RISK ASSESSMENT AND SPATIAL VARIABILITY IN GEOTECHNICAL STABILITY PROBLEMS by Pooya Allahverdizadeh Sheykhloo Copy rights by Pooya Allahverdizadeh Sheykhloo 2015 All Rights Reserved A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Mines in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Civil and Environmental Engineering). Golden, Colorado Date ________________ Signed: _____________________________ Pooya Allahverdizadeh Sheykhloo Signed: _____________________________ Dr. D. V. Griffiths Thesis Advisor Golden, Colorado Date ________________ Signed: ______________________ Dr. John E. McCray Professor and Head Department of Civil and Environmental Engineering ii ABSTRACT Reliability analysis has gained considerable popularity in practice and academe as a way of quantifying and managing geotechnical risk in the face of uncertain input parameters. The purpose of this study is to investigate the influence of soil spatial variability on the probability of failure of different geotechnical engineering problems, including block compression, bearing capacity of strip footings, passive and active earth pressure and slope stability problems. The primary methodology will be the Random Finite Element Method (RFEM) which combines finite element analysis with random field theory and Monte-Carlo simulation. The influence of the spatial correlation length of soil properties on design outcomes has been assessed in detail through parametric studies. Results from traditional Factor of Safety approaches have been compared with those from probabilistic analysis. In addition, the influence of the coefficient of variation of the input random variables and two different input random variable distribution functions on the probability of failure have been studied. The concept of a “worst case” spatial correlation length has been examined in detail, and was observed in all the geotechnical problems considered in this thesis. This “worst case” concept is of particular interest, because it could be used as a basis for design in the absence of good quality site-specific data. iii CONTENTS ABSTRACT .................................................................................................................. iii LIST OF FIGURES ............................................................................................................. vi LIST OF TABLES .............................................................................................................. xii LIST OF SYMBOLS ......................................................................................................... xiii ACKNOWLEDGMENTS .................................................................................................. xvi CHAPTER 1 INTRODUCTION ...................................................................................... 1 1.1 Objective of the Research ............................................................................ 4 CHAPTER 2 LITERATURE REVIEW AND METHODOLOGY ................................... 6 2.1 First Order Second Moment Method (FOSM) ............................................. 6 2.2 Point Estimate Method (PEM)..................................................................... 7 2.3 First Order Reliablity Method (FORM) ....................................................... 7 2.4 Even Trees and Historic Performance Methods ........................................... 7 2.5 Stochastic Finite Element Method ............................................................... 8 2.6 Monte – Carlo Simulations .......................................................................... 8 2.7 Random Finite Element Method (RFEM) .................................................... 9 2.7.1 Spatial Correlation Length ................................................................... 14 CHAPTER 3 JOINTLY DISTRIBUTED RANDOM VARIABLES METHOD ............. 16 3.1 Mathematical Background......................................................................... 16 3.1.1 Distribution of Sums ............................................................................ 17 3.1.2 Distribution of Quotients ..................................................................... 18 3.2 JDRVM Application in Geotechnical Engineering Problems ..................... 20 3.2.1 Shear Strength Example ....................................................................... 20 3.2.2 Undrained Slope Example ................................................................... 22 CHAPTER 4 BLOCK COMPRESSION PROBLEM ..................................................... 25 4.1 Parametric Study ....................................................................................... 29 iv 4.2 Probability of Failure (pf) .......................................................................... 33 CHAPTER 5 BEARING CAPACITY OF STRIP FOOTINGS ON WEIGHTLESS SOIL ....................................................................................................... 40 5.1 Bearing Capacity ....................................................................................... 40 5.2 Probability of Failure of the Bearing Capacity Problem ............................. 47 CHAPTER 6 EARTH PRESSURE ................................................................................ 51 6.1 Passive Earth Pressure ............................................................................... 51 6.1.1 Probability of Failure of the Passive Earth Pressure .............................. 54 6.2 Active Earth Pressure ................................................................................ 61 6.2.1 Probability of Failure of the Active Earth Pressure ............................... 63 CHAPTER 7 SLOPE STABILITY ANALYSIS .............................................................. 70 7.1 Undrained Slope........................................................................................ 70 7.1.1 Deterministic Layered Slope ................................................................ 71 7.1.2 RFEM Slope Stability Model ............................................................... 75 7.1.3 Highly Anisotropic Slopes ................................................................... 79 7.1.4 Slope Inclination Effect ....................................................................... 82 7.2 Drained Slope ........................................................................................... 89 7.2.1 Normal versus Lognormal Distribution ................................................ 96 CHAPTER 8 CONCLUDING REMARKS ..................................................................... 99 8.1 Concluding Remarks ................................................................................. 99 8.2 Recommendations for Future Research ................................................... 101 REFERENCES CITED ..................................................................................................... 102 v LIST OF FIGURES Figure 1.1 - The traditional approach for calculating the bearing capacity of a strip footing. ............................................................................................................. 2 Figure 1.2 - The probabilistic approach for calculating the bearing capacity of a strip footing. ............................................................................................................. 3 Figure 2.1 - Finite element analysis of James Bay Dike demonstrating a non-circular failure mechanism (Griffiths 2013). ................................................................ 10 Figure 2.2 - Development of multiple failure mechanisms with the same factor of safety. ... 11 Figure 2.3 - Random field generation (a) and mapping of finite element mesh (b). .............. 12 Figure 2.4 - Development of failure mechanisms in a highly variable soil modeled using the random finite element method (RFEM) (lighter zones are weaker). ........... 13 Figure 2.5 - Bearing capacity analysis with different special correlation length using RFEM (Griffiths and Fenton, 2001). ............................................................... 13 Figure 3.1 - Probability distributions function of the shear strength problem calculated with the JDRVM and Monte-Carlo methods. .................................................. 22 Figure 3.2 - Probability distributions function of the Factor of Safety of the undrained slope calculated with the JDRVM and Monte-Carlo methods. ......................... 23 Figure 4.1 - Loading and boundary conditions for the block compression problem. ............. 25 Figure 4.2 - Mesh used for the finite element block compression analysis. .......................... 28 Figure 4.3 - Variation of μqu with ϴ with different mesh density, V = 0.3............................. 29 Figure 4.4 - Typical random field realization and failure mechanisms for the block compression problem, μc′ = 100kPa and μtan ϕ′ = 0.577 (a) ϴ = 0.1, (b) ϴ = 5 ... 29 Figure 4.5 - Variation of normalized μqu with ϴ and V with a lognormal distribution for the input random variables, μc′ = 100kPa and μtan ϕ′ = 0.577. ............................ 30 Figure 4.6 - Variation of normalized μqu with ϴ and V with normal distribution for the input random variables, μc′ = 100kPa and μtan ϕ′ = 0.577. .................................. 31 Figure 4.7 - Variation of normalized μqu with ϴ for V = 0.2, 0.3, and 0.5 with normal and lognormal distributions for the input random variables, μc′ = 100kPa and μtan ϕ′ = 0.577. .................................................................................................