained by analyzing the expression for the probability of failure obtained for the case where the capacity follows a mixed lognormal distribution. This expression is presented in Equation 6.20 as: P ( S R) = P ( S > R) ( R = r ) P(R = r ) + P( S > R) R > r ) P(R > r ).dr LB LB 158 r= r LB > (6.20) The first part of Equation 6.20 can be evaluated analytically while the second part needs to be solved using FORM or SORM. Since an exact solution to the first part of the equation is available, the total uncertainty in the magnitude of the probability of failure is reduced significantly for a mixed distribution compared to truncated, uniform, or beta distributions (compare Figure 6.12 and Figures 6.8 to 6.11). The discrepancy between exact probabilities of failure and probabilities of failure estimated using FORM results from the assumption of a linear function to represent the non-linear failure surface (Figure 6.3a). The accuracy of the First Order Reliability Method (FORM) is expected to be good when the failure surface is nearly flat (low curvature) in the neighborhood of the design point (Der Kiureghian et al 1987). Principal curvatures of the performance functions corresponding to the cases presented on Figures 6.8 to 6.11 (truncated, beta, and uniform distributions) are plotted as a function of the LB LB

189 ratio of the lower-bound to the median or mean capacity on Figure 6.13 for the case where FS Median is equal to 3. 6 Mixed Lognormal Distribution 5 Reliability Index 4 3 FS Median 3.5 Numerical Integration FORM Ratio of Lower-Bound to Median Capacity Pf (FORM) / Pf (Numerical) FS Median Ratio of Lower-Bound to Median Capacity Figure Accuracy of FORM (mixed Lognormal Distribution, δ R = 0.4, δ S = 0.2). 159

190 Principal Curvature, Ks Truncated Normal Truncated Lognormal Uniform Beta Ratio of lower-bound to median (mean) capacity Figure Variation of the principal curvature with the lower-bound capacity for bounded capacity distributions (FS Median = 3.0, δ S = 0.2). The magnitude of the principal curvature increases from about zero (relatively flat performance function) at small ratios of lower-bound to median capacities to a maximum value of 0.7 for lower-bound capacities that are about 0.5 to 0.6 of the median capacities. The magnitude of the principal curvature drops for ratios of lower-bound to median capacities greater than 0.5 to 0.6. Geotechnical engineering systems generally have lower-bound capacities that are about 0.4 to 0.7 of the median or mean capacities. For this range of lower-bound capacities, and assuming that the load follows a lognormal distribution with a relatively small coefficient of variation (δ S = 0.2) compared to the coefficient of variation in the capacity, it is expected that the performance function describing failure will have curvatures ranging from 0.4 to 0.7 (Figure 6.13). The accuracy of FORM as indicated by the ratio of P f(form) to P f(numerical) is expected to decrease as the curvature of the performance function increases. The 160

191 variation of the ratio of P f(form) to P f(numerical) with the principal curvatures corresponding to the cases presented on Figures 6.8 to 6.11 is studied on Fig for the case where FS Median is 3.0. As expected, the accuracy of FORM tends to decrease as the curvature increases. For practical values of the principal curvature (0.4 to 0.7), the First Order Reliability Method overestimates the probability of failure by a factor of 1.5 to 2. A more detailed analysis involving all the data presented on Figures 6.8 to 6.11 is conducted to better understand the dependence of the accuracy of FORM on the magnitude of the principal curvature of the performance function. Results that correspond to different distributions for capacity, different ratios of lower-bound to median capacity, and different factors of safety are grouped according to the magnitude of the resulting reliability index and plotted on Figure Results indicate that the accuracy of FORM not only depends on the value of the principal curvature but also on the magnitude of the reliability index itself. The accuracy of FORM tends to decrease as the magnitude of the reliability index increases. In the cases analyzed on Figures 6.8 to 6.15, the load is assumed to follow a lognormal distribution with a relatively small coefficient of variation (δ S = 0.2). The accuracy of FORM in cases where the load has a relatively large coefficient of variation (δ S = 0.4) is studied on Figure 6.16 for cases where the capacity follows truncated normal and lognormal distributions with a coefficient of variation of 0.4. Results show that the ratio of the probability of failure calculated using FORM to the probability of failure calculated using numerical integration varies between 1 and 1.5 and increases as the factor of safety FS Median increases and as the ratio of the lower-bound to the median capacity increases. Comparison between Figure 6.16 (δ S = 0.4) and Figures 6.8 and 6.9 (δ S = 0.2) indicates that the accuracy of FORM increases as the uncertainty in the load increases relative to the uncertainty in the capacity. 161

192 Pf(FORM) / Pf(Numerical) Truncated Normal Truncated Lognormal Uniform Beta Principal Curvature, K s Figure Accuracy of FORM as a function of the principal curvatures of the performance function (FS Median = 3.0, δ S = 0.2). Pf(FORM) / Pf(Numerical) Reliability Index (1.0 to 2.0) Reliability Index (2.0 to 3.0) Reliability Index (3.0 to 4.0) Reliability Index (4.0 to 5.0) Principal Curvature, K s Figure Accuracy of FORM as a function of the principal curvatures of the performance function and the reliability index (δ S = 0.2). 162

193 4 4 Truncated Normal Truncated Lognormal Reliability Index 3 2 Exact FORM FSMedian Reliability Index 3 2 FSMedian Exact FORM Ratio of Lower-Bound to Median Capacity Ratio of Lower-Bound to Median Capacity Pf (FORM) / Pf (Numerical) FSMedian Pf (FORM) / Pf (Numerical) FSMedian Ratio of Lower-Bound to Median Capacity Ratio of Lower-Bound to Median Capacity Figure Accuracy of FORM (truncated distributions, δ R = 0.4, δ S = 0.4). 163

194 Principal curvatures of the performance function for the case where the coefficient of variation in the load is relatively large (δ S = 0.4) are calculated and plotted against the ratio of lower-bound to median capacity on Figure For ratios of lowerbound to median capacity ranging from 0.5 to 0.6, the curvature of the performance function has a maximum value ranging from 0.3 to For comparison, the maximum curvatures calculated in the cases where the load distribution had a coefficient of variation of 0.2 are double that value (K s = 0.7), but occur in the same range of ratios of lower-bound to median capacity (0.5 to 0.6). This decrease in the maximum curvature for relatively large coefficients of variation of the load (δ S = 0.4) is the reason why the accuracy of FORM increases (ratio of P f(form) to P f(numerical) decreases) in comparison to the case where the coefficient of variation was relatively small (δ S = 0.2). 0.4 Principal Curvature, Ks Truncated Normal (FS = 4.0) Ratio of Lower-Bound to Median Capacity 164 Truncated Lognormal (FS = 4.0) Figure Magnitude of the principal curvatures of the performance function for truncated distributions (FS Median = 4.0, δ S = 0.4, δ R = 0.4).

195 Results on Figure 6.16 (δ S = 0.4) and Figures 6.8 and 6.9 (δ S = 0.2) indicate that the accuracy of FORM is sensitive to the relative uncertainty in the load and the capacity. The maximum principal curvatures and the ratios of P f(form) to P f(numerical) corresponding to cases where the coefficients of variation of the load and the capacity are varied systematically are calculated and plotted on Figure Results are presented for a typical median factor of safety of 3.0 and a ratio of lower-bound to median capacity of 0.6. A lower-bound capacity that is equal to 0.6 of the median capacity is chosen because the value of the maximum principal curvature is expected to be maximum in the proximity of this ratio of lower-bound to median capacity (see Figures 6.13 and 6.17). The curves on Figure 6.18 indicate that the accuracy of FORM is expected to be the lowest for cases where the uncertainty in the load is small relative to the uncertainty in the capacity. As the uncertainty in the load increases, the accuracy of FORM increases with a corresponding decrease in the magnitude of the principal curvatures of the performance functions. The ratio of P f(form) to P f(numerical) has a maximum value of about 2.5 and a minimum value of about 1.2. These values correspond to maximum and minimum curvatures of about 0.8 and 0.15 respectively. The First Order Reliability Method is a widely used semi-analytical tool for estimating the probability of failure in reliability-based design analyses. The method is relatively simple compared to the Second Order Reliability Method and is computationally less expensive. Based on the results presented in this section, it is concluded that the First Order Reliability Method tends to overestimate the probability of failure in reliability problems where the distribution for capacity has a finite lower-bound capacity. The difference between probabilities of failure calculated using FORM to those calculated using numerical integration is relatively small for small ratios of lower-bound to median capacity and for designs that have relatively small factors of safety. For ratios 165

196 of lower-bound to median capacity and factors of safety that are representative of real geotechnical engineering systems, FORM tends to overestimate the probability of failure by a factor ranging from 1.5 to 2.5. Maximum Principal Curvature, Ks 1.0 Coefficient of Variation (Capacity) = 0.3 Coefficient of Variation (Capacity) = Coefficient of Variation (Capacity) = FS median = 3.0 r LB / r median = Coefficient of Variation (Load) Pf(FORM) / Pf(Numerical) Coefficient of Variation (Load) Figure Sensitivity of the maximum principal curvature and accuracy of FORM to the uncertainty in the load and capacity (truncated lognormal distribution). 166

197 Second Order Reliability Method A code was written in Matlab to implement the algorithm of the point-fitting Second Order Reliability Method that was proposed by Zhao and Ono (1999b) and described in Section The code is presented in Appendix D. The code is used in this section to calculate second order reliability indices for reliability-based design problems where the distribution of capacity is assumed to be bounded at the left-hand tail. The accuracy of SORM as indicated by the ratio of the probability of failure calculated using SORM to that calculated using numerical integration is studied on Figures 6.19 to Results correspond to cases where the capacity follows truncated normal and lognormal distributions (Figure 6.19), uniform and beta distributions (Figure 6.20), and mixed lognormal distributions (Figure 6.21). For all the cases analyzed, the load is assumed to be lognormally distributed with a coefficient of variation of 0.2. Results on Figures 6.19, 6.20, and 6.21 indicate that the probabilities of failure calculated using the Second Order Reliability Method are very close to the probabilities of failure calculated using numerical integration. The ratio of P f(sorm) to P f(numerical) varies between 0.94 and 1.04 (Figure 6.22). A detailed analysis of the results indicate a general tendency for SORM to slightly overestimate the probability of failure for small to intermediate ratios of lower-bound to median capacity and to slightly underestimate the probability of failure for large ratios of lower-bound to median capacities. The errors in probabilities of failure calculated using SORM are significantly smaller than those obtained using FORM and can be generally considered to be small for practical reliability-based applications. The Second Order Reliability Method can thus be considered as an efficient and accurate semi-analytical tool for calculating the probability of failure for reliability problems where the capacity distribution has a lower bound. 167

198 6 Truncated Normal Distribution Reliability Index FS Median Ratio of Lower-Bound to Median Capacity Truncated Lognormal Distribution Numerical Integration SORM Reliability Index FS Median Ratio of Lower-Bound to Median Capacity 168 Numerical Integration SORM Figure Accuracy of SORM (truncated normal and lognormal distributions, δ R =0.4, δ S =0.2).

199 6 Uniform Distribution 5 Reliability Index FS Mean Beta Distribution Ratio of Lower-Bound to Mean Capacity Numerical Integration SORM Reliability Index 4 3 FS Mean Numerical Integration 2.0 SORM Ratio of Lower-Bound to Mean Capacity Figure Accuracy of SORM (uniform and beta distributions, δ S = 0.2). 169

200 6 5 Mixed Distribution Model Numerical Integration SORM Reliability Index FS Median Ratio of Lower-Bound to Median Capacity Figure Accuracy of SORM (mixed lognormal distribution, δ S = 0.2) Pf(SORM) / Pf(Numerical) Ratio of lower-bound to median (mean) capacity Figure Accuracy of SORM (all distributions, δ S = 0.2). 170

201 6.4 SUMMARY In this chapter, the ability of semi-analytical tools to provide accurate reliability assessments for problems where the capacity distribution is bounded is investigated. Expressions and procedures that are required in applying the First Order Second Moment reliability method (FOSM), the First Order Reliability Method (FORM), and the Second Order Reliability Method (SORM) are presented. Probabilities of failure calculated using FOSM, FORM, and SORM are presented and compared to probabilities of failure calculated using numerical integration for a wide range of cases that are representative of practical geotechnical engineering designs. Based on the findings presented in this chapter, the following conclusions can be drawn: 1. The First Order Second Moment method results in poor estimates of the probability of failure. The method is incapable of reflecting the effect of the lower-bound capacity on the reliability since it only incorporates information about the means and standard deviations of the load and the capacity. 2. The First Order Reliability Method (FORM) is a widely used method for estimating the probability of failure. For the bounded distributions utilized in this chapter (truncated normal and lognormal, uniform, and beta), FORM tends to overestimate the probability of failure. 3. Maximum ratios of P f(form) to P f(numerical) occur at ratios of lower-bound to median capacity ranging from 0.5 to 0.6 and increase as the design becomes more conservative. The relatively large discrepancy between P f(form) and P f(numerical) for these cases results from increases in the curvature of the performance function at the most probable points. 171

202 4. The accuracy of FORM is affected by the magnitude of the reliability index and the relative uncertainty in the load and the capacity. The ratio of P f(form) to P f(numerical) increases as the reliability index increases and as the uncertainty in the load increases relative to the uncertainty in the capacity. For practical values of the coefficients of variation of the load and the capacity (0.1 < δ S < 0.5, 0.3 < δ R < 0.5), FORM will overestimate the probability of failure by a maximum factor ranging from 1.5 to The Second Order Reliability Method is an efficient and relatively accurate semi-analytical tool for calculating the probability of failure for reliability problems where the distribution for capacity has a lower bound. The ratio of P f(sorm) to P f(numerical) varies between 0.94 and 1.04 with a slight tendency for the higher ratios to occur at higher ratios of lowerbound to median capacity. 172

203 Chapter 7. Effect of Lower-Bound Capacity on the Reliability 7.1 INTRODUCTION In this chapter, the impact of having a lower-bound capacity on the reliability of geotechnical engineering systems is investigated. Initially, a mixed lognormal probability distribution is used to model the uncertainty in the capacity. The sensitivity of the probability of failure to variations in the ratio of the lower-bound to median capacity, the median factor of safety, and the relative uncertainty in the load and the capacity is then studied. In this initial analysis, the choice of a mixed lognormal distribution to describe the uncertainty in the capacity results from the wide-spread use of the lognormal distribution in conventional reliability analyses and because the mean and standard deviation describing the mixed lognormal distribution are approximately the same as for a non-truncated lognormal distribution. A variety of probability distributions (truncated normal and lognormal, uniform, and beta) are then utilized to study the effect of the distributional form of the capacity on the calculated reliability. Finally, a simple distribution-free bilinear model is introduced and used to calculate a slightly conservative estimate of the reliability index in terms of the ratio of the lower-bound to the median capacity, the median factor of safety, and the coefficients of variation of the load and capacity. In this simple model, the reliability index is calculated without the need for semi-analytical tools (FORM and SORM) or numerical integration. 7.2 EFFECT OF LOWER-BOUND CAPACITY To illustrate the effect of the lower-bound capacity on the reliability, the case where the load (S) follows a lognormal distribution and the capacity (R) follows a mixed lognormal distribution that is truncated at a lower-bound value (r LB ), is considered 173

204 (Figure 7.1). The choice of a lognormal distribution to model the continuous part of the probability distribution for capacity results from the wide-spread use of the lognormal distribution in conventional reliability analyses (Tang 1988 and 1990; Hamilton and Murff 1992; Tang and Gilbert 1993, API 1993; Horsnell and Toolan 1996; Bea et al. 1999; McVay 2000, 2002 and 2003; Kulhawy and Phoon 2002; Phoon et al. 2003; AASHTO 2004). With the probability distribution for capacity shown on Figure 7.1 and assuming a lognormal distribution for the load as is the convention (e.g., AASHTO 2004; McVay et al. 2000, 2002 and 2003; and Scott et al. 2003), the reliability is calculated as: Reliability = P(S < R) = P(S < R/r = rlb )P(R = rlb ) + ln(r = Φ ln(r + Φ LB LB ) ln(s ln(1 + δ ) ln(s ln(1 + δ median 2 S median 2 S ) ) ) ln(r Φ ) 2π LB ) ln(r ln(1 + δ 1 ln(1 + δ 2 R median 2 R rlb ) e )r P(S < R/r)f ) R 2 1 ln(r) ln(r ) median 2 2 ln(1+ δ R ) (r)dr (7.1) Probability Density Function Load Lower Bound Capacity Load and Capacity (kips) Figure 7.1. Reliability analysis using a mixed lognormal distribution for capacity. 174

205 where δ R is the c.o.v. accounting for uncertainty in the capacity; r median is the median capacity, δ S is the c.o.v. in the applied load; s median is the median load, and Φ() is the standard normal cumulative distribution function. A more general form of Equation 7.1 is obtained by normalizing the load and the capacity by the median capacity, r median : r ln r Reliability = Φ + rlb rmedian LB median r ln r Φ ln 2 ( + δ ) ln 1 LB median 1 FS S median ln 2 ( + δ ) ln 1 r ln () ln r 2 median e 2 ln( 1+ δ R ) d 2 r ( + ) r 2π ln 1 δ R rmedian S r LB ln ln 1 Φ rmedian 2 ln( 1+ δ R ) 1 FS median 2 r median () (7.2) where FS median is the median factor of safety and it is equal to the ratio of the median capacity to the median load. Hence, for given values of r LB /r median, δ R, and δ S, the reliability can be expressed entirely as a function of the median factor of safety. It should be noted that the median factor of safety differs from the factor of safety typically used in designs due to biases in the load and capacity and is related to it as follows: median design ( capacity median capacitydesign ) ( load load ) FS = FS (7.3) median design If conservative values of the load and capacity are used in calculating the design factor of safety, then the median factor of safety will be greater than the design factor of safety. 175

206 If there is no lower bound, that is r LB r median = 0, then the first term in Equation 7.2 is zero and the second term can be solved analytically, giving the following wellknown result (e.g., Ang and Tang 1984): Reliability = ln(fsmedian ) Φ if r LB = 0 (7.4) ln 2 2 [( 1+ δ )( + )] S 1 δ R When r LB r median > 0, an analytical solution is not available to Equation 7.2, so it must be solved using numerical integration or semi-analytical tools (FORM or SORM). In this chapter, the Second Order Reliability Method (SORM) is used consistently to calculate the reliability index for problems where the distribution for capacity is bounded at the left-hand tail. For purposes of comparison, the reliability can be converted into an equivalent reliability index, whether or not there is a lower-bound capacity, as follows: 1 ( Reliability) Reliability Index =β=φ (7.5) Curves showing the variation of the probability of failure and the equivalent reliability index as a function of the ratio of the lower-bound to median capacity are shown on Figure 7.2. The curves on Figure 7.2 represent the case were the uncertainty in the capacity (δ R = 0.4) is relatively large compared to the uncertainty in the load (δ S = 0.2). This example is representative of many geotechnical designs where the capacity is more uncertain than the load (McVay 2000, 2002 and 2003; Kulhawy and Phoon 2002; Phoon et al. 2003; AASHTO 2004). The primary conclusion from Figure 7.2 is that a lower-bound capacity can have a significant effect on the calculated reliability. For example, consider a typical case where the median factor of safety is 3.0. If the lower-bound capacity is anything greater than 176

207 0.6, the probability of failure is reduced by more than an order of magnitude compared to the case where there is no lower bound (Figure 7.2). The effect of the lower-bound capacity on the reliability is influenced by the magnitude of the median factor of safety; as the median factor of safety increases, the lower bound becomes more effective in reducing the probability of failure. 1.E-01 1.E-02 Probability of Failure 1.E-03 1.E-04 1.E-05 1.E-06 FS Median E-07 1.E Ratio of Lower-Bound to Median Capacity Approximation from Eq 7.8 Reliability Index 5 4 FS Median Most Probable Failure Point (Eq. 7.6) Ratio of Lower-Bound to Median Capacity Figure 7.2. Effect of lower-bound capacity on the reliability (δ R = 0.4 and δ S = 0.2). 177

208 The second conclusion from Figure 7.2 is that there is a threshold value for the ratio of the lower-bound to the median capacity, above which the lower bound affects the reliability (the knee in the curves on Figure 7.2). Below this threshold, the lower-bound capacity has essentially no effect on the reliability index. Above this threshold, the reliability index increases approximately linearly with increasing lower bound. The threshold decreases as the median factor of safety increases. A possible approximation for this threshold is the most probable failure point for the capacity in the case where there is no lower bound, which is given by Ang and Tang (1984) as: r LB,threshold r median r MPP,r r LB median = 0 = e ln ln(1+ δ [( )( )] β δ 1+ δ S 2 R ) R (7.6) The third conclusion from Figure 7.2 is that the shape of the curves for relatively large ratios of lower-bound to median capacities can be approximated analytically by assuming that the capacity is known and equal to the lower-bound value. For a load following a lognormal distribution, the reliability is bounded by the following expression: Reliability P(S < r LB ln(r ) = Φ LB ) ln(s median 2 ln(1+ δ ) S ) (7.7) A more general form of Equation 7.7 is obtained by normalizing the load and the capacity by the median capacity, r median such that: r LB 1 ln ln ( ) rmedian FSmedian Reliabilit y P S < r = Φ LB (7.8) 2 ln( 1+ δs ) 178

209 This bounding approximation is shown on Figure 7.2. As the ratio of the lowerbound to median capacity approaches one, the curves approach the analytical approximation in Equation 7.8. A noteworthy aspect of this approximation is that it does not depend on an assumed probability distribution for the capacity. To illustrate the magnitude of the effect of the lower-bound capacity, the median factor of safety that is required to achieve different levels of reliability are plotted on Figure 7.3 as a function of the ratio of the lower-bound to the median capacity. When the lower-bound capacity is greater than zero, trial and error is used with Equation 7.2 to find the median factor of safety that will give the specified reliability. Required Median Factor of Safety β = 4.0 β = 3.5 β = 3.0 β = Ratio of Lower-Bound to Median Capacity Figure 7.3. Required factors of safety for different lower-bound values (δ R = 0.4, and δ S = 0.2). The required median factor of safety decreases as the lower-bound capacity approaches the median capacity (Figure 7.3). To highlight the importance of the lowerbound capacity, consider a typical lower-bound capacity of 0.5 times the median capacity 179

210 and a target reliability index of 3.0. The required median factor of safety from a conventional reliability analysis is 3.7. However, if the lower-bound capacity is incorporated into the analysis, then the required median factor of safety is reduced to 2.9 while still maintaining the same level of reliability (β = 3). Results on Figure 7.3 indicate that resistance factors in a Load and Resistance Factor Design (LRFD), which control the median factor of safety, may need to incorporate information about the lower-bound capacity if they are to provide a consistent level of reliability. The results presented on Figures 7.2 and 7.3 are obtained by fixing the median capacity while varying the lower-bound capacity and the median load. Alternatively, for the purposes of considering LRFD, it is useful to perform an analysis where the magnitude of the lower-bound capacity is fixed relative to the median load, while the median factor of safety is varied. The same analysis can be repeated for different magnitudes of the lower-bound capacity (relative to the median load) to investigate the magnitude of the sensitivity of the calculated reliability index to the magnitude of the lower-bound capacity and the median factor of safety. In Figure 7.4, a fixed lower-bound capacity that is taken as a multiple of the median load is considered and the reliability index is plotted for different median factors of safety. Results are shown for ratios of lower-bound capacity to median load varying between 1.0 and 2.0. Both the median factor of safety and the ratio of the lower-bound capacity to the median load affect the reliability. However, results on Figures 7.2 and 7.4 indicate that the reliability index may be as or more sensitive to changes in the lowerbound capacity than to changes in the median capacity. Relatively small changes in the magnitude of the lower-bound capacity can lead to relatively large increases in the computed reliability indices. This result again supports the notion that LRFD codes may need to consider lower-bound estimates of capacity in addition to mean or median values. 180

211 5 Reliability Index 4 3 No Lower-Bound Capacity 2 Lower-Bound / Median Load = 1.2 Lower-Bound / Median Load = 1.5 Lower-Bound / Median Load = Median Factor of Safety Figure 7.4. Effect of median factor of safety and lower-bound capacity on reliability (δ R = 0.4, and δ S = 0.2). In all the cases presented above, the uncertainty in the load (c.o.v. = 0.2) was smaller than the uncertainty in the capacity (c.o.v. = 0.4). The variation of the reliability index is plotted as a function of the lower-bound capacity for the cases where the coefficient of variation of the load is varied on Figure 7.5. Different median factors of safety were chosen so that all the cases will have the same initial reliability index (2.50) when the lower-bound capacity is zero. The curves on Figure 7.5 show that the lowerbound capacity still affects the reliability index even when the uncertainty in the load is relatively high. However, the effect is smaller compared to the case where the uncertainty in the load is smaller. As the uncertainty in the load increases relative to that in the capacity, the tail of the load distribution dominates the probability of failure, making it less sensitive to the shape of the distribution for capacity. 181

212 5 Reliability Index 4 3 δ S Ratio of Lower-Bound to Median Capacity Figure 7.5. Effect of uncertainty in the load on the increase in the reliability (δ R = 0.4). The effect of the coefficient of variation of the capacity on the reliability index with a lower-bound capacity is shown on Figure 7.6. Results indicate that as the uncertainty in the capacity decreases, the reliability index becomes less sensitive to the lower-bound capacity. This result implies that the presence of the lower-bound capacity in problems where the uncertainty in the capacity is relatively small (δ R = 0.2) does not have a significant effect on the probability of failure. In addition, the convergence of the curves as the lower-bound capacity increases on Figure 7.6 indicates that the c.o.v. of the capacity, like the median capacity, becomes less important as the lower-bound capacity increases. These results in addition to the results presented on Figures 7.2 to 7.5 indicate strongly that the magnitude of the lower-bound capacity relative to the median capacity and median load tends to dominate the probability of failure which becomes less sensitive to the probability density function of the capacity. 182

213 6 Reliability Index δ R Ratio of Lower-Bound to Median Capacity Figure 7.6. Effect of uncertainty in the capacity on the increase in the reliability (FS median = 2.5, δ S = 0.2). 7.3 EFFECT OF THE PROBABILITY DISTRIBUTION FOR CAPACITY In this section, the effect of varying the distributional form for capacity on the calculated reliability is investigated by considering a variety of probability distributions that can accommodate a lower-bound capacity (truncated normal and lognormal, uniform, and beta). Truncated normal and lognormal distributions are defined by a coefficient of variation, a mean or median that is dictated by the factor of safety, and a lower-bound capacity that is a fraction of the median capacity. The beta distribution requires these parameters plus an additional parameter that defines the upper bound of the capacity distribution, which in this research is chosen to be 6 times the median load. The uniform distribution is defined solely by a median factor of safety and a lower-bound capacity that is a fraction of the median capacity. Once the lower-bound capacity and the median capacity of the uniform distribution are identified, the coefficient of variation is fixed. 183

214 The variation of the reliability index with the ratio of the lower-bound to median capacity for different distributions for capacity is plotted on Figure 7.7 for the case where the uncertainty in the capacity (δ R = 0.4) is relatively large compared to the uncertainty in the load (δ S = 0.2). The variation of the reliability index for the initial case where the capacity is assumed to follow a mixed lognormal distribution is also plotted on Figure 7.7 for comparison. Results corresponding to different distributions for capacity are plotted only for ratios of lower bound to median capacity that are greater than the threshold values discussed in Section 7.2. For ratios of lower bound to median capacity that are less than the calculated threshold values, the calculated reliability indices that are presented on Figure 7.7 correspond to the conventional case where both the load and the capacity are assumed to follow lognormal probability distributions and where the lower-bound capacity is assumed to be equal to zero. 6 5 Truncated Lognormal Truncated Normal Uniform Beta Mixed Lognormal Reliability Index FS Median Lower-Bound Capacity / Median Capacity Figure 7.7. Reliability calculations using different distributions for capacity (δ S = 0.2, δ R = 0.4). 184

215 Results on Figure 7.7 indicate that for a given ratio of lower-bound to median capacity, the probability distribution for capacity does not have a significant effect on the calculated reliability, especially as the median factor of safety increases. This result shows that the reliability of a design is affected more by the ratio of the lower-bound to median capacity than it is by the probability distribution used to model the capacity. Another important conclusion from Figure 7.7 is that reliability indices calculated using the mixed lognormal distribution constitute an approximate lower bound to the reliabilities calculated using other distributions. A true lower bound to the calculated reliabilities can be obtained by assuming that all the probability distribution for capacity constitutes a mass at the lower-bound capacity. This result is important because it indicates that the effect of the lower-bound capacity on the reliability can be higher than that shown on Figures 7.2 to 7.6. This observation together with the fact that the mean and standard deviation of a mixed distribution are relatively close to those of the nontruncated distribution makes the mixed distribution a powerful tool for modeling lowerbound capacities in distributions for capacity. To test the generality of the conclusions presented above, reliability calculations are performed for different combinations of coefficients of variations for the load and capacity, and the results are plotted on Figures 7.8 and 7.9. Figure 7.8 corresponds to the case where the uncertainty in the load (δ S = 0.1) is much smaller than the uncertainty in the capacity (δ R = 0.5) and Figure 7.9 corresponds to the case where the uncertainties in the load and the capacity are both large (δ S = δ R = 0.4). Results from Figures 7.8 and 7.9 indicate that the conclusions drawn from Figure 7.7 apply for a wide range of coefficients of variation in the load and capacity. For all the cases studied, the calculated reliabilities are governed by the ratio of lower-bound to median capacity and the median factor of safety, but much less dependant on the distribution for capacity. 185

216 Reliability Index Truncated Lognormal Truncated Normal Uniform Beta Mixed Lognormal 1 FS Median Lower-Bound Capacity / Median Capacity Figure 7.8. Reliability using different distributions for capacity (δ S = 0.1, δ R = 0.5). 4 Truncated Lognormal Truncated Normal Uniform Beta Mixed Lognormal 3 Reliability Index FS Median Lower-Bound Capacity / Median Capacity Figure 7.9. Reliability using different distributions for capacity (δ S = 0.4, δ R = 0.4). 186

217 7.4 SIMPLE BILINEAR MODEL FOR ASSESSING THE RELIABILITY A simple bilinear approximation is proposed to model the variation of the reliability index with the ratio of the lower-bound to median capacity (Figure 7.10). The model is defined by three points that can be evaluated without the need for numerical integration or semi-analytical approximations. First, the reliability index with no lowerbound capacity is calculated from Equation 7.4. The magnitude of this reliability index depends on the assumed probability density functions of the load and the capacity. The assumption of a lognormal distribution to model both the load and the capacity to calculate this reliability index is recommended and is consistent with conventional reliability analyses. Then a horizontal line is drawn from zero to the most probable failure point given by Equation 7.6. Finally, the reliability index with r LB / r median = 1.0 is calculated from Equation 7.8, and a straight line is drawn from the reliability index with no lower-bound capacity at the most probable failure point to this reliability index at r LB / r median = r LB /r median = 1.0 (Eq. 7.8) Reliability Index r LB = 0 (Eq. 7.4) r LB /r median = r MPP /r median (Eq. 7.6) Lower-Bound Capacity / Median Capacity Figure Proposed bilinear model for evaluating the reliability. 187

218 Bilinear approximations to curves describing the variation of the reliability index with the lower-bound capacity are plotted on Figure 7.11 for different levels of uncertainty in the load and capacity. Shown on the same figure are reliability curves corresponding to a mixed lognormal distribution for capacity. Results indicate that the proposed bilinear model provides a conservative but close approximation to the calculated reliability without the need for complicated computations. The proposed model provides an effective and simple tool for solving practical reliability-based problems in a realistic manner. The construction of the reliability curves on Figure 7.11 is relatively simple and can be performed by the practicing engineer without the need for a strong background in probability theory. 7.5 SUMMARY The major conclusion from this chapter is that a lower-bound capacity can cause a significant increase in the calculated reliability for a geotechnical design. The effect of the lower-bound capacity on the reliability is most pronounced when the uncertainty in the capacity is large compared to that in the load and when the degree of conservatism in the design is large. The reliability of a geotechnical system can be more sensitive to the lower-bound capacity than to the median capacity. This indicates that (1) resistance factors in a Load and Resistance Factor Design (LRFD) should incorporate information about the lower-bound capacity if they are to provide a consistent level of reliability and (2) reliability analyses will provide more realistic and useful information for decisionmaking purposes if they include information about lower-bound capacities. Finally, reliability assessments using different distributions for capacity indicate that the reliability of a design is affected more by the ratio of the lower-bound to median capacity than it is by the type of probability distribution used to model the capacity. 188

219 7 6 Mixed Lognormal Bilinear Model Reliability Index (a) δ R = 0.5, δ S = Lower-Bound Capacity / Median Capacity 5 Reliability Index (b) δ R = 0.4, δ S = Lower-Bound Capacity / Median Capacity Reliability Index 3 2 (c) δ R = 0.4, δ S = Lower-Bound Capacity / Median Capacity Figure Reliability approximations with the proposed bilinear model. 189

220 Finally, a simple bilinear model is proposed to relate the reliability to the lowerbound capacity as an approximation. Reliability assessments using the proposed model provide a close approximation to the reliability obtained using more complicated methods that are based on numerical integration or semi-analytical approximations (FORM and SORM). The proposed model provides an effective and simple tool for the practicing engineer to solve practical reliability-based problems in a realistic manner. 190

221 Chapter 8. Uncertainty in the Lower-Bound Capacity 8.1 INTRODUCTION In practice, the lower-bound capacity may not be known with certainty. For example, if physical models are used to estimate the lower-bound capacity, uncertainties will inevitably exist as a result of spatial variability in the properties of the soil and errors in the physical models used to predict the lower-bound capacity. If the lower-bound capacity is estimated from results of proof-load tests, uncertainties will emerge from inaccuracies in testing procedures. In this chapter, a mathematical framework is presented for incorporating information about uncertainty in lower-bound capacities in reliabilitybased analyses. Different approaches for modeling uncertainty in the lower-bound capacity are introduced and the effect of uncertainty on the reliability is addressed. 8.2 MODELING UNCERTAINTY IN LOWER-BOUND CAPACITIES In Chapter 7, it was shown that a mixed lognormal probability distribution is a practical alternative for modeling lower-bound capacities in reliability-based analyses. In all the cases analyzed in Chapter 7, the lower-bound capacity is assumed to be a deterministic known value. However, in practical applications, uncertainties in lowerbound estimates of capacity will inevitably exist and should be accounted for in reliability-based analyses. If uncertainty in the lower bound capacity exists, the probability of failure is calculated using the theorem of total probability as: 191

222 P f ( p f,r ) = f R (rlb)d(rlb ) (8.1) LB LB where p is the probability of failure given a lower-bound capacity (r LB ) and f (r ) f,rlb R LB LB is the probability density function describing the uncertainty in the lower-bound capacity. 8.3 EFFECT OF UNCERTAINTY IN LOWER-BOUND CAPACITIES ON THE RELIABILITY To illustrate the effect of uncertainty in the lower-bound capacity on the reliability, a case is considered where the lower-bound capacity is assumed to follow a lognormal distribution. The load (S) and the capacity (R) are assumed to follow lognormal and mixed lognormal distributions respectively with coefficients of variation of 0.2 and 0.4. An expression for the probability of failure for this case is given by: lnrlb λ r = ( ) 1 LB Pf pf,r φ d(rlb ) (8.2) LB 0 ξ r r LB LB where r LB λ and ξ r are parameters describing the lognormal distribution of the lower- LB bound capacity r LB, and φ() is the standard normal probability density function. The effect of uncertainty in the lower-bound capacity on the calculated reliability index is shown on Figure 8.1. The lower bound is assumed to follow a lognormal distribution with a median value (r LB,median ) ranging from 0.4 to 0.6 of the median capacity. For each median lower-bound value, the coefficient of variation is increased from 0 (no uncertainty) to 0.3; the resulting reliability index is then evaluated using numerical integration of Equation 8.2 and plotted on Figure 8.1. Results on Figure 8.1 indicate that for all ratios of median lower-bound to median capacity and for all levels of uncertainty in the lower bound, the calculated reliability 192

223 indices are greater than the reliability index obtained for the conventional case where the lower-bound capacity is assumed to be zero. For ratios of median lower-bound to median capacities that are relatively small, the reliability index is weakly affected by uncertainty in the lower-bound capacity. For larger ratios of median lower-bound to median capacities and for relatively larger coefficients of variation in the lower-bound capacity, uncertainty in the lower bound tends to decrease the reliability compared to the case where the lower bound is deterministic. 4 Median Lower Bound / Median Capacity 0.6 Reliability Index No Lower Bound Coefficient of Variation of the Lower-Bound Capacity Figure 8.1. Effect of uncertainty in the lower-bound capacity on the reliability (FS median = 3.0, δ R = 0.4, and δ S = 0.2). The curves on Figure 8.1 correspond to a typical case in geotechnical engineering where the uncertainty in the capacity (δ R = 0.4) is larger than the uncertainty in the load 193

224 (δ S = 0.2). An extreme but still practical case would be when the uncertainty in the capacity is much greater than the uncertainty in the load (δ R = 0.5 and δ S = 0.1). Curves are plotted on Figure 8.2 to show the effect of uncertainty in the lower-bound capacity on the reliability for this case. A median factor of safety of 3.0 is used in the analysis. Results on Figure 8.2 indicate that for the same median factor of safety and the same lower-bound capacity distribution, uncertainty in the lower-bound capacity affects the reliability in a greater degree compared to the previous case where the uncertainty in the capacity was smaller and the uncertainty in the load was larger (δ R =0.4 and δ S = 0.2). 7 6 Median Lower Bound / Median Capacity Reliability Index No Lower Bound Capacity Coefficient of Variation of the Lower-Bound Capacity Figure 8.2. Effect of uncertainty in the lower-bound capacity on the reliability (FS median = 3.0, δ R = 0.5, and δ S = 0.1). 194

225 8.3.1 Effect of the Degree of Uncertainty in the Lower-Bound Capacity To better understand the mechanism in which uncertainty in the lower-bound capacity affects the reliability, it is worth investigating how the shape of the mixed lognormal distribution for capacity is affected by uncertainty in the lower-bound capacity. When uncertainty in the lower-bound capacity exists, the cumulative distribution function of the capacity can be evaluated using the theorem of total probability as: CDF(r) = CDF(r rlb)f R (rlb)d(rlb) (8.3) LB For the special case where the lower-bound capacity follows a lognormal distribution and the capacity follows a mixed lognormal distribution, the cumulative distribution function (CDF) for capacity is given by Equation 8.4 as: CDF(r) = 0 lnr λ Φ ξ R lnrlb λ φ ξ rlb 1 r R rlb d(rlb) LB (8.4) The integral in Equation 8.4 can be evaluated analytically and expressed as: lnr λ lnr λ R r LB CDF(r) = Φ Φ (8.5) ξ R ξ rlb Equation 8.5 indicates that the CDF of capacity evaluated at a capacity r is equal to the CDF of r evaluated from the non-truncated lognormal distribution for capacity times a weighting factor that reflects the probability that the lower-bound capacity is less than the capacity r. For relatively high values of capacity (typically greater than the median capacity), the weighting factor approaches 1.0 and the CDF at r approaches the 195

226 CDF of the non-truncated lognormal distribution evaluated at r. For relatively small values of capacity, the two terms in Equation 8.5 approach zero and the CDF at r approaches zero. Values of capacity that are in the vicinity of the median lower-bound capacity are the most affected by uncertainty in the lower-bound capacity. The left-hand tail of the distribution for capacity typically governs the reliability of geotechnical designs. The effect of uncertainty in the lower-bound capacity is to allow for capacity values that are smaller than the median lower-bound capacity. Smaller capacity values will result for larger uncertainty in the lower-bound capacity. This will increase the region of overlap between the load distribution and the updated capacity distribution leading to a higher probability of failure compared to the case where the lower-bound capacity is deterministic. Cumulative distribution functions that describe the uncertainty in the capacity for cases where the lower-bound capacity follows a lognormal distribution that is characterized by ratios of median lower-bound to median capacity of 0.4 and 0.6 are shown on Figures 8.3 and 8.5 respectively. For each case, the coefficient of variation of the lower-bound capacity is varied to reflect different levels of uncertainty in the lower bound (Figures 8.4 and 8.6). The effect of uncertainty in the lower-bound capacity on the resulting distribution for capacity is evident on Figures 8.3 and 8.5. The left hand tails of the CDFs for capacity extend below the median lower-bound capacity (compared to the deterministic case) as a result of uncertainty in the lower-bound capacity. For the case with the smaller median lower bound (r LB,median / r median = 0.4), the degree of uncertainty in the lower-bound capacity does not affect the resulting capacity distribution significantly, which is the reason why the calculated reliability indices for this case are weakly affected by the degree of uncertainty in the lower-bound capacity (see Figure 8.1, r LB,median / r median = 0.4). 196

227 For the case with the larger median lower bound (r LB,median / r median = 0.6), the degree of uncertainty in the lower bound has a noticeable effect on the resulting capacity distribution and thus on the calculated reliability (see Figure 8.1, r LB,median / r median = 0.6) Effect of the Relative Uncertainty in the Load and the Capacity The degree in which uncertainty in the lower-bound capacity affects the reliability increases significantly as uncertainty in the capacity increases relative to uncertainty in the load (compare Figure 8.2 to Figure 8.1). Cumulative distribution functions for capacity for the case where uncertainty in the capacity is much larger than uncertainty in the load (δ R = 0.5 and δ S = 0.1) are plotted on Figure 8.7 for different levels of uncertainty in the lower-bound capacity. Plotted on the same figure is the probability density function for the load. Results on Figure 8.7 indicate clearly why uncertainty in the lower-bound capacity for this particular case has a significant effect on the reliability (see Figure 8.2, r LB,median / r median = 0.6). For the case with no uncertainty in the lower-bound capacity, the overlap region between the load and the capacity is extremely small resulting in a very small probability of failure (β = 6.3). As the uncertainty in the lower-bound capacity increases, the left hand tails of the distributions for capacity shift to the left resulting in an increase in this overlap region (decrease in β). The largest relative shift in the tail of the capacity distribution occurs for the case where the c.o.v. of the lower-bound capacity increases from zero (deterministic lower-bound capacity) to 0.1 resulting in a significant drop in the reliability index (β = 5.0). The relative shift between the capacity distributions decreases as the c.o.v. of the lower-bound capacity increases. This explains why the rate at which the reliability index decreases becomes smaller as the uncertainty in the lowerbound capacity increases (see Figure 8.2, r LB,median / r median = 0.6). 197

228 Cumulative Distribution Function Deterministic Lower Bound Lognormal Lower Bound (c.o.v.lb = 0.1) Lognormal Lower Bound (c.o.v.lb = 0.3) Ratio of Capacity to the Median Capacity Figure 8.3. Distribution for capacity with uncertain lower bound (δ R = 0.4, r LB,median / r median = 0.4). Probability Density Function Lognormal Lower Bound (cov= 0.1) Lognormal Lower Bound (cov= 0.3) Ratio of Lower-Bound to Median Capacity Figure 8.4. Probability distribution for the lower-bound capacity (r LB,median / r median = 0.4). 198

229 Cumulative Distribution Function Deterministic Lower Bound Lognormal Lower Bound (c.o.v.lb = 0.1) Lognormal Lower Bound (c.o.v.lb = 0.2) Lognormal Lower Bound (c.o.v.lb = 0.3) Ratio of Capacity to the Median Capacity Figure 8.5. Distribution for capacity with uncertain lower bound (δ R = 0.4, r LB,median / r median = 0.6). Probability Density Function Lognormal Lower Bound (cov= 0.1) Lognormal Lower Bound (cov= 0.2) Lognormal Lower Bound (cov= 0.3) Ratio of Lower-Bound to Median Capacity Figure 8.6. Probability distribution for the lower-bound capacity (r LB,median / r median = 0.6). 199

230 Deterministic Prior Mixed Lower Lognormal Bound Distribution Uncertain Updated Lower Distribution Bound (c.o.v. (covlb = 0.1) = 0.1) Uncertain Updated Lower Distribution Bound (c.o.v. (covlb LB = 0.2) = 0.2) Uncertain Updated Lower Distribution Bound (c.o.v. (covlb LB = 0.3) = 0.3) Capacity CDFs CDF or PDF Load PDF Ratio of Capacity or Load to the Median Capacity Figure 8.7. Effect of uncertainty in the lower-bound capacity on the reliability (r LB,median / r median = 0.6, FS median = 3.0, δ R = 0.5, and δ S = 0.1) Effect of the Median Factor of Safety Results on Figure 8.7 also indicate that the median factor of safety (which defines the distance between the load distribution and the capacity distribution) plays a role in determining the degree in which uncertainty in the lower-bound capacity affects the reliability. This effect is shown on Figure 8.8. The variation of the reliability index with the median factor of safety is plotted for two ratios of lower-bound to median capacities (r LB,median / r median = 0.5 and 0.7). For each ratio, two curves are plotted with one representing a deterministic lower bound and the other an uncertain lower bound with a coefficient of variation of 0.2. Two conclusions can be drawn from Figure 8.8: (1) for a given ratio of median lower-bound to median capacity, the drop in the reliability index due to uncertainty in the lower-bound capacity increases as the median factor of safety increases, and (2) the magnitude of the drop in the reliability index is larger for a larger median lower-bound to median capacity. 200

231 Reliability Index Median Lower Bound / Median Capacity = 0.5 Median Lower Bound / Median Capacity = 0.7 δ LB = 0 δ LB = No Lower Bound Median Factor of Safety Figure 8.8. Effect of median factor of safety on the degree in which uncertainty in the lower-bound capacity affects the reliability (δ R = 0.4, δ S = 0.2). 8.4 EFFECT OF THE PROBABILITY DISTRIBUTION OF THE LOWER BOUND CAPACITY In all the previous analyses, uncertainty in the lower-bound capacity was modeled using a lognormal distribution that is defined by a median (r LB,median ) and a coefficient of variation (δ LB ). In this section, the reliability is calculated for the case where the lowerbound capacity is assumed to follow a uniform distribution and compared to the reliability obtained for the original case where the lower bound was assumed to follow a lognormal distribution. The comparison is shown on Figure 8.9 for ratios of median lower-bound to median capacities ranging from 0.4 to 0.7 and for coefficients of variation in the lower-bound capacity ranging from 0 to

232 5 Median Lower Bound / Median Capacity Reliability Index Lognormally Distributed Lower Bound Uniformly Distributed Lower Bound 2 No Lower Bound Coefficient of Variation of the Lower-Bound Capacity Figure 8.9. Effect of the probability distribution of the lower-bound capacity on the reliability (FS median = 3.0, δ R = 0.4, and δ S = 0.2). Results on Figure 8.9 indicate that the form of the probability distribution of the lower-bound capacity does not have a significant effect on the calculated reliability. For coefficients of variation in the lower-bound capacity (δ LB ) ranging from 0 to 0.2, the discrepancies between calculated reliability indices are negligible. For coefficients of variation ranging from 0.2 to 0.3 and for relatively large ratios of median lower-bound to median capacity, calculated reliability indices for the uniformly distributed lower bound case tend to be smaller than reliability indices for the lognormally distributed lower bound case. The discrepancy between the calculated reliability indices results from differences in the shape of the left-hand tail of the resulting distributions for capacity in 202

233 each case. Cumulative distribution functions are shown on Figure 8.10 for the case where the coefficient of variation of the lower bound is equal to 0.3. Figure 8.10(a) and 8.10(b) correspond to ratios of median lower-bound to median capacity of 0.5 and 0.7 respectively. Results indicate that the distribution for capacity in the uniformlydistributed lower bound case is shifted to the left compared to that of the lognormallydistributed lower-bound case. These results help explain the discrepancy in the reliability indices presented on Figure Determinsitic Lower Bound Cumulative Distribution Function (a) r LB,median = 0.5 Cumulative Distribution Function Uniformly Distributed Lower Bound Lognormally Distributed Lower Bound (b) r LB,median = 0.7 r median Ratio of Lower-Bound to Median Capacity Ratio of Lower-Bound to Median Capacity Figure Distribution for capacity with uncertain lower bounds (δ LB = 0.3, δ R = 0.4, δ S = 0.2). 8.5 SUMMARY In this chapter, a mathematical framework is presented for including information about uncertainties in lower-bound capacities in reliability assessments. Practical probability distributions for lower-bound capacities are assumed and used to study the effect of uncertainty in the lower-bound capacity on the reliability. Based on the findings presented in this chapter, the following conclusions can be drawn: 203

234 1. For all ratios of median lower-bound to median capacity and for all levels of uncertainty in the lower-bound capacity, the calculated reliability indices are greater than the reliability index obtained for the conventional case where the lower-bound capacity is assumed to be zero. 2. For ratios of median lower-bound to median capacity that are relatively small, the reliability index is weakly affected by uncertainty in the lowerbound capacity. For larger ratios of median lower-bound to median capacities and for relatively larger c.o.v. s of the lower-bound capacity, uncertainty in the lower bound tends to decrease the reliability compared to the case where the lower-bound is deterministic. 3. The effect of uncertainty in the lower-bound capacity is to allow for values of capacity that are smaller than the deterministic lower-bound capacity. Smaller values for capacity will result for larger uncertainty in the lowerbound capacity. This will increase the region of overlap between the distribution of the load and the distribution of the capacity leading to higher probabilities of failure compared to the case where the lower-bound capacity is deterministic. 4. The degree in which uncertainty in the lower-bound capacity affects the reliability increases as uncertainty in the capacity increases relative to uncertainty in the load and as the median factor of safety increases. 5. For the same ratio of median lower bound to median capacity and coefficient of variation in the lower-bound capacity, the form of the probability distribution of the lower-bound capacity does not have a significant effect on the calculated reliability. 204

235 Chapter 9. Incorporating Lower-bound Capacities in LRFD Codes for Deep Foundations 9.1 INTRODUCTION Due to the implementation of a Load and Resistance Factor Design (LRFD) approach in American Association of State Highway and Transportation Officials (AASHTO) design specifications (e.g., AASHTO 1994 and 2004), there has been extensive research recently to develop resistance factors for foundations by performing reliability analyses with pile load test databases. Examples include the work by Barker et al. (1991), Withiam et al. (1997), Goble (1999), Liang and Nawari (2000), McVay et al. (2000, 2002 and 2003), Zhang et al. (2001), Kuo et al. (2002), Kulhawy and Phoon (2002), Phoon et al. (2003), Paikowsky (2003), and Withiam (2003). Similar work was conducted in the late 1980 s to support an LRFD code for offshore pile design (API 1993). Examples include Tang (1988 and 1990), Hamilton and Murff (1992), Tang and Gilbert (1993), Horsnell and Toolan (1996), and Bea et al. (1999). In all of the reliability analyses for deep foundations, the capacity is modeled using a lognormal distribution and the coefficient of variation for the capacity is relatively large, ranging from 0.3 to 1.0. A lognormal distribution, with a lower tail that extends to zero, does not capture the realistic possibility that there is a physical minimum or lower bound for the capacity. Incorporation of a lower-bound capacity into design for deep foundations is expected to provide a more realistic quantification of reliability for decision-making purposes (e.g., Fjeld 1977; Rodriguez et al. 1998; and Gilbert 2003) and a more rational basis for design (e.g., Bolton 1981 and Simpson et al. 1981). The objectives of this chapter are to propose an LRFD design-checking format that includes information on lower-bound capacities in addition to conventional design information. 205

236 9.2 INCORPORATING LOWER-BOUND CAPACITIES INTO LRFD Since a lower-bound capacity can have a significant effect on the reliability of a design, a reliability-based LRFD design code should include information on the lowerbound capacity. Two alternative formats are proposed here for including information about a lower-bound capacity in a LRFD design code: (1) a conventional design checking equation where the resistance factor is adjusted according to the lower-bound capacity and (2) a second design checking equation to include information about the lower-bound capacity Adjusted Resistance Factor for Lower-Bound Capacity The conventional design checking equation has the following general form: φ r γ s (9.1) R nominal S nominal where r nominal is the nominal capacity calculated using a design method, φ R is the resistance factor, q nominal is the nominal load for design, and γ S is the load factor. In order to incorporate the effect of a lower-bound capacity, this design checking equation is modified as follows: φ LB r γ s (9.2) R ( r ) nominal S nominal where the resistance factor, φ Rr ( LB ), is a function of the lower-bound capacity. When there is no lower bound, the reliability is calculated from Equation (9.3) as: 206

237 Reliability ( FS ) ln median = Φ if rlb = ln[ ( 1 δs )( 1 δ R )] (9.3) + + where FS median is the median factor of safety and δ S and δ R are the coefficients of variation of the load and the capacity respectively. If the nominal values of the load and the capacity are assumed to be equal to the mean values (the methods for predicting the load and capacity are unbiased), the required value for the resistance factor can be expressed in terms of the target reliability index as follows: φ R = Φ e β target ln γ S 2 2 [( 1+ δs )( 1+ δ R )] 1+ δ 1+ δ 2 S 2 R if r LB = 0 (9.4) The required resistance factor can also be expressed in terms of the required median factor of safety as: φ R γ 1+ δ 2 S S = 2 (9.5) FSmedian 1+ δr For a non-zero lower-bound, the required median factor of safety will decrease as the lower-bound increases. Since the lower-bound capacity does not affect the other variables in Equation (9.5), the required resistance factor when there is a lower-bound capacity, φ Rr ( LB ), can be expressed in terms of the conventional case as follows: φ R FS = FS median(r LB ( r ) R LB median(r = 0) LB ) φ (9.6) 207

238 where φ R is obtained from Equation (9.5) and corresponds to the case where there is no lower-bound capacity, FS median(r LB = 0) is the median factor of safety required for a target 2 βtarget ln[ ( 1+ δs )( 1+ δ )] reliability index if there is no lower-bound, 2 R FS median ( rlb = 0) = e ; and FSmedian( r LB ) is the median factor of safety required to achieve the target reliability index for a lower-bound value equal to r LB. Therefore, the increase in the resistance factor due to an increase in the lower-bound capacity is inversely proportional to the corresponding decrease in the required value for the median factor of safety. The variation of the ratio of the resistance factor incorporating a lower-bound capacity with the conventional resistance factor, φrr ( LB ) φ R, is shown as a function of the lower-bound capacity on Figure 9.1 for different target values of the reliability index. For reasonable values of the ratio of the lower-bound to median capacity, 0.4 to 0.9, the effect of the lower bound on the required resistance factor is significant. Consider a design with a target reliability index of 3.0 and a c.o.v. value for the capacity of 0.5. If the lower-bound capacity is 0.7 times the median capacity, then the required resistance factor in the design checking equation, Equation (9.1), will be more than double the conventional resistance factor (Figure 9.1(b)). For example, a conventional resistance factor of 0.25 would be increased to 0.6 in accounting for the lower-bound capacity Added Design Checking Equation for Lower-Bound Capacity An alternative code format would be to have two design checking equations: φ R φ r R nominal LB r LB γ OR γ S S s s nominal nominal (9.6) 208

239 2.0 (a) c.o.v. Capacity = δr = 0.3 β = β = 3 φ R(r LB ) / φ R 1.0 β = Ratio of Lower-Bound to Median Capacity (b) c.o.v. Capacity = δr = 0.5 β = 4 φ R(r LB ) / φ R β = 3 β = Ratio of Lower-Bound to Median Capacity Figure 9.1. Variation of the increase in the nominal resistance factor with the lowerbound capacity (c.o.v. Load = δ S = 0.15). 209

240 where the first design checking equation is the conventional equation and the second equation includes a resistance factor, φ, that is applied directly to the lower-bound R LB capacity. Providing that one or the other of the two equations is satisfied, a design will provide the specified level of reliability. The motivation for this form of the design checking equation is that the conventional approach is incorporated and does not need to be modified, whether or not there is a lower-bound capacity; the effect of a lower-bound capacity is reflected entirely in the second equation. The required value for φ R LB can be determined such that the target reliability is achieved by rearranging Equation (9.4) so that it is expressed in terms of the lower-bound capacity rather than the nominal capacity: φ R γ 1+ δ 2 = S S (9.7) 2 ( r r ) FS 1 δ LB LB nominal median(rlb ) + R Since the design method is assumed to be unbiased, the nominal capacity is equal to the 2 mean capacity (which is equal to FS 1+ ), and the lower-bound resistance factor median δ R given by Equation (9.7) can be expressed in terms of the ratio of the lower-bound to median capacity as: φ γ = δ (9.8) S R 1+ LB ( rlb rmedian ) FSmedian(r ) LB 2 S A plot of φr LB versus the lower-bound capacity is shown on Figure 9.2 for different target reliability indices. Approximations of φr LB as calculated using the simple bilinear reliability model that was proposed in Section 7.4 are also plotted on Figure 9.2 for comparison. The curves begin at values of the lower-bound capacity where the second design checking equation in Equation (9.6) governs. One advantage of this 210

241 approach with two design checking equations (Equation 9.6 versus Equation 9.1) is that φ R LB is not very sensitive to either the magnitude of the lower-bound capacity or the target reliability index (Figure 9.2). In fact, a conservative value of around 0.75 for could be used to cover a wide range of possibilities. 1.4 φ R LB φrlb β = 2 β = 3 β = 4 Approximation from Bilinear Model 0.2 (a) c.o.v. Capacity = δr = Ratio of Lower-Bound to Median Capacity 1.4 φrlb β = 2 β = 3 β = Approximation from Bilinear Model 0.2 (b) c.o.v. Capacity = δr = Ratio of Lower-Bound to Median Capacity Figure 9.2. Variation of the lower-bound resistance factor to account for a lower-bound capacity (c.o.v. Load = δ S = 0.15). 211

242 9.3 ILLUSTRATIVE EXAMPLES OF LOWER-BOUND CAPACITY IN LRFD In this section, two illustrative examples are provided whereby a lower-bound capacity is incorporated into a reliability-based design of a suction caisson foundation for a floating offshore facility for oil production and an onshore driven piles foundation supporting a pier of a bridge. The suction caisson is embedded in a layer of normally consolidated clay typical of soils in the Gulf of Mexico while the driven piles are embedded in layers of cohesionless soils Design of a Suction Caisson Foundation for an Offshore Facility The example offshore facility is a floating production system that is assumed to be representative of existing technology for deepwater oil production in the Gulf of Mexico. The offshore facility is fixed into place using mooring lines that are anchored to the seafloor using suction caissons. For the most heavily loaded suction caisson in the foundation system, the median vertical load at the Padeye is assumed to be equal to 500 kips. Due to the geometry of the structure and the metocean environment in deepwater, the uncertainty in the load is relatively small and can be reasonably characterized by a coefficient of variation of 0.15 (Gilbert et al. 2005). Conversely, the uncertainty in the capacity is relatively large (c.o.v. = 0.4) because the site is located in a frontier area with very little available data. The target annual reliability index for the design is set to 3.5. The conventional resistance factor (assuming a lower-bound capacity = zero) can be calculated from Equation (9.5) and is equal to The corresponding load factor is equal to 1.2. This corresponds to a required median factor of safety of 4.3. It should be noted that the median factor of safety differs from the factor of safety typically used in designs due to biases in the load and capacity and is related to it as follows: 212

243 median design ( capacitymedian capacitydesign ) ( load load ) FS = FS (9.9) median design Gilbert et al. (2005) report practical values ranging from 0.4 to 0.7 for the ratio of load median to load design and 1.3 for the ratio of capacity median to capacity design. Assuming a bias factor of 0.6 and 1.3 for the load and capacity respectively, the required median factor of safety of 4.3 is equivalent to a design factor of safety of about 2.0, which is typical for current design procedures used in the industry to design suction caissons By applying the conventional LRFD equation (Equation 9.1), a suction caisson with a nominal capacity (mean capacity) of 2308 kips is required to achieve the target reliability index of 3.5. The corresponding median capacity r median is equal to 2142 kips. Assuming normally consolidated clay with a shear strength intercept at the mudline of 100 psf and a shear strength gradient of 10 psf/ft (typical for Gulf of Mexico), a suction caisson with a diameter of 13 ft and a length of 77.5 ft will yield the desired capacity (2308 kips). An alpha value of 1 and an N value of 9 are assumed in the analysis. Curves showing the variation of the increase in the nominal resistance factor and lower-bound resistance factor with the lower-bound capacity are shown on Figures 9.3(a) and 9.3(b) respectively for the case where the coefficients of variation of the load and capacity are 0.15 and The curves correspond to a reliability index of 3.5. If a typical sensitivity value of 2.5 is assumed, a lower-bound capacity can be calculated using the lower-bound model described in Chapter 3. The lower-bound capacity is equal to 884 kips or 0.4 times the predicted capacity and 0.43 times the median capacity. From Fig. 9.3(a), the resistance factor accounting for the lower bound can be increased by 1.3 times the conventional value of 0.26: φ R( r LB = 0.42 r median ) = = The updated nominal capacity that is needed to withstand the applied load is 213

244 reduced to 1765 kips. A more economical suction caisson with the same diameter of 13 ft and a reduced length of 64 ft will yield the desired nominal capacity (1765 kips). The incorporation of the lower-bound capacity into this design has a significant effect on the level of conservatism required, reducing the required length from 76ft to 64ft while still providing the target level of reliability (β = 3.5) φ R(rLB) / φ R β = 3.5 (a) Ratio of Lower-Bound to Median Capacity β = 3.5 φ R (b) Ratio of Lower-Bound to Median Capacity Figure 9.3. Variation of the nominal resistance factor and lower-bound resistance factor with the lower-bound capacity (δ S = 0.15, δ R = 0.4). 214

245 Alternatively, the lower-bound resistance factor corresponding to a ratio of lowerbound to median capacity of 0.43 can be found from Figure 9.3(b) to be φ RLB = According to Equation (9.6), a proposed design is considered acceptable if it satisfies either one of the two proposed equations. The conventional LRFD equation yields a suction caisson with a diameter of 13 ft and a length of 76 ft. The second format requires a suction caisson that can yield a lower-bound capacity that is equal to ( ) / 0.86 = 697 kips. As expected a suction caisson with a diameter of 13 feet and a length of 64 feet will provide the required lower-bound capacity Design of Driven Piles for a Foundation of a Bridge In this section, a case study presented by Goble (1996) is used to illustrate how the proposed LRFD checking equation that incorporates a lower-bound capacity can be used in the design of a bridge foundation. The foundation to be designed supports an isolated bridge column that is subjected to a critical factored axial load of 10 MN. The piles to be used in supporting the bridge column consist of closed-ended steel pipe piles with an outside diameter of 355 mm and a length of 25 m. A simplified schematic of the soil profile at the site is shown on Figure 9.4. Depth (m) 6 Soil Description sandy silty clay medium - dense sand with fine gravel SPT N values 6 10 to medium coarse sand with some gravel 20 to 30 Figure 9.4. Soil profile for case study example (from Goble 1996). 215

246 The predicted (mean) axial capacity of a single driven pile is 1.8 MN as calculated by the API method (API 1993). If the c.o.v. values for the capacity (δ R ) and load (δ S ) are respectively 0.5 and 0.15 (which are typical for driven piles in cohesionless soils and for bridge loading conditions respectively), the mean bias (λ R ) for the capacity is 1.0, and the target reliability index is 3.0, then the conventional resistance factor (φ R ) is obtained from Equation (9.4) and is equal to The factored capacity per pile is then calculated from Equation (9.1): φ R r nominal = MN = 0.42 MN. Since each pile provides a factored capacity of 0.42 MN and the total factored load is 10 MN, the total number of piles needed to support the column load is 10 MN 0.42 MN = 24 piles based on the conventional approach. To account for the effect of a lower-bound capacity in this design, a lower-bound capacity can be calculated for each pile using the available soil profile (Figure 9.4) and the lower-bound model presented in Chapter 2 for driven piles in cohesionless soils. The lower-bound capacity is 0.9 MN, or 0.5 times the predicted capacity and 0.55 times the median capacity. From Figure 9.1(b), the resistance factor accounting for the lower bound can be increased by 1.7 times the conventional value of 0.23: φ Rr ( LB= 0.5rno min al ) = = The corresponding factored capacity per pile is then found from Equation (9.2): φ Rr ( = 0.5r ) r = MN = 0.72 MN. LB no min al nomin al to be Alternatively, the lower-bound resistance factor can be found from Figure 9.2(b) φ R = According to Equation (9.6), the target reliability will be achieved LB with the larger of the factored nominal capacity from the conventional equation, φ Rrnominal= 0.42MN, and the factored capacity based on the lower-bound capacity, φ r = MN = 0.72 MN. Therefore, the same factored capacity of RLB LB 0.72 MN for design is obtained using either version of the design checking equation (Equation 9.2 or 9.6), and the required number of piles to support the column load is

247 MN/0.72 MN = 14. Incorporating the lower-bound capacity into this design has a significant effect on the level of conservatism required, reducing the required number of piles from 24 to 14 while still providing the target level of reliability for each foundation. 9.4 SUMMARY Information about the lower-bound capacity can be incorporated into Load and Resistance Factor Design (LRFD) codes using two alternative formats that would not require substantive changes to existing codes. The practical significance of the proposed LRFD equations is illustrated using two examples related to the design of offshore and onshore deep foundations in cohesive and cohesionless soils respectively. Results indicate that the effect of the lower-bound capacity on the reliability can be easily reflected using the two proposed LRFD formats. 217

248 Chapter 10. Approaches for Updating Lower-Bound Capacities 10.1 INTRODUCTION Databases with load tests on driven piles and suction caissons provide evidence for the existence of a physical lower-bound capacity that can be calculated using information on the geometry of the foundation together with lower-bound site-specific soil properties. Estimates of lower-bound capacities for deep foundations can be further updated using information from: (1) soil resistance mobilized during the installation process, (2) static and dynamic proof load tests, and (3) historical record of the foundation under load. The objective of this chapter is to illustrate how such information can be utilized in a rational manner to update estimates of lower-bound capacities for foundation systems. Practical examples are then presented to illustrate the proposed concepts UPDATING LOWER-BOUND CAPACITIES OF INDIVIDUAL FOUNDATIONS Static Proof-Load Tests Traditionally, static proof-load tests have been used to verify the validity of design methods and construction procedures in foundation engineering. In the allowablestress design approach, the foundation is sized based on some empirical design method using a reduced factor of safety (typically 2.0) provided that it passes a proof-load test up to twice the design load (ASTM D1143, 1994). The observation that a foundation survives a static proof-load test indicates the minimum load-carrying capacity of the foundation, i.e. its lower-bound capacity. It does not reveal the actual capacity and 218

249 reliability of the proposed design. For a foundation that survives a specified proof-load test, the prior information about the lower-bound capacity of that particular foundation can be updated to reflect the result of the proof-load test. If a foundation passes a proofload test, the reliability of that particular foundation is expected to increase significantly due to the proof load, especially as the proof load increases. This assumes that the capacity that is measured in a proof-load test does not decrease with time. In structural reliability applications, attempts have been made to update the reliability of proof-loaded structural members of bridges and buildings by truncating the left-hand tail of the capacity distribution at the applied proof-load level (Fujino and Lind 1977; Grigoriu and Hall 1984; Fu and Tang 1995; and Stewart and Val 1999). Since the reliability of structural members is governed by uncertainty in the load, relatively high proof loads are required to provide a notable increase in the reliability of proof-loaded designs. In geotechnical engineering applications, the left-hand tail of the capacity distribution governs the probability of failure since the uncertainty in the capacity is generally larger than the uncertainty in the load. For a foundation that survives a specified proof-load test, the tail of the capacity distribution is essentially truncated at the proof-load level, reducing or even eliminating the probability that the capacity is less than the measured proof load. As a result, the reliability is expected to be strongly affected by the proof load even when the proof load is significantly smaller than the median capacity Dynamic Tests For driven piles, an estimate of the capacity at the end of driving (EOD) can be obtained by monitoring the blow counts that are required to advance the pile to its final depth of penetration. This estimate of pile capacity is typically obtained using dynamic wave equation models (Smith 1960; Thendean et al. 1996; Likins et al. 1996; and Likins 219

250 and Rausche 2004). GRLWEAP is a widely used program that utilizes the wave equation model to perform pile-driving analyses. Ultimate axial capacities of piles are expected to differ from estimated capacities at the end of driving due to pile setup (increase in capacity with time) or relaxation (decrease in capacity with time). Pile setup is typically observed in soft to medium clays, loose saturated silts, and loose to medium dense sands and silty sands (York et al. 1994). Pile relaxation has been observed in dense to very dense saturated fine sands, inorganic silts, and weathered bedrock formations (Yang, 1970; York et al. 1994; Morgano and White 2004). The change in pile capacity with time can be measured in the field using dynamic re-strike analyses in which the pile is re-driven and an updated pile capacity at the beginning of re-strike is computed with GRLWEAP. For both end of driving and beginning of re-strike conditions, the accuracy of the wave equation capacity predictions can be improved by updating the input parameters of GRLWEAP with measurements of acceleration and strain that are taken at the top of the pile during dynamic testing. The analysis is referred to as a CAPWAP signal matching process. For piles that exhibit strength increase with time due to soil setup, the estimate of pile capacity at EOD is expected to constitute a lower bound to the ultimate pile capacity and can be considered as a proof load for the tested foundation. If an increase in pile capacity is expected due to pile setup, the lower-bound estimate of capacity that is inferred from the driving information can be further updated with pile re-strike data. To explore the above hypotheses, GRLWEAP capacities at end of driving and CAPWAP capacities at beginning of re-strike for pile tests reported in databases analyzed by Thendean et al. (1996), Likins et al., (1996), and Likins and Rausche (2004) are digitized and plotted on Figure 10.1 versus measured capacities obtained from static load tests performed on the same piles. Measured capacities correspond to those obtained using 220

251 Davisson s method to reduce load test data (Davisson 1970). To account for uncertainty in the measured capacity due to test and interpretation errors (McVay 2002; Tang 1988), bounds corresponding to ±10 percent about the reported value are shown for each data point on Figure Measured Static Capacity (kn) Thendean et al (a) Measured Static Capacity (kn) GRLWEAP Capacity, EOD (kn) T CAPWAP / T St at ic < 1/ Best Match CAPWAP Capacity, BOR (kn) Figure Estimates of proof loads for piles using End of Driving (EOD) and Beginning of Restrike (BOR) data. 221 (b) Likins and Rausche 2004 Likins et al. 1996

252 The data on Figure 10.1(a) indicates that estimated pile capacities at EOD generally constitute a lower bound to the capacities measured using static load tests, with only 8% of the estimated capacities being higher than the measured capacities (Figure 10.1(a)). For this small percentage of piles, the observed behavior can be attributed to (1) the piles being driven in soils that promote relaxation, (2) uncertainty in the wave equation prediction of capacity at end of driving, and (3) inadequate setup time between the end of driving and time of static load test. A detailed analysis of the data points shown on Fig. 10.1(a) is not currently feasible due to the lack of comprehensive information regarding soil types, pile types, and time between the end of driving and static and dynamic tests for each data point in the databases. However the available data indicates that capacity at the end of driving reflects a lower-bound estimate of the ultimate measured capacity. Similar conclusions can be drawn from Figure 10.1(b) which shows pile capacities predicted using the CAPWAP signal matching method at the beginning of re-strike and measured capacities using static load tests. Only pile tests where dynamic re-strike tests were conducted at a much earlier time than static load tests are included in the analysis Historical Performance of Foundations under Load Successful performance of existing structures can provide information on proofload levels for structural members of older buildings and bridges (Stewart 1997; Stewart and Val 1999; and Faber et al. 2000). The same concept applies to foundation systems for existing offshore and onshore structures and bridges. If a foundation system can successfully withstand loads for a number of years of service, it can be deduced that the capacity of the foundation is higher than prior imposed loads. For example, the capacity of a deep foundation for a bridge that has been in service for a number of years is higher 222

253 than the dead and traffic loads that the foundation was subjected to in its design life. Similarly, observation of the successful performance of offshore foundations during severe environmental loading events indicates a minimum load carrying capacity of the foundations. The prior service loads act as a proof test of structural and geotechnical strength and will constitute a lower bound to the existing foundation capacity Uncertainties in Estimated Proof-Load Levels In practice, the proof-load level for a foundation capacity may not be known with certainty. If proof-load levels are estimated from static or dynamic load tests, uncertainties will emerge from inaccuracies in testing procedures, wave equation analyses, and definition of failure. If historical service load records are used to estimate the proof-load level of an existing foundation, uncertainties will result primarily from uncertainty in predicting the live loads. Zhang and Tang (2002), Likins and Rausche (2004), and Zhang (2004) report coefficients of variation ranging between 0.05 and 0.15 to represent uncertainties in capacities measured using static load tests. When proof load levels are estimated from dynamic tests, uncertainties will exist as a result of inaccuracies in the wave equation analysis used to predict capacity. Statistics showing the accuracy of the simple GRLWEAP wave equation analysis and the more sophisticated CAPWAP analysis with signal matching are summarized in Tables 10.1 and In Table 10.1, GRLWEAP capacities are compared to CAPWAP capacities at the end of driving with results indicating coefficients of variation ranging from 0.12 to 0.36 in the ratio of GRLWEAP capacity to CAPWAP capacity. In Table 10.2, CAPWAP capacities at beginning of restrike are compared to measured capacities from static load tests conducted on the same piles. The uncertainty in the ratio of CAPWAP capacities to static capacities can be represented by a c.o.v. of

254 Table Accuracy of Analysis using GRLWEAP (Rausche et al., 2004). * Capacity GRLWEAP / Capacity CAPWAP Pile Material No of Piles Mean c.o.v. Steel Concrete *GRLWEAP results are for Standard Wave Equation Analyses conducted at End of Driving * CAPWAP results obtained using Best Match approach also at End of Driving Table Accuracy of Analysis using CAPWAP (Likins and Rausche, 2004). * Capacity CAPWAP / Capacity STATIC Study No of Cases Mean c.o.v. Goble et al Likins et al Stress Wave Conferences All Studies *CAPWAP results are for Restrike analyses using Best Match approach * STATIC results are for Static Load Tests on the same piles 10.3 EXAMPLES OF UPDATING THE RELIABILITY OF FOUNDATIONS Updating with Static Proof-Load Tests The effect of proof-load tests on the reliability of foundations is influenced by the magnitude of the proof load relative to the median capacity and by the median factor of safety. Curves relating the reliability index and the probability of failure to the magnitude of the proof load that is applied to an example deep foundation are plotted on Figure The example considers a drilled shaft that is designed using a median factor of safety of 2.0 to support a load applied by a pier of a bridge. The curves on Figure 10.2 indicate that the reliability index for the proposed design increases from a low value of about 1.7 (for the case where the shaft is not proof-load tested) to a value of 3.5 provided that it successfully withstand a proof load that is greater than 1.5 times the design load (0.75 of the median capacity). A truncated lognormal capacity distribution is used in Figure 10.2 to calculate the updated probability of failure. 224

255 1.E+00 6 Probability of Failure 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 FS Median Reliability Index FS Median E Ratio of Proof Load to Median Capacity Ratio of Proof Load to Median Capacity Figure Effect of proof loads on the reliability (δ R = 0.4, δ S = 0.15). An important conclusion from Figure 10.2 is that proof loads can have a significant effect on the calculated probability of failure even when the proof load is significantly smaller than the median capacity. Load tests that are conducted using small proof loads are expected to be less expensive than conventional proof-load tests since they can be performed using a smaller loading system and in a shorter period of time Updating with Dynamic Load Tests Alternatively, the reliability of the drilled shaft can be validated using a dynamic re-strike test that is conducted using a relatively small proof load (relatively low energy imparted to the butt of the shaft) accompanied with a CAPWAP signal matching process. In this case, the estimate of the lower-bound capacity that is obtained from the dynamic test is assumed to be uncertain and can be modeled with a coefficient of variation of about 0.2 (Table 10.2). The effect of the uncertain proof load on the reliability of the proposed design is illustrated in Figure 10.3 for different median values and coefficients of variation of the proof load. 225

256 1 No Proof Load 4 Median Proof Load / Median Capacity Probability of Failure Reliability Index Median Proof Load / Median Capacity No Proof Load c.o.v. of the Proof Load c.o.v. of the Proof Load Figure Effect of uncertainty in the proof load (FS median = 2, δ R = 0.4, δ S = 0.15). For ratios of median proof load to median capacity that are relatively small, the probability of failure and the reliability index are weakly affected by uncertainty in the proof load. For larger ratios of median proof load to median capacity and for relatively larger coefficients of variation of the proof load, uncertainty in the proof load tends to decrease the reliability compared to the case where the proof load is deterministic. The impact of conducting a re-strike test on the design of the example drilled shaft is studied on Figure If the shaft is not dynamically tested, a median factor of safety that is greater than 3.0 is required to achieve a typical target reliability index of 3.0. If a re-strike analysis is designed such that a load of about 1.5 times the design load can be applied to the drilled shaft, results on Figure 10.4 indicate that the shaft can be designed using a reduced factor of safety of 2.2 while still maintaining the target reliability index of 3.0. This realistic analysis is based on the assumption that the estimate of the lower-bound capacity that is measured in the dynamic re-strike test is uncertain and is characterized by a c.o.v of

257 4 0 3 Reliability Index c.o.v. Proof Load Target Reliability Index 1 0 No Proof Load Median Proof Load / Median Load = Median Factor of Safety Figure Effect of uncertainty in proof loads on the design of foundations (δ R = 0.4, δ S = 0.15) UPDATING THE RELIABILITY OF FOUNDATIONS AT A SITE WITH LOAD TESTS When a limited number of proof-load tests are conducted on a small percentage of foundations at a site, Bayesian techniques can be used to update the probability distribution of the foundation capacity at the site (Sidi and Tang, 1987 and Zhang, 2004). In the updating process, the results of proof-load tests are typically used to update the middle of the capacity distribution (mean or median). Sidi and Tang (1987) and Zhang (2004) report that the level of the proof-load test has to be higher than the predicted mean (or median) capacities so that the updating process will have a significant effect on the reliability. Zhang (2004) recommends conducting 1 to 3 tests using proof loads that are larger than 1.5 times the predicted pile capacity so that the value of the proof-load test can be maximized. 227

258 Proof-load tests that are conducted up to 1.5 times the predicted capacity can be quite expensive. In addition, the likelihood of failing the foundation during the test increases significantly as the proof-load level increases. Since the reliability of a foundation is governed by the left-hand tail of the capacity distribution, it is hypothesized that results of a larger number of proof-load tests that are conducted up to a relatively small load compared to the median capacity (for example, 0.5 times the median capacity) can be effective in updating the tail of the capacity distribution. To test the above hypothesis, Bayesian techniques are utilized to update the lower-bound capacity (rather than the mean or median capacity) at the tail of truncated capacity distributions using results from load tests conducted at relatively small proof-load levels Bayesian Updating on the Lower-Bound Capacity To test the above hypothesis, it is assumed that the probability distribution describing the uncertainty in the capacity of foundations at a site is modeled with a truncated lognormal distribution that is characterized by a lower-bound capacity r LB, a median capacity r median, and a coefficient of variation δ R. The lower-bound capacity is assumed to be uncertain while r median and δ R are assumed to be deterministic. For simplicity, it is assumed that the prior distribution of the lower-bound capacity r LB is modeled using a probability mass function (example, Figure 10.5(a)). The statistics of the lower-bound capacity can be inferred from lower-bound capacity models using information from lower-bound soil properties. If n proof-load tests are conducted at a site using a proof-load level r proof, and if all the piles tested are able to withstand the proof load, the prior probability distribution of the lower-bound capacity at the site can be updated using Bayes Theorem such that: 228

259 P'' rlb (r LB ) = lnrproof λr lnrlb λr Φ Φ ζr ζr 1 P' r (rlb) LB lnr LB λr 1 Φ ζr n lnrproof λr lnrlb,i λr Φ Φ ζr ζr 1 P' r (rlb,i ) LB lnrlb,i λr 1 Φ ζr m i= 1 n (10.1) where P' ' LB ( r ) and ' ( r ) r LB P r LB LB are the prior and updated lower-bound probability mass functions respectively. The updated distribution of the lower-bound capacity is then used to calculate an updated estimate of the reliability of the foundations at the site Bayesian Updating Example To illustrate the proposed procedure, it is assumed that the distribution of pile capacity at a site where 1000 piles need to be driven follows a truncated lognormal distribution with a typical coefficient of variation of 0.4. The applied load follows a lognormal distribution with a coefficient of variation of With no information about lower-bound capacities, a median factor of safety of 3.5 is required to achieve a target reliability index of 3.0 at the site under consideration. However, the pile capacity at the site is expected to have a lower bound that is inferred from information on lower-bound soil properties. This estimate of the minimum capacity is described by the probability mass function presented on Figure 10.5(a). By incorporating information about lowerbound capacities using the distribution shown on Figure 10.5(a), the required median factor of safety reduces to 3.2 while still maintaining the same level of reliability. If load tests are conducted on 3% of the piles at the site using a relatively small proof load (0.6 times the predicted pile capacity), and if all the piles survive the proof 229

260 load, the distribution of the lower-bound capacity at the site can be updated using Equation 10.1 and is plotted on Figure 10.5(b). The effect of successful performance of the tested foundation under the proof load is to decrease the likelihood of the small lower-bound values and to increase the likelihood of the high lower-bound value. Using the updated lower-bound distribution, the median factor of safety required to achieve the desired reliability index of 3.0 is reduced to 2.5. By running successful proof-load tests of relatively small magnitude (0.6 of the predicted capacity) on 3% of the piles at the site, the required median factor of safety is reduced by about 30% while still maintaining the same level of reliability. Probability Mass Function /3 1/3 1/ Lower-Bound / Median Capacity Probability Mass Function Lower-Bound / Median Capacity (a) Prior Distribution (b) Updated Distribution Figure Prior and updated distributions for the lower-bound capacity (r proof /r median =0.6; n=30). 230

261 Discussion on the Design of Proof-Load Test Programs Based on the results of a load-test program, the use of lower factors of safety can be justified for the final foundation design. A less conservative design results in savings in installation and material costs and will lower the risk of foundation failure (e.g., buckling or refusal) during installation. The value of a particular load-test program is reflected in the difference between expected cost savings and costs associated with implementing the test program. The cost of implementing a load-test program depends on the expected number of piles to be tested and on the magnitude of the applied proof load. Different load-test programs can be implemented to justify the use of a less conservative design for deep foundations at a project site. Traditionally, static proof-load tests that are conducted on a limited number of test piles have been used to achieve this objective. When only a limited number of static proof-load tests are conducted at a site, results from reliability studies indicate that very high proof loads (3 times the design load) are required to achieve designs with acceptable target reliability indices (Zhang 2004). The cost of conducting load tests with large proof loads can be very high. Alternatively, less conservatism in the design can be achieved by designing test programs where a larger number of piles are proof-load tested to relatively low proofload levels (0.6 times the predicted capacity). The example design presented in Section illustrates this concept. Load tests that are conducted using small proof loads are expected to be less expensive since they can be performed using a smaller loading system in a shorter period of time compared to conventional proof-load tests. An extreme but still practical load-test program can involve proof-load testing all the piles at a site using a very small proof-load level (0.4 times the predicted capacity). A decision analysis that incorporates alternative load-test programs can be conducted in the design phase of a project to provide a rational basis for choosing the number of proof-load tests and the 231

262 magnitude of the proof load that would maximize the value of information of the test program SUMMARY The following conclusions are drawn from this chapter: 1. Static proof-load tests can have a significant effect on the reliability of a foundation even if they are not conducted to a high load level relative to the predicted capacity. If a foundation survives a proof-load test, the reliability of that particular foundation is expected to increase significantly as the proof load increases. 2. In soils that promote setup, an updated estimate of the lower-bound capacity of a deep foundation can be obtained using information from dynamic tests conducted at the End of Driving or at the Beginning of Re-strike. The uncertainty associated with the updated lower-bound estimate should be accounted for in the calculation of the updated reliability of the tested foundation. 3. Results from proof-load tests that are conducted to relatively low proof-load levels on a limited number of foundations can be used to update the distribution of pile capacity at a site. An example design indicates that the results of such proofload tests can be effective in significantly increasing the reliability of the foundations provided that an adequate number of foundations are proof-load tested. In summary, a decision analysis can be conducted in the design phase of a project to provide a rational basis for choosing the number and type of proof-load tests and the magnitude of the proof load that would maximize the value of information of a load-test program. 232

263 Chapter 11. Conclusions The objective of this research was to develop a methodology and a mathematical framework for incorporating information about lower-bound capacities in reliabilitybased analyses and to study the effect of lower-bound capacities on the reliability of geotechnical engineering systems. Evidence of the existence of a physical lower-bound capacity for deep foundations was found by analyzing databases of load tests conducted on driven piles and suction caissons. Information about lower-bound capacities, predicted capacities, and measured capacities was then integrated and used to calibrate realistic probability distributions for capacity. Multiple LRFD design-checking formats that include information on lower-bound capacities were proposed and practical approaches for updating information about lower-bound capacities were investigated. In this chapter, the research is summarized and the major conclusions reached from the work are presented, along with recommendations for future work EVIDENCE OF A LOWER-BOUND CAPACITY FOR DEEP FOUNDATIONS Two existing load test databases for driven piles in cohesive and cohesionless soils, in addition to a newly assembled database of load tests for suction caissons in normally consolidated clay, were analyzed to provide evidence of the existence of a lower-bound capacity for deep foundations. In this section, the conclusions that resulted from the analysis of these databases and recommendations for future studies are presented. 233

264 Conclusions Two pile-load test databases for driven piles in cohesive and cohesionless soils and one database for load tests on suction caissons in normally consolidated clays provide strong evidence for the existence of a physical lower-bound capacity. The lowerbound capacity can be estimated using simple and practical models based on site-specific soil properties and information about the geometry of the foundation. For driven piles in normally consolidated to slightly overconsolidated clay, the undrained shearing strength of the remolded soil is used to calculated an estimate of the lower-bound capacity. The ratio of the lower-bound capacity to the predicted capacity for a set of 34 piles in normally consolidated to slightly overconsolidated clays ranges from 0.3 to 1.0 and has an average of For driven piles in heavily overconsolidated consolidated clay, lower-bound capacities are calculated based on lower-bound estimates of the horizontal effective stress and interface friction angle between the pile and the soil. The ratio of the lower-bound capacity to the predicted capacity for a set of 11 piles in overconsolidated clay ranges from 0.4 to 0.85 and has an average of For driven piles in siliceous, cohesionless soils, an estimate of the lower-bound capacity is calculated using a variation of the API method, by replacing the lateral coefficient of earth pressure with the at-rest value and the soil-pile friction angle and endbearing capacity factors with the values for one-category less in density. For piles with oversized end plates, the lower-bound capacity is calculated assuming that the side resistance is equal to zero. The ratio of the lower-bound capacity to the predicted capacity for a set of 36 piles in siliceous, cohesionless soils ranges from 0.2 to 0.75 and has an average of For suction caissons in normally consolidated clay, estimates of the lower-bound capacity were obtained using a variation of the α-method by replacing the undisturbed 234

265 undrained shearing strength with the undrained shearing strength of the remolded soil and by using an α of 1 and an end bearing factor, N, of 9. In cases in which the top cap of the caisson was vented, the lower-bound side friction was calculated as the sum of the frictional resistance acting on the inner and outer walls of the caisson and the lowerbound reverse end bearing is assumed to act on the annulus of the caisson. In tests in which the top cap was sealed, the lower-bound side friction was calculated from the external skin friction and the lower-bound reverse end bearing was assumed to act on the full cross sectional area of the caisson. Data from a newly assembled database comprised of published laboratory model tests, centrifuge tests, and full scale field tests conducted in uplift on suction caissons in normally consolidated clays indicate that the ratio of the lower-bound capacity to the predicted capacity for a set of 25 suction caissons ranges from 0.35 to 1.0 and has an average of Recommendation for Future Work While these results correspond only to three databases for driven piles and suction caissons that are used primarily for the design of offshore foundations, the same principle is expected to apply to any load-test database that is used to develop design methods and resistance factors for geotechnical engineering systems. Future work in this area should include analyzing the lower-bound capacity in additional databases with load tests for a variety of foundation types. With these databases, models to calculate the lower-bound capacity should be developed, refined or calibrated REALISTIC PROBABILITY DISTRIBUTIONS FOR DEEP FOUNDATIONS Realistic and practical approaches for modeling lower-bound capacities were investigated to aid in providing a mathematical framework for including lower-bound 235

266 capacities in reliability assessments. Several types of probability distributions, including bounded, truncated, and mixed distributions, can accommodate a lower-bound capacity and were proposed in this research as candidate realistic probability distributions for capacity. The ability of Hermite Polynomial transformations to model general probability distributions for capacity was also studied and the order of the Hermite Polynomial that was needed to achieve adequate flexibility in modeling different tails of distributions of capacity was investigated. Several different types of probability distributions were then utilized to model the uncertainty in the capacity of driven piles and suction caissons based on information regarding measured, predicted, and lower-bound capacities. In this section the conclusions that resulted from the analysis of these databases and recommendations for future studies are presented Conclusions Bounded distributions (uniform, triangular, shifted exponential, shifted lognormal, and beta) have been rarely used in conventional reliability analyses in civil engineering. However, these distributions can accommodate a lower-bound capacity and serve as a likely alternative for modeling distributions for capacity. Since most reliability analyses in geotechnical engineering tend to utilize the conventional lognormal distribution to model the uncertainty in the capacity, the use of a shifted lognormal distribution to constrain the left-hand tail of the distribution for capacity at a certain lower-bound value seems plausible. However, because of the characteristics of a conventional lognormal distribution, the shifted lognormal distribution has a disadvantage in that the left-hand tail does not initiate from the value of the specified lower-bound capacity but rather from a larger value. The beta distribution, on the other hand, is widely used in different areas of civil engineering and has a main advantage in 236

267 that it has upper and lower-bound capacities and flexibility in the shape of the distribution. Truncated and mixed probability distributions also provide a simple approach for incorporating lower-bound capacities in reliability analyses. Truncated and mixed normal and lognormal distributions provide simple and practical alternatives for modeling the uncertainty in the capacity primarily because (1) conventional normal and lognormal distributions are commonly used in engineering design, and (2) the parameters describing them are the same as those of non-truncated distributions with the addition of one extra parameter, the lower-bound capacity (r LB ). When the coefficient of variation of the capacity becomes large and when the lower-bound capacity increases and approaches the mean or median capacity, the means and coefficients of variations of truncated and mixed distributions become different from the means and coefficients of variations of the conventional non-truncated distributions. This problem is less pronounced for mixed distributions. Results from analyses conducted using mixed normal and lognormal distributions indicated that for all ratios of lower bound to median capacity that are less than 0.9, the difference in the means of mixed and non-truncated distributions is less than 10%, which can be considered small compared to the difference in the means of truncated and non-truncated distributions (which can approach 30%). The fact that the means and standard deviations of the mixed lognormal distribution do not differ significantly from means and standard deviations of conventional non-truncated distribution for practical values of coefficients of variation of the capacity and practical ratios of lower-bound to median capacities indicates that results from statistical analyses that are published in the literature can be incorporated directly into the mixed probability model without having to re-analyze the raw data. 237

268 Hermite Polynomial transformations provide a flexible tool for modeling distributions for capacity with different arbitrary shapes. Results indicated that Seventh order Hermite Polynomials provide a very good fit to the left-hand tails of all distributions investigated in this study. These distributions included uniform, triangular, shifted exponential, shifted lognormal, beta, truncated normal and lognormal, and mixed normal and lognormal distributions. Fifth order Hermite Polynomials provided an acceptable fit to the left-hand tails of truncated distributions, shifted exponential distributions, and beta distributions, but a relatively poor fit for the tails of the uniform and triangular distributions in the close vicinity of the lower-bound capacity. Finally, third order Hermite Polynomials provided a relatively poor fit for almost all the cases analyzed in this study. Beta, truncated lognormal, and mixed lognormal probability distributions were utilized to model the uncertainty in the capacity of driven piles and suction caissons based on information regarding measured, predicted, and lower-bound capacities. A similar analysis was conducted with mixed seventh order Hermite Polynomial distributions to provide a more general and flexible model for the distribution for capacity. Seventh order Hermite Polynomials were used as a reference to which the models provided by bounded, truncated, and mixed distributions are compared. Results from the three sets of data indicated that the mixed lognormal distribution provided the closest approximation to the more general Hermite Polynomial distributions. Based on the results of these analyses, it is recommended that the mixed lognormal distribution be used to model the distribution for capacity of driven piles and suction caissons. 238

269 Recommendation for Future Work In this research, it was assumed for simplicity that data points in the three available databases constitute a set of statistically independent variables, following a certain probability distribution whose parameters are to be determined. A more general analysis that addresses any possible correlation in the data can provide a more realistic model for the uncertainty in the capacity and should be accounted for in future work. In addition, uncertainty exists in the values of the measured, predicted, and lower-bound capacities that were used to calibrate the parameters of the different probability models. Advanced techniques for calibration that are based on Bayesian analyses can account for the uncertainty in these values in future calibration efforts that aim at finding parameters of probability distributions for capacity EFFECT OF THE LOWER-BOUND CAPACITY ON THE RELIABILITY To investigate the effect of the lower-bound capacity on the reliability, different methods for calculating the probability of failure when there is a lower-bound capacity were explored. These included numerical and semi-analytical methods for calculating the probability of failure. The effect of having a lower-bound capacity on the reliability of geotechnical engineering systems is studied and the sensitivity of the probability of failure to variations in the ratio of the lower-bound to median capacity, the median factor of safety, and the relative uncertainty in the load and the capacity is investigated. The effect of uncertainty in the lower-bound capacity on the reliability is also included and addressed. In this section the conclusions that resulted from these analyses are presented. The ability of semi-analytical methods to provide accurate estimates of the probability of failure when there is a lower-bound capacity was investigated. Results indicated that the First Order Second Moment method (FOSM) results in poor estimates 239

270 of the probability of failure. The method is incapable of reflecting the effect of the lowerbound capacity on the reliability. Results obtained using the First Order Reliability Method (FORM) indicated that for practical values of the coefficients of variation of the load and the capacity (0.1 < δ S < 0.5, 0.3 < δ R < 0.5), FORM will overestimate the probability of failure by a maximum factor ranging from 1.5 to 2.5. Maximum ratios of P f(form) to P f(numerical) occur at ratios of lower-bound to median capacity ranging from 0.5 to 0.6 and increase as the design becomes more conservative and as the uncertainty in the load decreases relative to that in the capacity. Results obtained using the Second Order Reliability Method (SORM) were close to those obtained using numerical integration. Ratios of P f(sorm) to P f(numerical) varied between 0.94 and 1.04 with a slight tendency for the higher ratios to occur at higher ratios of lower-bound to median capacity. Results from a large number of reliability analyses with a lower-bound capacity indicated that a lower-bound capacity can cause a significant increase in the calculated reliability for a geotechnical design. The effect of the lower-bound capacity on the reliability was most pronounced when the uncertainty in the capacity is large compared to that in the load and when the degree of conservatism in the design was large. In addition, it was concluded that the reliability of a geotechnical system can be more sensitive to the lower-bound capacity than to the median capacity. This indicated that (1) resistance factors in a Load and Resistance Factor Design (LRFD) should incorporate information about the lower-bound capacity if they are to provide a consistent level of reliability and (2) reliability analyses will provide more realistic and useful information for decisionmaking purposes if they include information about lower-bound capacities. Reliability assessments using different distributions for capacity indicated that the reliability of a design is affected more by the ratio of the lower-bound to median capacity than it is by the type of probability distribution used to model the capacity. 240

271 Based on the observed variation of the probability of failure with the ratio of the lower-bound to median capacity, a simple bilinear model was proposed to relate the reliability to the lower-bound capacity as an approximation. Reliability assessments using the proposed model provided a close approximation to the reliability obtained using more complicated methods that are based on numerical integration or semi-analytical approximations (FORM and SORM). The proposed model provides an effective and simple tool for the practicing engineer to solve practical reliability-based problems in a realistic manner. In practice, the lower-bound capacity may not be known with certainty. Results of reliability analyses with an uncertain lower-bound capacity that was initially assumed to be lognormally distributed indicated that for all ratios of median lower-bound to median capacity and for all levels of uncertainty in the lower-bound capacity, the calculated reliability indices were greater than the reliability index obtained for the conventional case where the lower-bound capacity was assumed to be zero. For ratios of median lower-bound to median capacity that are relatively small (r LB, median / r median < 0.5), the reliability index was weakly affected by uncertainty in the lower-bound capacity. For larger ratios of median lower-bound to median capacities and for relatively larger c.o.v. s of the lower-bound capacity (δ R > 0.2), uncertainty in the lower-bound capacity tended to decrease the reliability compared to the case where the lower-bound is deterministic. Calculations using a uniformly distributed lower-bound capacity indicated that for the same ratio of median lower bound to median capacity and coefficient of variation in the lower-bound capacity, the form of the probability distribution of the lower-bound capacity does not have a significant effect on the calculated reliability. 241

272 11.4 INCORPORATING LOWER-BOUND CAPACITIES IN DESIGN OF FOUNDATIONS Since a lower-bound capacity can have a significant effect on the reliability of a design, a reliability-based LRFD design code for deep foundations should include information on the lower-bound capacity in the design-checking equations. In addition, information about lower-bound capacities can be updated with installation data, proofload data, and data regarding the historical behavior of the foundation under load. In this section, conclusions from analyses involving the use of lower-bound capacities in LRFD are presented. In addition, the implications of the use of different approaches for updating values of lower-bound capacities on the design of foundations are addressed and recommendations for future studies are presented Incorporating Lower-Bound Capacities into LRFD Two LRFD design-checking formats that include information on lower-bound capacities in addition to the conventional design information were proposed and methodologies for calculating resistance factors were presented. The first format is a conventional design checking equation where the resistance factor is adjusted according to the lower-bound capacity. In the second format, a second design checking equation is introduced to include information about the lower-bound capacity. The effect of the lower-bound capacity on design for both formats is reflected in the magnitudes of the resistance factors. One advantage of the second format with two design checking equations is that the lower-bound resistance factor φ is not very sensitive to either the magnitude of the lower-bound capacity or the target reliability index. In fact, a conservative value of around 0.75 for φ R LB could be used to cover a wide range of possibilities. R LB 242

273 Updating Lower-Bound Capacities The effect of different approaches for updating values of lower-bound capacities of deep foundations on the design was investigated. Results indicated that static proofload tests that are conducted on a particular foundation can have a significant effect on the reliability of the foundation, even if they are not conducted to a high load level relative to the predicted capacity. If a foundation survives a proof-load test, the reliability of that particular foundation is expected to increase significantly as the proof load increases. In soils that promote setup, an updated estimate of the lower-bound capacity of a deep foundation can be obtained using information from dynamic tests conducted at the end of driving or at the beginning of re-strike. The uncertainty associated with the updated lower-bound estimate can be large and should be accounted for in the calculation of the updated reliability of the tested foundation. Results from proof-load tests that are conducted to relatively low proof-load levels on a limited number of foundations can be used to update the distribution of pile capacity at a site. An example design indicated that the results of such proof-load tests can be effective in significantly increasing the reliability of the foundations provided that an adequate number of foundations are proof-load tested Recommendation for Future Work In future work, it is recommended that a thorough and comprehensive decision framework should be developed to provide a rational basis for choosing the number and type of proof-load tests, in addition to the magnitude of the proof load that would maximize the value of information of a load-test program. The framework should include realistic costs of the foundation that is being designed, the implementation of the proofload test program, and failure. Important factors that will affect the decision are the 243

274 expected number of foundations that will fail during the load test, the probability of failing this number of foundations, and the updated probability of failure of the foundation system as a result of the output of the test program. 244

275 Appendix A. Database of Load Tests on Piles in Cohesive Soils A.1 INTRODUCTION A database containing the results of 45 full scale pile load tests is described. The load tests are conducted on steel pipe piles embedded in cohesive soils. A detailed summary of each case study for which corresponding references are available is provided in the following sections. The summary includes information regarding pile geometry, soil properties, site preparation, pile driving, setup, and load testing. A.2 NORMALLY CONSOLIDATED TO SLIGHTLY OVERCONSOLIDATED CLAYS A.2.1 Pile # 1: Seed and Reese (1955) In 1955, Seed and Reese published the results of a series of pile load tests conducted on a 6 diameter steel pipe pile driven into 15 feet of normally consolidated clay. The tests were conducted at a site adjacent to the San Francisco-Oakland Bay Bridge on the east side of San Francisco Bay in California. The soil at the site consisted of 4 feet of fill on top of 5 feet of sandy clay overlying a 30 foot layer of organic silty clay containing shells. This layer is locally referred to as Bay Mud. The ground water is approximately 4 feet below the ground surface. In order to provide uniform soil properties at the site, a 12 inch steel casing was installed in the upper nine feet consisting of fill and sandy clay. The soil in the casing was then removed and the pile to be tested was lowered to the surface of the clay at the bottom of the casing and driven into the clay using a 150-lb drop hammer. 245

276 Unconfined compression tests were conducted to measure the undisturbed and remolded shear strength of the soil at the site. Undisturbed samples were obtained using a piston-type sampler in seamless steel tubes that were 30 inches long and 2.87 inches in diameter. A total of 23 unconfined compression tests were conducted on undisturbed samples whereas only 13 unconfined compression tests were conducted on remolded samples. Figure A.1 shows the variation of the undisturbed and remolded undrained shear strengths with depth. The average undisturbed and remolded strengths are 244 and 110 psf respectively indicating a soil sensitivity of about 2.2. The average unit weight of the undisturbed silty clay was 112 pcf whereas the average liquid and plastic limits were 41.5 and 23.5 respectively. Information about the soil properties at the site and the method of shear strength measurement is presented in Table A Undrained Shear Strength (psf) Pile Depth (ft) Undisturbed Shear Strength Remolded Shear Strength Figure A.1. Undisturbed and remolded shear strength profiles of Bay Mud. 246

277 Table A.1. Summary of soil properties at the site near San Francisco bay. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB Profile Thick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf Clay Clay Clay Clay UC-Pushed Tip UC-Pushed The pile was first tested 3 hours after driving was completed. After that, a series of tests were conducted at different time intervals ranging between 21 hours and 33 days after driving. In this database, the bearing capacity of the pile tested 33 days after driving is considered to be representative of the capacity of the pile at full setup, and will be referred to as Test 1 in the pile database. The test procedure involved loading the pile in increments of 500 lb to 1000 lb for the first few loads and decreasing the increments as the ultimate capacity was approached. The loading was stopped when the settlement increased markedly with small increase in the load. The results of the pile load tests conducted at different setup times are presented in Table A.2. Table A.2. Results from load tests conducted at different setup times. Time of Test Measured Capacity (lbs) 3 hours hours days days days days days

278 A.2.2 Piles # 2 and 3: McCammon and Golder (1970) Pile load tests were conducted on a 24-inch diameter steel pipe pile driven into a 200 foot thick layer of silty clay on the East bank of the Lower Arrow Lake in British Colombia, Canada. The upper 20 feet in the soil profile are soft, sensitive, and normally consolidated. Below that level, the soil is lightly overconsolidated and ranges in consistency from firm to very stiff. The undrained strength profile of the soil is shown on Figure A.2 and information about soil properties are presented in Tables A.3 and A.4. The undisturbed and remolded undrained shear strength of the soil at depths of 20 to 80 feet was measured using an insitu vane device. The soil in this depth range has a sensitivity of 2.6. The undisturbed undrained shear strength at depths greater than 80 feet was measured using unconfined compression tests conducted on samples obtained using 3 thin wall Shelby tubes that were pushed into the ground. The sensitivity of this layer was also assumed to be 2.6. Table A.3. Soil profile for Pile # 2 (length = 100 feet) Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB Profile Thick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf Clay UC-Pushed Clay Field Vane Clay UC-Pushed Tip Table A.4. Soil Profile for Pile # 3 (length = 158 feet) Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB Profile Thick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf Clay UC-Pushed Clay Field Vane Clay UC-Pushed Tip UC-Pushed

279 Undrained S hear S trength (psf) Undisturbed Vane Strength Undisturbed Unconfined Compression Remolded Vane Strength Pile 2 60 Depth (ft) Pile Figure A.2. Undisturbed and remolded shear strength profiles at British Columbia. One special feature of the test site is the presence of excess pore water pressures in the soil deposit. Water pressure as high as 13 feet above the ground level were measured at the test pile location. The trend of the variation of the excess water pressure head and the resulting effective stress with depth is shown on Figure A

280 Pressure Head (feet) Pressure (psf) Excess Pressure Head Hydrostatic Pressure Head Total Pressure Head 20 Total Stress Pore Pressure Effective Stress Depth (ft) Depth (ft) Figure A.3. Variation of the total stress, pore pressure, and effective stress with depth. The 24-inch pile was driven using a Vulcan OR single acting hammer that delivered approximately ft-lb of rated energy per blow. The weight of the ram was 9300 lb and the stroke was about 3.25 feet. The test pile was driven open ended to a depth of 100 ft. The clay inside the pile was then removed, the soil was cleaned out to one foot ahead of the pile tip, and a skin friction compression test was performed 26 hours after pile driving was completed. The loading system consisted of a large concrete slab with a capability of applying a 410 kips and a large number of concrete blocks each weighing 4 kips. The weights were jacked against the pile using a 1000-kip high-pressure hydraulic jack. 250

281 Nine days following pile installation, another load tests was conducted on the same pile to measure the increase in pile capacity due to increased setup time. This test will be considered as Test 2 in the pile database. The measured ultimate capacity was 424 kips. Following this set of tests, the pile was redriven to a depth of 153 feet, the interior cleaned up to one foot ahead of the tip of the pile, and another load test was conducted 30 hours after driving was completed. A 30 foot concrete plug was then tremied in the pile base and the pile was driven another 5 feet to a total depth of 158 feet. Three days after redriving, a load test was performed on the pile followed by another test after 170 days of setup. This test will be labeled as Test 3 in the pile tests database. The capacity of the 158-foot long pile 177 days after redriving was 840 kips compared to 710 kips for a setup time of 3 days. A.2.3 Piles # 4 to 7: Mansur and Focht (1956) Four pile load tests were conducted on steel pipe piles driven into an 80-foot layer of very compressible clay overlying sand at the site of the Morganza floodway control structure in Louisiana. The piles had diameters ranging between 18 and 30 inches and embedded lengths ranging between 60 and 70 feet. An 8 foot deep hole was excavated at each test location to eliminate any skin friction above the proposed base of the structure. The soil at the site consisted of an 80-foot layer of highly plastic compressible clay that is underlain by a layer of sand with a relative density of 80%. Both unconfined compression tests and unconsolidated-undrained triaxial compression tests were conducted on undisturbed samples taken from different depths in the clay layer. The results indicated an average undisturbed shear strength of 660 psf. A plot of the 251

282 undisturbed shear strength profile is shown in Figure. A.4. It should be noted that lab tests were only conducted on samples from depths that were less than 50 feet in the clay layer. Below that depth, the authors in the original paper show blow count values for depths exceeding 50 feet. For depths between 50 and 80 feet, the blow count values do not vary indicating uniform soil conditions. The sensitivity of the clay at the site is about It should be noted that the individual data points corresponding to the remolded strength tests were not presented explicitly. The clay layer had a water content ranging from 42% to 60%, a plastic limit ranging between 23% and 28%, and a liquid limit ranging from 60% to 100%. Undrained Shear Strength (psf) Prebored Sand Depth (ft) Pile 5 70 Piles 4,6,7 80 Figure A.4. Undisturbed shear strength profile at Morganza. 252

283 The steel pipe piles had an outer diameter ranging from 18 to 30 inches. Information regarding the diameter, embedment length, and wall thickness of each pile is presented in Table A.5. Piles 4, 6, and 7 had similar embedment lengths and were situated in the same vicinity on the site, whereas Pile 5 had a relatively shorter embedment depth and was situated about 1700 feet away from the other piles. Information about soil properties of the layers in which the individual piles were embedded in is presented in Tables A.6 through A.8. The piles were driven using a skid mounted, whirly type driver with 82-foot leads. A single acting hammer delivering 30,225 ft-lb of energy was used to drive the 24-inch and 30-inch piles whereas a single-acting hammer delivering 15,000 ft-lb of energy was used to drive the 18-inch pile. All piles were filled with concrete a few days after driving. A setup time ranging from 4 to 6 weeks was allowed prior to any load testing on the piles. The pile load tests were conducted by jacking the piles against concrete weights using hydraulic jacks mounted on steel pedestals. The piles were loaded in 40-kip increments at a rate of 3 kips per minute. The measured capacities from the 4 pile load tests are presented in Table A.5. Table A.5. Measured capacities and properties of the piles. Pile # Pile Diameter (in) Wall Thickness (in) Total Length (ft) Exposed Length (ft) 253 Depth of Casing (ft) Effective Embedment Length (ft) Measured Capacity (kips) / / / /

284 Table A.6. Soil profile for Pile # 4. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB SPT ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf N 60 Clay Clay UC+UU-Pushed SASI Clay UC+UU-Pushed Clay UC+UU-Pushed Tip UC+UU-Pushed Table A.7. Soil profile for Pile # 5. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB SPT ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf N 60 Clay Clay UC+UU-Pushed SASI Clay UC+UU-Pushed Clay UC+UU-Pushed Tip UC+UU-Pushed Table A.8. Soil profile for Piles # 6 and 7. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB SPT ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf N 60 Clay Clay UC+UU-Pushed SASI Clay UC+UU-Pushed Clay UC+UU-Pushed Tip UC+UU-Pushed A.2.4 Piles # 8: Hutchinson and Jensen (1968) A pile load test was conducted on a 35-cm (13.8 in.) steel pipe pile driven into soft to medium estuarine silty clay at the Port of Khorramshahr on the north bank of Shatt al Arab in Iran. The embedded length of the pile is 13.9 meters (45.6 ft). The silty clay is slightly overconsolidated and is irregularly banded with frequent layers of silt and fine sand. The variation of the undisturbed and remolded shear strengths with depth is shown on Figure A.5 while information about soil properties is presented in Table A.9. Both strengths were measured using the field vane device. The average undisturbed strength along the length of the pile is 662 psf whereas the average remolded strength is about

285 psf, indicating a sensitivity of about The liquid and plastic limit range between 40 to 57% and 18 to 27% respectively. The water content varies with depth and ranges between 20 and 40% with an average of 33%. The clay has an average density of 1.87 tons/m3 (116.8 pcf). The pile was driven using a drop hammer with a weight of 2 tons (4.4 kips). The pile load tests were conducted by jacking the piles against the reaction of a twin anchor using a hydraulic jack. The setup time for the test was 584 days. It is only known that the test duration was between 18 and 48 hours. The ultimate bearing capacity was measured as 90.4 kips. This pile will be referred to as Pile 8 in the pile test database. Undrained Shear Strength (psf) Undisturbed Shear Strength Remolded Shear Strength Depth (ft) Pile Figure A.5. Variation of undisturbed and remolded shear strengths with depth for the site in Iran. 255

286 Table A.9. Soil profile for Pile # 8 Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf Clay Field Vane Tip Field Vane A.2.5 Piles # 9, 10, and 11: Kirby and Roussel (1979) Pile load tests were conducted by Woodward-Clyde Consultants as a part of the Effective Stress Axial Capacity Cooperative Project (ESACC Project). The experiments included load testing of 4 piles at a test site on Hamilton Air Force Base in California. The piles had a diameter of 4.5 and were installed to 40 feet below ground surface. Two of the test piles were closed ended piles whereas the other two piles were driven open ended. One of the closed ended piles was jacked into place and will thus be excluded from the following analysis. The other three piles were installed using a drop hammer. The soils from the ground surface to a depth of at least 50 feet consist of highly plastic marine clay. The top 6 ft consist of a crust that has been desiccated by seasonal fluctuation of the water table. Undisturbed samples were obtained using a 3 diameter Osterberg sampler. The undrained strength profile as obtained from torvane test (upper 15 feet) and Unconsolidated-Undrained triaxial tests (below 15 feet) is shown on Figure A.6. Information about the soil properties at the site is presented in Table A.10. The water content increases rapidly from about 20% at the ground surface to about 95% at 15- ft depth. Thereafter, it decreases slightly with increasing depth and is approximately equal to the liquid limit. The liquid limit ranges from 75 % in the upper 6 feet to about 93 % below 6 feet. The plastic limit of the soil is essentially constant and is equal to 37 %. 256

287 Undrained Shear Strength (psf) Prebored 10 Depth (ft) Piles 9,10, UU-Triaxial Undrained Shear Strength Torvane Undrained Shear Strength Remolded (vane) Figure A.6. Variation of undisturbed and remolded shear strengths with depth for the Hamilton Base site. Table A.10. Soil profile at the Hamilton Base site (Piles # 9, 10, and 11) Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf Clay Torvane Clay UU-Pushed Tip UU-Pushed The piles were installed using a 500-lb hammer falling 30 inches and striking a 6.5-in. thick aluminum micarta cushion placed on the drive head. The installation time was about 30 minutes. A setup time of 18 days was allowed prior to load testing. Pore pressure measurements obtained during this period indicated full consolidation. Four 257

288 reaction piles were driven at 15-ft center to center spacing to act as a jacking force. The loading apparatus included a Duff-Norton model RC 50 H4.7 B cylinder and an Enerpac P14 hydraulic hand pump. The measured capacities were 10, 10.9, and 14.1 kips. A.2.6 Piles # 12, 13, 14, 15, 16, 17, 18, 19: Cox, McClelland, and Verner (1979) Four 14 in. diameter steel pipe piles were driven in strong under to normally consolidated clays at a land location that is 1 mile south of Empire, Louisiana. The soil profile consists of a 100 foot top layer of fine sand with layers of soft clay, underlain by a 77-foot layer of firm gray clay. Between a depth of 177 and 450 feet, a layer of stiff gray clay with numerous sand seems exist followed by a layer of dense fine sand. Load tests were conducted on each pile to get the axial capacity of the piles in both tension and compression. In order to isolate the tests to selected strata of soil, 18-in diameter conductors were driven to the top of test zones and then the soil plugs in each conductor were drilled out. The test piles were then driven in the conductors to penetrations of 40 or 50 feet below the conductors Three borings were drilled at the site location using a skid-mounted Failing 1500 rotary rig and a 3.5 in. IF drill pipe. Undisturbed samples were obtained using a 2.25 in. thin walled wire-line sampler that was driven with a 175-lb sliding weight dropped approximately 5 feet. Four test zones along the soil profile were chosen based on soil conditions. Information about the geometry of the piles and their location within the soil stratum is presented in Table A.11. The undrained shear strength of cohesive samples was determined using motorized miniature vane devices while the samples where still in the sampling tube. After extrusion, unconfined compression tests were conducted on 258

289 selected samples. Results of the unconfined compression tests showed signs of extreme sample disturbance and will not be presented herein. Undrained shear strength profiles for the four test Zones as determined from miniature vane tests are shown on Figure A.7. The reported sensitivity of the soil was about All test piles were driven open ended using a Vulcan 020 hammer operating at a half stroke of 1.5 ft. As mentioned earlier, 18 in. conductors with a 0.5 in. wall thickness were driven by the same hammer prior to the test pile installation to depths of 115, 205, 270, and feet. The conductors were cleaned out with water and were left standing full of water. After the installation of the conductors, the test piles were driven to embedment lengths of 50 feet for piles in Zones 1 and 2 and 40 feet in Zones 3 and 4. Compression tests followed by tension tests were conducted on all piles 4 to 10 days after driving. From this set of tests, only the compression tests will be included in this analysis and will be referred to as Tests 12, 14, 16 and 18 in the pile database. After 313 to 327 days, tension pull out tests were performed on the piles in Zones 1, 2, and 3 and a compression test was conducted on the pile in Zone 4. These tests will be referred to as Tests 13, 15, 17, and 19 in the pile database. The reaction frame used in the load tests consisted of three platforms. Two hydraulic jacks were used to transmit the load to the test piles. The average loading rate to failure was about 1 kip per minute. Loads at the top of the test section were measured with a strain gauge bridge at that level and with a backup reading of the jack pressure. The probable error in the overall accuracy of the measurement of the load at the top is the larger of 5 kips or 2 percent. Table A.12 contains the results of the relevant pile load tests, while information on the soil properties at the site is presented in Tables A.13 to A

290 Undrained Shear Strength (psf) Undrained Shear Strength (psf) Depth (ft) Pile 12,13 Depth (ft) Pile 14, Undrained Shear Strength (psf) Undrained Shear Strength (psf) Depth (ft) Pile 16,17 Depth (ft) Pile 18, Figure A.7. Variation of undisturbed shear strengths with depth for the Empire site. 380 Table A.11. Properties of the piles and soils at the Empire site. Wall Thickness Penetration Average Undrained Zone # Diameter (in) (in) Range, ft Shear Strength (psf) to to to to

291 Table A.12. Summary of the results of the tests at Empire Time of Test Zone # Tests ID Compression / Tension (Days) Measured Capacity (kips) 1 Test 12 Compression Test 16 Tension Test 13 Compression Test 17 Tension Test 14 Compression Test 18 Tension Test 15 Compression Test 19 Compression Table A.13. Soil profile for Piles # 12 and 13. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf clay MV-Driven Tip MV-Driven Table A.14. Soil profile for Piles # 14 and 15. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf Clay MV-Driven Tip MV-Driven Table A.15. Soil profile for Piles # 16 and 17. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf Clay MV-Driven Tip MV-Driven Table A.16. Soil profile for Piles # 18 and 19. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf Clay MV-Driven Tip MV-Driven

292 A.2.7 Pile # 20: Conoco Tension Pile Study: Bogard and Matlock (1998) and Audibert and Hamilton (1998). In 1987, Conoco completed an extensive laboratory and field study on the response of friction piles subject to tension loading in soft clay. The tests were conducted on a 30-in diameter steel pipe pile with a total length of 360 feet, an embedment length of 234 feet and a wall thickness of 0.75 inches. The tests were conducted at the West Delta Block 58A site situated in only 50 feet of water. The soil at the site was a deep and uniform deposit of soft marine clay that is characterized as medium to high plasticity clays and of recent geologic history with typical normal to under consolidated stress histories. The pile was driven in three sections. The first section (180 ft) was first swung into position and allowed to penetrate under its weight to a depth of 41 ft. The pile was then driven using a typical hammer to a penetration of 56 ft. The second pile section was then added and the pile was driven to a tip penetration of 145 feet. After that, the last section of the pile was added and the pile was driven to a depth of feet where a first static load test to failure was conducted in both tension and compression. The pile was driven again to a final tip penetration of 234 feet and was allowed to setup for four months (April 1984). At this time a series of static tension and compression tests were performed followed by cyclic loading to failure. One year later (April 1985), the same set of cyclic loading tests were conducted on the test pile, but this time preceded only by a static tension test to minimize cyclic degradation prior to cyclic testing. The last series of pile load tests was conducted in June 1986 and consisted of a static tension test followed 262

293 by cyclic loading to failure. Two days after cyclic testing the pile was completely pulled out from the soil deposit. Test results are presented in Table A.18. Three major field investigation efforts were done to characterize the soil at the West Delta 58A site. These where conducted in 1956, 1979, and 1981 respectively and consisted of a series of 5 borings. In the 1956 study, drilling was achieved using a skidmounted rotary drilling rig to obtain 3-in diameter pushed soil samples. No information were present regarding how specimens were sampled in the other investigations. However, it will be assumed that the samples obtained were 3-in diameter pushed samples. When plotted on a plasticity chart, the soil deposit was shown to consist mainly of high plasticity clays (CH). Some soil samples especially from a depth of 80 to 160 feet consisted of medium plasticity inorganic clays (CL). The PI increases from 30% at the surface to 60% at a depth of 50 feet, decreases again to 30% at a depth of 130 feet, only to increase to 70% at a depth of 240 feet. The specific gravity of the clay minerals ranged between 2.72 and 2.82 with an average of The submerged unit weight varied significantly with depth and ranged between 30 and 50 pcf. A summary of the soil properties at the site is presented in Table A.17. Table A.17. Soil profile for Pile # 20. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf Clay UU-Pushed Clay UU-Pushed Clay UU-Pushed Clay UU-Pushed Tip UU-Pushed

294 The undrained shear strength of the soil was measured using mini-vane tests, torvane tests, unconfined compression tests and unconsolidated undrained triaxial tests. The UU-triaxial tests gave results that were consistently greater than the MV-tests which were in turn consistently larger than the UC and torvane tests. The undrained shear strength profile as obtained using UU-triaxial tests is shown on Figure A.8. The remolded strength profiles as obtained from UU-triaxial and Mini-vane tests are plotted on the same figure. The soil sensitivity varied between 1 and 2 with an average of about 1.5. It should be noted that the individual data points corresponding to the remolded strength were not presented in the available references. Undrained Shear Strength (psf) Undisturbed Strength (UU) Remolded Strength (UU) Remolded Strength (MV) Depth (ft) 100 Pile # Figure A.8. Undisturbed and remolded shear strength profiles at the West Delta site. 264

295 In 1999, Jardine and Saldivar (1999) published a paper in which they included their own interpretation of the results of the West Delta pile tests. In the paper, they present a plot showing profiles of the total vertical stress, pore water pressure, and vertical effective stress that are present in the soil at the site. The data from the corresponding reference is re-plotted and presented in Figure A.9 and shows the presence of a significant amount of excess pore water pressure in the soil deposit. Pressure (psf) Total Stress Pore Pressure (Real) Pore Pressure (Hydrostatic) Effective Stess (Real) Effective Stress (Hydrostatic) 80 Depth (ft) Figure A.9. Profiles of total stress, pore pressure, and effective stress at the West Delta site (Jardine and Saldivar, 1999). 265

296 Table A.18. Results of load tests conducted at different setup times at the West Delta site. Time of Test Time After Driving Measured Capacity (kips) Dec, hours 460 April, months 970 April, months 1060 June, months 1100 A.2.8 Piles # 21 and 22: Stermac et al. (1969) Tension and compression pile load test were conducted on a in. diameter concrete filled steel tube driven 50.1 ft into a layer of silty clay and clayey silt in Metropolitan Toronto. The upper 8 ft of the deposit is silty sand with gravel and occasional clay seams. Underlying this layer down to 76 feet is a deposit of clayey silt with sand and traces of gravel. More plastic in its upper 10 to 15 feet, it becomes less plastic with depth to about 60 feet, where it again becomes more plastic. The water table is about 4 feet below the ground surface. The variation of the undisturbed undrained shear strength with depth is shown on Figure A.10. The undrained strength was measured using Unconsolidated-Undrained triaxial tests obtained from Shelby tube samples and had an average of about 1250 psf along the depth of the pile. A careful inspection of the samples indicated that the samples where disturbed. The undrained strength will thus be increased by 20% to represent the UU-undisturbed shear strength. The sensitivity of the soil ranges between 1.5 and 3. A summary of soil properties at the site is presented in Table A

297 Table A.19. Soil profile for Piles # 21 and 22. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB SPT ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf N 60 SISA SISA Clay UU-Disturbed Clay UU-Disturbed Tip UU-Disturbed Undrained Shear Strength (psf) Silty Sand Depth (ft) 30 Pile # 21, Figure A.10. Undisturbed undrained shear strength profile at the Toronto site. The pile was first tested in compression 42 days after driving followed by a tension test 5 days later. During the tests, each load was maintained for two hours or until 267

298 such time as the rate of settlement fell below 0.01 inches per hour, whichever was the shortest time. The pile was retested in compression and tension 461 and 465 days after driving. The measured capacities were 160 and 142 kips respectively. The compression test will be referred to as Pile 21 while the tension test will be referred to as Pile 22 in the pile test database. A.2.9 Pile # 23: Pelletier and Doyle (1982) Tension tests were conducted on a 30-in diameter steel pipe pile driven 74 ft into a layer of silty clay at a site in Aquatic Park in Long Beach, California. The pile was driven through a 42 inch casing from a depth of 190 feet to a depth of 264 feet. The clay in the depth of interest is moderately overconsolidated (OCR 2 to 3) and can be described as hard, silty clay to clayey silt. The unit weight of the soil ranged from 120 to 125 pcf and the ground water table was found at a depth of 14 feet. Both driven and pushed samples were obtained. Ten unconfined compression tests were performed on driven samples and seven unconfined compression tests were performed on pushed samples. The variation of the undrained shear strength with depth is shown on Figure A.11. The ratio of pushed-to-driven sample strength averaged about 1.6. Since no remolded shear strength data were available for the clay at the site, the lower-bound was conservatively calculated using the disturbed unconfined compression samples. A summary of soil properties at the site is presented in Table A.20. At a pile penetration of 263, the pile driving was stopped and the pile was subjected to a tension test. After the test, the pile was driven to its final depth of 268

299 penetration (264 ft). Sixty days after driving, the pile was tested statically in tension with loads applied in increments of 100 kips. The measured capacity was equal to 2338 kips. This pile load test will be considered as Pile # 23 in the pile test database. Table A.20. Soil profile for Pile # 23. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf Clay UC-Pushed Tip UC-Pushed Undrained Shear Strength (psf) Depth (ft) Pile # Unconsolidated Undrained - Pushed Unconfined Compression - Driven Figure A.11. Undisturbed and remolded shear strength profiles of Bay Mud. 269

300 A.2.10 Pile # 24: Darragh and Bell (1969) A Pile load tests was conducted on a in diameter steel pipe pile embedded in a 100 ft layer of slightly overconsolidated, medium to stiff fissured clays at a site near Donaldsonville, La. The water table at the site is shallow and situated at 2 to 4 feet below the ground surface. Unconsolidated-Undrained triaxial tests were used to determine the variation of the shear strength with depth (Figure A.12). All tests were performed on samples taken from a 2.5-in. stationary piston sampler. The tip of the pile was closed using a steel plate cut flush with the pipe. A compression load test was conducted on the pile 23 days after driving was completed, resulting in a measured capacity of 220 kips. 0 Undrained Shear Strength (psf) Pile # 24 Deoth (ft) Unconsolidated Undrained - Pushed 130 Figure A.12. Variation of the undisturbed undrained shear strength with depth at the Donaldsonville site. 270

301 Table A.21. Soil profile for Pile # 24. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf Clay UU-Pushed Clay UU-Pushed Clay UU-Pushed Tip UU-Pushed A.2.11 Pile # 25 (reference unavailable) Table A.22. Soil profile for Pile 25, Atchafalaya. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf Clay UC-Unknown Clay UC-Unknown Clay UC-Unknown Clay UC-Unknown Tip UC-Unknown A.2.12 Piles # 26, 27, and 28 (reference confidential) Table A.23. Soil profile for Piles 26 and 27, Venice. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB SPT ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf N 60 Clay UU-Triaxial Sand Clay UU-Triaxial Sand Clay UU-Triaxial Tip UU-Triaxial Table A.24. Soil profile for Pile 28, Venice. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB SPT ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf N 60 Clay UU-Triaxial Sand Clay UU-Triaxial Tip UU-Triaxial

302 A.2.13 Pile # 29 (reference confidential) Table A.25. Soil profile for Pile 29, Harvey, Louisiana. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf Clay UU-Triaxial Tip UU-Triaxial A.2.14 Piles # 30 and 31 (reference confidential) Table A.26. Soil profile for Pile 30, Eugine Island. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf Clay UU-Triaxal Sand Clay UU-Triaxal Clay UU-Triaxal Sand Clay UU-Triaxial Tip UU-Triaxial Table A.27. Soil profile for Pile 31, Eugine Island. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf Clay UU-Triaxal Sand Clay UU-Triaxal Clay UU-Triaxal Sand Clay UU-Triaxial Tip UU-Triaxial A.2.15 Pile # 32: Texas Coast (reference unavailable). Table A.28. Soil profile for Pile 32, Aquatic Park. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf Clay UC-Unknown Clay UC-Unknown

303 A.2.16 Pile # 33 and 34 (reference confidential) Table A.29. Soil profile for Pile 33, Eugine Island. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf Clay UU-Triaxal Silt Clay UU-Triaxal Clay UU-Triaxal SISA Clay UU-Triaxial Tip UU-Triaxial Table A.30. Soil profile for Pile 34, Eugine Island. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v Remolded Strength K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio s ur, ksf Clay UU-Triaxal Silt Clay UU-Triaxal Clay UU-Triaxal SISA Clay UU-Triaxial Tip UU-Triaxial A.3 OVERCONSOLIDATED CLAYS A.3.1 Pile # 35 and 36 (reference confidential) Table A.31. Soil profile for Pile 35, Port Arther, Texas. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio Clay Clay Mini-Vane Clay UC-Unknown Tip UC-Unknown Table A.32. Soil profile for Pile 36, Port Arther, Texas. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio Clay Mini-Vane Clay UC-Unknown Tip UC-Unknown

304 A.3.2 Pile # 37: Togrol (1973) Pile load tests were conducted on a in diameter steel pipe pile with a total length of 91.8 ft and an exposed length of 42.6 feet. The embedded 49.2-ft section of the pile was driven in a 19.7-ft layer of silts and highly plastic clays underlain by a 29.5-ft layer of low plasticity silt and sandy silt. The site is located east of the Alsancak harbor in Izmir, Turkey. The average unit weight of the upper layer is about 106 pcf. Field vane tests conducted in this layer gave average shear strength values of about 600 psf. The unit weight of the underlying layer increases to 125 pcf and the undrained shear strength as obtained from unconsolidated undrained tests increases to about 1400 psf. The pile was tested in compression 90 days after driving. The test load setup consisted of four reaction piles and the test load was applied in 10 equal increments. Each load increment was maintained for a minimum of half an hour. The measured capacity was equal to 165 kips. Table A.33. Soil profile for Pile # 37, Izmir, Turkey Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio Clay Field Vane Clay UC-Unknown Tip UC-Unknown A.3.3 Piles # 38 to 45: O Neill et al. (1981) Six tension tests and two compression tests were conducted on steel pipe piles having a in. diameter and a wall thickness of in. at a site in Houston, Texas. The piles penetrated to a depth of 43 ft and were closed at the tips by steel plates cut flush 274

305 with the periphery of the piles. The soils at the site can be seen to be very stiff, saturated, overconsolidated clays. The variation of the undrained shear strength with depth is shown on Figure A.13. The shear strength was measured using Unconsolidated-Undrained Triaxial tests on pushed samples. The piles were subjected to multiple testing at different setup times. The last series of load tests occurred at setup times ranging from 100 to 155 days and will be included in the database to represent the capacity of the piles at full setup. The load tests were conducted by applying load increments once an hour in an amount of about 10% of the expected failure load until plunging occurred. The applied load was measured with electronic load cells. A summary of the soil properties at the site is presented in Table A.33. Table A.34. Soil profile for Piles 38-45, Houston, Texas. Soil Layer Water Liquid Plasticity Liquidity Vert. Effective Undrained Method of Corrected s u / σ' v K LB δ LB ProfileThick, ft Cont, % Lim., % Index % Index Stress, ksf Strength, ksf s u measure. s u, ksf Ratio Clay UU-Pushed Clay UU-Pushed Clay UU-Pushed Clay UU-Pushed Tip UU-Pushed

306 Undrained Shear Strength (psf) Piles # 38 to 45 Depth (ft) Figure A.13. Undisturbed undrained shear strength profile for site in Houston. A.4 SUMMARY A database containing the results of forty five full scale pile load tests in cohesive soils is described. A detailed summary of each case study for which corresponding references are available is presented. A summary of the values of the measured, predicted, and lower-bound capacities for each case in the database is presented in Table A

307 Table A.34. Summary of measured, predicted, and lower-bound capacities. Normally Consolidated to Slightly Overconsolidated Overconsolidated Pile Diameter Length Load Test Measured Capacity API Capacity Measured Capacity API Capacity Lower-Bound Capacity # (in) (ft) Number* kips, (Davisson) kips (Unorrected) (Corrected) kips (Corrected) kips new

308 Appendix B. Database of Load Tests on Piles in Cohesionless Soils B.1 INTRODUCTION A database containing the results of 36 full scale pile load tests is described. The load tests are conducted on closed-ended steel pipe piles embedded in siliceous, coehesionless soils. A summary of soil properties for each case is provided in the following sections. B.2 DATABASE OF PILE LOAD TESTS B.2.1 Pile # 1 to 9: Arkansas River Table B.1. Summary of the properties of the soil at the sites of Piles # 1 and 2. Soil Layer Vert. Effective SPT Relative API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density δ' φ' N q K δ' LB φ' LB N q,lb K LB Sand M Sand M SISA M Sand D Sand D Sand D Table B.2. Summary of the properties of the soil at the Sites of Piles # 3 and 4. Soil Layer Vert. Effective SPT Relative API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density δ' φ' N q K δ' LB φ' LB N q,lb K LB Sand M Sand M SISA L Sand D SISA D Sand VD SISA VD Sand VD Sand VD

309 Table B.3. Summary of the properties of the soil at the sites of Piles # 5 to 9. Layer Vert. Effective SPT Relative Undrained API Predicted Capacity Lower-Bound Capacity Thick, ft Stress, ksf N60 Density Strength, ksf δ' φ' N q K δ' LB φ' LB N q,lb K LB L M D M M D D VD VD B.2.2 Pile # 10 to 18: Old River, LA Table B.4. Summary of the properties of the soil at the sites of Piles # 10 to 13. Soil Layer Vert. Effective SPT Relative Undrained API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density Strength, ksf δ' φ' N q K δ' LB φ' LB N q,lb K LB Silt VL Clay Silt M Sand M Silt M Clay SISA D Sand VD Sand VD Table B.5. Summary of the properties of the soil at the site of Pile # 14. Soil Layer Vert. Effective SPT Relative Undrained API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density Strength, ksf δ' φ' N q K δ' LB φ' LB N q,lb K LB Silt VL Clay Silt M Sand M Silt M Clay SISA D Sand D

310 Table B.6. Summary of the properties of the soil at the sites of Piles # 15 and 16. Soil Layer Vert. Effective SPT Relative Undrained API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density Strength, ksf δ' φ' N q K δ' LB φ' LB N q,lb K LB Silt VL Clay Silt M Sand M Silt M Clay SISA D Sand VD Sand VD Table B.7. Summary of the properties of the soil at the site of Pile # 17. Soil Layer Vert. Effective SPT Relative Undrained API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density Strength, ksf δ' φ' N q K δ' LB φ' LB N q,lb K LB Silt VL Clay Silt M Sand M Silt M Clay SISA D Sand D Table B.8. Summary of the properties of the soil at the site of Pile # 18. Soil Layer Vert. Effective SPT Relative Undrained API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density Strength, ksf δ' φ' N q K δ' LB φ' LB N q,lb K LB Silt VL Clay Silt M Sand M Silt M Clay SISA D Sand VD Sand VD

311 B.2.3 Piles # 19 and 20: Beech, R. (Ontario) Table B.9. Summary of the properties of the soil at the site of Pile # 19. Soil Layer Vert. Effective SPT Relative API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density δ' φ' N q K δ' LB φ' LB N q,lb K LB Sand D Sand D Silt D Silt D Table B.10. Summary of the properties of the soil at the site of Pile # 20. Soil Layer Vert. Effective SPT Relative API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density δ' φ' N q K δ' LB φ' LB N q,lb K LB Sand D Sand D Silt D Sand M Sand M B.2.4 Piles # 21 to 25: Ogeechee, River Table B.11. Summary of the properties of the soil at the site of Pile # 21. Soil Layer Vert. Effective SPT Relative API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density δ' φ' N q K δ' LB φ' LB N q,lb K LB Sand L Sand L Sand L Table B.12. Summary of the properties of the soil at the site of Pile # 22. Soil Layer Vert. Effective SPT Relative API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density δ' φ' N q K δ' LB φ' LB N q,lb K LB Sand L Sand L Sand M Sand M Sand M

312 Table B.13. Summary of the properties of the soil at the site of Pile # 23. Soil Layer Vert. Effective SPT Relative API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density δ' φ' N q K δ' LB φ' LB N q,lb K LB Sand L Sand L Sand M Sand M Sand M Sand D Table B.14. Summary of the properties of the soil at the site of Piles # 24 to 25. Soil Layer Vert. Effective SPT Relative API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density δ' φ' N q K δ' LB φ' LB N q,lb K LB Sand L Sand L Sand M Sand M Sand M Sand D Sand D B.2.5 Piles # 26 to 28: Helena, Ark Table B.15: Summary of the properties of the soil at the site of Pile # 26. Soil Layer Vert. Effective SPT Relative API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density δ' φ' N q K δ' LB φ' LB N q,lb K LB SISA VL Sand M Sand L Sand D Sand M Table B.16. Summary of the properties of the soil at the site of Pile # 27. Soil Layer Vert. Effective SPT Relative API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density δ' φ' N q K δ' LB φ' LB N q,lb K LB SISA VL Sand M Sand M

313 Table B.17. Summary of the properties of the soil at the site of Pile # 28. Soil Layer Vert. Effective SPT Relative API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density δ' φ' N q K δ' LB φ' LB N q,lb K LB SISA VL Sand M Sand M Sand M B.2.6 Piles # 29 to 32: Kansas City Table B.18. Summary of the properties of the soil at the site of Piles # 29 to 30. Soil Layer Vert. Effective SPT Relative Undrained API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density Strength, ksf δ' φ' N q K δ' LB φ' LB N q,lb K LB SISA L Clay SISA L SISA M Sand M Sand M Table B.19. Summary of the properties of the soil at the site of Piles # 31 to 32 Soil Layer Vert. Effective SPT Relative Undrained API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density Strength, ksf δ' φ' N q K δ' LB φ' LB N q,lb K LB Sand M Clay Clay Sand M Sand M Sand M B.2.7 Pile # 33: British Columbia Table B.20. Summary of the properties of the soil at the site of Pile # 33. Soil Layer Vert. Effective SPT Relative Undrained API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density Strength, ksf δ' φ' N q K δ' LB φ' LB N q,lb K LB Sand VL Sand M Clay Silt M Clay Sand VD Sand VD

314 B.2.8 Pile # 34: Florida Table B.21. Summary of the properties of the soil at the site of Pile # 34. Soil Layer Vert. Effective SPT Relative API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density δ' φ' N q K δ' LB φ' LB N q,lb K LB Sand M Sand L Sand M Sand M Sand M Sand B.2.9 Pile # 35: Japan Table B.22. Summary of the properties of the soil at the site of Pile # 35. Soil Layer Vert. Effective SPT Relative API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density d' f' N q K d' LB f' LB N q,lb K LB Sand VL Silt L Silt M Silt M B.2.10 Pile # 36: Michigan Table B.23. Summary of the properties of the soil at the site of Pile # 36. Soil Layer Vert. Effective SPT Relative API Predicted Capacity Lower-Bound Capacity Profile Thick, ft Stress, ksf N60 Density δ' φ' N q K δ' LB φ' LB N q,lb K LB Sand L Sand M Sand L Sand L B.3 SUMMARY A database containing the results of thirty six full scale pile load tests in cohesionless soils is described. A summary of the values of the measured, predicted, and lower-bound capacities for each case in the database is presented in Table B

315 Table B.24. Summary of measured, predicted, and lower-bound capacities. Pile Load Test Compression Plate Measured Capacity Predicted Capacity Lower-Bound Capacity # Number* or Tension kips, (Davisson) kips, (API) kips 1 89 C T C T C T C C T C T C T C C C T T C Oversized C Oversized C C C C T C Oversized C Oversized C Oversized C C C C C Oversized C C C * Olson and Dennis

316 Appendix C. Database of Pullout Tests on Suction Caissons in NC Clay C.1 INTRODUCTION A database containing the results of 25 published laboratory model tests, centrifuge tests, and full scale field tests conducted in uplift on suction caissons in normally consolidated clays is assembled and described. A detailed summary of each case study is provided in the following sections. The summary includes information about the geometry of the caissons, soil properties, setup, and load testing. C.2 DATABASE OF PULLOUT TESTS C.2.1 Tests # 1 to 7: Luke et al. (2003) Nine axial pullout tests were conducted at the University of Texas at Austin to quantify the components of axial capacity of suction caissons. The caissons are made of 4-inch diameter anodized aluminum tubes with a wall thickness of in. The length of the caisson is 3 feet and can be inserted to depths up to 2.82 ft. The caissons were pushed to full penetration with dead weights in three tests. In the other six tests, the caissons were inserted using a combination of dead weight (about 10 in.) and suction (full penetration). After installation, the caissons were allowed to setup for a period of 48 hours. Four pullout tests were conducted with a vented top cap. One of the four tests is excluded from the database since no reliable measurement of the uplift capacity could be obtained. In the other four tests, the top cap was sealed, and upon pullout, suction developed inside the top cap and the soil plug was pulled out of the deposit with the caisson. Rapid extraction tests (undrained conditions) were performed on the above eight tests at a rate of 5 to 20 mm/sec. The complete test program is presented in Table C

317 Table C.1. Test program and soil properties for the University of Texas at Austin study. Test # Cap Condition Installation Method Depth With DW (in) Total Embedded Depth (in) S u (average) (psf) S u (Tip) (psf) S ur (average) (psf) S ur (Tip) (psf) 1 Vented Dead Weight (DW) Vented DW + Suction Vented DW + Suction Sealed Dead Weight (DW) Sealed DW + Suction Sealed DW + Suction Sealed DW + Suction The kaolinite has a specific gravity of 2.58, a liquid limit between 54 and 58 and a plastic limit between 31 and 34. The test bed was prepared by allowing kaolinite slurry mixed at a water content of 164% to consolidate under its own weight to a final depth of 1.09 m (3.57 feet). Undrained shear strengths were determined using a T-bar penetration test using a bearing capacity factor of The c/p ratio for the undisturbed soil was about 2.0 and the sensitivity varied between 1.7 and 2.3. Profiles for the unit weight, effective stress, undisturbed strength, and remolded strength are presented in Figure C.1. Unit Weight (kn/m3) σ v (kpa) Undrained Shear Strength (kpa) Ratio Figure C.1. Properties of the soil in the test tank. 287

318 Tests results are presented in Table C.2. For the vented pullout tests, the alpha values taken as the average of the internal and external friction ranged between 0.5 and 0.7. These values were obtained by assuming an end bearing N value of 7.6. The peak side resistance was 25% higher in the single test were the caisson was inserted with dead weight rather than suction. In the sealed tests, analysis of the test results was affected by uncertainties in the values of the external skin friction, weight of adhered soil, and the value of the reverse end bearing (Luke et al., 2003). The external alpha value could not be deduced with confidence from the vented tests due to expected differences between the internal and external skin friction in those tests. The weight of the soil plug and the soil adhering to the outside wall of the sealed caissons were also uncertain due to infeasibility of accurate measurements after the completion of the tests. To get a better estimate of the external alpha value, pullout tests were performed on aluminum plates that were inserted into the test bed. Results from these tests indicate that alpha was between 0.46 and If values of alpha in this range are used together with values of W soil ranging from 90 to 110 N, the calculated N value would be between 13 and 21. Table C.2. Results of the tests (N = 7.6 (Vented) and N = 9.0 (Sealed)). Test # Cap Condition Drainage Condition Wsoil (lb) Q'tip (lb) Qside (lb) Alpha 1 Vented Undrained Vented Undrained Vented Undrained Sealed Undrained Sealed Undrained Sealed Undrained Sealed Undrained C.2.2 Tests # 8 to 10: Clukey and Morrison (1993) Centrifuge tests conducted at 100g were performed on cylindrical single-cell and multi-cell caissons with an L/D ratio of about 2. Only tests on single cells are included in 288

319 the suction caisson database. The prototype diameter and length of the caissons are 50 ft and 117 ft respectively. Actual embedment depths varied between 103 and 109 ft. The thickness of the caisson is not presented in the reference. Assuming that the caisson is made of steel, the thickness can be back calculated from information about the weight as 0.55 feet. The weight of one of the caissons as depicted in one of the figures in the reference is 4968 kips. The total unit weight of the soil as back calculated from the weight of the plug and its dimensions as depicted in the same figure is 108 pcf. A cylindrical bucket with an inside diameter of 2.94 ft was used as a test bed for the study. The bucket was filled with kaolin to a depth of 1.84 feet. The clay specimens were prepared by initially consolidating the samples in layers with a hydraulic shear press at 1-g conditions. After consolidating in layers, the samples were transferred to the centrifuge and accelerated at 100-g. After approximately 30 minutes to 1 hour following the attainment of the 100-g acceleration level, the caissons were installed using a combination of dead weights and suction. After installation, the centrifuge was decelerated to 1-g and the equipment for loading were installed and configured. The centrifuge was accelerated again to a 100-g level and the system was left to consolidate for about 24 hours. Monitoring of the pore pressure dissipation was done over the 24 hour period and indicated that 90 to 95 % of consolidation was achieved prior to any testing. Undrained shear strength measurements were generally done using a cone penetrometer just prior to loading of the caissons. The CPT measurements were calibrated to vane shear strengths. Based on experience, the authors propose reducing the undrained shear strength by 25% to correct for the method of shear strength measurement. Undrained shear strength profiles are presented in Figure C.2. The caissons were loaded in uplift by applying a constant rate of loading. Two single caissons were loaded to failure in times varying from 23 to 37 seconds corresponding to 2.6 to 4.1 days 289

320 on a prototype scale. One other pullout test was performed in a time varying between 118 and 176 seconds, corresponding to 13.7 to 14.7 days in a prototype scale. Figure C.2. Undrained strength profile for the soil in the Clukey and Morrison (1993) study. The measured soil uplift values are presented in Table C.3. The values represent the total measured force in uplift less the contribution from the weight of the caisson. One of the caissons was instrumented with strain gauges and pore pressure instrumentation, resulting in an average measured skin friction of 0.3 ksf. This corresponds to an alpha value of 0.53 if an average CPT undrained shear strength of 565 psf is considered versus an alpha value of 0.71 if a reduced average undrained shear strength of 424 psf is used instead. Attempts were made by the authors to back calculate the individual contributions of the internal and external skin friction from the same test using pressure measurements 290