11 ABSTRACT The self-boring pressuremeter test has the potential of providing the information necessary for calculating the deformation behavior of granular deposits. However, the stress path and loading orientation with respect to the depositional direction in the test is such that the stress-strain relationship obtained directly from the data following the existing procedures can be very conservative in manyfieldproblems of practical significance. A reasonably simple stress-strain model is proposed in this research accounting for inherent anisotropy for granular deposits and stress-path dependent soil behavior. The model is used to back-analyze self-boring pressuremeter tests and determine an optimum set of model parameters representing the behavior of the deposit. The exercise of back-analysis essentially involvesfittingthe response of the stress-strain model to the self-boring pressuremeter data by altering the model parameter by manual trial and error. To compute the model response in cylindrical cavity expansion, a commercially available explicit finite difference computer code (FLAC) is used. An implicit finite element code developed in this research can also be used for the purpose. To minimize the impact of non-uniqueness in the result, a-priori information about the bounds of values of some of the model parameters is used. Supplementary information about the state of packing of the deposit and small strain stiffness is obtained from a seismic piezocone penetration test carried out adjacent to the self-boring pressuremeter bore hole. The deformation behavior of an axisymmetric element for the model parameters predictedfromback analysis of self-boring pressuremeter test is found to agree with laboratory triaxial tests on undisturbed (frozen) samples in both compression and extension at several sites. The explicit finite difference computer code FLAC is again used to predict the response of an axisymmetric element. Further validation for the procedure proposed for estimation Abstract iii of deformation behavior of granular deposits from self-boring pressuremeter comes from the negligible value of the undrained strength obtained for datafroma mine site where there was a static liquefaction failure. The only reliable method available at present to estimate deformation behavior of granular deposits is extraction offrozensamples and laboratory testing. The cost of adopting this approach is often too great for routine use. Even in important projects economical options of estimating deformation behaviorfromindex tests such as SPT and piezocone penetration are thus adopted. These empirical procedures are usually very imprecise. The procedure based on back analysis of self-boring pressuremeter tests proposed in this research appears to provide a more reliable alternative than the approach based on index tests. At the same time, the approach is quite economical in comparison with the option of undisturbed sampling and laboratory testing. iv T A B L E OF CONTENTS ABSTRACT » TABLE OF CONTENTS iv LIST OF TABLES ix LIST OF FIGURES x LIST OF SYMBOLS AND ABBREVIATIONS xiii ACKNOWLEDGEMENTS xvi CHAPTER 1: INTRODUCTION 1 1.1 Rationale 1 1.2 Uncertainty in the Empirical Use of SPT and CPT 3 1.3 Possible Use of Self-boring Pressuremeter 4 1.4 Use of Self-boring Pressuremeter in Granular Deposits 5 1.5 Objectives and Organization 6 CHAPTER 2: SELF-BORING PRESSUREMETER TESTING IN GRANULAR DEPOSITS 7 2.1 Introduction 7 2.2 The Self-boring Pressuremeter 7 2.2.1 Typical SBPMT Data in Sand 13 2.2.2 Appropriate Equipment Geometry 15 2.3 Self-boring Pressuremeter Installation Procedure in Sand 17 2.3.1 Identification of Disturbance 17 2.3.2 Guidelines for Selecting Optimum Installation Related Variables 19 2.4 Self-boring Pressuremeter Test Procedure in Sand 21 2.4.1 Calibration 21 2.4.2 Cavity Expansion 23 2.5 Limitations in Existing Procedures for Deformation Analysis of Granular Materials 23 2.5.1 Limitations of the Approach Based on Laboratory Tests 23 2.5.2 Use of SPT and CPT 24 2.5.2.1 Use of CPT 24 v Table of Contents 2.5.2.2 Use of SPT 25 2.5.2.3 Limitations of Index Measurements 26 2.5.3 Model Calibration from SBPMT Data 2.6 Reliability of SBPMT Data 26 28 2.6.1 Random Error, Test Procedure and Equipment Related Uncertainties 29 2.6.2 Effect of Inherent Soil Variability on SBPMT 30 2.7 Interpretation of Cavity Expansion Data 31 2.7.1 Closed Form Solutions to Cylindrical Cavity Expansion 32 2.7.2 Numerical Solutions to Cylindrical Cavity Expansion 34 2.7.3 Limitations of Existing SBPMT Interpretation Procedures 36 2.7.3.1 Inherent Anisotropy 36 2.7.3.2 Stress Path Dependency 37 2.7.3.3 Small Strain Assumption 38 2.7.3.4 Rigid Plastic Assumption 39 2.7.3.5 Stress and Strain Level Dependency 40 2.7.3.6 Plane Strain Assumption 40 2.7.3.7 Numerical Stability 40 2.8 Summary 41 CHAPTER 3: NUMERICAL METHODS FOR DEFORMATION ANALYSIS 43 3.1 Introduction 43 3.2 Numerical Models 43 3.2.1 Calculation of Response of a Single Element 44 3.2.2 Calculation of Response of Expanding Cylindrical Cavity 44 3.3 Explicit Finite Difference 47 3.4 Implicit Finite Element 51 3.5 Modeling Incompressibility 54 3.6 Summary 56 CHAPTER 4: STRESS-STRAIN MODEL 58 4.1 Introduction 58 4.2 Assumptions 59 4.3 Stress-strain Behavior of Cohesionless Soils 60 4.3.1 Reversible Behavior 60 4.3.2 Post Yield (Irreversible) Stress-strain Behavior 61 4.3.2.1 Skeleton Response in Distortion 62 4.3.2.2 Yield and Failure in Distortion 64 4.3.2.3 Material Anisotropy in Distortion 65 vi Table of Contents 4.3.2.4 Causes of Anisotropy 67 4.3.2.5 Fabric and Stress Path Dependency of Effective Stress Friction Angle 67 4.3.2.6 Stress-strain Behavior in Isotropic Compression 68 4.3.2.7 Undrained ResponsefromDrained (Skeleton) Behavior 69 4.4 Proposed Stress-strain Model for Granular Soils 70 4.4.1 Modeling Elastic Stress-strain Behavior 71 4.4.2 Evaluation of Elastic Parameters 72 4.4.3 A Model for Plastic Distortion Behavior of Isotropic Material 73 4.4.4 Proposed Modification for Inherent Anisotropy 78 4.4.5 Modeling Isotropic Compression Behavior 79 4.4.6 Numerical Simulation of Element Mechanical Response 81 4.5 Estimation of Model Parameters for Distortion and Consolidation Mechanisms 83 4.6 Existing Information on Plastic Model Parameters 85 4.6.1 Parameters of Isotropic Compression Mechanism 85 4.6.2 Parameters of Distortion Mechanism 86 4.6.2.1 Parameters Related to Plastic Flow, X and u 86 4.6.2.2 Parameters r) and A n 89 4.6.2.3 Parameters R and n 90 F1 F P 4.6.2.4 Effect of Drainage on r) and A n 91 F1 4.6.2.5 Parameter for Inherent Anisotropy, m A 92 4.7 Model Performance 95 4.8 An Approximate Procedure for Model CalibrationfromLaboratory Tests 96 4.9 Validation of the Procedure 98 4.10 Sensitivity of Model Response to the Choice of Model Parameters 101 4.11 Geometric Non-linearity 105 4.12 Approximations in the Stress-strain Model 106 4.12.1 Isotropic Hardening in Distortion 106 4.12.2 Inability to Simulate Pure Rotation of Stress 107 4.12.3 Independence of Hardening Rules to Current Void Ratio 107 4.12.4 Isotropic Failure Criterion 107 4.12.5 No Inherent Anisotropy in Isotropic Compression Mechanism 108 4.12.6 Independence of the two Plasticity Mechanisms 108 4.12.7 Limitation of Non-associative Plasticity 108 4.12.8 Strain Softening in Drained Deformation 109 4.13 Summary 109 vii Table of Contents CHAPTER 5: Modeling Cylindrical Cavity Expansion 110 5.1 Introduction 110 5.2 Alternative Discretization Approaches 110 5.3 Numerical Tools 112 5.4 Adequacy of Spatial Discretization 112 5.4.1 Isotropic Linear Elastic Material 113 5.4.2 Isotropic Linear Elastic Perfectly Plastic Material 5.4.3 Elasto-plastic Material 115 117 5.4.3.1 Stress and Deformation Field 118 5.4.3.2 Alternative Schemes of Discretization 120 5.4.3.3 Boundary Effects 121 5.4.4 Small Deformation Assumption 121 5.4.5 Adopted Numerical Model 122 5.5 Sensitivity of Analytical Response of the Deforming Cavity to Model Parameters 5.6 Summary 123 126 CHAPTER 6. Site Description 128 6.1 Introduction 128 6.2 J-Pit 129 6.3 Fraser River Delta 131 6.3.1 KIDD#2 132 6.3.2 Massey Tunnel (South) 134 6.4 LLDam 135 6.5 South Eastern British Columbia Mine Site 137 6.6 Summary 138 CHAPTER 7: Interpretation of Self-boring Pressuremeter Tests 139 7.1 Introduction 139 7.2 Proposed Framework for Analyzing Self-boring Pressuremeter Data 140 7.2.1 Geostatic State of Stress 140 7.2.2 Model Parameters 141 7.3 Validation of the Proposed Procedure 143 7.3.1 J-Pit 143 7.3.2 KJDD # 2 147 7.3.3 Massey Tunnel (south) 150 7.3.4 Limitations of the Proposed Procedure 152 7.4 Application of the Proposed Procedure 155 Table of Contents 7.4.1 LLDam 7.4.2 South-eastern British Columbia Mine Site viii 156 1 5 8 7.5 Advancement of the State-of-the-art 160 7.6 Misfit Criterion 7.7 Index of Soil Behavior 164 7.8 Summary CHAPTER 8: Summary, Conclusions and Recommendations 8.1 Summary and Conclusions 8.2 Practical Applications 162 165 167 167 170 171 8.3 Avenues of Further Research BIBLIOGRAPHY APPENDIX 1: Compliance Matrices A l . 1 Elastic Compliance Matrix A l .2 Compliance Matrix in Distortion A1.3 Compliance Matrix in Isotropic Compression 173 185 185 185 187 A1.4 Stress-strain Relationship in Axisymmetry and Plane Strain 188 APPENDIX 2: Program for Computing Plane Strain Element Response 189 A2.1 Usage and Code Listing Text Box A2.1 Code Listing for Element Response APPENDIX 3: Program for Analyzing Cylindrical Cavity Expansion A3.1 Code and its Usage Text Box A3.1 Code Listing for Cylindrical Cavity Expansion Analysis Program 189 190 198 198 200 LIST OF TABLES Table. 2.1. Appropriate Jetting Variables for SBPMT in Sand Table. 4.1. Elastic Model Parameters Table. 4.2. Factors Influencing Parameters of Distortion Mechanism Table. 4.3. Model Parameters Associated with Plastic Flow Table. 4.4. Constant Volume Friction Angle of Sands Table. 4.5. Model Parameters. Triaxial Tests on Water Pluviated Fraser River Sand Table. 4.6. Particulars of Tests on Undisturbed Samples Table. 4.7. Model Parameters: Triaxial Tests on Undisturbed Samples Table. 4.8. Effect of Variation in the Values of the Model Parameters Table 5.1. Stresses at the Centroid of the Element Adjacent to the Cavity Table 5.2. Influence of Model Parameters on Computed Cavity Expansion Response Table 6.1. List of Abbreviations Table 7.1. Particulars of Laboratory Triaxial Tests on Undisturbed Samples Table 7.2. C in Studies Involving Back Analysis of SBPMT Table 7.3. Estimated Values of and K , and Flow Slide Potential SP X LIST OF FIGURES Fig. 1.1. Fabric Effect on Undrained Stress-strain Response of Syncrude Sand 2 Fig. 2.1. Schematic Details of the UBC SBPM Fig. 2.2. Strain Arms of the UBC SBPM 9 10 Fig. 2.3. UBC SBPM Electronics Fig. 2.4. Typical SBPMT Data in Sand 12 Fig. 2.5. SBPMT at Treasure Island Fig. 2.6. Detection of Pushed in Type DisturbancefromCPTU 16 19 Fig. 2.7. Comparison of Closed form Small and Large Deformation Solutions to Cylindrical Cavity Expansion 39 14 Fig. 3.1. Discretization Schemes for Simulation of Laboratory Element Tests Fig. 3.2. A Typical Element in an Axisymmetric Problem 44 Fig. 3.3. Use of Eight Node Rectangle to Simulate Triaxial Element Test 56 Fig. 4.1. Small Strain Behavior of Air Pluviated Sand Fig. 4.2. Drained Distortional Behavior of Sand 61 Fig. 4.3. Comparison of Three Failure Criteria in TZ Plane Fig. 4.4. Stress-strain Response of Air Pluviated Leightin Buzzard Sand Fig. 4.5. Stress-strain Behavior of Loose Sands in Triaxial and Isotropic Compression 65 45 62 66 69 Fig. 4.6. Loading and Failure Surfaces of the Distortion Mechanism and Stress Paths in TXC and TXE Fig. 4.7. Loading Surfaces for Distortion and Isotropic Compression Fig. 4.8. Computed Drained Response of a Plane Strain Element 75 Fig. 4.9. Computed Undrained Response of a Plane Strain Element 82 Fig. 4.10 Parameter "C" for Isotropic Compression 86 90 Fig. 4.11 4>' as a Function of Mean Effective Stress and Relative Density Fig. 4.12 Drained and Undrained Stress Paths in TXC 80 82 91 Fig. 4.13 Drainage Dependence of Th-ricv Fig. 4.14 Modeling Undrained Stress-strain Behavior of Fraser River Sand 92 Fig. 4.15 Simulation of Triaxial Tests on Undisturbed (Frozen) Samples Fig. 4.16 Simulation of Drained Triaxial Tests on Toyoura Sand 95 Fig. 4.17 Predicted and Measured Response in Plane Strain Tests Fig. 4.18 Effect of Variation of Parameter Values on Model Response -1 100 Fig. 4.19 Effect of Variation of Parameter Values on Model Response - II Fig. 4.20 Relevance of Assumption of Small Deformation 103 93 99 102 105 List of Figures xi Fig. 5.1. Spatial Discretization 111 Fig. 5.2. Numerical and Closed Form Solutions for Stress: Isotropic Linear Elasticity 114 Fig. 5.3. Numerical and Closed Form Solutions for Displacements: Isotropic Linear Elasticity 114 Fig. 5.4. Element Response in Triaxial Compression: Elastic-Perfectly Plastic Simulation 116 Fig. 5.5. Stress Distribution for Isotropic Linear Elastic Perfectly Plastic Material 117 Fig. 5.6. Stress Distribution for Elasto-Plastic Sand 119 Fig. 5.7. Deformation Computed for Elasto-Plastic Sand 119 Fig. 5.8. Simulation of Cylindrical Cavity Expansion with Alternative Discretization Schemes 120 Fig. 5.9. Response of Cylindrical Cavity Expansion for Different Domain Sizes 121 Fig. 5.10. Imprecision in the Assumption of Small Deformation 122 Fig. 5.11. Sensitivity of Cylindrical Cavity Expansion to Model Parameters -1 124 Fig. 5.12. Sensitivity of Cylindrical Cavity Expansion to Model Parameters - II 125 Fig. 6.1. Location of Test Sites 128 Fig. 6.2. Site Layout: J-Pit 130 Fig. 6.3. Range ofCPTU Data from J-Pit 131 Fig. 6.4. Surface Geology of Fraser River Delta 132 Fig. 6.5. Sampling and Testing Locations at KIDD # 2 and Massey Tunnel (south) 133 Fig. 6.6. Range of SCPTU Data from KIDD #2 134 Fig. 6.7. Range of SCPTU Data from Massey Tunnel (south) 135 Fig. 6.8. Sampling and In-situ Testing Locations at LL Dam 136 Fig. 6.9. Range of SCPTU Data: LL Dam 136 Fig. 6.10. SCPTU DatafromIron Tailings Impoundment: South-eastern BC Mine Site 138 Fig. 7.1. Analysis of SBPMT Data: J-Pit 144 Fig. 7.2. Predicted and Measured Undrained Triaxial Element Response: J-Pit 145 Fig. 7.3. Predicted and Measured Stress Paths: J-Pit 147 Fig. 7.4. Analysis of SBPMT Data: KIDD # 2 148 Fig. 7.5. Predicted and Measured Triaxial Element Response: KIDD #2 149 Fig. 7.6. Predicted and Measured Stress Paths: KIDD # 2 150 Fig. 7.7. Analysis of SBPMT Data: Massey Tunnel (south) 151 Fig. 7.8. Predicted and Measured Triaxial Element Response: Massey Tunnel (south) 153 Fig. 7.9. Predicted and Measured Stress Paths: Massey Tunnel (south) 154 Fig. 7.10. Analysis of SBPMT Data: LL Dam 157 Fig. 7.11. Range of Predicted Triaxial Response for LL Dam Deposit 157 Fig. 7.12. Stress Paths in Triaxial Tests Inferred from SBPMT at LL Dam 158 List of Figures Fig. 7.13. Analysis of SBPMT Data: South-eastern B C Mine Site Fig. 7.14. Plane Strain Element Behavior for South-eastern B C Mine Site Xlll LIST OF SYMBOLS AND ABBREVIATIONS B, B = Linear and non-linear strain displacements, respectively B i= 1,2,3 = Components of element body force C = Parameter of the plasticity mechanism for isotropic compression CPTU = Piezocone penetration test CyH = Elastic compliance tensor C y H * = Compliance tensor of the plasticity mechanism for distortion Dj, i=l,2,3 = Components of element damping force D 10 = Equivalent grain diameter for 10% passing D 50 = Median grain size Dgo = Equivalent grain diameter for 60% passing D = Relative density in percent = Elasto-plastic stiffness tensor = Elastic tangent shear modulus = Initial and current values of the tangential slope of the r| versus y Ii, i=l,2,3 = Invariants of the effective stress tensor J = Determinant of deformation gradient J , i= 1,2,3 = Invariants of the effective stress deviator tensor K, K = Linear and non-linear element stiffness matrices, respectively KQ = Coefficient of earth pressure at rest KQE = Elastic tangent shear modulus number K = Parameter of the plasticity mechanism for distortion = Coefficient of active earth pressure for Mohr-Coulomb material = Normalized SPT blow count for 60% rod energy i5 C R G E G P I , Gpr ; S P N (NJgo P10.P2.5jP0 S M P curve Effective cavity pressure at cavity strains of 10, 2.5 and 0 percent, = respectively P A = Atmospheric pressure P ; = Effective cavity pressure Rp = Parameter of the plasticity mechanism for distortion Rf = Friction ratio in CPT Rp = Radius of the plastic zone in cylindrical cavity expansion for MohrCoulomb Material SMP = Spatial mobilized plane Sy = Second Effective Piola-Kirchoff stress tensor S = Undrained shear strength SBPMT = Self-boring pressuremeter test SCPTU = Seismic piezocone penetration test u List of Symbols and Abbreviations xiv SPT Standard penetration test TXC Laboratory triaxial compression test TXE Laboratory triaxial extension test performed by applying an axial stretch U u ,u Hydrostatic pore water pressure 0 2 Penetration pore water pressure in CPTU: measured behind the tip and 3 behind friction sleeve, respectively Shear wave velocity W a Plastic work associated with isotropic compression mechanism c Radius of cavity before deformation Direction cosines of normal to spatial mobilized plane (SMP) % dEy, de/, d e / Total, reversible (elastic) and irreversible (plastic) strain increments, de/ respectively Strain increments due to distortion and isotropic compression mechanisms, respectively dYsMPs de SMP Components of principal strain increment vector parallel and normal to the SMP for the plasticity mechanism for distortion, respectively e e > e MAX> e M I N C O N S Current, maximum and minimum void ratios, respectively Void ratio at consolidation Coordinate bases e m A Parameter of the plasticity mechanism for distortion Principle bases n Elastic exponent % n Parameter of the plasticity mechanism for distortion P Exponent in the plasticity mechanism for isotropic compression P' q Effective mean normal stress t time u Displacement a Damping constant 5 Angle between the major principal stress and the bedding plane P T *« e « e , i=l,2,3 (i) Cone tip resistance corrected for unequal end area effect Kronecker delta Velocity strain tensor Major, intermediate and minor effective principal strains, respectively Measure of misfit between model response in cylindrical cavity expansion and SBPMT data 1) Stress ratio on SMP Tlcv 2f2 tan <J> C V Peak value of r| List of Symbols and Abbreviations xv Parameter of the plasticity mechanism for distortion A, Parameter of the plasticity mechanism for distortion Vv Parameter of the plasticity mechanism for distortion Elastic Poisson's ratio P o Total mass density o ' Original effective horizontal stress Vector of Cauchy stress Effective radial stress at elastic-plastic boundary in cylindrical cavity RP expansion for Mohr-Coulomb material Effective normal stress on SMP o. SMP Effective stress tensor Effective normal stress in radial (in axisymmetry) or "x" (in plane strain) direction (scalar) Effective normal stress in hoop (in axisymmetry) or "y" (in plane strain) direction (scalar) Axial normal effective stress (in axisymmetry) or that in the direction "z" of no deformation (in plane strain), scalar o ', i=l,2,3 = _/ _ (i) ° CONS ° HCONS _ / ° X = _ VCONS SMP Major, intermediate and minor effective principal stresses, respectively Original vertical effective stress Effective isotropic normal stress at consolidation Effective horizontal normal stress at consolidation Effective vertical normal stress at consolidation Shear stress on SMP Shear stress in the "x" direction on plane perpendicular to the "y" direction = (scalar) Effective stress friction angle Value of cb' in triaxial compression (TXC) Constant volume friction angle in terms of effective stress (j)' ^ T X C = 4>CV G):, = Spin tensor xvi ACKNOWLEDGEMENTS The writer would like to thank his supervisor, Dr. R.G. Campanella for his keen interest in this research and for personal help with the entire in-situ testing program. The writer also had the privilege of receiving technical help on an almost continuous basis during this study from Dr. P.M. Byrne, and Dr. J.M.O. Hughes for which he is grateful. In addition, the self-boring pressuremeter tests at J-Pit and LL Dam were carried out by Dr. Hughes, datafromwhich have been used in this study. Special thanks are due to Dr. D.L. Anderson, Dr. W.D.L. Finn and Dr. Y.P. Vaid for their invaluable comments about various technical aspects, and to Mr. Scott Jackson and Mr. Harald Schrempp for the assistance during the testing program of this research. In addition, the writer received assistance from Mohammed Ahmadi, Tim Boyd, Chris Daniels, Mike Davies, Scott Martens and Scott Tomlinson duringfieldtesting. This research was partly funded by the Canadian Liquefaction Experiment (CANLEX) Project, a cooperative undertaking of several universities, industrial participants and the Natural Sciences and Engineering Research Council (NSERC) of Canada. Thefinancialassistance from the University of British Columbia through the University Graduate Fellowship (UGF) program is also acknowledged. Valuable technical comments and suggestions on various aspects of this study were received from Dr. Jean BenoTt and Dr. Pedro DeAlba of the University of New Hampshire, Dr. Peter Cundall of Itasca Consulting Group of Minneapolis, Minnesota, Dr. Guy Houlsby of University of Oxford, Dr. Vaughan Griffiths of Colorado School of Mines, Mr. Mike Jefferies of Golder Associates, UK, Dr. Peter Robertson of University of Alberta and Dr. F. Tatsuoka of University of Tokyo. 1 CHAPTER 1 INTRODUCTION 1.1 Rationale To estimate the deformation of granular deposits under a loading history that the deposit is expected to undergo, one of the following approaches is adopted: • analysis of the mechanical response following principles of mechanics using an appropriately calibrated stress-strain relationship or • empirical use of index measurements from standard penetration test (SPT) or cone penetration test (CPT). A more precise result can be obtained from an analytical approach because unlike the empirical alternative, the effects of the state of packing, stress, strain, stress path and fabric on soil behavior can in principle be accounted for appropriately in the analytical approach. To adopt the analytical approach, knowledge of the stress-strain response of the material over a wide range of deformation is necessary. Such information is typically obtainedfromlaboratory tests, e.g., triaxial or plane strain. The stress-strain behavior measured in the laboratory depends to a great extent on the sample fabric (Fig. 1.1). As a result, testing of reconstituted samples in the laboratory instead of undisturbed specimens is not an acceptable alternative unless the fabric of the deposit can be reproduced in the sample preparation procedure. Extraction of undisturbed specimens of granular soils can be a very expensive and specialized undertaking and is therefore not often practicable. Consequently, there has been an increasing reliance on SPT and CPT. SPT has long been used as an index tool and is used primarily Chapter 1: Introduction 2 2.5 5 7.5 Shear Strain, % 10 Fig. 1.1. Fabric Effect on Undrained Stress-strain Response of Syncrude Sand (adapted from Vaid et al., 1995): e is the void ratio after consolidation CONS because of a vast experience with its application. The test is essentially of a dynamic nature and is of limited repeatability. Neither does it provide a detailed stratigraphic information like the CPT. However, SPT allows disturbed samples to be extracted during the testing process that can be used in determining grain size distribution and mineralogy. Since late 1970s and early 1980s, cone penetration testing with an electronic cone penetrometer is emerging as a preferred tool for site characterization because of its economy, spatial resolution and repeatability. However, the procedures for using CPT and SPT data in deformation analysis of granular deposits are quite comparable. Index measurements in these tests are typically used to estimate soil properties such as the small strain shear modulus and the effective stress angle of internal friction from empirical correlations. These estimates can then be used together with a simple isotropic linear elastic perfectly plastic stress-strain relationship in deformation analysis. As a result of the empiricism in this method, there can be a considerable uncertainty in the results. Occasionally seismic measurements are carried out during CPT. These measurements allow determination of small strain soil stiffness without empiricism. Chapter 1: Introduction 3 Another in-situ test, self-boring pressuremeter test or SBPMT, is not as commonly used as CPT or SPT mainly because the tool is comparatively new and the test is not as easily performed as CPT or SPT. Analysis of SBPMT data typically involves modeling the cavity expansion data measured in the test as a problem of cylindrical cavity expansion within a continuum. Among its distinct advantages are • the data can be analyzed from the fundamental principles of mechanics, • measurements pertain to a wide range of deformation at each depth of testing, • the test can be performed without causing significant disturbance to the surrounding medium, and • a larger volume of soil is sampled in each test compared to a typical specimen used in laboratory testing. An attempt has been made in this research to develop a procedure for deformation analysis of granular soils based on SBPMT. 1.2 Uncertainty in the Empirical Use of CPT and SPT In CPT and SPT, the soil deposit is subjected to a complex deformation history involving large deformation and the datafromthese tests are not generally amenable to analytical treatment. The index measurements in these tests (the cone tip resistance, q , in CPT and normalized blow T count, (N^go in SPT) relate empirically to the state of packing (see Bowles, 1996 for a survey of such correlations), which relates further to the effective stress angle of internal friction and the undrained shear strength (e.g., Seed et al., 1988; Fear and Robertson, 1995). These empirical relationships are in turn used in an analysis of the deformation response of the deposit. Chapter 1: Introduction 4 Since the soil adjacent to the probe is subjected to large deformation in CPT and SPT, the fabric effect is not expected to be adequately reflected in the data from these tests. In addition, the empirical correlations between the index measurements and the state variables are usually developed postulating stress-path independent isotropic soil behavior and are therefore of limited applicability. The fact that the correlations are usually not very precise unless site and problem specific relationship are available poses further problem with the adoption of this approach. A few procedures for interpretation of CPT data however rely on theory such as limit analysis (bearing capacity type formulations) and cavity expansion. Even these procedures are of limited applicability because they are usually based on the assumption of a very simple material behavior, e.g., isotropic rigid plastic. Noting further that the index measurements pertain to a certain value of average deformation in the surrounding deposit that is difficult to ascertain, data from these tests cannot be used to calibrate a stress-strain model even if the fabric and stress-path effects are discounted. 1.3 Possible use of Self-boring Pressuremeter As mentioned earlier, a detailed deformation analysis can be undertaken instead of adopting an empirical approach for computing the deformation of the earth structure or natural deposit. The exercise essentially involves calibration of an appropriate constitutive model from stress deformation data over a wide range of strain. Laboratory element test (e.g., triaxial, plane strain or simple shear) data from testing of undisturbed samples are commonly used for the purpose. SBPMT also provides stress-deformation data over a wide range of strain. A possibility therefore exists for the development of a calibration procedure for a realistic constitutive model based on SBPMT data. Since the test can be performed without causing a large disturbance to the Chapter 1: Introduction 5 soil adjacent to the probe during installation of the probe, the data are expected to reflect the fabric effect. If the constitutive model accounts for stress level and stress path dependency, an analytical approach based on calibration of such a model from SBPMT can provide a viable tool for deformation analysis of soil. The cone tip resistance and seismic measurements from seismic CPT carried out at an adjacent location can be used to minimize the impact of non-uniqueness associated with this approach. The procedure based on SBPMT is expected to be of precision comparable to that of a conventional analytical procedure based on laboratory tests on undisturbed samples without compromising economy. 1.4 Use of Self-boring Pressuremeter in Granular Deposits Many researchers have successfully used the self-boring pressuremeter to determine the effective stress angle of internal friction in plane strain (Lacasse et al., 1990; and Fahey at al., 1993) and the state parameter (Yu, 1994) in sand. A few studies also report successful derivation of the drained element stress-strain behavior over a wide range of deformation from SBPMT (e.g., Manassero, 1989; and de Souza Coutinho, 1990) representative of the stress path and loading direction (referred to the direction of deposition) imposed on the medium in the test. Therefore, the tool appears to be capable of providing reliable information of strength of granular soils. To the knowledge of this writer, no attempts have so far been made to predict drained or undrained mechanical behavior of granular materials over a wide range of deformation in monotonic loading from SBPMT for a stress path and loading direction not identical to that in the test. However, prediction of stress-strain behaviorfromSBPMT for a problem where the stress path and the direction of loading is not similar to that in an SBPMT appears to be feasible if an appropriate stress-strain relationship can be calibrated using SBPMT data. Such a stress-strain Chapter 1: Introduction ° model must be capable of accounting for stress path and fabric dependent mechanical behavior of granular soils internally. SBPMT data have been obtained from six research sites in western Canada under the auspices of the Canadian Liquefaction Experiment (CANLEX) project (In-situ Testing Group, 1994; Hughes In-situ Engineering, 1995; Hughes et al., 1995; and Bigger and Robertson, 1996). Undisturbed (frozen) sampling and laboratory testing have also been carried out at these sites. Thus, there is an opportunity to examine whether SBPMT data can be used to calibrate a stressstrain relationship that accounts for stress path and fabric. 1.5 Objectives and Organization The objective of this research is to device a practical but precise analytical procedure for estimating the mechanical response of an element of granular soil over a wide range of deformation using SBPMT and supplementary information from seismic CPT. The procedure essentially involves calibration of a simple yet realistic stress-strain relationship. The proposed procedure can be useful in predicting drained or undrained response of an earth structure or natural deposit at a reasonable expense. Monotonic response of anisotropic granular soils with stress, strain and stresspath dependent behavior is only considered in this study. The organization of the dissertation is as follows. Chapter 2 summarizes the current state of the art of self-boring pressuremeter testing in sand together with a critical review of the available procedures for SBPMT data analysis. Numerical tools for deformation analysis of continua that are relevant in analyzing SBPMT data are briefly described in Chapter 3 together with the advantages and limitations. Development of the constitutive model used in this study, its verification and pros and cons are discussed in Chapter Chapter I: Introduction I 4. Sensitivity of the predicted element response to the choice of model parameters has also been examined in Chapter 4. A numerical scheme for analysis of SBPMT data is devised and verified in Chapter 5. Sensitivity of model response in cylindrical cavity expansion to the choice of model parameters is also examined in Chapter 5. Relevant geotechnical details of the sites from which the data used in this study originated can be found in Chapter 6. Inverse modeling of drained SBPMT field data to calibrate the constitutive model is described in Chapter 7. To validate the proposed procedure laboratory triaxial element response is predicted using the stress-strain model calibrated using SBPMT data and the predicted response is compared with actual data from laboratory tests on undisturbed samples. Datafromthree CANLEX sites have been used in the validation process. Laboratory test datafromthe remaining sites are not available at present. SBPMT datafroma mine site in south eastern British Columbia with a documented case history of static flow failure are used to provide field validation by checking whether the response prediction from SBPMT is in qualitative agreement with the occurrence offlowslide. These details can also be found in Chapter 7. Major conclusions of this research are summarized in Chapter 8 together with suggestions for further research. CHAPTER 2 SELF-BORING PRESSUREMETER TESTING IN GRANULAR DEPOSITS 2.1 Introduction The present state-of-the-art of self-boring pressuremeter testing of granular deposits and data interpretation procedures are reviewed in this chapter. The salient details of the design of the probe used in this study and the test procedure are discussed and compared with equipment designs and test procedures suggested by other researchers. The commonly used analytical procedures for interpreting SBPMT data in sand are reviewed later. 2.2 The Self-boring Pressuremeter A pressuremeter test essentially involves application of pressure along a certain section of the wall of a borehole within a deposit and measuring the corresponding deformation. In an SBPMT, the borehole is created in the deposit by the probe itself through jetting, cutting or drilling. In this study jetting is only used to instal the probe in the layer of interest. The installation process Getting, cutting or drilling) can be optimized such that the disturbance and stress relief in the surrounding medium caused by the procedure is minimal. The self-boring pressuremeter probe (SBPM) used in this study is essentially a hollow steel cylinder with a cutting shoe at itsfront(Fig. 2.1). Two alternative designs of the cutting shoe used at the University of British Columbia (UBC) are shown in the figure. In the case of the central jetting system (shown attached to the SBPM probe in Fig. 2.1) the distance of the jetting nozzle from the edge of the cutting shoe can be easily altered to ensure minimum disturbance to the Chapter 2: Self-boring pressuremeter testing of sand 9 Electronic Cable Air Line Electronics: Jetting Rod A / D Board, amplifiers and microcontroller Transducer Housing: Internal Pressure, pore pressure and 6 strain arms Lantern and Membrane Transducer Housing: Load cell, temperature and 2 accelerometers MDBoog] SDoDPfj^y Dm Cutting Shoe Fig. 2.1. Schematic Details of the UBC SBPM surrounding soil due to probe installation. Such a flexibility is not available with the use of a jetting system of the "shower head" design, shown on the right of Fig. 2.1. To avoid the problem, a jetting system of the "central jetting" type was only used in this research. The annulus of the hollow cylinder consists of a central inflatable section and houses the transducers and downhole electronics. The diameter of the UBC SBPM is about 74 mm and the inflatable section of the probe is 450 mm long. The hollow section near the axis of the probe accommodates the jetting nozzle (or a cutter or a drill bit) and allows for circulation of the jetting (or drilling) fluid. The inflatable section can comprise a single inflatable cell (i.e., a monocell probe, such as the UBC probe) or one active cell (which is used to measure cavity deformation) and two guard cells (the 10 Chapter 2: Self-boring pressuremeter testing of sand tricell design, similar to type " E " or "GB" Menard Pressuremeter; Baguelin et al., 1978: p. 123). The expandable section is encapsulated in a rubber (as in case of the UBC probe) or an adiprene membrane (e.g., Hughes et al., 1977). The membrane is protected from the surrounding soil by an assemblage of steel strips, which has an appearance of (and also is referred to as) a Chinese Lantern. Compressed air (e.g., the probe used at UBC, and that used by Hughes et al., 1977), hydraulicfluidpressure (such as the weak rock self-boring pressuremeter, Clarke and Allan, 1989), or water pressure (Ajalloeian and Yu, 1996) is used to inflate the SBPM. Instrumented strain arms (Fig. 2.2) are often used to measure the deformation of the cavity at the central height (e.g, the UBC probe) or at several locations within the expandable section (e.g., Benoit et al., 1995). Operating range of the strain arms of the UBC SBPM is up to 8 mm with a resolution as small as Strain Arm in Retracted Position 0.78 mm thick Copper -Beryllium Strain Arm Plexiglass Contact 4 mm thick Phenolic Seat 2 mm thick Brass Seating Strain Arm in Normal Position Pressuremeter Body 6 Strain Arms @ 60° Copper-Beryllium Strain Arm in Normal Position 114 mm Nuts through 2 No.s 4X40 holes L 14 mm 131 mm 2 mm thick Brass Plate- Pressuremeter Body 6 mm |Long Section Fig. 2.2. Strain Arms of the UBC SBPM (Note: strain arms are mounted alternatively from uphole and downhole ends) Chapter 2: Self-boring pressuremeter testing of sand 11 0.006% in terms of cavity strains. The pressure expansion test in an SBPM is often approximated as a cylindrical cavity expansion, for which cavity strain is defined as the ratio of the radial deformation to the original radius of the cavity. Non contact sensors have also been used in some probe designs to monitor the borehole deflection (e.g., Clarke and Allan, 1989; Tani et al., 1995). The deformed shape of the cavity is sometimes approximately estimated by monitoring the volume of liquid pumped in (e.g., Ajalloeian and Yu, 1996) during the expansion process. The deformation measurement is usually performed at the inside of the flexible membrane. It is generally believed that the bedding error at the interface between the lantern and the borehole and the effect of compressibility of the lantern and the flexible membrane can be calibrated out by pressurizing the probe inside a rigid steel jacket. However, according to Tani et al. (1995) such a calibration does not account for bore hole roughness and separation of individual steel strips that form the Chinese Lantern from each other during cavity expansion. These researchers thus devised a non-contact deformation measurement system that directly monitors the movement of the steel strips of the lantern. The data acquisition (i.e., measuring and storing the history of cavity pressure, cavity deformation and pore water pressure) and testing sequence (i.e., inflation and deflation of the cavity and the rate of pressure application and removal) are automated in case of the UBC SBPM. However, in a few tests used in this research (those at J-Pit and LL Dam), cavity pressure was applied and reduced manually. All the tests in this research were carried out in a stress controlled manner. Schematic details of the electronics of the UBC probe are shown in Fig. 2.3 to explain the functioning of the UBC SBPM. During installation the uphole micro controller monitors the time, depth (from depth box), inclination of the probe (via two accelerometers), temperature (from 12 Chapter 2: Self-boring pressuremeter testing of sand Mud Flow and Pressure Air: 965 kPa (Inyjut signal) 120vi IA.C SMC NTT 202 Electronic Pressure Regulator Depth Pulses from Encoder RS232 9600 baud HCTL 2016 5v Microcomputer Motorola 68HC11 Microcontroller and 8 bit A/D converter AD 7543 12bitD/A Converter Hydraulic Ivalve switch] Air: up toi860 kPa^(To SBPM) LTC1290 12 bit A/D Converter 120v A.C. +15v -12v +5v power supply 15v D.C. Motorola 68HC11 Microcontroller +/-2.5vFromi Transducers 1 L Voltage Regulation Circuitry + 5v analog + 5v digital 3vEXC Amp. Typical Transducer To Transducers and Microcontroller Fig. 2.3. UBC SBPM Electronics (EXC: excitation voltage, Amp.: amplifier, "A" and "D' indicate analog and digital lines, respectively) downhole temperature transducer), vertical penetration resistance (from loadcell mounted on the SBPM body), mudflowand mud pressure (from uphole transducers). When the jacking is stopped, the uphole micro controller changes the data acquisition model from sounding to pore water pressure dissipation. In this mode the pore water pressure dissipation is monitored with time. The mode of data acquisition changes when afilecontaining the necessary instructions for performing Chapter 2: Self-boring pressuremeter testing of sand 13 a pressure expansion test is down loaded from the microcomputer to the uphole micro controller. During the pressure expansion test, the uphole micro controller monitors the pressure in the probe and the ambient pore water pressure via the downhole micro controller. The total cavity pressure transducer can measure pressure up to 2976 kPa and can resolve 0.7 kPa. The corresponding values for the pore pressure transducer are 1316 and 0.1 kPa, respectively. The deflections of the six strain arms are also monitored. The uphole micro controller also increases or decreases the total cavity pressure as per the instructions via the electronic pressure regulator. An air compressor supplies the compressed air for carrying out the pressure expansion test. At the end of the test the data (total cavity pressure, pore water pressure and deflections of the six strain arms) are stored and the data acquisition changes to the sounding mode when jacking of the probe starts again. Greater details on design and development of the UBC SBPM can be found in Campanella et al. (1990) and more information about the specifications of the transducers used at present in the probe is given by da Cunha (1994). For a comprehensive survey of alternative designs of the self-boring pressuremeter probe, reference may be made to Clough et al. (1990). 2.2.1 Typical SBPMT Data in Sand The terminology used in this dissertation is explained using a typical pressure expansion curve measured in an SBPMT at a sand site in the Fraser River Delta (Fig. 2.4). As mentioned earlier, during the expansion of the cavity the deformation at the cavity wall is monitored together with the corresponding gas pressure. The cavity strain calculatedfromthe average value of the cavity wall deformation measured with six strain arms is plotted against the corresponding gas pressure within the cavity, which is also called the total cavity pressure. As is evident, the cavity pressure was increased from zero to a value corresponding to point "B" constituting thefirstvirgin 14 Chapter 2: Self-boring pressuremeter testing of sand 1000 Ctj * CO Massey Tunnel S B P M T M245 depth 8.205 m 800 600 400 ( J 200 0 0 2F 4 6 8 Cavity Strain, % 10 12 Fig. 2.4. Typical SBPMT Data in Sand loading of the cylindrical cavity. In the process, the membrane started to deform as the total cavity pressure reached a value corresponding to point "A," also referred to as the "lift-off." If the installation process is perfect the lift-off gives the total horizontal geostatic stress. Following the first virgin loading, the total cavity pressure is held constant in the test until the deformation reached a value corresponding to point "C." The total cavity pressure is decreased to point "D" and increased again to point "E." A loop such as CD is usually referred to as the "unload-reload loop." The slope of an unload-reload loop provides approximate measure of the secant stiffness of over consolidated soil corresponding to the shear strain amplitude of the loop. The total cavity pressure is continuously decreased beyond "E." In sand, the cavity closes under in-situ water pressure, U . Sometimes the cavity expansion data are presented in terms of the effective cavity 0 pressure, which is obtained by subtracting pore water pressure from total cavity pressure. Chapter 2: Self-boring pressuremeter testing of sand 15 2.2.2 Appropriate Equipment Geometry If the length to diameter ratio of the self-boring pressuremeter is "sufficiently large," the pressure expansion data can be analyzed as a plane strain problem with no deformation in the vertical direction. However, if the probe is not long enough, the assumption of plane strain leads to erroneous interpretation. There is a considerable disagreement among researchers on the threshold value of length to diameter ratio above which the SBPM pressure-expansion can be analyzed as a plane strain problem without an appreciable loss of accuracy. Numerical studies of cavity expansion in a linear isotropic elastic material indicate that for a length to diameter ratio greater than 4, the deviation of the volume of the deformed cavity from that of a right cylinder is less than 10% (Hartman, 1974). There have been several other analytical attempts to address the issue based on more complex constitutive models (e.g., Houlsby and Carter, 1993; Yu and Houlsby, 1992; Salgado and Byrne, 1990; and Yan, 1988). The conclusions of these studies regarding the extent of approximation introduced by the assumption of plane strain depend upon the constitutive model and the model parameter of interest. Since a numerical study on the relevance of the plane strain assumption in modeling SBPMT is expected to be limited by the ability of the constitutive model to simulate the stress-path dependency, direct experimental studies (e.g., Basudhar and Kumar, 1995) appear to be of greater relevance. The data presented by Basudhar and Kumar suggest that the effect of end restraints is limited over a length of 2.3 times the diameterfromeach extremity of the expandable section of the probe, beyond which the shape of the deformed cavity remains essentially cylindrical for cavity strain less than 15%. Pass (1994) carried out a number of SBPMTs at Treasure Island near San Francisco, California in loose to medium dense sand (relative density between 30 to 60%). An SBPM with Chapter 2: Self-boring pressuremeter testing of sand 16 a length to diameter ratio of 7.4 was used in this study. The probe is rather similar to that shown in Fig. 2.1 except for the fact that it incorporates nine strain arms: at one fourth, one half and three fourth heights of the expandable section. Typical cavity expansion data measured in sand in this study are as shown in Fig 2.5. Letters "L," "M," and "U" in the figure denote lower, middle and upper strain arms, respectively. For clarity, only the virgin loading curves are shown in the figure. It is evidentfromthese data that in the absence of installation related disturbance the effect of end restrain is not apparent at 1.85 times the diameter awayfromthe ends of the expandable section for cavity strain less than 10%. These experimental studies indicate that SBPMT cavity expansion can be modeled as a plane strain problem if the length to diameter ratio exceeds a value of about 5. Therefore, data measured with UBC SBPM (length to diameter ratio of about 6) can be analyzed assuming plane strain without compromising accuracy if cavity strain does not exceed a value of about 10%. It should be noted that the conclusions about probe geometry are applicable only for an SBPM that estimates the deformed shape of the cavity by monitoring the cavity wall deflection at 800 0 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 Cavity Strain, % Fig. 2.5. SBPMT at Treasure Island: depth 9.8 m, depth of water table 1.5 m (modified from Pass, 1994) Chapter 2: Self-boring pressuremeter testing of sand 17 the central height of the expandable section. A calibration chamber study on SBPMT with volume measurement shows that for dense sand (relative density of about 85%) the effective cavity pressure at 10% cavity strain changes by 18 and 6% as the length to diameter ratio of the probe is increasedfrom5 to 10 andfrom10 to 15, respectively (Ajalloeian and Yu, 1996). 2.3 Self-boring Pressuremeter Installation Procedure in Sand The installation procedure of the SBPM needs to be optimized so that the disturbance and stress relief in the surrounding soil is not appreciable. In general, an optimum installation procedure may have to be developed for every site at the beginning of a self-boring pressuremeter testing program. For optimizing the installation process, any disturbance, if present, first needs to be identified. The installation procedure can then be altered until disturbance is minimized. 2.3.1 Identification of Disturbance When more material is removed during installation than is necessary to accommodate the volume of the probe, the material in the vicinity of the advancing probe may undergo a radial movement towards the probe. A stress-relief and reduction in the original relative density results from such a movement. Such a disturbance, referred to as "wash boring," leads to the creation of an oversize hole. If less material is removed than is needed as the probe advances through the deposit, a radially outward movement is imparted on a soil element near the tip of the probe causing an increase in the stress state and relative density in the vicinity. Such a disturbance is referred to as of the "pushed in" type. Its influence on SBPMT data is more significant for probe designs where the expandable section immediately follows the tip, as often is the case. Both types of installation-related disturbance essentially affect the initial part of the pressure expansion curve. 18 Chapter 2: Self-boring pressuremeter testing of sand As noted by da Cunha ( 1 9 9 4 ) , unless the disturbance is quite large, the pressure expansion curve is not affected appreciably at cavity strains greater than about 4 to 5 % . It is relatively easy to detect wash boring from the shape of the initial portion of the pressure expansion curve. In presence of excessive wash boring the lift-off tends to occur near the in-situ pore water pressure. Also the initial part of the pressure expansion curve tends to take an upward concave shape. On the other hand, in presence of the pushed in type of disturbance lift off may occur at a pressure larger than the actual geostatic horizontal effective stress depending on the probe geometry. However, the pressure expansion curve often remains convex upward. Therefore, it is difficult to detect the presence of pushed in type of disturbance only from visual inspection of the pressure-expansion curve since an independent estimate of the horizontal effective stress is seldom available. The existing procedures for detection of the presence of disturbance rely on the qualitative shape of the pressure-expansion curve measured in an SBPMT. Wroth (1982), for instance, suggests for a pressure-expansion curve relatively free from disturbance the quantity (Pio-P^sVCPioPQ) should be approximately 0 . 5 1 for sands, where the letter "P" denotes total cavity pressure and the subscripts are the values of cavity strain, da Cunha ( 1 9 9 4 ) , on the other hand, compares the shape of the measured curve with the expected model response. If the model response greatly differs from the measured response da Cunha suggests that the data are affected by installation related disturbance. Both these procedures have a common limitation that they may not be able to detect the pushed in type of disturbance for reasons outlined in the preceding paragraph. To avoid the difficulty, the vertical resistance during SBPM installation is estimated from the cone tip resistance and sleeve friction measured in a CPT performed at a location in the vicinity of the SBPMT in this research. Examination of SBPMT data with and without pushed in type of 19 Chapter 2: Self-boring pressuremeter testing of sand disturbance and CPT from three sand sites in the Fraser River Delta, KLDD #2, Laing Bridge (South) and Massey Tunnel, leads to the conclusion that the presence of pushed-in type of disturbance becomes discernible as the vertical resistance (force) encountered during SBPM installation exceeds 1.5 times the value estimated from CPT. The data from which the inference is drawn are presented in Fig. 2.6. The installation related variables are altered during SBPM insertion so as not to allow the vertical resistance to exceed 1.5 times that inferred from a nearby CPT. Wash boring can be avoided for instance by reducing the flow rate during jetting if in a cavity expansion test upward concave data are measured or lift off occurs near the estimated in-situ pore water pressure. 2.3.2 Guidelines for Selecting Optimum Installation Related Variables For a probe installed by jetting, the following factors affect the installation process: nozzle position, mud flow, and insertion speed. The scouring action can be increased by bringing the 40 Data used: M245, M225, K046, MSPM9401, MSPM9402, KDPM9401 KDPM9402 and LSBPM19 u .2 30 M I • Disturbed • Undisturbed H Undisturbed Installation 1 a 0 • if/ 2 0 7 Jill _ _ JTv* / i t <s % J9 B a I • 10 1 U i 00 , i , i 10 20 30 40 50 Observed Installation Resistance, kN 60 Fig. 2.6. Detection of Pushed in Type Disturbance of SBPMT Chapter 2: Self-boring pressuremeter testing of sand 20 jetting nozzle near the edge of the cutting shoe, by increasing the mud flow or by decreasing the insertion speed. The values of these parameters found suitable in this study are summarized in Table 2.1 together with the corresponding values suggested as appropriate by other researchers for probes similar to the UBC SBPM. Significance of the influence of probe design on installation procedure becomes clear from a comparison of the typical values of installation speed for a probe installed by jetting (4-10 mm/s: Table 2.1) with that for probes installed by cutting (0.08 to 0.6 mm/s: Bacchus, 1983; Bruzzi et al., 1986; Lacasse et al., 1990). For slightly aged medium dense sand a mud flow of 1.7 to 2.7 l/s is suggested by Bruzzi et al. (1986) for SBPMs installed by cutting. Table 2.1. Appropriate Jetting Variables for SBPMT in Sand Site (Sand Type) D o, 5 mm Massey Tunnel and KIDD # 2 (Fraser River) 0.14 D , % Nozzle Position mm Mud Flow, l/s Insertion Speed mm/s Reference 30 to 60 Oto 5 0.4 to 0.5 5 to 7.5 This study 20 to 30 0 0.4 to 0.5 5 to 7.5 This study Hughes, 1994 R A South Eastern BC Mine Site (Mine Tailings) 0.074 J-Pit (Syncrude) 0.20 30 to 60 Oto 5 Not available 5 to 15 Laing Bridge (Fraser River) 0.1 to 0.6 30 to 60 5 0.33 4.2 da Cunha, 1994 Treasure Island (Fill and Shoal) 0.9 -50 16 1.67 0.11 to 8 Pass, 1994 McDonald's Farm (Fraser River) 0.1 0.6 40 to 60 10 to 25 0.33 to 0.72 7 to 35 Hughes, 1984 Lulu Island (Fraser River) 0.130.17 50 to 60 20 0.9 to 1.3 4 to 15 Howie, 1991 Note: A. Offset of jetting tip behind the edge of the cutting shoe Chapter 2: Self-boring pressuremeter testing of sand 21 During SBPM installation, the probe is jacked down while the material scoured due to the jetting action is being removedfromnear its end. Due to wall friction, the soil elements in the vicinity of the probe undergo simple shear in the vertical direction (Houlsby, 1991). Usual magnitudes of vertical shear stress that develops during jacking of an SBPM in sand leads to volume contraction in the medium around the probe irrespective of the state of packing and depth of the deposit. To accommodate the contractive deformation, the jetting process must remove less material than that contained in a cylinder with the same diameter as that of the probe. The difference between the volume of material to be removed during the installation of a smooth SBPM and one with rough wall depends on the state of packing and stress within the deposit. Therefore, a universal procedure cannot be developed for undisturbed SBPM installation in granular deposits that can be used for all state of packing and stress. Hence, guidelines listed above should be considered approximate. However, the procedure developed in this research appears to be applicable for a wide range of loose to medium dense granular deposits. 2.4 Self-boring Pressuremeter Test Procedure in Sand Prior to performing cavity expansion, the probe needs to be calibrated for system compliance and stiffness. Usual procedures for calibration and cavity expansion is described below. 2.4.1 Calibration The calibration of the electronic transducers and the strain arms of the UBC SBPM is carried out before and after each testing program, following da Cunha (1994). The calibration curves for the electronic transducers and the strain arms are linear andfreefromhysteresis. Chapter 2: Self-boring pressuremeter testing of sand 22 In addition to the usual calibration procedure for the transducers, the SBPM needs to be calibrated from system compliance and stiffness. System compliance is attributable to the occurrence of the following as the SBPM is pressurized: • compression of the flexible membrane, • inflation of the air hose during inflation of the probe, • flattening of the Chinese Lantern, • separation of the steel strips of the Lantern and • bedding error at the interface between the Chinese Lantern and the soil. The system compliance is measured by pressurizing the probe inside a rigid aluminium jacket with an internal diameter the same as the outer diameter of the Chinese Lantern. The deformation measured in the process is subtractedfromthat of the pressure expansion curve measured in-situ. This procedure for estimating system compliance does not account for bedding error. Nor the deformation of the lantern against a rigid cylinder is expected to be the same as that against a wall of granular material. Thus, the method is approximate (Tani et al., 1995). However, no data are available at present showing the significance of these factors. To eliminate system stiffness, the probe is inflated in air and the measured pressures are subtracted from the total cavity pressure measured in-situ. The magnitude of corrections for system compliance and stiffness are relatively small for the UBC SBPM. For instance, in tests M225 and M245 at Massey Tunnel and K046 at KIDD # 2 the correction in total cavity pressure due to system stiffness was 25.6 kPa at cavity strain of 10%. The correction in the cavity strain at effective cavity pressure of 1000 kPa was about 0.06%. More details on the calibration of the probe for compliance and system stiffness for the UBC SBPM can be found in da Cunha (1994). Chapter 2: Self-boring pressuremeter testing of sand 23 2.4.2 Cavity Expansion A loading (and unloading) rate of 3.5 kPa/s was used in this study during inflation and deflation of the cavity. Such a rate was found to be slow enough not to result in a significance rise in pore water pressure during cavity expansion for frictional materials with a wide range of grain sizes, e.g., Syncrude Sand, South Eastern BC Mine Tailings and Fraser River Sand. Pass (1994) used a loading rate between 0.5 to 1.0 kPa/s in comparison. However, justification for adopting such a slow loading rate is not provided. To eliminate the effect of creep (Hughes, 1982; Howie, 1991) before performing an unloadreload loop the total cavity pressure is held constant in the SBPMTs reported in this study. Although da Cunha (1994) suggests an eight minute holding phase, a two minute holding phase was found sufficient in these tests. 2.5 Limitations in Existing Procedures for Deformation Analysis of Granular Materials A common framework of use of laboratory and in-situ test data in deformation analysis is briefly surveyed in the following sections. Since the use of SBPMT is not as widespread as CPT and SPT for granular soils, usual strategy of use of CPT and SPT is described first. How SBPMT data can enhance the precision of the approach is examined later. 2.5.1 Limitations of the Approach Based on Laboratory Tests Laboratory tests such as triaxial or plane strain provide stress-deformation data over a wide range of strain. The datafromthese tests are used to calibrate a stress strain model to derive the model parameters. The model parameters are then used to compute the mechanical response of a deposit when subjected to a deformation history using a numerical algorithm, e.g.,finiteelement Chapter 2: Self-boring pressuremeter testing of sand 24 orfinitedifference. Several laboratory tests are usually necessary to calibrate even a simple stressstrain model. The best results are obtained when tests are carried out on undisturbed samples that retain the in-situ fabric reproducing the geostatic stress as closely as possible. If a stress strain model is calibrated from such tests, the precision of the numerically computed deformations is limited only by the simplifying assumptions inherent in almost all stress-strain relationships. However, undertaking a comprehensive undisturbed sampling and laboratory testing program is usually very expensive for routine application. If reconstituted samples are used, fabric related uncertainty may affect the precision of the results to a great extent. Uncertainty in laboratory test data related to sample fabric have drawn extensive attention over recent years (see e.g., Mullilis et al., 1975). 2.5.2 Use of SPT and CPT The fabric related uncertainty in laboratory test data has lead to a great reliance on in-situ index measurements such as SPT ( N ^ (Seed, 1979), CPT (cone tip resistance and friction ratio, Robertson et al., 1986), and shear wave velocity measured in seismic CPT (Robertson et al., 1992). Although both CPT and SPT are essentially index tests, the former is fully automated and provides repeatability not usually obtained in an SPT. Also, unlike SPT, CPT provides almost continuous data with depth and can resolve soil layers as thin as a few millimeters. Consequently, procedures for using CPT and SPT data are considered separately. 2.5.2.1 Use of CPT The CPT is not currently amenable to analytical interpretation because the test procedures do not offer an easily tractable boundary value problem. Nevertheless, the strength parameters that Chapter 2: Self-boring pressuremeter testing of sand 25 affect CPT data and the nature of these effects are relatively well understood from a number of calibration chamber studies on CPT (e.g., Houlsby and Hitchman, 1988, Jamiolkowski et al., 1988). From these insights, empirical correlations between the index measurements and S and (b' have v been developed for a number of soils (see, e.g., Robertson and Campanella, 1986). These correlations have been refined for specific sites and have been used effectively for a number of years. However, attempts to develop precise correlations between CPT index measurements and Su and d>' using global data have been less successful and site specific relationships are often required. Nevertheless, the CPT is a very useful tool in identifying soil layering and determining the hydraulic characteristics of the layers. Also, indices such as q correlate quite well with the x state of packing of the deposit in drained cone penetration. Seismic measurements can be carried out during CPT. Datafromthese measurements provide precise estimates of average elastic strain properties, e.g., the shear modulus over specific depth intervals (Robertson et al., 1986). 2.5.2.2 Use of SPT Considerable experience with the use of SPT has been translated into a number of phenomenological correlations between (N,) ^ and soil strength based on field performance information. Some of these relationships perform very well globally. The correlation between the susceptibility of granular deposits to cyclic liquefaction and ( N ^ proposed by Seed (1979) can be cited as an example of such a relationship. However, attempts to use (NJgo as an appropriate index measurement in several types of deformation problems have lead to correlations that are rather imprecise (e.g., Seed et al., 1988, Stark and Mesri, 1992). Although ( N ^ appears to be a good index for the state of packing, detailed stratigraphic information cannot be obtained from the SPT. Among the advantages of the SPT is the fact that disturbed samples are usually recovered Chapter 2: Self-boring pressuremeter testing of sand 26 in an SPT. These samples can be analyzed to determine the grain size distribution and other indices, e.g., the Atterberg Limits. 2.5.2.3 Limitations of Index Measurements The main problem with the index measurements is that the measurements usually pertain to ranges of deformation at which the fabric effect is minimal. As indicated in the preceding discussion, the in-situ index measurements are generally perceived to be good measures of relative density. Correlations linking an index measurements to soil behavior are essentially stress path and fabric dependent. In principle, therefore, index measurements can be used provided that a site and problem specific correlation can be developed. However, necessity of even a few laboratory tests for developing such relationships can erode the economic advantage of in-situ tests. 2.5.3 Model Calibration from SBPMT Data Like laboratory tests, e.g., triaxial or plane strain, an SBPMT pressure expansion curve can be viewed as stress-strain data over a wide strain range. SPT and CPT, on the other hand, provide one single (index) measurement at a certain depth pertaining to a certain value of average deformation in the surrounding soil. However, unlike the laboratory element tests SBPMT in a homogeneous medium leads to the development of non-homogeneous stress and deformation fields. Interpretation of SBPMT is thus slightly more involved than laboratory element tests. Nevertheless, like laboratory tests SBPMT data can in principle be used to calibrate simple stress strain models. In fact, calibration of simple stress strain relationships is the essence of SBPMT interpretation procedures developed by Salencon (1969), Hughes et al. (1977), Carter et al. (1986), Yu (1992), Manassero (1988), and de Souza Coutinho (1989) for granular soils. Chapter 2: Self-boring pressuremeter testing of sand The exercise of SBPMT data interpretation usually involves calculation of model response in cylindrical cavity expansion for a given set of model parameters and refining the estimates of model parameters until an acceptable match is achieved between SBPMT data and model response. An index tool such as the piezocone may be useful as a supplement for obtaining the initial estimates of the model parameters to begin an iterative procedure to fit the model response to cylindrical cavity expansion to the SBPMT data. Since SBPMT data can be analyzed following the principles of mechanics, the applicability of results of interpretation is only limited by the capabilities of the selected stress-strain model. As in case of deformation analysis based on laboratory tests, the stress strain model calibrated from SBPMT can be used to estimate the mechanical response of an earth structure. Depending on the complexity of the constitutive model, and the number of independent measurements available for calibration, the problem can be under determined (when the number of independent unknown model parameters is more than the number of independent stress-strain measurements) or over determined. Usually sufficient number of independent measurements are not available even in case of very simple stress strain relationships. The problem therefore usually reduces to an under determined one. Non-linear nature of the inverse problem may lead to nonuniqueness (i.e., two different sets of model parameters may lead to virtually identical model response) unless extensive a-priori information about the model parameters is available to supplement the measured data appropriately. Thus, to be useful, the constitutive model needs to be simple with fewest possible model parameters. Ideally, the parameters should only depend on state variables, e.g., states of stress fabric and packing, and reasonable knowledge about their appropriate range of values should be in existence. Yet the model should capture the essence of mechanical response of soils. 27 Chapter 2: Self-boring pressuremeter testing of sand 28 In an SBPMT only one independent stress-strain measurement is carried out, while a typical laboratory element (e.g., triaxial) test data two independent stress-strain measurements - deviatoric and volumetric - are available. Therefore, model calibration from SBPMT data is usually slightly more under determined than that using laboratory element tests. Nevertheless, SBPMT has a few advantages over laboratory element test. If the probe can be installed without causing significant disturbance the measured response of soil would represent the behavior of the state variables actually existing in-situ. It should be noted that in a granular deposit, SBPMT data do not have to be absolutely free from disturbance to be useful. As pointed out by da Cunha (1994), in sand the effect of disturbance on interpretation of SBPMT is minimized by considering cavity expansion data within 4 to 10% range of cavity strain. The installation related disturbance affects SBPMT data below a cavity strain of 4%, while a finite length of probe (non cylindrical cavity expansion) can affect data at cavity strains above 10% depending on probe geometry. In addition, SBPMT is very economical in comparison with undisturbed sampling and laboratory testing. 2.6 Reliability of SBPMT Data Earlier applications of SBPMT in granular deposits lead to a mixed conclusion regarding the usefulness of the tool in such a material. Lacasse et al. (1990), and Fahey and Randolph (1984), for instance, successfully derived reasonable estimates of strength parameters of sand (e.g., shear modulus and 4>') and the effective horizontal geostatic stress, o ', from field SBPMT in H granular deposits. However, Bruzzi et al. (1986) failed to obtain reliable estimates of (J)' from SBPMT in sand. Robertson (1982) attempted to use SBPMT in cyclic liquefaction studies and failed to obtain reasonable results. As a result of partial success of SBPMT in sand, concerns have been raised about the general applicability of the tool in granular soils (Clarke and Gambin, 1995; Chapter 2: Self-boring pressuremeter testing of sand 29 Bruzzi et al., 1986). Consequently, SBPMT has largely been used in recent years to determine unload-reload moduli of sand (e.g., Koga et al., 1994). The virgin cavity expansion is often perceived to be of less value due to its sensitivity to factors that are difficult to control in sand, e.g., installation related disturbance (Clarke and Gambiri, 1995). Although objective studies on reliability of SBPMT tool vis. a vis. other in-situ testing tools can address the issue, such a study is rare. Uncertainty in systematically measured data such as those obtained from SBPMT arisesfrom(a) testing related factors and (b) natural variability of the deposit. Present knowledge on the significance of these factors in SBPMT and other in-situ tests is briefly discussed in the following. 2.6.1 Random Error, Test Procedure and Equipment Related Uncertainties In a recent study, Kulhawy and Trautmann (1996) estimate the lower bound of the coefficient of variation (CoV) for the SBPMT pertaining to the uncertainty in test procedure and random error (uncertainty related to these factors are commonly referred to as "repeatability") to be 5 to 18% with a mean value of 9%. The CoV is defined as 6/p, where p and a are the estimates of mean and standard deviation, respectively. They treated the unload-reload modulus in a calibration chamber study as the random variable of interest. Kulhawy and Trautmann estimate that in pre-bored pressuremeter tests a CoV of about 5% can be ascribed to equipment related uncertainty. To account for more diversity in the design of the self-boring pressuremeter probe, they suggest that 8% is a reasonable estimate of the CoV for equipment related uncertainty in case of the SBPMT. The CoV pertaining to the procedural uncertainty in SBPMT is similarly estimated to be about 15%fromthe corresponding value of 12% determined for a data-base of pre-bored pressuremeter tests. Thus, the inferred value of the CoV for the SBPMT, which is equal to the sum Chapter 2: Self-boring pressuremeter testing of sand 30 of squares of the CoVs pertaining to random errors, equipment design, and procedural factors, is between 15 and 25%. The corresponding range of values for the CPT and the SPT are 5 to 15% and 15 to 45%, respectively. Roy et al. (1997) analyzed SBPMT data from six sand sites and obtained a CoV between 9 and 12% considering data relatively free from installation related disturbance. The effective cavity pressure at 10% cavity strain and unload-reload modulus, both normalized to 100 kPa of effective horizontal stress, were considered as the random variables of interest in this study. It is therefore apparent that the repeatability and the equipment related uncertainty of the SBPMT following an optimal installation procedure is comparable to CPT, which is regarded as one of the most repeatable in-situ tests. The conclusion holds for both virgin loading (of which the normalized effective cavity pressure at cavity strain of 10% is an index) and unloadreload data (the normalized unload-reload modulus in the study of Roy et al. is an index of such data) measured in an SBPMT. 2.6.2 Effect of Inherent Soil Variability on SBPMT It also needs to be examined whether the SBPMT is as sensitive to the natural soil variability as other commonly used in-situ testing tools. In case of the pre-bored pressuremeter tests in sand, if only inherent soil variability is considered, the CoV ranges between 20 and 50% for the limit pressure (the effective cavity pressure at the steady state of cavity expansion) and 15 to 65% for the pressuremeter modulus (Phoon and Kulhawy, 1996). The pressuremeter modulus represents the slope of the linear section of the pressure - volume curve measured in a pre-bored pressuremeter test immediately following the initial concave upward portion with cavity pressure and volume plotted as abscissa and ordinate, respectively. The estimates of the CoV for the cone tip resistance in sand range between 20 to 60% (Phoon and Kulhawy, 1996). Roy et al. (1997) Chapter 2: Self-boring pressuremeter testing of sand examined SBPMT datafromsix sand sites in Western Canada and obtained a CoV between 7 and 37% for the effective cavity pressure at 10% cavity strain normalized for 100 kPa effective horizontal pressure. The corresponding range for similarly normalized unload-reload modulus was 12 to 33%. In comparison, the estimates of the CoV for the cone tip resistance normalized to 100 kPa effective horizontal pressure ranged between 20 and 50% at the same sites. In fact, a monotonic relationship between the estimates of the CoV and sample means pertaining to SBPMT measurements and the cone tip resistance was also apparentfromthese data. In other words, inherent soil variability affects the virgin loading as well as the unload-reload curves measured in an SBPMT and the cone tip resistance in a similar fashion. However, since the CoVs for the SBPMT measurements pertaining to natural soil variability are lower than that for the cone tip resistance, the spatial resolution of the SBPMT is expected to be less than the piezocone. 2.7 Interpretation of Cavity Expansion Data A deformation test very similar to in-situ cavity expansion is routinely performed in the laboratory hollow cylinder apparatus by researchers to derive an estimate of effective stress angle of internal friction in plane strain (Whitman and Luscher, 1962; and Wu et al., 1963) as well as to calibrate constitutive models of various complexity (e.g., Li and Pu, 1986; Juran and Mahmoodzadegan, 1989). As mentioned earlier, field SBPMT data have been used in a similar manner in calibrating a stress strain relationship to derive soil strength and deformation response. From the success of a number of studies involving use of the SBPMT in sand (Hughes et al., 1977; Carter et al., 1986; Manassero, 1989; Lacasse et al., 1990; de Souza Coutinho, 1990; da Cunha, 1994; Yu, 1994 and Roy et al., 1996), it can be suggested that the tool may be useful in estimating 31 Chapter 2: Self-boring pressuremeter testing of sand 32 the parameters of a constitutive model provided that the non-uniqueness in such an exercise of inverse modeling can be addressed effectively. Some researchers (e.g., Fahey and Carter, 1993) suggest that only unload-reload data from SBPMT should be used instead of virgin loading in sand. Their contention is that unload-reload data are relatively freefrominstallation related disturbance. Although in many studies plasticity models based on isotropic hardening have been applied to model unload-reload datafromSBPMT (e.g., Houlsby et al., 1986; Yu, 1996), stress strain behavior in unload-reload of sand is not usually captured by such a simple hardening rule (see, e.g., Lade and Boonyachut, 1982). A more comprehensive hardening rule may lead to a complicated stress strain relationship. Moreover, fabric related information cannot possibly be derivedfromunload-reload data. To avoid these problems, only virgin loading SBPMT data will be used in this research. Brief description of the recent developments in analysis of drained SBPMT data as a cylindrical cavity expansion process is briefly discussed in the following sections. A more comprehensive summary of historical development of analysis of cylindrical cavity expansion can be found in Ladanyi (1995), da Cunha (1994), and Ferreira, (1994). 2.7.1 Closed Form Solutions to Cylindrical Cavity Expansion Closed form solutions to cylindrical cavity expansion in an isotropic elastic-perfectly plastic Mohr-Coulomb material have been given by Salencon (1969), Hughes et al. (1977) and Carter et al. (1986) assuming deformation to be small. The plastic strain increments are assumed to be given by the dilatancy relationship of Rowe (1971). Approximate solution to the problem for such a material pertaining to large deformation has been given by Yu (1992). The model parameters for a Mohr-Coulomb material with Rowe's dilatancy relationship include the average values of the Chapter 2: Self-boring pressuremeter testing of sand 33 shear modulus and Poisson's ratio to describe the elastic response, the effective stress friction angle to describe failure and the constant volume friction angle to describe the plastic flow. The value of the effective horizontal geostatic stress is also treated as an additional unknown. As mentioned earlier, interpretation of SBPMT using these solutions involves varying the model parameters by trial and error until a reasonable match is obtained between the model response in cylindrical, cavity expansion and SBPMT data. The value of the constant volume friction angle can be found in the literature for several types of granular soils. Consequently, the estimate of this parameter is not needed from back analysis of SBPMT. Therefore, parameters excepting the constant volume friction angle and the effective horizontal geostatic stress only need to be derivedfromSBPMT. The average value of shear modulus correlates to the secant slope of the unload-reload loop (da Cunha, 1994, suggests that its value is about 0.7 times the unloadreload modulus corresponding to 20% unloading and reloading in terms of effective cavity pressure at cavity strain of about 5%). A value of 0.25 is often assumed for the average Poisson's Ratio (da Cunha, 1994). As has been demonstrated by da Cunha (1994), these simple solutions are very useful in deriving the peak effective stress friction angle pertaining to the plane strain problem. Ferreira (1994) gave a closed form solution for virgin loading in cylindrical cavity expansion in sands based on a two dimensional rigid plastic formulation for small deformation (p. 132-133, op. cit.). A hyperbolic relationship between the mobilized Mohr-Coulomb angle of internal friction in terms of the effective stress and total strain was assumed. The manner in which the dilatancy relationship of Rowe (1972) is implemented in this study requires the constant volume friction angle to be a function of cavity strain, which contradicts the usual definition of the quantity. In addition, stress and stress path dependent soil behavior and fabric effects are not accounted for in this isotropic two dimensional formulation. Chapter 2: Self-boring pressuremeter testing of sand 2.7.2 Numerical Solutions to Cylindrical Cavity Expansion Stress and strain level dependent soil behavior is usually accommodated in an analytical procedure adopting a numerical method such asfiniteelement orfinitedifference. Hartman (1974) implemented a non-linear stress and strain dependent hypo-elastic stress-strain model via finite element for analyzing pre-bored pressuremeter data. Although stress-dependency in elastic behavior can be accounted for in a hypo-elastic formulation, energy conservation is not guaranteed in such a model. Useful soil properties could not be derived in this study primarily because of the significant disturbance imparted to soil adjacent to the borehole wall. Manassero (1989) and de Souza Coutinho (1990) demonstrated that the deviatoric as well as volumetric response of an element undergoing distortion can be estimated from SBPMT data for small and large deformations, respectively. In these studies a two dimensional rigid plastic stress-strain model based on the Mohr-Coulomb failure criterion and the dilatancy relationship due to Rowe (1972) were calibratedfromvirgin loading data measured in SBPMT: Material behavior is assumed to be independent of the stress level in these simple formulations. The equilibrium and strain compatibility equations pertaining to cylindrical cavity expansion are solved via explicit finite difference. A reasonable value of the constant volume friction angle is first assumed. The deviatoric and volumetric responses of a soil element adjacent to the cavity are then derived in steps considering two adjacent points on the measured SBPMT pressure expansion curve. Sayed (1989) also derived the entire stress strain curve representative of an element undergoing distortionfromSBPMT data. A hypo-elastic stress-strain relationship was used in this study for developing a procedure for estimating the model parameters governing volumetric behavior from SBPMT. However, no validation of the procedure is provided. Also, the stressstrain relationship does not model the stress and stress path dependency and as in case of any hypo 34 Chapter 2: Self-boring pressuremeter testing of sand 35 elastic formulation, frame indifference is not guaranteed. Fahey and Carter (1993) also used a hypo-elastic stress level dependent stress strain relationship to derive stiffness behavior of sand and demonstrated that the stress-strain behavior of a single plane strain element can be predicted from SBPMT in principle. Roy et al. (1996) used a three dimensional stress-strain model that captures the stress path dependent behavior partially to analyze SBPMT data from two sites in Western Canada using the explicit finite difference computer code, FLAC (Cundall, 1992). The model parameters were then used to determine the element response in undrained triaxial extension. The predicted element response were compared with the laboratory measurements on undisturbed (frozen) samples extracted from locations adjacent to the SBPMTs. More recently, Yu (1994) implemented a three dimensional elasto-plastic stress-strain relationship using one dimensional finite element to derive the state of packing of a deposit. The stress strain relationship is based on a non associated flow rule and an isotropic hardening rule with the size of the loading surface being related to the initial state parameter of the deposit. The procedure was essentially devised to estimate the state parameter from the virgin loading curve measured in SBPMT. As has been observed by Roy et al. (1995), trying to estimate an approximate soil behavior index such as the state parameter is actually an under utilization of the SBPMT data particularly when it has been shown by researchers that the entire stress-strain behavior can be derived from analysis of SBPMT. Bahar et al. (1995) use SBPMT data together with laboratory data from isotropic compression and triaxial compression tests to calibrate an isotropic elasto-plastic three dimensional model for sand. The finite element method was used to analyze the SBPMT data. It is not clear Chapter 2: Self-boring pressuremeter testing of sand how the results of the procedure are affected by sample fabric of the laboratory specimens systematic validation for the procedure was provided either. 2.7.3 Limitations of Existing SBPMT Interpretation Procedures As is evident from the preceding discussion, the available analytical tools for inverse modeling of SBPMT data rely on isotropic stress-strain relationships, in a majority of which the stress level and stress path dependency in soil behavior are not modeled. In addition, some of these procedures have the capability of admitting large deformation, while the others are essentially small strain formulations. As a result, the inference drawn from back analysis of SBPMT using these procedures needs to be applied carefully. Limitations of the existing interpretation procedures are discussed in what follows. 2.7.3.1 Inherent Anisotropy As shown by may researchers (e.g., Oda, 1972, Arthur and Menezies, 1972), the fabric of sand in many depositional environments is anisotropic. Consequently, the mechanical response of such a deposit depends on the direction of loading vis. a vis. the direction of deposition during its formation (see, e.g., Vaid et al., 1996, Park and Tatsuoka, 1994). The effect of inherent anisotropy is usually erased with deformation. There is conflicting experimental evidence as to whether the fabric effect is still present at deformations as high as those at which peak effective stress angle of internal friction is measured (peak shear strain of about 2 to 3%). Nevertheless, a survey of the experimental studies (Lam and Tatsuoka, 1988; Tatsuoka et al., 1986; Been and Jefferies, 1986; Ergun, 1981; Haruyama, 1981; Shankariah and Ramamurthy, 1980; Reades and Green 1976; Lade and Duncan, 1973; Ramamurthy and Rawat, 1973; Green, 1971; Sutherland and Mesdary, 1969; Chapter 2: Self-boring pressuremeter testing of sand 37 Bishop, 1966) show that at such a deformation, the effect of fabric related anisotropy may not be very significant. More details on inherent anisotropy and its effect on peak effective stress angle of internal friction can be found in Chapter 4. It appears therefore that the peak angle of internal friction inferredfrominverse modeling of SBPMT data adopting an isotropic linear elastic perfectly plastic stress-strain model may be generally applicable in a deformation problem irrespective of fabric anisotropy. Laboratory element tests such as triaxial and plane strain invariably show that for sand samples prepared by air or water pluviation, the stress-strain behavior is stiffer at peak shear strains smaller than about 2 to 3% when the material is loaded with the major principal stress parallel to the depositional direction (see, e.g., Vaid et al., 1996, Park and Tatsuoka, 1994). Consequently, direct use of the strain level dependent mobilized value of the effective stress friction angle obtained from procedures suggested by Manassero (1989) and de Souza Coutinho (1990) may show lower mobilized strength than what may develop in a problem where the major effective principal stress is nearly parallel to the depositional direction for hydraulic or fluvial deposits. These procedures are also expected to over estimate volumetric contraction in distortional loading in such a problem. However, the error on this account is expected to become smaller with deformation. To the knowledge of the writer none of the existing interpretation procedures for analyzing SBPMT in granular deposits accounts for inherent anisotropy. 2.7.3.2 Stress Path Dependency The model parameters estimatedfrominverse modeling of SBPMT may not be directly applicable in a problem that is not of the plane strain type unless the stress-strain relationship is capable of modeling stress path dependency. For instance, the friction angles obtained by Juran Chapter 2: Self-boring pressuremeter testing ofsand 38 and Mahmoodzadegan (1989)froman analysis of a laboratory experiment similar tofieldSBPMT were higher than the corresponding triaxial values. Since the stress-strain relationship used in this study does not model stress path dependency, such a difference is as anticipated. Similar difference is expected in the results obtainedfromany other interpretation procedure that uses a two dimensional stress-strain relationship (e.g., Salencon, 1969; Hartman, 1974; Hughes et al., 1977; Carter et al., 1986; Manassero, 1989; Sayed, 1989; de Souza Coutinho, 1990; Yu, 1992; Fahey and Carter, 1993) to analyze SBPMT data. Results from these procedures are directly applicable only if the deformation problem is of the plane-strain type. On the other hand, Salgado and Byrne (1990) analyzed SBPMT employing a stress-strain that internally accounts for strain, stress and stress-path dependency. Their results are therefore applicable in a deformation problem involving distortional loading even if plane strain condition is not guaranteed. However, Salgado and Byrne did not account for inherent anisotropy in soil behavior. 2.7.3.3 Small Strain Assumption In order to ascertain the extent of approximation involved with the admission of strain displacement equations applicable in a small deformation problem Yu (1992) calculated the pressure expansion curves for cylindrical cavity expansion with and without using the assumption as shown in Fig. 2.7. A linear isotropic elastic perfectly plastic stress-strain relationship assuming Mohr-Coulomb failure criterion was used in these analyses with the effective stress friction angle of 30° and dilation angle of 0°. The results show that for the same set of model parameters, the calculated response of an expanding cylindrical cavity is softer in an analysis allowing large deformation than the corresponding results assuming deformations to be small. For the results presented in Fig. 2.7, the solution pertaining to large deformation diverges from that for small 39 Chapter 2: Self-boring pressuremeter testing of sand 5 4 O H 2 1 0 2 4 6 Cavity Strain, % 8 10 Fig. 2.7. Comparison of Closed Form Small and Large Deformation Solutions to Cylindrical cavity Expansion (adapted from Yu, 1992) deformation at cavity strains above 3%. However, the difference on this count does not exceed 4% for cavity strain less than 10%. As will be shown later, conclusions on this account is similar for a more realistic stress-strain relationship developed in this research. 2.7.3.4 Rigid Plastic Assumption Many of the interpretation procedures for SBPMT data neglect the elastic deformations altogether (e.g., Manassero, 1989; de Souza Coutinho, 1990). In another popular method for analysis of SBPMT due to Hughes et al. (1977) the elastic deformation in material undergoing plasticflowis neglected. The effective stress friction angle is consistently under estimated in such a procedure (Ajalloeian and Yu, 1996). Chapter 2: Self-boring pressuremeter testing of sand 40 2.7.3.5 Stress and Strain Level Dependency The stress dependency in soil behavior is accounted for in some of the existing interpretation procedures, e.g., Hartman (1974), Salgado and Byrne (1990, 1991), and Fahey and Carter (1993). However, none of the procedures based on closed form solutions to cylindrical cavity expansion accounts for stress or strain level dependent stress-strain behavior. Manassero (1989) and de Souza Coutinho (1990) account for strain level dependency but these procedures are based on stress level independent stress-strain relationships. 2.7.3.6 Plane Strain Assumption Cavity expansion in a non-homogeneous deposit cannot in general be analyzed as a plane strain problem. The plane strain assumption is inappropriate also in an undulating topography. Although such a problem can probably be interpreted if a three dimensional stress strain relationship (e.g., Salgado and Byrne, 1990, 1991) is used following well established principles of non-linear inverse theory (see, e.g., Parker, 1994), such a procedure needs extensive computation and may thus be too expensive to be financially viable at present. 2.7.3.7 Numerical Stability Manassero (1989) and de Souza Coutinho (1990) used explicit finite difference to solve the governing partial differential equation. The fact that explicit finite difference procedures are not unconditionally stable irrespective of the step size can be viewed as a major limitation of these procedures. Adoption of a smoothing strategy (e.g., filtering, running averaging orfittingthe SBPMT data to a polynomial) is therefore a must before attempting to derive the element mechanical behavior from SBPMT. Unlike Manassero, who uses afirstorder forward difference Chapter 2: Self-boring pressuremeter testing of sand 41 approximation for the derivatives, de Souza Coutinho uses a fifth or sixth order Runge Kutta scheme leading to much smoother results. However, a fifth or sixth order refinement of the derivatives may not guarantee a commensurate accuracy at the beginning of the stepping process. On the other hand, the explicitfinitedifference computer program FLAC ensures numerical stability by adopting very small time steps. 2.8 Summary To extract undisturbed samples for laboratory element tests retaining the in-situ fabric, which has a significant influence on the mechanical behavior of soils, can be prohibitively expensive particularly in granular deposits. Therefore, the exercise of calibration of a stress-strain relationship from laboratory tests on undisturbed specimens of granular soil to predict the deformation response is usually not a viable option. The existing procedures for estimating deformation response of granular soils from in-situ tests such as CPT and SPT are very economical. Parameters for very simple stress strain relationships such as isotropic elastic perfectly plastic Mohr-Coulomb model can be estimated from index measurements in these tests using empirical correlations. Due to empiricism the results can be very imprecise unless site, material and problem specific correlations are developed. Once development of such specific correlations is undertaken, the procedures based on CPT and SPT are no longer economical. Cavity expansion tests using a self-boring pressuremeter provide reliable stress-deformation data over a wide range of deformation. The measurements are representative of undisturbed soil if the probe is installed in the deposit without causing an appreciable disturbance. Thus, there is a potential of developing a procedure to calibrate a constitutive model to predict deformation behavior of soil over a wide strain range from SBPMT. Chapter 2: Self-boring pressuremeter testing of sand 42 A number of procedures are in fact available to derive the stress-strain behavior of a soil element over a wide range of deformation from back analysis of self-boring pressuremeter tests (e.g., Manassero, 1989; and de Souza Coutinho, 1990). However, due to the limitations in the postulated stress strain relationship, these procedures are applicable only • in a drained plane strain deformation problem, and • in a problem where the major principal stress is perpendicular to the depositional direction. As a result, these procedures usually under predict soil strength. Given the limitations of the existing procedures for interpretation of SBPMT data, there is a necessity to develop a realistic method for estimating the deformation behavior of soil using the self-boring pressuremeter. The main challenge in developing such a procedure is in devising a stress strain relationship that is • capable of capturing soil behavior irrespective of the stress path and stress level, • based on a number of parameters that are physically based and only depend upon material characteristics and state variables, and • for which there is a comprehensive a-priori knowledge regarding the bounds of values of a majority of the model parameters. Any single stress-strain relationship that the writer is aware of does not fulfill the criteria listed above. A constitutive relationship is developed later in this research by combining a number of available schemes for describing soil behavior after modifying them to improve their capabilities wherever appropriate. 43 CHAPTER 3 NUMERICAL METHODS FOR DEFORMATION ANALYSIS 3.1 Introduction To use a stress-strain relationship capable of accounting for stress, strain, stress-path and inherent anisotropy, the equation of motion (or the equilibrium equation in a static problem) and compatibility equations need to be solved numerically to calculate the deformation response of the medium. Following a short description of the problems at hand, two commonly used numerical procedures, viz., explicit finite difference and implicit finite element, are briefly discussed. The complications in a numerical analysis of a deformation problem involving nearly incompressible material (e.g., that pertaining to undrained loading) is examined later together with the available strategies to tackle such a problem. 3.2 Numerical Models Two types of deformation problem of a continuum are solved later in this research: calculation of the axisymmetric or plane strain stress-deformation response of a single element for a given set of model parameters and SBPMT pressure expansion response. The numerical models used in this research for tackling these problems are described below. As mentioned earlier, the exercise of calibration essentially involves refinement of the initial estimates (guesses) of model parameters by comparing the results of the calculated response to that observed in an actual test. Chapter 3: Numerical Methods for Deformation Analysis 44 3.2.1 Calculation of Response of a Single Element A possible finite element or finite difference discretization for both axisymmetric and plane strain elements is as shown in Fig. 3.1. While simulating the plane strain test, no deformation is allowed in the "z" direction while "z" is treated as the hoop direction in simulation of triaxial tests. Before the application of appropriate velocity at the nodes of the element, the element is brought to an equilibrium under the state of stress at consolidation. Both in plane strain and triaxial compression tests an equal velocity is applied to the top two nodes in the negative "y" direction. Velocity in the positive "y" direction is applied in simulation of extension tests. 3.2.2 Calculation of Response of Expanding Cylindrical Cavity Calculation of the response of an expanding cylindrical cavity essentially involves determination of the deformations of the elements such as that shown in Fig. 3.2. Coordinate directions x, y and z are aligned to the radial, vertical and hoop directions, respectively. Although shear stress T^is neglected in the computation for simplicity, it should be noted that such a stress may not necessarily vanish in an SBPMT. However, in a usual analysis of cylindrical cavity expansion o ', o ' and o ' give the three principal values of effective stress. Axisymmetry requires x y z Applied Velocity Fig. 3.1. Discretization Schemes for Simulation of Laboratory Element Tests 45 Chapter 3: Numerical Methods for Deformation Analysis Fig. 3.2. A Typical Element in an Axisymmetric Problem no deformation in the hoop direction. Plane strain conditions in the vertical direction allows no deformation in that direction, essentially leaving a one dimensional deformation problem in which only radial deformation is possible. The equation of equilibrium if the expansion process is slow for which the inertial effect is negligible, is given by do 7dx+(o '-o ')/x = 0 x x (3.1) z A numerical solution to the equilibrium equation ensuring compatibility of displacements for a given stress-strain relationship (with an assumed set of model parameters) and boundary condition gives displacement histories at several locations within the domain of interest. The main interest in solving the problem is to determine the effective cavity pressure for a radial velocity applied at the cavity wall. Alternatively, the displacements are computed for a value of the effective cavity pressure applied at the nodes adjacent to the expanding cavity. To solve the equilibrium equation numerically techniques such as finite element or finite difference can be used. These numerical methods are briefly described in the following sections. At the beginning of the deformation process in a cylindrical cavity expansion o ' is the y largest of the three principal stresses (in this study compressive stresses and strains are considered Chapter 3: Numerical Methods for Deformation Analysis 46 positive) if the coefficient of earth pressure at rest, KQ, is less than unity in case of a test carried out at a level site. However, the cavity pressure increases quickly and o ' becomes the major principal x stress in a nearfieldelement, and o 'and af become the minor and intermediate principal stresses, y respectively. The boundary conditions in the radial direction are o '=Pi at the cavity wall and x o '=o ' at a great distance from the cavity, where P and o ' are the effective cavity pressure and x H ; H the effective horizontal geostatic stress, respectively. In this study afixedouter boundary at a radial distance large enough to be of any significant influence on the states of stress and deformation at the cavity wall is adopted to simplify the problem. By taking the outer boundary at a great distance from the expanding cavity it is ensured that the stress-deformation response at the cavity wall is not affected by the postulated boundary conditions for a medium that contracts at small deformations and dilates as the deformation becomes larger. It is also assumed that the cavity expands as a right circular cylinder, i.e., the exercise is treated as a plane strain problem. The number of elements used in the horizontal direction should be large enough so that the distribution of stress and deformation in the entire domain is captured to the desired accuracy. Only a single layer of elements in the vertical direction is considered. As pointed out by Salgado and Byrne (1990), a more realistic approach would be to simulate the problem considering a number of elements such as that in Fig. 3.1 in the vertical direction to analyze a domain extending to a great distance above and below the expanding cavity. The horizontal boundaries may be treated as a stress boundary or afixedboundary in such a simulation depending upon how far the domain extends in the vertical direction. However, as pointed out earlier, natural variability of soil in thefieldwould render such a strategy too cumbersome to derive a meaningful inference from an SBPMT. Chapter 3: Numerical Methods for Deformation Analysis 47 3.3 Explicit Finite Difference A numerical tool implemented in computer code FLAC (Cundall, 1992) based on explicit finite difference is used in this study extensively. The formulation uses Updated Lagrange scheme together with Jaumann rate for stress (such an approach is sometimes referred to as the ULJ method: see, e.g., Bathe, p. 302) to solve a large deformation problem. The domain under consideration is discretized into four node quadrilateral elements. Each of these is treated as an assemblage of two three node triangles. The equation of motion, i.e., force is equal to mass times the acceleration, is solved for each node instead of the static equation of equilibrium, e.g., Eq. (3.1). The contribution to the nodal body force, B; , for a triangular sub-element "k" connected k to a given node is obtained from B k ; = o (n/ )SO) n ( )S( >)/2 1 ij 2 + j 2 (3.2) where Oy' is the Cauchy stress nj is the unit normal to a particular side of the triangle that connects to the node and S is the length of the side of the triangle. Superscripts "(1)" and "(2)" are used to denote the sides of a triangular sub-element that connects to a particular node contributing thereby to the nodal body force. All the contributions from the triangular subelements connecting to a certain node are added to obtain B . To ensure a symmetric displacement field in a problem { involving symmetric loading, two nodal body forces are calculated assuming two pairs of overlay triangles for each quadrilateral element. The contributions of these overlay triangles are similarly added for the node in consideration, as was done for the original set of triangles to obtain B . The ; nodal body force, B , isfinallyobtained by averaging the contributionsfromtwo overlay sets of ; triangles as Chapter 3: Numerical Methods for Deformation Analysis B =B ; i + 48 B /2 (3.3) : The applied forces at stress boundaries, S , are calculated in a similar manner only considering the ; nodes in the stress boundary. The gravitational force, G is obtained by lumping one third of the i5 masses of the triangles connected to a node. The damping force, D for a set of triangular sub k ; element is calculated assuming damping force to be proportional to the rate of change of kinetic energy from the following expression D, = - a l B ^ x f s i g n o f u U ^ ) (3.4) k where u|,. At/2 is the nodal velocity in direction "i" at time t-At/2, and a is a user defined constant. A dot on top of a symbol is used to denote time derivative. In a static problem a value of 0.8 is used for a to avoid oscillations. The total nodal force due to damping in direction "i," D , can now ; be calculated by summing D * over k for the node in consideration. D; is similarly determined for the overlay triangular sub-elements andfinallyD is obtained by averaging D and D . The nodal ( ; ; body forces are added to the nodal forces due to externally applied loads, gravity and damping to determine the nodal force F . The equation of motion can now be solved by approximating it as ; follows u,| Mt+At/2 where u |, +At/2 * ul Mt-At/2 +F. At/m (t) ' (3 s) ^ - ' J J is the velocity at time t+At/2 and m is thefictitiousnodal mass in case of a static problem. It should be noted that the nodal displacements are determined considering one node at a time. To guarantee no loss of accuracy in such a procedure, the time step, At, should be less than the smallest natural period of the system divided by TZ. Instead of adopting thisrigorousbut Chapter 3: Numerical Methodsfor Deformation Analysis 49 impractical approach FLAC assumes the time step to be equal to the time taken by the elastic compression wave to go across the shortest dimension of any element within the grid. The fictitious nodal mass in static analysis is determined such that At=l. Eq. (3.5) is again integrated to determine the nodal displacements at time t+At/2. Eq. (3.5) isfirstorder accurate since it uses velocity at instants of time that are shifted by At/2 from those for displacement and force. After the velocities and displacements for all the nodes have been determined, the nodal coordinates are updated if desired and the velocity-strain rate equations such as are used to determine the strain rates at the next instant of time. A comma in the subscript denotes a partial spatial derivative using spatial coordinates at the same instant of time (e.g., ,i = 3/dXj) unless mentioned otherwise. The partial derivatives in Eq. (3.6) are determined assuming linear variation of velocity between nodes. From the strain rates, the Jaumann rate of stress, 6$, is obtainedfroman appropriate stress-strain relationship. The corresponding Cauchy stress is then computed using °ij = V K V " ° i k k j ) w (3.7) where w u = (V^i) 7 2 (-) 3 8 The advantages and limitations of a procedure based on explicitfinitedifference (or finite element) are as follows. • Since the displacements arefrozenwithin a time-step (by virtue of ensuring that the elastic Chapter 3: Numerical Methods for Deformation Analysis 50 compression wave does not cross the smallest element dimension in the discretized grid), material non-linearity is accounted for without iteration at the nodal coordinates. • Since the global system of linear equations is not assembled, the procedure is economical in terms of computer memory requirement. • Large deformation is accommodated without much extra effort. • It can be used to model a physically unstable system. However, the procedure is inefficient particularly in situations where there is a large difference between the smallest and the largest natural periods of the physical system. Another problem is with the use of the Jaumann rate of stress in FLAC as explained in the following. A stress-strain relationship is difficult to formulate for large strain even for materials exhibiting simple mechanical response addressing all the unresolved issues (see, e.g., Naghdi, 1990; Rubin, 1994). Moreover, even if such a formulation can be developed it is difficult to devise an appropriate test to calibrate such a model given the fact that significant non-homogeneity develops in a test specimen of granular soil at large deformation. Some authors (e.g., Kleiber, 1989, p. 69) are therefore of the opinion that the constitutive relationships are not usually expected to accommodatefinitedeformations internally. In such a situation it can be argued that the Truesdell rate of the Cauchy stress, is a more accurate and convenient stress rate measure. The Truesdell rate of the Cauchy stress can be calculated from d U = °"ij" °ik j, - u o + o ^ u k i>k kj (3.9) and relates to the velocity strain rate by °ij = [ V - K j. 6 + ° j k i . i. jk 8 +0 S + 0 A) J/2}*u (3.10) Chapter 3: Numerical Methods for Deformation Analysis where D ep ijk] 51 is the elasto-plastic stress strain tensor for small deformation that transforms the Cauchy stress rate into the velocity strain rate and 8^ is the Kronecker delta. The determinant of the deformation gradient, J, can be shown to be equal to the ratio of the material density at the reference configuration to its current value. However, as pointed out earlier, for analysis of cylindrical cavity expansion over the range of deformations of interest (the cavity strain is normally less than about 10% in field tests) the difference between the solution allowing for large deformation and that assuming small deformation is small. Therefore the difference in results adopting the Jaumann or the Truesdell rate is not very significant. 3.4 Implicit Finite Element The implicit finite element method probably finds a wider application in numerical analyses in geotechnical engineering than the explicit finite difference approach described earlier. A brief description of a large strain formulation is as follows. In the updated Lagrange framework, the virtual work principle leads to the following integral form of the equilibrium equation (Bathe, 1982, p. 337) (3.11) where S (t+Atfj is the second Piola Kirchoff stress tensor at time t+At considering the configuration at time t as the reference, i.e., S (t At)ij + = ( ^ i ^ t ^ m J V ^ t ^ t M ^ t ^ n J P w / P ^ A t ) (3-12) Chapter 3: Numerical Methods for Deformation Analysis and the Green strain at time (t+At), e = Kt)i t At)ij + / a x (t)j (t+At)ij + a u 52 , is given by (t)j / a x ( »)i + 5 u (t)k / a x (t)i (t)k 5 u / a x ( (3.13) t)j) In Eqs. (3.11), to (3.13) subscripts within parenthesis are used to denote the time at which the quantity is derived, and x and u denote the coordinate and displacement of a material point respectively. Using incremental decomposition for stress and strain, i.e., S (t A.)ij = ( « ) u 0 + + s (t)ij a n d e (t)ij = ' * m * m + ( where over bar is used to indicate the non-linear part of Green strain, i.e., e linear part is given by e (t)iJ (t)jji (t)kii u (t)kJ 1 4 m m ) , while the by D ^ , ^ and 6e by 5e = (u +u )/2. Approximating S i m =u (t)ij 3 (t)ij the following linearized equation is obtained f ^ U ^ H n ™ * + («) v / V J v " ^(t+At) J °(ij) O C 8 ( A V P K ) ) (.) (t At)ij Q V + (t At) + (p-o V (3.15) where superscript within parenthesis indicate iteration number. Symbols A and 5 denote finite change and variation in a particular quantity and V denotes the element volume. 9t denotes the actual external virtual work, thus the right hand side of Eq. (3.11) represent the out of balance virtual work. The external virtual work can be approximated as 9t( t A t ) * + / f B (P-1) V,(t*At) (t At)i + 6 u (t At)i + d V (t At) + + / f S (t At)i + 8 u S (t At)i + d S (t At) <; (P-1) °(t*At) where f and f are applied body forces and surface traction, respectively. B s + ^.16) Chapter 3: Numerical Methods for Deformation Analysis 53 Now selecting an appropriate discretization scheme and interpolation functions to give approximate values of thefieldvariables (e.g., displacement) within an element and invoking the principle of virtual displacement for each nodal point for an element, a system of linear equations can be formulated. In a static problem, these equations assume the following form ( (') K + K («)) A U ( P > = R *.*r fi> F (317) ( p : i > where K and K are the linear and non-linear element tangent stiffness matrices, respectively, and R is the vector of applied external nodal loads and F is the vector of nodal forces equivalent to the element stress. The general form of these matrices and vectors are as follows K (0 = / V ® » D B d v o>. K (o = / S (.) TD (t) (t>B dV (t) (t) c.) v v and F (t) = /B v (o T (t) o dV (t) (t) (3.18) Matrices B and B are linear and non-linear strain displacement matrices. The exact form of these matrices depend upon the type of element used in the analysis and the algebraic form of the interpolation functions. D is the incremental stress-strain matrix representing material behavior and o is the vector of Cauchy stress. Superscript "T" is used to denote transpose of a matrix. After formulating the element stiffness matrices and load vectors, the equations are assembled to form a set of linear equations representing the entire domain at each iteration. The set of linear equations in unknownfieldvariables such as displacement for the entire domain is also referred to as the global stiffness equation. These are solved at each iteration to determine an estimate of nodal values of thefieldvariables of interest. From the estimated displacements, the strain and stress distribution across the domain is computed using the strain displacement Chapter 3: Numerical Methodsfor Deformation Analysis 54 relationship. The constitutive stress-strain relationship is then used to update the tangent stiffness matrices if needed. The global stiffness equation is reformulated and the estimates of the field variable of interest are refined. After refining the estimates to a specified tolerance,finalestimates of the stress and strain distribution and tangent stiffness matrix for thefirstiteration of the next time (or load) step is computed. The analysis can now proceed to the next time (or load) step. As mentioned earlier, the assumption thatfinitedeformation is adequately modeled by the matrix D is usually approximate. As pointed out in § 3.3, a more realistic approach may be to assume that D only represents small deformation material response. Under such an assumption the Truesdell rate of Cauchy stress becomes a better measure of stress rate than the Jaumann rate. Expressions similar to Eq (3.18) using Truesdell rate of the Cauchy stress can be derived (see, e.g., Klieber, 1989). 3.5 Modeling Incompressibility Often in a deformation analysis, the behavior of a nearly incompressible material needs to be simulated, e.g., a problem involving undrained loading of granular soils. In such a case, spatial discretization is sometimes not adequate even if it meets the convergence and completeness criteria. The difficulty is explained in the following with the example of an eight node (serendipity) element used in an undrained simulation of an axisymmetric problem. The conventional interpolation functions for such an element are u = Cj+c x + c y + c x +c xy+c y +c x y + c xy 2 2 3 2 4 5 2 6 7 2 8 v = c + c x + c y + c x +c 3xy+c y + c x y + c x y 2 9 10 11 12 2 1 14 2 15 16 2 ( 3 1 9 ) where C!,c ,...,c are unknown coefficients that are functions of geometry and deformation. 2 16 Chapter 3: Numerical Methods for Deformation Analysis 55 Coordinates x and y are assumed to be normal and parallel to the axis of symmetry and the corresponding displacements are given by u and v, respectively. Substitution of the constant volume condition for a small deformation problem, i.e., du/dx+3v/c)y+u/x=0 into Eq. (3.19) yields (3.20) If Gauss Legendre quadrature of the third order is used, Eq. (3.20) needs to be independently satisfied at nine different location within an element. This leads to nine independent constraints, viz. (2c +c ) = (3c 2 11 4 + C 15 = C c ) = (c +c ) = (3c 13 8 = 5 C l = C 14 3 = 7 C 6 = 0 (3.21) Noting that the number of degrees of freedom, f , for afinemesh entirely composed of elements E such as these is six (f = sum of the internal angle subtended at each node in radian divided by TT), E eight node rectangles in axisymmetric undrained analysis leads to an over determined problem and leads to "mesh locking" or "excessively stiff' simulation (Nagtegaal, et al., 1974). Analysis for a plane strain problem gives six independent constraints, c.f, Eq. (3.21). Thus in such a problem involving undrained loading an eight node rectangular element is adequate. Since there are nine degrees of freedom in a problem in which one eight node rectangle is used to determine the element behavior in a triaxial test (Fig. 3.3), such a scheme becomes barely adequate. For a more general problem, the issue is usually addressed by adopting a higher order element and/or as suggested recently by Yu and Netherton (1996), by using special interpolation functions. Since, the matrix formulation following Yu and Netherton (1996) calls for a number of Chapter 3: Numerical Methods for Deformation Analysis Applied' Velocity J i t 3 5,6 a I3> 56 Number .Identifying Nodal Degree(s) of Freedom 7,8 Node »»}>»»»>/» t/tf»}//M> Fig. 3.3. Use of Eight Node Rectangle to Simulate Triaxial Element Test modifications to the algorithm and the procedure is only developed for axisymmetric problems, it cannot be used at present in a general application. Marti and Cundall (1980) suggest a "mixed discretization" procedure to tackle the problem described above for a four node quadrilateral element that is essentially composed of a pair of three node triangular sub-elements, e.g., those used in FLAC. In mixed discretization, the isotropic strain and stress are computed for the quadrilateral and the strain and stress deviators are computed independently for the two triangular sub-elements. Mixed discretization scheme is incorporated in FLAC. Therefore, undrained triaxial element tests can be simulated using FLAC without compromising accuracy. 3.6 Summary Commercial numerical tools incorporating the algebra described in the preceding sections are available that easily accommodate user defined stress-strain relationship. Computer program FLAC can be cited as an example. Although such a tool can be used to estimate the deformation response of a single element or the response of cylindrical cavity expansion for a given set of model parameters, use of FLAC to solve these simple deformation problems can be viewed as an under- Chapter 3: Numerical Methods for Deformation Analysis 57 utilization of a versatile tool. Therefore, in addition to setting up a numerical model for computing deformation response of a single element and cylindrical cavity expansion using FLAC for a stressstrain relationship developed in the next chapter, two simple computer programs are also developed in this study. These programs are based on implicit finite element and can be used to solve the cylindrical cavity expansion problem and to calculate the stress strain response of a plane strain element for a user defined stress-strain relationship. 58 CHAPTER 4 STRESS-STRAIN M O D E L 4.1 Introduction A constitutive model for inverse modeling of virgin loading data from self-boring pressuremeter tests is proposed in this chapter. In addition to the capability of capturing the major features in the observed soil behavior using as few parameters as possible, such a relationship needs to fulfill the following requirements • invariance under superposed rigid body motion (i.e., the formulation should be based on stress invariants of the effective stress tensor), • should satisfy energy balance, and • plasticity formulations must include the possibility of neutral loading to satisfy uniqueness (Mroz, 1980; Mroz and Zienkiewicz, 1984). To minimize impact of non-uniqueness in the inverse modeling problem of SBPMT, a reasonable knowledge about the bounds of values of the model parameters is also necessary. It should also be noted that there is no rigid body motion in axisymmetric cavity expansion, e.g., pressureexpansion tests using a self-boring pressuremeter. Nevertheless, for a general application of the stress-strain relationship invariance under superposed rigid body motion needs to be ensured. The major features in the stress-strain behavior of cohesionless materials are reviewed first, following which details of the proposed model are taken up. Factors affecting the model parameters and the range of their appropriate values are summarized. The proposed model is verified using laboratory element tests. The proposed stress strain relationship is implemented Chapter 4: Stress-strain Model ^ through an explicitfinitedifference code, FLAC. Two implicitfiniteelement computer programs are also developed in this study to compute stress deformation response of a plane strain element and that of an expandable cylindrical cavity. The sensitivity of model response for a single element to the choice of parameters, appropriateness of the assumption of small deformation and the advantages and limitations of the proposed stress-strain model are discussed later. The convention followed in the symbolic representation of variables is as follows. Lower case Latin letters, e.g., i, j , k and 1 are used as tensorial indices that can assume values 1, 2 or 3. Unless otherwise mentioned, summation over repeated indices is assumed. Bold face lower case letters are used to denote vectors and bold face upper case letters denote second or higher order tensors. Compressive stresses and strains are considered as positive. 4.2 Assumptions The deformations are assumed to be small (c.f., § 3.3 of this thesis). The total strain increment, de^, is assumed to decompose into a reversible (or elastic) part, de/, and an irreversible (or plastic) part, de/: de = de^de? ;j (4.1) The plastic strain increment is in turn assumed to decompose as follows de? = d e ^ d e ; (4.2) where de/ and de/ are the contributions from the mechanisms modeling the plastic behavior in distortion and hydrostatic compression, respectively. The elastic and the plastic strain increments are calculated using a hyper-elastic formulation (in contrast with hypo-elastic formulations such as Chapter 4: Stress-strain Model ou those discussed in § 2.7.2, in hyper-elasticity energy conservation is guaranteed) and two strain hardening plasticity mechanisms. 4.3 Stress-strain Behavior of Cohesionless Soils Salient features of the observed stress-strain behavior of cohesionless soils are reviewed in this section. Since the constitutive model is expected to capture the major features of the observed behavior, such a review is necessary before a stress-train model can be proposed. The stress-strain behavior over a strain range at which deformation is recoverable and that pertaining to irrecoverable or post yield deformation are considered separately. 4.3.1 Reversible Behavior The stress-strain behavior at shear strains smaller than about 0.001% is essentially reversible and isotropic for several types of cohesionless material (Tatsuoka and Kohata, 1995; Yan and Byrne, 1990; Schmertmann, 1977; and Rowe, 1971) even when post yield material response is anisotropic. To illustrate the isotropy in small strain behavior, the stress-strain response of sand samples at similar void ratio, e, having an anisotropic structure is shown in Fig. 4.1 for different values of 6, the angle between the major principal effective stress, a ', and the bedding plane. (1) Bracketed subscripts are used in the symbols for the principal values of the effective stress and strain tensors to distinguish these quantities from vectors. Both theoretical consideration (Rowe, 1971) and experimental data (Hardin and Richart, 1963; Janbu, 1967; Yu and Richart, 1984) show that the elastic stress-strain behavior of cohesionless soils is affected by the stress level. The small strain stiffness also depends upon the void ratio of the deposit. Several studies indicate (Richart, 1977; Iwasaki and Tatsuoka, 1977; 61 Chapter 4: Stress-strain Model 6 Toyoura Sand # 2 Plane Strain Compression a3'= 150 kPa O C R = 5.33 • 5=45°, e=0.668 • 6=20°, e=0.661 A 5= 0°, e=0.660 0.001 0.003 0.002 Axial Strain, % 0.004 0.005 Fig. 4.1. Small Strain Behavior of Air Pluviated Sand (after Tatsuoka and Kohata, 1995) Kokusho and Esashi, 1981; Alarcon-Guzman, 1989) that for uncemented normally consolidated sands with medium compressibility the small strain shear modulus varies with (2.17-e) /(l+e). A 2 survey of these studies can be found in Roy, 1992 (p. 44). The loading rate on the other hand has only a niinimal effect on the elastic stress-strain behavior of cohesionless materials (Richart, 1977). It is therefore apparent that the small strain behavior of cohesionless soils can be modeled by an isotropic, stress-level dependent elastic formulation. 4.3.2 Post Yield (Irreversible) Stress-strain Behavior Stress-strain response of soils is largely irreversible at shear strains larger than about 0.001%. Such a response is referred to as post yield behavior and is modeled in this study by an elasto-plastic formulation. The main features of the post yield stress strain behavior of cohesionless soils are examined in this section. The response of the soil skeleton in distortional deformation is examined first. The behavior in isotropic compression is considered later. Chapter 4: Stress-strain Model e>z 4.3.2.1 Skeleton Response in Distortion Application of shear stress without changing the mean normal stress usually leads to the development of a significant volumetric deformation in cohesionless deposits when drainage is allowed. Such a behavior is primarily due to intergranular rearrangement and is not recovered upon removal of the shear stress. Volumetric deformation associated with the application of shear stress, commonly referred to as dilatancy, is illustrated in Fig. 4.2, which plots the deviator stress, i.e., the difference between the vertical and the horizontal stress, a , and the trace of the strain d tensor (which is equivalent to the volumetric strain at small deformation), e^, against the maximum Fig. 4.2. Drained Distortional Behavior of Sand (after Vesic and Clough, 1968): e void ratio after consolidation CONS is the 03 Chapter 4: Stress-strain Model shear strain, (=e -e ). e and e are the major and minor principal values of By. These data (1) (3) (1) (3) were measured in a series of drained triaxial tests on dry tamped Chattahoochee River sand carried out at constant effective mean normal stress (Vesic and Clough, 1968). Due to dilatancy, loose specimens behave in a contractive manner irrespective of whether the shear stress is small or large in comparison with the mean effective stress. In contrast, dense samples usually exhibit a contractive volumetric deformation at small shear stress while the specimen dilates at larger values of shear stress. An increase in the effective mean normal stress leads to an increase in the volumetric contraction for cohesionless soils. As indicated earlier, whether the behavior of a soil specimen is contractive or dilative depends on the magnitude of the shear stress relative to the mean effective stress. At small values of mean effective confining stress, p', loose samples may exhibit dilative behavior, while at large values of p' even dense specimens can exhibit contractive deformation when sheared. Micro mechanical studies of granular material (Taylor, 1948; Rowe, 1971; Matsuoka, 1974) show that whether the behavior is contractive or dilative is governed by a scalar measure of the ratio of shear stress to the normal stress. Such a measure is usually referred to as the stress ratio. Stress ratio in a two dimensional state of stress for Mohr-Coulomb material is usually defined as (o '(1) a (3)'y( (i) (3)')> where o ' is the minor effective principal stress. In a three dimensional state a /+0 (3) of stress this simple definition of stress ratio neglects the effect of the intermediate effective principal stress, o ' in its usual implementation. Below a critical value of the stress ratio, material (2) behavior is contractive and above the critical value a dilative response is observed. This critical state of stress is usually called the phase transformation (PT) state. Chapter 4: Stress-strain Model 04 4.3.2.2 Yield and Failure in Distortion Roscoe et al. (1963), and Poorooshasb et al. (1966) demonstrated experimentally that shear induced volume change occurs as the stress ratio exceeds its previous value. Since volume change due to dilatancy is essentially irreversible or plastic, stress ratio can provide a scalar measure for yielding for frictional materials. Consequently, the lines of constant stress ratio can be used as loading surfaces for sand in distortion. In stress-space plasticity, the loading surface is a surface in the six dimensional stress space defined in such a manner that yielding occurs when the stress increment has a non-zero component along the gradient to this surface. Also by definition, an admissible state of stress can only be on or inside the surface. Laboratory tests on granular materials also indicate that the plastic strain increments are not normal to the constant stress ratio lines, i.e., loading surfaces (Poorooshasb et al., 1966; and Tatsuoka and Ishihara, 1974). To ensure that an admissible state of stress can only be on or inside the loading surface, a hardening rule is introduced. The hardening rule allows the loading surface to evolve in terms of its size and location. Test data show that the loading surface for sand does not evolve indefinitely in pure distortion. In fact, there is an ultimate configuration of this surface. The loading surface remains unaltered once this configuration is reached. This ultimate configuration is usually referred to as thefailure surface. The failure surface can be evaluated by adopting an appropriate failure criterion. The Mohr-Coulomb failure criterion is often used in modeling soil behavior. According to this postulate, failure is triggered when (o '-a ')/(a '+a ') w Q) m (3) reaches a critical value, which is a material property. In reality, however, (o '-o ')/(o , +o ') depends also upon external factors / (1) (3) ( ) (3) such as stress path and can reach a higher value in plane strain than that in a triaxial test (see, e.g., Lee, 1970). As a result, a stress-strain model based on this postulate cannot capture stress-path dependency in soil behavior adequately. bS Chapter 4: Stress-strain Model Matsuoka (1974) extended the Mohr-Coulomb failure criterion to include the effect of o ' (2) on soil behavior. According to this postulate, failure is triggered when the ratio of normal to shear stress on the spatial mobilized plane (SMP) reaches a material specific value. The SMP is a material surface on which the particles are mobilized to a maximum extent on the average. More details on the concept of SMP will be provided in a later section. Lade and Duncan (1975) also developed an empirical failure criterion for a general three dimensional state of stress. Discrete element studies (Chang, 1993) and experimental data (Fig. 4.3) show that both Lade and Duncan (1975) and Matsuoka (1974) failure criteria performs rather satisfactorily for all stress paths. However, the Lade and Duncan criterion predicts a higher peak effective stress friction angle [d/ = the maximum value of sm {{a' -a' ^l(a' +a'f^^)}] l (l) Q (l) in triaxial extension than in triaxial compression. Due to a lack of conclusive experimental evidence to this effect the Matsuoka (1974) criterion is adopted in this study. For mathematical convenience, lines of constant stress ratio on the SMP are also used in this study as loading surfaces in pure distortion. 4.3.2.3 Material Anisotropy in Distortion Park and Tatsuoka (1994) carried out a series of drained plane strain compression tests on air pluviated sand samples isotropically consolidated to an effective stress of 78.5 kPa. As shown ATXC \S. y - Matsuoka Mohr-Coulomb ((J)'= 40°) NX" /— Lade-Duncan \C ^ - T X E • Air Pluviated Toyoura Sand (Lam and Tatsuoka, 1988) Fig. 4.3. Comparison of Three Failure Criteria in n Plane Chapter 4: Stress-strain Model oo in Fig. 4.4, stiffer response is observed for higher values of 6 due to anisotropy in material fabric. Several other studies employing sample preparation technique such as air or water pluviation show a similar behavior (e.g., Vaid et al., 1996; Park and Tatsuoka, 1994; Vaid et al., 1990; Mulilis et al., 1977; Lam and Tatsuoka, 1988; Arthur and Menezies, 1972; and Oda, 1972). The fabric effect is gradually destroyed with increasing deformation. However, the deformation at which the fabric effect becomes negligible is not clear. For instance, Been and Jefferies (1985) found that the fabric effect is insignificant at shear strains of about 2-5%. The data presented in Fig. 4.4, on the other hand, indicate that fabric effect is significant even at shear strain of 10%. The effect of fabric on the effective stress friction angle in examined later in greater detail. Chapter 4: Stress-strain Model 67 4.3.2.4 Causes of Anisotropy The term fabric is used to refer to the arrangement of particles and pore spaces in the soil (Mitchell, 1993). As demonstrated by Oda (1972) through a series of isotropically consolidated triaxial compression tests, anisotropy in fabric has a pronounced effect on the stress strain behavior of cohesionless materials. This study indicates that air pluviation tends to create a preferred distribution of contact normals along the depositional direction. As a result, the material response is stiffer when the major principal stress is aligned with the depositional direction and the softest response is observed when the major principal stress is perpendicular to the direction of deposition. On the other hand, for a sample prepared by tamping, such a preferred direction is much less pronounced. Water pluviation also leads to an anisotropic sample response (Vaid et al., 1996; Vaid et al., 1990). The anisotropy in fabric for a sample that did not experience a strain history subsequent to deposition is called inherent anisotropy. The spatial distribution of the contact normals tends to get oriented along the major principal stress when sand samples are sheared irrespective of the sample preparation method (Matsuoka, 1974; Oda, 1972). The strain induced anisotropy is called induced anisotropy. From discrete element studies and triaxial compression tests, Ishibashi et al. (1996) demonstrate that the behavior of an isotropic granular material may appear to be anisotropic in a test with a pronounced boundary effect. However, the boundary effect is not significant when the minimum sample dimension is greater than about 150 times the median particle diameter, D . 50 4.3.2.5 Fabric and Stress Path Dependency of Effective Stress Friction Angle As indicated earlier, stress-strain behavior of cohesionless materials is affected by the stress path. For instance, the peak effective stress friction angle, <b', and the apparent stiffness in a Chapter 4: Stress-strain Model os drained plane strain compression (PSC) test are higher than those in triaxial compression (TXC), especially for dense samples (Lee, 1970; and Tatsuoka et al., 1986). However, the relationship between (b' in TXC and triaxial extension (TXE) is not clear. For instance, Tatsuoka et al. (1986), Lam and Tatsuoka (1988) and Shankariah and Ramamurthy (1980) observed about 10% increase in the effective stress peak friction angle as 5 changesfrom90° to 0°. Lade and Duncan (1973), Reades and Green (1976), Green (1971), Ergun (1981) and Haruyama (1981) report an opposite trend. Bishop (1966), Sutherland and Mesdary (1969), and Ramamuthy and Rawat (1973) did not observe a significant difference between <b' in TXC and TXE. The conclusions of Been and Jefferies (1985) were also essentially the same. 4.3.2.6 Stress-strain behavior in Isotropic Compression The typical behavior of loose sand in triaxial compression and isotropic compression is shown in Fig. 4.5. As in a triaxial compression test, only afractionof the measured deformation is recovered upon removal of stress. In other words, like material response in pure distortion, behavior in isotropic compression is also elasto-plastic. However, typically the irreversible volumetric deformation in triaxial compression is considerably larger than that isotropic compression. In isotropic compression, loose deposits exhibit a softer response compared to a denser deposit. Angular poorly graded sands show a higher volumetric compression than a well graded sand of comparable mineralogy but rounded grains (Casagrande, 1965). Also, the deformation in isotropic compression is anisotropic for cohesionless soils with inherent anisotropy (Haruyama, 1981). The effect of anisotropy is more pronounced in case of the loose deposits (Negussey, 1984). Chapter 4: Stress-strain Model 69 Fig. 4.5. Stress-strain Behavior of Loose Sands in Triaxial and Isotropic Compression (after Lade and Nelson, 1987) 4.3.2.7 Undrained Response from Drained (Skeleton) Behavior Undrained response of an element of granular soil can be estimated from the skeleton (drained) response considering the appropriate kinematic constraint due to the compressibility of the pore fluid. However, in an undrained test the potential for the skeleton to dilate is suppressed almost entirely. As a result, the peak friction angle observed in a drained laboratory element test is usually higher than the corresponding undrained value. A boundary energy correction is thus needed to correlate test data with varying degree of drainage. Approximate analytical procedures for boundary energy correction have been proposed by Taylor (1948), Bishop (1954) and Poorooshasb and Roscoe (1961) for various laboratory element tests. These procedures essentially relate the peak effective stress friction angle determined in a drained laboratory element test to that Chapter 4: Stress-strain Model 7U pertaining to the corresponding undrained test. The procedures listed above are not formulated in terms of stress and strain invariants and may not guarantee invariance under superposed rigid body motion. In addition, due to the essential empirical nature of the hardening rule of the stress strain relationship used in this study, an empirical procedure will be used instead. 4.4 Proposed Stress-strain Model for Granular Soils A stress-strain relationship with the following components is proposed: • isotropic hyper-elasticity, • a non-associative anisotropic mechanism for the post yield behavior in pure distortion, and • a mechanism with associative flow rule for post yield behavior in isotropic compression neglecting inherent anisotropy. A strain hardening behavior is assumed to model irreversible behavior: a reasonable assumption as long as the state of deformation within an element is homogeneous (Loret and Luong, 1982). In order to simplify the formulation, it will be assumed further that the two mechanisms describing the plastic behavior do not interact with each other. In other words, strain hardening in the distortion (or the consolidation) mechanism does not necessarily lead to a strain hardening in the consolidation (or the distortion) mechanism. This simplifying assumption may not be very accurate for over consolidated materials (Loret, 1989). However, the interaction is not significant because shear strains due to the distortion mechanism are small in near hydrostatic stress paths. Although irreversible deformation is generated due to pure rotation of principal stresses, no attempt is made here to capture such a behavior. This limitation does not lead to any loss of accuracy when the stress-strain model is applied to cylindrical cavity expansion since in such a problem although there is a jump rotation of principal stresses (e.g., in a cylindrical cavity expansion problem with a K Q less Chapter 4: Stress-strain Model 71 than unity, a jump rotation occurs at the instance when the effective horizontal stress exceeds the magnitude of the vertical effective stress: where K Q is the at rest earth pressure coefficient), they do not rotate continuously. The elastic and plastic mechanisms are described in the following sections. 4.4.1 Modeling Elastic Stress-strain Behavior The elastic strains are evaluated from the following relationship de„' = C ^ ' d O u ' (4.3) where o ' is the effective Cauchy stress and the tangent elastic compliance tensor C H C =G ^ijkl E is given by 2v / e e ijkl ^ 1+V where G is the elastic tangent shear modulus, v is the Poisson's ratio and b represents E i} Kronecker's delta. The matrix form of Eq. (4.4) can be found in Appendix 1. Several hyper-elastic formulations for isotropic materials have been proposed in recent years to model stress dependent unload-reload behavior (Loret and Luong, 1982; Lade and Nelson, 1987; Molenkamp, 1988). These formulations ensure no energy generation or dissipation in closed unload-reload loops. Molenkamp (1988) expresses the tangent elastic moduli (the tangent elastic shear and the bulk moduli, G and K , respectively) as functions of the first invariant of effective E stress, E In contrast, according to Loret and Luong (1982) and Lade and Nelson (1987) the elastic moduli depend on Ij and the second invariant of stress deviator J , where 2 72 Chapter 4: Stress-strain Model Ij = o ' and H J (4.5) = I, -312 2 2 The second invariant of the effective stress tensor, I , is given by 2 i = {(V) -<V°ij'> 2 (4.6) /2 2 Although micro mechanical studies indicate that the exact nature of stress dependence of the elastic moduli depends upon material fabric, experimental data indicate that the elastic moduli are essentially independent of J (Santamarina and Cascante, 1996; Fioravante et al., 1994). In 2 addition, the formulation due to Molenkamp (1988) is almost identical to one of the most popular formulations for stress dependent elastic behavior in geotechnical engineering (Wilson and Sutton, 1948; Mmdlin, 1949). Thus, Molenkamp (1988) is adopted in this study. According to this model, G E is given by G (4.7) = K P [I /(3P )] B E GE A 1 A where K Q (elastic shear modulus number) and n (elastic exponent) are model parameters and P E E is the atmospheric pressure. It follows from the assumption that K E /G E A is constant, v does not depend on a '. The Poisson's ratio is an additional model parameter. The dependence of the small a strain modulus on void ratio, e, is accounted for by modifying the value of K G E during the deformation process in proportion with (2.17-e) /( 1 +e). 2 4.4.2 Evaluation of Elastic Parameters The elastic shear modulus number depends largely on soil type and the state of packing and the elastic exponent % appears to depend on mineralogy for granular materials. The Poisson's Chapter 4: Stress-strain Model l'i ratio usually varies between 0.11 to 0.23 (Hardin, 1978) and shows a slightly increasing trend with void ratio (Lade and Nelson, 1987). However, a value of 0.2 can be assumed for a large number of cohesionless soils irrespective of the void ratio. The elastic tangent shear modulus relates to the shear wave velocity, V , by s (4.8) where p is the total mass density («2 Mg/m ). Now if the state of stress and the value of n are 3 E known, the appropriate value of K G can be estimated from measuring the shear wave velocity by E carrying out a seismic piezocone penetration test. SBPMT is not especially suitable for the purpose because the tool usually can resolve strains no smaller than 0.006% (da Cunha, 1994; Pass, 1994), while the elastic parameters pertain to strains as small as 0.001%. From Eq. (4.8) the following relationship is obtained between the shear wave velocity and parameter K GE (4.9) GE Typical values of the elastic parameters, for level grounds r% and v, for a number of cohesionless soils are listed in Table 4.1. These values can be used in the absence of any material specific information. 4.4.3 A Model for Plastic Distortion Behavior of Isotropic Material Nakai and Matsuoka (1983b) developed an incremental stress-strain model from Micro mechanical studies. This simple model, which is based on an extension of the concept of stress ratio for a general three dimensional state of effective stress, will be used to evaluate the strain increment, de/. The ratio of shear and normal stress on a special plane, called the Spatial 74 Chapter 4: Stress-strain Model Table 4.1. Elastic Model Parameters Reference Location D ,% KQE Ottawa CI09 70 50 30 1260 1080 910 0.5 0.5 0.5 Cunning et al., 1994 Ditto Ditto Quiou Sand 59 53 995 845 0.7 0.7 Fioravante et al., 1994 Ditto Fraser River 38 750 0.5 Massey This study 50 950 0.5 KIDD Ditto 65 30 550 450 0.5 0.5 Cell 24 J-Pit Ditto Ditto Sand type Syncrude Sand R Mobilized Plane (SMP) is considered and as a result the model is often referred to as the SMP model. A little modification of the model makes it applicable for cohesionless soils with inherent anisotropy. The SMP model is based on non-associative strain hardening plasticity and the presence of a loading surface is evident. However, there is no explicit plastic potential function. The SMP is defined by Matsuoka (1976) such that at its intersection with the plane normal to the principal direction "k" of 0%, the ratio, (o '-a ')/(o '+a '), is maximized. The direction (i) (j) (i) (j) cosines, aj, of a unit normal to the SMP with respect to the principal directions are given by where o ' are the principal values of the effective stress tensor and the third invariant of the (i) effective stress tensor, I , is equal to the determinant of the effective stress tensor. The normal 3 (o SMP ) and the shear (x SMP o ) components of the effective stress tensor on the SMP are given by SMP =3yi,, T = I I /I -9l3 /I / s m p V 2 1 3 2 2 2 (4.11) Chapter 4: Stress-strain Model Whence, the ratio o SMP to x SMP '/:> , which is referred to as the stress ratio, r|, can be calculated as from r, = ^ 1 ^ / ( 9 1 3 ) - 1 (4.12) As mentioned earlier, constant r) lines give the loading surfaces, i.e., yielding occurs as (dr)/r3ajj'dOjj') becomes equal to or exceeds zero. A family of such loading surfaces in the effective principal stress space is shown in Fig. 4.6. Since the loading surfaces open out in the direction of the hydrostatic axis, this mechanism does not predict any plastic deformation for a stress path in that direction. Assuming the particles are mobilized to the maximum extent on the average along the SMP, it followsfromMicro mechanical considerations that the ratio d e SMP /dY MP ls a S linear function of rj. The components of the principal values, de , of the strain increment tensor of the mechanism s (i) for distortional plasticity, de£, normal and parallel to the SMP, de SMP and dy , respectively, are SMP given by Fig. 4.6. Loading and Failure Surfaces of the Distortion Mechanism and Stress Paths in TXC andTXE 76 Chapter 4: Stress-strain Model d e d YsMP rSMP " = Lv u c -(l) »2 h - d e-(2) ( 2 ) S «i > l ) ut a SMP 2 2+ + (( = dd e e d e (0 i' S a ((Dl ) h S S d e a (2) n S d The linear functional relationship between d£ /dY MP SMP TI = - X ( d e S M p i) a a n 2 + ( d S /d Y s M P d e ( D «4 S d e (2) S a i) ]° 2 (4.13) 5 is given by the following expression ) +u (4.14) where X and p. are model parameters. Micro mechanical considerations also lead to the following relationship between the component of strain tensor parallel to the SMP, YSMP> YSMP = A N D (4.15) Y exP 0 where u' is a model parameters Yo is given by Y =Y 0 + o i C log(I /I ) D where C is a model parameter and I and Yoi D 1 a r e m (4.16) l i e i; initial values of \ and y , respectively. 0 However the left hand side of Eq. (4.17) does not vanish with r| unless Yo vanishes with n: a condition not guaranteed by the relationship in Eq. (4.18). This difficulty is avoided by adopting an following mathematical form that closely resembles the shape of the TI-YSMP curve derived from micro mechanical consideration (Salgado, 1990) 1 = YsMp/n/Gpi+YsMP^uL-r] where (% is the slope of the r)-Y SMP curve and (4.17) is the value that is approached asymptotically by the stress ratio. Differentiation of Eq. (4.17) yields 11 Chapter 4: Stress-strain Model dr) = G d Y p T (4.18) S M P Gpj can be evaluated from = G (1 - R r , / r i ) PI where K s p F 2 F = ^ ( o ^ / P ^ O -R r,/r| ) F (4.19) 2 F , n and R are model parameters. More details on the derivation of Eqs. (4.18) and P F (4.19) can be found in Duncan et al. (1980), who used a hyperbolic expression mathematically identical to Eq. (4.17) for developing a hypo-elastic stress-strain model. The stress ratio at failure, %, is assumed to depend on I , thefirstinvariant of the effective stress tensor at failure, as follows 1F rj = n F F 1 -Ar|log{I /(3P )} 1F A ( 4 2 0 ) where r| and Ar) are model parameters. From Eqs. (4.14) and (4.18) one gets F1 de = (p-r))/(AG )dr) SMp (4.21) pT Assuming the principal directions of the strain increment and stress tensors to be the same, the direction cosines of de can be calculated from Eq. (4.10). Assuming further that the direction SMP of dy MP to be the same as that of x S de Substitution of d e i SMp + d Y s M P (o ' - O i S M P )/T s m p } (4.22) S s (j) = a {de , the principal values of dcy" can be calculated from and d y M P from Eqs. (4.18) and (4.21) yields S M P de s (i) SMP = a /G i p T {(p-ii)/A (o '-o + i S M P )/T S M P }dT 1 (4.23) Differentiation of Eq. (4.12) and substitution of the results into Eq. (4.23) yields an incremental Chapter 4: Stress-strain Model /» relationship between the principal strain increments and the effective stresses. Since the strain increment tensor in the principal stress space, de/ (= de /6 ), transforms to the strain increment ( s tensor in the coordinate axes via de,/ = N^N^de/, the following incremental stress strain relationship is obtained d e / = Cju'dou' The tensor, (4.24) is used to represent the direction cosine of the principal direction "i" with respect to the j-th coordinate. The matrix form of this relationship can be found in Appendix 1. 4.4.4 Proposed Modification for Inherent Anisotropy The model parameters X and u are not affected by the inherent anisotropy of the granular medium (Nakai and Matsuoka, 1983b). It follows from the discussion in § 4.3.2.5, that the quantity % may also not be significantly affected by inherent anisotropy. Therefore, Eq. (4.19) only needs to be modified to capture inherent anisotropy. Let the direction of deposition is given by a unit vector a, which has components a in the coordinate bases e^ i.e., ; a = c^e, (4.25) Since the coordinate bases transform into the principal bases by n = N^ej (4.26) The angle between the unit normal to the depositional plane and SMP, 0, can be calculated from cos0 = a » a = a ^ N - (4.27) '/y Chapter 4: Stress-strain Model where A=d^i . The components Ny of the transformation tensor in Eq. (4.28) are usually evaluated { by solving for the eigenvalues and eigenvectors of the effective stress tensor. For granular materials with inherent anisotropy, following modification to Eq. (4.20) is proposed Gp,. = G„ (1 - R r, / ri ) = n K 2 F F A s p (o / P ) (1 - R r, / r, ) np SMP A F 2 F (4.28) where the multiplier n depends on 0 and an additional model parameter, m , in the following A A manner for O°<;0< 45° n = 1 - ( m - l)(2cos 0 - 1) (4.29) 2 A A For 0 less than 45° n is set equal to unity. Gpx is updated at the end of each time step depending A on the current value of 0. 4.4.5 Modeling Isotropic Compression Behavior The SMP model would not generate any plastic strain if the stress tensor remains isotropic. However, as mentioned in § 4.3.2.5, such a deformation process may also lead to the development of plastic strain. To evaluate the component of plastic strain, de , associated with isotropic c compression, an isotropic associated plasticity model proposed by Lade (1977) is used. Negussey (1984) demonstrated from laboratory tests that when a specimen of granular soil with inherent anisotropy is subjected to an isotropic state of stress the deformation field is anisotropic. Since the strain increments de are usually quite small compared to de , discrepancy due to the use of an c 8 isotropic model may not be significant. The original isotropic formulation due to Lade is therefore used without modification in this study. The loading surface for this mechanism is given by 80 Chapter 4: Stress-strain Model (4.30) The relationship essentially represents a family of spherical surfaces in the effective principal stress space. The loading surfaces for the distortion (curve AOB) and consolidation mechanisms (curve AB) are shown schematically in Fig. 4.7. The mechanism is triggered as soon as dfc/da-do- equals or exceeds zero leading to the generation of plastic deformation: a process often referred to as isotropic consolidation. Calculation of de beginsfromthe following relationship between c the increment of plastic work associated with this mechanism, dW , and de c c dW„ = a,'del = y o u o..'AH8f /8o ' r (4.31) H where AH is a scalar multiplier. To evaluate AH, Lade used the following empirical relationship W c = CP (f /P j A c (4.32) P A where C and p are dimensionless model parameters. Differentiating Eq. (4.32) and noting that dfc/aoi/doy' = 2fc, the following relationship between de° and do/ is derived / . \ p_1 df af c 2f P c fi Loading Surface in Distortion — c —- ki' d o A Loading Surface in Isotropic Compression Fig. 4.7. Loading Surfaces for Distortion and Isotropic Compression (4.33) Chapter 4: Stress-strain Model Si The matrix form of Eq. (4.34) can be found in Appendix 1. The incremental effective stress-strain relationship can now be obtained from Eqs. (4.1), (4.2), (4.4), (4.24) and (4.33). 4.4.6 Numerical Simulation of Element Mechanical Response The elasto-plastic relationship developed above is used in computer program FLAC to compute the deformation problem for a single axisymmetric or plane strain element or cylindrical cavity expansion. Computed model responses for a single axisymmetric and plane strain elements are compared with the laboratory triaxial and plane strain test data later in this chapter to evaluate the proposed stress strain relationship. An alternative numerical tool based on implicit finite element method is also developed for computing the stress strain behavior of a plane strain element (see Appendix 2 for usage and code listing). A four node quadrilateral element is used. Both drained and undrained response can be computed from FLAC and the implicit finite element program. Appropriate volumetric constraint needs to be considered to derive the undrained response for a given set of model parameters pertaining skeleton response. Naylor (1974) is followed both in FLAC and the finite element code to account for the volumetric constraint corresponding to the compressibility of the pore fluid. To illustrate the procedure the stress in the element is computed from FLAC and the finite element program for isotropically consolidated drained plane strain compression and extension tests as shown in Fig. 4.8. The effective stress at consolidation is assumed to be 100 kPa. The model parameters used in this example are: K<j =720,'n =0;5, v=0.2, K =500, n =-0.5, R =0.92, ri =0.69, ArpO.05, m =2.0, E E SP P F F1 A C=0.0006 and p=0.9. The undrained response for the same model parameters and assuming apparent bulk modulus of pore fluid to be 1 GPa is as shown in Fig. 4.9. Back pressure is assumed 82 Chapter 4: Stress-strain Model 500 1^250 > 9 8 CO Q B CO o u j 1i *- u.u.rfP FLAC o Finite Element D -250 0.80 B i -0.80 -2.5 5.0 0 2.5 Axial Strain, % Fig. 4.8. Computed Drained Response of a Plane Strain Element to be zero in these analyses. The pore water cavitates at an axial strain of about 2.15% in the plane strain extension tests and response at larger deformation is not computed. 500 gga 250 [1 0 CO ° FLAC ° Finite Element -250 250 *5 1 0 OH •250 -5 -2.5 0 2.5 Axial Strain, % 5.0 Fig. 4.9. Computed Undrained Response of a Plane Strain Element Chapter 4: Stress-strain Model 83 4.5 Estimation of Model Parameters for Distortion and Consolidation Mechanisms An ideal procedure for determining the model parameters for the plastic mechanisms is briefly described in this section. Parameters C and p for the consolidation mechanism can be determined in the laboratoryfroman isotropic consolidation test on an undisturbed sample. The appropriate values of the elastic strain increments can be found using the elastic model parameters estimated following the guidelines listed in § 4.4.2. The strain increment associated with the dilatation mechanism, de/, is estimated by subtracting the elastic strain increments, de/,fromthe total strain increment, de . ;j W /P C A is then plotted against fc/P 2 A (=3p /P /2 2 A in isotropic compression) in log-log space for each load increment. The slope of the curve gives parameter, p, while the value of W /P evaluated using Eq. (4.33) at f / P C A c 2= A l gives parameter, C. The parameters for the distortion mechanism can be foundfromany laboratory test, in which n increases and stress and deformation histories are known. Since the triaxial test is probably the most common laboratory test, description of a procedure for model calibration from drained triaxial test data is given here. Calibration procedure for other types of laboratory tests, e.g., simple shear can be found elsewhere (see, for instance, Salgado, 1990). From the estimates of KQE, %, v, C and p, strain increments de/ and de/ are computed and subtractedfromthe total strain increments measured in a triaxial test to obtain de/. It may be noted that in a conventional drained triaxial extension test de/ vanishes. Following results are useful in model calibration from a conventional drained triaxial compression test, in which o ' is held constant: (3) d e / = da 7{2(l +v)G }, de a (1) E e (3) = -vde e ( 1} (4.34) Chapter 4: Stress-strain Model 84 Parameter evaluation of the distortion mechanism begins with the evaluation of histories °f % SMP and YSMP fr° m E t e s t data. Parameters u and X are then evaluated by plotting -de /dY MP SMP S against rj. The slope of the curve gives parameter X, while the value of r| at de /dY MP 0 gives = SMP S the value of p. Since Eq. (4.14) only depends on soil type and is not affected by sample fabric or density, for evaluating u and A testing of an undisturbed sample is not necessary. Triaxial drained test data on reconstituted samples can be used without any loss of accuracy. At least two drained tests are needed at two different effective mean normal stress at consolidation for evaluating the parameters r| and Ar). The stress ratio at failure, i.e., maximum F1 stress ratio, is plotted against the logarithm of the ratio of effective mean normal stress at failure and the atmospheric pressure. The slope of the curve gives parameter Ar) and the value of r) at failure corresponding to I /(3PjJ= 1 gives r) . F1 1F Rearrangement of Eq. (4.18) yields YSMP/ ! 1 = 1/Gp, + YSMP/TIULT- It is evident from this relationship that the intercept of the linear plot between y /r\ and YSMP gives 1/G . Since G SMP PI P I relates to model parameters K , n and n by (recall Eq. 4.28) SP A P G„ = n ^ o ^ / P j ' (4.36) To evaluate these parameters, values of G from at least two triaxial compression tests performed PI on undisturbed samples at different effective mean normal stress after consolidation are plotted against (as^p/P^) in log-log space. The slope of the curve gives the parameter n and the value P of G at o /P =l gives K xn . At least one triaxial extension test on an undisturbed sample is PI SMP A sp A needed for evaluating the parameter characterizing inherent anisotropy, m . Noting that m relates A A to n by Eq. (4.29), evaluation of 0 in triaxial compression and extension at the beginning of virgin A loading allows estimation of K SP and m . A Chapter 4: Stress-strain Model 85 4.6 Existing Information on Plastic Model Parameters As pointed out in the preceding section, two triaxial compression, two isotropic compression and one triaxial extension tests on undisturbed samples are needed for an adequate characterization of a granular deposit at a certain depth. Such an elaborate laboratory testing program is usually not feasible due to resource constraints. Available information often includes data from undrained laboratory element tests on undisturbed samples or in-situ self-boring pressuremeter tests, which are not conventionally used in calibration of constitutive models. Such data, which usually do not provide sufficient information for model calibration, can nevertheless be used if reasonably precise information about the bounds of values of the model parameters is available for the material under consideration. Existing information on the bounds of values of the model parameters for the distortion and consolidation mechanisms is summarized for several soils in the following. The factors affecting the model parameters are also discussed. 4.6.1 Parameters of Isotropic Compression Mechanism The parameter C in Eq. (4.33) is mainly affected by the relative density of the material and grain compressibility. The exponent, p, on the other hand, is affected mainly by the grain compressibility and is rather insensitive to the state of packing. The effect of sample fabric on C and p is not investigated. Linear relationships between log C and relative density depending on grain compressibility are apparentfromisotropic compression test data on a number of granular materials (Fig. 4.10). Due to the absence of a more precise information, these relationships are used in this study. Analyses of isotropic compression test data used in constructing Fig. 4.10 also shows that for parameter p values of 0.9 and 0.65 can be assumed for granular materials with medium and high compressibility granular soils, respectively. 86 Chapter 4: Stress-strain Model v Crushed Granite (Lade, 1977) O Fraser River Sand (Stedman, 1997) o Glass Beads (Haruyama, 1981) • Ottawa Sand (Negussey, 1984) • Pointed Rock Material (Lade, 1977) • Sacramento R. Sand (Lee and Seed, 1967) • Toyoura Sand flsriihara, 1993) 40 60 Fig. 4.10. Parameter "C" for Isotropic Compression 4.6.2 Parameters of Distortion Mechanism Major factors affecting the parameters of the distortion mechanism are listed in Table 4.2, in which the factors affecting the parameter is indicated by shading. It appears from the table that parameters X, u, r) , Ar], R and n are affected by only a few factors. The following discussion F1 F P is intended to provide guidelines for selecting appropriate values of these parametersfromminimal material specific information. Once the approximate values of these parameters are identified, model calibration simplifies greatly because iteration over the remaining model parameter, K and S P m , is only necessary. A 4.6.2.1 Parameters Related to Plastic Flow, X and pt Nakai and Matsuoka (1983b) demonstrated that Eq. (4.14) does not depend on void ratio or inherent anisotropy of the soil. Thus, the relationship only depends on soil type and the Chapter 4: Stress-strain Model Table 4.2. Factors Influencing Parameters of Distortion Mechanism Parameter u K n Function Soil Type Density Fabric Drainage Reference Plastic Flow Nakai and Matsuoka, 1983b Failure Salgado, 1990 Hardening modulus P RF m A This study Anisotropy parameters associated with the correlation, X and u, can be easily evaluated from a suitable drained element test, e.g., triaxial or plane strain, on reconstituted specimens. From available test data, these parameters have been evaluated as listed in Table 4.3 for several sands. Like the constant volume friction angle (i.e., the steady state value of the mobilized effective stress friction angle), (bcv, u appears to be affected only by mineralogy of the soil and not Table 4.3. Model Parameters Associated with Plastic Flow Soil Type X H Reference Erksak Sand 0.83 0.30 Jefferies: personal communication Fraser River Sand 0.77 0.39 da Cunha, 1992, Vaid et al., 1996 Hilton Mines Tailings 0.80 0.40 Jefferies: personal communication Ottawa C109 0.85 0.26 Ditto Syncrude Tailings Sand 0.85 0.29 Ditto Ticino Sand 0.87 0.33 Ditto Toyoura Sand 0.90 0.27 Ditto; Nakai and Matsuoka, 1983b Chapter 4: Stress-strain Model 88 by grain characteristics. Since <j> is known for a variety of soils, some of which have been cv summarized in Table 4.4, a correlation between (b and u can be of a practical value if tenable. cv Appropriate values of (b^ for a few additional types of sand can be found in Salgado (1990) and Been et al. (1987). The correlation between <b (in degrees) and u is as follows cv Table 4.4. Constant Volume Friction Angle, Mineralogy and Grain Characteristics of Typical Granular Soils Soil Grain shape, Mineralogy e D , 1 0 mm MIN / e MAX 4>cv, Reference degree 1.06/ 0.69 33±2 Kuerbis and Vaid, 1989 0.61/ 1.1 33 Vesic and Clough, 1968 0.53/ 0.75 29±2.5 Been et al., 1991 0.33/ 0.14 0.72/ 1.10 31.5±2 Vaid et al., 1996; Thomas, 1992 Q 0.85/ 0.65 0.58/ 0.75 30 Sladen et al., 1985 Lornex Angular, Q and F 0.30 / 0.11 0.68/ 1.08 32.5± 2.5 Castro et al., 1982 Ottawa C109 Subround, Q 0.40/ 0.30 0.50/ 0.82 30 Syncrude Sand Subangular to Subround, Q 0.20/ 0.09 0.67/ 0.96 29.5±2 Vaid et al., 1996 Ticino Subround, Q, Tr M 0.55/ 0.36 0.60/ 0.89 31 Beenetal., 1987 Brenda Mines Angular, 85% Q, 15% M Chattahoochee River Subangular, Q, TrM Erksak Subround, 73% Q, 22% F Fraser River Subround, 40 % Q, 11 %F Leighton Buzzard 0.47/ 0.21 Sasitharan et al., 1994 Ishihara, 1993 30±1 0.66/ Subangular, 75% 0.19/ 0.87 0.11 Q, 25% F Note: e ^ and e ^ denote the maximum and the minimum void ratio and 60 and 10% of the particles are finer than T)^ and D i , respectively. Abbreviations F, M , Q and Tr are used to denote feldspar, mica, quartz, and trace, respectively. Toyoura 0 89 Chapter 4: Stress-strain Model u = -0.252 +0.019<b cv (4.37) ( The correlation coefficient for Eq. (4.37) is 0.664. Examination of Tables 4.3 and 4.4 indicates that unlike <b , X is affected by angularity of the grains as well as the mineralogy. Consequently, an cv attempt to correlate X and cb leads to a correlation coefficient of only 0.032 and such a cv relationship is therefore statistically untenable. 4.6.2.2 Parameters T) and ATJ F1 In triaxial compression tests, the stress ratio r) relates to d/, by F rj = 2 2tan(b'/3 (4.38) / F v Been and Jefferies (1985) demonstrated that <b' is not significantly affected by the sample fabric and is primarily affected by the relative density of the sample. Using (b instead of (J)' in Eq. cv (4.38) the constant volume stress ratio, r | , can be evaluated. From an examination of a large cv number of triaxial tests on several types of sand, Bolton (1986) concluded that the difference between the peak (or failure) value of the friction angle and (b correlates well with the relative cv density. Fig. 4.11 presents the relationships proposed by Bolton between the angle of internal friction in triaxial compression in terms of effective stresses, 4>TXC> and mean effective normal stress at failure, p ', together with the data used in developing the correlations. The interpolated F relationships are shown using solid lines while extrapolated relationships are shown by broken lines. In the absence of material specific information, r),, Ar) can be estimated from Eq. (4.38) and Fig. 4.11 using an appropriate value for (b , such as those in Table 4.4. cv 90 Chapter 4: Stress-strain Model PF'/PA Fig. 4.11. <b' as a Function of Mean Effective Stress and Relative Density (after Bolton, 1986) 4.6.2.3 Parameters R and n F P The parameter Rp primarily depends on the relative density of the deposit. Since the parameter pertains to post yield soil behavior, the effects of stress path and fabric is expected to be niinimal. Experience with the use of a similar model (Srithar, 1994) appears to indicate that a value of 0.75 can be assumed for Rp for very dense granular soils and a value of 1 may be appropriate for very loose deposits. Thus, in the absence of material specific information, R F linearly interpolated from a value of 1.0 at D =0 to 0.75 at D =100%. R R Previous experience with the distortion mechanism used in this study (Salgado, 1990; Srithar, 1992) indicate that n ranges between -0.3 and -0.6 for sands. In this study, a number of P laboratory triaxial tests on several sands could be simulated using n of -0.5. This value will be P used in this study in the absence of a more precise information. 91 Chapter 4: Stress-strain Model 4.6.2.4 Effect of Drainage on T) and Ar) F1 As pointed out in § 4.3.2.7, the value of <j>' measured in a laboratory element test allowing drainage is usually higher than the corresponding value from an undrained test. The difference between the value of <|>'in a drained test from that obtained in an undrained test is systematic in nature and is therefore not ascribable to experimental error. The physical reason for such a difference is also well understood (see, e.g., Taylor, 1948; Bishop, 1954; Poorooshasb and Roscoe, 1961). To illustrate the point, triaxial compression stress paths for Kogyuk 280/5 sand are plotted in Fig. 4.12. Similar comparison for other granular materials also leads to the same conclusion irrespective of stress path, sample fabric and confining pressure. Since the parameter r) relates F directly to the peak friction angle in triaxial compression by Eq. (4.38), model parameters r)j and At) are expected to be affected by the drainage condition as well. To account for the effect of drainage on these parameters the following empirical framework is proposed. Fig. 4.13 shows a plot between rh-rirjv measured in drained tests against the corresponding values in undrained tests. It is apparentfromthe plot that the relationship between the drained and 500 0 200 400 600 0 (o /+c /)/2,kPa (1 200 400 (3 Fig. 4.12. Drained and Undrained Stress Path in TXC 600 92 Chapter 4: Stress-strain Model undrained values of r j r | r (% cv can be expressed as the following linear correlation ~ ^Cv)Undrained = ® ^ (% (4.39) ~ ^Cv) Drained The correlation coefficient, R , for the relationship in Eq. (4.39) is 0.94. Thus, appropriate values 2 of %, and Ar| can be obtained by scaling the abscissa of Fig. 4.11 according to the relationship in Eq. (4.39) depending on drainage condition. As is apparentfromFig. 4.13, the relationship in Eq. (4.39) is does not dependent on the stress path, sample fabric and soil type. 4.6.2.5 Parameter for Inherent Anisotropy, m A In order to determine the manner in which the parameter m is affected by the effective A confining pressure and relative density, a series of undrained triaxial data on reconstituted Fraser River Sand was simulated with the stress-strain model (Fig. 4.14). These tests were carried out on water pluviated samples after removing thefractionpassing #200 sieve (Thomas, 1992). # • • • • • O O 0.20 0.15 g 0.10 k 0.05 0 0.1 0.2 0.3 0V-11cv)]'Drained 0.4 Alaska 140/5: TXC, MT, 1 Alaska 140/10: TXC, MT, 1 Fraser River: TXC, US, 2 Kogyuk 280/5: TXC, MT, WP, 1 Leighton Buzzard: TXC, MT, 1 Ottawa 530/0: TXC, MT, 1 Sacramento River: PSC, AP, 3 Syncrude: TXC, US, 2 Note: AP: air pluviation MT: moist tamping US:frozensample WP: water pluviation 1: Jefferies, personal communication 2: UBC 3: Lee, 1970 Fig. 4.13. Drainage Dependence of T]r \cv r 93 Chapter 4: Stress-strain Model 700 600 — Test Data - - Simulation Isotropic Consolidation Pressure, kP 500 / / ii /i / / / Citt) s\ /A /' '/' /' iJ II / / ' / ' / ' / 300 / / / 400 & CO / / W / / j 1 7 1 7 £ 200 Q 0 -100 <m$>' I -200 -300 5. 0 -5.0 Axial Strain, % -2.5 D RC = 59% 5 .0 0 2.5 Axial Strain, % Fig. 4.14. Modeling Undrained Stress-strain Behavior of Fraser River Sand (D = relative density at consolidation) RC Model parameters excepting K SP and m were estimatedfromthe guidelines summarized earlier A in § 4 . 4 . 2 and 4 . 6 . For K G typical values of shear wave velocity measured in the Fraser River E Delta were used. Parameters K and m were derivedfromfittingthe model to the test data by SP A trial and error. The derived model parametersfromthe exercise are noted in Table 4 . 5 . The results indicate that m is not significantly affected by state of packing and confining pressure. A Additional triaxial datafromtwo sites were also analyzed using the proposed stress-strain model described earlier to investigate the effect of soil type and sample fabric on m . The deposit A at J-Pit was created by spigotting of natural sand, while the test data of KIDD # 2 represent a river Chapter 4: Stress-strain Model 94 Table 4.5. Model Parameters: Triaxial Tests on Water Pluviated Fraser River Sand D R C , % K p S Ar) RF C m A 40 800 600 0.645 0.0275 0.90 0.00045 3.00 59 900 800 0.680 0.0450 0.85 0.00028 3.00 Remaining parameters are assumed not to depend on D . They are: v - 0.2, n = 0.50, n = -0.5, X = 0.77, u = 0.39 and p = 0.9. RC E P channel deposit. These samples were extracted using in-situ freezing and therefore reflect the insitu material fabric. Particulars of the tests are summarized in Table 4.6. Fig. 4.15 presents the test data and the calculated response afterfittingthe model by trial and error to the data as earlier. The model parameters for the computed response are summarized in Table 4.7. From these results it can be tentatively concluded that (a) m is not appreciably affected by A the state of effective stress and packing and (b) a value of 2.0 is appropriate for hydraulically deposited sand. However, the stress strain model needs to be applied more extensively to ascertain whether these conclusions be can refined further. Table 4.6. Particulars of Tests on Undisturbed Samples Site Tests J-Pit KIDD Note: o ' e coNs ®MAX/®MIN ° VCONS ° HCONS kPa kPa Test Reference FS26C213 FS26C214 0.708 0.698 0.960 /0.670 196 196 98 98 TXC Vaid et al., 1996 T X E Ditto K94F3C3A K94F3C4B2 0.950 0.946 1.077 /0.715 144 156 72 78 T X E Addendum to T X C Vaid et al., 1996 V mNs and o ' H r n K c are the vertical and the horizontal effective stress at consolidation Chapter 4: Stress-strain Model 95 Axial Strain, % Fig. 4.15. Simulation of Triaxial Tests on Undisturbed (Frozen) Samples: J-Pit (J) and KIDD # 2 (K) 4.7 Model Performance Although the results of the preceding exercise indicate a satisfactory performance of the proposed model, systematic discrepancy of the following nature can be identified from Figs. 4.14 and 4.15. The deviatoric response of the proposed model is stiffer especially at small strain. In addition, a larger negative pore water pressure is predicted in anisotropically consolidated triaxial extension tests. Table 4.7. Model Parameters: Triaxial Tests on Undisturbed Samples m Site Tests J-Pit FS26C213, FS26C214 500 300 0.540 0.020 0.85 2.5 2.0 KIDD K94F3C3A, K94F3C4B2 950 600 0.635 0.023 0.91 5.0 2.0 R K p S F CxlO 4 A Remaining parameters (independent of D ) are: v = 0.2, n = 0.5, n = -0.5, and p=0.9; for JPit, X=0.85, u=0.29 and for KIDD #2, A=0.77, u=0.39. RC E P Chapter 4: Stress-strain Model Regarding the stiffer deviatoric response of stress-strain model, it should be noted that the deformations in these tests simulated here were measured externally. Recent research (e.g., Tatsuoka and Kohata, 1995) indicates that data from laboratory tests, in which deformation measurements are performed externally leads to an underestimation of stiffness at small strains primarily due to bedding error. In the stress-strain relationship developed earlier unloading in pure distortion is handled by allowing the loading surface to shrink following the material state in the effective stress space. Consequently, in an anisotropically consolidated triaxial extension test performed under decreasing mean normal effective stress (e.g., K94F3C3A and FS26C214: Table 4.6), in which the ratio of horizontal to vertical effective stresses at consolidation is less than 1, elastic material response is predicted as long as the stress ratio decreases leading to the development of a relatively large negative pore water pressure in the model response. Such a discrepancy can be avoided by the adoption of an appropriate scheme of kinematic hardening, e.g., Lade and Boonyachut (1982). A similar approach was not used in this study to preserve simplicity. 4.8 An Approximate Procedure for Model Calibration From Laboratory Tests It may not be possible to calibrate the stress strain relationship in a rigorous manner following the procedure described in § 4.5 due to non availability of sufficient laboratory test data. To tackle the problem, an approximate procedure making a minimal use of material specific information is suggested in this section for an approximate calibration of the stress-strain model. The essential details of the approach are as follows. • The procedure calls for a reasonable knowledge about the states of packing, and geostatic effective stress and grain compressibility. The state of packing is easily estimated from 96 ter 4: Stress-strain Model 97 CPT while SBPMT data at very small deformation can be used to estimate the state of geostatic effective stress. Depending on whether the deposit is primarily comprised of angular or rounded grains, grain compressibility of the deposit is low or high. Using shear wave velocities from seismic CPT and an estimate of the geostatic state of effective stress the elastic shear modulus number is determined. The Poisson's ratio is assumed to be 0.2 in the absence of a more precise information. A reasonable value of the elastic exponent can be assumedfromthe knowledge of mineralogy. A value of 0.7 can be assumed for carbonate sands, otherwise a value of 0.5 can be used. From the knowledge of state of packing and grain compressibility, C and p can be estimated. t) can be estimated from the peak value of (o '-o 3 ')/(o , '+o ') measured in a triaxial a) F ( ) ( ) (3) test. To analyze plane strain laboratory element test, r\ can be calculatedfromEq. (4.12). F From the measured value of %, <b TXC can be calculated (using Eq. 4.38 or following Salgado, 1990, for plane strain element test data). Since p '(the value of mean normal F effective stress at which r) is measured) is also knownfromthe laboratory element test F data, Ar) can be estimatedfromFig. 4.11. It is desirable to determine X and ufromtesting reconstituted specimens in the laboratory. If such a test cannot be carried out values listed in Table 4.3 and Eq. (4.37) can be used for guidance. For hydraulically deposited granular materials a value of 2 can be assumed for m and a A value of -0.5 can be assumed for n in the absence of more precise information. P Chapter 4: Stress-strain Model • K S P 98 can now be estimated from fitting the stress-strain model to measured stress- deformation data over a wide range of strain (e.g., laboratory triaxial test) by adjusting its value by trial and error. 4.9 Validation of the Procedure In this section the approximate calibration procedure for the stress strain model developed earlier and described in the preceding section is put to test using laboratory data. Lam and Tatsuoka (1988) report a series of triaxial compression and extension tests on air pluviated Toyoura sand. A series of plane strain compression tests was also performed on samples with 8 ranging between 0° and 90°. The soil has a minimum and maximum void ratio of 0.605 and 0.977, respectively. The median grain size is 0.16 mm and the uniformity coefficient is 1.46. A drained triaxial compression test and an extension tests on dense specimens are used to calibrate the stressstrain model. The model parameters obtained by fitting the response of a single axisymmetric element to the test data by trial and error are: K = 1250, K E S P = 200, R = 0.78, r) = 0.820, Ar] F F1 = 0.12, u = 0.27, A=0.90, € = 0.0001, p = 0.9 and m = 2.0. The results are plotted in Fig. 4.16. A The slight discrepancy in the deviatoric response at e i ) - e greater than about 2% arisesfromthe ( (3) fact that in the proposed stress-strain model the effective stress friction angles in triaxial compression and extension are the same. In contrast, the test data shown in Fig. 4.16 indicates a higher friction angle in triaxial compression than that in triaxial extension. The stress-strain model can be readily adapted by assuming rj , to be a function of 6, as has been done for G F n (Eq. 4.29). However, as pointed out earlier, a consensus is yet to emerge that the effective stress friction angle is indeed affected by inherent anisotropy to a significant extent. 99 Chapter 4: Stress-strain Model —Data Simulation D » 90% RC ^a' Extension: C O N S = 392kPa 0 -0.5 Compression: 0 ^ = 98 kPa 1 -1.0 -1.5 1 0 2 3 £(1)"^{3)> ••• 4 5 % Fig. 4.16. Simulation of Drained Triaxial Tests on Toyoura Sand (data from Lam and Tatsuoka, 1988): H/W (Height to Width ratio of Cubic samples) = 2.0 The prediction of the plane strain responses using the calibrated model together with the corresponding laboratory test data for 6=90° and 8=0° are shown in Figs. 4.17a and 4.17b, respectively. Comparison of Figs. 4.16 and 4.17 shows that the stress-strain model is successful in capturing the stiffer response in plane strain. The predicted values of o 7a ' in plane strain (1) (3) at large deformation are also larger than the corresponding values in triaxial tests. Except for an over prediction of the o 7o ' ratio and a discrepancy on account the assumption of isotropy in (2) (3) effective stress friction angle, the performance of the proposed stress-strain model and the Chapter 4: Stress-strain Model 100 101 Chapter 4: Stress-strain Model approximate calibration procedure in simulating the aniostropy and stress-path dependency appears to be reasonable from the results presented in Fig. 4.17. 4.10 Sensitivity of Model Response to the Choice of Model Parameters A sensitivity study is undertaken to develop an understanding about the model behavior when the available estimates of the parameters are imprecise. Such an exercise also provides useful information about the possible imprecision in the calibrated model when necessary and sufficient laboratory element test data are not available. Values of the model parameters used in the analyses are: K =900, n =0.5, K =300, n = GE E SP P -0.5, RF=0.90, r] =0.70, Ar)=0.10, u=0.29, A=0.85, C=0.0005, p=0.9 and m =2.0. An isotropic F1 A consolidation pressure of 100 kPa was used. Individual model parameters (except rj and An) F1 were changed by ±10% and the computed responses are presented in Figs. 4.18 and 4.19. r\ and F1 An were considered coupled: n =0.77 and An=0.135 were used as the upper bounds; and F1 r) =0.63 and Ar)=0.065 were assumed to be the lower bounds. Results of these calculations are F1 presented in Figs. 4.18 and 4.19. The conclusions of this exercise are summarized in Table 4.8. Since some of the model parameters can vary over a wider range (e.g., K S P and C) than the others (e.g., X and u: whose approximate range is substantially covered by the ±10% variation), qualitative inference on the relative importance of a certain model parameter should not be drawn from the model response allowing ±10% variation in the value of model parameters. Such an inference can only be drawn after development of an extensive experience with the use of the model, which is unavailable at present. To illustrate the point the responses for upper and lower bound values of Kgp (« 1500 and 50, respectively) are plotted as a shaded band in Fig. 4.18. Similar plots for parameter C using 0.0025 and 0.01 as upper and lower bounds are presented in Chapter 4: Stress-strain Model 102 Chapter 4: Stress-strain Model 103 104 Chapter 4: Stress-strain Model Table 4.8. Effect of Variation in the Values of the Model Parameters Response Strain v Deviatoric Elastic * n K E Rp n S P A T Failure Elastic Plastic X Yl u C p m A * PlaStiC Isotropic Compression r\ & Ar| P A T • A * • T T Note: * and • indicate a stiffer and softer response with an increase in parameter value Fig. 4.19. The importance of the influence of K from these results. S P on triaxial element behavior becomes evident A similar exercise is not undertaken for the other parameters of significance(r) , X, and u) because the range of their usual values is substantially covered in the F1 ±10% variation assumed in the earlier calculations. In model calibration following § 4.8, u and X are essentially derivedfromlaboratory triaxial test data on reconstituted samples and n is found directlyfromtests on undisturbed samples F without any loss of accuracy. Since in the suggested procedure K S P is derivedfromfitting the model to laboratory element tests by trial and error, its estimate is expected to be reasonably precise because • the values of r\ , u and X used in the exercise are accurate and • the remaining model parameters have less significant effect of model response. Fl Another limitation of a sensitivity study of this nature is that in this exercise a single axisymmetric element response similar to an isotropically consolidated laboratory triaxial compression test is computed. The extent of influence of each model parameter on material response is however expected to differ slightly depending on stress path and loading direction. Chapter 4: Stress-strain Model 105 4.11 Geometric Non-linearity As pointed out earlier, FLAC can accommodate large deformations by updating the finite difference grid with the progress of the deformation process. In this section the importance of geometric non-linearity on element mechanical behavior due to large deformation is examined. For illustration, drained triaxial tests are simulated using the following model parameters KQ =720, E nE=0.5, Ksp=240, n =-0.5,Rp=0.92, r) =0.685, Ar)=0.048, u=0.39, A=0.77, C=0.0006, p=0.9 and P F m =2.0. Such values are typical of a loose deposit of sand, e.g., that at Massey Tunnel. The model A response in triaxial compression and extension for a sample consolidated to a vertical and horizontal effective stress of 100 and 50 kPa, respectively, is presented in Fig. 4.20. For simplicity, the axial strain is assumed to be simply equal to the axial deformation divided by the original axial -50' 0.2 -0.81 -5 1 i -2.5 " 1 ! 1 0 2.5 Axial Strain, % 1 1 5 Fig. 4.20. Relevance of Assumption of Small Deformation Chapter 4: Stress-strain Model 106 dimension of the element. It should also be noted that the trace of the strain tensor, e^, gives the volumetric strain when deformation is small. It is apparent that the for strain not exceeding 5% the model response admitting geometric non-linearity is virtually identical to that obtained from small deformation analysis. Thus no attempt was made in the implicit finite element code to accommodate large deformation. However, unless indicated otherwise, while using FLAC nodal coordinates are allowed to be updated. 4.12 Approximations in the Stress-strain Model It has been demonstrated in the preceding that the stress-strain model developed in this chapter reasonably captures the stress-path dependency and anisotropy usually observed in laboratory testing of cohesionless soils. However, for an appropriate use of the model, the following approximations inherent in the formulation need to be noted. 4.12.1 Isotropic Hardening in Distortion Existence of a single loading surface is assumed while developing the model. The size of the loading surface is given by the previous value of r\. Unloading is assumed if r| decreases below this value and soil behavior is postulated to be elastic. The limitations of this simple scheme have been pointed out in § 4.7. Although the implication on this account may not be great while calculating monotonic response, such a simple isotropic hardening rule is not capable of modeling cyclic behavior accurately. In cyclic loading a more elaborate hardening rule needs to be adopted. As pointed out earlier, Lade and Boonyachut (1982) have proposed one such scheme. 107 Chapter 4: Stress-strain Model 4.12.2 Inability to Simulate Pure Rotation of Stress The constitutive model described in this chapter is intended for use in analyzing SBPMT data to estimate the undrained monotonic behavior of cohesionless soils. In such a problem there is no pure rotation of effective stress tensor. Thus, possibility of pure rotation of effective stress tensor was ignored in the development of the stress-strain model, i.e., within a time step (or deformation increment) the change in Gpj with 0 is neglected. In the present implementation of the model, the value of 0 at the beginning of a time step is used to calculate the current value of G PT In addition, experimental data show that in pure rotation of principal stress, the assumption of coaxiality of plastic strain increment and the effective stress is violated (see, e.g., Matsuoka and Sakakibara, 1987). Since the proposed stress-strain model assumes coaxiality, the formulation is not expected to simulate the deformation process due to pure rotation of stress accurately. 4.12.3 Independence of Hardening Rules to Current Void Ratio The model accounts for the hardening of elastic response due to a change in void ratio during the deformation process by treating the elastic moduli as a function of void ratio. Change in void ratio may have an analogous effect on G . However, due to a limited knowledge about the PI effect of void ratio on the model parameter, K , it was not treated as a function of void ratio. SP 4.12.4 Isotropic Failure Criterion In addition, as pointed out earlier, parameter %, (and also Ar)) is treated as isotropic. Some test data suggest (e.g., Lam and Tatsuoka, 1988) that this parameter is affected in a similar manner by material anisotropy as G . However, a large amount of test data (e.g., Been and T Chapter 4: Stress-strain Model 108 Jefferies, 1985) indicates that the parameter r| is not significantly affected by material anisotropy. F1 On the other hand, some researchers (e.g., Lade and Duncan, 1975) suggest the failure stress ratio in triaxial compression is lower than the corresponding value in triaxial extension. Since the issue does not appear to be resolved yet, the parameter r) (and consequently At]) is treated as isotropic. F1 4.12.5 No Inherent Anisotropy in Isotropic Compression Mechanism The plasticity model for isotropic compression does not account for inherent anisotropy: an approximation not adequately validated. However, the deformation ascribable to this mechanism is usually much smaller in comparison with that due to the distortion model. As a consequence, the discrepancy on this account may not be significant. 4.12.6 Independence of the two Plasticity Mechanisms Also, the plasticity models for distortion and isotropic compression are assumed to be independent. In other words, hardening in one mechanism can take place irrespective of whether loading or unloading is taking place in the other. As pointed out earlier, this simplifying assumption may lead to some discrepancies in simulating the behavior of over consolidated deposits. 4.12.7 Limitation of Non-associative Plasticity Admission of the non-associative plasticity in itself draws criticism from some researchers particularly for stress increments in the wedge region: the region in stress space that lies between the loading and the plastic potential surfaces (Hashiguchi, 1991). On the other hand, Nova (1991) showed that to describe the liquefaction phenomenon, the stress-strain model must be nonassociative. Although two of the most commonly used plasticity models for frictional materials, Chapter 4: Stress-strain Model 109 viz., Nakai and Matsuoka (1983) and Lade and Duncan (1977) assume non-associative flow, the debate on the admissibility of non-associativity does not appear to be resolved yet. 4.12.8 Strain Softening in Drained Deformation The proposed stress-strain relationship is not capable of accounting for significant strain softening in drained deformation. Although laboratory drained element test data such as triaxial and plane strain often show significant strain softening especially for dense granular soils, such a behavior is primarily due to non-homogeneity in the deformation field. Some types of soils however may exhibit strain softening in drained deformation due to structural collapse or significant grain crushing triggered as soon as the state of stress reaches a critical point in the effective stress space. The stress-strain model developed earlier is not expected to capture such a behavior. 4.13 Summary A hyper-elastic plastic stress-strain model for anisotropic frictional material is proposed in this chapter. The parameters are physically based and approximate values of many of the model parameters are available. The rigorous procedure for the evaluation of the model parameters strictly from test data has been outlined. However, often such an elaborate laboratory testing program is not feasible. Consequently, necessary and sufficient test data for calibration of the model are usually not available. To tackle the problem, an approximate procedure making a minimal use of material specific information has been used for the calibration of the stress-strain model. Laboratory stress-strain data have been used to validate the approximate calibration of the stress-strain model developed earlier. A similar procedure for SBPMT data is proposed and validated later in this research. 110 CHAPTER 5 MODELING CYLINDRICAL CAVITY EXPANSION 5.1 Introduction To compute the stress and displacement around a cylindrical cavity, a finite length of which is under pressure, using the finite difference or the finite element methods, the medium needs to be discretized appropriately. In the study outlined in the following sections a number of alternative schemes have been evaluated to identify an appropriate numerical model for analyzing SBPMT data. Another aspect of numerical analysis of SBPMT - whether or not it is necessary to adopt a numerical tool capable of handling large deformations - is also investigated in this chapter. A sensitivity study is then undertaken to examine qualitatively the extent of non-uniqueness in the results of back-analysis of SBPMT data for the numerical model. For validation of the numerical schemes closed form solutions are utilized wherever possible. 5.2 Alternative Discretization Approaches As pointed out in Chapter 2, the SBPMT pressure expansion curves can be analyzed as a plane strain problem for probes of a similar geometry and design as that of the UBC SBPM. Two alternative approaches can be adopted to tackle the cylindrical cavity expansion problem. One approach is to consider the medium discretized as shown in Fig. 5.1a as an axisymmetric deformation problem allowing no nodal vertical movement, while in the other, a horizontal slice is analyzed in plane strain allowing no nodal displacement in the hoop direction at the boundary (Fig. 5.1b). For simplicity, the external boundary in the "x" direction is considered fixed. Infinite Chapter 5: Modeling Cylindrical 111 Cavity Expansion la Coordinates: x = radial, y = vertical, z = hoop Fixity Applied displacement or pressure 0.037 m 0 Is X a -7.413 m- Coordinates: x = radial, y = hoop, z = vertical Applied displacement Fig. 5.1. Spatial Discretization elastic boundary in the radial direction at the far field boundary nodes is a better choice for the boundary condition. However, from the results presented in the following sections it becomes apparent that the assumption of fixity at the farfieldboundary does not lead to any appreciable loss of accuracy for the problem at hand. In fact, the distance of the assumed outer boundary from the expanding cavity is so large that the outer boundary has virtually no influence on the computed stresses and displacements at the cavity wall. Since the discretization schemes presented in Fig. 5.1 are essentially the same, correct implementation of these schemes is expected to give identical results. Chapter 5: Modeling Cylindrical Cavity Expansion 112 5.3 Numerical Tools An explicit finite difference computer code FLAC is used in the analyses of cylindrical cavity expansion. Essential details of FLAC have already been described in Chapter 3. Small as well as large deformation analyses are carried out with FLAC using the discretization schemes shown in Fig. 5.1. Unfortunately, the code is not very efficient for a problem such as that at hand. The long and narrow geometry of the problem calls for a very small time step that increases the computation time. Another computer program has been developed for use in the analysis of SBPMT using the constitutive relationship described in Chapter 4. The static equilibrium equation is solved by implicit finite element in the code. Radial pressure is applied at the cavity wall in very small increments. Since the loading increments are very small, no iteration scheme is included in the code. The discretization scheme of Fig. 5. la is only programmed and geometric non-linearity is neglected. Four node quadrilateral elements with conventional shape (interpolation) functions are used in this procedure. The choice of four node quadrilateral element is primarily motivated by the fact that it allows direct comparison of the results obtainedfromthe implicitfiniteelement program with thatfromFLAC. Three point Gauss Legendre Quadrature is used to approximately integrate the internal strain energy over the volume df an element. A portion of the code, which is listed in Appendix 3, is essentially a simple adaptation of the relevant modules found in Smith and Griffiths (1988). Instructions on the use of the program can also be found in Appendix 3. 5.4 Adequacy of Spatial Discretization To establish the adequacy of the schemes of spatial discretization the numerically computed displacement pattern and stress distribution are compared with the theoretical predictions for an Chapter 5: Modeling Cylindrical Cavity Expansion 113 isotropic linear elastic material in this section. The stress fields obtained from the computer codes are then compared with the closed form solutions for an isotropic linear elastic perfectly plastic material assuming deformations to be small as an additional check. Subsequently, the performance of the stress strain model of Chapter 4 in modeling cylindrical cavity expansion is examined. The small and large strain solutions from FLAC are also compared to find whether the small strain assumption is accurate enough in a typical SBPMT data analysis while using the proposed stress strain relationship. 5.4.1 Isotropic Linear Elastic Material When the deformation is small, the radial effective stress, o ', the tangential effective stress, x a ', and the radial displacement, u, for a linear isotropic elastic cylinder of infinite thickness with 2 internal radius a, and subjected to an internal effective pressure of P are given by ; ° ' = a ' (P.-o ')a /x , o ' = o '-(P -o ')a /x and u = (P o ')a /(2G x) 2 x H + 2 2 H 2 H ; 2 H 2 i+ H E ( 5 1 ) where x is the radial coordinate and o ' is the free field horizontal effective stress. H The explicit finite difference (FLAC) and implicit finite element solutions for stresses for the scheme of Fig. 5.1a using an elastic shear modulus of 30 MPa and o ' = 50 kPa are shown in H Fig. 5.2 together with the closed form results. The displacement fields calculated numerically are compared with the closed form solution in Fig. 5.3. The stress and displacement fields are plotted in Figs. 5.2 and 5.3 for at two different values of the cavity strain. The radial extent of the discretized domain in these analyses is 7.413 m. Adequacy of the discretization scheme is evident from these plots over the entire range of deformation normally obtained in a typical SBPMT cavity Chapter 5: Modeling ? Cylindrical Cavity 0 114 Expansion -0"'P""f| D D|...l...»£J..m..l^[J 3 0 o ': a ': o ': o ': o ': o,': est x z o • • x z x o theory theory finite element finite element FLAC FLAC x, m Fig. 5.2. Numerical and Closed Form Solutions for Stress: Isotropic Linear Elasticity expansion ( - e ^ 10%). Cavity strain, - e ^ is given by iw'a, where u ^ is the radial deformation at the cavity wall and a is the original radius of the cavity. To determine the effect of boundary condition on the results of the numerical simulation, additional analyses were carried out with -ezwaii = 4% 1 E i i IB x, m m 2 HgzwaU = 9% « a a a m BI =s —Theory • Finite Element • FLAC M. x, m Fig. 5.3. Numerical and Closed form Solutions for Displacements: Isotropic Linear Elasticity Chapter 5: Modeling Cylindrical 115 Cavity Expansion discretized domain extending in the radial direction to 3.676 and 14.977 m. Comparison of these results with those using a domain extending to 7.413 m indicates that the boundary effects on the simulated pressure expansion curve is negligible (Table 5.1). The results indicate that the numerical model of Fig. 5.1 is adequate for an isotropic linear elastic material for both explicit finite difference and implicit finite element analyses. 5.4.2 Isotropic Linear Elastic Perfectly Plastic Material The soil elements adjacent to an expanding cylinder of an elastic perfectly plastic MohrCoulomb material of infinite thickness starts to undergo plastic flow as the ratio of the major and minor effective stresses exceeds a material specific value. When the deformation is small, the radius, Rp, that envelopes the material undergoing plastic flow is given by R = a[P, = { o ' { l s i n < | » ' ) } ] P H l/(1 + - (5.2) N) where N = (1-sin <b')/(l+sin (J)'). Following are the analytical expressions for the stresses in the material undergoing plastic flow when deformation is small Table 5.1. Stresses at the Centroid of the Element Adjacent to the Cavity (R = 0.041 m) Radial Extent, m zwaU E £ zwall "= = 4% 9% FLAC Finite Element FLAC Finite Element a ', kPa o ', kPa a ',kPa o ', kPa o ', kPa o ', kPa o ', kPa o ', kPa 2496 2471 2471 -2395 -2371 -2371 2449 2448 2451 -2341 -2342 -2344 5350 5296 5296 -5248 -5196 -5196 5424 5435 5437 -5323 -5334 -5337 x 3.676 7.413 14.97 : z x z x z x z 116 Chapter 5: Modeling Cylindrical Cavity Expansion o' = x where '(R /x)'- anda; N O R P p = Na ' (5.3) x is the effective radial stress at a distance of R from the center of the expanding cavity. P In the elastic zone that extends from x=Rp to x=°°, the stress field is calculated using Eq. (5.1) after replacing o^,' by P and R by a, i.e., s °/ =V P + (°RP' - V X V * ) 2 «>d ° z ' = V " ( ° R P ' - a ' ) ( R / x ) H p 2 (5.4) It should be noted that the stress field around the cylindrical cavity of infinite thickness is only affected by the failure criterion and not by the flow rule. The displacement field in such a deformation process, on the other hand, depends on the flow rule as well. In the analyses a Mohr-Coulomb material with <j>'=36° is approximated by the following model parameters: KG =720, %=0, V=0.2, K =240, np=0, r| =G.69, Arj=0, A=0.77, u=0.39, Rp=0, E sp F1 C=0 and m =l. The axisymmetric element response for such a material in a deformation process A as that in a laboratory triaxial test is shown in Fig. 5.4 using an isotropic consolidation stress of 100 kPa. The computed stress over a portion of the domain adjacent to the cavity for a geostatic 5» 300 Axial Strain, % Fig. 5.4. Element Response in Triaxial Compression: Elastic-Perfectly Plastic Simulation Chapter 5: Modeling Cylindrical Cavity Expansion 11'/ horizontal effective stress of 50 kPa is shown in Fig. 5.5 together with the corresponding closed form solution. The results indicate a good agreement between the results computed numerically and the corresponding theoretical values. It appears from these results that the numerical model for cylindrical cavity expansion of Fig. 5.1 captures the behavior of an isotropic linear elastic perfectly plastic material reasonably. 5.4.3 Elasto-plastic Material In this section the performance of the stress-strain relationship of Chapter 4 is examined using the numerical model for cylindrical cavity expansion shown in Fig. 5.1. Since there is no closed form solution for materials exhibiting stress dependent mechanical behavior, the performance of the numerical tools developed in this research for analyzing SBPMTs cannot be Fig. 5.5. Stress Distribution for Isotropic Linear Elastic Perfectly Plastic Material Chapter 5: Modeling Cylindrical Cavity Expansion 118 checked against any theoretical results for the proposed stress strain relationship. To indirectly find whether the performance of the numerical procedures is satisfactory while using the stress strain relationship of Chapter 4 it is checked whether the • explicitfinitedifference (FLAC) and implicitfiniteelement procedures give the same result to the stress and deformationfieldsaround the cavity, • alternative schemes of Figs. 5.1a and 5.1b lead to an identical result for an inherently anisotropic material. In addition, it is examined whether for the discretization schemes the boundary effect becomes negligible while using the proposed stress strain model. Results for model parameters typical of a loose deposit of sand is only presented in this section. However, the conclusions for parameters representative of sands at other states of packing is largely similar. The values of the model parameters used in this exercise are: K =720, n =0.5, GE E v=0.2, K =240, n =-0.5, r] =0.69, Ar|=0.05, A=0.'77, u=0.39, R =0.92, C=0.0006, p=0.9 and SP P F1 F m =2. The vertical and horizontal effective geostatic stresses are assumed as 100 and 50 kPa, A respectively. 5.4.3.1 Stress and Deformation Field Figs. 5.6 and 5.7 present the stresses and deformation over a portion of the discretized domain adjacent to the expanding cavity computed using FLAC and implicitfiniteelement using discretization scheme of Fig. 5. la. Deformation is assumed to be small in the exercise while using FLAC. With the compression positive sign convention, in axisymmetric deformation the radial strain, e , is equal to du/dR, while the hoop (or the circumferential) strain e = -u/R. The results x 2 from explicitfinitedifference and implicitfiniteelement methods are quite comparable. Chapter 5: Modeling Cylindrical Cavity 1 Expansion 22 Element FLAC: Axisymmetric | x, m Fig. 5.6. Stress Distribution for Elasto-Plastic Sand n 0.1 Q-D D pi n 0.2 x, m •—B r-1 0.3 i-i 11=1 l_J 0.4 5 — l—' 0.5 B Finite Element FLAC: Axisymmetric 1 0 0.1 0.2 0.3 0.4 0.5 x, m Fig. 5.7. Deformation Computed for Elasto-Plastic Sand 120 Chapters: Modeling Cylindrical Cavity Expansion 5.4.3.2 Alternative Schemes of Discretization In an axisymmetric problem FLAC assumes "x", "y" and "z" to be the radial, hoop and vertical coordinate directions, respectively (referring to Fig. 5.1a). On the other hand, in a plane strain problem, the code assumes coordinate direction "z" to be aligned along the direction in which no deformation is allowed. Consequently, "x" and "y" become the two horizontal directions (referring to Fig. 5. lb). Thus for an orthotropic material characterized with m =2, the vector a A becomes [0 1 0] and [0 0 1 ] , respectively, for discretization schemes detailed in Figs. 5.1a and T T 5.1b. The results obtained from the corresponding explicitfinitedifference simulation are shown in Fig. 5.8 together with thefiniteelement solution. In all these simulations geometric non-linearity is neglected. Comparison of these results shows that the performance of the proposed stress strain relationship is reasonable in modeling cylindrical cavity expansion in both explicitfinitedifference and implicitfiniteelement methods. 500 400 % 300 200 100 0 S I • O Finite Element F L A C : Plane Strain F L A C : Axisymmetric 8 10 Cavity Strain, % Fig. 5.8. Simulation of Cylindrical Cavity Expansion with Alternative Discretization Schemes Chapter 5: Modeling Cylindrical l^1 Cavity Expansion 5.4.3.3 Boundary Effects Numerical solutions to cylindrical cavity expansion for the stress strain model of Chapter 4 are obtained for discretized domains extending to radii of 7.41 and 14.97 m to examine the extent of the effect of the fixed outer boundary. The results are presented in Fig. 5.9. Discretization schemes of Fig. 5.1 are used in the calculation once again. Examination of the results indicate that the computed responses with outer boundaries at 7.41 and 14.97 are virtually identical to each other. This demonstrates the adequacy of the domain extending to 7.41 m radius. 5.4.4 Small Deformation Assumption A comparison of the large deformation solutions to cylindrical cavity expansion with the corresponding results neglecting geometric non linearity is shown in Fig. 5.10 using the stress strain relationship of Chapter 4 and numerical models of Fig. 5.1. The results indicate that the numerical TjO-l 200 7.4 m: • Finite element O FLAC 100 14.9 m: O Finite element * FLAC Cavity Strain, % Fig. 5.9. Response of Cylindrical Cavity Expansion for Different Domain Sizes Chapter 5: Modeling Cylindrical Cavity 122 Expansion 500 400 at 22 300 200 eft o i e 1001 0C 0 Large Strain • Plane Strain • Axisymmetric Small Strain •Plane Strain o Axisymmetric 8 10 Cavity Strain, % Fig. 5.10. Imprecision in the Assumption of Small Deformation solution assuming deformations to be small may lead to an over estimation of the effective cavity pressure by about 5% for -e^^ 10%. These conclusions are in agreement with those of § 2.7.3.3 wherein linear elastic perfectly plastic material was considered. Use of the implicitfiniteelement computer program developed in this research (or FLAC without updating nodal coordinates) in analyzing cylindrical cavity expansion is therefore approximate. However, the loss of accuracy on this account is not significant. 5.4.5 Adopted Numerical Model It is apparentfromthe preceding discussion that any of the alternative analytical tools and discretization strategies discussed above leads to results of reasonable accuracy for a material whose constitutive behavior can be modeled by the stress-strain model developed in Chapter 4. However, in the analyses presented later in this research, discretization scheme shown in Fig. 5.1b 123 Chapter 5: Modeling Cylindrical Cavity Expansion is used with FLAC allowing nodal coordinates to be updated with the progress of the deformation process. In the computation, the domain is first brought to equilibrium under the best estimates of the state of geostatic effective stress. A radial velocity is imposed on the nodes adjacent to the cavity wall in such a manner that the unbalanced force is within a tolerable limit at all stages of deformation after applying the appropriate kinematic boundary condition. 5.5 Sensitivity of Analytical Response of the Deforming Cavity to Model Parameters The sensitivity of the model response in cylindrical cavity expansion is studied in the following. The exercise is expected to provide qualitative insight into the extent of non uniqueness in back analysis of SBPMT using the stress strain relationship developed earlier. Following values of the model parameters are used as reference: KG =800, n =0.5, v=0.2, E E K =500, n =-0.5, n =0.75, Ar|=0.08, A=0.77, u=0.39, R^O.85, C=0.0004, p=0.9 and m =2. SP P F1 A The geostatic vertical and horizontal effective stresses are assumed to be 100 and 50 kPa, respectively. The reference value of each model parameter (except n i and Ar|) is altered by ±10% F and the computed response is presented together with the model response for the reference values (Figs. 5.11 and 5.12). r) and Ar| are treated as coupled: r| =0.825 and Ar|=0.118 are used as the F1 F1 upper bound, while rj =0.675 and Ar)=0.043 are used as the lower bound. Also investigated is FI the sensitivity of the model response in cylindrical cavity expansion to the choice of geostatic horizontal effective stress, o ', by varying its value by ±10% with respect to the reference. Since H the uncertainty involved with the estimate of the geostatic vertical effective stress is usually small, sensitivity of the model response to its choice is not investigated. The conclusions of the exercise (summarized in Table 5.2) are similar to those for the sensitivity study undertaken earlier for a single axisymmetric element undergoing a deformation 124 Chapter 5: Modeling Cylindrical Cavity Expansion — Reference - - - +10% 8 10 0 2 Cavity Strain, % 10% 8 Fig. 5.11. Sensitivity of Cylindrical Cavity Expansion to Model Parameters - 1 10 Chapter 5: Modeling Cylindrical 125 Cavity Expansion 2 4 6 Cavity Strain, % 8 10 — Reference • - - +10% - - -10% 0 2 4 6 8 Cavity Strain, % 10 Fig. 5.12. Sensitivity of Cylindrical Cavity Expansion to Model Parameters - II Table 5.2. Influence of Model Parameters on Computed Cavity Expansion Response K S P Rp n P r) and Ar) F1 A C p m A Note: * and • denote stiffer and softer response with increasing parameter value, respectively Chapter 5: Modeling Cylindrical Cavity Expansion 126 process similar to the conventional laboratory triaxial compression test except for parameter n . P However, this parameter does not appear to have a significant influence over either the element response or that in cylindrical cavity expansion. Nevertheless, as indicated earlier, the sensitivity of model response is expected to depend on the stress path of numerical simulation. Since some of the model parameters can vary over a wider range (e.g., K S P and C) than the others (e.g., A and u: ±10% variation substantially covers their approximate range), qualitative inference on the relative importance of a certain model parameter should not be drawn from model response for ±10% variation in the value of the parameter with respect to the reference value. Such an inference can only be drawn after an extensive experience is accumulated with the use of the model. The experience of the writer with the use of the stress strain relationship in modeling cylindrical cavity expansion is not comprehensive at this moment. Nevertheless, to provide an approximate guidance on the qualitative importance of the individual model parameters, the model response for the upper and lower bound values for K s p («1500 and 50, respectively, for sand) and C («0.0008 and 0;0002, respectively, for medium dense sand of medium compressibility) are also plotted as shaded areas in Figs 5.11 and 5.12. These results illustrate the important influence of K , r| (and Ar)), A and u on model response in cylindrical cavity expansion. SP F1 5.6 Summary Appropriate discretization schemes for cylindrical cavity expansion have been identified by comparing the computed response with closed form solutions for isotropic linear elastic and isotropic linear elastic perfectly plastic media. Two alternative numerical tools, one based on explicitfinitedifference (FLAC) while the other on implicitfiniteelement, used in the computations gave virtually identical results. As Chapter 5: Modeling Cylindrical Cavity Expansion 127 indicated in Chapter 3, the FLAC model can at least partially account for geometric non-linearity, while deformation is assumed to be small in the implicitfiniteelement computer program. On the other hand, the implicitfiniteelement computer program was found to execute in less time than FLAC for the numerical models for cylindrical cavity expansion. Model response in cylindrical cavity expansion have also been computed using FLAC with and without the small strain assumption. Comparison of these results indicate that assumption of small deformation leads to about 5% over estimation of effective cavity pressure provided that e zwaii^ 10%. Usually a discrepancy of such a magnitude is not considered significant. To calculate the response of an expanding cylindrical cavity later in this research, FLAC will be used allowing for geometric non-linearity together with the scheme for spatial discretization of Fig. 5.1a. However as demonstrated in the preceding sections, virtually identical results can be obtained using the implicitfiniteelement computer program developed in this research. 128 CHAPTER 6 SITE DESCRIPTION 6.1 Introduction In-situ and laboratory test data have been obtained from several sites in Western Canada (Fig. 6.1). Each of these sites is quite distinct in terms of the geologic and depositional environments and the confining stresses encountered in the zone of interest. Short descriptions of the near surface geology of the test sites can be found in the following sections. A list of abbreviations used to designate site characterization activities can be found in Table 6.1. Out of the tests carried out at the sites described in thefollowingsections, datafromSCPTU, SBPMT and laboratory triaxial tests on samples extracted by groundfreezingare only used in this research. Study Sites 1. J-Pit 2. Massey Tunnel (South) 3. KIDD #2 4. L L Dam 5. South-eastern BC Mine Site Logan Lake (4) ® ^Vancouver (2,3) South-eastern BC® Mine Site (5) Fig. 6.1. Location of Test Sites 129 Chapter 6: Site Description Table 6.1. List of Abbreviations Abbreviation Activity Abbreviation Activity CPTU Piezocone Penetration Test LDS Large Diameter Sampling DCPT Dynamic Cone Penetration Test SBPMT Self-boring Pressuremeter Test DMT Dilatometer Test SCPTU Seismic CPTU FPS Fixed Piston Sampling SPT Standard Penetration Test Geo Geophysical Logging (Gamma-Gamma) SPT/E SPT with Energy Calibration 6.2 J-Pit The operation at the facilities of Syncrude Canada Ltd. Near Fort McMurray, Alberta involves extraction of crude petroleum from naturally occurring oil sand. The tailings primarily comprisefinequartz sand (D «0.15 mm) obtained after the extraction of crude petroleum. These 50 tailings are deposited underwater in an abandoned pit by hydraulic means. The deposition was carefully carried out so as to provide a very loose foundation for undertaking a full scale experiment on static liquefaction in Phase III of the CANLEX Project. An 8 m high earth embankment and a 10 m high compacted sand retaining structure upstream of the embankment was constructed on the loose foundation and tailings sand was quickly deposited behind the compacted sand with an intention to trigger static liquefaction in the loose foundation. An extensive testing program was undertaken to characterize the foundation deposit at J-Pit. The layout of the site and the location of sampling and in-situ testing are shown in Fig. 6.2. The cone tip resistance, q , in x the foundation deposit at J-Pit was found to vary between 0.8 and 3.0 MPa and a friction ratio, Rf, between 0.5 and 1.0% was measured (Fig. 6.3). Water table was encountered at a depth of about 130 Chapter 6: Site Description Fig. 6.2. Site Layout: J-Pit 1.25 m during the self-boring pressuremeter testing. Also shown on the q versus depth plot are T the contours of expected relative density, DR, for medium compressibility sands using the following relationship (Jamiolkowski et al., 1988) q where q is in kPa, D T R T = 172 o ' ° v 5 1 exp (0.0273 D ) R (6.1) is in percent and o ' is the geostatic effective vertical stress in kPa. No v seismic measurements were carried out in the vicinity of the location of ground freezing. However, 131 Chapter 6: Site Description q , MPa .0 5 0 T % 1.25 U ,m 2 2.5 0 15 Notes: • U denotes penetration pore water pressure measured behind the tip in meter of water column • Solid circles denote the depths of SBPMTs that are analyzed later. • Range of CPTU data are from tests at locations shown in Fig. 6.2. 2 Fig. 6.3. Range of CPTU Data from J-Pit the shear wave velocities at several other locations within the foundation deposit were found to vary between 100 and 140 m/s. Iravani et al. (1995) give more geotechnical details of this site and the details of the field experiment can be found in Lefebvre et al. (1995). 6.3 Fraser River Delta Data from two sites in Fraser River Delta are used in this study (Fig. 6.4). The Fraser River Delta extends about 15 to 23 km from a gap in the pleistocene uplands to the strait of Georgia. Evidence of paleoseismic liquefaction have been detected at several locations in the sand layers of the Fraser River Delta (Clague et al., 1992). The deltaic deposits reach a maximum known thickness of236 m and overlie Pleistocene glaciogenic deposits. The upper-most layer of the delta consist of a complex sequence of distributary channel sands capped by intertidal and flood plane Chapter 6: Site Description silt to the west and peat to the east. These deposits in turn overlie a unit of seaward dipping silt and sand deposited on delta slope. More details on local geology can be found in Monahan et al. (1995). The location of sampling and in-situ testing at KTDD # 2 and Massey Tunnel (south) are shown in Fig. 6.5. A majority of the sampling and testing at these sites was undertaken in Phase H of the CANLEX Project and described in more detail by In-Situ Testing Group (1995 , 1995 ). a b 6.3.1 KTDD #2 KTDD # 2 is situated about 8 km south of Vancouver. At this site a siltyfinesand layer is found below a 1.5 m thick desiccated silt cap. Fine to medium Fraser River Sand is found at a depth of 6 to 8 m which extends to a depth of about 17 m. The depth of water table is affected by tidal effects at KIDD # 2. Water table was encountered about 1.5 m below ground surface at the Fig. 6.4. Surface Geology of Fraser River Delta (after Clague et al., 1992) 133 Chapter 6: Site Description • FPS/Geo OLDS • SBPMT • SCPTU • SPT/E O Vibrocore KD9302 • ^ KDPM9401 Ground Freezing # KD9301 /KD9306 KDPM9402 &K046 MSSC 9402^ KD9401 • M225 • Ground Freezing MSSC 9406„ •MSPM9401 • MSSC9401 M245, MSSC9405 \ « • Magnetic North MSSC* 9404 .LS20 Massey Tunnel (south) MSPM9402 ISSC 9403 Fig. 6.5. Sampling and Testing Locations at KIDD # 2 and Massey Tunnel (south) time of SBPMT K046, the data from which are analyzed later in this research. The range of SCPTU data measured at KIDD # 2 are shown in Fig. 6.6 together with the contours of D for R medium compressibility sand obtained from Eq. (6.1). The sand unit at this location is less than 4500 years old (Monahan et al., 1995). 134 Chapter 6: Site Description q ,MPa T Rp% U ,m 2 V ,m/s s Note: solid circles denote the depths of SBPMTs analyzed later Fig. 6.6. Range of SCPTU Data From KIDD # 2 6.3.2 Massey Tunnel (South) Massey Tunnel is situated about 16 km south of Vancouver, BC. The stratigraphy at Massey Tunnel includes layers of clean loose to medium dense sand between 7 and 15 m depth: the zone of interest. The depth of water table was about 2.2 m during SBPMT M245. Data from this test are analyzed later in this research. At this site water table is affected by tidal fluctuations. Fig. 6.7 shows the range of SCPTU measurements at this site together with the contours of D for R medium compressibility sand. Radio Carbon dating of organic materials found in a vibrocore sample from this site yields an age of less than 100 years for the sand layer of interest. 135 Chapter 6: Site Description q ,MPa T R^/o U ,m 2 V ,m/s s Note: solid circles denote the depths of SBPMTs analyzed later. Fig. 6.7. Range of SCPTU Data from Massey Tunnel (south) 6.4 L L D a m LL Dam is a part of the Highland Valley Copper Mine system located approximately 15 km west of Logan Lake, BC. This dam is essentially a compacted clay till core, center line constructed tailings dam. The deposits in the upstream side - where the tests analyzed in this research (SBPMT3: Fig. 6.8) were carried out - are primarily a uniform (D «0.14 mm with 7 to 9 % fines), 50 double cycloned, loose to medium dense uncompacted underflow tailings sand. Although no information on grain compressibility is available as yet from this location, it appears from the angularity of tailings sand generated in a similar operation found at a nearby location of Lornex (Enos et al., 1982) that an assumption of high compressibility is appropriate for the deposit. Water table was struck at depth of about 3.69 m during SBPMT3. The depth of water table is however 136 Chapter 6: Site Description SCPTU4> Geo" SBPMT2 •<> 15 cm CPTU ~SBPMT3 2 o Ground Freezing LDS To Waterline (-17 m from Center) SBPMT1* r CPTU3 • CPTU1 • Geo / • SCPTU20 SPTl Fig. 6.8. Sampling and In-situ Testing Locations at L L Dam found tofluctuateby about 1.25 m at this site. The range of CPTU measurements within the zone of interest are plotted in Fig. 6.9 for the tests shown in Fig. 6.8. Since for the same values of D R and o ', q for high compressibility sand is about 85 percent of the corresponding value for a v x q ,MPa x Rp% U ,m 2 V ,m/s s Note: solid circles denote the depths of SBPMTs analyzed later. Fig. 6.9. Range of SCPTU Data: L L Dam Chapter 6: Site Description / medium compressibility deposit (Robertson and Campanella, 1986), the relationship in Eq. (6.1) was modified for plotting the relative density contours in Fig. 6.9. The shear wave velocities from SCPTU4 are not available. As a result, the shear wave velocities measured in SCPTU20 are only plotted in the figure. More details about this site can be found in Bigger and Robertson (1996). 6.5 South Eastern British Columbia Mine Site The active iron tailings pond, where the data used in this study were measured, is essentially an impoundment supported by an upstream constructed tailings dam. The original starter dyke is founded on primarily competent ground comprising glacial till and sand and gravel. The spigotted iron tailings are typically cohesionless soil offinegradation with 50% or more passing # 200 sieve of which about 15% is of "clay sizes," i.e., less than 0.002 mm. Specific gravity of the iron tailings is as high as 4.2. The tailings are generated in a crushing operation and are therefore expected to be angular. Assumption of high compressibility is thus appropriate for the deposit. Cone tip resistance between 0.25 and 2.5 MPa and friction ratio between 1.25 and 2.5% was measured at this site (Fig. 6.10). Very loose weak zones are evident within the tailings as indicated by the relative density contour shown in Fig: 6.10. It is interesting to note that the shear wave velocities at this site (range: 130 to 245 m/s) do not generally reflect the loose state of packing of the deposit and are as high as those at Massey Tunnel The ground water table is struck at a depth of 5.0 m during the SBPMT performed recently at this site, data from which are analyzed later in an illustrative example. A sudden flow failure occurred in 1991 in a section of the tailings impoundment at this location during an incremental rise designed to provide sufficient storage for additional tailings. 138 Chapter 6: Site Description gi'—t-> i I i -> i I i ^- i i ; 1 1 Note: • solid circle shows the location of the SBPMT analyzed later • U is the penetration pore water pressure measured behind friction sleeve in meter of water 3 Fig. 6.10. SCPTU Data from Iron Tailings Impoundment: South-eastern BC Mine Site 6.6 Summary Extensive site characterization programs have been carried out atfivedifferent sites. The deposits at these sites represent fluvial processes ranging from spigotting of tailings to sediment deposition in a natural river channel. A procedure is proposed in Chapter 7 to derive the stress strain behavior of cohesionless soils over a wide range of deformation from SBPMT making use of seismic data from SCPTU and other existing information of rather generic nature. Data from the sites described in this chapter, which represent a wide range relative densities (from dense sand at KTDD # 2 to very loose silt at South eastern BC Mine Site) and compressibility (from subround Fraser River Sand to angular iron tailings) are used later to validate and illustrate the proposed procedure. 139 CHAPTER 7 INTERPRETATION OF SELF-BORING PRESSUREMETER TESTS 7.1 Introduction An approximate calibration procedure for the stress strain relationship of Chapter 4 is proposed in the following section. The procedure is primarily based on back analysis of SBPMT data. Supplementary information from seismic measurements in SCPTU and existing correlations between some of the parameters and q and relative density are also used. T Model parameters representative of the layers exhibiting the stiffest and the softest SBPMT response are derived later using in-situ test data from J-Pit, KIDD # 2 and Massey Tunnel (south) following the proposed approach. These parameters are then used to calculate the response of a single axisymmetric element in a deformation process similar to laboratory triaxial tests. The calculated element response is compared with the corresponding laboratory triaxial test data on samples extracted by ground freezing to validate the proposed procedure for model calibration. To check whether the procedure can be applied in a practical situation where little quantitative information is available over and above SBPMT and SCPTU data, the in-situ test data from L L Dam are analyzed. Very little material specific information is available at present from this location other than the SBPMT and SCPTU data. Undrained plane strain element response is also estimated for a tailings dam site in Southeastern British Columbia from in-situ tests to check whether the results are in agreement with a documented case history of static liquefaction at this location. This exercise is undertaken to provide a field validation for the approximate calibration procedure. Chapter 7: Analysis of Self-boring Pressuremeter Tests 140 7.2 Proposed Framework for Analyzing Self-boring Pressuremeter Data The proposed approximate procedure to calibrate a constitutive model from SBPMT using supplementary measurements from a cone penetration test and existing information about the appropriate values of the model parameters is detailed below. Although the procedure is used together with the stress-strain model described in Chapter 4, the principle can be used with any constitutive model so long as a reasonablefirstguess can be made about the values of the model parameters to begin an iterative procedure to fit the model to the SBPMT data. In fact, such a procedure has been used to back analyze SBPMT in clay (Jefferies, 1988) as well as in sand (da Cunha, 1992) employing other stress strain relationships. The iterative procedure is essentially by manual trial and error at present. However, algorithms such as steepest descent (Parker, 1994, p. 311) can in principle be employed to automate the procedure. To keep the trial and error procedure tractable, a majority of the model parameters is estimated as outlined in the following sections. These estimates are kept unaltered in the subsequent trial and error exercise. Adherence to these guidelines may not be necessary when more precise and pertinent information is available. In the analyses presented later, the entire domain is assumed to be represented adequately by a single set of model parameters for simplicity. In other words, the variability in the horizontal direction is ignored. Also, unless indicated otherwise the following guidelines and discussions are applicable to uncemented granular soils. 7.2.1 Geostatic State of Stress A reasonable estimate of the geostatic state of stress is necessary in order to derive a reliable estimate of the model parameters. If perfect installation of the probe can be ensured the lift off pressure can provide a good estimate of the total horizontal pressure within the deposit prior Chapter 7: Analysis of Self-boring Pressuremeter Tests 141 to cavity expansion. However, when the probe is pushed in during installation, lift off may occur at a total stress considerably higher than the total geostatic horizontal stress. The extent by which lift off pressure exceeds the original total horizontal stress depends on the geometry of the probe. To determine whether pushed-in type of disturbance is present, the vertical installation resistance measured during installation of the self-boring pressuremeter can be used as outlined in Chapter 2. The total geostatic vertical pressure can be found from a reasonable estimate of the unit weight of the deposit and the hydrostatic pressure. As mentioned in § 2.2.1, the total cavity pressure as the cavity is retracted back to its original configuration gives the hydrostatic pressure in sands. It should be noted that the ratio of the original effective horizontal stress to effective vertical stress, i.e., the coefficient of earth pressure at rest (KQ) is believed to be related to geologic history. A value of 0.35 to 0.55 is deemed appropriate for KQ for normally consolidated cohesionless deposits, while a value greater than unity is sometimes encountered in heavily over-consolidated sand (Lambe and Whitman, 1979, p. 100; Wood, 1990, p. 316). Adoption of a value of KQ considerably higher than 0.5 should thus be justifiablefromthe geologic history of the deposit. 7.2.2 Model Parameters The elastic parameters are estimated herefromseismic measurements carried out in an seismic CPT performed at a location adjacent to the SBPMT following § 4.4.2. An approximate estimate of K Q can also be obtainedfromthe secant slope of an unload-reload loop in an SBPMT E in the absence of relevant seismic data following Bellotti et al. ( 1 9 8 9 ) . A majority of parameters pertaining to the irreversible behavior is determinedfroman estimate of the state of packing. These parameters are R , r| and Ar). Limited experience with F F1 the use of the proposed stress-strain model indicates that n can be approximately taken as -0.5 for P Chapter 7: Analysis of Self-boring Pressuremeter Tests 142 granular soils. If the grain compressibility of the deposit is also known parameters C and p can be reasonably estimated. Sand, silt or even clay sized cohesionless particles obtained from crushing operations in the mining industry are usually angular and therefore can be regarded as compressible. In contrast, the particles in a fluvial deposit of granular material (e.g., deposits of Fraser River Sand found at KIDD # 2 and Massey Tunnel) are often sub angular to sub round. Therefore such deposits are usually of medium compressibility. A CPT can provide a good estimate of the in-situ state of packing if the grain compressibility of the deposit is known. Parameter p depends upon mineralogy and X depends on both mineralogy and grain compressibility. These parameters have been evaluated in Chapter 4 for a number of soils. For other soil types X and u should be determined first from a laboratory test such as triaxial extension on a reconstituted sample. Application of the stress-strain model of Chapter 4 in modeling laboratory triaxial compression and extension tests (§ 4.6.2.5) indicates that a value of 2.0 is appropriate for m for hydraulically A deposited sand. Thus, the following approach can be adopted before undertaking trial and error back analysis of SBPMT data: • Assume a value of 0.5 for KQ in the absence of a strong geologic evidence of over consolidation or ageing. From a reasonable estimate of the unit weight of the deposit and water table depth, the state of stress prior to cavity expansion can now be evaluated. • Estimate X and u from element test(s) on reconstituted samples or from available data. • Decide whether the deposit is of high, medium or low compressibility from qualitative information on grain shape. Reasonable estimates for parameters C and p are now available from § • 4.6.1. Estimate the state of packing from cone tip resistance. Reasonable values of K , Rp, r) , SP F1 and Ar| can now be estimated from § 4.6.2. In the absence of reliable information of the Chapter 7: Analysis of Self-boring Pressuremeter Tests 143 existing state of packing prior to cavity expansion, a reasonable value can be assumed for the relative density to start the fitting procedure that should be refined iteratively. • Assume a value of 2.0 for m and a value of -0.5 for n A P An estimate of Kgp is obtained by fitting the calculated response of the stress strain relationship of Chapter 4 in cylindrical cavity expansion to the SBPMT data. The exercise essentially involves changing the value of K S P until a reasonable match between the computed model response is satisfactory. For the same deposit Kgp increases with relative density although not quite in a similar fashion as K G . E Experience indicates that the K S P to ratio can vary between 0.1 to 0.4 depending on the type of sand if the deposit is loose. The ratio is about 1 for dense sand. 7.3 Validation of the Proposed Procedure The approximate procedure for calibrating the stress strain model from in-situ tests has been used with datafromthree sites in this section. To validate the procedure the inferred material behavior is compared with laboratory measurements on undisturbed samples. 7.3.1 J-Pit The SBPMT data exhibiting the stiffest (test no. Cant 12: carried out at 5.18 m depth, SBPMT1) and the softest (test no. Cant 11: at 4.53 m depth, SBPMT1) response were analyzed. These data represent medium dense and loose states of packing, respectively. The measured SBPMT data together with the model response after thefittingprocedure are shown in Fig. 7.1. The match between the model response and SBPMT data is quite good at cavity strain greater than 4%. At smaller deformation the match is poor due to installation related disturbance. According to da Cunha (1992), it is only necessary to fit the model response to SBPMT data between cavity Chapter 7: Analysis of Self-boring Pressuremeter Tests Cavity Strain, % Fig. 7.1. Analysis of SBPMT Data: J-Pit strains of 4 to 10% to derive reasonable estimates of model parameters. The model parameters estimated for Cant 12 are as follows: {^£=500, n =0.5, v=0.2, K =300, n =-0.5, r) =0.624, E SP P F1 ArpO.057, A=0.85, u=0.29, R =0.87, m =2.0, C=0.00028 and p=0.9. The corresponding values F A for Cant 11 are: ^=400, K =175, rj =0.517, Ar|=0.009, A=0.85, Rp=0.98, and C=0.0006. The sp F1 remaining parameters are the same as those for Cant 12. For Cant 11 and 12 the effective vertical stresses prior to cavity expansion are taken as 44 and 54 kPa, respectively, and K„ is assumed to be 0.5. The undrained responses of axisymmetric elements in compression and extension holding horizontal stress constant for the model parameters estimatedfrominverse modeling of SBPMT are shown in Fig. 7.2. For loose sand effective vertical stresses at consolidation of 102 and 22 kPa are used respectively in compression and extension. The corresponding values for the dense layer are 24 and 20 kPa. The effective horizontal stress at consolidation is taken to be one half times the effective vertical stress and a value of 1.5 GPa is assumed as the bulk modulus of the pore fluid. The triaxial laboratory test data for loose and medium dense specimens extractedfromJ-Pit by groundfreezingfor states of stress at consolidation the same as those assumed above are also shown in Fig. 7.2. Samples for tests FS5C1B31 and FS5C1B33 were extractedfrom3.66 m depth. Chapter 7: Analysis of Self-boring Pressuremeter Tests 145 Specimens for FS26C211 and FS6C2B24 were extracted from 3.23 and 2.95 m depths respectively. These depths are not identical to those at which SBPMTs were carried out at J-Pit. In fact, to partially circumvent the problem associated with spatial variability due to non availability of SBPMT and laboratory test datafromthe same location, the stiffest and the softest response of an axisymmetric element inferred from SBPMT have been compared with the stiffest and the softest laboratory triaxial data in this exercise and those undertaken later. Other particulars of the laboratory tests are summarized in Table 7.1. The computed element response for the loose layer with the laboratory measurements in tests FS5C1B33 and FS5C1B31 are in reasonable agreement Axial Strain, % (a) Loose (b) Medium Dense Fig. 7.2. Predicted and Measured Undrained Triaxial Element Response: J-Pit Chapter 7: Analysis of Self-boring Pressuremeter Tests Table 7.1. Particulars of Laboratory Triaxial Tests on Undisturbed Samples Site Test No. e CONS e E J-Pit KIDD #2 Massey FS5C1B31 FS5C1B33 0.811 0.811 FS26C211 FS6C2B24 0.708 0.680 K94F1C70 K94F1C23 0.835 0.886 K94F3C4B2 K94F2C2A 0.946 0.932 M94F4C42 M94F4C43 0.982 0.983 M94F6C7A M94F6C5B 0.914 0.908 MAx/ MTN 0.930/ 0.550 1.077/ 0.715 1.102/ 0.715 ° VCONS kPa a HCONS Test Reference kPa 44 204 22 102 TXE TXC Vaidetal., 1996 Ditto 40 48 20 24 TXE TXC Ditto Ditto 150 120 75 60 TXE TXC Ditto Ditto 156 145 78 72 TXE TXC Addendum to Vaid et al., 1996 110 112 55 56 TXE TXC Vaid et al., 1996 Ditto 124 117 62 58 TXE TXC Ditto Addendum to Vaid et al., 1996 except for the fact that in compression a higher dilation rate is predicted from SBPMT. Also, the derived model parameters for the loose layer indicate a lower relative density than that measured in the laboratory. Since the void ratio estimated in the laboratory is essentially an average value for the entire specimen and does not account for variability within the specimen, such a value may not always represent material behavior. The mechanical response of a loose sand specimen is usually governed by the minimum local value of void ratio within the sample. The computed response for the medium dense layer and'the measurements in laboratory tests FS26C211 and FS6C2B24 are also comparable. The stress history for triaxial elements in numerical simulation and in the laboratory are also quite comparable (Fig. 7.3). The stress paths shown in Fig. 7.3 representing compression and extension tests do not originate from the same point because the samples used in these tests are not consolidated to the same states of initial effective stress. Chapter 7: Analysis of Self-boring Pressuremeter Tests 50 100 147 150 200 0 50 (a /+c /)/2,kPa (1 100 150 200 (3 (a) Loose (b) Medium Dense Fig. 7.3. Predicted and Measured Stress Paths: J-Pit 7.3.2 KTDD #2 Results of simulation of SBPMT datafroma loose ( 1 4 . 3 7 m depth) and a dense layer (16.375 m depth) at KTDD # 2 are presented in Fig. dense layer are as follows: K GE 7.4. The model parameters estimated for the = 1 1 0 0 , n =0.5, v=0.2, K = 1 0 0 0 , n =-0.5, r) =0.79, ArpO.10, E A=0.77, u=0.39, Rp=0.82, m =2.0, C=0.00028 A SP and p=0.9. P F1 The effective vertical and horizontal stress are assumed to be 155 and 108.5 kPa, respectively. Comparison of the vertical resistance encountered during the installation of the self-boring pressuremeter with the cone tip resistance measured at an adjacent location following the approach outlined in § 2.3.1 does not indicate that the probe was pushed in the ground during the installation process to cause a high value of KQ. Such a value of KQ is however consistent with the corresponding values reported by Sully (1991) and da Cunha ( 1 9 9 4 ) for medium dense to dense deposits in the Fraser River Delta. The model parameters for the layer at 14.37 m depth are: KG =750, K = 4 0 0 , E s p r) =0.710, Ar|=0.06, F1 Chapter 7: Analysis of Self-boring Pressuremeter Tests 1400 1200 1000 . cit 2a | 800 OH & 600 I 400 200 0 0 2 4 6 Cavity Strain, % 8 10 Fig. 7.4. Analysis of SBPMT Data: KIDD # 2 A=0.85, Rj=0.98, and C=0.0006. Other parameters are the same as those for 16.375 m depth. The effective vertical and horizontal geostatic stresses at 14.37 m depth are assumed to be 137 and 68.5 kPa, respectively. It should be noted that the quality of fit is slightly affected by the simplicity of the interpretation procedure described in § 7.2.2. In this simplified approach estimates for some of the model parameters are not iteratively refined. However, as shown later, the quality of fit is comparable or better than those obtained in many other studies on interpretation of SBPMT. The calculated axisymmetric element responses for the model parameters estimated from back analysis of SBPMT data from KIDD # 2 using 1.5 GPa as the bulk modulus of pore fluid are 149 Chapter 7: Analysis of Self-boring Pressuremeter Tests shown in Fig. 7.5. For the loose sand the effective vertical stress at consolidation in compression and extension are taken as 145 and 156 kPa, respectively. The corresponding values for the dense layer are 120 and 150 kPa, respectively. The effective horizontal stress at consolidation is one half times the effective vertical stress. The corresponding laboratory triaxial data for loose and dense specimens extracted by in-situ freezing from the site are also shown in Fig. 7.5. Samples K94F1C23, K94F1C70, K94F2C2A and K94F3C4B2 were extracted from depths of 12.78, 16.46, 600i /1 Data(K94FlC70 \ 11 ii andK94FlC23) /l /A Prediction /'i\ Data(K94F3C4B2 andK94F2C2A) Prediction 400 60 GO Ji CO I /1 /1 /1 /1 /1 /1 * 200 / / f * s f i i i y f 0 - • * -200 1.0 0.5 m _£. ._S...l. t \ \ \ 1 \ I 5 0 /\ / \\\ \ 1 1 i \ \ f N i -0.5 •1.0 -5 i i J -2.5 2.5 (a) Loose -5 -2.5 i i /I I V A \\ A \Y () i 2.5 Axial Strain, % (b) Dense Fig. 7.5. Predicted and Measured Triaxial Element Response: KTDD # 2 5 l f>U Chapter 7: Analysis of Self-boring Pressuremeter Tests 13.20 and 14.80 m, respectively. Other particulars of these tests can be found in Table 7.1. A higher pore water pressure is predicted in compression for loose sand while a lower pore water pressure is computed in extension for the dense layer. Otherwise the stiffest and the softest computed response are in approximate agreement with the corresponding laboratory triaxial data. A possibility of an exact match between the measured triaxial response and those inferred from SBPMT data can be virtually precluded given the natural variability of the deposit. Fig. 7.6 shows the observed triaxial stress paths and those inferred from SBPMT. The comparison again reveals approximate similarity. 7.3.3 Massey Tunnel (south) SBPMT data from a loose (14.205 m) and a medium dense layer (13.205 m) from Massey Tunnel (south) are interpreted in a manner similar to that used for J-Pit and K I D D # 2. The (0 /+0 ')/2,kPa (1 (a) Loose (3) (b) Dense Fig. 7.6. Predicted and Measured Stress Paths: K I D D # 2 Chapter 7: Analysis of Self-boring Pressuremeter Tests 151 measured data and the model response to cylindrical cavity expansion for the optimum values of the model parameters are plotted in Fig. 7.7. The effective vertical stresses prior to cavity expansion for the medium dense and the loose layers are assumed to be 128 and 142 kPa with a K„ of 0.5. Parameters estimated from the exercise for the medium dense deposit are as follows: KG =850, n =0.5, v=0.2, K =500, n =-0.5, r) =0.75, An=0.08, 1=0.11, u=0.39, R =0.85, E E SP P F1 F m =2.0, C=0.00028 and p=0.9. For the loose layer Kc =650, K =220, r| =0.69, An=0.05, A E SP F1 ^=0.9 and C=0.0006 are used. Values of the remaining model parameters were taken as the same as those for the medium dense layer. The simplified procedure outlined in § 7.2.2 was adhered to in the back analysis and estimates of some of the model parameters were not iteratively refined. As mentioned earlier, a better quality of fit between the model response in cylindrical cavity expansion and SBPMT data is expected if the estimates of all model parameters are iteratively 1000 800 22 600 CO % 400 200 0 0 2 4 6 Cavity Strain, % 8 10 Fig. 7.7. Analysis of SBPMT Data: Massey Tunnel (south) Chapter 7: Analysis of Self-boring Pressuremeter Tests refined in inverse modeling. 152 However, such an approach is cumbersome unless the fitting procedure is automated. Uniqueness of the solution also is not guaranteed in such a procedure. The undrained triaxial compression and extension behavior is computed using the model parameters estimated above for the medium dense layer and effective vertical stress at consolidation of 117 and 124 kPa, with the corresponding values in the horizontal direction being half as much. Triaxial response is also computed for the model parameters representative of the loose layer using a'vcoNs ofl 12 in compression and 110 in extension. The corresponding values in the horizontal direction are one half times the values in the vertical direction. The bulk modulus of pore fluid is taken as 1.5 GPa for both medium dense and loose layers. The computed triaxial behavior for the medium dense and the loose layers are plotted in Fig. 7.8. The corresponding laboratory triaxial test data for loose and medium dense sand for undisturbed samples extracted by ground freezing from Massey Tunnel are also shown in Fig. 7,8. Specimens M94F4C42, M94F4C43, M94F6C7A and M94F6C5B were obtained from 10.36, 10.54, 11.79 and 10.95 m depths. Other particulars of the laboratory tests can be found in Table 7.1. The responses of the loose and medium dense deposits inferred from SBPMT are qualitatively similar to the corresponding laboratory measurements. The stress paths in the computed triaxial response and those measured in the laboratory are also comparable for both loose and the medium dense layers (Fig. 7.9). However, a slightly higher rate of dilation is predicted from SBPMT. Consequently a stiffer deviatoric response is predicted for both loose and medium dense layers. 7.3.4 Limitations of the Proposed Procedure The procedure proposed in § 7.2 for estimating monotonic mechanical response of cohesionless depositsfromSBPMT making supplementary use of in-situ seismic measurements and 153 Chapter 7: Analysis of Self-boring Pressuremeter Tests 600 D ata(M9^IF6C7A anid M94F6C5B) - - - Pi ediction Data (M94F4C42 andM94F4C43) Prediction 5a1400 / / / / •/ / CO CO / / / | 200 / fi 0 -200 ~-— \ >. i \ I i \ i . 7_> * 1 1 \ i \i i \ ! N \ N \i / \ \ \ \ \ * V -0.5 i I \ V ' ' * ' » ' \ ' f "^-^ Vf 4 \ ^ \ V s \ \ \ \ \ -1.0 5 -2.5 () 2.5 5 -2.5 0 2.5 Axial Strain, % (a) Loose (b) Medium Dense Fig. 7.8. Predicted and Measured Triaxial Element Response: Massey Tunnel (south) existing information on the appropriate values of the model parameters has been applied to three different sites. Except for a few mismatches between the triaxial behavior inferred from in-situ tests and laboratory test data, the predicted mechanical responses are found to be generally comparable to those measured in the laboratory. The deviations are not of a systematic nature. As noted earlier, a perfect match can be virtually precluded between response predicted from SBPMT and that measured in the laboratory due to spatial variability. Nevertheless, it should be noted that the Chapter 7: Analysis of Self-boring Pressuremeter Tests (o '+o )/2,kPa , (1) (3) (a) Loose (b) Medium Dense Fig. 7.9. Predicted and Measured Stress Paths: Massey Tunnel (south) procedure is approximate and the following factors may affect the accuracy of the results. • In the finite difference (or finite element) discretization for computing the response of cylindrical cavity expansion all the elements are assigned the same set of model parameters. This may not be appropriate in a highly variable deposit. • Estimates of D and R from a nearby SCPTU are also approximate because of spatial variability. • The correlations between C and <b 'and D (Figs. 4.10 and 4.11, respectively) are imprecise. • Knowledge about the exact nature of influence of the state variables (i.e., states of effective R stress and packing), mineralogy and grain characteristics on the values of some of the model parameters, e.g., n , n and m is limited at present. E • P A To keep the back analysis procedure simple, estimates of some of the model parameters are Chester 7: Analysis of Self-boring Pressuremeter Tests 155 not iteratively refined. The precision of the approximate model calibration procedure of § 7.2.2 is also limited by the capability of the stress-strain model developed in Chapter 4. Consequently, the comments of § 4.12 impose further limitations to the proposed calibration procedure. Nor is the procedure recommended for SBPMT in • shallow layers in an undulated topography (The problem of back-analysis of SBPMT from the shallow layers of a site with undulated topography cannot be approximated as an axisymmetric one unless the axis of the self-boring pressuremeter matches one of the principal directions of the state of geostatic effective stress. Although SBPMT data from a site with highly undulated terrain can in principle be analyzed with a three dimensional numerical model, such an approach may be difficult for the manual trial and error procedure developed earlier in this chapter.), and • cemented sand deposits (in this case the stress strain model of Chapter 4 has to be modified to account for a cohesion intercept in effective stress space). 7.4 Application of the Proposed Procedure The procedure proposed in § 7.2 is used with in-situ test datafromtwo different tailings dam sites. No laboratory element test data are availablefromthese sites pertaining to undisturbed samples that can be used for further direct validation of the approach. The exercise involving insitu test datafromthe L L Dam site at the moment serves the purpose of an illustrative practical application at a site from which little information is available in addition to the in-situ tests. Undisturbed (frozen) samples have been extractedfromthe site and laboratory testing of these specimens is scheduled to take place in the near future. The results of analysis of LL Dam SBPMT Chapter 7: Analysis of Self-boring Pressuremeter Tests 156 data can thus be viewed as a "class A" prediction of the future laboratory tests. There is a well documented case history of staticflowfailure at the other site in South-eastern British Columbia. SBPMT data from this site are analyzed to provide afieldvalidation of the proposed procedure. 7.4.1 L L Dam As mentioned earlier, no information is available from this site about the grain characteristics (and therefore about the compressibility) of the deposit. However, information from a similar mining operation from a nearby location of Lornex (Enos et al., 1982) indicates that the material is likely to be angular and consequently compressible. The interpretation of the SBPMT is therefore based on the assumption of high compressibility. As earlier, the SBPMT data showing the stiffest (8.53 m depth) and the softest (5.48 m depth) response are analyzed representing medium dense and loose states of packing, respectively. The estimated model parameters for the suffer layer are: K<j =480, n =0.5, v=0.2, K =200, n =-0.5, r] =0.79, ArpO.065, A=0.8, u=0.4, E E SP P F1 R =0.86, m =2.0, C=0.0008 and p=0.65. In the absence of material specific information, the F A values of X and u of Hilton Mines Tailings (see Table 4.3) are used. Hilton Mines tailings are believed to be of similar compressibility to the tailings sand found at the L L Dam site because the milling operations at these locations are similar. An estimate of 4>cv for tailings sandfroma copper mine operation at Lornex (situated near LL Dam) is also used in back analysis of SBPMT. The following parameters are found appropriate for the loose layer: K Q ^ I O , K =120, r| i=0.719, sp F Ar)=0.029, R^O.95 and C=0.002. The remaining parameters were taken as the same as those for the medium dense layer. For the stiffer and the softer layers the effective vertical stresses prior to cavity expansion are assumed to be 110 and 80 kPa, respectively, with Ko=0.5. The SBPMT data and the model response for the parameters listed above are plotted in Fig. 7.10. Chapter 7: Analysis of Self-boring Pressuremeter Tests 157 600 ed 2a I 400 I a 200 °0 2 4 6 8 10 Cavity Strain, % Fig. 7.10. Analysis of SBPMT Data: L L Dam The estimated model parameters are then used to predict the expected range of element undrained behavior of undisturbed samples in triaxial compression and extension as shown in Fig. 7.11. In these calculations the same stress states are assumed at the end of consolidation as those in the analyses of SBPMT. A value of 1.5 GPa is assumed for the bulk modulus of pore fluid. The corresponding triaxial stress paths are shown in Fig. 7.12. The results are qualitatively similar to typical triaxial undrained behavior of loose to medium dense sands. In spite of the fact that very little material specific information is available at present for the deposit at L L Dam, the stress strain 158 Chapter 7: Analysis of Self-boring Pressuremeter Tests D ——Medium Dense Loose 50 50 0 100 150 (o /+c /)/2, kPa (1 200 (3 Fig. 7.12. Stress Path in Triaxial Tests Inferred from SBPMT at LL Dam response over a wide range of deformation could be inferred from SBPMT data following the proposed procedure. 7.4.2 South-eastern British Columbia Mine Site SBPMT and SCPTU were conducted at a location within the vicinity of a static liquefaction flow failure. Although no element isotropic compression test data are available from this site that the writer is aware of, the assumption of high compressibility appears to be appropriate given the fact that the tailings are generated from a crushing operation at a nearby plant. Limited information available from the location indicates that the saturated unit weight of the iron rich tailings found at this site can be as high as 25 kN/m . In the analysis of the cavity expansion data at a depth of 3 5.7 m, the vertical effective stress prior to expansion of the cavity is taken to be 130 kPa with Ko=0.5. As in case of LL Dam, A and u are assumed to be 0.8 and 0.4, respectively. From back analysis of the stiffest SBPMT cavity expansion response measured at this location, the following model parameters were derived: K =565, ^=0.5, v=0.2, 1^=60, n =-0.5, r| =0.66, Ar|=0, GE P F1 Chapter 7: Analysis of Self-boring Pressuremeter Tests Cavity Strain, % Fig. 7.13. Analysis of SBPMT Data: South-eastern BC Mine Site A=0.8, u=0.4, RjpO.99, m =2.0, C=0.002 and p=0.65. As mentioned earlier, the measured shear A wave velocity at this site (140 to 245 m/s) is much higher than what is typical of a very loose uncemented normally consolidated deposit of quartz or feldspar sand. As a result, the value of listed above is higher than expected. The SBPMT datafrom5.7 m depth and the model response for the parameters listed above are shown in Fig. 7.13. Fig. 7.14 presents the undrained plane strain element response holding o (3) constant for the set of model parameters estimatedfromin-situ tests. In compression and extension pore water pressure virtually equal to the effective confining pressure is generated resulting in a significant reduction in inter-granular contact. From a limited amount of laboratory test data on the material at this location it is apparent that the tailings are essentially without cementation. As a result, virtually the entire shear strength of the material is lost when the inter-granular contacts are lost due to the generation of pore water pressure. Such a behavior is also observed in the laboratory in monotonic triaxial compression and extension tests on very loose specimens of granular soils (see, e.g., Ishihara, 1993). The fact that both in compression and extension, the material exhibits negligible residual shear strength in undrained loading is in qualitative agreement with the incidence of static liquefaction in the past at this location. Chapter 7: Analysis of Self-boring Pressuremeter Tests 160 Axial Strain, % Fig. 7.14. Plane Strain Element Behavior for South-eastern BC Mine Site The performance of existing correlations for estimating undrained strengthfromin-situ index measurements, e.g., SPT ( N j ) ^ and CPT tip resistance at this site also may be of interest. The upstream constructed tailings dam was analyzed using a value of undrained strength of 8 kPa for the liquefied tailings. The value of the undrained strength was inferredfromavailable (NJ^ and CPT data following Seed and Harder (1990). The stability analysis using the residual strength of the tailings gave a misleading and unconservative factor of safety of 2.0 in static undrained loading. 7.5 Advancement of the State-of-the-art The current state-of-practice of back analysis of SBPMT is essentially based on isotropic linear elastic perfectly plastic material behavior (Salencon, 1969; Hughes et al., 1977; Carter et al., 1986). As mentioned earlier, the results of this approach are applicable only in plane strain problems. Also, the fact that the derived properties pertain to a certain average value of Chapter 7: Analysis ofSelf-boring Pressuremeter Tests deformation, which must characterize the entire deforming domain, presents additional difficulty in their application. To illustrate the point, cylindrical cavity expansion response from Carter et al. (1986) for ( b ' ^ c y (i.e., a dilation angle of zero) is plotted as the contractant-dilatant boundary in Fig. 7.13. The only possible inference from the fact that the SBPMT measurement is below the response for d / ^ c v is that a very large deformation is needed to mobilize a value for the effective stress friction angle equal to 4> . cv It can also be concluded qualitatively from the result that the deposit is contractive and expected to exhibit strain softening in undrained distortion (Hughes et al., 1997). No quantitative estimate of the drained or undrained strength of the layer is possible in this case because the average deformation measure for the entire deforming domain is difficult to evaluate. In contrast, the procedure proposed in § 7.2 gives quantitative estimates of soil strength. However, the conclusions of the procedure developed in this study are in qualitative agreement with those from the much simpler Carter et al. (1986) approach. The approaches proposed by Manassero (1989) and de Souza Coutinho (1990) on the other hand aim at deriving the monotonic behavior of sand over a wide range of deformation instead of isolated soil properties such as (J)'. Therefore, these approaches may appear to be very similar to the procedure proposed in § 7.2. However, as pointed out in Chapter 2, unlike the procedure proposed in this research, adoption of Manassero (1989) and de Souza Coutinho (1990) may lead to a considerable under estimation of soil strength due to the rigid plastic, stress path and stress level independent isotropic constitutive assumption. In fact, the writer is aware of no earlier attempts of deriving the stress-path and loading direction independent estimates of the mechanical response of granular soils from self-boring pressuremeter. 161 162 Chapter 7: Analysis of Self-boring Pressuremeter Tests 7.6 Misfit Criterion In this section the lack of fit between model response in cylindrical cavity expansion and the pressure expansion curve measured in field SBPMT is quantified using the following approximate measure for misfit 10 (1) where P is the effective cavity pressure measured in the SBPMT and at -c^air^n % (n = 4, 5, 6, 7, n 8, 9, 10) and P is the corresponding value for model response. It should be noted that C 0 = n indicates a high-quality match between SBPMT data and model response. It needs to be emphasized that the simple misfit criterion devised above cannot be viewed as the means of ensuring uniqueness in the solution to the inverse modeling problem. The value of C is only used as a guide to assess whether the model could befittedto the data satisfactorily (by determining whether the value of C could be minimized) and to compare the quality offitobtained in different curvefittingexercise (by identifying the result that gives the minimum value of Q. Data between cavity strain of 4 and 10 % are only considered because as noted by da Cunha (1994), at smaller deformation installation related disturbance can affect the SBPMT data, while at higher values error due to the assumption of cylindrical cavity expansion becomes more pronounced. Table 7.2 presents typical upper bound values of C obtained in this study and those obtained in a few other studies involving back analysis of SBPMT. It is evident from the comparison that the quality of fit obtained in this research is comparable or better than those in other studies of similar nature. Nevertheless, there is an opportunity for improvement in the quality offitwhile using the stress-strain relationship developed in this research by refining the estimates Chapter 7: Analysis of Self-boring Pressuremeter Tests Table 7.2. C in Studies Involving Back Analysis of SBPMT in Sand c Stress-strain Model Approach Reference 0.03 Chapter 4 Manual inverse modeling This study Fraser River Sand (KIDD #2) 0.05 to 0.10 Chapter 4 Manual inverse modeling This study Fraser River Sand (Massey Tunnel) 0.06 to 0.10 Chapter 4 Manual inverse modeling This study LL Dam 0.05 to 0.10 Chapter 4 Manual inverse modeling This study 0.02 Chapter 4 Manual inverse modeling This study 0.04 to 0.12 Isotropic linear elastic perfectly plastic Manual inverse modeling da Cunha (1994) Ticino Sand (calibration chamber data) 0.20 Non-linear elastic plastic log-log (see Notel) Alluvial sand: Perth, Australia 0.15 Non-linear elastic perfectly plastic Manual inverse modeling Fahey and Carter (1993) King's Lynn Sand 0.22 Isotropic linear elastic perfectly plastic log-log Fahey and Randolph (1984) Soil Type J-Pit South Eastern BC Mine Site Fraser River Sand (Laing Bridge) Note: 1. 2. Yu (1994): see Note 2. In the log-log approach model parameters are related to slope of the effective cavity pressure versus cavity strain plot in the log-log space. Fig. 7, op. cit., without correction for finite length of cavity (in the measurements a probe with length to diameter ratio of 6 was used for which no correction is necessary as per discussions in Chapter 2). of parameters such as n , n , and p during the back analysis. The values of these parameters are E P kept unchanged at present because of lack of experience with the use of the stress strain relationship used in this research and to keep the iterative procedure simple. Chapter 7: Analysis of Self-boring Pressuremeter Tests 164 7.7 Index of Soil Behavior Soil behavior indices have great practical value in identifying problem deposits. Since most of the parameters of the stress strain relationship of Chapter 4 either vary over a limited range of values or do not have a very pronounced influence on the mechanical behavior of granular soils, the values of and K S P are significant and can be viewed as approximate indices of the stability of hydraulic deposits of sand. Examination of the results indicates the following. • Low values of K S P consistently lead to the development of greater pore water pressure in undrained monotonic distortional loading. As a result low values of the parameters can be viewed as an indicator of potential flow slide hazard. • Contrary to the suggestions of some researchers (e.g., Robertson et al., 1 9 9 2 ; Fear and Robertson, 1 9 9 5 ) higher values of small strain shear modulus (i.e., higher values of K Q ) E do not necessarily indicate a lower flow slide potential. For example, the shear wave velocity in the slide mass at the South-eastern BC Mine Site is higher than that measured at J-Pit. However, aflowslide developed at the South-eastern BC Mine Site, whereas an attempt to trigger aflowslide at J-Pit was not successful (Lefebvre et al., 1995). A large data-base is already in existence comprising in-situ small strain datafrommine sites that support the conclusion that the small strain stiffness is not an unambiguous index of soil behavior. However, examination of small strain in-situ test data is in the scope of another research. • In fact, if the remaining factors are kept unchanged, an exercise of inverse modeling of the given SBPMT data set with increasing small strain shear modulus would lead to a smaller value of K K G E /K S P S P and consequently would indicate a higher flow slide hazard. The ratio of may thus be viewed as a more sensitive index of flow slide potential than K . S P Chapter 7: Analysis of Self-boring Pressuremeter Tests Table 7.3. Estimated Values of and K , and Flow Slide Potential SP Remarks K P Flow Slide Potential Flow slide could not be triggered at J-Pit State Site 165 S J-Pit Medium Dense Loose 500 400 300 175 1.7 2.3 Low Medium KIDD #2 Dense Loose 1100 750 1000 400 1.1 1.9 Very low Low Massey Tunnel (south) Medium Dense Loose 850 650 500 220 1.7 3.0 Low Medium LL Dam Medium Dense Loose 480 410 200 120 2.4 3.4 Medium Medium South-eastern BC Mine Site Very Loose 565 60 9.4 High Flow slide occurred at this location However, more research is necessary before conclusive guidelines can be developed along these lines. A summary of the values of K G and K E S P for the sites examined earlier can be found in Table 7.3 together with qualitative conclusions about their flow slide potentials. 7.8 Summary A procedure is developed to estimate the monotonic stress-strain behavior of cohesionless soils from SBPMT using supplementary small strain information (from shear wave velocity) and an estimate of relative density from SCPTU together with the existing knowledge about the appropriate values of some of the model parameters. The procedure has been applied to three different sites - J-Pit, K I D D # 2 and Massey Tunnel (south) - representing different sand types, depositional environment and state of packing to estimate the undrained triaxial element stressstrain behaviorfromin-situ tests. Comparison of the estimated element behavior with laboratory Chapter 7: Analysis of Self-boring Pressuremeter Tests 166 test data on undisturbed samples extracted from the sites via ground freezing indicates a reasonable agreement between the estimated and measured element response providing a direct validation of the proposed procedure. It has been demonstrated that the procedure can be used in situations where very little information is available over and above the in-situ test data to draw reasonable inference regarding monotonic stress strain behavior over a wide range of deformation. In-situ test data from LL Dam have been used in this "class A" prediction. Undisturbed samples have already been extracted from this site for future laboratory testing. The laboratory test data can be compared to the results obtainedfromback analysis of SBPMT later. SBPMT datafroma mine site in South-eastern BC have also been analyzed following the proposed procedure. Negligible residual strength results for the model parameters derived in this exercise in both compression and extension for a plane strain element in a deformation process keeping the minor principal stress constant. The results are in agreement with the incidence of static liquefaction at this site providing further validation to the proposed approach. The procedure can thus be used to estimate the behavior of cohesionless deposits over a wide range of deformation. The proposed approach for characterizing cohesionless deposits is quite economical in comparison with the available technology of extraction of undisturbed samples of cohesionless soil and laboratory testing and the results are more precise than the alternative empirical approaches (e.g, Stark and Mesri, 1992) based solely on in-situ index tests. 167 CHAPTER 8 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS 8.1 Summary and Conclusions The purpose of this study was to develop an economical and practical procedure for estimating the deformation behavior of granular deposits from in-situ tests. The only reliable approach that is available at present for this purpose, namely, extraction of high quality undisturbed samples and laboratory testing, is often prohibitively expensive for saturated sand below the ground water table. In-situ cavity expansion tests using a self-boring pressuremeter provide stress-strain data over a wide range of deformation and the test can be performed in the ground without causing appreciable disturbance to the surrounding soils. In addition, a comparatively larger volume of soil is sampled and the test provides reliable data amenable to analytical interpretation. As pointed out in Chapter 2 of the dissertation, the existing methods of interpretation of SBPMT data employ simple stress-strain relationships that only account for the effect of the state of packing on dilatancy. The results from these procedures are therefore not applicable in deformation problems of a general nature. For a wider application, SBPMT data must be interpreted employing a stress-strain relationship that accounts for inherent anisotropy, stress, strain and stress path dependency. All of these factors are known to affect the mechanical behavior of granular soils to a significant extent over and above the state of packing. Details of the numerical tools that can be used to analyze cylindrical cavity expansion are summarized in Chapter 3. A stress-strain model is proposed for cross anisotropic frictional materials in Chapter 4 of the dissertation taking cognizance of the effects of state of packing, inherent anisotropy, stress, Chapter 8: Summary, Conclusions and Recommendations 168 strain and stress path dependency. An effort was made to keep the model as simple as possible because inverse modeling of SBPMT data using a constitutive model with a large number of model parameters can pose an extremely difficult problem. To minimize the impact of non-uniqueness on the inverse modeling exercise, the existing knowledge regarding the appropriate values of the parameters of the proposed model is also summarized in Chapter 4. Two alternative numerical tools have been developed, compared and validated in Chapter 5 for analyzing SBPMT data using the proposed stress strain model. One of these tools is based on a commercially available explicit finite difference software, FLAC, while the other code is developed in this study based on implicit finite element. A procedure based on the stress-strain relationship developed in Chapter 4 is proposed in Chapter 7 to derive stress strain behavior of granular soils from back analysis of SBPMT. In addition to SBPMT data, the procedure makes use of the following • SCPTU data to estimate the small strain properties and relative density, • a qualitative a-priori knowledge about the grain compressibility of the deposit, • correlations between model parameters r| , R and C and relative density developed from F F existing information, • existing information on parameters m , n and n , and • laboratory test data on reconstituted samples to estimate parameters X and u. A E P The back analysis procedure essentially involves fitting the model response in cylindrical cavity expansion to the observed SBPMT data by manual trial and error. Stress deformation data measured in SBPMT cavity expansion test over a range of cavity strain between 4 and 10% is used in the back analysis. SBPMT data pertaining to smaller values of cavity strain are often affected by installation related disturbance and is therefore not considered. The proposed back analysis Chapter 8: Summary, Conclusions and Recommendations 169 procedure for SBPMT data is only developed for problems • involving monotonic loading and • in which a continuous rotation of stress is not expected. SBPMT data from several sand sites in Western Canada are analyzed following the proposed procedure in Chapter 7. As described in Chapter 6, these sites present a range of hydraulic depositional environment and state of packing. To provide validation for the proposed procedure, undrained response of axisymmetric elements using the model parameters derived from analyses of SBPMT are compared with laboratory triaxial test data on undisturbed samples extracted from three sites via ground freezing as part of the CANLEX Project. To illustrate the use of the proposed procedure, SBPMT data from a tailings dam site near Lornex, British Columbia are analyzed. Reasonable a-priori estimates of the initial state of packing of these deposits were available during the analyses from piezocone penetration data. However, the grain compressibility of these deposits is inferred from the mechanical process in which the grains are generated. No additional material specific information is available at present from this site. Although no laboratory data are available as yet to verify these results, the inferred element behavior as in qualitative agreement with the expected triaxial stress strain response of loose to medium dense deposits of granular soils. These results only demonstrate that the proposed approach can be used reasonably in a situation where little material specific information is available over and above the SBPMT and SCPTU. Undisturbed samples have been extracted from this site in Phase rv of the CANLEX Project. It will be of interest to compare the predictions of undrained behavior of axisymmetric elementsfromSBPMT to laboratory triaxial data on undisturbed samples as they become available. Chapter 8: Summary, Conclusions and Recommendations 170 To provide a field validation, the procedure is used for data from a tailings dam site in South-eastern British Columbia, where there has been a recent incidence of static liquefaction flow failure. The initial state of packing was again estimated from a seismic piezocone penetration test carried out near the SBPMT hole and the grain compressibility was inferred from the mechanical process in which the grains are generated. The undrained response of a plane strain element was then computed using the model parameters derived from back analysis of SBPMT. The results show negligible residual strength both in compression and in extension. The interpretation of SBPMT following the proposed procedure is therefore in good agreement with the incidence of static liquefaction at this site. It can be concluded from the results of the validation exercise of Chapter 7 that • monotonic deformation response of an earth structure can be estimated reasonably from SBPMT following the proposed procedure for a cross anisotropic frictional material in a problem where significant rotation of stress is not expected and • the procedure can be used to calibrate the stress strain relationship of Chapter 4 reasonably in a practical problem where very little quantitative material specific information is available over and above SBPMT and SCPTU. 8.2 Practical Application The procedure can be useful as it is in a number of problems of practical importance involving monotonic loading. One such problem pertains to the design of tailings dam and coal dumps. Although the importance of performing undrained deformation analysis is quite well understood, the existing procedures based on in-situ index tests, e.g., SPT and CPT that are usually Chapter 8: Summary, Conclusions and Recommendations utilized in the mining industry for this purpose are not as accurate as might be required in large projects of high risk. Usefulness of the proposed procedure in undrained analysis of tailings deposits is illustrated in Chapter 7 with an example involving a mine site in South eastern British Columbia. A statically induced flow slide occurred in a section of the tailings dam at this location in spite of the fact that undrained analysis making use of an existing SPT based correlation that virtually constitutes the current state of practice gave a factor of safety of 2.0 for the structure. On the other hand, the stress-strain response of a plane strain element from SBPMT following the procedure developed in this study shows negligible undrained residual strength both in compression and extension: a result that is in good agreement with the observed flow failure of the structure. It needs to be emphasized that the philosophy of model calibration using self-boring pressuremeter data proposed in this research is not limited to the stress-strain model developed in Chapter 4. Procedure for calibrating other stress-strain models following this approach can be extended for any other appropriate stress-strain model provided that the issue of non-uniqueness in the solution to the inverse modeling problem can be handled satisfactorily. 8.3 Avenues of Further Research A logical sequel to the present research is to investigate the possibility of application of SBPMT in site response analysis during an earthquake or other cyclic loading. As in a problem involving monotonic loading of granular soils that has been addressed in this research, to derive cyclic behaviorfromSBPMT, a stress-strain relationship needs to be devised that in spite of being simple can account for the influence of the state of packing, inherent anisotropy, stress and stress path on soil behavior. The stress strain model presented in this research can form a basis for such 171 Chapter 8: Summary, Conclusions and Recommendations 172 a model of more comprehensive capability with • a modification following Matsuoka and Sakakibara (1987) to account for pure rotation of stress and • replacing the simple isotropic hardening scheme of the present stress-strain relationship with a kinematic (e.g., Lade and Boonyachut, 1982) or mixed hardening approach to make the formulation suitable for a cyclic loading problem. Such a stress strain relationship can potentially be calibrated from SBPMT data that include virgin loading as well as unload reload sequences following a procedure similar to that proposed in this research. The possibility of automating the inverse modeling procedure for SBPMT data to derive model parameters also needs to be investigated. Well established techniques of non-linear inverse theory, e.g., steepest descent, can in principle be applied. Since the forward problem (i.e., calculation of pressure expansion response for a given set of model parameters) no longer requires high computer overhead (typically, on a personal computer one forward problem can be solved in tens of seconds at present for the stress strain model used in this study), automating the inverse modeling problem appears viable. 173 BIBLIOGRAPHY Ajalloeian, R., and Yu, H.S. 1996 Chamber studies of the effects of pressuremeter geometry on test results in sand. Research Rep. 141.09.1996, University of Newcastle, NSW 2308, Australia. Alarcon-Guzman, A , Chameau, J.L., Leonards, G.A., and Frost, J.D. 1989. Shear modulus and cyclic undrained behavior of sands. Soils and Foundations, 29(4), 105-119. Arthur, J.R.F., and Menezies, B.K. 1972. Inherent anisotropy in a sand. Geotechnique, 22, 115-128. Bacchus, R.C. 1983. An investigation of the strength deformation response of naturally occurring lightly cemented sands. Ph.D. Dissertation, Department of Civil Engineering, Stanford University, CA, USA. Baguelin, F., Jezequel, IF., and Shields, D.H. 1978. The pressuremeter andfoundation engineering. Trans Tech Publishers, Clausthal, Germany. Bahar, R., Cambou, B., Labanieh, S., and Foray, P. 1995. Estimation of soil parameters using a pressuremeter test. The Pressuremeter and its New Avenues, Ballivy, G., ed., Balkema, Rotterdam, the Netherlands, 65-72. Basudhar, P.K., and Kumar, D. 1995. Performance studies of cavity expansometer: A monocell pressuremeter. The Pressuremeter and its New Avenues, Ballivy, G., ed., Balkema, Rotterdam, the Netherlands, 73-80. Bathe, K-J. 1982. Finite Element Procedures in Engineering Analysis. Prentice-Hall, Inc., Englewood Cliffs, NJ, USA. Been, K , and Jefferies M.G. 1985. A state parameter for sands: reply to discussion. Geotechnique, 35, 127-132. Benoit, J., Atwood, M.J., Findlay, R.C., and Hilliard, B.D. 1995. Evaluation ofjetting insertion for the self-boring pressuremeter in soft clays. Canadian Geotech. J., 32, 22-39. Bigger, K.W., and Robertson, P.K. 1996. Detailed site characterization: Highland Valley Copper Mine, B.C. CANLEX Phase IV Rep., Dept. of Civil Engrg., Univ. of Alberta, Edmonton, Canada. Bishop, A.W. 1985. Shear characteristics of a saturated silt, measured in triaxial compression: correspondence. Geotechnique, 2, 43-45. Bibliography 174 Bolton, M.D. 1986. The strength and dilatancy of sands. Geotechnique, 36, 65-78. Bowles, J.E. 1996. Foundation analysis and design. McGraw Hill, New York, US A. Bruzzi, D., Ghionna, V , Jamiolkowski, M., Lancellotta, R., Manfredini, G. 1986. Self-boring pressuremeter in Po river sand. Proc., Pressuremeter and its Marine Application, STP 950, ASTM, Philadelphia, PA, 283-302. Campanella, R.G., Stewart, W.P., and Jackson, R.S. 1990. Development of the UBC Self-Boring Pressuremeter. Proc. Ill Int. Symp. on Pressuremeters, British Geotechnical Society, Oxford, pp. 65-72. Carter, J.P., Booker, J.R., and Yeung, S.K. 1986. Cavity expansion in cohesive frictional soils. Geotechnique, 36, 349-358. Chang, CS. 1993. Powders and Grains 93, Thornton, C , ed., Balkema, Rotterdam, the Netherlands, 105-110. Chen, W.F., and Han, D.J. 1988. Plasticityfor Structural Engineers, Springer-Verlag, New York. Clague, J.J., Naesgaard, E., and Sy, A. 1992. Liquefaction features on the Fraser River Delta: evidence for prehistoric earthquakes? Canadian J. of Earth Sciences, 29, 1734-1745. Clarke, B.G., and Allan, P.G. 1989. A self-boring pressuremeter for testing weak rock. Proc, XII int. conf. on soil mech. and foundation engrg, Rio de Jeneiro, 1, 2/15, 211-213. Clarke, B.G., and Gambin, M. 1995. Pressuremeter testing in onshore ground investigations. Rep., Submitted to Committee TC16, Int. Society of Soil Mech. and Foundation Engrg. Clough, G.W., Briaud, J.L., and Hughes, J.M.O. 1990. The development of pressuremeter testing. Proc. Dl Int. Symp. on Pressuremeters, British Geotechnical Society, Oxford, pp. 25-45, 1990. Cundall, P.A. 1992. FLAC ver. 3.0: user's manual, Itasca Consulting Group, Minneapolis, Minnesota, USA. da Cunha, R.P. 1994. Interpretation of selfboring pressuremeter tests in sand. PhD. Dissertation, University of British Columbia, Vancouver, Canada. de Souza Coutinho, AG.F. 1990. Radial expansion of cylindrical cavities in sandy soils: application to pressuremeter tests. Canadian Geotech. J., 21, 131'-748. Bibliography 175 Dawson, R.F., Morgenstern, N.R., Gu, W.H. 1994. Liquefactionflosslides in western Canadian coal mine waste dumps-Phase II: case histories." Rep., Energy, Mines and Resporces, Ottawa, Canada. Duncan, J.M., Byrne, P.M., Wong, K.S., and Marby, P. 1980. Strength, stress-strain and bulk modulus parameters for finite element analyses of stresses and movements in soil masses. Rep. No. UCB/GR/78-02, Dept. of Civil Engrg., University of California, Berkeley, USA. Eckersley, J.D. 1984. Flow slides in stockpiled coal. Proc., IV Australia-New Zealand Conf. on Geomechanics, 2, 607-611. Enos, J.L., Poulos, S.J., France, J.W., and Castro, G. 1982. Liquefaction induced by cyclic loading. Rep., Project 80696, National Science Foundation, Washington D.C., USA. Ergun, M.U. 1981. Evaluation of three-dimensional shear testing. Proc, X Int. Conf. on SoilMech. and Foundation Engrg., Stockholm, Sweden, 1, 593-596. Fahey, M., and Carter, J.P. 1993. Afiniteelement study of the pressuremeter test in sand using non linear elastic plastic model. Canadian Geotech. J., 30, 348-362. Fahey, M . , and Randolph, M. F. 1984. Effect of disturbance on parameters derived from selfboring pressuremeter tests in sand. Geotechnique, 34, 81-97. Fear, C.E. 1996. In-situ testing for liquefaction evaluation of sandy soils. Ph.D. Dissertation, Department of Civil Engrg., University of Alberta, Edmonton. Fear, C.E., and Robertson, P.K. 1995. Estimating the undrained strength of sand: a theoretical framework. Canadian Geotech. J., 32, 859-870. Fioravante, V., Jamiolkowski, M., and Lo Presti, D.C.F. 1994. Stiffness of carbonic Quiou sand. Proc, XIII Int. Conf. on Soil Mechanics and Foundation Engrg,, 1, 163-167. Hardin, B.O. 1978. Stress-strain behavior. Proc, Earthquake Engrg. and Soil Dynamics, ASCE, 1, 3-90. Haruyama,M. 1981. Anisotropic deformation-strength characteristics of an assembly of spherical particles under 3 -D stresses. Soils and Foundations, 21 (4), 41 -5 5. Hashiguchi, K. 1991. Inexpedience of the non-associative flow rule. Int. J. Numerical and Analytical Methods in Geomechanics, 15, 753-756. 176 Bibliography Houlsby, G.T. 1979. The work input to a granular material. Geotechnique, 29, 354-358. Houlsby, G.T., and Hitchman, R. 1988. Calibration chamber tests of a cone penetrometer in sand. Geotechnique, 38, 39-44. Houlsby, G.T., arid Carter, J.P. 1993. The effects of pressuremeter geometry on the results of tests in clay. Geotechnique, 43, 567-576. Howie, J.A 1991. Factors affecting the interpretation and analysis of full displacement pressuremeter tests in sands. PhD. Dissertation, University of British Columbia, Vancouver, Canada. Hughes, J.M.O., Roy, D., and Campanella, R.G. 1997. Preliminary assessment of the stability of mine tailingsfromself-boring pressuremeter data. Proceedings, 50th Canadian Geotechnical Conference, Ottawa. Hughes, J.M.O., Wroth, CP., and Windle, D. 1977. Pressuremeter tests in sands. Geotechnique, 27,455-477. In-situ Testing Group (Dept. of Civ. Engrg., Univ. of BC). 1995". General site characterization at Massey Tunnel in Deas Island, British Columbia. CANLEX Rep., Dept. of Civil Engrg., University of Alberta, Edmonton, Canada. In-situ Testing Group (Dept. of Civ. Engrg., Univ. ofBC). 1995 . General site characterization at KIDD # 2 in Richmond, British Columbia. CANLEX Rep., Department of Civil Engrg., University of Alberta, Edmonton, Canada. b Iravani, S., Hoffmann, B.A., Fear, C , Cyre, G., Lefebvre, M.E., Natarajan, S., Stahl., R.P., and Robertson, P.K. 1995. Phase III Activity 111-3 A general site characterization, CANLEX Rep., Department of Civil Engrg., University of Alberta, Edmonton, Canada. Ishihara, K. 1993. Liquefaction and flow failure during earthquakes. Geotechnique, 43, 351-415. Ishihara, K., Tatsuoka, F., and Yasuda, S. 1975. Undrained deformation and liquefaction of sand under cyclic stresses. Soils and Foundations, 15(1), 29-44. Iwasaki, T., and Tatsuoka, F. 1977. Effects of grain size and grading on dynamic shear moduli of sands. Soils and Foundations, 17(3), 19-35. Bibliography 177 Jarniolkowski, M., Ghionna, V.N., Lancellotta, R., and Pasqualini, E. 1988. New correlations of penetration tests for design practice. Penetration Testing 1988, de Ruiter, ed., Balkema, Rotterdam, the Netherlands, 1, 263-296. Jefferies, M.G. 1988. Determination of horizontal geostatic stress in clay with self-bored pressuremeter. Canadian Geotech. J., 25, 559-573. Jefferies, M.G. 1993. Nor Sand: a simple critical state model for sand. Geotechnique, 43, 91-103. Kokusho, T., and Esashi, Y. 1981. Cyclic triaxial test on sands and coarse materials. Proc, X Int. Conf. on Soil Mechanics and Foundation Engrg., Stockholm, 1, 673-676. Kleiber, M. 1989. Incremental Finite Element Modelling in Non-linear Solid Mechanics. Ellis Horwood Limited, Chichester, UK. Kuerbis, R.H., and Vaid, Y.P. 1990. Undrained behavior of clean and silty sands. Proc, Discussion Session on Influence of Local Conditions on Seismic Response, XII Int. Conf. on Soil Mech. and Foundation Engrg., Rio de Jeneiro, 91-100. Kulhawy, F.H., and Trautmann, C H . 1996. Estimation of in-situ test uncertainty. Uncertainty in the geologic environment: from Theory to Practice, Geotech. Special Publication No. 58, ASCE, 1, 269-286. Lacasse, S., D'Orazio, T.B., and Bandis, C. 1990. Interpretation of self-boring and push-in pressuremeter tests. Proc, III Int. Symp. on Pressuremeter and its Marine Application, Oxford, Thomas Telford, UK, 273-285. Lade, P.V. 1977. Elasto-plastic stress-strain theory for cohesionless soil with curved yield surfaces. Int. J. of Solids and Structures, London, 13, 1019-1035. Lade, P.V., and Boonyachut, S. 1982. Large stress reversals in triaxial tests on sand. Proc, IV Int. Conf. On Numerical Methods in Geomechanics, Edmonton, 1, 171-182. Lade, P. V., and Duncan, J.M. 1973. Cubical triaxial tests on cohesionless soil. J. Soil Mech. and Foundation Engrg., ASCE, 99, 793-811. Lade, P.V., and Nelson, R.B. 1987. Modelling the elastic behaviour of granular materials. Int. J. Numerical and Analytical Methods in Geomechanics, 11, 521-542. Lafeber, D., and Willoughby, D.R. 1971. Fabric symmetry and mechanical anisotropy in natural soils. Proc, Australia-New Zealand Conf. On Geomechanics, Melbourne, 1, 165-174. 178 Bibliography Lam, W.,-K., and Tatsuoka, F. 1988. Effects of initial anisotropic fabric and a on strength and deformation characteristics of sand. Soils and Foundations, 28(1), 89-106. 2 Lee, K L . 1970. Comparison of plane strain and triaxial tests on sand. J. Soil Mech and Foundation Engrg., ASCE, 96, 901 -923. Lee, K.L., and Seed, H.B. 1967. Drained strength characteristics of sands. J. Soil Mech. and Foundation Engrg., ASCE, 93, 117-141. Lefebvre, M.E., HofBnann, B.A., and Natarajan, S. 1995. Phase III - liquefaction event instrumentation report, CANLEX Rep., Department of Civil Engrg., University of Alberta, Edmonton, Canada. Lo Presti, DC.F., and O'Neil, D.A. 1991. Laboratory investigation of small strain modulus anisotropy in sands. Proc., Int. Symposium on Calibration Chamber Testing, Huang, A.-B., ed. Elsevier Science Publishing Co., Inc., New York, 213-224. Loret, B. 1989. Geomechanical applications of theory of multimechanisms. Geomaterials: Constitutive Equations and Modeling. Drave, F., ed,, Elsevier Applied Science, London. Manassero, M., 1989. Stress-strain relationships from drained self-boring pressuremeter tests in sands. Geotechnique, 39, 293-307. Marti, I, and Cundail, P. 1980. Mixed descretization procedure for accurate modeling of plastic collapse. Int. J. Numerical and Analytical Methods in Geomechanics, John Wiley, New York, 6, 129-139. Matsuoka, H. 1976. On the significance of the spatial mobilized plane. Soils and Foundations, 16(1), 91-100. Matsuoka, H. 1974. A microscopic study on shear mechanism of granular materials. Soils andFoundations, 14(1), 29-43. Matsuoka, H., and Sakakibara, K. 1987. A constitutive model for sands and clays evaluating principal stress rotation. Soils and Foundations, 27(4), 73-88. Mitchell, J.K. 1993. Fundamentals of Soil Behavior, John Wiley, New York. Mindlin, R.D. 1949. Compliance of elastic bodies in contact. J. Applied Mechanics, Trans., ASME, 71, A-259 to A-268. Bibliography 179 Molenkamp, F. 1988. A simple model for isotropic non-linear elasticity of frictional materials. Int. J. Numerical and Analytical Methods in Geomechanics, 12, 467-475. Monahan, P.A, Luternauer, J.L., and Barrie, J.V. 1995. The geology of the CANLEX Phase II sites in Delta and Richmond, British Columbia, Proc, 48th Canadian Geotech. Conf, Vancouver, 1, 59-67. Mroz, Z. 1980. On hypoelasticity and plasticity approaches to constitutive modeling of inelastic behavior of soils. Int. J. Numerical and Analytical Methods in Geomechanics, 4, 45-55. Mroz, Z., and Zienkiewicz, O.C. 1984. Uniform formulation of constitutive equations for clays and sands. Mechanics of Engineering Materials. Desai, C.S., and Gallagher, R.H., eds. John Wiley and Sons Ltd., London, 415-448. Mulilis, J.P., Seed, H.B., Chan, C.K., Mitchell, J.K., and Arulanandan, K. 1977. Effects of sample preparation on sand liquefaction. J. Geotech. Engrg, ASCE, 103, 91-108. Naghdi, P.M. 1990. A critical review of the state of finite plasticity, J. of Applied Math, and Phys. (Zeitschrift fur Angewandte Mathametik und Physik), Birkhauser Verlag, Basel, Switzerland, 41, 315-394. Nagtegaal, J.C., Parks, D M . , and Rice, J.R. 1974. On numerically accurate finite element solutions in the fully plastic range. Computer Methods in Applied Mech. and Engrg., 4, 153-177. Nakai, T., and Matsuoka, H. 1983 a. Constitutive equation for soils based on the extended concept of "spatial mobilized plane" and its application tofiniteelement analysis. Soils and Foundations, 23(4), 87-105. Nakai, T., and Matsuoka, H. 1983b. Shear behaviors of sand and clay under three-dimensional stress condition. Soils and Foundations, 23(2), 26-42. Negussey, D. 1984. An experimental study of the small strain response of sand. Ph.D. Dissertation, University of British Columbia, Vancouver, Canada. Nova, R. 1991. A note on sand liquefaction and soil stability. Constitutive Laws of Engineering Materials: Recent Advances and Industrial and Infrastructure Applications, Desai, C.S., Krempl, E., Frantziskonis, G., and Saadatmanesh, H., eds., ASME Press, New York, 153156. Bibliography ieu Naylor, D.J. 1974. Stresses in nearly incompressible materials by finite elements with application to the calculation of excess pore pressures. Int. J. Numerical and Analytical Methods in Geomechanics, 8, 443-460. Oda, M. 1972. Initial fabrics and their relations to mechanical properties of granular material. Soils and Foundations, 12( 1), 17-36. Park., C , -S., and Tatsuoka, F. 1994. Anisotropic strength and deformation of sands in plane strain. Proc., XUI Int. Conf. on Soil Mechanics and Foundation Engrg., New Delhi, India, 4, 1-4. Pass, D.G. 1994. Soil characterization of the deep accelerometer site at Treasure Island, San Francisco, California. M.S. thesis, University of New Hampshire, Durham, NH, USA. Phoon, K.K., and Kulhawy, F.H. 1996. On quantifying inherent soil variability. Uncertainty in the geologic environment:fromTheory to Practice, Geotech. Special Publication No. 58, ASCE, 1, 326-340. Poorooshasb, H.B., Holubec, I., and Sherbourne, A.N. 1966. Yielding and flow of sand in triaxial compression: Part I, Candian Geotech. J., 3, 179-190. Poorooshasb, H.B., and Roscoe, K.H. 1961. The correlation of the results of shear test with varying degree of dilation. Proc, V Int.Conf. on Soil Mech. and Foundation Engrg., 1, 297-304. Ramamurthy, T., and Rowat, P.C. 1973. Shear strength of sand under general stress system. Proc, VUI Int. Conf On Soil Mech. and Foundation Engrg., Moscow, Vol 1.2, 339-342. Reades, D.W., and Green, G E . 1976. Independent stress control and triaxial extension tests on sand. Geotechnique, 26, 551-576. Richart, F.E., Jr. 1977. Field and laboratory measurements of dynamic soil properties. Proc, Dynamical Methods in Soil and Rock Mechanics, Prange, B., ed., Balkema, the Netherlands, 1, 3-36. Robertson, P.K. 1982. In-situ testing of soil with emphasis on its application to liquefaction assessment. PhD. Dissertation, University of British Columbia, Vancouver. Robertson, P.K., and Campanella, R.G. 1986. Guidelines for use, interpretation and application of the CPT and CPTU, Soil Mechanics Series No. 105, Department of Civil Engrg., University of British Columbia, Vancouver, Canada. Bibliography 181 Robertson, P.K., Woeller, D.J., Kokan, M., Hunter, J., and Lutemauer, J. 1992. Seismic techniques to evaluate liquefaction potential. Proc, 45th Canadian Geotech. Conference, Toronto, 5:1-5:9. Roscoe, K.H., Schofield, A.N., and Thurairajah, A. 1963. Yielding of clays in states wetter than critical. Geotechnique, 13^211-240. Rowe, P.W. 1971. Theoretical meaning and observed values of deformation parameters for soil. Proc, Roscoe Memorial Symposium, 143-194, G.T. Foulis and Co., Henley-onThames, UK. Roy, D. 1992. Site response and liquefaction analysis during earthquakes. M.S. Thesis, University of Idaho, Moscow, ID, USA. Roy, D., Campanella, R.G., Byrne, P.M., and Hughes, J.M.O. 1996. Strain level and uncertainty of liquefaction related index tests. Uncertainty in the geologic environment: from Theory to Practice, Geotech. Special Publication No. 58, ASCE, 2, 1149-1162. Roy, D., Campanella, R.G., Byrne, P.M., and Hughes, J. 1996. State parameter from selfboring pressuremeter tests in sand. Discussion, J. Geotech. Engrg., ASCE, 122, 413. Roy, D., Hughes, J.M.O., and Campanella, R.G. 1997. Reliability of Self-boring Pressuremeter in Sand. Proceedings, 50th Canadian Geotechnical Conference, Ottawa, Canada. Rubin, M. 1994. Plasticity theory formulated in terms of physically based microstructural variables - Part I. theory, Int. J. Solids and Structures, Pergamon Press, UK, 31, 2615-2634. Salencon, J. 1969. Contraction quas-statique d'une cauite a symetric spherique ou cylindrique dans un milieu elastoplastique. Annales des ports et Chaussees, 4, 231-236. Salgado, F.M. 1990. Analysis procedures for caisson-retained island type structures. PhD. Dissertation, Department of Civil Engrg., University of British Columbia, Vancouver. Salgado, F.M., and Byrne, P.M. 1991. A three dimensional constitutive elasto-plastic model for sands following the 'spatial mobilized plane concept.' Computer Methods and Advances in Geomechanics, Beer, G., Booker, J.R., and Carter, J.P., eds., Balkema, Rotterdam, the Netherlands, 675-679. Salgado, F.M., and Byrne, P.M. 1990. Finite element analysis of pressuremeter chamber tests in sand. Proc, HI Int. Symp. on Pressuremeter and its Marine Application, Oxford, Thomas Telford, UK, 209-219. Bibliography 182 Santamarina, J.C., and Cascante, G. 1996. Stress anisotropy and wave propagation: a micromechanical view. Canadian Geotech. J., 33, 770-782. Sasitharan, S., Robertson, P.K., Sego, D.C., and Morgenstern, N.R. 1994. State-boundary surface for very loose sand and its practical applications. Canadian Geotech. J., 31, 321-334. Schmertmann, J.H. 1977. Effect of shear stress on the dynamic bulk modulus of sand. Final Rep. to the Waterways Experiment Station, Corps of Engineers, US Army, Contract DACW 3976-M-6676. Seed, H.B. 1979. Soil liquefaction and cyclic mobility evaluation for level grounds during earthquakes. J. Geotech. Engrg, ASCE, 105, 201-255. Seed, H.B., Seed, R.B., Harder, L.F., and Jong, H.-L. 1988. Reevaluation of the slide in the Lower San Fernando Dam in the earthquake of February 9, 1971. Rep. No. UCB/EERC88/04, University of California, Berkeley, USA. Seed, R.B., and Harder, L.F., Jr. 1990. SPT-based analysis of cyclic pore pressure generation and undrained residual strength. Proc, H. Bolton Seed Memorial Symp., Duncan, J.M., ed., 2,351-376. Shankariah, B., and, Ramamurthy, T. 1980. Strength and deformation behaviour of sand under general stress system. Proc, III Australia-New Zealand Conf. on Geomechanics, Wellington, 1, 207-212. Smith, I.M., and Griffiths, D. V. 1988. Programming the finite element method. John Wiley & Sons, Chichester, UK. Srithar, T. 1994. Elasto-plastic deformation and flow analysis in oil sand masses. PhD. Dissertation, Department of Civil Engrg., University of British Columbia, Vancouver. Stark, T.D., and Mesri, G. 1992. Undrained shear strength of liquefied sands for stability analysis. J. Geotech. Engrg., ASCE, 118,1727-1747. Stark, T.D., and Olson, S M . 1995. Liquefaction resistance using CPT andfieldcase histories. J. Geotech. Engrg, ASCE, 121, New York, USA, 856-869. Stedman, J.D. 1997. Effects of static shear and confining pressure on liquefaction resistance of Fraser River Sand. M.A.Sc. Thesis, University of British Columbia, Vancouver. 183 Bibliography Stokoe, K.H., II, Lee, J.N.-K., and Lee, S.-H. 1991. Characterization of soil in calibration chambers with seismic waves. Calibration Chamber Testing, Huang, A.-B., ed., Elsevier Science Publishing Co., Inc., New York, 363-376. Sully, J.P. 1991. Measurement of in-situ lateral stress during full-displacement penetration tests. Ph.D. Dissertation, University of British Columbia, Vancouver. Sutherland, H.B., and Mesdary, M.S. 1969. The influence of intermediate principal stress on the strength of sand. Proc, VII Int. Conf. on Soil Mech. and Foundation Engrg., Mexico City, 1, 391-399. Tani, K., Nishi, K., and Okamoto, T. 1995. A new measuring method of borehole wall displacement for pressuremeter tests. The Pressuremeter and its New Avenues, Ballivy, G., ed., Balkema, Rotterdam, the Netherlands, 373-377. Tatsuoka, F , and Ishihara, K. 1974. Yielding of sand in triaxial compression. Soils and Foundations, 14(2), 63-76. Tatsuoka, F , Sakamoto, M., Kawamura, T., and Fukushima, S. 1986. Strength and deformation characteristics of sand in plane strain compression and extension at extremely low pressures. Soils andFoundations, 26(1), 65-84. Tatsuoka, F., and Kohata, Y. 1995. Stiffness of hard soils and soft rocks in engineering applications. Rep., Institute of Industrial Science, University of Tokyo. Taylor, D.W. 1948. Fundamentals of Soil Mechanics, John Wiley, New York. Thomas, J. 1992. Static, cyclic and post-liquefaction undrained behaviour of Fraser River Sand. M.A.Sc. thesis, Dept. of Civil Engrg., University of British Columbia, Vancouver. Vaid, Y.P., Chung, E.K.F., and Kuerbis, R.H. 1990. Stress path and steady state. Canadian Geotech. J., 27, 1-7. Vaid, Y.P., Sivathayalan, S., Eliadorani, A., and Uthayakumar, M. 1996. Laboratory testing at UBC. CANLEX Rep., Department of Civil Engrg., University of Alberta, Edmonton, Canada. Vesic, A.S., and Clough, G.W. 1966. Behavior of granular materials under high stresses. J. Soil Mech. and Foundation Engrg, ASCE, 94, 661-688. Bibliography 184 Vick, S. 1991. Inundation risk from tailings dam flow failure. Proc, IX Pan Am. Conf. on Soil Mech. and Foundation Engrg., 3, 1137-1158. Wilson, G., and Sutton, J.L.E. 1948. A contribution to the study of the elastic properties of sand. Proceedings, U International Conference on Soil Mechanics and Foundation Engineering, Rotterdam, 1, 197-202. Yan, L. 1986. Numerical studies of some aspects with pressuremeter tests and laterally loaded piles. M.A.Sc Thesis, University or British Columbia, Vancouver, Canada. Yan, L., and Byrne, P.M. 1990. Simulation of downhole and crosshole seismic tests on sand using the hydraulic gradient similitude method. Canadian Geotech. J., 27, 441-460. Yu, P., and Richart, F.E., Jr. 1984. Stress ratio effects on shear modulus of dry sands. J. Geotech. Engrg, ASCE, 110, 331-345. Yu,H.S. 1994. State parameter from self-boring pressuremeter tests in sand. J. Geotech. Engrg., ASCE, 120, 2118-2135. Yu, H.S. 1992. Expansion of a thick cylinder of soils. Computers and Geotechnics, Elsevier Science Publishers, England, 14, 21-41. Yu, H.S., and Netherton, M.D. 1996. Performance of displacement finite elements for stress analysis of incompressible materials. Research Rep. No. 143.09.1996, Department of Civil Engrg. and Surveying, The University of Newcastle, NSW 2308, Australia. 185 APPENDIX 1 COMPLIANCE MATRICES Al.I Elastic Compliance Matrix In terms of orthonormal coordinates, 1-2-3, the stress-strain relationship in Eq. (3.11) can also be written as follows: « > } where {de^} = [de e n de,/ de,/ de = [C ]{do,'} (All) e de^ de ] , {do^} = [do,/ e e 12 T 13 do ' da ' do ' do^' 22 33 12 do '] , and the elastic compliance matrix [C ] is given by the following equation T e 13 [C ] e = ± E 1 - V - V 0 0 0 - v 1 - V 0 G 0 - V - V 1 0 0 0 0 0 0 1 +v 0 0 0 0 0 0 1+V G 0 0 0 0 0 1 +v (A1.2) Superscript "T" indicates transpose operation. A1.2 Compliance Matrix in Distortion The matrix form of the stress strain relationship in the distortion mechanism is given by Nakai and Matsuoka (1983b). The relevant details are as follows. Eq. (3.26) can be written as {de } = {X } d* s (i) s (A1.3) 186 Appendix 1: Compliance Matrices where {de } = [de 8 (i) s (1) de de s (2) ] , and vector {X } is given by Eq. (3.25). The increment in 8 T (3 8 stress ratio relates to the stress increments as follows dr, = [ Y ] { d ' } (A1.4) s 0ij where [Y ] is a 1 x6 matrix given by 8 I I +I I (o ' o ' ) - I I ( o ' o ' - o ' ) 2 2 3 y 3 + 1 3 22 + 3 3 1 2 22 33 23 I I (o ' o ')-I I (o 'o '-o ' ) 2 1 3 3 3 + 1 1 1 2 3 3 n 1 3 I I I I (o ' o ')-I I (o 'o '-o ' ) 2 [Y ] = 2 s is n i 2 3 3 + 1 3 u + 2 2 1 2 1 1 2 2 1 2 (A1.5) -2I I o '-2I I (o 'o '-o 'o ') 1 3 12 1 2 23 13 33 12 -2I I o '-2I I (o 'o '-o 'o ') 1 " 2 I 3 23 1 i 3°i3' - W I 2 2 13 12 W n " 23 W ) The plastic distortional principal strain increment vector {de } transforms to the orthonormal 8 (i) coordinate system as follows: N 2 N 2 N 2 N 2 N 2 N 2 1M N {de.*} = [Z]{de } = s ^31 iN 22 ^23 (i) N„N 1 2 N N 1 3 1 2 N N ^21 ^22 ^31^32 N N N3 2N ^33 ^22^23 2 3 2 1 d e (l) s d E (2) S d £ (3) S 2 N N N N„ N N 1 3 32 N ^33 2 (A1.6) iN N N 3 3 3 1 where N is the direction cosine of the principal direction "i" w.r.t coordinate direction "j". This y matrix can be obtained by solving the characteristic equation for the stress tensor and back substitution. It may be noted that calculation of [Z] is computationally less intensive in the HaighWestergaard stress space (see, e.g., Chen and Han, 1988: p. 70). Combining Eqs. (3A.3), (3A.4) Appendix 1: Compliance 187 Matrices and (3 A. 6) the following relationship between strain and stress increments is obtained {de/} where {de/} = [de de s n = [C ]{do/} de^ de s 8 22 (A1.7) s de^ de ] and the 6><6 compliance matrix, [C ], in 8 8 12 8 T s 13 distortional plasticity is given by the following equation. Eq. (3 A.7) is identical to the relationship in Eq. (3.26) and [D ] is the matrix representation of C *. 8 iia [C ] s 6 x 6 = [Z] {X } 8 6 x 3 [Y ] (A1.8) s 3 x l 1 > < 6 Matrix dimensions for Eq. (3 A. 8) are shown as subscripts for convenience. A1.3 Compliance Matrix in Isotropic Compression Expansion of Eq. (3.37) yields {de } = [C ]{do '} c where {de/} = [de de c n c 22 ij de ° de ° de 33 12 e 23 de ] , and the compliance matrix [C ] is given by c Cpf ">- > 2 u ' V a u ' V c p (2p-l) 2 a a a a ii'°u' T c 13 a O n ' (A1.9) c fj n' 0 33' 2o 'o ' 2o 'o ' 2o 'o ' 2o 'o ' n 1 2 22 ° 2 2 °33' 2 2 ° 1 2 0 ° 2 2 a 2 ° 2 3 2 a n 2 0 °33'°12' 0 3 ' °2 ' 3 12 n 22 2 3 23 33'°23' 33'°12' 2 0 2o 2o 'o ' u , 2 12 1 2o 'o n 2 0 2 a 1 3 22'°1 3 33'°13 23 2o 'o 13 ' 2o 'o 13 12 2 2a 2o 'o ' 12 23 2 23 * 23 3 u l o x 2 a n 2 °23'°13' 2o x , 2 "13 (ALIO) Appendix 1: Compliance A1.4 188 Matrices Stress-strain Relationship in Axisymmetry and Plane Strain The incremental stress-strain relationship in axisymmetry is obtained from the corresponding three dimensional relationship such as *h W u = C (Al.ll) da by treating 1, 2 and 3 as the radial, hoop and the axial directions, respectively. The components of effective stress and strain tensors with subscripts 12 and 23 vanish in axisymmetric deformation. The incremental stress-strain relationship in plane strain the components of the strain tensor in a given direction vanish. Consequently, the corresponding shear stresses also vanish. For instance, if no deformation is allowed in coordinate direction, 3, de = de = de^ = do ' = do^' 33 J3 13 = 0. Imposing these constraints a 3x3 system of linear equations of the following type is obtained. de dV' n >de = 22 de (A1.12) do ' [C'V 22 do ' 12 12 The components of [C*] can be calculated from the components of compliance tensor C , which ijkl in essence is a 6*6 matrix. Denoting the components of the 6x6 matrix as C , C ,..., C , the u 12 a components C* , C* ,..., C* are given by n C n 12 =C -C n 33 C /C , C = C - C C /C , C = C 13 31 33 1 2 C 21 C i - C ^ C j/C , c = 2 C i 3 = 3 33 2 2 C i - C C /C , C 4 43 31 33 = 3 2 12 =C C 42 13 32 33 1 3 1 4 - C C /C , C 23 C — 22 -C 23 43 32 33 C /C , c 32 33 3 3 2 4 -C C /C , 34 33 - C^ C / C , —C -C M 13 34 43 33 C /C . 34 33 The normal effective stress in the direction in which no deformation is allowed can be calculated from do ' = - (C d o ' + C do ' + C do ')/C . 33 31 u 32 22 34 I2 33 189 APPENDIX 2 PROGRAM FOR COMPUTING PLANE STRAIN E L E M E N T RESPONSE A2.1 Usage and Code Listing A FORTRAN 77 code for computing element response in plane strain can be found in the following text box. Four node quadrilateral element with conventional interpolation functions is used. Such an element is not suitable for undrained deformation analysis of multiple plane strain elements or a single axisymmetric element. The code must be modified to incorporate at least eight node (serendipity) quadrilaterals for multiple element plane strain simulation or fifteen node triangles for axisymmetric geometry in undrained analysis. No iteration is performed to redistribute the unbalanced force and it is the responsibility of the user to select a small enough vertical displacement increment applied to the nodes on the top horizontal surface so that the results do not change if a smaller increment is chosen. The code can be used for both drained and undrained plane strain element tests by setting the apparent value of the bulk modulus of pore fluid, K , F appropriately. For a drained test K , is set to zero. In an undrained test K = nKB/(l-B) for soils F F with negligible grain compressibility, where B is Skempton's pore water pressure parameter and K is the bulk modulus of the soil skeleton. The bulk modulus of de-aired water is 2* 10 kPa. 6 The input is from a free format ASCII file containing the following variables in the given order: o VCONS ', o HCONS ', vertical displacement increment, no of increments, C, p, porosity, K Q , Rp, Ar), rfci, m^ Ksp, X, u, n E & np, v, P^ K , and intervals in number of displacement increments after F which results are sought. The output is printed to a user specified file (overwritten if already existing). 190 Appendix 2: Program for Computing Plane Strain Element Response or co co z o o CNJ CNJ o o * CL CL OO CD cn h— OO c o tu CD o CU H r— CNJ CO OO r— LU ••— LU •r— —J —I i-H C\J 40 Ll -< O "< I J 3 : I •< •<. I * + * LT> OO TJ C L _c to z or o o _i •< <JD Q z Q : -< - - . •< t Oct U J C£. L U i C L Oct —' — C Q Z CSI x >- r-J II - <c -< < C || _ o -~< X I X o O Q . C L C L Q . L L O I I —I I OO I <u-<2:2- -<-<-<or Q-uj o CE: * * * * * * * * , r-H I I I I o z a c z z o + ' + * O O — i < c\j co ^a- l •—' • l I Q - _ - U J I n : C£. 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OC L u : a : r-i UQ Q - -UIO * CO o - CO o <h~ >Q . u c r : O —- I — u z CJ3 -d>—< o co 1—t CD —I . - • n : CD O —' —' 0 • < <-3 t—i • Ctt Oct - <—1 o LU L U o CD O • < - < - —' •— I CNJ —I > >- Z3 : m z • o i : 3 < Q CD n n z c • • DZ 3= LU o s » o 1 1—• • « — c t t . oct ctt• * C I ""D 1 — < • L U I LU L U O O ' CO I o : * c u --a t— act 3Z * LU LU CO t—• •' - LU I LU LU ( U J CO » •< CO >— CO CD —• • t — • C D • LU UJ LkJ CO CO — ' — ' CO J Z z: o c r t LO D o 1 ui 1 1— —' <_> " 192 Appendix 2: Program for Computing Plane Strain Element Response CO z CO fc* HE v» X H- CO CO Lf> ** * 1 DT CM C5 ZD CD *»«• I CO CD * CO fc* • *«f fc* ' LU o x 1 CO 3 " fc* ZD O w fc* + CM h- w X fc* r-H ' II II CO o >- X z 1 CJ o . fc* I CNJ CM co DZ > x CO ZD | | u_ -< i—< « o z z *A vy v» o o o O oo o o — f c « * V * V * h- * — * Ctt CO "*«.>- O 1 1 Io fc* fc* fc* ' „ II II II CO CO DZ X —^ CO ZD o O CO 1 fc* GC >- >- CO II OO DZ I * * ZD C_3 — ' O fc*fc*h - CO fc* -— • CDwCD i—i CO CO X ZD 1 fc* + CO >- CO 1 >x OO X CO DZ ZD CD ro CD O >• 1 •—i 2£ o o -— «»* w h * + c o ^ O | x OO DZ 1 CO II II CO 1 CO CO r o CO DZ DZ > - X X >ZD ZD 1 1 u| 1 LU CD z z CD z z —< fc* fc*CO fc* fc* M O Il_ o o o I u - Z | | Z W W | Z • I-H fc* | u_ •—i II CD II CO I I DZ DZ ZD ZD - l—l t—I >~ x < I 1 | I Lu ^ - Z Z L U C D C D Z Z ^ t Z U J U _ LU i o • • OO r-H r»H CD x >— WW Q i I Io CO II x >L UZ o t-i ft O to CM CO o O t-H CD x fc* «•* OO CO « o o o L U CO — * i z LU O r-J CM I CO I ro I o o o «-< CM — CM CM " < -< I/) O O X > - N ^ I **—..— I I I ' U o o o O CD • • , _ O CO : fc* II D CO CO fc* CO CO X >- «—' CO CO x >- I I I I fc* fc* CO fc* fc* L Q I '—> r - j L U CD I DZ o * X' Z ZD CD fc* r - j CO fc* CO r-j • X fc* r-g fc*<T>1fc* >CO z DZI ZD CD DU DZ i—H ** fc* CO 1 DZ r»J ZD CO O CD X z 1 O 1 1DZ * -< DZ >- Cu X —1 fc*CM CD <_> t DZ i— DZ CO ZD CD O | CJ 1 DZ I^J o u * DZ CM I/) , • CJ >CO O. fc* O C i fc*fc* z fc*r o I fc* w» ZD 1— CM DZ fc* * >- « X •< 1 DZ CO fc* X ZU- t DZ DZ DZDZ fc*o + >- TD •r- > - O + C OO • X O - DZ O • ZD **C I CD L u I CD I x CD >— : 1 at ( Z X + + + + + + + CO fc* CO fc* <J0 i x i— w LU • CO Ctt CO - r-t - O CO CD - < * f It CD • CO ^ fc* 1 X rj) fc* c o • L U <a + w * * o o W t CO z fc*CO LUDZ * 1— >~ >5 fc* + : c t t a : c u c t t D z t 3 C M c o > - ' I I I | D X V » l / ) 0 e CM CD I—" CO CO ZD X ZD o S Z O CD C u LU -< LU Z i 1 CO | DZ CO 1 OJ • * ZD >->—••-< CD CD ) CO Ctt : Ctt fc* O Z fc* h - — ,, o CO •< a: fc* (JD CO _i fc*fc* >- CO CD 1 CD DZ ^ -< DZ DZ Cu * L_ 1 co < : 1 r-H CL r o z fc* 1 Lu 1 -< LU O DZ <C *4— 1 -< ZD o 1— _ 1 X CO > - LU O DZ LUy~ fc* •< t > " 1 Lu CM DZ DZ f DZ DZ1 c : z c u DZ >I o Ctt —1 DZ * C u •< -u» CD • < • < t— •< —J h u LU c u •<. U J C u cu CO O >- c o 1 I O I L_ fc*fc* fc* >X CU es Ofc*Q- DZ * DZ CO CM FA | FA CM ETJ - Q_ C CM <t CM V H f O f CD - J: v» N D D I CD O ZD CO O Z D : CD fc* U J CD C CD J 1 fc* CO L O CD DZ ZD L U II CD fc* CD Z Z CO • — II DZ CD t-H CM DZ : fc*Z D — O CO fc* CJ Lu : t * H CM I ZD L U i I CD i—< • • fc* CD r-t CM I 1 I CD CD CD' i — i o o o o o I > - CO O O x fc* B • CO — O r-t CO CM CM CO CO : >- x >- fc* I X > - r I I I I I I C D CD CD I H II i H O II ro ro tl CM : >- x I I >- I I ' I Z Z Z L u j t ^ . - i . ^ — . • fc*fc*fc*OOfc***fc*fc*CD I o CD CO -»«. DCO fc* o 11 fc*a CO CM CO X X o t—1 CM >• n CO I DZ ZD 1 Z Lu CD fc* fc* CO fc* •<z 1 193 Appendix 2: Program for Computing Plane Strain Element Response -< zc Q_ I -< X •fc «-• CD 1 z: ZD oo LU + CO CD + CO CD 1 to 1 | C\J CNJ CNJ CNI *—t * * — SC CO II -< < V» < i - CO CD II W CM . . i i—I I - —> I I * . ,-H | : +J « O —- — V* O CO CO CO -fc C D OD c - < - < (O * *—* H - hCrT a w O O O I-H co co CO II II H i W < | HI CD ZC ZD L U LU i * • CO i I 11 I • — i Z D ' - < _J CNJ—• coCO O h— L i _ •— CL Z * L D t-H w «-• + CNJ CNJ CNJ * -K i — ' CNJ CNJ (/) X > I CO + I + I V* V* V* v» oo L U * m i - z - o i—i i—• •—•O - < Q£ • co co co CD- + v> w II tl Q <z< + t-H * L=U * CD I I z s: ! O • —I CNJ -<c I O- O ZC o o * CNJ * T 1 + LU SC O CD u_1 zc I CJ — O* f— */» I ZC 3 t K ZD _ J CZ3 — <C O 1 * *••> w * * v> ZD O Q X -fc CNJ N +I co a CO t SC ZD ZD J J CO -<C - < • < |— I I I r—i * w I CD ^ O ZD CD CD CD • L U L U Q «—< •—I 4 CO CO CO ' — i v . w ^ f*) i r--. r - . 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ZD i O I II CO I I I O O X «• ZD Q Z I I Z L L I~H CNJ CO I I CD CD ro x ro ro < LLlV>l/)WWWV)Wl w tf> co z: t II C (O •r- O II CO CNJ CNJ CO X >~ CO CO s: >- x i-H o CL oo v» O O O ^ SC O^D O t O - -fc O SC • * O CO •—< o — or * co ^ ( I s: s. • + r~i CO O CNJ CNJ >x CL l-L 1 w LU V > CJ L L CD — i I 1 C O CDL V » » v» —< L CNJ1 tl CD CNJ1 C O CD « • > •—* CO1 CD CO1 CO CD V» •— L s:I 1 L L —I O 1 I o s: o LU 11 L II 194 Appendix 2: Program for Computing Plane Strain Element Response r-H I ro > CO CO : ctt ctt CO Ctt 1— CO o CM CO Ctt 1— CO cj ro ro ro Ctt Ctt i— CO CO CJ CJ * * * * DZ DZ DZ DZ ZD ZD ZD ZD a CD CD O I ZD Z D I o ez> + + CM CM I ro I : ctt oct „ CM oct 1— CO CJ Q CM CO CO OO < «H Ctt 1— CO o o o O O O o CD CD CD o T-t CM c O CM CM CM Ctt Ctt Ctt rtj •:t C E x— ZD ZD : i ZD Z D > - > : Z D ZD ) o o o CD C D . o i 4-J * o CN CO CJ fc* CD 1 ro II o c o o ** ** 4- + r-t CM CM CM act .act * CM * 1 DZ ZD Q r-H * II .--> * || „ II CM r o r-H < CM CO I I :I ** * •<»< -fc II i CM CO i I N I M: I -fc O O DZ O i CM^s-CO NT i CJ ( J CJ O I I t I I I * * I I• •fc*fc***fc*fc*r-tfc***fc*c: I HCM-J-i-tCM^a-COr-I CM " H H H C M M C M r t < - ' t - f O U U U O O U U U U * * * * fc* fc* * * * * * * * * * * * * 3Z : z> ** :** fc* fc* I | Q < : 3 Z DZ ^ fc*fc*fc*** fc*fc*fc*fc*fc*fc*fc* ^ • r o c o c o ^ - r o c o c o c o ^ r e n : ctt •<f— TD ' CO T J I CM CO C O ( I N N (J I ^ * CM CM CM z: r o CO CO c o CM ZD ZD ZD O Q CD C J C J o C J C J CJ> o CJ 1 r— CD- I r-H C fc* fc* II fc*fc*fc* II (J c; <a ZD ZD Z D Z D r a O C D O O '— i *r * * * XX „ O fc* + *r CM act 1 1—1 CM CO 1 1 1 1 DZ DZ DZ >-t >- I DZ ZD ZD ZD ZD Z Z O CD a a 1 „ o fc* + co CM ctt fc*fc*fc*fc* CM r o -sr <—• < ZZ) ZD H> -—- — Q O O O O CJ c j CM Ctt h CO CJ Ctt Ctt Ctt Ctt t— 1— 1— CO CO CO CO o CJ CJ CJ \— >- fc*fc* * * * * fc*fc*fc*fc* + CM + CO + + ctt CC 1— OO CO o CJ 1 II ro Ctt CM Ctt 1— CO o fc*fc* fc* 1fc* * fc*-fc DZ DZ CM CO z : o o o o > - ZD ZD ZD ZD Q D • Q 1 z z z CM CM CM CM CO ^J" . r-t CM CM CO Lt-t_i . r-HI C IM C O >—I CM ~ w ZD : ro Ctt ro OCt Ctt CttI L O I CJ • fc* Ctt h CO o CM Ctt 1— CO CJ CM CM 1 DZ DZ DZ ZD ZD ZD O CD CD CM DZ ZD Q : ctt ctt t 1 fc*fc* -fc fc* ,* CM fc* | fc** | fc* |W fc* W-fc fc* * z : * * * * o CM Ctt 1— CO o fc*fc*fc* + CM + CO + + Ctt Ctt Ctt cct «+- r-t CM act h— CO CJ CM CM CM 1 1 1 DZ DZ DZ ZD ZD ZD Q CD Q CM CM CM CM ZD ZD ZD ZD O CM Ctt 1— CO o fc*fc* * * ** Q CM Ctt i— CO o )CM COCO CO•( t fc*fc*fc*fc* + CM + CO + + fc*fc*fc*fc* 4+ CM + CO + <a- •fc*fc* Q * DZ DZ DZ DZ ZD ZD ZD ZD Q O Q CD ro ca i— CO CJ fc*fc*fc*fc* fc*fc*fc*fc* r o r o CO CO CO CO Ctt h~ co CJ CO CO r o r o — r i ro ro CM ro Ctt h~ CO o CO at J— CO CJ r-tW-fHC\J-4-{*)HC\J-4CO Q U U O O U U U U U U u _ LL. 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Z D ZD ZD : fc* fc* fc* fc*CM fc* O Q O I C D C D C D II I I II II II * * o 1 DZ O ZD O fc*Lz U o * M Z UJ z c h - C D CO CO CO fc*fc*-fc fc* o Ctt Ctt -»— C D C D C D CO CO CO CM C D DZ DZ r - J CO o o r-H t-H r-H r-H CD CD r-H 1 Z D Z D Z D Z D Z D Z D Z D Z D Z D Z D Z D Z D I Q O Q C D Q O O O C D C D Q O I ^ o t— UJ ** O ZD r~ o i-fc* ,w fc«».*^*»».fc*fc*fc*****fc*fc*fc*fc* -fc^-fc-fc-fc-fc-fc-fc-fc-fc-fc* r-H CM CO • o CD I > ICO ^I LO DZ DZ DZ r-tCMCO^--—tCMCO^T—• CM CO ' CO ( r-Hr-— il Ir-tCMCMCMCMCO ^ o • •fc o o CD O t - DZ L U — • O DZ * l Z D Z D Z D Z D Z D Z D Z D Z D Z D Z D Z D Z D Q Q Q O Q Q C D O Q C D O Q o Z D Z D Z D Z D Z D Z D Z D Z D Z D Z D Z D Z D C D Q O O Q C D C D O Q C D O C D fc<»**fc*fc*fc***fc*fc*fc*fc*fc*fc* II I CO ^ II II O O CD o II CO O o II o I I II CM c o ro cc: at _ _ _ ZDZDZDLL-QiCttQCtCttCttCttCttOetO-: Q O O U J Q Q Q — . I — I— I— I— I— I— I— r - H fc*fc*fc*CO****fc*OCOCOCOCOCOCOCOCOCO _ l z u o u u u o u u o t oct act Ctt I I 195 Appendix 2: Program for Computing Plane Strain Element Response C O UO LO -30 1 LO LO CO — -~«-l CO COCTi •JOCO COCO •< -<CO CTl <o 0 CO CO I ' * • . " O H C M H C O CO •— •—' ts ,-H o 2 E c r > O i Q . 1—4 CL C L IH OI Z Z co CO > <JC D O CO » MD j ICONI .—I r-H <JD < CO <JO < r-H CO < I X ) CT) < L O cn ( o < L O CD < • H CSJ c CO * CM . H i—I CNJ Q . Q LO Z 2 <JD • < - a I— I— CD LO C N J C O f cz> co - < J D U D ^ - ' — "3"CTl CO CM " (COCOf—I.—if—tcnr--ro — > CO — t* * * - ^ L O C J N C M >Ot—iwwwCMUDi-HCO CO CO CO CO C O CM * CM CM C O — c r . <JD C O C L ( rto o rC-O. I D z Z l CM V LT) < < »cricricocMr~4^rt—ico I - C \ J ( O Q - Q - Q - r O r v . C r i - — r H C D Z Z Z H O N O . CM CM CM I—I CL C L cn LO rs CM T CO CM I-H CTI -vt CTi ro CTi CTl LO CO r-CTi ro I-CO I-H CO L O I—I t-H N H LD o r o ^ L O CO — tt — i I LO «3-0 U l DCMCOCOCOt—<CO^T<: II O f - C i CTl r - . r—. T-H <JD -iH 00 oo CM «aCO to CO C O O P s ^ us A I H o r o r—tr—ii—IU3CT1LOC0-— -— * CTi O c o CM v M n o cn co o ro L O ro CM O . H Q . ^ I N C O C f i " - ' Z Z Z a O ^ H S Q - l l < < < C M P N C D r H Z Z l O H ti ww — — m CO i—I CM CM N.^owi/)i/)HfNirosr<< CO CO | CM CO — i i CM CO — i l CM CO i—I CM COCMCO«^-LOr-HCMCO^-LO— Or-Hr^i Ht Hr-Hi Hr-HCMCMCMCMCMi I Qu Q<<<<<<<<<0<<<<<<<<<<<<0<<<<<<<<<<<<< C/JCOCOCOCOCOCOC/)COOCOCOCOCOCOCOCOC/5COCOCOC^ CO CO CO CO ii u* I H* I I |r-*t—ICMCMCMOI-HI—it—II—.CMCMCMCMOr-Hr-H'r—!•—II—ICMCMCMCM CL L CH L CQ L .i Q . o . a I O Q -zQ z- Qz- Q . — I ' C oL .C a z. z 3- s: 2 : 2: : 1^ • < • < • < - I C L CL* C L CL" C L C L C L C L 3:2:3:2:3:3:2:1:3: + -< >- C L | CO CM CM CM CM 1 2~ 2~ ZD ZD O O >CNJ CM v» CM CM CM V* CM X | CO 1 1 * 1 1 * 21 2: >- 3: 21 2: >- 3" >- >- X ZD X X CO -X ZD ZD X ZD ZD ZD O O CO CD Q O CO CD CO CO 0 1 «** *•* CM 2: >X X X >• >- >>- X + ZD >- CO CO CO CO CO CO CO CO CO CO CO >~CD v» *»» «/» z CO *»> *-* L U «»•> •X X X X ZC 0 X >- O >- 0 0 0 -< CO CO CO CO 1— >~C L C L CO CO CM *»> CM CM CM 00 j 1 ST 2: + + + + + + + + + + + * * * * * * * **A ^ * * * * * -iH + CJ CO O X I I 2: 2! ZD O O CJ 1— CD CO CM || || C J II CD CM | t ar I— 2: 2: %<» ZD ZD CD CD CJ ZD ZD v» * I CO v» LL CM CO I-H CJ CJ CJ *»> v » II * * * r-H CM CO r-H CM 3: ZD O CM •fc >~3: X ZD CO CD CM . 1 * 3: >ZD X O CO « •fc > or H CnM II r O i^t i II n i ID Ir II s Hj!^ C M r O ' X f L J!^ O J JL D, C L CL" C L CL CL CL CL C L CL CL CL I 2 : 2 : 2 : 2 : 2 : 2 : 2 : ^ 2 : 2 : 2 : 2 : 2 : C L CL CL CL CL C L CL CL CL CL CL CL 2: 2 : 2 : 2 : 2 : ^ 2 : 2 : CM + CJ CJ CJ CJ CJ CJ 0 0 «•» *•» II II II CM CO II -iH 11 II >• +-> •fc c or cu > LU E O LU LU CO LU + + + CM CO r-H CO CM CO •^r CM CM CM CM CO CO CO ro CJ CJ CJ CJ CJ CJ O VT w r-H CJ CJ CJ CJ O to 0 LL LU r-H T J or CZ 0 •Nf CD CD z LU CO CO O E tz 1—1 •sT CJ CJ CO CD «3 CO a. z L U 1— ZD O or CO Z D CO CO • — < r-H UJ LU CO 2: —1 -< LU or r-1 O Q *-< II II - II I C O ZD -<. COCO-< - • • w • I-HT-OI Z r-H *r-HCMOi - -Ht-HCMCMOr-H| w QT QjCO —J•— I -iH t-H r-Hl r-H I — 'rt- * r-H ww 21 1—1 or ro -fc 1 LU CM i £ CD L U L U > I I CZ oO C L LO 2 : ^JCD - D r w r o O CM LU cn II II Z C u CM * II CM CO < u or CJ L U O CO «3 2: c or 4-> UJ 0 CJ CJ CJ O tr> ¥T 1) X E 1 2: ZD O * + 4~ c 4J CD (O •fc •fc rM CO CO CO CO CO «yy •fc •fc •fc •fc •fc O X >- O r - j CO CO CO CM CM CM CO c 0 LU CM CM CM CM CO CO CO co •=J" t-H t-H r-Ht-H— t 1— r1 w *r» CM CO * * * * r-J O O CM CM 1 2: 21 DT ZD 2: ZD ZD CD ZD CD O Q CL * r H C M r O ^ L f l ^ D r H C N j r O ^ - L D ^ D t CO *r»—» CM CM CM CM CM CM L U L U I X CO CO CD CO C J LU * «C O W r W W W CM L CL CL CL C L C_j L CD I— co > •*£ 2 : 2: 1 — 2 : : 2: 21 DZ D LU O < < O " c o c D o r c D c o c o c D c o c o c o c o c _ O 1 W — CL I 196 Appendix 2: Program for Computing Plane Strain Element Response -< CSJ * C U CM M -< T • CM • -— CJ +• < : - -"3 C J -fc -< — ^ C M H ^ CM • CM * CM • - * * - < r-H w CJ O — C J •< •< C J • < O *-3 • < i< r-H *— r-t o Q •-3 -< O O : _< z : O : >- O Z O •< I-H C D i — z i -< LU U J <J LU O a: o ^ II H fl W H II II CM CM tl C M H H J <c o • • • • II II r^<Cr-Hr-HCMCMi<'—I - < C i — « " " " D -—• r-3 w ; ST - CM + r-t Q I « Z I LU o z LU C J r-H —I O < U UJ Z > LU I I r-H r-t r-t I— O < J O J L J < < r-H r-H r-t < <J < 0 0 ^ O <c Q z •-3 L U - o •<CQ + J CD U +J U * cu a » t/> QJ H_> ro c •r-O L_ O o «< M O tt H Z Z - < O Q Z CO C J U J - r-l O LL. + r-H • Z LU + r-H > ^ CC — ' Z r-H O L U w r-H + r-H O Z -fc • L U + o • •>Z CI w -fc ' _ ^ CD S I -fc 31 x g - C M Cr*. — Z D Cu Z D * o w z —t : r-Hr-H II I! II ~D - 10 — 'fc r-3 — I I ^ — i I — i I C3 Cv. >—' r-t «a- CU CM CO 3 " I Z Z D Z D Z D Z D Z : Z2 : Z •C<C 1— Z 3 a: r o O Q , 0 O 0 —' O CD O +; * -fc -fc CD CO C D CD E < < " < < C D C D C r j C C Z D - < < C - < • < • ' ' ' z II II I I II II II I I II ^ ^ ^ ^ ^ ^ ^ - s ^ - v x - v r-H L u <-H i - H r-H r-H CM CM CM CM + Z U II r H C V J C T T H C M f O ' t :3! — - — ••—' ' — — ' ' 1 — < O Q Q O Q Q O Q Q r-t r-t [| - J Z O r-. Q • - Q_ Q_ Q O Z M — w O O U ^.C£.OLC£.C£.C£.C£.OtCC O O U O O U O O U O O U O O O O Q O O O Z U U U U J .—(/) -fc * -O -0 - < CD _J \— * + -< DZ LU z 1— ZD O cr* ZD CO ZD DZ CD CD DZ •< Z t— •< DZ I.J OB LU Z <C •""D •< I -< LU O Of" O O O u_ II ZD UH CD O C£L CD •<£ U J CO ^ z ZD TD rz CO Lu *"D CD -fc •<. -< CO —1 DZ Z fl II -< LU O O CXL CD 0 O 0 II X CM O O LU -< + X X -*D CJ ZD z 1— z 0 0 z <J LU i D C C O ZD co - r-t I _ J < LU O cr* O I I 197 2: Program for Computing Plane Strain Element Response -< —. o -< /—. X ZD CO O DZ ZD CO 1 -< * ^ »—i •"D •"3 Z O o o •< -< X+ CO H* II O "D z + r— o •""3 o II CD Q_ II o 1 CO ZD ZD 1 CO CO o o H* CJ Q ZD CO II an x •< o 11 o CD O Q O CO -fc -< -< * * -X o o w w •< X+ I H* H< > Z Z _l w O O w . ZD CO w Z O r-H O H^H r-H : ~ C J r-H w . + : r-H O —' < II • II Z O O i CM CD ' h— O -fc > w ; •-H CO I X o ZD CO •< r"H t > • II o ^ ' o • < I— > ^ — w Z w C 3 X « - H t— T J o - CO • X —• ^ w r-H CQ ZD O -< 4- - < > - O II *-< X w o 3CL, II ZD CO II II X II r— —O CQ M ZD *—• LU CD ' -<t—t>-CDCO>-LJ_ZD t— S Z W ^ - . ^ H H Z H — - I I l| O X CO Q . O i - i > r-H ZD LL. Z I— X •—< ^ CD Q t—• L U Z D Q . O Q _ O O C D w w z CQ Q C_J •—' O O L U 1 i CO O : X w > ZD O > CO Q I . O O II CO o : t I ac — i i t• * CD • < I -fc ZD L U ; : -fc CO CrT • - a : v : ^ o z x * l < L U " r-H ZD O ZD " i C J O r-H CO CD CO : CD * Z -fc : a > Z CQ < hZD L U Z < I > LU || u t~. =*• _ CL 1 — 1 o o CD LO o ro * * * * * * * * * * JD CL - CO LU X X —I •r O CQ O -*C I- 4J M LU (O CD OC E •<• ro CrT C. O <L> > 1/1 I CO OC I— O Z I— LU O Z — >— —* I— " LL < L' L CD L U t-H O CO -< w r-H w — I I LU h~ CO Q i ZC Z iri » •< - H Z ZD ce :x II O LU O C£ O Cd _J -<C CJ CO LU -fc * -fc ^ - N O Z r - H * -fc -fc •fc CD L O —I L U CM II : •< o x o co -fc QC • O " • ""D ^ I —' -fc LL LU O > ' h—i L U CD C J O C O O O Z D Q C L U O Z Z O O CO •—• - O Z • -r-Hr-H * • w or CO • < i — ZD L U Z CJ CO C £ ' — i Q w - •< -< CO ii n o M I o CO z ^ LU Z UJ Z) CD r-. ZD < : o ) z ) LU > ** * * *** * * * * ZD O CC CO ZD i/i * Z -< w CO — * * * * * * * * * * * w O O -fc O w L U CO w CM CO II CO II CO ^ CQ - O < -H .- CC _JI -<I CDULUJ Or-t *-tw^ ^>—w<C * Z - J —I —I CD * O <C H< -< z * C J C J O C J LU * Z ra • * z || z o - t-H - —t -<C LU a: 5^ O CM = - II OC >-H - ra o LU — CD O — i—i C J ro ZD < LU _ J r - H w C M O I— r-H —I O CD Z II - < O w O i—i T ( j o •< u x 198 APPENDIX 3 PROGRAM FOR ANALYZING CYLINDRICAL CAVITY EXPANSION A3.1 Code and its Usage A FORTRAN 77 code for computing the response of an expanding cavity can be found in the following text box. Only those routines that have not been used in the code listed in Appendix 3B are included here. Four node quadrilateral element with conventional interpolation functions is used. A single row of axisymmetric elements is used with the vertical movement is restrained at all nodes. The stress-strain relationship proposed in Chapter 3 is used. No iteration is performed to redistribute the unbalanced force and it is the responsibility of the user to select a small enough vertical displacement increment applied to the nodes on the top horizontal surface so that the results do not change if a smaller increment is chosen. Some of the routines used in this program are taken from Smith and Griffiths (1988). The input isfromafreeformat ASCII file containing the following variables in the given order: number of elements (NRE), original cavity radius [RAD(l)], outer radius [RAD(NRE+1)], ratio of the radial dimensions of an element and the corresponding value for the adjacent element towards the cavity (RAT), final effective cavity pressure to which analysis is needed (PRS), number of increments of effective cavity pressure (NSTP), number of pressure increments after which output is sought (NPRN), o ', o ', C, p, porosity, K ^ , Rp, Ar|, T) v H FU m , K , X, u, n , n , v, P , A SP E P A Kp, and intervals in number of displacement increments after which results are sought. The output is printed to a user specifiedfile(overwritten if already existing). At the top of thefilesthe original geometry is printed after which once every NPRN increments of effective cavity pressure the Appendix 3: Program for Analyzing Cylindrical Cavity Expansion 199 following results are printed: effective cavity pressure, cavity strain, incremental nodal displacements, and strains and stresses at the centroid of each element. It should be noted that the strains and stresses follow extension positive convention and are printed in the following order: e , x ey, Yxy, e a ', o ', v x y and o '. 2 Appendix 3: Program for Analyzing Cylindrical 200 Cavity Expansion «— CJ + oo — o. n ~ 31 CJ CD CM - t—. i — . - — oo oo O •—i O Q Q. HH || 4-> EL 1— <u o + Z LU L u z CD -< o L U -< at Ot z at z z z * CM D X o CO 1 o LU o sv II o o CM ^r o 4c -) u u at z Cu o o LU h - O CD CM O CD L U o CM • < CO L u L u oe: Of* o O O o — ( Z T —t CM + * 11 3 Cu B II o z c u •<. CO 3 «—« t—t LU CD z o o II o Cu Cu CO z Ctt Ctt 1 — h - CD c o CO CO oc CO 1 — o CO o o o Q o Q o Q_ C u o > <a O /-N z crt II II CO z • o C a : • < act CO •(- O —J O * C J Z i _ 1 cet - < CD O C O i i I oI ' DZ • O h-H I CD — J O • LU Q O CO •—• I O O CM CO t— - i . O -CD O O I LU LU O LU < UJ*"D < '• C J - O CO Ctt Ctt C J C J — I LU U J < -< • i O O T3 • ' t Q_ - CD • -—- CD Z I CM —i Z D > - CD • DT 3 " CO i at H o . z '— ' 'o'—i . C D II CO > _ J > | -< • I U-i r-H ~ •D Ul I L U -—- I CD CD ZD — 2 1 —J , CD —J —J —I -— o U J £ L M • U J Z L J Ul Ul. Ul ""D i—• I co i CD M c o < I — I i—. o oo o I • c o —> * _ . U J M • ^ O | • U J CM CO • CO CO >—. Q COO — u . - C u CD m O > CD Q i .-. CD LU LU L U : LU C J ctt cet > - Ctt •—t • DZ Ctt I CJ 3 CD: c o CD ' • - < C u C u C t t ' - s T — • — l ~D L J _ D Z I— D Z Z M U J > J II i • at * II 41 3 w Ctt Lu o CU »-l * w L U CO CM L U - LU L U >—> i ctt C J a C r=c. - I— i o Cu o a ^ "£ Z to •< CQ^£4C Z r-t r-t w>0—I—"O—I—I—l-J—1— E •—i a CD Z L U ZD r-4_l_l—J—J-J— — OO CU —'— t•O »— t t— Oi — iCDr-H tZOi t_ Cu CO cet HCO • D — - < f-j — Cu z .-. O ctr — ' — CO CO CC c u + !_•_. i—I i CO • • Ctt O L U t— i—i L U c o I— -X. oo O * at z o o - *<r a —• H - L U L U <C O Q Q c o crt • •HOrHO; * o CJ LU a. u < Q r -< T » - — •< Get 3 1 ZD ZD I— I— I— OO DZ 3Z Q=t I I I CD —I —I Q * * * * * * o CO — ••Z GO Ctt > ^ * CO • CO <^ Q - J •o o: -<r : «+- J = * — o o 'I WCOLT 3 3* t II Ul — r -r-* *UH « <: * ctt CD * oCtt *** Cu * * * O o ' Ctt CM • LU I— UJ DZ •< CC o crt Ht h DZ CD CO CO CO Ul o CM 0 0 * I Q X -< 1 z Ctt O DZ CO Q Lu at DZ 1 DZ 1 Ctt •<£ CM CM Z 1 DZ Z CO -~- I w C O Z 3 M crt < O i CD CO *—' ^ 3 • : CO DZ r-H - i o — ZD DZ 1 DZ CO *~* Lu z Ul Qt LU O | | Z * DZ DZ - < —. h - ZD >- LU DZ OR DOF CJ * N T LO DZ • * CO | s U _JJ DZ Lu 1 LU 1 C u DZ Z o r-- ai : LU LU - LU | DZ LU • O at t C u Ctt DZ L C J •< ^ C u DZ Ctt -u> | LU CD Q . C o_ LU C u CO - Z at 3 I— CJ 0) E a> r-H L: i •< < . c 0— <u CM h- a CO ZD ZD r— Ul Ul +J - •< CO _J U ( i — h - E (0 <o c: ' X ZD —• — co 3 : Ctt D Z | | x \— •<. 3 : 3 : 3 : O z CD _ l z O ZD <u Ul DZ z CJ 4-J CO - < O ^ o CM 1 | L U x> -< Q • LU 1 *sT C D - - UJ CO - ZD CO Ctt -< Lu Z CD Lu zc CO | O C u at D Z — - O CO ^ at D Z ^j- O O DZ •<. 3 I a LU 1 1— LU Cu Cu 1 DZ -<C zc Cu - < 1 DZ DZ 1Ul I I o < c CO •<c o || ^ ^ CO 3: Z h - o- Lu •<. zc Q z ZD Z CD 1-4 l — i DZ at ^ - _ J ^ DZ oo z 3Z cu CO CD D ^ C u •< CD Ul cu on i DZ Lu DZ ALP OA CD _J LU O CD -< * * o > CD •et CO * * CO SA ERI (IL DZ II Lu z r-t -• -r4-> roU-a. r-* * r— C > (O »+- rtJ : JZt O JZ1 i O I (U • O N E • r - OI » II CO t/) •«r DZ •< O CO CM CD Cu <c o o * CD UJ> >- PA, OOR CM Ul Ul IM at z „• CD CO 'VI VHd CJ O CO *r CO BT( OAD ID CO Q I DZ CD 11 3 o tu T _AL * i—. - < at c o O DZ Cu - < CO r - t < •<. CO Ul 1— CO CM LU Uu i Zl * O- ^ Lu Lu - U J • U II o D O t-H * * * * H < LO UD Z < uiZ Z ui a, u_ _ L _ ^ CD O M w ^ II || L U c r t L u c t t u i c u c u O C D : Q . c t o . a o o z z ' * * * * * * ( o •o I ^ O n o r M O II • U J U J CM CO CD O •< —J —J ^ r-H Z LU D— f-4 - c j •< •< < I < X I I • L u Cu Cu O l - J —I CO : D Z - < • < - < DC i 1 I crt CM -X I L U Crt C D 3 Ctt + • < • z oct cet —I t i l l II z LU r-t at crt l — ( — Cu 3 : DZ '—' u i ZD O ZD L U Cet O C O CD C O o ; w —I Appendix 3: Program for Analyzing Cylindrical 201 Cavity Expansion - c ce z o o D <+- O - E y E CC >• (0 CD u CM >—' i C c e LU t-H CM • ra o I— t— s c —• I r-r O O + CD LU 3L- CNJ* ' r-H II 1-4 ce ^ —- L L o o o l l —• r—•- . o i - II * t-H * * Z r o CK: CNJ r o « y — • i— O a + ^ L L O L L O C CL , CL DB .. I — ra c o ZD O l > i C L t H i II C D I— I— —- Z O O O N^ CO OC O - Z •w w —- O (— QJ Crf CO O QJZ3 D X ^ + •r-CO fr--H r-t ""3 H +r-t-— V ) O — * r H i i1 r- 3 w + . r-H , 3 OL —* — ; r-t U w 3 : r-H r-H — O r-. U * r-t r-. LU L L '-N — II o o •— ^ OII — r a ^ o ZD I D t ' o- o r a w o r - . z z C J CNJ C D •—I mO cu < Z H I o o QJ ^ CN •Jy -I U D LL.ULU a: r H r-t O CO LUw—-U-l LU ||flw Lwwl ULU L— ULU LUO Z Oilf\| »II Q || LULU LU Qi CO Q OO O rH O3 ^ co < — > O O —• CO E -a -C <OT3 ZJ CD Z Z M T II r-HO CL -O « H O ra Or-lOftr-IS^CJCJUJ i x c o c o > c o c o m u u j CNJ CM r-H O• - CM »—. o t-H CL --H C L . ^ . CD wOO O C C O O + CO Z 0£ i— o r r— I—1 W co CD ra L OL O 1-H co CD X « 2" •8 2 o O C •CO Ora -PO CO 2Nii - < t-H ZJ - < <D r-. O O ro ^ - i < - O r-H — I I I -< * Q I O LO co LU II II QJ S_ CJ CD + —J O _ J ce > CL CJ - < rr--H * HO r-H ** 0 0 + co + + L- o o CD C D I Z C L >~ CD • ra ce ce i LU LU CD '— CJ L U CD CD CD I T X CD Z or - LU CL o Z ^ •ll Ce ce CJ • CL CD •—• Z ll ra o - LU O O ^ I— - _J ZD Z D > 2: I] Z t-t -ti a. Z ) II • L L I CM > — 1 • o C-H 1 <c CM 2: CJ 1 II < >• -r2 : ra 0 cd • ce H - . C O < Z - l - J CD — « r-H o 1 — 1 —' -< -< o C J C J CD • Z _ i t - i 1-1 ce o : LH . t-H 2: - CD LU NJ£ c e o c • — —• I— o > > * ^ DZ - t-H Z t-H ' —I c e I— •< 2; L L * ce t— Z II LU c e c j a I— co : C J —1 —1 C J LU L L - LU i — i LU CU J I—I* L C L •—• O •< . r-H —J LL _J —I •< -< z«_l•—_J 3CDJSCJO • o o O QJ Z _ J _ ) 0 0 — i O w c e c e - < i "< H • L . r a ' — . E • CO O CTOz • . w . _i c O O CO o Zo rtJo •w <w CJO — •<-<I o—* fO © — o I— co o r CL +J O . y-y crT C J LU co CC . -H ; r— CJ I ra c i —• - OC =» — tO• •— w O r-H CO CO O O •zc r-t - O C O • — • — f CL CLce co< •< O• u LO*-* OO CO wC— O CL CC — |O CO C I— O CO 1O 5 / <— I h — OO n > • Z D • CO CM O O _l oB r-H OO DT . r-H C o wCD x z c y i' o oCJ — - —• O — I— 1 C O r* -< : :: r-i- w< r-H f. • r-H i CL CL L O C L j: < L. •«H-CO CO O LU U_ CO CD O • r-HOO—ICOO—I—Jr-Hl i 4-> O >—I C O t r-H O - O » - t CM r-H £ O— r — < CI CD. - o CD CJ CJ t—. Appendix 3: Program for Analyzing Cylindrical - 202 Cavity Expansion o D_ -4-) * I O CD C u C u C u O LULU LU : < < i±j :a o Qo o ll 11 ll CO ll .Hr-lrHrHC^C\JC\JIM II CJ z • CD CD C D O o o CrT CrT CrT cr: cr: • O oO O O O O O Q • o oO O O O o o z \ o <_> u u o u C J C J L U : at atc r : E-r-H r—t r-H r-t CU . )CQ 3 3 > C J O *—t • II r - t —I — -- — rt CO 3 ww> U * • t— ' t O - H M + I _l r-l CM Z r-« Z •—• Z L • < C u II II . — < — - < OO OO •—• z z > LU o r-t o n L U a z z II L u > Q - r r - M r - t Z Q t — ' V) < < - 1 .—.•—.Q-<CJ-<'—"CO—I" -< — i i at o L U err o z a: 3 L L L U o 3 o * * *-fc * CJ) LU z Lu Z (_ ai E =J c JD CD -t-> c cu E O <u O CJ> cu CD CrT o err o o o J= CJ ro CD L. o Cu 4 LU LO Q •*-> CJ CrT >• LU CD CD Z o Cu LU CD -< cu o cu fO < e a fl H + h Z CO + —I o — r-l II . CO w : cu — . CO CO • < co Cu —• CO m I LU || • Z r—I O - Cu LU O SC LU . L U I—* ZD C u z t—t >i H LU II z C u O HH h-H w w cr: u_ : i—i o a 3 • r-H ^H w• 1/1 wO r-H O ww -H O * o O O ta co co i * to c a > t_ C D H CU I > • r-t -"D II ' ZD Z D CD Z D Cu + • < 'II LU CM >-D =^ *~D CTt *-D Z — • t O t—t CU I Q_ O O D II - II r — ' II Z= ' t Z Q • Z H M M T C U O cu z Cu —• + — * * O < : C J LU 0 ' — i I— ^ CD CD CTi n M M Q Q. I Z II Cu —' ^D CO II L u CO CO CO L O r-H CD CO O Cu —I +J > o O — < —i — •> crT + o + + Lu Z CU CD CD >- o B CrT z c LU Z CU L U 4-> -U> C J CD u CO r-l O CU I— O '—• * DZ ZD CD Z c l_ cOJu :w —-mi err : > co :o ZD CO * + + LU o z C u DZ -fc crT CD ZD C u Z || 11 >-> C J L U CD CM CO LU CJ V) J Z 1— DZ DZ DZ 4-> L U z ZD ZD ZD 1_ Z Z Z LU z • a u. 1— ZD o err CO * O o o o -o o o or E o 4— * * * * * ar + CO DZ ZD Z II D "fr DZ ZD Z
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Mechanical properties of granular deposits from self-boring pressuremeter tests Roy, Debasis 1997
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Title | Mechanical properties of granular deposits from self-boring pressuremeter tests |
Creator |
Roy, Debasis |
Date Issued | 1997 |
Description | The self-boring pressuremeter test has the potential of providing the information necessary for calculating the deformation behavior of granular deposits. However, the stress path and loading orientation with respect to the depositional direction in the test is such that the stress-strain relationship obtained directly from the data following the existing procedures can be very conservative in many field problems of practical significance. A reasonably simple stress-strain model is proposed in this research accounting for inherent anisotropy for granular deposits and stress-path dependent soil behavior. The model is used to back-analyze self-boring pressuremeter tests and determine an optimum set of model parameters representing the behavior of the deposit. The exercise of back-analysis essentially involves fitting the response of the stress-strain model to the self-boring pressuremeter data by altering the model parameter by manual trial and error. To compute the model response in cylindrical cavity expansion, a commercially available explicit finite difference computer code (FLAC) is used. An implicit finite element code developed in this research can also be used for the purpose. To minimize the impact of non-uniqueness in the result, a-priori information about the bounds of values of some of the model parameters is used. Supplementary information about the state of packing of the deposit and small strain stiffness is obtained from a seismic piezocone penetration test carried out adjacent to the self-boring pressuremeter bore hole. The deformation behavior of an axisymmetric element for the model parameters predicted from back analysis of self-boring pressuremeter test is found to agree with laboratory triaxial tests on undisturbed (frozen) samples in both compression and extension at several sites. The explicit finite difference computer code FLAC is again used to predict the response of an axisymmetric element. Further validation for the procedure proposed for estimation of deformation behavior of granular deposits from self-boring pressuremeter comes from the negligible value of the undrained strength obtained for data from a mine site where there was a static liquefaction failure. The only reliable method available at present to estimate deformation behavior of granular deposits is extraction of frozen samples and laboratory testing. The cost of adopting this approach is often too great for routine use. Even in important projects economical options of estimating deformation behavior from index tests such as SPT and piezocone penetration are thus adopted. These empirical procedures are usually very imprecise. The procedure based on back analysis of self-boring pressuremeter tests proposed in this research appears to provide a more reliable alternative than the approach based on index tests. At the same time, the approach is quite economical in comparison with the option of undisturbed sampling and laboratory testing. |
Extent | 10704585 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-04-20 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0050293 |
URI | http://hdl.handle.net/2429/7419 |
Degree |
Doctor of Philosophy - PhD |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1997-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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