UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Analysis procedures for caisson-retained island type structures Salgado, Francisco Manuel Goncalves Alves 1990

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1990_A1 S24.pdf [ 16.97MB ]
Metadata
JSON: 831-1.0050494.json
JSON-LD: 831-1.0050494-ld.json
RDF/XML (Pretty): 831-1.0050494-rdf.xml
RDF/JSON: 831-1.0050494-rdf.json
Turtle: 831-1.0050494-turtle.txt
N-Triples: 831-1.0050494-rdf-ntriples.txt
Original Record: 831-1.0050494-source.json
Full Text
831-1.0050494-fulltext.txt
Citation
831-1.0050494.ris

Full Text

ANALYSIS PROCEDURES FOR CAISSON-RETAINED ISLAND TYPE STRUCTURES  by FRANCISCO MANUEL GONCALVES ALVES SALGADO j  B.Sc., Technical University of Lisbon, Portugal, 1972 M.A.Sc., University of British Columbia, Vancouver, Canada, 1981  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES Department of Civil Engineering  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA May, 1990  © FRANCISCO MANUEL GONCALVES ALVES SALGADO, 1990  In  presenting  degree freely  this  at the  thesis  in  partial  University  of  British Columbia,  available for reference  copying  of  department publication  this or  thesis by  for  his  fulfilment  and study. scholarly  or  of this thesis for  her  purposes  representatives.  C - W l  Date  DE-6 (2/88)  M a y  l£  ,  requirements that the  agree that  for  Columbia  1 5 3 0  an  advanced  Library shall make  permission for  may  be  granted  by  It  is  understood  the that  financial gain shall not be allowed without  ETnjG-? D E C K I N G  T h e University of British Vancouver, Canada  the  I agree  I further  permission.  Department of  of  it  extensive  head of  my  copying  or  my  written  ABSTRACT  This thesis is concerned with the analysis of large offshore gravity structures used for oil exploration and recovery in the Beaufort Sea. Because of the high ice loads and the water depths involved, these structures comprise a large steel box infilled with a sand core for stability. One such structure was subjected to severe ice loading in April 1986 causing portions of the sand core to liquefy and bring the structure to a near failure condition. This structure was heavily monitored and thus serves as a case study against which the proposed analysis procedure can be checked. The behaviour of these soil-structure systems is highly complex depending upon the characteristics of the soil, the structural elements and the soil-structure interface.  In this thesis a three-dimensional Finite  Element computer program with soil, interface and structural elements is developed. Emphasis is placed on the three-dimensional stress-strain constitutive law both in terms of its ability to model observed laboratory response as well as the determination of the constitutive law parameters from in situ testing. The results obtained in terms of displacement, acceleration and zones of liquefaction by the analysis were then compared with the field measurements obtained during the April 1986 ice load event.  The good agreement  obtained between predicted and observed response is a validation of the proposed procedure.  ABSTRACT  ii  TABLE OF CONTENTS  iii  LIST OF TABLES  xiii  LIST OF FIGURES  xiv  NOMENCLATURE  xxv  ACKNOWLEDGEMENTS  xxxv  CHAPTER 1 - INTRODUCTION  1  1.1 Purpose  1  1.2 Scope  4  CHAPTER 2 - 3-D CONSTITUTIVE MODEL FOR SANDS FOLLOWING THE CONCEPT OF THE SPATIAL MOBILIZED PLANE  11  2.1 Introduction  11  2.2 2-D Constitutive Model for Sand Following the Concept of the Mobilized Plane  16  2.2.1 Brief Description of the 2-D Constitutive Model  16  2.2.1.1  Yield Criterion  18  2.2.1.2  Flow Rule  20  2.2.1.3  Hardening Rule  21  2.2.2 Brief Development of the 2-D Model in the Cartesian ... System of Coordinates 2.3 Discussions on the Theories of the "Compounded Mobilized Planes", CMP, and the "Spatial Mobilized Plane", SMP  29  33  2.4 3-D Constitutive Model for Sands Following the Concept of the Spatial Mobilized Plane  34  2.4.1 Description of the 3-D Model  36  2.4.2 Development of the Plastic Constitutive Matrix {CP} ...  37  2.4.2.1  Yield Criterion  38  2.4.2.2  Flow Rule  41  2.4.2.3  Hardening Rule  .  43  2.4.2.4  Summary of the Basic Equations of the SMP Model ..  45  2.4.3 Development of the SMP's Plastic Constitutive Matrix in the Cartesian System of Coordinates 2.4.3.1  Relationship Between Increments of Plastic Principal Strain, Ael? and Ae| Mp and  2.4.3.2  Relationship Between Increments of Plastic Principal Strain, Ae?, and Increment of Stress Ratio on the SMP, ^(XSMP//°SMP^  2.4.3.3  Relationship Between Increments of Plastic Cartesian Strain {Ae^} and Increments of Plastic Principal Strain {Aef}  45 47  51  2.4.3.4  Relationship between Increments of Plastic Cartesian Strain {AeP} and Increment of Stress Ratio on the SMP, A(T SM p/o SM p)  2.4.3.5  Evaluation of the Increment of the Stress Ratio on the SMP as a Function of the Increments of Cartesian Stress {Ao}  53  Evaluation of the Plastic Constitutive Matrix ....  56  2.4.3.6  2.4.4 Evaluation of the Elasto-Plastic Constitutive Matrix {CeP}  56  2.4.4.1  Loading and Unloading Constitutive Matrix  2.4.4.2  Implementation of the Modified SMP Model into  2.4.4.3  57  Finite Element Form  58  Load Shedding Formulation  59  2.5 Review of the Assumptions Considered in the Modified SMP Model  61  2.5.1 Summary of the Assumptions Used in the Modified SMP Model  61  2.5.2 Discussion of the Assumptions Regarding the Direction of the Increments of Principal Strain  63  2.5.3 Discussion of the Assumptions Regarding the SMP Failure Criterion  65  2.6 Disadvantages of the SMP Model  66  CHAPTER 3 - PROCEDURES FOR THE EVALUATION OF SOIL PARAMETERS FOR USE IN THE MODIFIED SMP MODEL. VERIFICATION OF THE MODIFIED SMP MODEL  69  3.1 Introduction  69  3.2 Evaluation of Soil Parameters for Use in the Modified SMP Model from the Standard Triaxial Test  70  3.3 Verification of the Modified SMP Model Against Observed Laboratory Test Data  71  3.3.1 First Level of Verification of the Modified SMP Model. Calibration with Simple Shear Test Data on LeightonBuzzard Sand  72  3.3.1.1 Soil Parameters for Leighton Buzzard Sand (e0=.53) for Use in the Modified SMP Model  72  3.3.1.2 Calibration with the Simple Shear Data Reported by Stroud  72  3.3.2 Second Level of Verification of the Modified SMP Model. Predictions of Simple Shear and True-Triaxial Test Data on Ottawa Sand  79  3.3.2.1 Soil Parameters for Ottawa Sand (Dr=87%) for Use in the Modified SMP Model and the Hyperbolic Model  82  3.3.2.2 Predictions of the Simple Shear Test Data Reported by Vaid, Byrne and Hughes  83  3.3.2.3 Predictions of the Post-Workshop Test Data of the 1980 Workshop of McGill University  87  3.A Conclusions  96  CHAPTER 4 - INTERFACE ELEMENTS  99  4.1 Introduction  99  4.2 Brief Review of the Existing Interface Elements  99  4.3 Description of the "Thin" Interface Element  100  4.4 Determination of Soil Parameters  105  4.5 Implementation of the "Thin" Interface Element into Finite Element Formulations  Ill  4.6 Performance Studies of the "Thin" Interface Element  Ill  4.6.1 Closed Form Solution of a Soil-Pipe system  Ill  4.6.2 Retaining Wall Study  116  4.7 Conclusions  130  CHAPTER 5 - EVALUATION OF SOIL PARAMETERS FROM THE PRESSUREMETER TEST IN SAND 5.1 Introduction  132 132  5.2 Evaluation of the Maximum Shear Modulus for Sand From the Unload Shear Modulus Obtained from Pressuremeter Tests  135  5.2.1 Assumed Stress-Strain Relations for Sand Upon Unloading  136  5.2.2 Analysis Procedure  141  5.2.2.1 Loading Phase  143  5.2.2.2 Unloading Phase  147  5.2.3 Results 5.2.4 Validation of the Proposed G * / G m a x > 0 Chart 5.3 Evaluation of Soil Parameters from the First Time Loading Part of the Pressuremeter Test 5.3.1 Finite Element Predictions of Pressuremeter Chamber Tests  150 150 156 157  5.3.2 Evaluation of Soil Parameters from the Pressuremeter Test  165  5.3.2.1 Numerical Verification of the Method Proposed by Manassero  166  5.3.2.2 Procedures for the Evaluation of Soil Parameters for Use in the Modified SMP Model from Pressuremeter Test Data  168  5.4 Summary  174  CHAPTER 6 - EVALUATION OF SOIL PROPERTIES FOR USE IN THE ANALYSIS OF THE MOLIKPAQ STRUCTURE AT THE AMAULIGAK 1-65 SITE ... 177 6.1 Introduction  177  6.2 Brief Description of the Site Investigation and Construction Sequence. Type of Sand Used in the Berm and Core of the Molikpaq  178  6.3 Evaluation of the Soil Parameters of Erksak 320/1 Sand for Use in the Analysis of the Molikpaq  180  6.3.1 Evaluation of the Moduli Used in the Analysis  182  6.3.1.1 6.3.1.2  Evaluation of the In Situ Void Ratio vs Depth .... 183 Evaluation of the In Situ Maximum Shear Modulus versus Depth 190  6.3.1.3  Evaluation of the Young's Modulus  197  6.3.1.4  Evaluation of the Bulk Modulus  199  6.3.1.5  Evaluation of the Plastic Shear Parameter G and the Flow Rule Parameters for Use with the Modified SMP Model  201  6.3.2 Evaluation of the Peak Friction Angle and the Peak Stress Ratio on the SMP  204  6.3.2.1  Evaluation of the Peak Friction Angle  204  6.3.2.2  Evaluation of the Peak Stress Ratio in the SMP ... 210  6.3.3 Predictions of the Drained Triaxial Tests on Erksak 320/1 Sand  212  6.3.4 Selection of Soil Types and Soil Parameters to Use in the Molikpaq Analysis  217  6.4 Evaluation of the Liquefaction Resistance Curves for Erksak 320/1 Sand  222  6.4.1 Review of Available Cyclic Loading Triaxial Tests on Erksak Sand  222  6.4.2 Evaluation of Liquefaction Resistance Curves for Erksak 320/1 Sand Based on No Static Bias  224  6.4.3 Discussion on the Past-History of Cyclic Loading and Drainage Conditions at the Amauligak 1-65 Site  231  6.4.4 Pore Pressure Rise  235  CHAPTER 7 - 3-DIMENSIONAL FINITE ELEMENT ANALYSIS OF THE MARCH 25 AND APRIL 12, 1986 ICE LOAD EVENTS  239  7.1 Introduction  239  7.2 Ice Loading Function Used in the Analysis  241  7.3 3-Dimensional Modelling of the Molikpaq  244  7.3.1 3-Dimensional Structural Model of the Molikpaq  244  7.3.2 3-Dimensional Finite Element Mesh Used in the Analysis  244  7.3.3 3-Dimensional Soil Model Used in the Analysis  247  7.4 3-Dimensional Analysis 7.4.1 Construction Phase Analysis 7.4.2 3-D Analysis of the Static Ice Load Event of March 25, 1986 7.4.3 3-D Analysis of the Dynamic Ice Load Event of April 12, 1986 7.4.3.1  Liquefaction Assessment  251 252 253 262 262  7.4.3.2  Pore Pressure Rise Assessment  266  7.4.3.3  Acceleration Assessment  273  7.4.4 3-D Analysis for the Settlement Assessment  278  7.5 2-Dimensional Analysis of the Dynamic Ice Load Event of April 12, 1986  286  7.5.1 Description of the Key Parameters Studied in the 2-D Analysis  286  7.5.2 Conclusions From the 2-D Analysis  288  7.6 Conclusions  289  CHAPTER 8 - SUMMARY AND CONCLUSIONS  290  8.1 3-D Constitutive Law for Sands  290  8.2 Evaluation of Stress-Strain Parameters of Soils from Laboratory and/or In Situ Testing  293  8.2.1 Evaluation of Soil Parameters from Laboratory Tests .... 293 8.2.2 Evaluation of Soil Parameters from the Pressuremeter Test 8.2.3 Evaluation of Soil Parameters from Laboratory and In Situ Testing 8.2.3.1 Summary of the Procedures Followed to Evaluate the Moduli Used in the Analysis 8.2.3.2 8.2.3.3  294 295 296  Summary of the Procedures Followed to Evaluate the Failure Parameters Used in the Analysis  297  Summary of the Procedures Followed to Evaluate the Liquefaction Resistance Curves for Erksak 320/1 Sand  297  8.3 Interface Elements  298  8.4 Summary of the Analysis Procedure and of the Results Obtained from the Analysis  299  REFERENCES  305  APPENDIX 2  320  2.1 Evaluation of the friction angle <f>13 for b-values varying from 0.0 to 1.0  323  2.2 Relationship between principal stresses and, the normal and shear stress on the SMP. Extrapolation of these relationships to the increments of plastic strain space .. 329 2.3 Fundamental relationship between cartesian stresses and principal stresses. Extrapolation of these relations to the increments of plastic strain space  332  2.A Development of A(x SM p/o SM p) in terms of Ao^ Ao3, Ao3 .... 336 2.5 Relations between increments of principal stress and increments of cartesian stress in the 3-Dimensional stress space  348  2.6 Evaluation of the plastic constitutive matrix {Cp} of the SMP model  379  2.6.1 3-Dimensional  380  2.6.2 2-Dimensional  381  2.6.3 Axisymmetric  _  2.7 Load shedding formulation to use with the modified SMP model 2.8 Discussion of the assumptions regarding the direction of the increments of principal strain, based on data derived from hollow cylinder tests APPENDIX 3 3.1 Procedures for the Evaluation of Soil Parameters for Use in the Modified SMP Model from the Standard Triaxial Test 3.2 Evaluation of Soil Parameters for Leighton-Buzzard Sand (e0 = .53) for Use in the Modified SMP Model 3.3 Evaluation of o l t o 2 , o 3 , AeP, AeP, AeP and o x from the Simple Shear Data on Leighton-Buzzard Sand Reported by Stroud (1971)  385 386 397 419  422 436  446  3.4 Evaluation of Soil Parameters for Ottawa Sand (Dr = 87%) for Use in the Modified SMP Model  459  3.5 Strain Softening Formulation for use in the Modified SMP Model  474  APPENDIX 4  478  4.1 Implementation of the "Thin" Interface Element Into the Finite Element Formulation  480  4.2 Load Shedding Formulation for Interface Elements  488  4.3 Extension of Matsuoka-Nakai Failure Criterion to Granular Soils with Cohesion and Friction  491  APPENDIX 5  496  5.1 Brief description of the method proposed by Manassero (1989)  498  5.2 Development of Manassero's incremental equation  501  5.3 Assessment of the peak friction angle <J>PS and dilation angle v for Leighton-Buzzard sand (e0 = .53), based on simple shear test data reported by Stroud (1971) and Budhu (1979)  506  5.4 Relationship between KGe and Gjjy  513  5.5 Evaluation of soil parameters for use in the modified SMP model from pressuremeter test data  516  APPENDIX 6 6.1 K 0  534 Assessment  6.2 Assessment of Soil Parameters for Erksak 320/1 Sand Based on the Drained Triaxial Test Data Reported by Golder Associates (1986) APPENDIX 7  546  545 559  7.1 Evaluation of the Number of Equivalent Cycles of the Ice Loading Function  562  7.2 2-Dimensional Structural Models  568  7.3 Evaluation of the 3-D Load Vector Used in the Analysis ... 573 7.A Procedures Followed in the 3-D Analysis of the Dynamic Ice Load Event of April 12, 1986  577  7.4.1 Liquefaction Assessment  578  7.4.2 Porewater Pressure Rise Assessment  581  7.4.3 Acceleration Assessment 7.5 Evaluation of the Moduli and Load Vector Used in the Settlement Analysis 7.6 2-D Finite Element Analysis of the Dynamic Ice Load Event of April 12, 1986  583 587 591  7.6.1 Introduction  592  7.6.2 Verification of the Structural Model Used in the 2-D Ice Loading Analysis of the Molikpaq Structure  592  7.6.3 Study of the Influence of the Interface Element Type and the Value of the Angle of Friction, 6, Used in the Analysis  593  7.6.4 Study of the Influence of the Method Used to Redistribute the Shear Stress of the Liquefied Soil Elements  598  7.6.5 Study of the Influence of the Stress-Strain Law Used in the Analysis  603  7.6.6 Conclusions  610  LIST OF TABLES Table 3.1  Summary of Soil Parameters for Use in the Modified SMP Model  70  3.2  Soil Parameters for Leighton-Buzzard Sand (e0=.53) for Use in the Modified SMP Model  73  3.3  Soil Parameters for Ottawa-Sand (Dr=87%) for use in the Modified SMP Model  82  3.4  Soil Parameter for Ottawa-Sand (Dr=87%) (Evaluated by Duncan (1980) and Used in the Hyperbolic Model)  83  5.1  Soil Parameters for Leighton-Buzzard Sand (e0=.53) for Use in the Modified SMP Model  157  5.2  Comparison Betwen Plastic Soil Parameters  172  6.1  Index Properties of Erksak 320/1 Sand (Reported by Golder Associates (1986))  182  6.2  Hyperbolic Soil Parameters - Erksak 320/1 Sand  216  6.3  Modified SMP Soil Parameters - Erksak 320/1 Sand  216  6.4  Summary of Multi-Year Ice Loading Events (Spring, 1986) (After Jefferies and Wright, 1988)  231  7.1  Maximum Residual Excess Porewater Pressure (kPa)  269  7.2  Magnitude and Time of Maximum Acceleration  274  Figure 1.1  Beaudril Mobile Artie Caisson "Molikpaq"  2  1.2  Schematic Cross-Section of Molikpaq and Berm  3  1.3  Molikpaq Site Locations  4  2.1  Analysis of Laboratory Tests on Sand Carried out With the Hyperbolic Model: (a) Standard Triaxial Test (Byrne and Eldridge, 1982) ..  13  (b) Simple Shear Test (Present Study)  13  2.2  Mohr-Coulomb, Lade and Matsuoka-Nakai Failure Criteria  15  2.3  (a) 2-Dimensional Mobilized Plane  17  (b) Evaluation of <t>m, o^p, and t^p  17  2.4  Matsuoka-Nakai 2-D Failure Criterion  19  2.5  (a) Relationship Between C^Mp/oMp) and -(AeMp/AyMp). Toyoura Sand (after Matsuoka, 1974)  23  (b) Relationship Between (tMp/oMp) a n ( * ~eMP^^MP* Toyoura Sand (after Matsuoka, 1974)  23  2.6  2.7  2.8  Hyperbolic Relationship Between (Tvp/oMp) a n d T^p* Toyoura Sand (after Matsuoka, 1983)  26  (a) Mohr Circle of Stresses  30  (b) Mohr Circle of Increments of Plastic Strain  30  (a) Three 2-Dimensional Mobilized Planes  35  (b) Development of Three Mobilized Friction Angles in the x, o Stress Space (c) Spatial Mobilized Plane 2.9  35 35  Matsuoka-Nakai and Mohr-Coulomb Failure Criteria. (a) Projection on the Octahedral Plane  40  (b) 3-Dimensional Stress Space  40  - <f>]jx ) with b-Value  2.10  Variation of  2.11  (a) Relationship Between (tsMP^SMP^ a n d Toyoura sand (after Matsuoka, 1983)  42 ~^SMP^SMP^ 44  (b) Relationship Between (tsmP^°SMP^ a n < ^ ^SMP* Toyoura Sand (after Matsuoka, 1983) 2.12  44  Variations of Abi (i = 1,2,3) with 'b-Value' (a) i=l; (b) i=2; (c) i=3  2.13  49  Predicted and Measured Simple Shear Data (a) x z x versus r z x > ( b ) e v  v e r s u s  Tzx!  (2a  J  /a 1 +a 3 )  versus y 2.14  2.15  50  (a)  Shear Failure During Loading  60  (b)  Shear Failure During Unloading  60  (c) Shear and Tension Failure During Unloading Rotation of Principal Axes During Simple Shear Tests on Leighton-Buzzard Sand (e0 = .64). Data reported by Roscoe (1970).  60  (a) Definition of Angles  64  £ and x  (b) Virgin Loading Test Data 2.16  3.1  64  (c) Virgin Loading, Unloading and Reloading Test Data .. 64 Variation of Friction Angle <f> With 'b-Value' for Leighton-Buzzard Sand 67 Predicted and Measured Simple Shear Data on LeightonBuzzard Sand (t/s versus 7). (a) o v = 48 kPa; (b) o v = 72 kPa; and (c) o y = 172 kPa  3.2  Predicted and Measured Simple Shear Data on LeightonBuzzard Sand, (a) t z x versus y z x ; (b) e v versus x z x ••••  3.3  Predicted and Measured Simple Shear Data on LeightonBuzzard Sand (o x /o xo versus y). (a) o = 48 kPa; (b) o v = 72 kPa; and (c) o v = 172 kPa  75 76  77  3.4  Predicted and Measured Simple Shear Data on LeightonBuzzard Sand (o2/s versus y). o v = 48 kPa  3.5  a) Stress Paths Used to Generate Data Base for Modelling  80  b) Grain Size Distribution of Ottawa Sand  80  3.6  Stress Paths Used for Predictions  81  3.7  Stress-Strain Behaviour of Ottawa Sand in Drained Simple Shear (after Vaid, Byrne and Hughes, 1980)  3.8  3.9  3.10  3.11  3.12  78  84  Predicted and Measured Simple Shear Data on Ottawa Sand. a) t z x versus r z x  86  b) e v versus f z x Predicted and Measured Constant Mean Stress Test Data on Ottawa Sand, b = 0.2, o m = 10 psi.  86  a) Predictions with the Hyperbolic Model  89  b) Predictions with the Modified SMP Model  89  Predicted and Measured Constant Mean Stress Test Data on Ottawa Sand, b = 0.5, o m = 5 psi. a) Predictions with the Hyperbolic Model  90  b) Predictions with the Modified SMP Model  90  Predicted and Measured Constant Mean Stress Test Data on Ottawa Sand, b = 0.5, o m = 20 psi. a) Predictions with the Hyperbolic Model  91  b) Predictions with the Modified SMP Model  91  Predicted and Measured Constant Mean Stress Test Data on Ottawa Sand, b = 0.8, o m = 10 psi. a) Predictions with the Hyperbolic Model  92  b) Predictions with the Modified SMP Model  92  .3.13  3.14  Predicted and Measured Proportional Loading Test Data on Ottawa Sand, = 10 psi. a) PL1 Test, Predictions with the Hyperbolic Model  94  b) PL1 Test, Predictions with the Modified SMP Model ...  94  c) PL2 Test, Predictions with the Hyperbolic Model  94  d) PL2 Test, Predictions with the Modified SMP Model ...  94  Predicted and Measured Reduced Triaxial Test Data on Ottawa Sand, = 20 psi. Reduced Triaxial Compression Test (RTC). (a) Predictions with the Hyperbolic Model  95  (b) Predictions with the Modified SMP Model  95  Reduced Triaxial Compression Test (RTE).  3.15  4.1  (c) Predictions with the Hyperbolic Model  95  (d) Predictions with the Modified SMP Model  95  Predicted and Measured Circular Stress Path Test Data on Ottawa Sand. o m = 10 psi; x o c t = 4.2 psi. a) Predictions with the Hyperbolic Model  97  b) Predictions with the Modified SMP Model  97  "Thin" Interface Element: (a) 2-Dimensional; (b) 3-Dimensional  101  4.2  Bonding and Debonding Modes  102  4.3  Slip and No Slip Modes  104  4.4  a) Comparison of Hyperbolic and Actual StressDisplacement data (after Clough and Duncan, 1971) ... 110 b) Transformed Linear Hyperbolic Plots for Interface Tests (after Clough and Duncan, 1971) 110 c) Evaluation of 6 and C from Direct Shear Tests  110  4.5  Soil Pipe System  112  4.6  F.E. Meshes and Soil Properties Used for the Soil Pipe Closed Form Solution  114  4.7  Soil Pipe System. Closed Form Solutions and F.E. Predictions 2-D and 3-D: (a) o r /p Q ; (b) t/p0  4.8  Retaining Wall Field Study: (a) Retaining Wall Instrumentation; (b) Wall Positions  118  4.9  a) Grain Size Distribution of Silty Sand  119  115  b) Index Properties and Strength Parameters of Silty Sand 4.10  4.11  119  a) Earth Pressure Measurements Versus Depth  121  b) Inferred Earth Pressure Coefficient, K, versus Depth  121  c) Relationship Between P/p0 and Displacement at Each Depth a) Cross Section Illustrating Retaining Wall and Backfill  121 123  b) Finite Element Mesh and Soil Properties Used in the Analysis  123  Comparison Between Earth Pressure Measurements and F.E. Predictions at Rest Condition, d = 0.0 cm  125  Comparison Between Earth Pressure Measurements and F.E. Predictions at Active Condition, d = 1.6 cm  126  Comparison Between Earth Pressure Measurements and F.E. Predictions at Active Condition, d = 8.4 cm  127  Variation of Horizontal Wall Pressure Distribution with Wall Movement and Interface Friction Angle, 6 (after Clough and Duncan, 1971)  129  5.1  Pressuremeter Unload Modulus, G*  134  5.2  Loading and Unloading in a Conventional Triaxial Path (After Negussey, 1984)  137  4.12 4.13 4.14 4.15  5.3  Comparison Between E and Various Initial Modulis E^ (After Negussey, 1984)  5.4  Measured and Computed G m a x Values. (After Yu and  137  Richart, 1984)  140  5.5  Assumed Unload Stress-Strain Behaviour  140  5.6 5.7  G/G  m a x V e r s u s Shear Strain Sketch of the Loading and Unloading Response of the Pressuremeter  143  5.8  Stress State After Pressuremeter Loading  144  5.9  Plain Strain Axisymmetric Finite Element Mesh  149  5.10  Chart for Determination of G m a x Value  151  5.11  Relationship Between G m a x  5.12  0  from the Measured G*  142  and G r c .  (a)  Chamber Test, Ideal Installation (Camkometer) .... 153  (b)  Chamber Test, Self-Bored (Camkometer)  153  Relationship Between G m a x > 0 and G^. (a)  In Situ, Self-Bored (Camkometer)  155  (b)  In Situ, Self-Bored (PAF-79)  155  5.13  Pressuremeter Chamber Test Set Up  159  5.14  Axi-Symmetric Domain Used in F.E. Analyses  160  5.15  Predicted and Observed Response at Face of Pressuremeter. (a) o z = 200 kPa. (b) o z = 90 kPa  5.16  Displacement at Inner and Outer Boundary  163  5.17  Predicted Displacement Patterns. (b) o z = 45 kPa  164  5.18  Pressuremeter and Simple Shear Data Versus Shear Strain, y  (a) o z = 200 kPa;  162  164  5.19  5.20  (a) Variation of b-value with Shear Strain, y  171  (b) Variation of (f/b-value) with y  171  (a) Variation of (o r )f ace . ^v^face shear strain, y  and  ^G^face  w i t h  173  (b) Variation of (e r )f a c e and (eQ)face with y  173  6.1  (a) On site investigation and construction sequence .... 179  6.1  (b) Soil Conditions at the Amauligak-165 Site  6.2  Grain size distribution of Erksak 320/1 sand (after  179-A  Golder Associates, 1986)  181  6.3  State parameter and steady state line  18A  6.A  "Mean" values of q c in the core, berm and foundation (after Jefferies and Livingstone, 1985) Location of cone penetration tests carried out in  186  the Molikpaq core  187  6.5  6.6  In situ state parameter, \J>, versus depth  6.7  In situ void ratio, e c versus depth  6.8 6.9  Variation of maximum shear modulus, G m a x with depth .... 191 Relationship between cone bearing, q c and maximum shear modulus, G m a x (after Robertson and Campanella, 198A) ... 192  6.10  Variation of (q c /N 60 ) with mean grain size (after Seed and DeAlba, 1986)  6.11  188 189  19A  Variation of maximum shear modulus number, (KG m a x ) a v with depth  196  6.12  Maximum shear modulus number, K G m a x versus void ratio, e c 198  6.13  Young's modulus numbers, K E m a x and KE versus void ratio, e c  200  6.1A  Bulk modulus numbers, KB u r and KB versus void ratio, e c  6.15  Plastic shear number, KG versus void ratio, e r  202 203  L  6.16  Flow rule relationship for Erksak 320/1 sand  205  6.17  Relationship between state parameter, and peak friction angle, <f>p (after Golder Associates, 1986)  207  6.18 6.19  Relationship between peak friction angle, <pF and log10 ( ( o ) / P a ) f o r E r k s a k 32 m i °/1 s a n d "Mean" values of q c in the berm before placement of the core (after Jefferies and Livingstone, 1985)  208  209  6.20  Variation of ($j)av and (A<j>)av with depth  6.21  Relationship between (tsmP//°SMP^F a n d l°Sio ((°SMP^F^a^ for Erksak 320/1 sand 213  6.22  (a) Relationship between (TSMP/'0SMP^ i a n d 320/1 sand (b) Relationship between A(t SM p/o SM p) 320/1 sand  6.23 6.24 6.25 6.26(a) 6.26(b) 6.27 6.28  6.29  Variation of (("tsMP^SMP^ i^av a n d with depth  and  211  for  E r k s a k  214 ^  for  E r k s a k  214  ^^TSMP/'0SMP^av 215  Predictions of drained triaxial tests on Erksak 320/1 sand using the hyperbolic model  218  Predictions of drained triaxial tests on Erksak 320/1 sand using the modified SMP model  219  Soil types and parameters for use in the Molikpaq analysis with the hyperbolic model  220  Soil types and parameters for use in the Molikpaq analysis with the modified SMP model  220  Soil types and finite element layers used in the Molikpaq analysis  221  Relationship between stress ratio causing liquefaction in 15 cycles and modified cone tip resistance for sands and silty sands (after Seed and DeAlba, 1986)  225  Relationship between cyclic stress level and the number of cycles to cause liquefaction (after Been, 1988)  226  6.30  Relationship between stress ratio causing liquefaction, in 5 to 1000 cycles, and modified cone tip resistance for Erksak 320/1 sand  227  6.31  Variation of average (qc)x with depth  229  6.32  Liquefaction resistance curves for Erksak 320/1 sand ... 230  6.33  Effect of previous cyclic loading on porewater pressure development in (a) medium dense, and (b) loose sand (after Finn et al., 1970)  233  6.34  Effect of previous loading history on liquefaction resistance (after Seed et al., 1988)  233  6.35  Residual porewater pressure rise as a function of the number of cycles to liquefaction  237  7.1  Idealized Ice loading function used in the analysis .... 242  7.2  Actual ice loading function  243  7.3  2-Dimensional sketch and properties of the Molikpaq's steel caisson  245  7.4  3-Dimensional structural model of the Molikpaq's steel caisson and properties used in the analysis  246  7.5(a)  3-Dimensional F.E. mesh used in the analysis  248  7.5(b)  Cross-section of the 3-Dimensional F.E. mesh along the core center line  248  7.6  Soil types and properties used in the analysis  250  7.7  2-Dimensional finite element mesh and structural model used in the construction analysis 254 (a) Displacements of the structure after construction .. 255  7.8  (b,c,d) Distributions of stress with depth, computed from 3-D and 2-D finite element analysis of the construction phase: (a) o z versus depth; (b) o x versus depth; and (c) t z x versus depth 7.9  Possible ice pressure distributions during the ice load event of March 25, 1986  255-A 257  7.10 7.11  7.12  Location of the instruments used to monitor the ice load event of 25 March, 1986  258  3-Dimensional caisson deformations due to the ice load event of 25 March, 1986. Comparisons between field observations and predictions  259  Deformation profile in the core and berm due to the ice load event of 25 March, 1986. Comparisons between field observations and predictions  261  7.13  Liquefaction resistance curves used in the analyses .... 264  7.14  3-Dimensional liquefaction assessment for (a) (b) (c) (d)  layer layer layer layer  no. no. no. no.  6 5 4 3  265 265 265 265  7.15  Plots of the observed settlement and the computed liquefiable soil  267  7.16  Location of piezometers at the Amauligak 1-65 site  268  7.17  Excess porewater pressure values versus time computed at the locations of pieszometers El, E2 and E3  271  Comparison between the excess porewater pressure values versus time, measured and computed at the location of piezometer El  272  7.18  7.19  Location of accelerometers at the Amauligak 1-65 site .. 275  7.20  Comparison between the acceleration values versus time, measured and computed at the location of accelerometer no. 841  276  Comparison between the acceleration values versus time, measured and computed at the location of tiltmeter no. 706  277  7.22  Volumetric strains induced by cyclic stresses and liquefaction (after Tokimatsu and Seed, 1987)  280  7.23  Relationship between (N 1 ) 60 and depth  7.21  281  7.24  Comparison between the settlement, measurements and predictions of the top of the core surface  283  7.25  Location of the inclinometer used to measure the residual horizontal displacements  284  7.26  Comparison between the residual horizontal displacements, measured and computed at the location of the west side inclinometer 285  NOMENCLATURE A  = area  Acc  = accelerometer  A(t)  = peak acceleration at time t  a  = direction cosines of the normal stress, Og^p, in relation to principal direction i, (i = 1,2,3)  a  =  (o1-o3)/(o1+o3) = sin(<J>13)  bj  =  (bi)c^ = direction cosines of the shear stress tg^p in relation to principal direction i, (i = 1,2,3)  = direction cosines of the increment of plastic shear strain, Ar| MP , in relation to direction i, (i = 1,2,3) b  = b - value = (o2-o3)/(ax-o3)  B  = bulk modulus  B  e  = B u r = elastic or unload/reload bulk modulus  B  t  = tangent bulk modulus  C  = cohesion  Ca  = cohesion (interface)  tCe]  = elastic constitutive matrix  [CeP]  = elasto-plastic constitutive matrix  [CP]  = plastic constitutive matrix  CMP  = compounded mobilized planes  CPT  = cone penetration test  C  = uniformity coefficient  u  d  = displacement  D.  = relative density  D10  = effective grain size  D50  = medium grain size  E  = Young's modulus  E  =  max  m  &ximum Young's modulus  Ejj  = normal modulus (interface)  e  = void ratio  ec  = consolidated void ratio (in situ)  (e c ) av  = average consolidated void ratio (in situ)  e0  = consolidated void ratio (laboratory)  ess  = void ratio at steady state  ecc 1  = void ratio at steady state (defined at 1 kPa)  {f)^g  = nodal load vector for load shedding  f0  = frequency  FS  = factor of safety  G  = shear modulus  Gqjj  = maximum shear modulus obtained from cross hole seismic tests  Gj-jjj  = maximum shear modulus obtained from down hole seismic tests  Gjjjj  = maximum shear modulus in the horizontal plane  G^  = initial shear modulus (interface element)  G0  = the initial slope of the hyperbolic relationship between and rzx  Gp  = plastic shear modulus parameter  Gpt  = tangent plastic shear modulus parameter  Gp^  = initial slope of the hyperbolic relationship between (t  G_„„ IuclX G„„„ . IildA » U  SMP /0 SMP )  a n d  ^IMP  = G Q = elastic or maximum shear modulus 6 = maximum in situ shear modulus  Grc  = maximum shear modulus from resonant column tests  Gs  = secant shear modulus (also used as specific gravity)  Gt  = tangent shear modulus  Gyn  = maximum shear modulus in the vertical plane  G*  = equivalent elastic shear modulus evaluated from unload/reload pressuremeter tests data  H  = height = principal stress invariants (i = 1,2,3)  I„,I, x. y r  = moments of inertia  k  = earth pressure coefficient  k0  = earth pressure coefficient 'at-rest'  [K]  = stiffness matrix  K  = modulus number  KB (or K b ) = bulk modulus number KB e  = KB u r = elastic or unload/reload bulk modulus number  Kc  = a[/o\  KE (or K e )  = Young's modulus number  KEmax  = maximum Young's modulus number  KG  = shear modulus number  KGe  = K G m a x = elastic or maximum shear modulus number  (KG m a x ) a v  = average maximum shear modulus number  KGp  = plastic shear modulus parameter number  Kj  = stiffness number (interface element)  Kjj  = normal modulus number  Kg£  = initial tangent stiffness (interface element)  Kgt  = tangent stiffness (interface element)  UR  = unload/reload modulus number  ^a ^max  = parameter to define G m a x (Chapter 6)  L  = length  K  = direction cosine of x-axis in relation to principal direction i(i = 1,2,3) mi  = direction cosine of y-axis in relation to principal direction i(i = 1,2,3)  m  = bulk shear modulus exponent  MP  = mobilized plane (2-D)  MSL  = mobilized stress level  (MSL) max ni  = maximum mobilized stress level = direction cosine of z-axis in relation to principal direction i(i = 1,2,3) shear and Young's modulus exponent; also normal direction to interface.  n np  = plastic shear modulus parameter exponent  N  = standard penetration resistance; also number of cycles; also yield stress ratio (a1/a3)  W  6 0  = standard penetration resistance (hammer energy = 60%)  OU.o  = normalized standard penetration resistance  Nfi  = number of cycles to liquefaction  P  = earth pressure = initial earth pressure 'at-rest'  P  a  = atmospheric pressure  Qc ( o r Cqc>i  = cone penetration test end bearing  R (or) r  = radial distance  r  = interface shear direction:  = normalized (or modified) cone end bearing  ^face  =  current pressuremeter radius  R0  = initial pressuremeter radius  Rp  = plastic radius  Rp  = failure ratio  s  =  SBP  = self-boring pressuremeter  SMP  = Spatial Mobilized Plane  SL  = stress level  SRL  = stress ratio level  su  = residual shear strength  t  =  U  = displacement; also static pore pressure  Ug  = generated porewater pressure  U_  = displacement at the boundary between plastic and elastic zones  Ur  = relative displacement  X(t)  = the amplitude of the static displacement at time t correspondent to one-half cycle of load/unload  a  = angle between the o 3 direction and the horizontal direction  a^  = anisotropic factor  dp  = disturbancy factor  aPD  = G*/G ' max  P  = angle between the  y  = e x -e 3 = maximum shear strain  fjjp  = shear strain on the Mobilized Plane (2-D)  (oj+o,)/2; also interface shear direction  (O1-O3)/2; also interface thickness  direction and the horizontal direction  = plastic shear strain at which e^p is a minimum  foct  = octahedral shear strain  fSMP  = shear strain on the Spatial Mobilized Plane (3-D)  r  = unit weight of soil  r'  = unit weight of submerged soil  ra  = unit weight of dry soil  Tsat  = unit weight of saturated soil  rw  = unit weight of water interface friction angle  6  A  = increment  Ab1 A(i ,x)  =  Mi.y)  = increment of the angle of rotation of the principal axes, i, (i = 1,2,3) with y-axis  A(i,z)  = increment of the angle of rotation of the principal axes, i, (i = 1,2,3) with z-axis  AU  (bi)  - (bi) AeP  = increment of the angle of rotation of the principal axes, i, (i = 1,2,3) with x-axis  cyclic developed porewater pressure  cy  = increment of plastic shear strain on the Mobilized Plane (2-D) ^Imp  = increment of plastic shear strain on the Spatial Mobilized Plane  Aegp  = increment of plastic normal strain on the Mobilized Plane (2-D)  Ae|MP  = increment of plastic normal strain on the Spatial Mobilized Plane (3-D)  {A o}LS  =  AO,  = cyclic variation in total mean normal stress  AT  = over-shear stresses to be shed  racy LS  load shedding stress vector  A(T  S M P  /OSMP)  Aip  = decrease in failure stress ratio for a 10 fold increase in (o S M p ) F decrease in friction angle for a 10 fold increase in o, ra strain  e  principal strain in direction i, (i = 1,2,3) elastic principal strain in direction i, (i = 1,2,3) plastic principal strain in direction i, (i = 1,2,3)  le)  strain vector  (e e)  elastic strain vector  (ep)  plastic strain vector  e  MP  normal strain on the Mobilized Plane (2-D)  SMP  normal strain on the Spatial Mobilized Plane (3-D)  e e  r  Cer)face  radial strain radial strain at the face of the pressuremeter circumferential strain  (e 5  e face  circumferential strain at the face of the pressuremeter volumetric strain  40 0  volumetric strain associated with 400 cycles of load angle defined by the vertical direction and the direction of the major principal strain increment, Aej  I  slope of Matsuoka's flow rule, and 1st hardening rule ss  slope of steady state line intercept of Matsuoka's flow rule intercept of Matsuoka's 1st hardening rule poisson's ratio; also used as dilation angle  max  maximum dilation angle  (NOTE: All o stresses below are considered to be effective) = stress  a  = principal stress in direction i, (i = 1,2,3) {o'}  = stress vector = normal stress in direction of wave propagation =  o' ar  (o;  + op/2  = boundary radial chamber stress = confining stress  c  = o,-o,  °d o, m o'm^  = mean normal stress  °MP  = normal stress on the Mobilized Plane (2-D)  = initial mean normal stress  = normal stress to interface  N o;  = initial pressuremeter radial stress  a  = normal stress in direction of particle vibration  al  = radial stress face  RP SMP (OSMP^ xo a  = radial stress at the face of the pressuremeter = radial stress at the outer radius of the plastic zone = normal stress on the spatial mobilized plane (3-D) = normal stress on the spatial mobilized plane at failure = initial cartesian stress in the x-direction (simple shear test) = vertical stress  = in situ vertical stress o, vo (o./Oj) = failure stress ratio max = maximum shear stress = (o1-o3)/2 •(Tav/o^0) or (xcy/o^0) or (Tea/o^0) = cyclic stress ratio causing eq' "vo liquefaction  Tf  = failure shear stress  x^  = shear stress prior to unloading  Tjjj  = mobilized shear stress  T^jp  = shear stress on the Mobilized Plane (2-D)  (xMp/oMp)  = stress ratio on the Mobilized Plane (2-D)  ^TMp/°MP^ult  =  (T m p /O m p ) f  asymptotic value of the stress ratio on the Mobilized Plane (2-D)  = failure stress ratio on the Mobilized Plane (2--D)  N  shear stress to cause liquefaction in N cycles  T  oct  octahedral shear stress  t  SMP  shear stress on the Spatial Mobilized Plane (3--D)  t  ( t s m p / O s m p ) =stress ratio on the Spatial Mobilized Plane (3--D) ^TSMP^°SMP^ult  =  asymptotic value of the stress ratio on the Spatial Mobilized Plane  ^TSMP/°SMP^F  =  failure stress ration on the Spatial Mobilized Plane (3-D)  =  ^SMP^SMP^ i  failure stress ratio on the Spatial Mobilized Plane at (o S M p ) F = 1 atmosphere  Tuit  = asymptotic shear stress  t15  = shear stress to cause liquefaction in 15 cycles  <p  = angle of internal friction  <t>  cv  = angle of internal friction at constant volume  <J>F  = failure friction angle  <f>|s  = failure friction angle (plane strain)  <J>p  = failure friction angle (triaxial conditions) failure friction angle (for o 2 ^o 3 and o J ^o 1 )  1 3  = 13  failure friction angle (defined by principal stresses o. and 1 „ \  <f>F ***  = failure friction angle (defined by principal stresses o 2 and °3> V  <|>F  = failure friction angle (defined by principal stresses o x and  12  o2)  <t>m  = mobilized friction angle  A  = mobilized friction angle (defined by principal stresses o x and o 3 )  m 1- 3  A  33  = mobilized friction angle (defined by principal stresses a 2 and o 3 )  <pm« '1 2  = mobilized friction angle (defined by principal stresses and o 2 )  <()p  = peak friction angle  <(>£  = residual friction angle  <f>1  = peak friction angle at the effective mean normal stress of 1 atmosphere  X  = angle defined by the vertical direction and the direction of the major principal stress increment, Aox  \|)  = angle defined by the vertical direction and the direction of the major principal stress, o x  \|)  = state parameter (chapter 6)  ui(t)  = angular frequency at time t  ACKNOWLEDGEMENTS  I am extremely thankful to my supervisor, Professor P.M. Byrne, for his guidance, ideas, encouragement and enthusiastic interest throughout my research. Despite an overflown timetable Professor Byrne has always been 100% available to clarify my ideas and strengthen my arguments. I would also like to express my grateful appreciation to Professors Y.P. Vaid, W.D.L. Finn, D.L. Anderson and K.W. Savigny for their critical comments and for being flexible in their schedule in meeting my deadlines. The valuable suggestions received from Professor Vaid are much appreciated. In addition, I extend my thanks to: Professor P.K. Robertson and J.A. Howie for many stimulating discussions on the pressuremeter tests; as well as Mr. M.G. Jefferies for the valuable discussions and the data on the Molikpaq study; Dr. M.K. Lee for his assistance in implementing the hyperbolic stress-strain law into 3-D F.E. formulation; my friends and colleagues Dawit Negussey, Mustapha Zergoun, Afzal Sulleman, Upul Atukorala, Li Yan, Blair Gohl, (Yoge) M. Yogendrakumar, Alberto Sayao and (Wije) W. Wijewickreme, whose cheerfulness and comments have carried me through moments of factor of safety = 1.0; Mrs. Kelly Lamb for typing my thesis with such ability and speed and for being flexible in her time table to meet my deadlines. I am grateful for the financial support provided by the University of British Columbia, the National Science and Engineering Research Council, the Graduate Research Engineering and Technology (G.R.E.A.T.) award sponsored by the Science Council of British Columbia and Golder Associates (Vancouver, British Columbia). Finally I would like to thank my daughter Marta, my parents Luciano and Etelvina and brother Ze' whose encouragement, faith and support made this thesis a reality.  CHAPTER 1 INTRODUCTION  1.1 Purpose This thesis is concerned with the development and evaluation of analysis procedures for caisson-retained island type structures deployed in the Beaufort Sea for oil exploration and recovery. Offshore exploratory oil drilling has been carried out through the years from either piled platforms or gravity platforms, the type being dependent essentially on the depth of water, the characteristics of the foundation soil and the loading, wave or ice.  In the Beaufort Sea one of  the governing factors for the type of platform to be used is the presence of ice, which covers the sea for at least 3/4 of the year. Loads developed on stationary structures by movements of the ice can be very high and have a dynamic as well as static component.  Because of the high ice loads,  gravity platforms comprised of artificial sand islands have been used in shallow waters.  In deeper waters, however, such an approach is not viable  due to the very large volume of fill required.  To overcome this problem,  two of the major oil companies (Esso Resources Canada and Gulf Canada Resources Inc.) have investigated the concept of an artificial caissonretained island.  The first monolithic caisson-retained island deployed in  the Beaufort Sea was a mobile arctic caisson called Molikpaq, shown in Fig. 1.1, and owned by Gulf Canada Resources Inc.  The Molikpaq has been  described in several publications including: Bruce and Harrington, 1982; McCreath et al., 1982; Fitzpatrick and Stenning, 1983; Jefferies et al., 1985; Stewart and Brakel, 1986; Jefferies and Wright, 1988; and Jefferies et al., 1988.  The platform consists of a steel caisson containing ballast water tanks, with a simply supported steel deck. In plan the caisson is almost a square as shown in Fig. 1.1 with an outside dimension of 111 metres. The central void, which is approximately 72 by 72 metres in plan is filled with sand to provide sufficient mass to resist the large horizontal ice loads. The overall height of the structure is 33.5 metres and the height of the sand fill core is approximately 21.0 metres.  As shown in Fig. 1.2 the  steel caisson with its sand core rests on a submerged sand berm, which in turn rests on a prepared area of the sea floor. The thickness of the sand  Figure 1.1 Beaudril Mobile Artie Caisson "Molikpaq" (after Jefferies et al., 1988)  110m  ORIGINA SEA 6EC w  SOFT SURFICIAL SEDIMENTS  FIRM FOUNDATION  Figure 1.2 Schematic Cross-Section of Molikpaq and Berm (after Jefferies et al., 1988) • >  berm at a particular site depends on the water depth and on the strength characteristics of the sea floor deposits.  Soft sea floor sediments are  removed and replaced by a sand fill subcut prior to constructing the berm. The caisson is then placed on the berm and filled with sand to provide a stable drilling platform.  After completion of exploratory drilling at a  site the core sand fill is removed, the ballast water pumped out, making the caisson reusable for another site. The Molikpaq was first deployed in October 1984 at Tarsiut P-45, see Fig. 1.3 for location, where the ice-loads mobilized were considered to be quite modest (Jefferies and Wright, 1988). In September 1985, the Molikpaq was moved to Amauligak 1-65. At this site during the Winter 1985/86 the ice loads, which were dynamic in nature, were quite large and caused part of the sand core fill to liquefy and the platform to come close to limiting stability (Jefferies and Wright, 1988; and Jefferies et al., 1988). The behaviour of these soil-structure systems is highly complex depending upon the characteristics of the soil and structural elements as  A  Figure 1.3 Molikpaq Site Locations (after Jefferies et al., 1988)  well as the construction sequence and the type of loading.  Important  aspects of their behaviour can be obtained from field observation as well as laboratory centrifuge tests.  In addition, much can be learned from  analytical modelling of these structures, particularly when such models can be calibrated with known field behaviour.  This thesis is concerned with  the development of such a model and in comparing its predictions with field measurements.  1.2 Scope A sophisticated modelling of Molikpaq type structures require a three dimensional  (3-D) computer program for non-linear analysis of soil-  structure interaction problems. Besides structural elements, the following is required: • Soil elements with an appropriate 3-D constitutive law for soil.  • Assessment of stress-strain parameters of the soils from laboratory and field tests. • Interface elements. • Analysis procedure to assess the static response of the Molikpaq during the fill construction phase and moderate ice loading phases. • Analysis procedure to assess the dynamic response of the Molikpaq during high ice loading phase. The above topics are briefly discussed next. a)  3-D Constitutive Law for Sands Due to the 3-D nature of the problem, not only in geometric terms but  also in the terms of the loading, the stress paths mobilized in the core and berm sand fills during the construction phase and the ice loading phases, a 3-D constitutive law for sands that can model its shear, dilation and principal stress axis rotation characteristics is required.  The  three parameter dilatant elastic plastic stress-strain model for sands developed by Byrne and Eldridge (1982) was used in preliminary analysis after extending it to 3-dimensions.  It was soon found, however, that  although this formulation is able to model adequately the shear and dilation characteristics of the sand when subject to the triaxial stress path, it could not model the shear, dilation and deformation under principal stress rotation characteristics of the sand when subject to the simple  shear  stress path, a path which  is considered  to be more  representative of the stress change induced by the horizontal ice load movement on the Molikpaq structure. To satisfy the above requirements, a 3-D model for sands following the concept of the Spatial Mobilized Plane (SMP) (Matsuoka, 1974,1983) was developed and implemented into Finite Element (F.E.) form.  The performance of this model is evaluated by comparison with laboratory measurements obtained from triaxial tests, true triaxial tests, simple shear tests and pressuremeter chamber tests, and also by comparison with in-situ measurements obtained from field tests.  b)  Evaluation of Stress-Strain Parameters of Soil from Laboratory and In Situ Testing Extensive in situ testing is carried out in the core, berm and founda-  tion of the Molikpaq, any time this structure is deployed at a new site in order to assess the quality and strength characteristics of the foundation soil and of the sand fills used (Jefferies et al., 1985).  The in situ  testing consists mainly of cone penetration tests (CPT), self-boring pressuremeter tests (SBP) and shear wave measurements by downhole and crosshole methods.  In addition, laboratory testing is also carried out on  samples obtained from those fills to complement the in situ testing. A special effort was made in this thesis for the development of procedures to evaluate soil parameters for use in the two analytical models described earlier.  It will be shown that soil parameters can be obtained  from the following three sources: i)  Laboratory tests  ii) Pressuremeter tests iii) Laboratory and cone penetration tests. Particular attention in this thesis is paid to the pressuremeter test. This test can yield useful information about the in situ stress-strain behaviour of soil during loading and unloading.  However, because the  stress field induced by the SBP is not homogeneous, a rational analysis and  interpretation of the SBP test data requires that it be analyzed using selected stress-strain relations.  In addition, it is important that such  analysis and interpretation be checked against experimental data under controlled conditions before application to in situ field conditions.  A  review of the existing methods to infer soil parameters from the unloading, and the first time loading pressuremeter test data indicate that: • A detailed analysis method, that considers both the stress and void ratio changes induced by pressuremeter loading and the nonlinear stress-strain response upon unloading, to infer the maximum in situ shear modulus, G from the unload pressuremeter shear modulus, G*, had not been max, o developed. Herein such a procedure is developed and checked against both laboratory and field data. • The evaluation of soil parameters from the first time loading part of the pressuremeter tests in sand has been restricted for many years to the evaluation of the peak friction angle, Only recently, Manassero  and the dilation angle, v.  (1989) proposed a method that allows the  complete plane strain nonlinear stress and volume change response of sand to be obtained from pressuremeter pressure-expansion data. was  analytically  verified  This method  herein against Finite Element generated  pressuremeter data which was computed using the 3-D SMP model developed in this thesis.  Procedures to evaluate soil parameters for use in this  3-D model from the pressuremeter test data were also developed.  These  procedures used Manassero's method after expanding it to take into account the intermediate principal stress, o 2 . Particular attention was also focussed on the evaluation of soil parameters from the data obtained from both cone penetration and laboratory  tests.  The in situ void ratio, e c > was the key parameters used to link  the laboratory test data with the CPT data.  Soil moduli such as the  Young's modulus, E, the shear modulus, G, and the bulk modulus, B, are highly dependent on the consolidated void ratio, ec»  The in situ void  ratio, e c was evaluated from the in situ state parameter, \j), which was pre-obtained from the CPT cone bearing, q c , following the procedures developed by Been et al. (1986).  Once e c was known, the in situ moduli  were estimated from existing laboratory data.  c)  Interface Elements Interface elements were considered necessary to model the contact  between the Molikpaq steel structure and the sand fill.  A 3-D interface  element following the concept of Desai's 'Thin' element (Desai et al., 1984), was developed and implemented into the finite element formulation. Procedures for the evaluation of soil parameters for this interface element were also developed.  Its performance was evaluated by comparisons with  available closed form solutions.  In addition, both the 'Thin' element and  the SMP model predictions were compared with earth pressure measurements on a 10 m retaining wall field test.  These F.E. studies were considered  necessary to check the procedures followed in the construction analysis of the Molikpaq, since there were no earth pressure measurements during the core construction phase of this structure.  d)  Static Assessment To assess the static response of the Molikpaq upon gravity loading  (construction phase) and moderate ice loading the following procedures were followed in the analysis:  The construction of the berm and core was simulated in the 3-D analysis by placing the above fills in one single layer.  Although the ideal  approach is to "analytically construct" these fills in layers, that procedure was not followed due to the large band width of the system of equations.  However, the stresses so obtained from the 3-D analysis (along  the cross-section oriented in the direction of the ice load) were compared with the stresses obtained from 2-D analysis in which construction in layers was simulated.  It was found that the stresses obtained from both  2-D and 3-D analyses were in reasonable agreement. • Moderate Ice Loading Phase On March 25, 1986 the Molikpaq structure was subject to moderate ice loads of about 110 MN.  Based on data reported by Jefferies and Wright  (1988) , this event was considered to be a static ice load event. Therefore 3-D static analysis simulating the ice load conditions of March 25, 1986 were carried out and the model used in the analysis was calibrated against the reported field behaviour.  e)  Dynamic Assessment On April 12, 1986 the Molikpaq structure was subject to severe dynamic  ice loads. To analyse this event a 3-D finite element dynamic program with an appropriate stress-strain law is required.  To date, however, such a  program does not exist. Adequate 2-D finite element dynamic programs do exist, such as the program RICEL developed by Yogendrakumar and Finn (1987). This program was used in 2-D dynamic and pseudo-dynamic analysis of the Molikpaq's response  to the above ice load event by Finn et al, (1988), who showed that the Molikpaq's system damping was very large and consequently no significant dynamic amplification occurred. Hence the response of the structure can be estimated from pseudo-dynamic or pseudo-static analysis which do not consider inertia forces. The 3-D dynamic assessment was carried out using 3-D pseudo-static analysis, following an approach in which the response of the Molikpaq structure to a number of ice load cycles was inferred from the displacements and stresses computed from a single static loading cycle.  The  proposed procedure is outlined later in Chapter 7. The stress cycles obtained from the pseudo-static analysis were used to compute the potential for liquefaction by comparing these stresses with the liquefaction resistance of the sand fills.  This liquefaction resist-  ance was developed based on the cone penetration resistance, q c , of the fills and on the chart proposed by Seed and DeAlba (1986).  However,  because this chart is valid for earthquakes of magnitude 7.5 or 15 significant load cycles and because the Molikpaq was subject to a substantially larger number of cycles than 15 (Jefferies and Wright, 1988), an extrapolation of this chart for a larger number of cycles was required and this was considered herein, as is described later in Chapter 6.  CHAPTER 2 3-D CONSTITUTIVE MODEL FOR SANDS FOLLOWING THE CONCEPT OF THE SPATIAL MOBILIZED PLANE  2.1 Introduction The Molikpaq caisson retained island represents a good example of a 3-D geotechnical problem.  The 3-D aspects are evident not only from the  3-D geometry of the structure but also from the 3-D aspects of the ice movement which can strike the structure from any horizontal direction. For these reasons soil elements within the core and berm fills can be subjected to many different stress paths.  Constant stress ratio conditions (o1/o3 =  1/K0) are likely to develop during the construction phase in the soil elements located on the centreline of the fills (K0 is the earth pressure coefficient at rest). Simple shear conditions are likely to develop in the sand fills during the ice loading phase.  A rigorous solution of the  problem requires an adequate constitutive law that can model the shear, dilation and principal stress axis rotation characteristics of sand in a 3-D stress space. The hyperbolic model which was developed by Duncan and Chang (1970) and Duncan et al. (1980) was used in preliminary analysis after extending it to 3 dimensions and implementing it in the 3-D F.E. code 'NONSAP' (Bathe et al., 197A).  In this model the stress-strain curves are assumed to be  hyperbolic as first proposed by Kondner (1963) and Kondner and Zelasko (1963) and are characterized by a tangent Young's and tangent bulk moduli that vary with both stress level and relative density. This model was later modified and expanded by Byrne and Eldridge (1982) with an additional dilatant parameter based on Rowe's (1962, 1971) stress-dilatancy theory to account for the dilation characteristics of the  sand material when subject to a triaxial stress path as shown in Fig. 2.1 (a).  However, this model can not predict adequately the shear, dilation  and principal stress axis rotation characteristics of sand when subject to the simple shear stress path. Predictions using the hyperbolic model (with and without dilation parameters) of the simple shear results in Ottawa sand (Vaid, Byrne and Hughes, 1981) are presented in Fig. 2.1(b).  It may be  seen that the dilation effects of the test correspondent to D r = 72.3% were reasonably modelled (with dilation parameters), however, the predicted shear stresses are too low.  This is due to the Mohr-Coulomb failure  criterion used in the hyperbolic model.  This criterion assumes that the  friction angle, <f>, at failure is constant regardless of the stress path to failure, i.e. the influence of the intermediate principal stress, o 2 , is not considered. Since the simple shear stress path is most likely to occur in the Molikpaq sand fill during the ice loading phase it was decided to review the existing  3-D constitutive models  for sand to select the most  t  appropriate one. A brief review is presented below. The yield criterion used in the existing plasticity based constitutive models is considered here to be one of the key issues for the selection. The other important issue is the capability of the constitutive law to model adequately the stress-strain dilatancy behaviour of sand in a 3-dimensional space. These issues are briefly discussed next. Wroth (1984), Matsuoka and Nakai (1985) presented good reviews of the 3-dimensional failure criteria for sands most used in practice.  These  consist of the following: i)  Extended Mohr-Coulomb, defined by <p  = constant 3  ii) Lade (1972), Lade and Duncan (1975) defined by I^Ij = constant iii) Matsuoka-Nakai (1974,1985), defined by IjIj/Ij = constant  Stress Difference , 0 5 - c r , - k g / c m *  Volumetric Stroin, e v - %  o o o o © bi K> o Mf CDt  00  c  NJ  m1 > i — 3 CL P i t( Dli—• H* iea ra-c or1i Mn(0 i — c n H n* O 1 o—'3 Ml1 0 0o r NJ p n>-to cr — • >—> o M cr,—VP r> t ,— V' i O vO —^ to V! H-C OH 3 rt •a P3 (C wt roa-r tB (n C E/T3PL O ( >H 3 Pt > >-1> 1 CO H H Cu* 0 3) (D X aW+ Hr 0>1o h— D> , — v i-t *o H M H ( D H ( MD W rt C CD^ ( 3D SO rt to crt ( ra t> 31 X C (D Ho-(B &rt w 3 a.rt »  ••  zr  Volumetric S t r a i n , € v - % O  —  CO 00  zr n  Q — -» ro  CO  -n & Q 3' ro O X I ro ro CD oj ro  ex  o  —  i ro  i oj T"  i  i cn -4 CU V N O w W O (H § Ui s o OI c*  §  Shear S t r e s s , r - k P a <T> O 1a m o  0* 9  3 a <8 JO OI o to> 0> -el\> ->9<M OI O OI I o « -4-- oo § s § 3s--"| *  -siZA?  II  n cn 3 ?  I  where I l f I 2 and I 3 are the known three principal stress invariants. A sketch showing these criteria projected on the octahedral plane is presented in Fig. 2.2.  It may be seen that both Lade and Matsuoka-Nakai  failure surfaces coincide with the Mohr-Coulomb criterion for triaxial compression tests.  The Matsuoka-Nakai and Mohr-Coulomb criteria also  coincide for triaxial extension tests, whereas the Lade curve does not. Nevertheless Lade's and Matsuoka-Nakai failure criteria are very similar. Of the three criteria, Matsuoka-Nakai' s was chosen here for two reasons:  (a) it was initially developed from theory and not from curve  fitting of experimental data (Wroth, 1984); and (b) it appears to predict experimental data best (based on the proceedings of the Cleveland workshop on constitutive equations for granular non-cohesive soils (Saada and Bianchani, 1987)). Based on the above discussion and because Matsuoka's 3-D flow rule following the concept of the Spatial Mobilized Plane (SMP) considers the stress-strain dilatancy behaviour of sand in a 3-Dimensional stress space (Matsuoka, 1983) his SMP model was selected here with some modifications to make it more practical and, take into account the rotation of the principal stress axis of sand when subject to the simple shear stress path. A detailed description of this model and its implementation to 3-D, 2-D and axisymmetric F.E. codes is presented in this chapter.  Before describing  this model, however, it was felt that a brief description of Matsuoka's 2-D constitutive model which is based upon the Mobilized Plane (MP) concept should be presented first because its development served as a basis for the more complex 3-D SMP model.  Figure 2.2 Mohr-Coulomb, Lade and Matsuoka-Nakai Failure Criteria  2.2 2-D Constitutive Model for Sand Following the Concept of the Mobilized Plane The concept of a single mobilized plane for 2-D constitutive models was first developed by Murayama (1964).  The term "Mobilized Plane" (MP)  refers to the plane where the mobilized stress ratio between the shear stress and normal stress on the plane,  x  M p/° M p»  is  a  maximum. The 2-D  representation of this plane is shown in Fig. 2.3(a). The plane forms an angle of 45° + (<J>m/2) with the major principal stress plane, where <p^ is the maximum angle of friction that is mobilized.  This <f>m angle can be  easily obtained by constructing the Mohr's circle for the current principal stresses o x and a 3 as shown in Fig. 2.3(b). Based on the above "Mobilized Plane" concept Matsuoka developed 2-D and 3-D constitutive models.  A brief description of the 2-D constitutive  model is presented below. 2.2.1 Brief Description of the 2-D Constitutive Model The 2-D model for sand following the Mobilized Plane concept is an elasto-plastic model.  As in any model of this type its constitutive  matrix, {C6^} , which relates the increments of strain {Ae} with the stresses {o} , is elasto-plastic and is composed of two components, an elastic component {Ce } and a plastic component {Cpr} related by the following equation:  {C ep } = {Ce) + {CP}  (2.1)  The elastic component is defined by Hooke's constitutive law and the plastic component is based on a yield criterion, a flow rule, and a hardening rule. What makes this model different from other elasto-plastic  NORMAL STRESS  Figure 2.3  (a) 2-Dimensional Mobilized Plane; (b) Evaluation of <j>m, o M p, and t M p  models existing in the literature is the way these three components are defined.  2.2.1.1 Yield Criterion The yield criterion of the model describes the stress conditions causing elastic or plastic strains and is composed of the yield and failure surfaces described below. A family of yield surfaces in the ( x M p . schematically shown in Fig. 2.A.  s t r e s s  space is  These yield surfaces are given by the  following equation developed by Matsuoka and Nakai (1974,1985)  t  = MMP' D/°mi. MP  tan  (m< U =  (2.2)  K  The "current" yield surface corresponding to the stress state at a point in a mass of soil, is defined by the maximum stress ratio mobilized at the point during its history of loading. Assuming that at a given time of loading the "current" yield surface is yield surface A (K = K^) (see Fig. 2.4), than inside this yield surface (K £ K A ) only reversible (elastic) strains occur.  This corresponds to an unloading condition.  Outside the yield surface A (Kg > K^) both reversible and irreversible (plastic) strains occur and when this happens the yield surface moves from line A to line B.  This corresponds to a loading condition.  The limit or  bound of the yield surfaces is called failure surface and no further loading is possible outside this surface.  The failure surface is defined  by the following equation:  T  (2.3)  FAILURE  <£ f  YIELD  SURFACE  K = K F =tan(</> F )  SURFACE  K=K c=tan((<£ ° YIELD C  mo)c  SURFACE  K = KB=tan((<£ YIELD  )  mo  )B)  SURFACE  K =KA=tan((<£  mo)A  )  NORMAL STRESS  Figure 2.4 Matsuoka-Nakai 2-D Failure Criterion  where: = the failure friction angle of the sand r (rMr 1.T,/owr.)_ = the failure stress ratio MP F 2.2.1.2 Flow Rule The flow rule of the model relates the plastic strain increment ratio, defined on the current mobilized plane (or yield surface) with the stress state at the point. Murayama and Matsuoka (1973), Matsuoka (197A) developed the following relationship between the shear-normal stress ratio  o n  Mobilized Plane and the increment ratio (Ae^p/Af^p)  (2.A)  where A,ji are soil parameters evaluated as is shown in Fig. 2.5(a) (pg. 23) and the stresses  and o„„, and the increments of strain Ar u n and AeUT, MP MP Mr MP  are evaluated as is described later in section 2.2.2 (pg. 29). This relationship was verified by data derived from triaxial tests (compression and extension), under constant mean principal stress, o^, and a plane strain test under constant o 3 , on Toyoura sand as shown in pg. 23, Fig. 2.5(a). It may be seen that the data points plot in a straight line despite the initial void ratio, e^, which ranged from ,6A to .89 and the mean normal stress, o , which ranged from 100 kPa to 396 kPa. m It should be noted that the above increments of strain, AeWT, and Ar„_ MP MP were treated initially as increments of total strain, however in later  publications Matsuoka (1987) refers to these quantities as increments of plastic strain and therefore, these are designated from now on as Ae^p and a  4 -  From eq. (2.A), Ae^ can be obtained as follows: Mr  (2.5)  To develop an incremental stress strain law it is necessary to relate the term  with the increment of the stress ratio A^p/o^p) on the current  Mobilized Plane (or yield surface), and that is done by combining the above flow rule with the hardening rule described below.  2.2.1.3 Hardening Rule Two different types of hardening rules were developed by Matsuoka which resulted in two different 2-D stress strain laws. - 1st Hardening Rule Matsuoka  (197A) developed the following relationship between the  stress ratio (TMp/°Mp)  an<  *  normal-shear plastic strain ratio  on the Mobilized Plane: MP = X (- 'MP) + u' MP 'MP  3  where: X  = same soil parameter used in eq. (2.A)  li'  = soil parameter  (2.6)  This relationship was also verified by data derived from triaxial tests (compression and extension), under constant mean principal stress, om,  on Toyoura sand as shown in Fig. 2.5(b). The initial void ratios, e^,  of these tests varied from .68 to .89 and the mean normal stress, a , from m  100 kPa to 398 kPa. Again the fit with the laboratory data is seen to be good. Combining the flow rule given by eq. (2.A) with the hardening rule given by eq. (2.6) the following equation was obtained by Matsuoka (1983) T T MP , , MP . — = Cm* - yHn — + \i °MP ^  (2.7)  where: = plastic shear strain at which  is a minimum  Differentiating eq. (2.7) with respect to the stress ratio (Tjjp/0^p) the following incremental stress-strain law is obtained:  ( w v - ^  a 4 = fa e —x p^ (  >  Ar  , , MP MP  (2.8)  (T  MP/OMP)_U Designating G^ = y^/(n-ji') exp ( ^, ), the above equation can be rewritten as follows:  (2.9)  I .0 0.9 0.8 0.7 0 . 6  - -  0.5 -0.4 -0.3 -0.2  LEGEND  - -  0. I -0 -0.3-0.2-0.1 "AeMP / A X m p  TEST TYPE O TC X TE A PS + TC • TE • TC <7 T E  e  0 . 89 0. 6 4 0. 66 0. 68 0. 6 8 0. 6 8 0. 68  (kPa) 100 300 N.A.  196 196 396 396  WHERE TC = T r i o x i o l Compretsion TE = Triaxial Extension PS = Plane Strain  (e^ = 200lrRi)  -0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 "^mpZ/mp  Figure 2.5  (a) (b)  Relationship Toyoura Sand Relationship Toyoura Sand  Between (t m p /o m p ) and -(AeMp/ArMP). (after Matsuoka, 1974,1983) Between (t M P /o M p ) and -eMP/TMp(after Matsuoka, 1983)  where G^ is designated here as the plastic shear modulus parameter which relates the increment of plastic shear strain with the stress ratio increment on the current mobilized plane (or yield surface). Equation (2.9) represents Matsuoka's first incremental stress strain law. - 2nd Hardening Rule Matsuoka (1987) developed a model for soil where the rotation of principal stresses is taken into account.  The flow rule of the model is  the same as that described by eq. (2.4). However, the hardening rule was developed based on the assumption that the relationship between the ratio of the shear stress and vertical stress, T /O , with the shear strain, xz z y , is hyperbolic and given by the following equation: X2  v  1  ^ x z ^ F  ^ m / V  xz " Go (xxz/a z F - (xxz/Oz)  f  .  ,  m  where: Go = the initial slope of the relation between (txz/oz) and rxz. (T /O )„ = the failure stress ratio xz z F After some manipulations, Matsuoka shows that eq. (2.10) can be rewritten as: , sm(4J sm(<f> ) sm(2a) 1 F m f'xy = n —/T~\ • , , s— Go sin(L) J F - sm(d> m)  ,\ (2.10a)  the failure friction angle *m  the mobilized friction angle  a  the angle between the o 3 direction and the o x direction.  To account for the "shear" on the Mobilized Plane and the "rotation" of the principal stress axis, eq. (2.10a) is differentiated with respect to <f>m and a, respectively. The details are given by Matsuoka (1987). When a = 45°, eq. (2.10a) becomes , sin(<j> ) sin(<J> ) T = 7T- • r-rrV ' G q sin(<J>F)-sm(<J>m) where: y  = the maximum shear strain =  (2.10b)  e1-e3.  Based on the above, Matsuoka (1987) concludes that G can be obtained o from either the simple shear test (eq. 2.10a) or the triaxial test (eq. 2.10b).  This approach, however, was not followed here as is explained  below. Matsuoka (1983) has also shown that the relationship between the stress ratio  anc  * t*16 plastic shear strain  on the Mobilized  plane is hyperbolic as is shown in Fig. 2.6, which is based on data obtained from triaxial tests (compression and extension) on Toyoura sand. Because this relationship is considered to be more fundamental and easier to implement in a 3-D space than the relationships given by eqs. (2.10), it was decided to develop the plastic hardening of the model as is described below.  .0 C T m = 3 9 2 kt •  comp.  •  ext.  2  i - D —  a 2  o. 0 . 5 2  3  0  0  .0  2.0  3.0  Figure 2.6 Hyperbolic Relationship Between ("tMp/Otfp) Toyoura Sand (after Matsuoka, 1983).  4.0  T^pt o CT>  Following similar procedures as outlined by Kondner (1963) and Kondner and Zelasko (1963) the following equation representative of the hyperbolic relationship on the Mobilized Plane is obtained:  rP 'MP —  x MP — = °MP G  1 pi  P rMP ^MP^MP^lt  (2.11)  where: G . = the initial slope of the stress ratio - strain curve pi and (twt,/owt,) = the Jasymptotic value of the stress ratio which is r MP MP ult related closely to the failure stress ratio (T„_,/OV._.) T,. J MP MP F The plastic shear strain increment, Afj^p is obtained from eq. (2.11) and is given by:  (2.11a)  This equation represents the hardening rule of the proposed model and is the substitute for Matsuoka's 1st and 2nd hardening rules which were given earlier by eqs. (2.9) and (2.10), respectively. Since the proposed hardening relationship is assumed to be hyperbolic the procedures developed by Duncan et al. (1980) to evaluate the tangent Young's modulus, E , for the hyperbolic model were followed here and by  analogy give the tangent shear plastic parameter G . This parameter is pt considered to be dependent on both the normal stress, o^,, and stress ratio level, SRL, and given by the following equation:  (2.12)  where: G. Pi " K G p ( ° M P / P a ) n P KG = the plastic shear number np  = the plastic shear exponent  Pa  = the atmospheric pressure = a parameter that relates the asymptotic value of the stress ratio, (^p/^p^ult' w i t h  the  stress ratio,  ^mp^mp^F'  by the equation:  (T  MP / o MP } F  =  VW'Wult  (2.13)  and the stress ratio level, SRL, is given by the equation:  SRL = ( x M p/° M p )/(T M p /o M p ) F or  (2.14)  i.e., SRL relates the "current" yield surface defined by tan(^m) with the failure surface defined by tan(<£p). At failure SRL = 1.  2.2.2 Brief Development of the 2-D model in the Cartesian System of Coordinates A brief development of the 2-D stress-strain , constitutive law is presented here since its development will give an insight into the more complex development of the 3-D stress-strain constitutive law which will be described later. The development of the 2-D model consists briefly of the following: Relationships between the shear stress,  normal stress, o^p, on  the Mobilized Plane and^the principal stresses  and o 3 can be obtained  from the Mohr circle plot shown in Fig. 2.7(a).  From this figure the  following relationships are obtained:  T  MP  =  (0  i" 0 3> / 2  cos  <t>m (2.15)  °MP =  ( o  i  +  °3  ) / 2  "  (°i"°3)/2  s i n  *  m  Assuming that the direction of the increments of plastic principal strain coincide with the direction of the principal stresses Matsuoka (1983) obtains the increments of plastic strain as is shown in Fig. 2.7(b). From this figure the following relationships are obtained,  Ar  MP  =  (Ae  ? "Ae3}  COS<P  m  (2.16) Aegp = (Aep + Aep)/2 - (AeP - Aep)/2 sin<f>m  Figure 2.7  (a) Mohr Circle of Stresses; (b) Mohr Circle of Increments of Plastic Strain  Substituting the values of sin<f>m (OJ+OJ)  =  (o 1 -o 3 ) / (OJ+OJ ) and cos<|>m  =  2Tla1a3/  in the above equations and solving for AeP and AeP the following  equations are obtained:  AeP = Ae P p + (c^/o,)*" A r P p / 2 (2.17) Aep = Ae p p - (o./o^i" A r P p / 2  From Fig. 2.7(b) the following relationship between increments of plastic strain, {AeP} and increments of plastic principal strain are obtained:  AeP  = (AeP + AeP)/2 - (AeP - AeP)/2 cos2a  AeP z  = (AeP + AeP)/2 + (Aex - Ae3)/2 cos2a  (2.18)  ArP = (AeP - AeP)sin2a 21X  Since A e P p = (vi - T M p / ° M p ) ArjJpA (see eq. (2.5)), and Ar P p = 1/G pt A(x^ p /o^ p ) (see eq. (2.11)), then, manipulating the above equation, relationships between increments of plastic strain, {AeP}, and the increment of stress ratio on the Mobilized Plane can be obtained.  The manipulation  of these equations is not presented here since only a brief insight to the model is intended at this stage.  Nevertheless the increments of plastic  strain would be given by equations of the following form:  A e  x  =  G ~  A e  z  =  g T  a  4  =  G T  pt  ( f  pt pt  x>  A ( T  A ( t  M P  m p  A ( t  / o  / o  m p  M P  m p  / o  )  )  M P  ( 2  -  1 9 )  )  where the terms f , f , f can be obtained as described above. X z zx To completely define the stress-plastic strain relation it is necessary to develop a relationship between the increment of the stress ratio on the Mobilized Plane, A(x^/o™) , and the increments of stress, Ao , Ao and ' MP MP ' x' z AT  .  zx Such development will not be carried out here because it will be carried out later for the 3-D Spatial Mobilized Plane.  Nevertheless that  relationship would be given by an equation such as the following: A(T M I ,/O M 1 ,) = (TMX)Ao MP MP x  + (TMZ)Ao  z  + (TMZX)AT  zx  (2.20)  where TMX, TMZ and TMZX are terms that will be defined later in Section 2.4.3.5. Substituting this equation into eqs. (2.19), the plastic strain-stress relation shown below can be obtained: {AeP} = [CP] {Ao}  (2.21)  The above equation completes the development of the plastic constitutive matrix of the model.  As described earlier the elastic strain-stress  relation is given by:  {Aee} = [Ce] {Ao}  (2.22)  where [C ] is Hooke's constitutive law. Therefore, to obtain the complete strain-stress relation, designated also as elasto-plastic strain-stress relation, eqs. (2.21) and (2.22) are added to give  {Ae} = {Aee + AeP} = [Ce] {Ao} + [CP] {Ao} = [C ep ] {Ao}  (2.23)  The extension of the 2-D Mobilized Plane concept to 3-D is discussed next.  2.3 Discussion on the Theories of the 'Compounded Mobilized Planes' (CMP) and the 'Spatial Mobilized Plane' (SMP) The concept of the 2-D Mobilized Plane was later expanded to 3-D by Matsuoka and Nakai (1974).  From their work, two theories were developed.  The theory of the 'Compounded Mobilized Planes' (CMP) and the theory of the 'Spatial Mobilized Plane' (SMP). These two theories are well described by Matsuoka (1983) and only a brief discussion is presented here. In the CMP theory the 3-D stress-strain constitutive model is developed based on three 2-D mobilized planes which are defined based on the three pairs of principal stresses (o Jt o 2 ), (o2,o3) and (Oj.Oj) as shown in pg 35, Fig. 2.8(a), and, the correspondent mobilized friction angles, <p , m i2 <b and <b are obtained as shown in Fig. 2.8(b). 23  m  !3  To evaluate the deformations at a point representative of a mass of soil, the same 2-D stress-strain constitutive law is used independently for each of the three mobilized planes, and the principal strain e^ in the direction i (i = 1,2,3) is obtained by a linear summation.  In the SMP theory the 3-D stress-strain constitutive law is based upon a single 3-D mobilized plane as shown in Fig. 2.8(c), and the deformations are obtained directly from stress-strain relationships developed for this plane. The first theory, CMP, was favoured by Matsuoka and in 1987 he introduced the hyperbolic hardening rule (described earlier in section 2.2.1.3) in order that the rotation of principal axis is taken into account.  The  constitutive law is 2-D and is expanded to 3-D following the CMP concept described above. The second theory, SMP, was favoured here because this concept is considered to be more fundamental, since the stress-strain relations are obtained directly for a 3-D stress space and the 2-D constitutive relations can be obtained from the 3-D constitutive relations by imposing the necessary boundary conditions as will be described later. A detailed description of the SMP model is presented next.  2.4 3-D Constitutive Model for Sand Following the Concept of the Spatial Mobilized Plane A constitutive model based on one 2-D mobilized plane, defined by the major and minor principal stresses (o: and o3), has been presented in section 2.2. In this section a 3-D model based on the Spatial Mobilized Plane (SMP) (Matsuoka, 1983) is presented. This model is also based on a single plane, however, this time the plane is defined in a 3-D space by the three principal stresses (olt o 2 and o3). The concept of the 2-D model described earlier will be closely followed for the development of the 3-D model and includes several devia-  (a)  Figure 2.8  (a) Three 2-Dimensional Mobilized Planes; (b) Development of Three Mobilized Friction Angles in the T, O Stress Space(c) Spatial Mobilized Plane  tions from Matsuoka's 3-D SMP model.  As will be shown, these deviations  allow adequate modelling of the 3-D shear, dilation and rotation characteristics of sand when subject to several stress paths including simple shear. A detailed description of the 3-D formulation will be presented first, and the 2-D and axisymmetric formulations which are obtained from the 3-D formulation by imposing boundary conditions will be presented after.  2.4.1 Description of the SMP (After Matsuoka, 1983) The state of stress on a soil element can be characterized by its three principal stresses o l t o 2 , and o 3 .  Based on these stresses three  Mohr circles can be constructed as shown earlier in Fig. 2.8(b) and three mobilized friction angles <p m  13  , <b m  111  12  and <b m  1U  obtained. Based on these J3_  mobilized friction angles a 3-D plane is geometrically developed in principal stress space and intercepts the principal stress axes 1, 2, and 3 at the points A, B and C as shown earlier in Fig. 2.8(c). This plane (ABC), which is defined by Matsuoka as "the plane on which the soil particles are most mobilized on average in the 3-D stress space", is designated as Spatial Mobilized Plane (SMP). The SMP is characterized by a normal direction, n, normal stress, o S M p , and shear stress, t S M p. The direction cosines of the normal to the SMP are given by the following equation: °1 °2  °3  1'2  a. = ( — ? ) i o i (o 1 o J + o 2 o 3 + a 3 a l  (2.24)  where: o^  = principal stress in direction i (i = 1,2,3)  a.  = cos(n,i), direction cosine in relation to principal direction i  and the normal stress, ° S M p and the shear stress t S M p on the SMP are given by the following equations:  °SMP =  +  +  (2,25)  l  and x  = ((a x -o 2 ) J a 1 2 a J 2 + (o 2 -o 3 ) 2 a 2 2 a 3 2 + (0,-0^ 2 a 3 2 a i 2 )i' 2  (2.26)  where: a^  (i = 1,2,3) are given by eq. (2.24).  Assuming that the principal plastic strain increments, Ae?, have the same direction as the principal stresses o^, it follows that the increment of the plastic normal strain, Aej^p, and the increment of the plastic shear strain, ^TgMp»  the SMP are given by the following equations: AeP  = AePaj2 + Ae^a, 2  AeP2a32  (2.27)  and 23J Ar P M p /2 = ((AeP-AeP)2a12a22 + (AeP-AeP)2a22a32 + (AeP-AeP)21a^ 3 a i:  (2.28)  Because the above assumption is one of the key assumptions of the model, it will be discussed in detail in section 2.5 of this chapter. 2.4.2 Development of the Plastic Constitutive Matrix [CP] The plastic component of the 3-D constitutive matrix is based on the yield criterion, flow rule and hardening rule described below.  2.4.2.1 Yield Criterion The 3-D failure criterion of the SMP model is the same as the 2-D failure criterion described earlier except that this time the yield and failure surfaces are defined based upon the three principal stresses o a , o 2 and o 3 . The yield and failure surfaces are given by the following equations developed by Matsuoka and Nakai (1974,1985).  • Yield Surfaces t / o Q M p = 2/3 (tan* mo SMP SMP i2  + tan<f> + tan*rao )*" = K mo 23 i3  (2.29)  • Failure Surface = 2/3  ' W W F  (tan  *F12  + tan  *F23  +  W F J  1  "  • KF  (2  '30)  where:  <(> m i2  m  , <p 23  and <j> m !3  are the mobilized friction angles  and <J>„ , <(>„ 12  and  2 3  are the failure friction angles. 13  Earlier attempts made by the writer to predict measurements obtained from simple shear tests on Leighton-Buzzard sand (Stroud, 1971) and from true-triaxial tests on Ottawa sand (Yong and Ko, 1980) (Workshop soil modelling, McGill University) indicate however that the peak failure stress ratio, (xgMp/°sMp)p» failure, (°SMp)p»  an<  dependent on the normal stress on the SMP at * that a better agreement with the laboratory data was  obtained if the failure stress ratio was expressed by the following equation:  (ZSMP, , SMP  SMP  - 4,182, 1 O 8 i , ( ^ £ F ) p SMP a  (2.31)  where T  SMP ( ). = failure stress ratio at (o,,^),, = 1 atmosphere. SMP F °SMP 1 and ^ QWN  A( ) = decrease in failure stress ratio for a 10 fold increase in °SMP ^ s m p V  A sketch of the failure surface, projected on the octahedral plane and in the 3-D stress space is presented in Fig. 2.9(a) and (b) respectively. The 3-D Mohr-Coulomb failure surface is also shown in the figures and it may be seen that the Mohr-Coulomb and Matsuoka-Nakai failure surfaces coincide whenever the triaxial stress path is followed (compression or extension) but differ for any other stress path. To show the influence of the intermediate principal, o 2 , on the failure friction angle <p  13  = sin'  1C(a  relationship was developed between <p that was developed by Bishop (1966).  1 -a 3 )/(a  13  1 -a 3 )),  better known as <j>, a  and b-value which is a parameter  The relationship is given by the  following equation, which was obtained from eq. (2.30) (see Appendix 2.1) .  tan<  * „ 3 , b(l+a)+(l-a) (1-b) P p ^ ~ F 2 (1+a) (b+b2)+ (l-a) (2-3b+b2)  where: b  = (o2-o3)/(o1-o3)  a  = (0^03)7(0^03) = sin <|>F  ,,  M O H R - COULOMB  \N \  MATSUOKA-NAKAI  (a)  Figure 2.9 Matsuoka-Nakai and Mohr-Coulomb Failure Criteria: (a) Projection on the Octahedral Plane; (b) 3-Dimensional Stress Space  K  = constant = 2V2/3 tan <J>„  *  * 1 3  (obtained from eq. (2.30)  tx Designating  * 1 3  by  *  i.e. the failure friction angle corresponding  to triaxial conditions, then: *  tx  <t>„  *  i.e., <f>„  " 13 «  = </)„  13  F  <J>* f 13  e  for  b = 0  or b = 1  for 0 < b < 1  is the failure friction angle, defined by o x and a 3 , for o2?oj  and OJ^OJ.  Values of  *  tx were computed for different values of <f>„ and for 13 *  different values of the b-value using eq. (2.32), and the differences * tx between <f>„ and <(>„ evaluated. The results are presented in Fig. 2.10. M3 * * tx It may be seen that the value (<f>„ -<(>„) is equal to zero when b = 0 or b * 13  *  = 1 (triaxial compression or triaxial extension) and has values > 0, when 0 tx < b < 1. The highest difference occurs when <f>„ = 50° and b is about .25. r This behaviour has been observed by many researchers based on laboratory measurements from tests on sand using true-triaxial, plane strain (triaxial and simple shear), hollow cylinder and other devices. 2.A.2.2 Flow Rule The 2-D flow rule described earlier by eq. (2.A) has been extended to the SMP by Matsuoka and Nakai (197A) and Matsuoka (1983) who show that eq. (2.A) can be re-written as:  T  SMP/oSMP = X ( - A e S M P / A 4 p )  +  ^  (2  '33)  b-VALUE Figure 2.10 Variation of (<j>p  - <J>jjx) with b-Value.  where: X,)i are the same soil parameters earlier defined for the 2-D model. This relationship was verified by data derived from triaxial tests (compression and extension) on Toyoura sand as shown in Fig. 2.11(a)  It  may be seen that the same intercept (^ = .20) and about the same slope (X = 1.12) is obtained from the 3-D flow rule as compared with the 2-D flow rule, (p. = .20 and X = 1.20), shown earlier in Fig. 2.5(a).  2.A.2.3 Hardening Rule The 2-D hyperbolic hardening rule described earlier by eqs. (2.11) is extended here to the SMP by re-writing this equation as follows:  (2.3A)  This relationship is shown to be verified by the triaxial test data (compression and extension) on Toyoura sand, presented in Fig. 2.11(b). To evaluate the tangent plastic shear parameter, G  » eq. 2.12, which  was derived earlier for the 2-D model is used here, i.e. G . = pt 2 Gpi. (1-R„ SRL) , however in the 3-D model, G . , R„ and SRL are defined on F pi F the SMP by the following equations: G  pi  = KG  p  R  F  =  SMP /0 SMP ) F /(T SMP / °SMP ) ult  (T  (  °SMP/Pa)nP  (2  '35)  (2  '36)  -  A 6  SMP  / A  /SMP  1 .0  cr m = 392kN/m 2  a.  S  0.5  )  Figure 2.11  SMP(0/o)  (a) Relationship Between ( x S M p/o S M p) and -(AeSMp/ArSMp). Toyoura sand (after Matsuoka, 1983). (b) Relationship Between ( T S M p / o S M p ) a n d rSMP' T ° y ° u r a S (after Matsuoka, 1983).  a n d  and  SRL = ( T SM P/° SM P)/ (X SMP /A SMP ) F  (2  '37)  2.4.2.4 Summary of the Basic Equations of the SMP Model The basic equations of the SMP model are presented below.  Yield Criterion ^SMP'WF  =  (t  SMP / o SMP^ ~ A(T SMP / °SMP )  1o  (2  «XO ^ S M P ' ^  '38)  Hyperbolic Hardening Rule a  4 p  = r pt : A< W  W  ( 2  '39)  Flow Rule A e P M p = l / X ( H - (x S M p /o S M p )) A r P M p  2.4.3 Development of the SMP's Plastic Cartesian System of Coordinates  (2.40)  Constitutive Matrix in the  A brief development of the plastic constitutive matrix {CP} is presented below. Details will be given after. The following relationships will be developed:  •  Ae  i = fa  (Ae  SMP«  A  4p>  The relationship f is developed in section 2.4.3.1 (pg. 47)  (2  '41a)  Ae  i  = f  The relationship  b ^  W  (2  W  'Alb)  is obtained by substituting eqs. (2.39) and (2.40) into  eq. (2.41a). This is described in section 2.4.3.2 (pg. 51).  {Aep} = f (Ae?) c 1  (2.41c)  where {AeP} = increments of plastic strain. The relationship f is developed in section 2.4.3.3 (pg. 51)  {AeP}  = f  d ^  w  w  (2  *  -Ald)  The relationship f^ is obtained by substituting eq. (2.41b) into eq. (2.41c). This is described in section 2.4.3.4 (pg. 53).  A (  W°SMP)  = f  e  ({Ao})  (2  -Ale)  where {Ao} = increments of stress. The relationship f is developed in section 2.4.3.5 (pg. 53).  • {AeP} = [CP]{Ao}  where the plastic constitutive matrix,  (2.41f)  is obtained by substitut-  ing eq. (2.41e) into eq. (2.41d). This is described in section 2.4.4 (pg. 56).  To develop the above relationships the procedures described by Matsuoka (1983) will be followed here except for a few deviations that will be outlined.  2. A. 3 .1 Relationship Between Increments of Plastic Principal Strain, Ae? and Ae| Mp and A ^ M p Matsuoka (1983) showed that the relationship between Ae? and, AeP SMP and Ay^p is given by the following equation (see Appendix 2.2):  A A ee  P-AeP i " SMP + a. I  2  (2.42)  where: a b. l  = the direction cosines of the normal stress °i~°SMP T  SMP  (see eq. 2.24)  a. = the direction cosines of the shear stress Tg^p (see Appendix 2.2)  (2.43)  To establish the above eq. (2.42) the following two assumptions were considered by Matsuoka: a)  That the direction cosines of Ae^p are the same as the direction cosines a. I  b)  that the direction cosines of Afg^p are the same as the direction cosines b. I  Earlier attempts made by the writer to predict laboratory measurements obtained from simple shear tests (data published by Stroud, 1971, on Leighton-Buzzard sand), and from true-triaxial tests (data used in the  workshop soil modelling competition, McGill University, on Ottawa sand, Yong and Ko, 1980), indicate, however,  that the assumption regarding the  direction cosines b^ was not in agreement with the published laboratory data.  Deviations between the b.values obtained from the stresses, o. , and 1 l  the b^ values obtained from the increments of plastic strain, Ae?, were observed to be stress path dependent and a function of the 'b-value' = (o 2 -a 3 ) / (o 1 -o 3 ).  These deviations of b^ are given by the following  equation: Ab. = (b.) L  X  - (b.) i A P 'i i Ae.  (2.44)  O.  where: (b.)  i  = direction cosines of the shear stress direction  (b.) = direction cosines of the increment of plastic shear 1 Ac? direction  The variation of Ab^ with the 'b-value' are shown in Fig. 2.12. Based on this figure eq. (2.41) is re-written in a different form as follows:  Ae  i =  Ae  SMP  +  <v  A  v  a  4p  (2.45)  Predictions of the simple shear test data reported by Stroud, using Ab.= 0, and Ab. f 0 (obtained from Fig. 2.12) are shown in Fig. 2.13. It  Figure 2.12 Variations of Abi (i = 1,2,3) with 'b-Value*. (a) i=l; (b) i=2; (c) i=3  o a V)  £  60+-  U) 40 o 20 a>  &  -tr  0  0  2  4  6  8  Shear strain , % LEGEND * measurements predictions (Abj * 0) predictions (Ab ; = 0)  0  2  4  6  8  Shear strain , %  Figure 2.13 Predicted and Measured Simple Shear Data, (a) t z x versus r z x ; (b) e v versus r z x 5 ( C ) U O J / O J + O J ) versus f  may be seen that good agreement with the measured data is obtained when A b ^ 0.  When Ab^ = 0 the predictions of the shear stress, and volumetric  strain, versus shear strain are in fair agreement with the measured data, however, the prediction of 2os/(OJ+OJ) versus shear strain is extremely high, and inadequate.  Therefore the use of eq. (2.45) instead of eq.  (2.42) in the SMP formulation is considered to be justified.  2.4.3.2 Relationship Between Increments of Plastic Principal Strain, Ae? and Increment of the Stress Ratio on the SMP, Substituting the values of Af P M p from eq. (2.39) and Aej^p from eq. (2.40) into eq. (2.45) the following equation is obtained: U - (T SMp /o AE?  1  = [(  ;  M P  S M F  A  )  b -Ab  ) +  2a  I)]  i  t G  (2.46)  p  SMP  where i = 1,2,3.  Designating the term in square brackets by M^ then eq. (2.46) can be rewritten as: Ae? = [M.] 1  G  p  A(^) SMP  (2.47)  2.4.3.3 Relationship Between Increments of Plastic Cartesian Strain {AeP} and Increments of Plastic Principal Strain {AeP] Now that a relationship between Ae? and, AeP^p and AfP^p has been established a relationship between the cartesian components of plastic strain {AeP} and Ae? is needed. l Matsuoka (1983) assumes that the direction cosines JL , nu and n^ (i = 1,2,3) which relate cartesian stresses with principal stresses, are the  same direction cosines that relate increments of plastic cartesian strain with increments of plastic principal strain.  Based on the above assump-  tion, the following equations are obtained (see Appendix 2.3):  Aexp = . -I, Ae? i. ii 1=1,3 0  p  Ae =  AeP =  i=l ,3  i=l,3  Ae? l m. l  Ae^ n. i  Ay 22 =  I  i=l,3 p Aryz  2  i=1.3  A 7" zx i-1.3  Ae? fi.m. l ii  Ae? l m.n. ii  (2.48)  Ae? l n.S. ii  The above assumption is considered to be correct for stress paths such as the triaxial or other paths where there is no rotation of principal stress axes.  However simple shear test data reported by Roscoe (1970),  Stroud (1971) and Wood et al. (1979) on Leighton-Buzzard sand show that the above assumption is not strictly correct. A review of these is carried out later in section 2.5.  From this review it is concluded that during the  initial stages of all tests the angle \J> (defined by the vertical direction and the direction of the major principal stress, o x ) , and the angle E (defined by the vertical direction and the direction of the major principal strain increment, Aex, can diverge considerably but when failure is approached these angles start converging and at failure the deviation between the angles \|) and £ is not significant. Based on the above it is concluded that the assumption considered by Matsuoka (1983) is reasonable and can be used here to relate the increments of plastic cartesian strain with the increments of plastic principal strain as presented by eqs. (2.48) above.  2.4.3.4 Relationship Between Increments of Plastic Strain, {AeP}, and Increment of Stress Ratio on the SMP The relationship between the increments of plastic strain, {AeP} and the increments of the stress ratio on the SMP, A(x S M p /o S M p) are given by the following equations which were obtained by substituting eq. (2.47) into eq. (2.48):  P  i  t  <;MP  AeX = ( Z M.«.») fr A(-^) G i=l,3 p SMP Aep y  Ae  p  2 = ( Z M.m. )G A(-^) 1 1 i=l,3 p °SMP  , i SMP, = ( Z M.n. ) jr A(? SMP i=l,3 l l 2  Af;  &  =  Af;  . Z, _ M.i.m.) i l l Gp A(^MP) 1=1,3 °SMP  (  , —1 A( A /SMP, M.m.n.) ) 1 1 1 G i=l,3 p °SMP  Y* = - ( Z  Af zx  1  <?MP  = ( Z M.n.JL) A(-^-) i=l,3 1 1 1 G p SMP (2.49)  2.4.3.5 Evaluation of the Increment of the Stress Ratio on the SMP as a Function of the Increments of Stress, (Ao) The last step required to develop {Cp} consists of developing a relationship between A ( T S M p / o S M p ) and the increments of stress, {Ao}. The approach described by Matsuoka (1987) to account for the "shear" on the 2-D Mobilized Plane, MP, and the "rotation" of the principal stress axis is followed below after adapting it to the 3-D Spatial Mobilized Plane, SMP. The stress ratio on the SMP is given by the following equation which is obtained from eq. t_ M P °SMP  =  (2.25) and eq. (2.26):  [(0,-0,) I a 1 2 a 2 2 +(a 2 -o 3 ) J a J 2 a3 2 +(o3-o 1 ) J a 3 2 a 1 J ] 1/2 (2.50) o1a12+o2a22+o3a32  To account for the "shear" on the SMP, the following steps were taken: • Differentiating Eq. (2.50) with respect to the principal stresses, o^(i = 1,2,3), and the direction cosines, a^, of the normal to the SMP the following equation is obtained (see Appendix 2.4):  A ( T c m / o c m n ) =1 fJ(Ao., Aa.) smp smp i l smp smp  (2.51)  Eq. (2.24), which relates a^ with o^ is differentiated with respect to o^, and relationships between Aa^ and Ao^ are obtained. Substituting these in eq. (2.51) the following equation is obtained (see Appendix 2.4) A( _sra E) =  smp  ( T S M 0 B l ) A O i  + (XSM0B2)AO2 + (TSM0B3)Aos  (2.52)  where the terms TSM0B1, TSM0B2 and TSM0B3 are as described in the Appendix. To account for the "rotation" of the principal stress axis the following relationships were considered: (o -o.)cos(i,x) + x cos(i,y) + x cos(i,z) = 0 J x i ' xy zx x cos(i,x) + (o -o.)cos(i,y) + x cos(i,z) = 0 J yx y l yz  (2.53)  x cos(i,x) + x cos(i,y) + (o -o.)cos(i,z) = 0 J zx zy z l cos2(i,x) + cos2(i,x) + cos2(i,z) = 1  where  cos(i,x), cos(i,y) and  cos(i,z) are the direction cosines of  direction i (i=l,2,3) with respect to directions x,y and z, respectively.  In the above, the first three equations were obtained from the fundamental relationships between cartesian stresses and principal stresses and the fourth equation from the known relation for direction cosines (see Appendix 2.5). By differentiating the above equations, equations for the increments of principal stress, Ao^ and for the increments of the angle of rotation of principal stress axis, A(i,x), A(i,y) and A(i,z) are obtained as functions of the increments of the stresses, {Ao} (see Appendix 2.5).  Since "i"  above has values of 1, 2 and 3, then three systems of 4 equations with 4 unknowns, are obtained.  Solving these equations yields values of Ao^,  which take into account the rotation of the principal stress axis, in terms of the increments of the stresses, {Ao}. The above procedures are presented in detail in Appendix 2.5.  The  final results are as follows:  (2.54)  where: Q . , Q . , Q . , Q _ .,Q_ . and Q . are terms described in Appendix tci' ^yi' ocyi ^yzi zxi 2.5. Substituting the values of Ao^ from eq. (2.54) into eq. (2.52) a relationship between A(x M p / o Q M P ) and {Ao} is obtained and given by:  A(T  SMP/oSMP  (TMX)Ao + (TMY)Ao + (TMZ)Ao  + (TMXY)Ax + (TMYZ) AT xy  y  z (2.55)  + (TMZX) At zx  where:  TMX =  I (TSMOBi) (Q^) i=l,3  TMXY =  I (TSMOBi) ( Q ^ ) i=l,3  TMY =  S (TSMOBi) (Q^) i=l,3  TMYZ =  I (TSMOBi) (Q i=l,3  TMZ =  X (TSMOBi)(Q .) • T <-i 21 1=1,3  TMZX =  )  (2.56)  X (TSMOBi)(Q .) • i o ZX1 1=1,3  2.4.3.6 Evaluation of the Plastic Constitutive Matrix Substituting eq. (2.55) into eq. (2.49) a relationship between increments of plastic strain {AeP} and increments of stress {Ao} is obtained and given by the following equation:  {AeP} = [CP]{Ao}  (2.57)  where {CP} is the plastic constitutive matrix of the SMP model. This matrix is given in detail in Appendix 2.6.  2.4.4  Evaluation of the Elasto-Plastic Constitutive Matrix [ c e p ] As described earlier the strains in the SMP model are composed of two  components.  The plastic strains are given by the above eq. ( 2 . 5 7 ) and the  elastic strains by the following equation:  Ae® = [Ce]{Ao}  (2.58)  Q  [C ] is defined using an incremental linear elastic and isotropic law. Isotropy is a convenient assumption since it lowers the number of elastic parameters to two. The two parameters selected here are the shear modulus, G, and the bulk modulus, B.  Both moduli are considered to be dependent on  the mean normal stress and evaluated in the formulation by the following equations:  where:  G = KG P (o /P ) n a m a  (2.59)  B = KB P (o /P ) m a m a  (2.60)  KG and KB are the shear modulus and bulk modulus number, and n and m are the shear modulus and bulk modulus exponent.  To obtain the complete strain-stress relation, eq. 2.57 and 2.58 are added to give  {Ae} = tCep]{Ao}  (2.61)  Procedures for the evaluation of both the elastic and plastic parameters for use in the modified SMP model will be given later, in Chapter 3 (from laboratory test data), in Chapter 5 (from pressuremeter test data), and Chapter 6 (from laboratory and cone penetration test data).  2.A.A.I Loading and Unloading Constitutive Matrix One of the major advantages of using an elasto-plastic constitutive matrix  is that it is very easy to model the loading and unloading  characteristics of the sand material in a F.E. formulation.  A soil element is considered to be in a loading stress path whenever the stress ratio level (SRL) of the element, in the current load step, is higher than the SRL of the previous load step, and for this condition the full [C e p ] matrix in eq. (2.61) will be used in the analysis. However, if the current SRL is smaller or equal to a previous SRL then the soil element is considered to be on an unloading stress path.  The plastic component  [CP] of the total constitutive matrix [C ep ] will be dropped and only the £  elastic component [C ] given by eq. (2.58) will be used in the analysis. 2.A.4.2 Implementation of the Modified SMP Model into Finite Element Form To analyze the response of the Molikpaq structure to ice loading, the modified SMP model was implemented in the 3-D computer code '3DSLB1. Unfortunately because the required computer memory to analyze the Molikpaq exceeds the existing UBC computer (Amdahl) capacity of 1 megaword, this model could not be used in the 3-D analysis.  However, the modified SMP  model was implemented in the 2-D computer code '2DSLB' and 2-D plane strain analysis of the Molikpaq were carried out. The modified SMP model was also implemeted in axisymmetric form in the computer code '2DSLB' to analyse pressuremeter test data obtained in the fills of the Molikpaq.  Both the 2-D plane strain and axisymmetric  formulations were obtained from the 3-D formulation by imposing the corresponding boundary conditions.  The details are presented in Appendix  2.6. Formulation for "load shedding" was also included in the above two computer codes and this is briefly discussed next.  2.A.4.3 Load Shedding Formulation During the ice loading on the Molikpaq structure the soil elements adjacent to the loaded wall can undergo shear failure due to loading (see Fig. 2.14(a)) and the soil elements underneath the base of the structure on the side of the loaded wall can undergo shear and/or tension failure due to unloading (see Figs. 2.14(b) and 2.14(c)). In the F.E. formulations (3-D and 2-D) earlier described, whenever a soil element reaches failure the G p shear parameter is defaulted to a prescribed low value so that the element does not absorb any additional significant shear stresses during subsequent load steps. This approach will work provided the soil element is being subject to increasing normal stresses.  Its effectiveness depends on the magnitude of  the load increment to failure, see Fig. 2.14(a), on how low the G p parameter is defaulted to and on how small the subsequent load increments will be. This approach, however, will not work if the normal stresses on the element decreases during the load step, because the element stresses will stay practically the same violating the failure criteria. To solve the problem, Zienkiewicz et al. (1968), Byrne and Janzen (1984) proposed a stress redistribution technique called "stress transfer" or "load shedding" by which the element overstresses are redistributed to the adjacent stiffer soil or structural elements. Briefly this "load shedding" technique consists of the following: (a) Evaluation of the over-shear stress, Ax^ s to be shed in terms of increments of cartesian stress, {Ao}TC (i.e. Ao L5  X  , Ao LS  y  .., AT LS  ). LS  (b) Correct the current stresses {a} of the element overriding the failure criterion by -{Ao}^. (c) Default the shear modulus of this element to a low value.  LEGEND = A = s t r e s s point at l o a d increment ( K ) B = s t r e s s point at l o a d increment ( K + 1)  A t .LS  a m o u n t of s h e a r s t r e s s to be s h e d d e d  A a ^ s = a m o u n t of n o r m a l s t r e s s to be s h e d d e d  Figure 2.14  (a) Shear Failure During Loading; (b) Shear Failure During Unloading; (c) Shear and Tension Failure During Unloading.  (d) Compute nodal loads  a r e  equivalent to the overstress.  Perform an additional load step of analysis with the load vector This will redistribute the load to the adjacent elements.  By doing the above, the quantity {Ao}^g which is in violation is taken from the element and is redistributed to the adjacent elements in such a way as to satisfy both equilibrium, compatibility  and the failure  criterion. The implementation of the above procedures in the modified SMP formulation is presented in Appendix 2.7.  2.5 Review of the Assumptions Considered in the Modified SMP Model The assumptions considered in the formulation of the modified SMP model were described while the model was presented.  A summary of these  assumptions is presented below.  2.5.1 Summary of the Assumptions Used in the Modified SMP Model 1st Assumption:  The elastoplastic constitutive matrix [C e p ] is e  composed of two components, an elastic component [C ] and a plastic component [cP]. 2nd Assumption: The elastic constitutive matrix [C ] is assumed to be isotropic.  3rd Assumption:  The increments of elastic principal strain are  assumed to have the same direction as the increments of principal stress.  4th Assumption:  The increments of plastic principal strains are  assumed to have the same direction as the principal stresses.  5th Assumption:  The Spatial Mobilized Plane is assumed to be the  plane in which the soil particles are most mobilized on average in the 3-D stress space.  6th Assumption:  The relationship between the stress ratio  and the ratio of the increments of plastic strain AepUT,/Arp „ is assumed to r SMP SMP be given by the following equation.  T  SMP /a SMP  = X(  -AeSMP/AiMP)  +  »  The above relationship, which is designated as flow rule, is assumed to be independent of both the initial void ratio e^ and the initial confining, a ^.  7th Assumption:  The relationship between the stress ratio  t  smp^°smP  and the plastic shear strain Tg^p i s assumed to be hyperbolic and given by the following equation:  Ar  SMP " G  A(t  SMP / o SMP )  This equation represents the hardening rule and it assumes that the shear parameter, G , is dependent on both the normal stress, stress ratio level, SRL.  an<  i  8th Assumption: X  SMP /o SMP  reac  hes  The sand material fails when the stress ratio  a  limiting value, which is given by the equation  =  9 3 Vtan '*i»F  t  cmp  + tan2  ^3F  + tan  '*i»F  = K  F  and it is assumed that the above equation is valid for any stress path. From the above assumptions the ones regarding the direction of the increments of principal strain (Assumptions #3 and #4) and the assumption regarding the SMP failure criterion (Assumption #8) are considered to be the most pertinent and are discussed below.  2.5.2 Discussion of the Assumptions Regarding the Direction of the Increments of Principal Strain As described, simple shear conditions are likely to develop in the sand fills of the Molikpaq structure during ice loading.  Therefore the  simple shear data on Leighton-Buzzard sand reported by Roscoe (1970) will be briefly discussed below. The rotation of principal axes with shear strain during a typical simple shear test is presented in Fig. 2.15.  The directions of the  principal axes of stress, increments of strain and increments of stress are defined as is shown in Fig. 2.15(a) in which if), £ and x are the angles with the vertical made by the directions of the major principal stress o x , major principal strain increment Aex, and major principal stress increment Ao,, respectively.  Virgin loading test data is shown in Fig. 2.15(b) and,  virgin loading, unloading and reloading test data is shown in Fig. 2.15(c). Based on these test results, Roscoe (1970) concluded the following:  cV  ae,  Co)  (b)  0-2  X  .>;•  60'  »' f / r  40'  <r  20'  (C)  «(=X)  _L _1_ 0-05 oor Q.09 OJOJ OjCJ 0.07 0-09 o.l I -ViOGtU 10ADING H l-UML0MINfr4 V« RELOADING » 20 -40'  ^ V  Figure 2.15 Rotation of Principal Axes During Simple Shear Tests on Leighton-Buzzard Sand (e0 = .64). Data reported by Roscoe (1970). (a) Definition of Angles \|>, £ and x> (b) Virgin Loading Test Data, and (c) Virgin Loading, Unloading and Reloading Test Data.  a)  For monotonically increasing stresses the principal axes of increments of strain (£) and of stresses  coincide as the sand was sheared,  except for the earliest stages of the test before the sample developed its minimum void ratio (Min.V.R.) (see Fig. 2.15(b)).  This fact  reinforces Assumption #4 because after Min.V.R. the deformations are essentially plastic. b)  At no stage of a virgin loading test did the axes of increment of strain (£) and increment of stresses (x) coincide.  However, if after  monotonic increase of the shear stress this stress was reduced and then increased again the angles £ and x coincide (see Fig. 2.15(c)) indicating elastic behaviour. This fact reinforces Assumption #3. However, to fully validate the above assumptions, other stress paths with rotation of principal axis should be addressed. For that the research work carried out by Symes et al. (1982,1984,1988) on Ham river sand and Sayao (1989) on Ottawa sand using the hollow cylinder torsional apparatus are briefly discussed in Appendix 2.8.  The main conclusions are as  follows: For the "continuous rotation tests" carried out by Symes et al. and Sayao, with increasing or decreasing values of \|) but with constant values of stress ratio a./a,, b-value and, mean normal stress o it is concluded 1 3 ' ' m that the deviations between the angle \p (stresses) and the angle £ (increments of strain) can be quite significant.  The same conclusions  apply for the "continuous variation in b-value tests" carried out, by Sayao, with increasing or decreasing b-value but with constant values of \j), a,/a, 1 3 and o . This indicates that Assumption #4 is not valid for these two m types of tests. However Assumption #4 is shown to be valid for the hollow cylinder tests where a stress path to failure was followed (except for the  early stages of the tests) such as the "initial anisotropic tests" carried out, by Symes et al. and Sayao, with increasing o 1 /a 3 but with constant values of \p, b-value and o^.  The same conclusions apply for the  "proportional loading tests" carried out by Sayao with increasing o^ but constant o 1 /o 3 , \p and b-value. Since for the case of the Molikpaq the stress path that matters is a failure stress path, it is concluded therefore that Assumption #4 is considered to be adequate enough for the Molikpaq analysis.  2.5.3 Discussion of the Assumption Regarding the SMP Yield Criterion As discussed the SMP yield criterion developed by Matsuoka and Nakai (1974,1985) assume that at failure, the stress ratio TgMP/'0SMP regardless of the stress path followed up to failure.  c o n s t a n t  The above also  implies that the failure friction angle <{>13 defined by the principal directions o x and o 3 is the same for the triaxial stress path in compression or extension.  Based on experimental data reported by some reseachers,  however, the above have not been verified. As an example, the variation of <f>13 with b-value obtained from the data reported by Arthur et al. (1977) ispresented in Fig. 2.16. This data was obtained from true-triaxial tests on Leighton-Buzzard sand (e0 = .52).  In the figure the <fi13 obtained from  the simple shear results on the same sand (e0 = .53) reported by Stroud (1971) is also presented. Applying Matsuoka-Nakai  failure criterion equation with the <p  13  obtained from the triaxial compression test (b-value = 0) the dashed line No. 1 is obtained.  Following the same procedure with the <f>13, values  obtained from the simple shear test (b-value = .33) and triaxial extension test (b-value = 1) than the dashed lines No. 2 and 3 are obtained, respectively.  55 n  1  /  50 / fO -cT  c o  o  45  1  1  1  i  LEGEND• •  4  s  Measurements'  /  / /  • - true triaxial A r t h u r et a l . ( l 9 7 7 )  .  } /  t  / /  o - simple s h e a r Stroud (I971)  ' —  / /  ' //A 40  Qi  Predictions  //  Matsuoka - N a k a i failure c r i t e r i u m =  »  o» <c  using  Q 35  1  30 0.2  i  0.4 0.6 b - value  ( D from simple shear ( D from t r i a x i a l extension  1  0.8  from t r i a x i a l compression  1.0  Figure 2.16 Variation of Friction Angle (j> With 'b-Value' for Leighton-Buzzard Sand.  o  It may be seen from the figure, that for this set of data, the Matsuoka-Nakai failure criterion underestimates the <f>13 for b > 0 if the triaxial compression data point is used. From the above it is concluded that although the Matsuoka- Nakai failure criterion represents an improvement as compared with the MohrCoulomb failure criterion, which does not show any increase of <f>13 with the b-value, it can not account for the different values of <j>13 for the cases of triaxial compression and extension.  2.6 Disadvantages of the SMP Modified Model The main disadvantage of the proposed formulation is that it involves a non-symmetric stiffness matrix [K] and therefore it requires a solver routine for non-symmetric system of equations.  This fact in itself is not  a problem because routines to solve this type of systems are available. However, the required computer memory is considerably larger than that required for the standard symmetric banded system of equations.  This dis-  advantage became relevant for the case of the 3-D FE mesh used in the analysis of the Molikpaq (Chapter 7) because a 2.5 megaword memory capacity was required and this is larger than the existing 1 megaword memory capacity of the current UBC computer (Amdahl).  Therefore the modified SMP  model could not be used in the 3-D FE analysis of the Molikpaq and only 2-D FE analysis could be carried out with this model.  For the 3-D FE analysis  the hyperbolic model (Duncan et al., 1980) which uses a symmetric stiffness matrix was used instead.  This disadvantage however is considered to be a  temporary one since computers with larger memory capacity than the UBC Amdahl are used in other other technological fields and hopefully soon will be available to solve special geotechnical problems such as the Molikpaq study.  CHAPTER 3 PROCEDURES FOR THE EVALUATION OF SOIL PARAMETERS FOR USE IN THE MODIFIED SMP MODEL. VERIFICATION OF THE MODIFIED SMP MODEL  3.1 Introduction This chapter is concerned with the following aspects which are important to any constitutive model:  a)  Development of procedures for the evaluation of soil parameters for use in the model.  b)  Verification of the model against observed laboratory test results. The laboratory data selected here was obtained from the following three sources: • Data reported by Stroud (1971) for Leighton-Buzzard sand using the Cambridge Simple-Shear Apparatus Mark 7. • Data reported by Vaid, Byrne and Hughes (1981) for Ottawa sand using the UBC Simple-Shear Apparatus. • Data provided for the workshop on "Limit Equilibrium Plasticity and Generalized  Stress-Strain  in Geotechnical  Engineering", McGill  University, and published by Yong and Ko (1980). The data consists of true triaxial test results on Ottawa sand. The simple shear test receives particular attention here because the shear, dilation and deformation under the principal stress rotation of sand when subject to the simple shear stress path is considered to be representative of the stress change induced by the horizontal ice loading on the Molikpaq sand fills.  3.2 Evaluation of Soil Parameters for Use in the Modified SMP Model from the Standard Triaxial Test The soil parameters required for the modified SMP model can be divided into two main groups: i)  Elastic parameters  ii) Plastic parameters  In all, 11 parameters are used in the modified SMP model.  These  parameters are summarized below in Table 3.1.  Table 3.1 Summary of Soil Parameters for Use in the Modified SMP Model Type Elastic  Hardening Rule Plastic Flow Rule  Parameter  Description  KGe  Elastic shear modulus number  n  Elastic shear modulus exponent  KBe  Elastic bulk modulus number  m  Elastic bulk modulus exponent  K G  P  Plastic shear number  np  Plastic shear exponent  P  Flow rule intercept  X  Flow rule slope  SMP /o SMP ) 1Failure stress ratio at 1 atmosphere Decrease in one log cycle of ( TSMP /'0SMP^F A(TSMp/oSMp) (t  Failure  R  F  Failure ratio  The above soil parameters can be evaluated from: (a) laboratory test results, (b) pressuremeter test results, and (c) laboratory and cone penetration test (CPT) results.  The procedures to evaluate the soil parameters from the standard triaxial test are presented in Appendix 3.1.  Procedures to evaluate soil  parameters from pressuremeter test results are presented in Chapter 5 and from laboratory and CPT results are presented in Chapter 6 where the soil parameters for use in the Molikpaq analysis are obtained.  3.3 Verification of the Modified SMP Model Against Observed Laboratory Test Data To check the modified SMP model's formulation and capabilities the following two levels of verification were carried out: 1° Level of Verification (Calibration) This consists of an evaluation of soil parameters from the results of a particular laboratory test on sand and calibration with the measured results obtained from the same test.  This will allow a check of the  following: a)  that the procedures described earlier to evaluate soil parameters are correct; and  b)  that the formulation used in the model is also correct.  2° Level of Verification Evaluation of soil parameters from the results of a compression triaxial test (b-value = 0) and/or extension triaxial test (b-value =1) on a particular sand and prediction of the observed laboratory data obtained from other tests (simple shear, true-triaxial) on the same sand consolidated to the same void ratio. This will allow a check of the capability of the model to predict the response of sand when subject to various stress paths, using soil parameters that were determined from the standard triaxial test.  3.3.1 First Level of Verification of the Modified SMP Model. Calibration with the Simple Shear Test Data Reported by Stroud on LeightonBuzzard Sand The data reported by Stroud (1971) for Leighton-Buzzard sand using the Cambridge simple-shear apparatus Mark 7 (SSAM7) was selected here to calibrate and verify the 3-dimensional formulation of the modified SMP model because: • The SSAM7, which was developed by Stroud, gives information on all three principal stresses, o l t o 2 and o 3 during the simple shear test. • A gradual rotation of the axes of principal stress and strain occur during this test. • Simple shear conditions are likely to develop in the fills of the Molikpaq structure during ice loading.  3.3.1.1 Soil Parameters for Leighton-Buzzard Sand (eff=.53) for Use in the Modified SMP Model The Leighton-Buzzard sand tested by Stroud (1971) is a coarse rounded quartz sand graded between No. 14 and No. 25 BS sieves with 60-65% passing no. 18 BS sieve.  The sand samples were prepared with a void ratio e 0 = .53  (Dr = 87%). The soil parameters for Leighton-Buzzard sand for use in the modified SMP model are summarized in Table 3.2 and were obtained as described in Appendix 3.2.  3.3.1.2 Calibration with the Simple Shear Data Reported by Stroud From the data reported by Stroud (1971) the tests carried out with constant vertical stress, o = 48 kPa, o = 76 kPa and o = 172 kPa were z z z selected here to be predicted. In the numerical analysis the initial values for the horizontal stresses, a , (the direction of shear) and o , x y' (the direction of the intermediate stress) were taken equal to .44 owhich were the values reported by Stroud.  Table 3.2 Soil Parameters for Leighton-Buzzard Sand (e0=.53) for use in the Modified SMP Model  Elastic Parameters  Plastic Parameters  KGe  = 620  n  = 0.63  KBe  =580  m  =0.60  Plastic Shear Modulus Parameters  KG  np  = -.48  Flow Rule Parameters  Vi  = .20  X  =1.20  Failure Parameters  p  = 335  T f SMP^  SMP 1 a t m '  =  A(; S M P ) = .08 SMP (Rp)av = .957  Using the soil parameters shown in Table 3.2 the following predictions were carried out: (a) t/s versus y (presented in Fig. 3.1) where: t = (ox - o3)/2 = shear stress s = (ox + O3)/2 = mean normal stress and  y = ( e 1 - e 3 ) = shear strain  (b) x  versus y (presented in Fig. 3.2(a)) ZX zx where: xzx = cartesian shear stress y = cartesian shear strain ' zx  (c) e versus y (presented in Fig. 3.2(b)) v zx where: e = volumetric strain v (d) a /o versus y (presented in Fig. 3.3) x xo where: a = the initial horizontal stress in the x-direction xo o = the mobilized stress, due to shear, in the x-direction x (see Appendix 3.3 for the evaluation of afrom the reported laboratory data) (e) o2/s versus y (presented in Fig. 3.4).  The above predictions were carried out using both the 3-D and 2-D plane strain formulations of the modified SMP model, both gave exactly the same predictions. The overall agreement between the predictions and the measured data is seen to be good except for the ° x / ° x 0 versus y predictions.  These  differences are attributed to deviations of the measured vertical stress, o^, from the assumed constant vertical stress boundary conditions of the tests.  Based on the above, it is concluded that the procedures described  in Appendix 3.1 to evaluate soil parameters and both the 3-D and 2-D formulations of the model have been verified.  MEASUREMENTS  Figure 3.1 Predicted and Measured Simple Shear Data on Leighton-Buzzard Sand (t/s versus r)• (a) a = 48 kPa; (b) o„ = 72 kPa; and v (c) o v = 172 kPa  Measurements Predictions  0", -172^0  a}  SHEAR STRAIN . y % 'ZX  W  o—i  SHEAR S T R A I N , y 2X(°/»)  Figure 3.2 Predicted and Measured Simple Shear Data on Leighton-Buzzard e Sand, (a) x ^ versus r^; v v e r s u s Tzx  S h e a r Strain  S h e a r Strain  S h e a r Strain Figure 3.3 Predicted and Measured Simple Shear Data on Leighton-Buzzard (c) o = 1 7 2 ° k V e r S U S  r)  '  U )  °V  = 48 kPaJ  (b)  °v  = 72 kPa;  a n d  Shear  strain ,  /%  Figure 3.4 Predicted and Measured Simple Shear Data on Leighton-Buzzard Sand (o2/s versus y) . o v = 48 kPa  -j °°  3.3.2 Second Level of Verification of the Modified SMP Model. Predictions of Simple Shear and True-Triaxial Test Data on Ottawa Sand The true triaxial test data on Ottawa sand (Dr = 87%) used in the 1980 workshop on McGill University (data reported by Yong and Ko) is used here, as is described below, together with the simple shear test data reported by Vaid, Byrne and Hughes (1981) on the same sand (D = 72.3% and 92.7%) to further verify the modified SMP model. The above test data was subdivided here into two sets of data: • Data Base from which the soil parameters for use in the model were obtained.  The stress paths used to generate the data base are presented  in Fig. 3.5(a) and the grain size distribution of the Ottawa sand used in the tests is shown in Fig. 3.5(b).  It may be seen that the tests con-  sisted of conventional triaxial tests (compression (CTC) and extension (CTE)), constant mean stress triaxial tests (compression (TC) and extension (TE)) and hydrostatic compression test (HC). This data is the same data base (or pre-workshop data) used in the 1980 workshop at McGill University. • Data Used for Predictions.  The stress paths considered for the predic-  tions were subdivided into the following five groups: - Simple shear test which can be characterized by a b-value = .30 (see Fig. 3.6(a)), - Constant mean stress triaxial tests with b-values of .2, .5 and .8 (see Fig. 3.6(a)), - Proportional loading tests, PL1 and PL2 (see Fig. 3.6(b)), - Reduced triaxial tests, RTC and RTE (see Fig. 3.6(b)), and - Circular path test (see Fig. 3.6(c)). A detailed description of these tests is given later in the predictions section.  a)  •J5 o„ * Grav«<  .5 M 100  Sand Coarja (o Pint Silt mafliuffl us. lanaira sis SIII1 2 2 J i i5 3 3 0 o i i s 1 z J-  or P'nes CUy  \(  rt  b) -  20 I \  k  •2* t — Z © o' o* o* Grain dumaip. mm  Figure 3.5  (a) Stress Paths Used to Generate Data Base for Modelling; (b) Grain Size Distribution of Ottawa Sand.  (b)  Deviatoric  Plane , C  oc l  = constant  Figure 3.6(a,b,c) Stress Paths Used for Predictions  For comparison purposes, predictions of the above tests using the hyperbolic model (Duncan et al. (1980)) are also included here because the hyperbolic model was used in the 3-Dimensional analysis of the Molikpaq.  3.3.2.1 Soil Parameters for Ottawa Sand (0^=87%) for Use in the Modified SMP Model and the Hyperbolic Model The soil parameters for use in the modified SMP model were evaluated from the Data base as described in Appendix 3.3.  A summary of the soil  parameters is given in Table 3.3. The soil parameters for Ottawa sand (D = 87%) for use in the r hyperbolic model are presented in Table 3.A and were evaluated by Duncan and published by Yong and Ko (1980). Table 3.3 Soil Parameters for Ottawa Sand (Dr = 87%) for Use in the Modified SMP Model KG n KB m  = 1640 = 0.49 = 2578 = 0.25  Elastic Parameters  KG np  = 190 = -.50  Plastic Shear Modulus Parameters  \i X  = .25 = 1.10  Flow Rule Parameters  C ^ ) !1 = .935 SMP SMP  A(^ ) - .62 SMP (R F ) av  - .97  Failure Parameters  Plastic Parameters  Table 3.A Soil Parameters for Ottawa Sand (D^ = 87%) (Evaluated by Duncan (1980) and Used in the Hyperbolic Model) Parameter K *UR n Cohesion, c m  * R  f  Average Value 400  Range ± 120  2000  -  0.85  -  0.5 psi 700 0.50 43° .86  -  -  ± 3° -  3.3.2.2 Predictions of the Simple Shear Test Data Reported by Vaid, Byrne and Hughes (1980) The simple shear tests on Ottawa sand were carried out using the UBC simple shear apparatus described by Finn and Vaid (1977), which is a Cambridge type of apparatus similar to that developed by Roscoe (1953). The tests were carried out under drained conditions at a constant vertical confining stress, o = 200 kPa. The test results are shown in Fig. 3.7 for z a range of relative densities.  The tests were strain controlled and one  test (Dr = 92.7%) exhibited strain softening. In order to take into account the strain softening behaviour of Ottawa sand the modified SMP formulation was expanded with two additional soil parameters.  The details are given in Appendix 3.5 and follow similar  procedures as outlined by Carter and Yeung (1985) after some adaptations to the SMP formulation.  Shear S t r a i n , y ( % )  Figure 3,7 Stress-Strain Behaviour of Ottawa Sand in Drained Simple Shear (after Vaid, Byrne and Hughes, 1980)  Predictions of the simple shear test results were carried out using both the modified SMP model (with and without strain softening parameters) and the hyperbolic model.  Because the relative density corresponding to  the Data base was 87%, only the measured simple shear test data for D r = 72.3% and 92.7% will be considered here to bound the predictions. The initial stresses used in the analysis consisted of: vertical stress, o = 200 kPa, which remained constant; and o = o =86.5 kPa which z x y assumes a K 0 = .A3.  This K 0 value represents the average value of that  reported by Stroud (1971) (K0 = .AAO) and, Wood, Drescher and Budhu (1979) (K0 = .A25) which were obtained from the experimental values recorded in the elaborately instrumented Cambridge simple shear apparatus. The laboratory results together with the predictions obtained with the two models are presented in Fig. 3.8. It may be seen that up to peak shear stress the predictions obtained with the modified SMP model for D r = 87% lie very close to the laboratory measurements (shear and volume measurement) corresponding to D r = 92.7%. After peak, the predictions taking into account the strain softening effects are in good agreement with the laboratory measurements.  On the other hand, if strain softening is not  considered,' the mobilized shear stresses xzx maintains a constant value after the peak shear stress is reached. The predictions evaluated here using the soil parameters developed by Duncan with the hyperbolic model are also presented in Fig. 3.8 and show that this model underestimates the failure strength of the Ottawa sand when it follows the simple shear stress path.  This is related with the Mohr-  Coulomb failure criteria that is used in the hyperbolic model. Regarding the volumetric predictions, the hyperbolic model does not predict any  200-i  a) PREDICTIONS (Dr=87%) G  0  MODIFIED  SMP  N o St r o In S o ^ t a n l n g  A  &  MODIFIED  SMP  With Strain Softening  X 10  15  30  20  SHEAR STRAIN , v  HYPERBOLIC  %  < (/) O  b)  cc  I— UJ  3  o >  10  15  20  SHEAR STRAIN, y  ' 'zx  '  %  Figure 3.8 Predicted and Measured Simple Shear Data on Ottawa Sand, (a) t versus y z x . (b) e v versus y ^  volume changes since its formulation does not take into account volume changes mobilized by the shear component. As described in the beginning of this chapter even using the version of the hyperbolic model developed by Byrne and Eldridge (1982) which takes into account the volume changes due to dilation, the failure strength of the Ottawa sand for this particular stress path was well underestimated.  This was one of the reasons why the  modified SMP model was developed.  3.3.2.3 Predictions of the Post-Workshop Test Data of the 1980 Workshop of McGill University The test equipment used is a flexible, fluid cushion cubical device with stress control, which is known as cube or true-triaxial device. The vertical axis, z, and the horizontal axes x and y are principal stress axes, which means that shear stresses cannot be applied with this device. The stress paths of the tests the investigators were asked to predict in the 1980 workshop of McGill University were given earlier in Figs. 3.5(b), (c) and (d). These tests can be divided into four groups: a)  constant mean stress tests  b)  proportional loading tests  c)  reduced triaxial tests  d)  circular path test  The predictions of the test results will be presented following the above order together with a brief description of each test and comments on the quality of the predictions by the two constitutive models.  a)  Constant Mean Stress Tests  Four tests where the o and the b-value were kept constant were m performed:  a,) b = .2 and a 1  Aoy = 2/3 Aoz.  = 1 0 psi. y  m  During this test Ao 6  x  = 1/3 Ao , and z'  a.) b = .5 and o = 5 psi. During this test Ao = Ao . 2 r m " x z a.) b = .5 and a = 20 psi. During this test Ao = Ao . 3 r 6 m x z a.) b = .8 and o = 1 0 psi. During this test Ao = .5 Ao and 4 r b m x z Ao = 1.5 Ao .  y  z  The test results and the predictions from the two models are presented in Fig. 3.9 to Fig. 3.12. The predictions obtained with the modified SMP model are in good agreement with the measured results for both the initial phase and the failure phase of the tests.  Test with b=.5 and  o  m  =  2 0 psi was the only  exception where 1 psi difference of t ^ at failure is noted. The predictions obtained by Duncan with the hyperbolic model show that the higher bound of the predicted range is in good agreement with the tests  with b=.2 and b=.8, but that the predictions for the tests b=.5 (o =5 and m 20 psi) reach failure conditions much sooner than the laboratory measurements indicate. Mohr-Coulomb  The reasons for this are most likely related with the  failure  criteria  used  in  the hyperbolic  model, which  underestimates the failure friction angle for stress paths with b values different than 0 (triaxial compression) or 1 (triaxial extension).  qb'cr to psi  /•Measured Predicted Range Unlood-Re'ood^  5  f»-Unlood-Reiood LL.  •2  "I 0 . +1 Principal S t r a i n s , %  +2  (a) Constarrl Meon Stress = 10 psi , b = 0.20 (b)  •10  r 10  8 a  N  - 8  (/)  00  -6  u O  D a) JZ 00 •4 —  cn -4 — o "O a) JZ o 2 o O  -1  0  Horizontal S t r a i n - X %  Figure 3.9  tn  6 -  L.  CD  -2  8 -  V) <D  JZ  0  V) CL to  00 <u 6  10n  Legend 4H  PREDICTIONS •  QJ JZ D O  MEASUREMENTS  o i -2  1  r  o  - 1  Horizontal S t r a i n - Y %  o  1  2  Vertical S t r a i n - Z %  Predicted and Measured Constant Mean Stress Test Data on Ottawa Sand, b = 0.2, o m = 10 psi. (a) Predictions with the Hyperbolic Model, (b) Predictions with the Modified SMP Model  51 "^ct-P r b  c£ CT = 5psi Meosured  Predicted^ Rcnge ^Unlood-Re!ocd -0.5 0 0.5 Principal Strains, %  (a) Constant U«an Stress = 5 p» , b = 0.50  (b) 4  r  Legend PREDICTIONS •  MEASUREMENTS  "D <r> JZ _o ~o o 0  1  —r~  -2  -1  0  Horizontal S t r a i n - X %  0  i i < — > ' > i  1  2  H o r i z o n t a l Strain—Y %  o 1 Vertical S t r a i n - Z  Figure 3.10 Predicted and Measured Constant Mean Stress Test Data on Ottawa Sand: b = 0.5, o m = 5 psi. (a) Predictions with the Hyperbolic Model, (b) Predictions with the Modified SMP Model  -1  Principal  0  I  Strains , %  (a) Constant Mean Stress = 20 ps , b = 0.50  (b) 14  14-  14 -I —  V)  12-  Q. 10-  v>  0 U~) o a> 00  n  8  •a a) .c "o O  -6  -4 -2 0 Horizontal S t r a i n - X %  0 2 Horizontal Strain-y %  o«—• 1 • , 0 2 4 Vertical S t r a i n - Z %  Figure 3.11 Predicted and Measured Constant Mean Stress Test Data on Ottawa Sand, b = 0.5, o m = 20 psi. (a) Predictions with the Hyperbolic Model, (b) Predictions with the Modified SMP Model  T  OCT" PSi  TI2  °bcT l|C) p s i  Meosured Predicted Range  PrincipaI Strains, %  (a)  (b) r8  8  -6  £oo  w 0) L. -i  4  PREDICTIONS MEASUREMENTS  - 2  o a> JZ 00  Q)  JZ oo •a 0)  "O  j.z  -4  i  -2  Horizontal S t r a i n - X %  0  1  CD  JZ 2  a  "o O  o  - 6  6  00  u , o  Legend •  i/i a.  CO a (/) tn 0)  2  0 Horizontal S t r a i n - y %  o42  Vertical S t r a i n - Z %  Figure 3.12 Predicted and Measured Constant Mean Stress Test Data on Ottawa Sand, b = 0.8, o m = 10 psi. (a) Predictions with the Hyperbolic Model, (b) Predictions with the Modified SMP Model  b)  Proportional Loading Tests Two proportional loading tests with  =  10 P s i were carried out.  In the first Aox = Aoy = 1/8 Ao z and was designated PL.. In the second & i Ao x = AOy = 1/2 Aoz and was designated PL,. The test results together with the predictions are presented in Fig. 3.13. The predictions by the two models of the PL1 test underestimate considerably the failure phase of the test, and no apparent reasons can be offered since the b value = 0. The predictions of the PL2 test indicate that the predictions of e z  strain by the modified SMP model and the high bound of the hyperbolic model are in very good agreement up to a e = .3% and after that the laboratory z  data seems to deviate from its initial trend.  Regarding the e x and e  strains, both models predict strains with the wrong sign as compared with the laboratory measurements and again no apparent reason can be offered to justify that deviation. c)  Reduced Triaxial tests  Two reduced triaxial tests with (o ). =20 psi were carried out. One mi was a reduced triaxial compression test, RTC, with b=0, Aoz=0 and decreasing Ao = Ao , and the other a reduced triaxial extension test RTE with & x y b=l, Ao x = AOy = 0 and decreasing Aoz. The test results and predictions are shown in Fig. 3.14. The predictions of the RTC and RTE tests carried out by the two models are in good agreement with the laboratory measurments. d)  Circular Path Test This test, which was performed with a constant o m = 10 psi, is  composed of 3 phases. In the first phase the sand is loaded in shear until  OCT-PS"  l  ^Measured  Measured Predicted Range Proporti'cncl Loading Test (PL1)  m-0 i • •  fredictei Pcrge  Proportiono Loading Test (PL 2) m =0 Unload-Reioad -0.5  Unlocd-Reload C  0.5  Principal Strains, %  OCT""5'  Principal Strains %  -0.5  0  0.5  Principal Strains %  Figure 3.13 Predicted and Measured Proportional Loading Test Data on Ottawa Sand, o ^ = 10 psi: (a) PL1 Test, Predictions with the Hyperbolic Model, (b) LLl Test, Predictions with the Modified SMP Model, (c) PL2 Test, Predictions with the Hyperbolic Model, (d) PL2 Test, Predictions with the Modified SMP Model  (a)  Measured  Measured S H Predicted Rcnge  Predicted Range Reduced Tricxiai Extension Test  Reduced Triax'iol Compression Test  Unload-Re lood 2 1  m = 1.0  m=0 -Unload-Reload 0  1  Unload-ReloadJ; VjrUniocd-Reload '< IT- • . . i , . . . i . - 2 - 1 0 I 2 Principal Strains, % i  2  Principal S t r a i n s , %  'OCT'PSI' r 10  Legend PREDICTIONS •  -2  0  2  Principal Strains %  MEASUREMENTS  - 4 - 2  0  2  Principal Strains %  Figure 3.14 Predicted and Measured Reduced Triaxial Test Data on Ottawa Sand, o ^ = 20 psi. Reduced Triaxial Compression Test (RTC): (a) Predictions with the Hyperbolic Model, (b) Predictions with the Modified SMP Model. Reduced Triaxial Compression Test (RTE): (c) Predictions with the Hyperbolic Model; (d) Predictions with the Modified SMP Model.  a T o c t = 4.2 psi is developed. The second phase is characterized by maintaining a constant x . = 4.2 psi while the three stresses o , o and o & oct ^ x' y z vary in magnitude, and share at different times the directions of the principal stresses a l t a 3 and o 3 .  Although the name of the test (circular  path) seems to imply that a gradual rotation of the principal stress axis is taking place, in reality that is not the case, and what takes place is an instant flip of the principal directions at different times during the test.  This phase is terminated when the "apparent" angle of rotation 0  reaches a value of 420°.  During the last phase of the test the sand is  unloaded from xoct. = 4.2 psi r ^ to xoct. = 0. The results and the predictions corresponding to the 2nd phase of the test are shown in Fig. 3.15. The predictions obtained by Duncan using the hyperbolic model are in fair agreement with the laboratory measurements.  On the other hand the  predictions carried out with the modified SMP model, which are in fair agreement with the laboratory measurements during the first (1/3) of the second phase of the test, show a poor agreement for the rest of the test. However, because this circular path test is not representative of the stress paths that occur in the sand fills of the Molikpaq structure during either the construction or ice loading phases, explanations for the above differences were not investigated at this time, but will deserve consideration in the future by the writer.  3.4 Summary and Conclusions Procedures for the evaluation of soil parameters from laboratory test results for use in the modified SMP model has been presented in this chapter. This was followed by a verification of the proposed model against  0 - deg rees  8 - degrees 120  240  3 60  4 80  •  T 1  1  120  240  1  O o  •  •  • • O  o O  •  %  -11  o-r  •  -Pi.  O Mecsured • Predicted  o 0  4 60  3 60  1 r  o  o •  •  y  •  •  -1.0 L + 1.0  • •• o -1.011  I  I L ' 240 360 fl-degrees  120  I I  430  120  0 240  0  •  360  6 - degrees  1  430  Figure 3.15 Predicted and Measured Circular Stress Path Test Data on Ottawa Sand. o m = 10 psi; x o c t = 4.2 psi. (a) Predictions with the Hyperbolic Model. (b) Predictions with the Modified SMP Model  observed laboratory test results.  Based on the overall good agreement  between the predictions and the reported data the following is concluded: (1) The procedures described in this chapter are adequate to evaluate soil parameters for use in the modified SMP model. (2) Both the 3-D and 2-D plane strain formulations of the proposed model can reproduce well the reported simple shear test data on LeightonBuzzard sand. This indicates the following: (a) The model takes into account the gradual rotation of the axes of principal stresses and strains that occur during that test. (b) The 2-D formulation which is derived from the 3-D formulation by applying the appropriate boundary conditions give a good prediction of the intermediate principal stress o,. (3) The overall good predictions of the simple shear and true-triaxial tests on Ottawa sand, with the exception of the circular path test, further indicate that the proposed model is able to predict the behaviour of that sand with reasonable accuracy for the stress-paths of practical importance. (A) Because the circular path test is not representative of the stress paths that occur in the sand fills of the Molikpaq structure during either the construction or ice loading phases, the reasons for the poor predictions of this test by the modified SMP model were not investigated herein.  CHAPTER 4 INTERFACE ELEMENTS  4.1 Introduction The behaviour of the interface between the sand fills used in the core and berm and the Molikpaq steel structure are important aspects that will effect the behaviour of the soil-structure system under load. In this chapter a brief review of the existing interfaces elements available in the literature is presented.  From this review an interface  element designated as "thin" element is selected due its simplicity, concept and performance.  Next this "thin" element is described together  with methods for the evaluation of the soil parameters used in the constitutive laws for the element. Procedures for the implementation of the thin element to 2-D and 3-D F.E. codes, are also described and include the implementation of load shedding techniques for elements that failed in tension or shear. Finally, the performance of the "thin" interface element is assessed by comparing F.E.  results with the closed form solutions of a  pipe-soil system and with earth pressures measured on a 10 meter retaining wall field study.  4.2 Brief Review of the Existing Interface Elements A good review on this topic is given by Desai (1981) and Desai et al. (1984) and a brief summary is presented here. The first interface elements described in the literature are the pinended element (Anderson and Dodd, 1966), the spring element (Ngo and Scordelis, 1967) and the zero thickness joint element (Goodman et al., 1968).  From these, the one that received most attention and has been  used in F.E. analysis, (Clough and Duncan, 1971) and others, is the interface element proposed by Goodman.  This element's formulation is  derived on the basis of relative nodal displacements of the solid or structural elements surrounding the interface.  Because there is a lack of  physical basis for this zero thickness interface element and because it requires a formulation with a special element stiffness matrix, other types of interface elements have been developed since then.  Of these elements,  the one derived by Desai et al. (1984) was found most desirable due to its simplicity and excellent performance.  This element with a finite but thin  thickness was designated as a "thin" interface element.  This element is  described next.  4.3 Description of the "Thin" Interface Element The "Thin" element is treated essentially like any other solid isoparametric F.E. element, except for a few differences described below. Its behaviour involves a finite thin zone as shown in Fig. 4.1(a) (2-D element) and Fig. 4.1(b) (3-D element). The two basic differences between the "thin" element and the standard isoparametric solid element are:  (1) The "Thin" element uses an anisotropic constitutive law, where the two moduli used to described its behaviour, normal modulus, E.., and shear .N modulus, G, are independent of each other. The E^j modulus is used to characterize the normal response of the element when subject to compressive loads (bonding mode, i.e. normal stress, Ojj > 0) or tensile loads (debonding mode, o^ £ 0) as shown in Fig. 4.2. When o. NT > 0 the E„ N modulus is equal to a value that characterizes its  Figure 4.1 "Thin" Interface Element: (a) 2-Dimensional; (b) 3-Dimensional  -cr. + 0",N  I  \&zzzzzzzzzzzz£. E n ( bonding)  E n (debonding) - c r .N  + 0~, N  Figure A.2 Bonding and Debonding Modes  compressive behaviour, and when o^ £ 0 the E^ modulus is equal to a small value in order that the element does not absorb any more tensile stresses during the subsequent load increments. The G modulus is used to characterize the shear response of the element when subject to shear loads as shown in Fig. 4.3.  When the  absolute value of the mobilized shear stress, |t I , is smaller than the m resistant shear stress, t d the element is in a "no slip" mode and the G modulus is equal to a value that characterizes its shear behaviour, and when it i £ t„ the element is in slip c mode and the G modulus is defaulted m R to a small value in order that the element does not absorb any more shear stresses during the subsequent load increments. Recommendations concerning the values of the moduli E„ and G and the interface failure parameters M angle of friction, 6, and cohesion, C , are given later in Section 4.5 of this chapter together with the constitutive models that are generally used in practice. The advantage of using an uncoupled pair of moduli is that in the case where the interface element fails in shear (slips) the low shear modulus does not also imply a low E N inferring a tension failure, i.e. the element behaves as cross-anisotropic.  If a coupled pair of moduli was used instead  the element will behave isotropically, which means that when it fails in shear the element will soften in both shear and axial directions since the moduli E and G are linked this time by the Poisson's ratio, v. (2) The other difference from the standard isoparametric solid elements is that the constitutive matrix [D] that relates the strains and stresses of each element, is defined in the local coordinate system (n, r, s) of the  I -T m  j^N . J  i  ZZZZZZZZZZZL  4  *  T  l m l >Tf  « T  I m I < Tf  n  I  I > Tf  G( no slip)  G (slip) -r  L  G (slip) -TV  r  f  + T  = cr N t o n 8 + C  Figure 4.3 Slip and No Slip Modes  element rather than the global coordinate system (x,y,z) used in the standard solid element's formulation (see Fig. 4.1).  4.4 Determination of Soil Parameters: Constitutive Modelling The soil parameters required to define the characteristic behaviour of the interfaces can be divided into two groups: shear parameters and normal parameters.  • Evaluation of Shear Parameters For the evaluation of the shear modulus, G, the interface friction angle, 6, and cohesion, C a > the test most recommended in the literature is the direct shear test.  These tests are performed using a standard direct  shear machine, where the soil is compacted in the upper half of the shear box and the lower half consists of a specimen of the structure under consideration (steel, concrete, wood, etc.).  Several tests are generally  carried out at different normal stresses, o^, to simulate the expected range of stresses in situ. The measured relative displacements between the upper and lower halves of the shear box are assumed to characterize the shear interface response upon the applied stresses.  Other devices also  used are the torsion ring shear and the not so common multi-degree-offreedom shear device (Desai, 1981), where translational and rotational modes can be mobilized. From the results obtained from the direct shear test the following relationships can be obtained: a)  Shear stress, t, versus relative displacement, u r .  b)  Failure shear stress t, versus normal stress, o„. I N  From relationship (a) the shear modulus, G, can be obtained, and from relationship (b) the strength parameters 6 and C will be evaluated as Si  described next.  • Evaluation of the Shear Modulus, G To evaluate the shear modulus, G, the procedures recommended by Clough and Duncan (1971) combined with the procedures recommended by Desai (1984) will be followed. Clough and Duncan assume that the relationship between the shear stress, x, and the relative displacement, u r , is hyperbolic as shown in Fig. 4.4(a) and that the initial tangent stiffness,  is given by the  following equation:  K  si = Vw ( S )n  where: K  I n  = dimensionless stiffness number = stiffness exponent = unit weight of water  Pa  = atmospheric pressure  To obtain the above terms, several direct shear tests are carried out at different normal stresses and plots of (u^/x) versus u^ are developed as shown in Fig. 4.4(b).  From these transformed plots, and following steps  that are very similar to the steps followed to evaluate the parameters Kg and n used for the standard hyperbolic model (Duncan and Chang, 1970; Duncan et al., 1980), the parameters K T and n are evaluated.  As for the case of the standard hyperbolic model, the shear behaviour of the interfaces is assumed to be dependent on the stress level, and the tangent stiffness, K  is given by the following equation:  Kst. = Ksi. (1 -  Rfx  2  (A.2)  where: R  f  T f T  ult  failure ratio = T f / T u ^ t failure shear stress = C + o„ tan 8 a N the asymptotic shear stress = inverse of the slope of the transformed plot  m  mobilized shear stress  To evaluate the value of the initial shear modulus G^ the equation proposed by Desai (1984) will be followed:  (4.3)  G. = K . • t 1 si where: t  = interface thickness  and the tangent shear modulus, G^ of the interface is given by:  G =K •t t st  or G  t  = G  i(1"Rf  f?  (4.4)  Substituting the values of eqs. (4.1) and (4.3) into (4.4)  and to be consistent with the formulation used in Chapter 2, the tangent shear modulus G is given by  (4.5)  where the shear modulus number, K_ is given by the following equation:  Klrj = K_(y i w /Pa) • t  (4.6)  • Evaluation of Interface Thickness, t As described by Desai  (1984) the quality of simulation of the  interface behaviour will depend on a number of factors such as physical and geometric properties of the surrounding media, nonlinear material behaviour and the thickness, t, of the "thin" interace element.  If the thickness is  too large in comparison with the length, L, of the interface (see Fig. 4.1), the "thin" element will behave essentially as a standard solid element.  If it is too small, computational difficulties may arise.  The  choice of the element thickness, t, is therefore an important issue. Based on parametric studies in which the predictions from various thickness, t,  were compared with direct shear test observations, Desai (1984) concluded that if the ratio t/L is within the range 0.01 to 0.1 than satisfactory agreement between the predicted results and the laboratory measurements was obtained.  • Evaluation of 6 and Ca The values of the interface friction angle, 6, and cohesive, Ca, are easily evaluated from the plot of the failure shear stress, t^ versus the normal stress o N as it is illustrated in Fig. 4.4(c).  • Evaluation of Normal Parameters Regarding the normal modulus, E^, the literature is not to clear how to evaluate it. Based on the assumption that the structural and geological media do not overlap at interfaces, generally a high value has been assigned for this modulus.  Desai believes, however, that there is no  physical ground for such an assumption, which can create numerical instability problems.  In reality, the normal properties of the interface must be  dependent upon the characteristics of the thin interface zone as well as the state of stress and properties of the surrounding elements.  Based on  Desai's experience he concludes that satisfactory results can be obtained by assigning the interface normal component the same properties as those of the adjacent soil elements.  This concept was followed in this thesis with  satisfactory results as will be shown later when the predictions are carried out.  Therefore in the formulation, E^, is considered to be  dependent on the normal stress, o„, and given by the following equation:  NORMAL STRESS  32.5 Ib./sq. in.  £ 10 0 -  16.2 Ik./iq. in.  6.5 Ib./sq. in.  0 02 0 03 Rclolivt Ditplocamanl, in  0.04  w> </> v a a>  x: «A «  ^  »  '5  N o r m a l s t r e s s , cr, N  Figure 4.4  (a) Comparison of Hyperbolic and Actual Stress-Displacement data (after Clough and Duncan, 1971) ; (b) Transformed Linear Hyperbolic Plots for Interface Tests (after Clough and Duncan, 1971) ; (c) Evaluation of 6 and Cfl from Direct Shear Tests  E  N =  P A  (  ?!  ) N  ( 4  -  7 )  where: normal modulus number, generally taken = Kg (Young's modulus number) of the adjacent material n  normal modulus exponent, generally equal to the Young's modulus exponent of the adjacent material  4.5 Implementation of the "Thin" Interface Element into the Finite Element Formulation The "Thin" interface element was implemented in the 3-D and 2-D F.E. formulation codes '3DSLB' and '2DSLB'.  The required F.E. formulation is  rather simple and is described in detail in Appendix 4.1. A load shedding formulation for the "Thin" element was also developed and implemented into the above two computer codes. The details are presented in Apendix 4.2.  4.6 Performance Studies of the "Thin" Interface Element 4.6.1 Closed Form Solution of a Soil-Pipe System In order to gain confidence with these formulations, predictions of a 'dosed form solution were made.  A closed form solution of a soil-pipe  system was developed by Burns and Richards (1964) in their study of "Attenuation of Stresses for Buried Cylinders".  A schematic view of the  idealized soil-pipe system is shown in Fig. 4.5.  The pipe, which is  treated as a linear elastic material has a radius of .84 m and is encased in a homogeneous soil which is also linear elastic.  The external bound-  aries are located four radii from the pipe centre as shown. The boundary and loading conditions are also shown in Fig. 4.5.  For this soil-pipe  Pressure, p 0  Figure 4.5 Soil Pipe System  system, Burns and Richards developed two closed form solutions correspondent to the two extreme frictional cases: (a) completely bonded; and (b) completely frictionless. cases were predicted.  The results from these two extreme frictional  Because these two extreme cases are not representa-  tive of sandy soils an additional frictional case was also considered: (c) interface friction angle 6 = 14°. For this case no closed form solution is available. Since the problem is plane strain, the 2-D F.E. code 2DSLB was used in the analysis. In addition, and to test the capability of the 3-D interface element, 3-D F.E. analyses were also carried out using the computer code 3DSLB. The F.E. meshes for both the 2-D and 3-D analysis are shown in Fig. 4.6 together with the soil, interface and pipe properties. To insure that the 3-D analyses were carried out under plane strain conditions the width, b, of the 3-D F.E. mesh was assigned a value b=l unit length. In addition, movement was restricted in the width direction (see Fig. 4.6(b)).  The  interface thickness, t, was assumed to be L/10, where L = side length of the interface elements. The closed form solutions together with the F.E. predictions for both 2-D and 3-D analysis are shown in Fig. 4.7.  It may be seen that the  predictions obtained are in very good agreement with the closed form solutions for both the bonded and frictionless cases.  The 3rd solution  where a intermediate friction 6 = 14° was used appear reasonable since both normal and shear pressures are in between the two extreme cases. To note that the 2-D and 3-D results were exactly the same and therefore a degree of confidence is established for the implementation of the "thin" interface element in both the 2-D and 3-D F.E. computer codes.  PROPERTIES SOIL TYPE  MATERIAL  \J  tan&  .30  10.0  1  Pipe  E (kPa) 20.7 x 10'  2  Soil  18 x 10i  .33  10.0  3  Interface •Bonded •Frictionless •Friction  18 x 10J  .33  10.0 0.0 .25  Figure 4.6 F.E. Meshes and Soil Properties Used for the Soil Pipe Closed Form Solution  .5 .4 .3  £ ^ b  tani = 0.25  (a)  .2 .1 1 i  "  Bonded  o  .0  9  8  0 .7 0 6 0 5 0  Frictionless  J  I  L  -I—•  10 20 30 4 0 50 60 70 80 9 0 0° , ( f r o m s p r i n g line towards crown)  LEGEND: -o Closed form solution -x  Bonded  F.E. predictions 2 - D and 3 - D  ( b )  tan<£ = 0.25  0  10 20 30 40  50 60 70  80 9 0  0 ° , ( f r o m spring line towards c r o w n )  Figure 4.7 Soil Pipe System. Closed Form Solutions and F.E. Predictions 2-D and 3-D: (a) or/po; (b) x/po  4.6.2 Retaining Wall Study Introduction The analyses presented above are of interest because they allowed a check on the performance of the "Thin" element against a closed form solution situation.  However that example does not represent the conditions at  the interface between the Molikpaq steel structure and its sand fills. Since no measurements of earth pressures were carried out in the structure fill interface during the core placement phase at the Molikpaq's two sites (Tarsiut and Amauligak), the writer looked elsewhere for a case study that could resemble such a situation. The experimental study of earth pressures developed in a 10 in retaining wall carried out by Matsuo et al.  (1978)  represents a good field case to further test the described "thin" element and at the same time test the performance of the two nonlinear stressstrain models, the hyperbolic (Duncan et al., 1980) and the modified SMP model. From the results obtained in the field and in the F.E. analysis, discussions will be made regarding the following points: (a) the importance or not of using the "thin" interface element instead of using the standard solid element for this given situation; (b) comparisons between the two nonlinear stress strain models and a third point, (c) assessment of the coefficient of earth pressure K 0 , which as will be shown later (Chapter 7) will play an important role in the F.E. predictions of the Molikpaq upon ice loading. • Description of the Retaining Wall Field Test A detailed description of the problem is given by Matsuo et al. (1978). herein a brief description is presented.  A schematic section of the retaining wall is shown in Fig. A.8(a). This wall which is 10 m high and made of concrete is laterally supported by three oil jacks which are installed between the retaining wall and the wall of an adjacent building, and, is bottom supported by a hinge which allows the wall to stay in a state of rest position (vertical position) or to rotate to an active state position (inclined position) as shown in Fig. 4.8(b).  To measure the earth pressures mobilized during the above two \  positions, twenty load cells were employed which were located between the retaining wall and five pressure receiving plates as shown in Fig. 4.8(a). This way the earth pressure was evaluated in detail at five locations. Three different field tests were carried out using three different types of backfill materials. One using a silty sand backfill and the other two using slags produced from iron manufacture plants.  From these three  field tests only the first will be analyzed here since in the Molikpaq the fill material used is sand. • Characteristics of the Silty Sand Fill The silty sand fill was placed in lifts until a height of 10 m was achieved.  The average unit weight of this fill was about 19 kN/m3 and the  water contents ranged between 5% and 8%. The parameters describing the silty sand fill are given by Matsuo et al. (1978) and are reproduced in Fig. 4.9.  It is understood that the  strength parameters Ca and 6 given in that figure represent the mean values of many data obtained by direct shear tests in which soil samples of 10 cm in diameter with 5.5% to 6.7% in water content and a unit weight of = 19 kN/m3 were used.  Unfortunately, laboratory data curves of shear stress  versus shear strain were not published by Matsuo and therefore values for  .155  WAS  rn-rr^  <D  ri,  a  © _L  sr ® © © © © ©  )  WALL CP THE EXISTING BUILDING OIL JACK RETAINING WALL LOAD CELL EARTH PRESSURE CELL PRESSURE RECEIVING PLATE  WALL  j'—i \l 1 \ 1 i .  BACK-\ FILL  W (a) (a) (b)  STATE AT REST ACTIVE STATE  Figure 4.8 Retaining Wall Field Story: (a) Retaining Wall Instrumentation; (b) Wall Positions  G R A I N SIZE  (mm)  BACKFILL PROPERTIES Silty Sand s  2.69  opt(%)  11.9  G u  rdmax  Strength Parameters  Figure 4.9  1.94  (t/ra3)  C <t>  2.3 (t/m») 27°  (a) Grain Size Distribution of Silty Sand; (b) Index Properties and Strength Parameters of Silty Sand  the moduli used in the F.E. analysis were based on the published work by Byrne et al. (1987) and on the writer's experience on sands with similar characteristics of strength.  • Assessment of the Quality of the Field Measurements The field measurements of the earth pressures mobilized at the "at-rest" or K q position and at several inclined positions of the wall are shown in Fig. A. 10(a), where d is the displacement at the top of the wall. The inferred earth pressure coefficients, K, for d = 0 , d = 1.6 cm, and d = 8.A cm, are shown in Fig. A.10(b).  As shown, highest values for K were  computed as expected for the "at-rest" position, d=0, and ranged from .7A at a depth of 1.0 m to .28 at a depth of 5.0 m with an average value (K) = ,A7.  On the other hand, when the wall is rotated and d = 1.6 cm, the K  values ranged from .09 at 5.0 m depth to .A at 9.0 m depth with an average value (K) = . 25 . If the wall is further rotated to d = 8. A cm, the K av values further decreased to a (K) = .11. av The data presented above in Figs. A. 10(a) and (b) show that as the displacement, d, increases the K values decrease as expected.  To further  assess the quality of the field measurements in Fig. A.10(c), the plot developed by Matsuo et al. (1978) is shown where the ratio between the earth pressure for different wall rotated positions, p, and the initial earth pressure at rest, p 0 , for different depths, is plotted against the displacements inferred for these depths.  It may be seen that the active  state for the whole backfill was reached for values of d ranging from .3 to .8 cm (or d/H ranging from .003 to .008, where H = 10 m is the wall height).  This is in good agreement with the results of tests performed by  Terzaghi (193A) which showed that the active conditions on a rough wall was  Earth pressure coefficient, K  0  0  0.2  0.4  0.6  0.8  a "  P. : I N I T I A L E A R T H PRESSURE AT REST  •  0j8  4  •  %  OJB  c  8 SAND  h -  10  .o  A ft A 0.4 • 0 o A 0.2 •o • •  o 6  (a) SILTY  (P/Po) 1.0  2 1  1.0  8.4 \  1.6  0 = d (cm)  DEPTH 9 7  o— o— A •  5 1  3  A  o D • •• • I 2 I 4 >6 8  1  0  DISPLACEMENT A T E A C H DEPTH  Figure A.10  (rn)  (a) Earth Pressure Measurements Versus Depth; (b) I n f e r r e d Earth Pressure Coefficient, K versus Depth; (c) Relationship Between P/po and Displacement at Each Depth  10  (cm)  i—1 NJ  reached for a value of d/H = .001A for dense sand, and a value d/H = .0084 for loose sand. Now that the measured earth pressures carried out by Matsuo et al. (1978) are considered to be reliable, then F.E. analysis were carried out to assess the reliability of the different F.E. element types and constitutive model types.  • F.E. Analysis A cross-section showing the geometric conditions at the site is given in Fig. 4.11(a) and the F.E. mesh together with the soil parameters used in the analysis is shown in Fig. 4.11(b). Three field conditions were analyzed: (a) at rest condition (d/H = 0) and, active conditions (d/H = .0016) and (d/H = .0084). To analyze these conditions, a simulation of the in situ sequence of construction was carried out by placing the different soil layers shown in Fig. 4.11 in 10 layers and following the analytical procedures described by Byrne and Duncan (1979).  During the construction sequence the nodes  located on the wall were not allowed to move in the horizontal direction. At the end of the construction phase the stresses obtained were stored and  used as initial streses for the active phase were this time those nodes were allowed to move in both horizontal and vertical directions being only constrained by the movements of the stiffer beam member used to represent the wall. Each of the above conditions was analyzed with different sets of element types and constitutive law types. carried out. These are described below:  In all, three F.E. studies were  MODIFIED SMP MODEL  HYPEREOLIC MODEL SOIL  SOIL  PARAMETER BACKFILL FOUNDATION K £; n  400; .5  Kg; m  240; .25 900; .25  ~  —  PARAMETER  1500 ; .5 KSEI  —  N  INTERFACE MODEL  SOIL • INTERFACE BACKFILL FOUNDATION PARAMETER MATERIAL 410; .5  1540; .50  n  400; .50  KSe; m  600; .25 2250; .25  K^; n  165; .50  ^GD' N P  270; .5  1030; -.50  —  —  RP  .9  .9  R  F  .3  •I  27"  40°  (T SM p /a SM p )j.48  .79  6  27O  Aip  0  0  A( t  .0  —  —  C (kPa)  22.6  .0  SMP /a SMP )0.0 21.3 C (kPa)  .0  C a (kPa) 22.6  1.0  .4  K„  1.0  .4  K»  —  —  —  U  .25  .25  —  —  —  \  1.0  1.0  R  .1  i*  F  0.0  1.0 —  —  Figure 4.11 (a) Cross Section Illustrating Retaining Wall and Backfill; (b) Finite Element Mesh and Soil Properties Used in the Analysis  In the first study, the wall soil interface was represented by the "thin" interface element using an elastic perfect plastic model and the backfill soil represented by standard solid elements using the hyperbolic model. In the second analysis both interface and backfill were represented by standard solid elements using the modified SMP model and in the third analysis both interface and backfill were represented by standard solid elements, this time using the hyperbolic model. It should be noted that the modified SMP model's formulation was expanded with an additional parameter, the cohesion, c, in order that the strength of the backfill material was properly characterized in the analysis using this model. The details are given in Appendix 4.3.  • F.E. Results The results obtained in the F.E. analyses are given in Figs. 4.12 to 4.14 together with the field measurements observed by Matsuo and his co-workers.  From the comparisons between the field measurements and the  F.E. predictions the following conclusions are made: (1) All the combination of element types and constitutive model types give an almost identical earth pressure distribution for the "at-rest" condition (see Fig. 4.12).  These results are considered to be in  agreement with the field measurements with the exception of the "atrest" earth pressures computed at the depths of 5 and 7 m where the field measurements are overestimated and underestimated respectively by approximately 15 kPa. (2) The F.E. results obtained for the active conditions (see fig. 4.13 and 4.14) are shown to be in agreement with the field measurements, especially the earth pressures computed by the modified SMP model.  EARTH 0  20  PRESSURE,kPa 40  60  Field  3  80  Measurements  — Hyperbolic  4  100  —  SMP  Mode!  Model  i £ UJ a  5  6  —  7  -  8  — "Thin" Interface-  9  --  Figure 4.12 Comparison Between Earth Pressure Measurements and F.E. Predictions: At Rest Condition, d = 0.0 cm.  DEPTH , m  HOQ c <D 1-1  13 O O (D 3 0> H* cn O 3 a  cd  rt (D rt cd H <-3 w no fl> H o 3 r+ O-3 ' Hr+ T3 H-t-S O (D 3 W W C p. i-t (D II 1s i— n> • 0> <7> W 0 •i >  o K  cd  S  ( D 3 r W+ PJ 3  E A R T H P R E S S U R E , KPa  DECREASE IN E A R T H P R E S S U R E FROM AT R E S T C O N D I T I O N , kPa  (3) From this particular case study it seems that standard solid elements when used with adequate stress-strain constitutive laws are adequate to model interface behaviour and there is no the need for a "thin" interface type of element.  • F.E. Parametric Studies Carried Out by Others (Influence of 6) It was the intention of the writer to carry out F.E. analysis where the angle 6 would vary from 6=0 (smooth wall) to 6=</> (rough wall) to show its importance, but such study has been done by Clough and Duncan (1971) and herein only a brief description of their study together with their main conclusions is presented instead.  Their analytical study comprised a 10  foot retaining wall. The interface between the wall and the soil backfill was represented by the zero thickness element proposed by Goodman, following the hyperbolic model described in section 4.4.  The soil properties  used in the analysis together with the results obtained are reproduced in Fig. 4.15.  It may be seen that the variation of 6 from a smooth wall  condition (6 = 0) to a rough wall condition (6 = <p) makes little difference in the results. The results obtained by Clough and Duncan (1971) show that the active condition was reached over the entire height of the backfill when the outward movement at the top, d, has become equal to .0023 H which is also in good agreement with the results of tests performed by Terzaghi (1934) for a medium dense sand.  Their results also show that the initial  assumed value of the coefficient of earth pressure K 0 = .43 decreases considerably as the wall moves away from the backfill. Average K values of .25 to .28 can be inferred, using the equation presented below, from their results when d/H = .0023.  Backfill HI  Parameter 12)  Symbol (3)  Medium-dense sand backfill  Unit weight, In pounds per cuoic foot Coefficient of earth pressure at rest Cohesion intercept, in pounds per square foot Friction angle in degrees Primary loading modulus number Unloading-reloading modulus number Modulus exponent Failure ratio Poisson's ratio  Wall-backfill interface  Friction angle tn degrees Stiffness number Stiffness exponent Failure ratio  Base-backflU interface  Friction angle in degrees Stiffness number Stiffness exponent Failure ratio  Y K  Value (4)  100 0.43  0  c  0 35 720 900 0.5 0.8 0.3  »  K  nK u r  Rr  V  s  nKl */  i n  *r  Varies Varies Varies Varies :4  75,000 0.5 0.9  Properties Used by Clough and Duncan (1971)  ro  -0 « <2 to. 54 m * S 6 X to. X a iu a H (  i  •  1  1  1  3) "  >  -  -  -  100 200 300 400 HORIZONTAL WALL PRESSURE, p»l -o  ACTIVE ROTATION rougft -oil (i-44)  i IOO 200 300 400 HORIZONTAL WALL PRESSURE, pif  1'  I H • 10*  L*9tnd — ClotticM Theory o o q FlniU Element  I00 200 300 400 HORIZONTAL WALL PRESSURE, ptl 1) That the interface was smooth. For this analysis 4-0.1', Kj-1.0, n-0.0, and 2) That the wall-friction angle was 2/3 of n-1.0, and R£-0.9 3) That the wall-friction angle was equal to n-1.0, and Rf»0.9  For this analysis 4=2ft'. Kj-40.000. For this analysis 4-35°, Kj-75,000,  Figure 4.15 Variation of Horizontal Wall Pressure Distribution with Wall Movement and Interface Friction Angle, 6 (after Clough and Duncan, 1971)  °H K = — K0 °H0  (4.22)  where: K0  = earth pressure coefficient at rest (= .43)  o„ = horizontal wall pressure for d = 0 o ou = horizontal wall pressure for d > 0 H Similar K values were recorded by Matsuo et al.  as was described  previously.  4.7 Conclusions From the material presented in this Chapter, the following can be concluded: (1) The F.E. results show that an excellent agreement with the closed form solutions of a soil-pipe system developed by Burns and Richards (1964) is obtained when the "thin" element is used in both the 2-D and 3-D F.E. analysis (plane strain conditions). (2) The field measurements carried out by Matsuo et al. (1978) and the F.E. predictions carried out by the writer, are in good agreement with the results of tests performed by Terzaghi (1934) and with the analytical work carried out by Clough and Duncan (1971).  The field  measurements show that the coefficient of earth pressure at rest, K 0 varies from a maximum K 0 = .74 at 1.0 m depth to a minimum K 0 = .28 at 5.0 m depth.  If the wall is allowed to rotate away from the backfill  then the coefficient of earth pressure, K, decreases considerably to  an average (K)  = .11 which corresponds to a movement of the top of  the wall of 8.A cm or .84% of the wall height. (3) F.E. studies carried out by Clough and Duncan (1971) show that the predicted earth pressures on a 10 ft. retaining wall are only slightly affected by the interference friction angle 6 when it varies from a smooth wall conditions (6=0) to a rough wall condition (6=<j>). (4) F.E. predictions of the earth pressures measured on a 10 m retaining wall by Matsuo et al. (1978) show that adequate results are obtained using standard solid elements with an appropriate stress-strain models, such as the hyperbolic model (Duncan et al., 1980) or the modified SMP model both expanded with load shedding capabilities. There is no need for a special interface element.  CHAPTER 5 EVALUATION OF SOIL PARAMETERS FROM THE PRESSUREMETER TEST IN SAND  5.1 Introduction Analytical predictions of the response of sand masses to applied loads requires a suitable stress-strain law whose parameters are adequately defined. While these parameters can be determined from laboratory tests on sand samples it is very difficult to recover and test undisturbed samples and determine meaningful parameters for in situ conditions.  Due to the  above, any time the Molikpaq structure is deployed at a new site, extensive in situ testing is carried out in the hydraulically placed core and berm fills. The Self-Boring Pressuremeter (SBP) Test is one of the in situ tests that was performed at the Amaulikpaq 1-65 site to evaluate soil parameters representative of the sand fill. However, because the stress field induced by the SBP is not homogeneous, a rational analysis and interpretation of the SBP test requires that it be analyzed using selected stress-strain relations. In addition, it is important that such analysis and interpretation be checked against experimental data under controlled conditions before application to in situ field conditions. This chapter is concerned with the evaluation of soil parameters from the SBP test in sand, and is subdivided in the following two sections: a)  Evaluation of the maximum shear modulus, G from the unload-reload max loop of the pressuremeter test.  b)  Evaluation of soil parameters from the first time loading part of the pressuremeter test.  These two topics are briefly discussed below.  • Evaluation of G max One of the soil parameters that can be derived from the SBP is the equivalent elastic unload shear modulus, G*, which is obtained from the slope of the unload-reload pressuremeter loop as shown in Figure 5.1. G*, however, is not equal to the maximum shear modulus, G at the original ^ max,o stress due to expansion of the pressuremeter as well as high shear strains close to the face of the pressuremeter. r  G  is a fundamental soil max,o  parameter that is essential for dynamic analysis of soil structures and is also one of the elastic parameters used in the modified SMP model. Previous researchers (Robertson (1982) ; Robertson and Hughes (1986) ; and Belloti et al. (1989)) have proposed methods for correcting the measured G* to obtain G based upon an average stress and strain in the plastic 1T1&2C y o zone.  In the first part of this chapter a more detailed analysis  considering the complete variation in the stress and strain state is presented.  The method considers both the stress and void ratio changes  induced by pressuremeter loading and the nonlinear stress-strain response upon unloading. The results are presented in the form of a chart that allow Gmax,o to be determined from the equivalent elastic unload modulus, ^ G*, for a wide range of loading and unloading conditions.  The analysis  procedure is checked with laboratory and field data and the results are found to be in good agreement provided factors to account for disturbance and anisotropy are considered. • Evaluation of Soil Parameters from the First Time Loading Part of the Pressuremeter Test The evaluation of soil parameters from the first time loading part of the pressuremeter tests in sand have been restricted for many years to the  Figure 5.1 Pressuremeter Unload Modulus, G*.  evaluation of the peak friction angle <p and the dilation angle v. Ladanyi (1963), Vesic (1972), Wroth and Windle (1975), Hughes et al. (1977), Robertson (1982) , and Robertson and Hughes (1986), proposed procedures to determine <p by assuming that the sand around the pressuremeter behaves as a plane strain linear elastic-perfect plastic material.  The main differ-  ences between these methods is the way that volume changes due to shear are taken into account. Only recently, Manassero (1989) proposed a method that allows the complete plane strain nonlinear stress and volume change response of sand to be obtained from pressuremeter pressure-expansion data. This method can be used to determine parameters for the proposed stressstrain model as will be discussed later in this chapter.  5.2 Evaluation of the Maximum Shear Modulus for Sand From the Unload Shear Modulus Obtained from Pressuremeter Tests The shear modulus G* obtained from the pressuremeter test is unlikely to be equal to the maximum shear modulus Gmax because of the stress and ^ strain changes caused by the pressuremeter. correcting6 the measured G* to G  max  An analysis procedure for  is presented herein. The method is r  based upon an elastic-plastic analysis of the pressuremeter domain to determine the stresses in the domain prior to unloading, and a nonlinear elastic analysis to determine the displacement at the pressuremeter face upon unloading, which in turn is used to compute the equivalent pressuremeter shear modulus, G*.  By comparing the computed G* with G m a x for  various levels of applied radial stress prior to unloading, and for various amounts of unload, a chart is generated from which G*/G can be obtained ° max depending on the applied pressuremeter loading conditions.  Because the stress-strain relations used to model the behaviour of sand during unloading are an important factor in the analysis these are described prior to presenting the analysis and results.  5.2.1 Assumed Stress-Strain Relations for Sand Upon Unloading Upon unloading it is assumed that the initial shear modulus is the maximum shear modulus, G , and that the unloading curve is nonlinear and max ° hyperbolic.  Justification for these assumptions is presented in Fig. 5.2  and 5.3 from Byrne et al. (1987) based on triaxial tests by Negussey (1984).  Figure 5.2 shows that the Young's modulus upon unloading is  nonlinear with strain, and Fig. 5.3 shows that the initial unload modulus is equal to the maximum modulus obtained from resonant column tests. Since the Young modulus, E and the shear modulus, G are. related through the Poisson's ratio, v, it is reasonable to assume the same behaviour for the shear modulus. The above indicates that the observed unload response of the pressuremeter can yield the in situ G value if appropriate modifications for nici3C f o stress and strain levels are applied as discussed below. • Gmax and Stress Level Hardin (1978) rproposed that G max for sand can be expressed as r r follows: G = A • F(e) • Pa • (o'/Pa)0,5 max m where: o' m Pa  i i i 1/3(oJ+oJ+oJ)  = the mean effective stress  atmostpheric pressure  (5.1)  300 r LEGEND• • Primary Looding  200 -  A  Unloading  +  Re-Loading  A (E J  ' ij0A0iNG"22OMPa  B (Ei)  R£-LOAOlNG*340MF0  o 0.  C (E  ' 'uNL3AOINGS460MPA Dr » 50%  £  o-j = 250KPa * Emoi,Resonant Column ° ^initiol • Triaxial  0.2  Figure 5.2 Loading and Unloading in a Conventional Triaxial Path (After Negussey, 1984)  0  100  200  3 0 0 400 cr'( KPa)  Figure 5.3 Comparison Between E m (After Negussey, 1984)  500  600  and Various Initial Modulis Ej_  and the parameters A and F(e) depend on particle shape and void ratio, e, as follows: F(e) = (2.17-e) 2/(l+e) -| A = 700 F(e) = (2.97-e) V(l+e) A = 326  n  Rounded sand  (5.2)  Angular sand  (5.3)  Hardin and Black (1966) and Hardin (1978) concluded that G was max independent of deviator stress or stress ratio level depending only on However, more recent test data presented by Yu and Richart (1984) indicates 1 that Gmax depends on an average effective stress, oav that is somewhat e different to a'. In addition, Gmax , also depends on the stress ratio, m i i a 1 /a 3 . Their proposed equation is:  0 5 1 5 Gmax = A • F(e) • Pa • (o* av/Pa) ' (1-0.3k n ')  (5.4)  where: j1 av a P  = (a' + o')/2 a p = the normal effective stress in direction of wave propagation = the normal effective stress in direction of particle vibration  and kn = (o11/o -1) 3)m 3 3-l)/((o 1 13/omax where: i (o,/o.) is the failure stress ratio. 1 3 max  (5.5)  In the above eq. (5.A)  may also be considered as the average  effective stress in the plane in which the strains are induced and therefore in the pressuremeter analysis carried out herein  =  in which o 1 and o' are the effective radial and circumferential stresses, r 6 Equation (5.A) is in good agreement with the results of resonant column tests as shown in Fig. 5.A. It indicates that for a given sand at a given 6  void ratio,' e,' the maximum shear modulus, G max will increase with increased average effective stress, but will decrease with increased i i stress ratio, o,/o,. There will be a 30 percent reduction in G max in zones 1 3  where the stress ratio is a maximum, i.e. where the strength of the sand is fully mobilized. Upon unloading, the sand is assumed to respond in a nonlinear elastic manner as shown in Fig. (5.5).  In the analysis the unload stress-strain  curve is assumed to be hyperbolic with the secant and tangent shear modulus given by:  Gs = Gmax (1 - SL)  (5.6)  G. t = Gmax (1 - SL)*  (5.7)  where: Gmax = the maximum shear modulus, obtained from eq. (5.A) SL = the stress level, which is given by:  SL = (Tl - t)/(Tl + T f )  (5.8)  SAND COMPRESSION EXTENSION  OTTAWA • *  BRAZIL • *  TOYOURA *  Figure 5.4 Measured and Computed G m a x Values. (After Yu and Richart, 1984).  Figure 5.5 Assumed Unload Stress-Strain Behaviour.  where: r^  = the shear stress prior to unloading  t  = the current shear stress  t,  = the shear stress at failure  These shear stresses are shown in Fig. 5.5. The secant shear modulus, G as defined by eq. (5.6) implies a modulus s reduction with stress or strain level as shown in Fig. 5.6. The computed values of modulus reduction from the stress-strain unload-reload loops of Fig. 5.2 are also shown in Fig. 5.6 and are in reasonable agreement with eq. (5.6). Also shown in the figure are the average upper and lower bounds described by Seed et al. (1986).  It may be seen that the equation chosen  lies within the bounds specified by Seed et al. 5.2.2 Analysis Procedure A brief description of the procedures followed in the analysis is presented below. A sketch of the pressuremeter loading and unloading phases is shown in Fig. 5.7.  These two different phases are treated in the analysis as two  separate cases as is described next. From point A to point C the pressuremeter is loading and at point C the stresses mobilized in the sand domain prior to unloading are computed herein using available closed form solutions. From point C to point D the pressuremeter is unloading.  A finite  element axisymmetric plain strain analysis was used herein to evaluate the inward movement, u, of the pressuremeter face at point D.  1.0 Equation  0.8 S  E o »  6  0.6 o 0.4  Data point from Fig. 2  0.2  S e e d et a l . ( l 9 8 6 )  0 icr4  io"3  lo - 2 SHEAR  Figure 5.6 G/G  STRAIN  io~' ( % )  Versus Shear Strain.  Figure 5.7 Sketch of the Loading and Unloading Response of the Pressuremeter.  i.o  Assuming that the soil behaves elastically during unloading, the pressuremeter unload shear modulus, G*, is evaluated using the following equation:  G* = (Aor)face/((2*u)/Ro)  (5.9)  where: (Ao r )^ ace = the decrease in pressure at the pressuremeter face from point C to point D. Rq  = the initial pressuremeter radius  By comparing the computed G* with the in situ maximum shear modulus, G , for various levels of applied radial stress prior to unloading and mdx p o for various amounts of unload, a chart is generated from which G*/G ° max,o can be obtained depending on the applied pressuremeter loading conditions. A detailed description of the procedures followed in the analysis for the loading and unloading phases is presented next. 5.2.2.1  Loading Phase  Prior to loading, at point A (see Fig. 5.7) the in situ maximum shear t modulus Gmax,o is evaluated using& eq. i (5.4). Since a'r = a'0 = a„, i.e., i ai 1 /a 3 = 1 (in the horizontal plane), it follows that:  G  m a v « max,o  = A  '  ' P a 0(o]/Pa)°'5  (5.10)  During loading the stresses in the sand domain change as shown in Fig. 5.8.  Initially the radial stress a' increases while the circumferential  (a)  2 (b)  Figure 5.8 Stress State After Pressuremeter Loading (Elastic-Plastic Model).  stress, OQ decreases.  However, once the failure envelope is reached (at  point B) and a plastic zone develops, o' commences to increase in the y plastic zone and the average effective stress  = (o^+Og)/2 increases as  shown in the figure. As described, the stresses prior to unloading, at point C (see Figs. 5.7 and 5.8) were computed using a closed form solution. The deformations were assumed to occur under plane strain and follow an elastic-plastic stress-strain law. The closed form solutions followed in the analysis has been described by Gibson ad Anderson (1961) , Ladanyi (1963) , Vesic (1972) and Hughes et al. (1977), and herein only the selected equations will be presented. In the plastic zone, which is defined by Mohr-Coulomb failure criterion, the radial and circumferential effective stresses o' and o' are linked ' r 8 by = a\/a\ = tan2 (45 + <f>/2) - N  (5.11)  where: <j>  = the angle of internal friction (assumed to be constant in the analysis)  N  = the failure stress ratio  The outer radius of the plastic zone, R^, (see Fig. 5.8) is given by:  E  p - Rface < ( ^ ) f a c e / < = ; ( H 3 i n « ) ) ( 1 + s i n * ) / ( 2 s i n * '  where: face = the current pressuremeter radius  (5.12)  = the current effective radial stress at the pressuremeter face  The stresses in the plastic zone, (r £ R^), are given by:  °r = °R ' ( V P  r ) 1 N  a' = o'/N 8 r  (5,13)  (5.14)  where the radial stress at the outer radius of the plastic zone, o' is K P given by:  P  = o 0 (l+sin<f>)  (5.15)  Outside the plastic zone or within the elastic zone (r > R ), the P stresses are given by:  2 2 o' r = a'o (1 + (Rp/r )sin<f>)  (5.16)  o' = a'0 (1 - (Rp/rJ)sin<f>)  (5.17)  The above equations describe the stresses induced by expansion of the pressuremeter and these will be used in eq. (5.4) to compute G m a x prior to unloading.  In addition, there may be additional changes in G^^ due to  shear induced volume changes and this will be addressed next. Based upon Hughes et al. (1977), the shear strain distribution f as a function of r in the plastic region is given by:  r  = (R / r ) ( n + 1 ) (U /R ) ( n + 1 ) P P P  (5.18)  where: Up  = the displacement that takes place at the interface between the plastic and elastic zones and is given by:  U p = (R /2G) * o 0 • sinf  n = (l-sin\)) / (l+sin\))  (5.19)  (5.20)  and v  = the dilation angle  Assuming that the dilation angle is constant with shear strain, the volumetric strain is given by:  e v = -y sin\)  (5.21)  and the change in void ratio is given by:  Ae = (1 + e 0 )e v  (5.22)  This change in void ratio was included in eq. (5.4) when computing the value of G m a x prior to unloading.  5.2.2.2 Unloading Phase Upon unloading the whole domain is assumed to behave in a nonlinear elastic manner. However, because the average stress, (o'+o')/2, the stress  ratio, o'/o' and the shear strain, f, prior to unloading are different at r o every point within the domain, G m a x will be different throughout the domain.  In addition the appropriate shear modulus will reduce with the  level of unloading in accordance with eqs. (5.6) or (5.7). Consequently, although the material is assumed to be elastic upon unloading the state of stress is not homogeneous in the elastic zone and hence it is not appropriate to use closed form elastic equations to compute stress changes upon unloading.  Herein a finite element analysis using a plane strain axisym-  metric domain as shown in Fig. 5.9 was used, following the procedures described next: • The initial stresses o^ and o^ in the soil elements were computed for a given pressuremeter stress, (°j)f a c e  us  i n g the closed form solutions  described above. • The shear strains and consequent changes in void ratio were computed from eqs. (5.18) and (5.22), using G = 1/2 G max p o • From these stresses and void ratio changes, G values were evaluated max for each element based upon eq. (5.4). • The stress at the face of the pressuremeter was then unloaded in a series of small steps and a tangent shear modulus corresponding to the average shear stress in each element was computed in accordance to eq. (5.7). • The inward displacement at the face of the pressuremeter (A u )f a c e was computed for each step of unloading (A° r )f a c e  summed to allow the  complete unloading response to be determined. • The equivalent modulus G* was computed at various stages of unloading using eq. (5.9) and compared with G which was computed from the max y o initial stress and void ratio state, using eq. (5.10). The ratio G*/G was then determined for a range of (Ao ), /(d1),. ratios. max,o ° r face r face  f I  R 0 - 38 mm  100 R,  xsqi  J2T  I  P  rM>  '  t S t. 7?  Figure 5.9 Plain Strain Axisymmetric Finite Element Mesh. -t-  The process was then repeated using a range of Co'), values. This r race i allowed G*/G to be computed as functions of both (o'), /o„ r 0 and max,o r face (Aor)_face/(o'), r face as described next, 5.2.3 Results Based on the Self-Boring Pressuremeter field tests carried out at the Amaulikpaq 1-65 site the following range of stress ratios was considered appropriate and was used in the analysis:  3 and  * «°;w°o) * 12  0.0 S ((A orJface - / ( o r' face ) - ) * 0.60 The  results  obtained  from  the  analysis  for  the  different  loading-unloading conditions are presented in a form of a chart in Fig. 5.10 where the factor a p  = G*/G is plotted against the stress ratio c max, o °  i 1 (o')_ 0 for the different stress ratios (Ao )_ r face/o. r face/(or), face i The analyses were carried out over a range of o 0 values as well as a range of void ratio values (.4 < e 0 < .7) and the results were found to be  insensitive to these variables. It was also found that shear induced void ratio effects on Gmax were less than 5% for all loading ° conditions shown in Fig. 5.10. Dilation angles ranging between 0° for loose sands and 16° for dense sands were considered. 5.2.A Validation of the Proposed G*/G  Chart  The proposed analysis presents a method of determining the in situ Gmax , o value from the secant modulus G* from the unload-reload  <Acr r ) F o c e  0.010  II  12  (QFOC,  Figure 5.10 Chart for Determination of G,,,^ 0 from the Measured G* Value (after Byrne, Salgado and Howie I 1990)  pressuremeter loops which considers the variation in stress-strain and void state imposed by the pressuremeter.  The results are expressed in terms of  a single parameter a^which is obtained from the chart of Fig. 5.10 and allows Gmax, o to be determined as follows: Gmax.o = a — P  (5.23)  The method is based upon analytical concepts and idealized soil behaviour and its validation requires comparison with measured data under controlled conditions before application to in situ field conditions. Belloti et al. (1989) present both pressuremeter and resonant column and shear wave velocity tests for both laboratory and field conditions which allow an evaluation of the proposed chart. Their data is used in the evaluation that follows. ^max  va  ^-ues  w e r e  computed from the pressuremeter data using the chart  of Fig. 5.10 and compared with G 6 r velocity tests.  max  from the resonant column or shear wave  The comparison from the "ideal" pressuremeter chamber  tests in which the pressuremeter was inserted prior to placing the sand is shown in Fig. 5.11(a) where it may be seen that G values from the max resonant column tests, G r c , are on average higher than those from the pressuremeter test by a factor of 1.25.  Pressuremeter tests involve  strains in the horizontal plane whereas resonant column tests involve strain in the vertical plane. Bellotti et al. (1989) based on tests (Knox, 1982; Stokie and Ni, 1985; Lee, 1986) indicate that the anisotropic factor a. = G„TT/GtIII = 1.2 in which G,rII = the maximum shear modulus in the vertical A VH HH VH plane and G H „ = the maximum shear modulus in the horizontal plane. Thus  200-1  200-i  160-  160-  a. 120-  o 120Q. E  o  ii  80H  40-  o  CHAMBER TEST IDEAL INST. (CAMKOMETER) 40 80 120 160 Gmox.o (m?Q)  80-  CHAMBER TEST SELF-BORED (CAMKOMETER)  40-  i  Figure 5.11 Relationship Between G m f l X  i  40  80  Gmox,o I m  0  120 P o  160  >  and G r c .  (a) Chamber Test, Ideal Installation (Camkometer) (b) Chamber Test, Self-Bored (Camkometer) (After Byrne, Salgado, and Howie, 1990)  the predicted G^^values from the pressuremeter are in good agreement with the expected G max values for strains in the horizontal plane. c A similar comparison for "self-bored" pressuremeter chamber tests is shown in Fig. 5.11(b).  It may be seen that the G values from the max  resonant column test, G r c > are on average 1.75 times higher than those from the pressuremeter test.  This indicates that the process of self-boring  introduces a disturbance factor, a^ = 1.75/1.25 = 1.4. Comparisons of G  x  values computed from self-boring pressuremeter and  crosshole (CH) seismic tests for field conditions are shown in Fig. 5.12(a) and (b). It may be seen that the G ^ values exceed those computed from the pressuremeter by a factor of 1.58 for the Camkometer and 1.43 for the PAF-79 probe. If the disturbance factor a^ for Camkometer is taken as 1.4, then a. A = 1.58/1.4 = 1.13. The comparison with laboratory and field data indicates that the proposed method can be used to predict the in situ G value provided ni&x | o corrections are made for disturbance and anisotropy.  The maximum shear  modulus for horizontal loading G ^ can be obtained from pressuremeter tests as follows: G  HH " ^  a  D  (5  '2A)  in which G* is the secant modulus from the pressuremeter unloading loop, otpis the factor from the proposed chart shown in Fig. 5.10 and a^ is the disturbance factor = 1.4 for the Camkometer. The Gmax value for vertical loading, G„ with & VHTT, which corresponds ^ seismic crosshole (CH) or downhole (DH) can be expressed as follows:  Figure 5.12 Relationship Between G j n a X f 0 and G c h . (a) In Situ, Self-Bored (Camkometer). (b) In Situ, Self-Bored (PAF-79).  G  VH  = G  CH " G DH " G HH a A  =  ^ VA  (5  '25)  in which a^ is an anisotropic factor. Belloti et al. (1989) suggest a^ = 1.2.  Yan and Byrne (1989) based upon hydraulic gradient model tests and  shear wave velocity measurements found a^ = 1.1, and the test data analyzed herein shows that: a^ = 1.25 (Chamber tests, 'ideal' installation) and a^ = 1.13 (in situ tests, 'self-bored'). Based on the above four a. values an average value (°^) av  =  1.17 is obtained.  This value together with the  proposed chart will be used later (Chapter 6) to evaluate values of G ^ from the SBP tests carried out at the Amauligak 1-65 site. shown that G  It will be  „ values obtained as described here are in good agreement  with G w u values obtained from the cone penetration test (CPT) using VH  empirical correlations.  5.3 Evaluation of Soil Parameters from the First Time Loading Part of the Pressuremeter Tests in Sand The evaluation of soil parameters for use in stress-strain models for sand from the first time loading part of the pressuremeter tests have been restricted for many years to the evaluation of the peak friction angle <p and the dilation angle, v.  However, Manassero (1989) proposed a method  that allows the complete plane strain nonlinear stress and volume change response the sand to be obtained from pressuremeter pressure-expansion data. To verify numerically Manassero's method, F.E. pressuremeter test analyses under plane strain and infinite outer boundary conditions were carried out using the modified SMP model.  Manassero's method was then  applied to the F.E. generated pressuremeter response and the stress-strain  and volume change response for an element at the face of the pressuremeter obtained by his method was compared with the F.E. predictions at the face using the modified SMP model. First, however, before the above analysis were carried out, the modified SMP model was validated against known and controlled pressuremeter chamber test data. This is presented next.  5.3.1 Finite Element Predictions of Pressuremeter Chamber Tests Leighton-Buzzard sand has been tested under controlled laboratory conditions by many researchers.  Simple shear tests on this sand were  reported by Stroud (1971) and pressuremeter chamber tests on the same sand are reported by Jewell et al. (1980). Stroud's data was used earlier to evaluate soil parameters for the modified SMP model and predictions by this model of Stroud's data were shown to be in very good agreement (see Chapter 3).  To further validate  the modified SMP model, F.E. predictions of the pressuremeter chamber tests reported by Jewell et al. were carried out herein using the SMP model. The soil parameters used in the analysis are presented in Table 5.1 below.  Table 5.1 Soil Parameters for Leighton-Buzzard Sand (e0 = .53) for Use in the Modified SMP Model Elastic Parameters  KGe = 620, n = .63 KBe = 580, m = .60  Plastic Parameters: • Hardening parameters  KGp = 330, np = -.48  • Flow rule parameters  li = .20, X = 1.20  • Failure parameters  (t  SMP /o SMP ) i  A(T  SMP / O SMP ) Rp = .957  = =  ,862  -08  A description of the pressuremeter chamber conditions and the results are presented next.  Pressuremeter Chamber Tests A detailed description of the pressuremeter chamber tests is given by Jewell et al. (1980). The test apparatus is shown in Fig. 5.13. The main features of the chamber are that independent horizontal and vertical boundary stresses can be applied to the sides and base of the sand domain using the membranes (1) and (2), and that the lateral movements of the sides were monitored.  Nine pressuremeter tests were reported by Jewell et al.  initial radial stress of 90 kPa was used for all tests. stress was controlled by a pressure applied at the base.  An  The vertical Tests were  conducted using vertical pressures of 200, 90 and 45 kPa and a constant radial outer boundary stress, o r > of 90 kPa. Finite Element analyses of the pressuremeter chamber tests were carried out using the axisymmetric mesh shown in Fig. 5.14 which simulates the geometry and boundary conditions used in the laboratory tests. To study the influence of different boundary conditions, analyses were also carried out using a plane strain axisymmetric domain with an outer boundary at a distance R = .467 m which corresponds with the chamber test as well as an outer boundary which simulates an infinite radius. The finite element mesh used for this analysis is shown in Fig. 5.9 For all the analyses the radial pressure at the inner boundary was increased in small increments and the stresses and deformations examined in the domain. The predicted and observed pressuremeter response in terms of radial stress at the face (a )_  , and circumferential strain at the face,  riftff fffftr tf ft tr  1y  ^  ¥ 1  >  c <r Q Z  O CD UJ or 3 CO  cr  UJ iUJ E 2 LJ E cr O 3  to  UJ E E o. in Q <x  cr  cn oin cn tu  cr  tr CL  UJ  y-  z <  H  V)  z o o  ¥  CONSTANT  VERTICAL PRESSURE  BOUNDARY  426.8mm Figure 5.1A Axi-Symmetric Domain Used in F.E. Analyses.  (£g)^ a c e is shown in Fig. 5.15.  It may be seen that there is generally  good agreement between the predicted and observed response, provided the actual boundary conditions of the chamber test are used in the analysis. The observed response is somewhat softer than computed during the initial loading stage for o = 200 kPa (Fig. 5.15(a)), but is in good agreement z with the compute response for o = 9 0 kPa (Fig. 5.15(b)). The observed z  response for a = 45 kPa was not published by Jewell et al. and therefore z comparisons with the computed results are not presented. The computed responses for a plane strain condition corresponding with the outer boundary at: (i) the chamber test location; and, (ii) at infinity are also shown in Fig. 5.15(a) and are seen to be significantly softer and stiffer, respectively in the high stress range than the computed using the actual boundary conditions. Computed and observed displacements of the inner and outer boundary movements are shown in Fig. 5.16.  It may be seen that for any selected  level of inner boundary movement the computed displacement at the outer boundary are significantly higher than the observed values for the test carried out at o = 200 kPa. However at higher stress levels the computed ratio between the inner and outer boundary movements, i.e. the slope of the line, is in good agreement with the measured values.  It may be also seen  that the computed values of displacement at the outer boundary are highly dependent on oz , being much lower for the lower oz values. Jewell et al. do not show outer boundary movements for these lower a^ stresses but they can be inferred to be much lower from their computation regarding dilation. The strong influences of the vertical boundary stresses, o , on the displacements can be understood from the computed displacement patterns shown in Fig. 5.17.  It may be seen that for o = 200 kPa, upward z  a)  Analysis A  O  & C h a m b e r Test B o u n d o r y C o n d i t i o n s •• P l a n e S t r o i n , C h a m b e r Test Outer Boundary Conditions O P l a n e S t r a i n , I n f i n i t e Outer B o u n d a r y  b)  0  •  2  4  -O  6  CIRCUMFERENTIAL  Laboratory  Measurements  Test B5 A n a l y s i s , C h a m b e r Test Boundory Conditions  8  10  STRAIN , €  12 9  14  (%)  Figure 5.15 Predicted and Observed Response at Face of Pressuremeter. (a) o z = 200 kPa. (b) o 2 = 90 kPa.  2.0  r-  o o  E E  o Laboratory measurements o Analysis  m  >•  CC < Q  1.5-  ID O CD CH  Ld hZ>  o  s Iz Ld  1.0-  Ld  (<_> _J CL  CO  I .0 0. I  I  2  3  4  5  6  7  DISPLACEMENT AT INNER BOUNDARY, mm  Figure 5.16 Displacement at Inner and Outer Boundary.  Figure 5.17 Predicted Displacement Patterns. (a) o z = 200 kPa; (b) o z = 45 kPa  displacement at the bottom boundary is predicted while for o = 45 kPa z downward displacement is predicted.  These vertical movements greatly  affect the predicted horizontal movements, particularly at the outer boundary and suggest that there could be considerable error in applying a plane strain analysis to these tests as was done by Jewell et al. to compute dilation effects. The analysis indicates that the chosen stress-strain model can accurately predict the pressuremeter response under controlled chamber test conditions.  In this case the parameters for the model were first obtained  from simple shear tests. In practice, what is needed is the reverse, i.e., given the pressuremeter response, what are the appropriate stress-strain model parameters for the in situ sand. One could vary the model parameters in a finite element analysis of the pressuremeter and by trial and error determine a set of parameters that gives a best fit to the observed pressuremeter response.  However, Manassero (1989) has presented a method  of obtaining plane strain stress-strain response from pressuremeter tests. A numerical verification of this method using the F.E. data presented in Fig. 5.15(a) for plane strain infinite outer boundary conditions is presented next. 5.3.2 Evaluation of Soil Parameters from the Pressuremeter Test This section is concerned with the evaluation of soil parameters from pressuremeter test data and is subdivided into the following parts: (a) A numerical verification of the method proposed by Manassero (1989) is carried out first. (b) Procedures for the evaluation of soil parameters for use in the modified SMP model from pressuremeter test data are presented last.  5.3.2.1 Numerical Verification of the Method Proposed by Manassero Manassero's method is briefly described in Appendices 5.1 and 5.2. This method was applied to the finite element generated pressuremeter response for plane strain conditions with the outer boundary at infinity. As described, Manassero's method yields the stress-strain and volume change response for an element at the face of the pressuremeter and this is compared in Fig. 5.18(a) and (b) with the finite element prediction at the face using the modified SMP model.  It may be seen that Manassero's method  predicts a stress-strain and volume change response that is in amazing agreement with the modified SMP model. His model predicts values of <f>pS = 45.5° and \> = 13° while the computed values by the modified SMP model are rf)Ps = 45.5° and \) = 14°. An assessment of the actual values for <f>^S and v •P P based on laboratory data is presented in Appendix 5.3. From this assessment values of  46° and v = 15° were obtained and are considered to be  representative of the failure strength and dilation characteristics of Leighton-Buzzard sand for the pressuremeter test stress conditions analyzed herein.  It may be seen that the set of values computed by both the  modified SMP model and Manassero's method agree extremely well with the expected values.  The stress-strain and volume change obtained from simple shear test data and predictions of these using the modified SMP model are also shown in Fig. 5.18(a) and (b).  It may be seen that the predictions are in very  good agreement with the measured data, however, the stress-strain and volume change response computed by the modified SMP model for the pressuremeter and simple shear stress paths is not quite the same. The difference is due to the greater increases in mean stress during the pressuremeter test as shown in Fig. 5.18(c).  The higher confining stresses in the  Shear Strain (%) 2  u V  4  6  8  I  I  1—  10  J  rmnr  a a  (a) PHCSSURCMCTER  QUA  10", » 200KPol  (Plone srrotn infinite Boundary end.)  ec  <s -a V .a r a M O  computed (mod. SUP model) Q  predicted (Uonossero's model)  51UPLE SHE AR  OATA I a, s 172kPo)  measured predicted (mod. SUP model)  s  <i  o •c V  E  9 £  Shear Strain (%)  600 n  S3  J©-  c j3  400  «  +t> t>  C  j3  (c)  T t y «x * * 200-  o4  0  6  i  10  Shear Strain (%)  Figure 5.18 Pressuremeter and Simple Shear Data Versus Shear Strain, y.  pressuremeter leads to lower dilation and hence lower mobilized friction in agreement with the data. The excellent agreement between the Manassero and modified SMP predictions indicates that parameters for the proposed SMP model can be determined from the pressuremeter test data using Manassero's method.  This is  described in detail in the next section.  5.3.2.2 Procedures for the Evaluation of Soil Parameters for Use in the Modified SMP Model from Pressuremeter Test Data The modified SMP model is a 3-D stress-strain model that requires two different types of soil parameters: (a) elastic; and (b) plastic.  • Evaluation of Elastic Soil Parameters From Unload-Reload Pressuremeter Test Data The elastic components are specified in the modified SMP model by tangent shear and bulk moduli, G^ and B^ defined as follows:  G. = KG Pa (o /Pa)n t e m  (5.27)  and B = KB Pa (o /Pa)m t e m where o = (o,+o,+o,)/3, and KG , n, KB and m have been described earlier 1 3 3 m e' e (Chapters 2 and 3). To evaluate KG , the G*/G (or G*/G1II.) chart developed in section r e max,o HH 5.2 will be used here as follows.  The maximum shear modulus for horizontal loading, G^, is evaluated first from the unload reload shear modulus, G*, using the chart presented in Fig. 5.10 together with eq. (5.24), which is repeated below:  G  HH  =  ^  a  (5  D  '28)  Manipulating the above equations (see Appendix 5.4), the following equation is obtained: KGe = (Pa/(Ko ov ) ) 0 , 5 G„ tin u O  (5.29)  where o = the in situ vertical stress, vo To evaluate KBg the following relationship is obtained (assuming that the exponent m = 0.5):  KB = (2KG (l+\>) )/(3 (l-2v)) e e  (5.30)  Assuming that a Poisson'e ratio value of v = 0.1 is representative of the elastic behaviour of sand, it follows from Eq.  KBe = KGe  (5.30) that  (5.31)  • Evaluation of Plastic Soil Parameters from First Time Loading Pressuremeter Test Data Before the method proposed by Manassero is used herein to evaluate plastic soil parameters, it is necessary to develop an additional relationship to estimate the changes of the intermediate principal stress, a 2 during the pressuremeter cavity expansion for the following reasons:  (a) The modified SMP model is a 3-D stress-strain model, i.e. considers the three principal stresses a l t a 2 and o 3 . (b) The method proposed by Manassero is a 2-D plane strain model, i.e. a 2 is not considered. Such a relationship was developed earlier in Appendix 5.3 and is presented in Fig. 5.19 where the variation of b-value = (oa-o3)/(o1-o3) with shear strain, y = ej-e3 is presented.  Because the above relationship  is hyperbolic a relationship between (f/b-value) and y was developed and is presented in Fig. 5.19(b).  It may be seen that a straight line is obtained  for y £ 0.5%. From this figure, the following equation is obtained  b-value = y/(1.2  + 2.83y  )  (5.32)  From the above equation the value of the intermediate principal stress o2 = o v  is easily obtained and given by:  °2 = *  v  = (o -o0)((e -e0)/(1.2+2.83(e -efi)) + ofl r o r o r o o  (5.33)  The above equation is valid for the case when a =o lt otherwise o 2 =o r which is the measured stress. Since Manassero's method computes values of e and o Q from the r y measured o and e a pressuremeter data (see Appendix 5.1) substituting these r y values in eq. (5.33) the value for a 3 can be evaluated. To validate the above, Manassero1s method was applied together with eq. (5.33) to the finite element generated pressuremeter response for plane strain conditions with the outer boundary at infinity for the case of K q = .45.  The computed values of (o )_ , (o ) and (o)„, , by the r r face' v face 0face' J modified SMP model, versus shear strain (f)f ace f° r a n element at the face  LEGEND: Simple S h e a r  Data  Pressuremeter K 0 = 0.45 K 0 = 1.00  Data  Q)  b)  ( b-value) = /(1.2+2.83y)  4  6  8  10  0/  7 ( °)  (a) Variation of b-Value with Shear Strain, y. (b) Variation of (f/b-value) with y.  of the pressuremeter is presented in Fig. 5.20(a) together with the corresponding values predicted by Manassero model and eq. (5.33).  It may  be seen that an excellent agreement is obtained. A comparison between the computed values of ( e r ) f a c e  (£e^face  versus shear strain (Y)f ace i-s presented in Fig. 5.20(b) together with the values predicted by Manassero's model.  As expected the agreement is very  good. Since the soil parameters used in the F.E. analysis are known the data shown in Fig. 5.20 was used to back predict these same soil parameters. This was carried out in Appendix 5.5, and a comparison between the soil parameters used in the F.E. analysis and the backpredicted from the data generated by Manassero's model and eq. (5.33) are presented in Table 5.2.  Table 5.2 Comparison Between Plastic Soil Parameters  Hardening Parameters Flow Rule Parameters Failure Parameters  Deviation (%)  Used in the F.E. analysis  BackPredicted  KGp  335.0  325.0  np  -.480  -.500*  Y  0.200  .195  -2.50  X  1.20  1.220  +1.67  (ISME)I  0.862  0.854  -.93  0.080  0.076  -5.00  0.957  0.976  -2.98 not applicable  °smp A(^smE)  °smp R  F  +1.95  a)  LEGEND « C o m p u t e d , m o d . S M P model P r e d i c t e d , Manassero's model and eq.(5.33)  O A x  cr r , € r a e' € 9  b)  Figure 5.20  (a) Variation of (° r )f ace » ^v^face  a n d  (o 5  e face  (b) Variation of (e r )f ace and ( e 0 ) f a c e with y.  w i t h  r  '  It may be seen that very good agreement is obtained. Based on the above comparisons, it is concluded that, theoretically, soil parameters for use in the modified SMP model can be obtained using the method proposed by Manassero. However, from a practical point of view, the proposed method needs further validation to assess the influence of initial disturbance that might occur at the beginning of in situ SBP tests. Because the method proposed by Manassero (1989) is a recent method and because the SBP tests at the Amauligak 1-65 site were carried out under partially drained conditions (Jefferies, 1988) the first time loading data obtained from these tests was not analyzed herein to infer first time loading soil parameters for use in F.E. analysis of the Molikpaq.  These  soil parameters will be estimated on the basis of both cone penetration test (CPT) data and laboratory data as will be described in the next chapter.  5.4 Summary (a) A procedure for analyzing the unloading response of the pressuremeter has been presented.  The analysis considers the effects of change in  the average stress (o'+o')/2, the stress ratio o'/ol, and shear r 0 r 0 induced volume change on the maximum modulus. Results of the analysis are presented in a chart which allows the in situ, G to be * max,o computed from the equivalent elastic shear modulus G* taking into account both the level of pressuremeter loading and unloading. The predicted G r  max  values from pressuremeter chamber and field  tests using the proposed chart are compared with values obtained from resonant column and crosshole seismic test and are found to be in good  agreement provided factors are included to account for disturbance and anisotropic effects. (b) Finite element predictions of the pressuremeter chamber tests reported by Jewell et al. (1980) were carried out to further validate the modified SMP model.  The soil parameters were obtained from simple  shear test data reported by Stroud (1971) on Leighton-Buzzard sand the same sand used in the pressuremeter chamber tests. The results indicate generally good agreement between computed and  observed  pressure-deflection  relations  at the face of the  pressuremeter provided the actual boundary conditions of the chamber tests are modelled.  The measured response is a little softer at the  initial stages of loading. This may be due to disturbance. The computed displacement patterns in the chamber tests are sensitive to the vertical stress oapplied at the base of the chamber and indicate that plane strain conditions did not prevail in the chamber tests. (c) A procedure for analyzing the first time loading response of the pressuremeter has been developed by Manassero (1989). His method was applied to the finite element generated pressuremeter response for plane strain conditions with the outer boundary at infinity.  An  excellent agreement was obtained between the stress-strain and volume changes, predicted by Manassero's method, and computed by the modified SMP model. (d) Soil parameters for use in the modified SMP model can be determined from poressuremeter test data using Manassero's method provided that: (i) elastic parameters for the model are estimated first from the unloading response of the pressuremeter using the proposed G*/G  chart; and (ii) that Manassero's method is expanded to take into account the intermediate principal stress, o 2 . (e) From a practical point of view (i.e. to interpret in situ self boring pressuremeter tests) the method proposed by Manassero needs further validation to assess the influence of initial disturbance that might occur at the beginning of these tests.  CHAPTER 6 EVALUATION OF SOIL PROPERTIES FOR USE IN THE ANALYSIS OF THE MOLIKPAQ STRUCTURE AT THE AMAULIGAK 1-65 SITE  6.1 Introduction The soil types and their stress-strain parameters used in the analysis of the Molikpaq structure at the Amauligak 1-65 site are described in this chapter.  Two different stress-strain models were used in the Finite  Element (F.E.) analysis of the response of the structure during both the construction and ice loading phases. The hyperbolic model was used in 3-D and 2-D F.E. analysis and the modified SMP model was used only in 2-D F.E. analyses. As will be described later in Chapter 7 these analyses included: (a) a static assessment of the response of the Molikpaq structure during the construction and moderate ice loading phases; and (b) a dynamic assessment of the response of the Molikpaq structure during high ice loading phases. The soil parameters were estimated on the basis of both in situ test and laboratory data.  The in situ test data at the Amauligak 1-65 site  consisted of cone penetration test (CPT) data (Gulf Canada Resources Inc., 1986) and self-boring pressuremeter (SBP) data (Thurber Consultants, 1986). In situ test data obtained at the Tarsiut 1-45 site were also used in the assessment of soil parameters because the in situ conditions at Tarsiut are considered to be reasonably similar to those at Amauligak. The laboratory data used in the assessment consisted of monotonic drained and cyclic undrained triaxial tests (Golder Associates, 1986 and  This chapter is divided into three main sections:  In the first section, a brief description of the on site investigation and construction sequence of the berm and core fills is presented together with a description of the sand used. In the second section, the procedures followed to obtain the appropriate soil parameters for use in the two stress-strain models are presented. Finally, in the third section the procedures followed to obtain the liquefaction resistance curves of the core and berm fills for use in the dynamic assessment are presented.  6.2 Brief Description of the On Site Investigation and Construction Sequence. Type of Sand Used in the Berm and Core of the Molikpaq A good description of Island construction in the Canadian Beaufort Sea (including the Molikpaq) is given by Jefferies et al. (1988). Herein only a brief summary of the investigation and construction sequence at the Amauligak 1-65 site is presented and consists of the following (see Fig. 6.1(a)):  1)  A detailed geotechnical investigation program including sampling, in situ testing and laboratory testing was carried out in order to provide parameters for foundation design (Jefferies and Livinstone, 1985).  This investigation showed that the foundation soil conditions  at the site consist of the following (see Fig. 6.1(b)):  • very soft to soft silty clay 9 m thick, followed by • compact silty sand 5 m thick, followed by • compact to very dense fine sand 35 m thick, followed by  178-A • stiff to very stiff clayey silt 10 m thick, followed by • very hard, frozen sand (permafrost).  Based on MacKay (1972) permafrost as much as 300 to 400 m thick should exist in the southern Beaufort Sea.  Herein considerations of time-  dependent deformations of the permafrost foundation are not considered because the Molikpaq is a temporary structure for oil exploration purposes.  However, if this structure becomes a permanent structure  for oil recovery purposes, then time-dependent deformations (creep) of the permafrost should be accounted for in the long term stability of the structure.  2)  The weak superficial sediments (silty clay) were removed;  3)  The excavation was backfilled and the berm was constructed in several "lifts" by both the "pump-out" and the "bottom discharge" methods. These two methods are well described by Jefferies et al. (1988). Verification of adequate berm density was carried out and consisted of nine CPT's. Molikpaq;  Then the berm was levelled prior to set down of the  2.  EXCAVATE SOFT SURFICIAL SEDIMENT (SUBCUTTING  5.  C O R E FILLING  n f i n T T a  t ™ ::::::  •.•.-............ /  /  >> > 3.  CONSTRUCT BERM  6.  :  ">  ^::::::::::::::::::  :::::::::::::::::::::  VERIFICATION  Figure 6.1 (a) On site investigation and construction sequence (after Jefferies et al., 1985)  WATER TABLE • - 3 . 0 M I — T O P OT CORE E L E V . I . 8 N >  410m +5  0 CAISSON SLOPE  i  -5  -10  £AND|cone  -15  I'.8  -20  -25  SAND B E R M ORIGINAL^ S E A B E D _  SlOYCLAyC«» movant) SILTY  SANO  -30 -35 -40 -45 -50 -55  F I N E SANO (UNFROZEN)  -60 -65 -70 -75  -BO CLAYEY S I L T (UNFROZEN)  r  SAND I F R O Z E N )  Fig. 6.1(b) Soil conditions at the Amauligak 1-65 Site (after Jefferies and Livinstone, 1985).  -85 -90 -95M  ^ >  4)  The Molikpaq was positioned over the berm and setdown by filling its ballast water tanks;  5)  The caisson core was hydraulically filled with sand using a moveable (horizontally and vertically) discharge pipe;  6)  A verification program was carried out from the surface of the sand core.  20 CPT's together with 3 SBP holes (20 tests) were carried out  to verify that the core and the berm had adequate densities.  Determination of the fill quality used in the berm and core was obtained from grain size testing on representative samples of sand. Based on available information the Erksak 320/1 sand is considered to be the type of sand representative of the berm and core fills used at the Amaulikpaq 1-65 site.  It is a uniform, brown fine to medium sand and its  grain size distribtuion is shown in Fig. 6.2.  Index properties of the  Erksak 320/1 sand were reported by Golder Associates (1986) and are presented in Table 6.1.  6.3 Evaluation of the Soil Parameters for Erksak 320/1 Sand This section is subdivided into two subsections.  In the first, the  procedures for the assessment of the different moduli used in the hyperbolic and the modified SMP models are presented.  In the second, the  procedures for the evaluation of the failure angle of internal friction used in the hyperbolic model and of the failure stress ratio on the SMP used in the modified SMP model are presented.  SIZE millimetres  75 40 20 10  BOULDER COBBLE SIZE SIZE  OF SIEVE  5  I | coarse 1 medium I fine GRAVEL SIZE  OPENING f.in\ m i c r o n s  2 5 125* 630 315 160 60  co°r§6  • | lmgdiyrp 1 fine SAND SIZE  M.l.T.  SILT  SIZE  CLASSIFICATION  Figure 6.2 Grain size distribution of Erksak 320/1 sand (after Golder Associates, 1986)  CLAY  SIZE  Table 6.1 Index Properties of Erksak 320/1 Sand Reported by Golder Associates (1986) PROPERTY  DESCRIPTION (or) VALUE  Mineralogical composition (visual inspection)  Quartz, calcite, feldspar and mica  Medium grain size, D 5 0 (vim)  320  Effective grain size, D 1 0 (jam)  200  Uniformity coefficient, C u = D 6 0 / D 1 0  1.6  % passing #200 sieve  0.8  Specific gravity, G g  2.67  Average sphericity  0.67  Average roundness  0.47  Particle shape  Subrounded  .3.1 Evaluation of the Moduli Used in the Analysis The void ratio is a fundamental soil parameter.  Soil moduli such as  the Young1s modulus, E, shear modulus, G, and bulk modulus, B, are highly dependent on the consolidated void ratio, e c > and therefore in situ moduli may be estimated once the in situ void ratio is known by combining it with data from laboratory testing or using it with empirical correlations. The procedure followed for estimating the in situ void ratio is presented below.  6.3.1.1 Evaluation of the In Situ Void Ratio Versus Depth The in situ void ratio was evaluated from the in situ state parameter, \f>, which was obtained from the CPT cone bearing, q c > as it will be described. The state parameter concept was developed by Been and Jefferies (1985) , and is a quantitative measure of the state of the sand that combines both the effects of void ratio and effective mean normal stress in a unique way for each sand. As shown in Fig. 6.3, the state parameter, \|), is a measure of the distance of the consolidated void ratio, e c > from the steady state line. The steady state line represents a condition of zero dilation during shear (Castro, 1969). Once the in situ state parameter, \|>, and in situ effective mean normal stress, a', are known, the in situ void ratio e can be m c evaluated. To evaluate the in situ state parameter the procedures outlined by Been et al. (1986) were followed.  Based on chamber test data Been et al.  conclude that \p can be estimated from CPT data using the following equation: , q -o f = - M n [(-^HVK] m  (6.1)  where:  o' m Kq  = (8.1 - In Xss) = 11.2 = CPT cone bearing ("mean" qc value was considered) = effective mean normal stress = a' v (1+2Ko)/3 = .AO (based on K q study presented in Appendix 6.1)  a' v  = effective vertical stress (based upon r, = 9.5 kN/m3) b  u  = static pore pressure (based upon water level = 3.0 m below top  m  qc  of core, estimated from CP.T soundings)  VOID RATIO, e  e s s l = 0.875 +  Steady state line  XS5= 0 . 0 4 6  '°g,0  Figure 6.3 State parameter and steady state line  { c r  m  )  a m  K  = Total mean normal stress = o1 + u m =  +  ( 8  -  2 3  -  3  ss and Xgs  = slope of steady state line = 0.046 for the Erksak 320/1 sand (Golder Associates, 1986)  Using the "mean" q c value for the core and berm fills shown in Fig. 6.4, which is based upon 20 CPT's carried out in the fills of the Molikpaq (see Fig. 6.5 for CPT location) , a profile of the state parameter versus depth was evaluated and is presented in Fig. 6.6.  Based on the \p values  shown in this figure the in situ void ratio was then obtained following the steps described below: 1)  The in situ void ratio on the steady state line, e is evaluated ss using the following equation (see Fig. 6.3)  e = e - X log., (o') ss ssx ss 6 1 0 m where: e g g  (6.2)  = 0.875 (Golder Associates, 1986) is the void ratio on the  steady state line correspondent to o' = 1 kPa, and the effective mean m normal stress, o^ was computed as described in equation (6.1). 2)  The in situ void ratio, e c > is then evaluated from the equation:  e = e + \p c ss r  (6.3)  The resulting values of the in situ void ratio versus depth are shown in Fig. 6.7.  In order to define different soil types to be used in the  analysis, the profile shown was divided into four depth segments each of  10  Q c ,MPa 20 30 40  50  <A ka> a>  E LxJ  a< iter  z> V)  LU £T O O  £ o LU  CD  X I-  NOTE » LAYERS SHOWN ARE THE L A Y E R S USED IN T H E F I N I T E E L E M E N T ANALYSIS .  CL  UJ  O  Layer 0  Figure 6.4 "Mean" values of q c in the core, berm and foundation (after Jefferies and Livingstone, 1985)  SECTION  i  A-A  i  Figure 6.5 Location of cone penetration tests carried out in the Molikpaq core  STATE PARAMETER -0.2  -0.1  •  0.0  0.1  0.5  INSITU VOID RATIO 0.6  0.7  I  i  0.8  i r\ • \ i •jI  (ec)av.= 685 •j  " CORE  - 1 0 -  V)  0)  El  -20  Ld o  •  £ cr tn  o  o  y  i/  => Ld  j > (ec)av.= 660  -  - 3 0 -  (ec)av.=.620  "BERM \lIif  o  _J Ld CD H I— Q_  -40-  LLJ  Q  -50-  ll 11 II ( \ \ / \  1 1 1 1 1 I  FOUNDATION ,(ec)a».=. 680  5 Figure 6.7 In situ void ratio, e versus depth  which was assigned an average in situ void ratio, (e ) , as shown in the c av e figure. It may be seen that ( c ) a v range from .620 in the berm to .685 in the upper 17.5 m of the core.  In the transition zone of the core and berm  (ec)av = .660 and for the foundation soil (ec)av = .680. 6.3.1.2 Evaluation of the In Situ Maximum Shear Modulus Versus Depth The in situ maximum (low strain) shear modulus, G , was determined max from empirical correlations between void ratio, CPT data, pressuremeter tests, and from direct in situ measurements of shear wave velocity. The values of G as a function of depth obtained from the various max methods described below are shown in Fig. 6.8.  • Hardin and Drnevich (1972): The shear modulus Gmax is a function of the in situ void ratio and effective mean normal stress and is given by the following equation: (2.973 - e )2 o' ~r' m 1'^ G = 320 =— (—) rp max 1 + ec a p *a  where p  a  (6.4)  is the atmospheric pressure,  This equation together with the in situ void ratios from Fig. 6.7 were used to estimate the Hardin and Drnevich values.  • Robertson and Campanella (1984): The G m a x values were obtained from the "mean" cone bearing resistance q c values of Fig. 6.4 following the relationship proposed by Robertson and Campanella (1984) which is shown in Fig. 6.9.  SHEAR MODULUS G m o x , MPa 0  40  80  120  160  200  Figure 6.8 Variation of maximum shear modulus, G m a x with depth  C O N E B E A R I N G , q£ . bars  Figure 6.9 Relationship between cone bearing, q c and maximum shear modulus, G (after Robertson and Campanella, 1984)  Seed and Idriss (1970): *  The shear modulus,' Gmax is 6given by J the  equation: Gmax = 21.7 (KJ p (—) 2 max *a p  where  (6.5)  depends upon the density or normalized standard penetration  resistance of the sand, (N1)60.  Tokimatsu and Seed (1987) suggest the  following relationship:  ^max  =  20[(N 1 ) 60 ] 1 ' 3  The (N 1 ) 60 values were obtained as follows.  (6.6)  First the (N)60 values  were computed using the chart proposed by Seed and DeAlba (1986) that correlates the penetration ratio q c /(N) 6 0 with the mean grain size D 5 0 . This chart is shown in Fig. 6.10. For the Erksak 320/1 sand with D s o = .32 mm, q c /(N) 6 0 = A. 6.  The q c values of Fig. 6.A were used to obtain (N)60  values as a function of depth and these were modified to (Nx)6 0 values using the following equation proposed by Liao and Whitman (1986):  (N^o =  v  * (N)60  (6.7)  The computed values of G m a x vs depth using these (N 1 ) 60 values and Eqs. (6.5) and (6.6) are shown in Fig. (6.8).  • G from Self-Boring Pressuremeter Data: max _  The self-boring °  pressuremeter data obtained at the Amauligak 1-65 site were analyzed using the method described in Chapter 5 and are shown in Fig. 6.8.  The shear  1  1  1  1  I  I  I  V x  o  <0  2 \ 5  —  —  o  x  WL 0 X  o  o  y t  O  a:  5  3  —  » 2 a> 0_ I  X  v  X  X  o  —  +  ^  —  V A  O  Erksak  320 / 1 sand  O J o m i a l k o w s k i et o l . , I 9 B 5 X M u r o m a c h i and K o b a y o s h i , 1982  X  & 0 V +  —  1 tsf = 96 KPo  0 0.01  X  0  0 x  1  I  1  0.02  0.05  0.1  I s h i h o r o o n d K o g o , 1981 Ro be r t s o n , 1 9 8 2 M i t c h e I I , 1985 H o r d e r et o l . , 1984 q in t s f 1 C 1 1  0.2  0.5  —  —  I  I  Mean Grain S i z e , D 5 0 - mm Figure 6.10 Variation of (q c /N 60 ) with mean grain size (after Seed and DeAlba, 1986)  S  modulus G m a x was determined from the secant unloading modulus G* using the chart developed in Chapter 5 and shown earlier in Fig. 5.10. Gmax values determined by of J Golder Associates (1987) from analysis J SBP data from TARSIUT 1-45 are also shown in Fig. 6.8. • Seismic Determination of Gmax : The shear moduli deduced from downhole and crosshole measurements of shear wave velocities at the TARSIUT 1-45 site reported by Golder Associates (1986) are shown in Fig. 6.8. It may J be seen from the G  max  data shown in Fig. 6.8 that there is °  considerable scatter. The solid line, which closely follows the Robertson and Campanella prediction represents the average G  of all the data shown  in the figure and was used in the analysis. • KGmax Versus Void Ratio In the analysis carried out with the modified SMP model the maximum shear modulus was obtained using the following equation which has been already described in Chapter 2: n Gmax = KGmax Pa (^) Pa  From this equation KGmax is obtained as follows: ^ o* KG = G /(Pa(^)n max max Pa  (6.8)  Assuming that n = .5 (common value for sands), and using eq. (6.8) and the G values shown in Fig. 6.8 by values versus depth J the solid line, KG r max max were obtained. From these values, average (KG ) values were obtained max av and are presented in Fig. 6.11. Combining the values of ( e c ) a v a n < *  MAXIMUM SHEAR MODULUS NUMBER ,(KGmax) 0  - 1 0 -  -j  200  400  1 1 I  GV =  600  l—j  800  1000  1200  1400  a-v  CORE  V) ©  "S3 =  -20-  KGmaxa)v=.785  0 £ Z) 01  LU  Dd  -30-  BERM  O O  LU CD JZ  -40-  t— Q_  LU  o  -50-  x)av=850  FOUNDATION  -60-  Figure 6.11 Variation of maximum shear modulus number, (KG_„„) . IllelJt d V  (KGmax)av a relationship between KGmax and ec is obtained and shown in Fig. e 6.12.  In this figure the points representative of the core and berm  (points A, B, C) define a curve that is thought to be representative of KGmax vs ec for the Erksak 320/1 sand. It may be seen that point D,j which j c is representative of the foundation soil was not taken into account to define that relationship, since the data available for the foundation soil is based on 1 single CPT test and also it is thought that the foundation soil is located too far away from the ice loading location for its importance to be relevant in the analysis.  6.3.1.3 Evaluation of the Young's Modulus Two Young's moduli were used in the analysis carried out with the hyperbolic model. One is the tangent Young's modulus on first loading, E , and the other the unloading/reloading or maximum Young's modulus, E  x>  The Young's modulus, E^ was evaluated in the analysis using the following equation developed by Duncan et al. (1980):  n E. (1 - SL R_)' t = KE Pa (o'/Pa) m r  (6.9)  where: KE  = Young's modulus number  n  = Young's modulus exponent = failure ratio, the ratio of the strength from the Mohr-Coulomb  R  F  SL  criterion to the strength from the hyperbola = stress level, the ratio of the mobilized deviator stress  1400-  1200  |  1000-  ID  X  D  800-  600-  400 -  LEGEND A = P o i n t R e p r e s e n t a t i v e of  CORE  B = Point Representative of TRANSION(CORE-BERM) C = P o i n t R e p r e s e n t a t i v e of B E R M 200-  0.5  D = Point R e p r e s e n t a t i v e of FOUNDATION  0.6  VOID RATIO , ec  0.7  0.8  Figure 6.12 Maximum shear modulus number, KG__„ versus void ratio, e UlaJv  The soil parameters KE, n and R^ as a function of void ratio, e c are shown in Fig. 6.13 and were obtained using the drained triaxial test data on Erksak 320/1 sand reported by Golder Associates (1986).  The relevant  data for that evaluation is presented in Appendix 6.2. The maximum Young's modulus, E m a x was evaluated in the analysis using the following equation: E = KE Pa (o'/Pa)n max max m  (6.10)  To evaluate the maximum Young's modulus number, KE , the equation ° max that relates the Young' s modulus with the shear modulus was adapted to the parameters KE max and KG max, i.e. r KEmax = 2 (1+\j)KGmax  (6.11)  The relationship between KE and e is shown in Fig. 6.13 and was ^ max c obtained using equation (6.11), assuming a Poisson's ratio value of \> = .2 and using the values of KG  TH612C  versus e shown in Fig. 6.12. c  6.3.1.4 Evaluation of the Bulk Modulus As for the case of the Young's modulus, two bulk moduli were used in the analysis.  One is the tangent bulk modulus on first loading, B t > and  the other the unloading/reloading bulk modulus, B  . Both B t and B u r  depend on void ratio and effective mean normal stress and were evaluated in the analysis using the following equation:  ra Bu(or Bur) = KB (or KBur) Pa (^) r Bl  (6.12)  4000-1  LEGEND A = Point R e p r e s e n t a t i v e of  CORE  B = Point Representative of TRANSION(CORE-BERM)  3500-  C = P o i n t . R e p r e s e n t a t i v e of  BERM  D = Point R e p r e s e n t a t i v e of F O U N D A T I O N 0 6'3.<B Data x  point ( t r i a x i a l  test-63)  3000-  o £  "D O  2500  KE max  cn UJ  CO 3  z  2000  in  ZD  Q O  1500  V)  O O 1000  O 01 500  <H  0.5  0.6  0.7  VOID RATIO , ec  0.8  where: KB  = bulk modulus number (first loading)  KB u r = bulk modulus number (unloading/reloading)  a' m m  = effective mean normal stress = bulk modulus exponent  The tangent bulk modulus parameters, KB and m for Erksak 320/1 sand are shown in Fig. 6.14 and were determined from the triaxial isotropic compression test data reported by Golder Associates (1986), following the procedures outlined by Byrne and Eldridge (1982).  The relevant data for  this evaluation is presented in Appendix 6.2. The unloading/reloading bulk modulus number, KB u r was evaluated from B^/B relationships which were obtained from test data on Ottawa sand r t ur provided by Negussey (1987). The relationship between  and e c for the  Erksak 320/1 sand is also shown in Fig. 6.14. 6.3.1.5 Evaluation of the Plastic Shear Parameter Gp and the Flow Rule Parameters for Use With the Modified SMP Model As described in Chapter 2 the plastic shear parameter, G^, is evaluated in the analysis by the following equation: G = KG (o'/Pa)np (1 - R_ SL) p p m r  (6.13)  The soil parameters KG^, np and R^ as a function of void ratio, e c > are shown in Fig. 6.15 and were obtained from the drained triaxial test data reported by Golder Associates outlined earlier in Chapter 3.  (1986), following the procedures  HOO-i  LEGEND A = Point R e p r e s e n t a t i v e of  CORE  B = Point Representative of TRANSION(CORE-BERM) C = Point R e p r e s e n t a t i v e 1200  D = Point R e p r e s e n t a t i v e of 0 6 3 = D a t a Point ( t r i a x i a l  of  BERM  FOUNDATION test-63)  1000-  X D  800-  600  400-  200-  0.5  0.6  0.7  VOID RATIO , ec  0.8  U00-  LEGEND 0 6 3 * Data  Point ( t r i a x i a l  test-63)  1200  O .  1000-  cc  UJ  m o: < 800CO o (— (A <  600  400  200-  0.5  0.6  VOID RATIO , e  0.7  0.8  The Flow Rule relationship for Erksak 320/1 sand is shown in Fig. 6.16.  From this figure values of X = .97 and yi = .25 were obtained. The  relevant data used to obtain the above quantities is given in Appendix  6.2.  6.3.2 Evaluation of the Failure Friction Angle and the Failure Stress Ratio on the SMP Relationships between the moduli and void ratio have been established in the previous section. The two additional soil parameters needed are the failure friction angle, <(>-,, to use with the hyperbolic model, and the r failure stress ratio on the SMP (t^/o.^),,, to use with the modified SMP SMP SMP F model.  The procedures followed to evaluate these two soil parameters are  described below. 6.3.2.1 Evaluation of the Failure Friction Angle In the analysis carried out, the failure friction angle was defined using the following equation:  * F « 4>x - M log10(o;/Pa)  (6.14)  where: <px  = peak friction angle at the effective mean normal stress of 1 atmosphere  A<f>  = decrease in <p„ r stress  for a ten-fold increase in effective mean normal  1.2-1 1.1-  10.9-  PL.  a CO D V P-. 33 Ol  0.80.7-  AAD  0.6-  0.7  0.4  O  0.3  • X  0.1  —6-0.1  0.6  A  .2  -0.2  •  Test Test Test Test Test  0.5  0.0  0.1  0.2  0.3  0.4  0.5  p P - AeSMP /Ar SMP  Figure 6.16 Flow rule relationship for Erksak 320/1 sand  -  01 02 61 62 63 0.8  fo o Ln  The above equation is based on the equation developed by Duncan et al. (1980), to define <p p . The only difference is that, herein, o^ is used to define the terra A<f> rather than o^ which is used by Duncan et al. The two soil parameters  and A<f> were evaluated as follows.  A relationship between failure friction angle, <J>F and state parameter, \J), for several sands including the Erksak 320/1 sand, was developed by Golder Associates (1986) and is presented in Fig. 6.17. Using the in situ state parameter values shown in Fig. 6.6 together with the corresponding in situ effective mean normal stress, o', a plot of <!>„ versus log1n (o'/Pa) i 0 m r m was developed and is shown in Fig. 6.18. Data points evaluated from the drained triaxial test data reported by Golder Associated (1986), together with data points representative of the berm fills prior to placement of the core are also included in Fig. 6.18. These berm values were obtained using the "mean" q c data shown in Fig. 6.19. Void ratio values, e c , correspondent to the triaxial test and the berm and core fills are also shown in Fig. 6.18.  It may be seen that the field  and laboratory data plot in agreement with each other and the data points representative of the berm fills before and after the placement of the core show that the failure friction angle decreases with o', as would be m expected, and converge towards a residual friction angle <J>D = 33°. A K similar trend is also observed for the core material and for the three triaxial tests carried out at the lower void ratios.  LEGEND' •  Kogyuk  x  Isserk  o O t h e r B e a u f o r t S a n d s ( S S L estimated Castro (1969) a  S P T Correlation Sands  v  B j e r r u m et a l . , ( l 9 6 l ) Valgrinda S a n d  +  Toyoura  • Erksak  Sand 320/1  C u r v e used in <p' p e a k e v a l u a t i o n  -e-  32 -  28  A * A  -  24 -0.3  -0.2  -0.1  0  S T A T E P A R A M E T E R , i// Figure 6.17 Relationship between state parameter, \J> and peak friction angle, <f> (after Golder Associates, 1986)  M O  Figure 6.18 Relationship between failure friction angle, <f>F and log 10 ((o,;)i/Pa)) for Erksak 320/1 sand  to  O co  V) CD  "a?  E  LLI  Q  O _J  LJ CD  Q_ Ld Q  -12  -14  12  18  qc , mPa  Figure 6.19  "Mean" values of q c in the berm before placement of the core (after Jefferies and Livingstone, 1985)  Based on the results shown in Fig. 6.18 values of <t>1 and A<|> were evaluated for each data point and in turn a relationship of <p versus depth was developed.  (A<f>)  and A<J>  1  Based on these values, average (<f>a)  and  values, corresponding to the same depth intervals used to evaluate  6i V  the average in situ void ratios (e )  , were evaluated and are shown in  C FL V  Fig. 6.20.  It may be seen that (<f>i)av range from 35.7° for the upper 17.5  m of the core to 39.7° for the berm, while the corresponding  range  from 2.8° to 6.1° for the same sand zones. 6.3.2.2 Evaluation of the Peak Stress Ratio on the SMP In the analysis carried out with the modified SMP model the peak stress ratio on the SMP was defined using the following equation developed earlier in Chapter 2.  ^SMP'WF  =  ' W W *  " A ( T SMP / °SMP ) l o g io((°SMP ) F / P a )  (6  '15)  where: (x  SMP/°SMP) 1  = t h e  f a i l u r e  s t r e s s  ratio on the SMP at (°sMp)p  =  1  atmosphere. A ( x S M p / o S M p ) = the decrease in ( T S M p / a S M p ) f o r  (aivra)p SMr r  =  a  ten-fold increase in  (o1 ) v SMP'f the effective normal stress on the SMP at failure.  The two soil parameters (T P M T ,/O c m t 1,) . and A ( T C L U M / O C V M ) were evaluated r SMP SMP SMP SMP as follows:  "  -10-  V)  a)  E  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1"i •  CORE </> L.  E o £ QC Z> 00  o ZD  00 LJ - 3 0 Crl O CJ  -  BERM  3 Ld CD - 4 0 X I— a. Ld Q  Ld  ac  • • •  -20  i  - 3 0 --  BERM  O O ^  o  _J Ld  CD - 4 0 Q_ Ld O  -50-  -60  CORE  -  jl)  • • •  -20-  -10  ! 1 1 1  • • • • • • •  !• FOUNDATON  FOUNDATION i i i i ii  30 32 34 36 38 40 42 44 46 48 50  ( 0, )av. , degrees  • • • • • •  -60-  2  3  4  5  6  7  8  9  ( A 0 )av. , degress  Figure 6.20 Variation of ($j)av and (A<f>)av with depth  10  Using the drained triaxial test data on Erksak 320/1 sand reported by Golder Associates (1986), the relationships shown in Fig. 6.21 between ( T S M p / o S M p ) F and log j 0 ((o^Mp)F/Pa) were obtained, and from it, values of ^XSMP/'0SMP^ 1  anc  * A^TSMP^°SMP^  w e r e  evaluated for each test. Using these  values and the values of <pi and Aif>, earlier obtained for the same tests, linear relationships were obtained between (TgMp/°SMP^i a n d A(T  SMP/OSMP)  anc  *  A<  ^'  These  Finally using the (<f>i)av a n < 3  t w o va  anc  * between  relationships are shown in Fig. 6.22. ^ues  s  ^own in Fig. 6.20 together with  the relationships shown in Fig. 6.22 the values of ((t.m/o.u.),) and bMr SMr 1 av A(x ol . r ./0 o „_,) shown in Fig. 6.23 were obtained. SMP SMP av 6.3.3 Predictions of the Drained Triaxial Tests on Erksak 320/1 Sand In  the previous  two  sections  (6.3.1 and  6.3.2) the following  relationships were established: (a) void ratio versus depth; (b) moduli versus void ratio; (c) A(T  SMP /°SMP )  V E R S U S  DE  and A<f> versus depth; and (d)  i  and  PTH-  To validate the soil parameters obtained from these relationships, predictions of the drained triaxial tests on Erksak 320/1 sand were carried out using the hyperbolic model with the soil parameters shown in Table 6.2  and the modified SMP model with the soil parameters shown in Table 6.3.  0.9 Ut  ai oi  0.8-  P4  s  V)  0.7-  0.6  0.5  100 ({o  SMP } F )/P .  Figure 6.21 Relationship between (TsMP/'0SMP^F for Erksak 320/1 sand  and  l°8io ^^SMP^F^^3^  Figure 6.22(a) Relationship between ("tsMP^SMP^ 1 for Erksak 320/1 sand  and  A<f> Figure 6.22(b) Relationship between M^SMP^SMP^ for Erksak 320/1 sand  anc  * A<f>  -10  CORE  CORE  -10-  v) a)  E  -20-  -20-  UJ  LLJ  o £  o  £  QC  3  DC  in  UJ  QC  -30-  BERM  o o £ O  CD  X I— Q_ Ld Q  3 UJ (/> - 3 0 QC O CJ  BERM  £  O _j UJ  CO  -40  X J— Q_  -40-  UJ  Q  -50-  -50-  -60  —i  0.2  1  0.4  r~  0.6  FOUNDATION  -60-  0.8  0.00  0.05  0.10  0.15  0.20  ^SMP^SMP^av  Figure 6.23 Variation of ((TSMP/'C,SMP^ i^av a n d ^^SMP^SMP^av  w i t h  de  Pth  Table 6.2 Hyperbolic Soil Parameters - Erksak 320/1 Sand  Test # e c  n KB KB m KE KE max ur  r  F  •l  A<f>  <t>  P  .50 230 550 .38 .80 34.4 2.0 33.8  .740 720 1260  02  .717 960 1710 .50 250 610 .38 .80 35.5 2.8 34.4  61  .572 1320 4000  .50 378 1130 .36 .90 45.6 10.8 45.6  62  .552 1648 4600  .50 573 1750 .40 .87 46.4 11.6 39.5  63  .583 1160 3600  .50 474 1420 .38  CD U3  01  44.8 10.0 38.6  Table 6.3 Modified SMP; Soil Parameters - Erksak 320/1 Sand  Test # e KGmax n KB c ur  m KG np P  RF  CSM\A C°SMP ) u TSMP  SMP  X  1  01  .740 525  .5 550  .38 575 -.56 .95  .64  .04  .25 .97  02  .717 710  .5 610  .38 620 -.56 .94  .68  .072  .25 .97  61  .572 1670 .5 1130  .36 910 -.56 1.00 1.01  .340  .25 .97  62  .552 1920 .5 1750  .40 980 -.56 .94 1.04  .362  .25 .97  63  .583 1500 .5 1420  .38 840 -.56 .94  .313  .25 .97  .98  The predictions obtained with the hyperbolic model are presented in Fig. 6.24. It may be seen that the shear behaviour up to peak stress ratio is very well predicted by this model.  However, as may be seen the volum-  etric dilative strains were not predicted.  This is because the hyperbolic  model used in the analysis does not consider dilatant effects. The predictions obtained with the modified SMP model are presented in Fig. 6.25.  It may be seen there is very good agreement between the  measured and computed values of both shear and volume change responses up to the peak stress ratio.  However, since strain softening parameters were  not considered for the predictions, past the peak the predictions are not in good agreement with the measurements of those tests that do show strain softening.  6.3.4 Selection of Soil Types and Soil Parameters to Use in the Molikpaq Analysis The soil types and parameters for use in the Molikpaq analysis for both the hyperbolic and the modified SMP models are tabulated in Fig. 6.26(a) and (b) respectively and were obtained following the procedures described below. Based on the profile of.the in situ void ratio versus depth evaluated earlier, and the existing water level at the time of the April 1987 ice loading event (4.8 m below core surface), five soil types were identified as shown in Fig. 6.27.  Also shown in the figure are the locations of the  different soil layers used in the F.E. analysis. To obtain the different soil parameters shown in Fig. 6.26(a) and 6.26(b), the (e ) value of each soil type obtained from Fig. 6.27 was c av used together with the following figures:  1600-1  U00-  1200  1000  MEASUREMENT 800  600  400  O  PREDICTION TEST-62 (ec=.552 , Sig3=400)  A  PREDICTION TEST-63 (ec=.583 , Sig3=400)  +  PREDICTION TEST-02 (ec=.717 , Sig3=250)  X  PREDICTION TEST—61 (ec=.572 , Sig3=100)  O  PREDICTION TEST-01 (ec=.740 , Sig3=250)  200-  2  4  6  8  10  12  14  16  18  20  22  24  26  28  14  16  18  20  22  24  26  28  AXIAL STRAIN %  61  63 62 02 01  12  AXIAL STRAIN %  Figure 6.24 Predictions of drained triaxial tests on Erksak 320/1 sand using the hyperbolic model  MEASUREMENT  10  12  O  PREDICTION TEST-62 (ec=.552 , Sig3=400)  A  PREDICTION TEST-63 (ec=.583 , Sig3=400)  +  PREDICTION TEST-02 (ec=.717 , Sig3=250)  X  PREDICTION TEST-61 (ec=.572 , Sig3=100)  O  PREDICTION TEST-01 (ec=.740 , Sig3=250)  14  16  18  20  22  24  26  28  20  22  24  26  28  AXIAL STRAIN %  61  ai t—  63  O >  02 01  62  CO  10  12  14  16  18  AXIAL STRAIN %  Figure 6.25  Predictions of drained triaxial tests on Erksak 320/1 sand using the modified SMP model  SOIL  TYPES  SOIL  ^INTERFACES  SOIL ® SOIL  Elevation from MSL (m) T+l-8 ---3.0 ---15.7  rs)  —  20.0  SOIL Q  ---40.0 SOIL© J--60.0  NOTE •• NOT TO SCALE  First Loading Parameters Soil Avg. Type Void No. Ratio te cV 5 .685 4 .685 3 .660 2 .620 1 .680  A<t>  r kN/m3 K 0 15.6 9.7 9.7 9.7 9.7  Repeated Loading Parameters  Degrees  0.4 0.4 0.4 0.4 0.4  35.7 35.7 36.7 39.7 36.5  2.8 2.8 3.8 6.1 3.6  KE 900 900 920 1000 900  n R F kB .50 .80 290 .50 .80 290 .50 .80 295 .50 .80 370 .50 .30 295  m ^max .38 1600 .38 1600 .38 1880 .38 2690 .38 2100  n RF .50 0.0 .50 0.0 .50 0.0 .50 0.0 .50 0.0  750 750 850 1030 750  m .38 .38 .38 .38 .38  Figure 6.26(a) Soil types and parameters for use in the Molikpaq analysis with the hyperbolic model Plastic Parameters Hardening Flow Parameter Rule Yield Parameters Param.  Elastic Parameters Avg. Soil Void Type Ratio No. (e c J av r kN/m 5 .685 15.6 4 .685 9.7 3 .660 9.7 2 .620 9.7 1 .680 9.7  K0  KG  .40 .40 .40 .40 .40  670 670 790 1120 880  max n .raur m .50 .50 .50 .50 .50  750 750 850 1030 750  KG  P  .38 645 .38 645 .38 685 .38 760 .38 650  np -.56 -.56 -.56 -.56 -.56  U .25 .25 .25 .25 .25  T X ( SMP) SMP 1 .97 .685 .97 .685 .97 .715 .97 .815 .97 .710  MJsej SMP .070 .070 .105 .180 .095  R  F  .94 .94 .94 .94 .94  Figure 6.26(b) Soil types and parameters for use in the Molikpaq analysis with the modified smp model.  0-1  I (ec)av.=.685 SOIL (5) «  -10-  i (ec)av.=685 SOIL (4)  CORE  v> Q) <D £ -20 -  i (ec)av.=.660  SOIL (3)  LAYER 7 LAYER 6 LAYER 5 LAYER 4 LAYER 3  Ld  U £  LAYER 2  cc ZD (/) Ld  Cd O O  -30-  £ 3 Ld  m _l_ 1— Q_ Ld Q  (ec)av.=.620 SOIL (2)  BERM  LAYER 1 —40 -  -50-  0.  FOUNDATION (ec)av.=.680 SOIL (1) i  r- " i  "  50 0.55 0.60 0.65 0.70  (ec)av.  !  0.75 0.80 SOIL TYPE NO.  LAYER 0  F. E. LAYER NO  Figure 6.27 Soil types and finite element layers used in the Molikpaq analysis  • Fig. 6.12 to obtain KG max 6.13 to obtain KE and KE • Fig. max • Fig. 6.14 to obtain KB and KB ur ° n -r • Fig. 6.15 to obtain KG 5 P • Fig. 6.21 to obtain <|>1 and A<p ).)1and • Fig. 6.23 to obtain (TSMP /o SMP  A(T  SMP/oSMP  6.4 Evaluation of Liquefaction Resistance Curves for Erksak 320/1 Sand Most of the geotechnical engineering experience with cyclic loading has been developed over the past 30 years based on field data obtained from both earthquake and wave storm events as well as from laboratory tests simulating these events.  This experience has shown that the liquefaction  resistance of sandy soils depends essentially on the following major factors: • Density and grain size characteristics of the soil. • Static stress conditions prior to cyclic event. • Amplitude and number of cycles of loading. • Past history of cyclic loading. • Drainage conditions.  These factors will be discussed next together with a review of the available cyclic loading laboratory tests on Erksak 320 sand.  6.4.1 Review of Available Cyclic Loading Triaxial Tests on Erksak Sand A series of repeated loading triaxial tests were carried out by Golder Associates (1987) on samples of sand obtained from the Molikpaq core after  the dynamic ice load event of April 1986. A review of the above data indicates the following. The data available for predicting liquefaction resistance were from tests carried out with a static shear stress bias corresonding to consolidation ratios K^ = o[/a'3 in the range 2 to 3. Based on the static bias developed by the ice loads, which will be discussed in detail later in Chapter 7, these high K^ values seemed, at first, to be appropriate. However, liquefaction occurred in none of the tests. This fact is contrary to what happened in the field where liquefation was triggered in the soil elements adjacent to the loaded wall. The above contradiction is discussed below. The soil elements adjacent to the loaded wall would initially have a static bias. However in contrast to the triaxial test conditions where the shear stress bias is imposed, the elements of the core that experience a significant increase in porewater pressure will gradually lose their bias during subsequent cyclic loading because the shear stresses initially carried out by these elements will be transferred to other parts of the structure.  This stress transfer  (or reduction in static bias) will  ultimately allow a soil element adjacent to the loaded wall to achieve 100% porewater pressure rise and liquefy. It is understood that additional testing was carried out by Golder Associates where no static bias was considered.  Unfortunately that data  was not available for this study. Based on the above, the liquefaction resistance curves for Erksak 320/1 sand were evaluated based on curves available in the literature for no static bias rather than the available laboratory data reported by Golder Associates.  The procedures followed for that evaluation are described below.  6.4.2 Evaluation of Liquefaction Resistance Curves for Erksak 320/1 Sand Based on no Static Bias The liquefaction assessment chart proposed by Seed and DeAlba (1986) and shown in Fig. 6.28 was used to evaluate the liquefaction resistance of Erksak 320/1 sand. This chart shows the relationship between the modified cone tip r resistance (q ). c 1 and the cyclic stress ratio t eq/o'vo causing liquefaction in 15 cycles and is appropriate for sands and silty sands. The correlations are based on a large body of field data on liquefaction due to earthquake shaking.  It may be seen that the resistance is strongly  dependent on the mean grain size D s o of the sand and the % of fines. For Erksak 320/1 sand D K n = .32 mm and % fines £ 5%. The shear stress, T »o  . is  ©q  the uniform shear stress that would have the same effect in 15 cycles of shaking as the actual irregul ar shear stress, and  is the effective  overburden pressure. The April 1986 ice event involved a much larger number of cycles than 15 and therefore it was necessary to develop a relationship between the shear stress T^ to cause liquefaction in N cycles and T 1 5 which caused liquefaction in 15 cycles. Such a correlation was developed by Been (1988) based on both published data and data obtained by Golder Associates (1984) as shown in Fig. 6.29. Based on the charts shown in Fig. 6.28 and 6.29 the liquefaction resistance curve for the different soil layers used in the analysis were obtained as follows: 1)  The relationship between stress ratio causing liquefaction and (qc)j for 15 cycles and D s o = .32 and % fines £ 5 was obtained from Fig.  6.28.  40 80 120 160 200 240 2 Modified Cone Penetration R e s i s t a n c e , ( q c ) , T o n s / F t  Figure 6.28 Relationship between stress ratio causing liquefction in 15 cycles and modified cone tip resistance for sands and silty sands (after Seed and DeAlba, 1986)  8  TTTT  1  1—I  I I II I I  1  1 | I I  LEGEND  1. 6  Saa^cJ <=t a! (ifB 3) Seea' f/979.) - D r -S4'/. JSatsd - Dr- - <9 2%. (jarga and M^Kay (l^e>4) Or - So'/. Shi bate* et tzl 0t7Z) Toyoura jSand (D r « 3!'/.) Ishiharc* <f WaianciLt (tH7(=-) Miigcia Sand {C Golder K'oyyuk 35o sand (D r - So'/.) (bolder (t<?e>4) IsstsrU 2/o sand (D r •»So-/.)  I . 4  1. 2  r  ' B 4'/.)  = C y c l i c s t r e s s r a t i o to c a u s e l i q u e f a c t i o n in N c y c l e s  0  0.8  0.6  0 . 4  I 1 I M  I I 10  I  I  I I I I II I  N U M B E R OF C Y C L E S TO Figure 6.29.  J  I  M i l l  100  1000  LIQUEFACTION  Relationship between cyclic stress level and the number of cycles to cause liquefaction (after Been, 1988)  to to  cr>  2)  Cyclic shear stress ratios, T n / T 1 5 for N = 5, 15, 100, 500 and 1000 cycles were obtained using Fig. 6.29.  3)  The relationship, obtained in (1), for N = 15 cycles was scaled by the shear stress ratio T n / T 1 5  obtained in (2) using the following  equation:  (6.16) where: Ni  = 5, 100, 500, 1000 cycles  These relationships are presented in Fig. 6.30. A)  The appropriate (q c ) x for each layer was obtained from the "mean" q c profile shown earlier  in Fig. 6.4 and the normalizing equation  proposed by Liao and Whitman (1986): q = Mq (Pa/o' )»'» Xx c vo vo A plot of of the average (q  (6.17)  ((qc)i)av f°r  eac  h layer is presented  in Fig. 6.31. 5)  Using the C(<lc)i)av values corresponding to each soil layer, and using Fig. 6.30 relationships between the stress ratio, x /o1 , and the eq vo number of cycles causing liquefaction was obtained for all the soil layers used in the analysis as is shown in Fig. 6.32. These relationships represent the liquefaction resistance curves for  Erksak 320/1 sand. In the above evaluation the effects of both the past-history of cyclic loading and drainage conditions at the Amauligak 1-65 site were not taken into account due to unavailability of data.  Nevertheless its possible  influence on the liquefaction resistance curves is briefly discussed next.  0.4  0  40  80  120  i  160  l  200  240  MODIFIED C O N E PENETRATION RESIStANCE , (qc) Tsf  Figure 6.30 Relationship between stress ratio causing liquefaction in 5 to 1000 cycles and modified cone tip resistance for Erksak 320/1 sand  MODIFIED CONE RESISTANCE , (qc) mPa o  11 11 111 11 >  -10-  GO Q) "a> - 2 0 -  O £ cn  ZD  00 Ld  CH  -30-  O CJ UJ CD  10  15  20  LAYER 7 ((qc)l)av =4.50 LAYER 6 ((qc) l)a v.—4.50 j LAYER 5 ((qc)l)av.=5.30 i LAYER 4 ((qc)l)av.=5.50 LAYER 3 ((qc)l)av.=7.80 — . ... .. ,1 1 LAYER •1 ((qc)1)av.=13.1 ii ii LAYER 1 ((qc)l)ov.=14.3 • •  -40-  •  Q_ Ld Q -50-  L,4YER 0 ((qc)l)av.—10.8  -60  Figure 6.31 Variation of average (qc)l with depth.  ^av crvo ,  Elev.  Loyer Loyer Loyer L oy er  7 6 5 4  Lo yer 3  2E  ce m  0.2  LU  (m)  + 1.8  - 3.0 - 7.25  -  - 11.5  - 15.75 - 20.0 •  Loyer 2 Layer I FOUNDATION  -30.0 " -40.0-  ( Layer O )  0.  J  I L 10 15  100  500 1000  NO CYCLES TO LIQUFACTION  Figure 6.32 Liquefaction resistance curves for Erksak 320/1 sand  to  LO  O  6.4.3 Discussion on the Past History of Cyclic Loading and Drainage Conditions at the Amauligak 1-65 Site  • Past-History of Cyclic Loading The Molikpaq structure while stationed at the Amauligak 1-65 site was subject to several ice loading events as described by Jefferies and Wright (1988). A summary of these is presented in Table 6.4. It may be seen that  Table 6.4 Summary of Multi-Year Ice Loading Events (Spring, 1986) (after Jefferies and Wright, 1988)  Date/Time Peak Ice Load (MN)  Failure Mode* and Direction  Normalized** Dynamic Amplitude  Comments  March 7 15:30-17:43  230  Crushing at north west (N.W.) faces  16%  -  March 8 17:32-18:37  320  crushing at north west (N.W.) faces same flow  26%  —  March 25 13:00-16:00  110  viscous flow/ buckling at at north face  0%  —  April 12A 08:00-08:45  >500*** crushing followed by ridge break out at east face  45%  April 12B 13:02-13:51  210  crushing at south east (S.E.) face  0%  May 12 03:01-03:26  250  crushing at north face  20 to 45%  June 25 05:31-05:44  130  crushing at west face  Notes:  25%  Liquefaction in the core adjacent to east face -  -  -  * for failure mode definition see Jefferies and Wright (1988) ** 0% indicates static ice loading conditions; 45% indicates dynamic ice loading conditions with an amplitude of ±45% of the average ice load *** Maximum ice load greater than this value (not recorded due to instrumentation failure)  prior to the event under study herein (April 12) the Molikpaq structure was subject to two major dynamic ice loading events. Peak loads of 230 and 320 MN were applied to the N.W. faces of the structure on March 7 and 8 respectively.  The possible significance of these two events was discussed by  Finn et al (1988), their main points are summarized below. Based on simple shear tests Finn et al. (1971) showed that previous cyclic loading had a significant effect on the rate of porewater pressure generation and liquefaction resistance.  Generally, cyclic loading which  generates small shear strains and does not disrupt the structure of the soil, results in greatly increased resistance to liquefaction and a slower rate of porewater pressure development in subsequent loading. Typical effects of previous cyclic loading are illustrated in Fig. 6.33(a) for a medium dense sand and in Fig. 6.33(b) for a loose sand. In each case a virgin sample was subjected to undrained cyclic loading in a simple shear test until the porewater pressure reached 50% of the effective confining pressure.  The samples were then drained and again subjected to  cycles of undrained loading of the same amplitude as previously. Although the void ratio had changed by only a very small amount, there was a marked increase in liquefaction resistance and a much slower rate of porewater pressure development than previously.  The liquefaction resistance of the  dense sand increased from about 17 to 145 cycles and the resistance of the loose sand from about 25 to 65 cycles. Recent investigations by Seed and Lee (1988) have confirmed these earlier conclusions.  They found increases  in liquefaction resistance between 25-35% due to previous loading (see fig. 6.34). This phenomenon is routinely taken into account in the determination of the resistance of North Sea sands to wave loading.  Generally, in  I  {  A = no prior loading history 2.0  TEST 70  (4);,=  ls  CZrcyz >,  2 0  B = prior loading history' V  t .22 V ^  to oj  /  (eo-.  i s  c m i  1 c  / so  eo  soo  NUMtER or cycles  Figure 6.33 Effect of Previous Cyclic Loading on Porewater Pressure Development in (a) Medium Dense, and (b) Loose Sand (after Finn et al., 1970).  Figure 6.34 Effect of Previous Loading History on Liquefaction Resistance (after Seed et al., 1988)  testing these materials, samples are first subjected to a considerable number of small strain cycles to simulate the effect of the milder summer storms before the samples are subjected to the large amplitude stresses and strains considered typical of the winter storms (Bjerrum, 1973). Because test data was not available to determine the possible increase in liquefaction resistance of the sand core of the Molikpaq due to past load events of March 7 and 8, 1986, the curves shown in Fig. 6.32 were not modified. • Drainage Conditions For stability reasons the water level inside the core of the Molikpaq was maintained at an elevation of -3.0 m below mean sea level (msl). This lowering of the water table was developed by a series of water pumps located around the inside perimeter of the structure near the core berm interface. In order to assess correctly the liquefaction resistance and (or) the porewater pressures rise developed in the fills during the dynamic ice loading event of April 12, 1986, the drainage effects developed by those water pumps should be taken into account.  A rigorous solution of the  problem would require a 3-D consolidation analysis in which both generation and dissipation are considered to occur simultaneously. considered in the present study.  That was not  Nevertheless it is considered that the  liquefaction resistance for the soil located near the core berm interface and adjacent to the perimeter of the structure should be higher than that shown in Fig. 6.32. To summarize the liquefaction resistance curves presented in Fig. 6.32 do not take into account the following factors:  a)  initial effects of the static bias developed by the ice loading,  b)  past history of cyclic loading, and  c)  drainage conditions. In addition, the possibility of seasonal frost at the top of the core  sand fill was also not considered herein. Based on the above it is considered that the curves shown in Fig. 6.32 represent a lower bound to liquefaction resistance. This will be taken into account, later in Chapter 7, when assessing the results obtained from the analysis.  6.A.A Pore Pressure Rise The liquefaction resistance curves evaluated in the previous section were used to assess the soil elements that liquefy, i.e. the soil elements in which the ratio between the generated porewater pressure and the effective overburden pressure, u /o' = 1. ^ g vo To evaluate the porewater pressure rise within the sand fill material during the dynamic ice load event of April 1986, two different models were considered.  One to estimate the increase of the residual porewater  pressure and the other to estimate the cyclic porewater pressure. • Residual Porewater Pressure Model: The following equation proposed by Seed et al. (1976) was used in the analysis to predict the residual pore pressure, u /o' = 2/TT sin"1 (N/N n ) 1 / 2 a g vo £ where: u  = the generated porewater pressure  (6.18)  o' r vo = the effective overburden pressure N  = the number of cycles  N^  = the number of cycles to cause liquefaction  a  = an exponent taken to be 0.7  This equation which is valid for cases where there is no static bias such as horizontal ground, is plotted in Fig. 6.35. As discussed earlier the static bias will be initially present during ice loading and could perhaps cause a significantly different initial rise in porewater pressure than assumed by the use of Eq. 6.18.  This was  considered by examining the pore pressure rise data from the Golder tests with K c values in the range 2 to 3 which corresponds to a high static bias. The data from these tests are also shown on Fig. 6.35 and indicate that the initial porewater pressure rise at low values of N/N^ would be a little faster than predicted by Seed's curve.  However this is compensated for in  the analyses herein, by using an Nj value corresponding to zero static bias which is lower. At higher ratios of N/N^, the forced static bias of the Golder tests curtails the rise of porewater pressure.  However, in the Molikpaq as  discussed earlier the static bias drops as the soil softens and hence the porewater pressure will continue to rise. It was therefore considered that Seed's model, used in conjunction with N^ based upon zero static bias, would give a reasonable estimate of the residual porewater pressure rise in the Molikpaq.  • Cyclic Pore Pressures:  Data from the Golder cyclic triaxial tests  described above, indicated that the transient or cyclic component pore pressure could be approximated by the following equation:  I .0  0.8  -  0.6  -  b° \  D> 3  h< tr LU  en  z> 0.4 -  CO CO UJ  tr a.  LLI  tr  o  0.2  Q. 0  0.2  0.4 CYCLIC  0.6  0.8  1.0  R A T I O , N/N^  Figure 6.35 Residual porewater pressure rise as a function of the number of cycles to liquefaction  Au = Aa - 0.7 AT . cy m oct  (6.19)  where: Aucy  = cyclic developed pore pressure  Aom = cyclic variation in total mean normal stress J cy ATQct  = 2/3 [((Ao1-Ao2)/2)2 + ((AOj-AOj)/2)2 + ((A03-A0J/22]1'2 = cyclic variation in octahedral shear stress  CHAPTER 7 3-DIMENSIONAL FINITE ELEMENT ANALYSIS OF THE MARCH 25 AND APRIL 12, 1986 ICE LOAD EVENTS  7.1 Introduction On April 12, 1986 the Molikpaq structure, while stationed at the Amauligak 1-65 site in the Beaufort Sea, was subject to high ice loading (maximum ice load > 500 MN) , dynamic in nature, causing vibrations which were sufficiently severe to induce partial liquefaction of the sand core. Since this is the principal element by which the Molikpaq resists the driving forces of the ice, the ability of the platform to withstand such load conditions was almost compromised (Jefferies and Wright, 1988). To study this complex soil-structure geotechnical problem, five research  groups,  three  in  Canada  (Golder  Associates, Calgary; EBA  Engineering Consultants Ltd.; and the University of British Columbia (UBC)), and two in the United States (University of Southern California (USC); and Wooward-Clyde) were commissioned in 1987, by Supply and Services Canada, to predict the geotechnical response of the Molikpaq to this ice loading event.  The predictions included the assessment of porewater  pressures, accelerations, horizontal displacements and settlement. The writer, due to the topic of his Ph.D. research programme, was part of the UBC research team, which was led by W.D. Finn. The final UBC report was based on the work of several investigators and comprised the different studies listed below. a)  3-Dimensional and 2-Dimensional Structural Model of the Molikpaq. Work carried out by D.L. Anderson, P.M. Byrne and F.M. Salgado.  b)  Assessment of Soil Properties to be used in the Static and PseudoStatic Analysis. Work carried out by P.M. Byrne and F.M. Salgado and presented earlier in Chapter 6.  c)  Assessment of Liquefaction Resistance Curves to be used in the Analysis.  Work carried out by P.M. Byrne, F.M. Salgado and B.  Stuckert. d)  Assessment of soil properties to be used in the dynamic and pseudodynamic  analysis.  Work  carried  out  by W.D. Finn and M.  Yogendrakumar. e)  3-Dimensional and 2-Dimensional Static and Pseudo-Static, Finite Element Analysis of the Molikpaq. Work carried out by P.M. Byrne and F.M. Salgado.  f)  2-Dimensional, Dynamic and Pseudo-Dynamic, Finite Element Analysis of the Molikpaq. Work carried out by W.D. Finn and M. Yogendrakumar. This chapter is divided into three sections and references to the UBC  report are made, when necessary, by referring to the appropriate investigators . An assessment of the ice loading function used in the analysis is presented in the first section. The structural model, finite element mesh and stress-strain law used in the 3-Dimensional analysis are described in the second section. The procedures followed in the 3-Dimensional analysis are presented in the third section together with both the results obtained and the available field measurements. To study the influence on the results of some key parameters, including the stress-strain law used in the analysis, several 2-Dimensional analysis were also carried out.  7.2 Ice Loading Function Used in the Analysis Ice cover is a major environmental feature in the Beaufort Sea. Ice loads used for design are influenced by two major factors: risk of exposure to thick ice and type of the artificial island (Jefferies et al., 1988). During the ice event of April 12, 1986 the ice was approximately 1.5 m thick (Jefferies, 1987).  The failure pressure of thin ice against a  stationary structure whose width is 100 m (such as the Molikpaq) is about 1 MPa.  For this type of structure (vertical ice/island interface) the ice  failure consists of pure crushing whereby intact ice is fractured to a granular material which is then extruded from the failure zone. This type of crushing of ice produces systematic load cycling with an amplitude in the order of 50% of the peak load and frequencies in the order of 1 Hz (Jefferies and Wright, 1988). The given ice loading function for use in the analysis is presented in Fig. 7.1.  It should be noted that this is an idealized version of the  actual ice load which is shown (part of it) in Fig. 7.2.  Because the  actual load was made available after the analysis had been carried out, the following assessment applies only to the idealized load version. It may be seen that from time 8:10:00 to 8:17:51 the ice load increased slowly reaching a peak value of about 59 MN and a trough value of 22 MN.  From  8:17:51 to 8:21:45 the ice load increased very rapidly to a peak value of 397 MN and a trough value of 161 MN.  Between 8:21:45 to 8:26:00 the peak  and trough load remained constant at 397 MN and 161 MN respectively. Subsequently, the load increased rapidly to a maximum load of 500 MN, however, between 8:27:00 and 8:29:10 a data gap is observed. After 8:29:10 the ice load decreased rapidly.  241-A In the analysis the above ice loading function was treated as follows:  a)  8:10:00  to 8:17:51 loading: this low cyclic load level was ignored;  and  b)  8:17:51 to 8:21:45 loading: This phase of the ice loading with its variable dynamic amplitude which lasted for 290 cycles was scaled to 60 equivalent cycles of a uniform dynamic amplitude of 118 MN which is the amplitude of the steady cyclic phase from 8:21:45 to 8:26:00. This greatly simplifies the analysis while preserving the effects of the more complicated load. The scaling technique used is presented in Appendix 7.1 and follows the procedures described by Seed and Idriss (1982).  c)  8:21:45 to 8:26:00 loading: The ice loading function was kept the same and 250 cycles of amplitude = 118 MN are inferred for this interval.  d)  8:26:00 to 8:27:00 loading: The 40 cycles of amplitude = 127.5 MN were scaled to 90 cycles of amplitude 118 MN as discussed in Appendix 7.1  MINIMAL  600  -  o < o  _J  LU  O <  400 -  </) < UJ  200  -  08:10  08.20  08:30  T I M E ( 86/04/12) ,HRS.-MIN  Figure 7.1 Idealized ice loading function used in the analysis  to to  CHANNEL NUMBER CHRNNEL DESCRIPTION G 00,  197 ERST FRCE LORD  STRRT DRTE STRRT TIME  •  12 Rpr 1386 RB:19:R8  /AM 500-  400  300  200  N3 -t-  LO  Time Figure 7.2  in  Minutes  Actual ice loading function.  Based on the above, the given loading function was treated in the analysis as a load with a static component = 280 MN and a dynamic component comprising 400 uniform cycles with amplitude = 118 MN.  7.3 3-Dimensional Modelling of the Molikpaq Descriptions of the structural model, finite element , (F.E.) mesh and stress-strain law used in the 3-Dimensional (3-D) F.E. analysis are presented below.  7.3.1 3-Dimensional Structural Model of the Molikpaq The (3-D) structural model of the Molikpaq's steel caisson was developed by D.L. Anderson (see Section 7.1 of this chapter) and is described below. A 2-Dimensional sketch of the Molikpaq's steel caisson is shown in Fig. 7.3. The caisson acts as a large beam-column and the overall section member properties are also presented in Fig. 7.3.  In order to take into  account both the torsional response and the differing "hoop" forces between the top and bottom of the caisson, Anderson proposed the 3-D structural model as shown in Fig. 7.4.  It may be seen that the 3-D structural model  of the caisson comprises two ring beams connected by truss members.  The  properties of the members of the 3-D structural model are also given in Fig. 7.4 and were chosen so as to match the lateral, torsional and axial stiffnesses of the caisson section shown in the previous Fig. 7.3.  7.3.2 3-Dimensional Finite Element Mesh Used in the Analysis The ice loading was treated in the analysis as being perpendicular to its eastern face and therefore because of symmetry only half of the domain was modelled in the analysis. The ice loading and the 3-D F.E. mesh, which  Caisson Properties  I  =  375 m*  y  =  132 m*  Shear areas:  j*  =  134.5 m*  A  A  s  3.81 m».  A  X  =  7.2 m  X  I y  = 12.4 m = 1.0 m 2  X  E  = 1.8 m J  y  = 200,000 MPa  Figure 7.3 2-Diraensional sketch and properties of the Molikpaq's steel caisson  3D Structural Model Properties Area  Shear Area  Member  A ml  V m1  *  J m*  1 2 3 4  1.31 2.50 4.0 4.0  0.33 0.67 0* 0  0.60 1.20 0 0  46 88 0 0  I  x' m" 54 104 200 0  * 38 74 200 0  *A zero shear area indicates shear deformation is neglected E = 200,000 MPa G - 80,000 MPa Figure 7.4 3-Dimensional structural model of the Molikpaq's steel caisson and properties used in the analysis  comprises 675 soil elements and 204 beam elements, are shown in Fig. 7.5(a). The soil elements were modelled by 3-D brick isoparametric finite elements. The total number of degrees of freedom is 2448. The mesh is 250 meters long in the x direction, 100 m long in the y direction (direction perpendicular to the ice load direction) and 63.5 m in the vertical or z direction. Nodes located on the vertical boundary planes were not allowed to move in the direction perpendicular to those planes, but were free to move in the other two directions.  Nodes on the foundation base were  assumed fixed. The interface between the steel caisson and the sand fills was considered in the analysis as shown in Fig. 7.5(b) where part of the cross-section of the 3-D finite element mesh along the core center line is shown.  The interface elements which were treated as standard 3-D soil  brick elements were assigned a thickness, t = 5 cm and an angle of internal friction, 6 = 20° (Broms, 1966).  7.3.3 3-Dimensional Soil Model Used in the Analysis The stress-strain law used in the analysis is presented in this section which includes a description of the different types of moduli and failure criteria used in the analysis. The complex nonlinear stress-strain relations of the soil elements shown in Fig. 7.5 were modelled in the 3-D analysis, using the hyperbolic model (Duncan et al., 1980) by a 2 parameter incremental elastic and isotropic law using a tangent Young's modulus, E, and a tangent bulk modulus, B, described in Chapter 6.  The tangent Young's modulus depends  upon the initial consolidated void ratio, e c , the mean effective stress, o', m and the mobilized shear stress level, MSL. The tangent bulk modulus, B, depends upon e c and  In addition, both moduli were considered to be  ICE LOAD DIRECTION  Figure 7.5(a) 3-Dimensional F.E. mesh used in the analysis  W <-  -> E  Figure 7.5(b) Cross-section of the 3-Dimensional F.E. mesh along the core center line  dependent upon the loading condition, i.e. whether it is a first time loading or an unloading/reloading condition.  The soil parameters used in  the analysis for the two different loading conditions are presented in Fig. 7.6 and were evaluated as described earlier in Chapter 6. In order to trigger the appropriate moduli for the different loading conditions, an additional soil parameter was included in the stress-strain model used in the analysis.  This parameter, designated as (MSL)max is  defined as the maximum mobilized stress level during the loading history of the soil elements.  The mobilized stress level, MSL, is defined in the  analysis as follows:  MSL =  ^mobilized failure  (?>1)  When MSL < (MSL) the soil elements will respond with the unload/ r max reload or repeated loading parameters shown in Fig. 7.6.  Otherwise the  first loading parameters shown in Fig. 7.6 will be used and the soil elements will follow a hyperbolic stress-strain law. The first time loading parameters were used in the analysis of the construction of the berm and core, and at the end of this loading phase an average value of (MSL) max = .7 was computed and subsequently used for all soil elements. Upon ice loading, MSL first drops in value due to the increase in horizontal normal stress and therefore, in this ice loading phase, the repeated loading soil parameters were used in the analysis.  At higher  levels of loading, MSL increases again and for MSL > (MSL)^^^ a hyperbolic stress-strain law was used in the analysis.  SOIL  TYPES  EUvotion from MSL ( m )  SOIL RT)  ^^^ I N T E R F A C E S  SOIL © SOIL (3)  -r+1.8 3.0  ---20.0  SOIL <2> ---40.0 SOIL©  ---60.0 N O T E : NOT TO  SCALE  First Loading Parameters Soil Avg. Type Void r No. Ratio kN/ K0 3 (e c>av ra 5 A 3 2 1  <t>i  Repeated Loading Parameters  A <p  Degrees  .685 15.6 0. A 35.7 2.8 .685 9.7 0. A 35.7 2.8 .660 9.7 0. A 36.7 3.8 .620 9.7 0. A 39.7 6.1 .680 9.7 0. A 36.5 3.6  KE  n  900 900 920 1000 900  .50 .50 .50 .50 .50  .80 .80 .80 .80 .80  KB  m ^max  290 290 295 370 295  .38 .38 .38 .38 .38  n R F KB u r ra  1600 .50 0.0 750 .38 1600 .50 0.0 750 .38 1880 .50 0.0 850 .38 2690 .50 0.0 1030 .38 2100 .50 0.0 750 .38  Figure 7.6 Soil types and properties used in the analysis  During analysis of the construction phase, the 3-D Mohr-Coulomb failure criterion was used.  However, during the ice loading phase, a 2-D  failure criterion was used.  The 2-D Mohr-Coulomb failure criterion was  selected because the shear T in the direction of ice loading was zx considered to be most effective in controlling strains and displacements and in generating pore pressure. When a soil element fails in shear according to the Mohr-Coulomb failure criterion (MSL = 1) a low value of the tangent Young's modulus given by E* = 0.03 B t was used. This corresponds to a Poisson's ratio, \) = 0.495. The soil may also fail in tension. A soil element is considered to be failing in tension whenever o^ is less than or equal to zero. Low values of the tangent Young's modulus, E*, and tangent Bulk modulus, B*, given by B* = B /10 and E* = 0.03 B* were used. A soil element is considered to have liquefied whenever the cyclic stress ratio (Ax /a' ) is 6greater than or equal to the cyclic resistance J zx vo e ^ ratio (x /o') n . Once an element liquefies, its shear stiffness is eq o 2 ^ considered to be very small.  7.4 3-Dimensional Analysis The following 3-D assessments of the Molikpaq response to different load conditions were carried out: • static assessment during the construction phase of the berm and core; • static assessment during the moderate ice loading phase of the event of March 25, 1986; • pseudo-static assessment during the high ice loading phases of the event of April 12, 1986;  • static assessment of the settlement phase after the ice loading event of April 12, 1986. The procedures followed in the analysis are described next.  7.4.1 Construction Phase Analysis The ideal approach to simulate the construction of the berm and core fills is to "analytically construct" these fills by layers as recommended by Kulhawy et al. (1969).  To do that, the nodes of the F.E. mesh shown  earlier in Fig. 7.5 need to be numbered along horizontal planes, starting from the bottom.  However, such a numbering procedure leads to a system of  equations with a large band width which in turn requires a large computer time for its solution. To overcome the problem, the nodes were numbered along vertical planes, which led to an acceptable band width. However, this restricts the modelling of the sand fills to a gravity "turn-on" type of construction, which simulates deployment in 1 single placement. To minimize the differences between the results that would be obtained by the gravity "turn-on" approach and the ideal approach described above, the following procedures were followed in the analysis. a)  All soil elements were initialized, prior to applying the gravity loads, with moduli based on the following effective stresses: vertical stress, o' = 1/2 (r'h) and horizontal stresses o' = o' = 1/2 (K0n o1), ' z x y z where h is the distance from the centroid of each element to the top of the sand fill. in Fig. 7.6.  The unit weight, y' , and K 0 values used are shown  b)  During the gravity loading the moduli of the soil elements were reformulated at mid-step following the procedures described by Duncan et al. (1980).  Since field observations of the stresses mobilized at the end of construction were not carried out, it was considered important to compare the results obtained from the gravity approach against 2-D plane strain construction analysis where the sand fills were built in 7 layers. The 2-D analyses were carried out using the finite element mesh and the structural model shown in Fig. 7.7. This mesh is an exact replica of the cross-section, along the E-W core centre line, of the 3-D finite element mesh presented earlier in Fig. 7.5. The results of the horizontal and vertical displacements of the structure obtained from the construction analysis are presented in Fig. 7.8(a). It may be seen that the computed vertical displacement of the structure by the "gravity method" is 182 mm (3-D analysis) and 193 mm (2-D analysis) , and by the "layer method" (2-D analysis) is 71 mm. This is because higher shear stresses, T  , are computed by the "gravity method" at the fill-  structure interface than the "layer method".  Nevertheless the horizontal  displacements of the structure, and the vertical displacement of its base relative to the structure inside corner, computed by all analyses are in good agreement as is shown in the figure. The results of the stresses o', o' and T versus depth obtained from r z x zx all the above construction analyses (3-D and 2-D following the gravity method and 2-D following the layer method) are presented in Fig. 7.8 (b,c,d).  It may be seen that the results obtained by the two, 2-D  Beorriv^  1 H  1  72m  PROPERTIES FOR 2-D STRUCTURAL MODEL CONSTRUCTION PHASE (See Appendix 7.2) A = 0.042 mVm  I = 82 m*/m  E = 200,000 MPa  Location, H = 13.5 m  Figure 7.7  2-Dimensional finite element mesh and structural model used in the construction analysis  en  J>  construction methods and the 3-D gravity method agree quite well (except for the T stresses at the fill-structure interface). Based on the above, zx the computed 3-D construction stresses were considered appropriate for use as the initial stress condition in the ice loading phase analysis that are considered in the next section.  7.A.2 3-D Analysis of the Static Ice Load Event of March 25, 1986 Prior to the dynamic ice load event of April 12, 1986, the Molikpaq structure was subject to several other ice load events as described earlier in Chapter 6.  Because the event of March 25, 1987 was the best documented  ice event, several 3-D analyses were carried out to compare with the available field measurements and therefore calibrate the soil-structure para-  («xe»>.4 fc*4h€ v.^iCQl oJi'5^kaccrvirni  SWW)  Geometric SCAUE  Fig. 7.8(a) Displacements of the structure after construction.  DEPTH BELOW CORE SURFACE,meters.  * Oo H<« < 3 rt » n 3* > 1 nrtu a C C CD Wf Ot Trt3 Q. H(0 3O »3* rt3*>a or>  3" hi  .. p.  3 ^ Hn r+  DEPTH BELOW CORE SURFACE,meters. v-ssz  parameters for use subsequently in the 3-D analysis of the dynamic ice load event of April 12, 1986. The procedures followed in the analysis are described below. It is understood that the ice load event of March 25, 1986 was characterized by a static ice load of approximately 110 MN ( see Chapter 6, Table 6.A) applied to the north face of the Molikpaq structure. The load center line was at 18.5 m above the base of the structure or 1 m below mean sea level.  To simulate this ice load, the two 3-D ice pressure distributions  shown in Fig. 7.9 were considered in the analysis because of the uncertainties about the actual ice pressure distribution.  The 3-D load vector  associated with these two pressures was computed as is described in Appendix 7.3. Prior to applying the above load vector, all soil elements were initialized with the stresses that were computed at the end of the construction analysis. Based on preliminary 3-D results the mobilized stress level, (MSL), (see eq. (7.1)), computed at the end of loading (ice load = 110 MN) , decreased in value from the MSL evaluated at the end of construction. Therefore the unload/reload soil parameters shown in Figure 7.6 were used in the analysis which were carried out using one single increment of load = 110 MN. The results obtained using the two different pressure ice load distributions are presented and compared below with the field observations which consisted of displacement measurements carried out, during and after the loading, with the inclinometer and extensometers shown in Fig. 7.10. The measured 3-D caisson deformations due to the 110 MN ice load are shown in Fig. 7.11 together with the results obtained.  From the two sets  NORTH — >  NORTH — >  110 MFK  Possible ice pressure distributions during the ice load event March 25, 1986  PL AH - DECK LEVEL  SECTION <W> TYPICAL 5  :s xTmi  Figure 7.10 Location of the instruments used to monitor the ice load event of 25 March, 1986  Center Line  NORTH  ICE  LOBD  7 \ l /  EAST  LEGEND *  Caisson Prior to Loading Measured with Extensometer Inferred Deflected Shape  DISPLACEMENT SCALE — 10 mm — A — Prediction (Triangular Pressure) — P r e d i c t i o n (Rectangular Pressure) N>  Figure 7.11 3-Dimensional caisson deformations due to the ice load event of 25 March, 1986. Comparisons between field observations and predictions.  of predictions shown in the Figure, the results obtained using the triangular pressure distribution are considered to be the ones that best model the inferred deflected caisson shape (see Fig. 7.11).  It may be seen that the  3-D deflected shape of the caisson is adequately modelled by the analysis except for the NE corner, where the results obtained indicate movements in the opposite direction from the measurements. The measured deformation profile in the core and berm due to the 110 MN ice load is shown in Fig. 7.12. These deformations were measured by the in-place inclinometer located at the centre line and approximately 3 m from the loaded face (see Fig. 7.10). The results obtained from the analysis using the triangular pressure distribution of the ice load are also shown in Fig. 7.12.  Because there  were no nodes at the inclinometer location, displacements were computed at 0 and 6 m from the face and an average value was obtained to assess the displacements at the inclinometer location (see Fig. 7.12). It may be seen that the average of the computed results agree extremely well with the field measurements and both show a remarkable difference between the shear behaviour of the core and the shear behaviour of the berm. This is in agreement with the maximum shear modulus profile selected for use in the analysis and presented earlier in Chapter 6. Based on Jefferis, M.G. (1987), the in-place inclinometer returned to its pre-ice loading position after unloading. This indicates that the core and berm sand fills responded elastically during the ice load event of 25 March, 1986. This is in agreement with the unload/reload set of parameters used in the analysis. Based on the quality of the predictions shown above, it is concluded therefore that the soil parameters shown earlier in Fig. 7.6 are adequate  HORIZONTAL DISPLACEMENTS , m m •30-28 - 2 6 - 2 4 - 2 2 - 2 0 -18 -16 -14 -12 -10 - 8  -6  -4  -2  0  LEGEND MEASUREMENTS P R E D I  C T I O  M  S  AVERAGE OF PREDICTIONS  Figure 7.12 Deformation profile in the core and berm due to the ice load event of 25 March, 1986. Comparisons between field observations and predictions.  for use in the subsequent analysis of the dynamic ice load event of 12 April 1986.  7.A.3 3-D Analysis of the Dynamic Ice Load Event of April 12, 1986 The procedures followed in the 3-D analysis of the above ice load event are described below. Prior to applying the ice loads, all soil elements were initialized with the stresses mobilized due to the gravity loads. The ice load shown earlier in Fig. 7.1, was treated in the analysis as a rectangular pressure on the east face of the caisson.  A triangular  pressure was not considered because during the ice event of April 12, 1986 the full lateral perimeter of the caisson was surrounded by ice which moved from an eastern direction. The 3-D dynamic assessment of the response of the Molikpaq to the ice loading  was  subdivided  herein into three separate  assessments: (a)  liquefaction assessment; (b) porewater pressure rise assessment; and (c) acceleration assessment.  7.A.3.1 Liquefaction Assessment The 3-D response of the caisson to series of cyclic ice loading pulses was determined by computing the static response to one-half cycle of load/ unload.  The cyclic stress ratios so computed were assumed to be the  dynamic values corresponding to a dynamic amplification factor of 1. This is in close agreement with the findings of the 2-D dynamic analysis carried out by Finn and Yogendrakumar (see section 7.1 of this chapter). Once these cyclic stress ratios were obtained the potential for liquefaction was assesed by comparing these stresses with the liquefaction  resistance stress ratios of the Molikpaq sand fills.  These resistance  stresses are shown in Fig. 7.13 and were developed earlier in Chapter 6 based on the cone penetration resistance of the fills, q c , and an extrapolation of the chart developed by Seed and DeAlba (1986). These procedures are described in detail in Appendix 7.A (section 7.A.1). A brief summary is presented below. (1) The east face of the caisson was loaded to 397 MN (see Fig. 7.1). During this phase the soil elements were considered to be drained. (2) Next the east face of the caisson was unloaded back to 297 MN (i.e. one-half cycle of load/unload) and the cyclic stress ratio, evaluated for each soil element by the following equation: a  AT  zx vo  T  - x ZX,  ZX,„, '39 7  ""a 7 9  (7.2)  z279  The numbers 279 and 397 indicate the ice load levels in MN. (3) The soil elements that would liquefy were determined by comparing the /o1 , with the cyclic resistance ratio, J zx vo T /o' , obtained from Fig. 7.13. av vo °  above cyclic stress ratio, AT J  (A) The Young's modulus and bulk modulus of the liquefied elements were now assigned their default values.  The east face of the caisson was  loaded again from 0 to 397 MN and the static analysis repeated. (5) Steps (2) to (A) were repeated for different stages of loading (i.e. different number of cycles), as is described in Appendix 7.A, and the liquefaction assessment updated.  The 3-D liquefaction assessment at the end of the given ice loading is presented in Figs. 7.1A(a) to 7.14(d).  It may be seen that the 3-D  l  av  cr,vo  Elev.  fm)  + 1.8  -  - 3.0 - 7.25  - 11.5  - 15.75 -  0.2  20.0 •  -30.0 FOUNDATION ( Loyer 0 )  -40.0-  0.1  I  I I  5  10 15  100  500 1000  NO CYCLES TO LIQUFACTION  Figure 7.13 Liquefaction resistance curves used in the analyses  f  to  o>  Figure 7.14 3-Dimensional liquefaction assessment: (a) layer no. 6; (b) layer no. 5; (c) layer no. 4; and (d) layer no. 3.  liquefied zone is concentrated around the loaded face and has its greatest horizontal extent near the top where the load was applied.  This is in  agreement with the actual liquefaction areal extent as is shown in Fig. 7.15 where is plotted the observed settlement of the core surface (along the E-W cross-section of the 3-D F.E. mesh) and the computed liquefiable soil for the same location  Since the observed settlement was due to the  dissipation of the porewater pressure from the liquefied soil zone it is concluded  that  a good  liquefaction prediction was obtained by the  analysis.  7.4.3.2 Pore Pressure Rise Assessment During cyclic loading two kinds of porewater pressure are generated in saturated sands. One is cyclic in nature and the other is residual. The procedures followed to assess the soil elements that liquefy (i.e. the soil elements in which the ratio between the generated residual porewater pressure and effective overburden pressure, U /o^ = 1), were described in the previous section. The porewater pressure model developed by Seed et al. (1976) was used to evaluate the residual porewater pressure rise (i.e. to evaluate the ratios Ug/o' vo < 1).  The procedures followed are described in detail in F  Appendix 7.4 (section 7.4.2) where are also described the procedures followed to evaluate the cyclic porewater pressure, AU^. A comparison between the results obtained and the field porewater pressures developed during the ice load event of April 12, 1986 is presented next. The location of the piezometers used to monitor the porewater pressures is shown in Fig. 7.16.  It may be seen that the piezometers El,  oasEweo 6Fm_eMEKrr V  A'V V  4  ice load COMfVTfD LiC60ETiM6LE SO>L-  s  0  &EOAAE.TR ic I I I  I  25  50m  SETILFMEAJT  i  Figure 7.15 Plots of the Observed Settlement and the Computed Liquefiable Soil.  I 2Jr\  t o CT« -4  WATER TABLE » -3.0 I—TOP or CORE ElEV 1.8 VX/////0  •10  i c *///////////////////  •5  0  -5  -10  -10.5 -15.5  -15  -20.0  SAND BERM  DEWATER1NG PORT  -20  -25 ORIGINAL SEABED •• -30 -35 -40 SILTY SAND -45 -50 -55 -60  -65 -TO -75  -BO CLAYEf Ell T -65 (UNFROZEN) -90  I  SAND (FROZEN)  -®5m  SECTION A-A  25.0  29.0  u  •trr MOLIKPAQ CENTRE  J  DIRECTION • OP ICE MOVEMENT  ^-LOCATION OFIPIEZO'-CTERS •  \ SCALE  PLAN  Figure 7.16 Location of Piezometers  0  10 20m  E2 and E3 are located near the loaded face and the piezometers W1, W2 and W3 are located near the trailing face. Before a comparison between the measurements and the computed values is made, a description of the drainage conditions in the core is briefly described below because of its importance in the interpretation of the results. For stability reasons the water level inside the core was maintained at an elevation of -3.0 m below mean sea water level. This lowering of the water table was developed by a series of water pumps located around the inside perimeter of the structure near the base of the caisson (see Fig. 7.16). In order to assess correctly the porewater pressures developed during the dynamic ice loading event, the drainage effects developed by those water pumps should be taken into account.  However, as described earlier,  drainage was not considered in the analysis, but its possible effects should be kept in mind in the interpretation of the results presented below. The maximum residual excess porewater pressures recorded, and predicted by the 3-D analysis following the procedures described earlier are presented in Table 7.1.  Table 7.1 Maximum Residual Excess Porewater Pressures (kPa) El  E2  E3  W1  W2  W3  Measured  £150*  £190*  £40*  =10  =20  =20  Predicted  148  196  103  *Maximum values not registered. Data gap between time 8:27 and 8:29 **No significant values computed.  o**  The excess porewater pressures versus time computed at the locations of piezometers El, E2 and E3 are presented in Fig. 7.17 and a comparison between the measured excess porewater pressure versus time and computed at the location of piezometer El are presented in Fig. 7.18. It may be seen from the results presented in the above table and figures that the maximum residual excess porewater pressures computed from the analysis for the locations of piezometers El and E2 are in good agreement with the field measurements, which indicate that liquefaction was developed at these two locations. The results also indicate that the prediction of the time to liquefaction, at the location of piezometer El, is not correct. The computed time is 8:21 while the recorded time was 8:26.  This fact is not unexpected  because as described earlier the liquefaction resistance curves used in the analysis do not take into account the effects of the initial static bias, the effects of drainage and the effects of the previous dynamic ice loading events that took place at the Amauligak 1-65 site during the time period March 7-8, 1986.  In addition, the analyses were carried out using the  idealized ice loading function presented earlier in Fig. 7.1 which is somewhat different from the actual ice loading of April 12, 1986 presented in Fig. 7.2. The results also indicate that the predictions of the residual porewater pressure for the location of piezometer E3 is substantially higher than the maximum measured value before the data gap took place. This can also be explained by the above considerations. Regarding the piezometers Wl, W2 and W3 no significant residual porewater pressure values were computed from the analysis which is in agreement with the low values measured at those locations (see Table 7.1).  D Q_  E2  200-  DC  3 00 CO Ld  DC  150  Q_  RESIDUAL P.W.P. CYCLIC P.W.P.  DC Ld  S DC  O  E1  LEGEND 1/ jj 0 *  E3  100-  II  III I I I I I II I I II I I I I I I II  Q_  (/) CO Ld  o  ' // Xy ' * • /*XX x/* * / X / // // /*  I  X  50-  II  mrnmmp---  '  /  /  //Z *  /  *  V v *' I I •/ ' I'/' Lf' /' r  t  1/1 ~7  TIME .minutes  \  •  '  '  15  Figure 7.17 Excess porewater pressure values versus time computed at the locations of pieszometers El, E2 and E3  '  Figure 7.18 Comparison between the excess porewater pressure values versus time, measured and computed at the location of piezometer El  Based on the results shown above it is concluded that the procedures followed in the analysis to compute the residual porewater pressures are adequate.  However, future analysis should be carried out with the actual  ice loading and taking into account the effects of the initial static bias, drainage and previous dynamic loading since they can be very important for the evaluation of the exact time to liquefaction.  7.4.3.3 Acceleration Assessment The procedures followed in the analysis to evaluate the accelerations developed by the ice loading function shown in Fig. 7.1, are described in this section. The 3-D response of the caisson to series of cyclic ice loading pulses was determined by computing the static response to one-half cycle of load/unload.  The amplitude of the displacements so computed were assumed  to be the dynamic values corresponding to a dynamic amplification factor of 1.  In addition, assuming that the response to cyclic loading of a  particular frequency was harmonic, the peak acceleration at time t, A(t), was computed by the following equation:  A(t) = X(t) • wCt)2  (7.3)  where: X(t) = the amplitude of the static displacement at time t correspondent to one-half cycle of load/unload u) (t) =the angular frequency at time t The procedures followed in the analysis to evaluate the function X(t) are described in detail in Appendix 7.4 (section 7.4.3).  The above procedures were followed for the evaluation of the accelerations.  The results obtained are presented below together with available  field measurements. The field accelerations developed during the ice load event of 12 April 1986 were monitored with the accelerometers located as shown in Fig. 7.19. However, the available field data is restricted to the accelerations measured, with accelerometer no. TM706 located on the loaded wall, and with accelerometer No. 841 located at centre line on the top of the core. The magnitudes and time of maximum acceleration, measured, and predicted by the 3-D analysis following the procedures described earlier are presented in Table 7.2.  Table 7.2 Magnitude and Time of Maximum Acceleration TM-706  ACC-841  %g  Time  %g  Time  Measured  £10.5  £8:25:36  £5  8:22:30  Predicted  17.0  8:23:00  6.0  8:23:00  A comparison between the accelerations versus time, measured at the location of ACC-841, and computed at this location are presented in Fig. 7.20 and the measured and computed accelerations correspondent to the location of TM-706 are presented in Fig. 7.21. It may be seen from the results presented in the above table and figures that the predictions of the accelerations for the location of ACC841 underestimate the field measurements around the time 8:20 but agree  ACCcLEROkCTER  TM 70S  TOP o r CORE E L . -H.8™ ACCELEROMETER  TM 71 4  I •  f 5  -  S m  / ////  //////y/  ACCELEROMETER 84 1  ///Att*v///y//Ay///// I \ +0.5m  '  1  k € A N  •10 •5 S E A  L E V E L  0 -5 -10  ^///Z>\  ACCELEROMETER 836 * ACCELEROMETER  840  i  Si.NO SERM  Slop?  VACCELEROk«TER  ^ -15 -20  838  O R I G I N A L SEABED  •-  -25 -30 -35  SILTY  -40  SAND  -45 -50 -55 -60  -65 -70 -75  -80 -85 SAND  (FROZEN)  -90  SECTION A - A  SIGN CONVENTION — ACCELEROfcCTER  TM 713  SCALE  PLAN  Figure 7.19  Location of accelerometers  jejim  15-  10-  - 1 0 -  -15-  -20 H  1  0  5  Figure 7.20  •  1 10  «——'  •  1 15  Comparison between the acceleration values versus time, measured and computed at the location of accelerometer no. 841  1  •  •  20  25  TIME .minutes to  J Qi  Figure 7.21 Comparison between the acceleration values versus time, measured and computed at the location of tiltmeter no. 706  TIME .minutes  NJ —I  'quite well with both the maximum acceleration value recorded and the time of its occurrence. On the other hand the predictions of the accelerations for the location of TM-706 agree quite well with the field measurements around time 8:20, but both the maximum acceleration recorded and its time of occurrence are not well predicted by the analysis. This is not unexpected because the computed values of the accelerations were based on the assumption that the response of the Molikpaq to cyclic loading was harmonic which is an extremely crude assumption.  In addition an idealized loading function was  used in the analysis,  7.4.4 3-Dimensional Analysis for the Settlement Assessment After the ice loading phase was completed the east face of the caisson was unloaded to zero ice load and the settlement assessment carried out as follows. The cyclic shear stresses and strains from the ice loading event induce plastic volumetric strains which cause a rise in porewater pressure and subsequently settlement of the sand core. Because testing of the sand core material from which such plastic volumetric strains could be evaluated was not performed, estimates of such strains were made based on the work developed by Tokimatsu and Seed (1987) and Byrne (1990), as is described below. The procedures followed were subdivided into two parts. In the first, an assessment of the volumetric strains associated with earthquakes of magnitude = 7.5 (or 15 cycles) was made based on the work reported by Tokimatsu and Seed.  In the second part an assessment of the volumetric  strains associated with a larger number of cycles than 15 was made based on the work developed by Byrne.  • Assessment of Plastic Volumetric Strains Associated with 15 Cycles of Load A chart showing the expected volumetric strains as a function of both cyclic stress ratio and (N 1 ) 60 value was developed by Tokimatsu and Seed and is presented in Fig. 7.22. Their values are based on laboratory tests and field experience during earthquakes. The material of the core has a density corresponding to a (N 1 ) 60 value in the range of 8 to 12 as is shown in Fig. 7.23 and hence from Fig. 7.22, the likely volumetric strains associated with 15 cycles of load, if liquefaction is triggered, would be in the range 2 1 ' 2 to 3%. If liquefaction is not triggered, the cyclic shear strains and the volumetric strains are likely to be small and may be neglected.  • Assessment of Volumetric Strains Associated with a Larger Number of Cycles than 15 As described earlier, 400 equivalent cycles of load of magnitude 118 MN were assessed for the time period starting at 8:10 and finishing at 8:27(see Fig. 7.1).  As shown in the figure a data gap took place between  8:27 and approximately 8:29 and therefore the total number of equivalent cycles, N, is unknown but is expected to have been > 400. Based on the work developed by Byrne (1990) the ratio between the volumetric strains associated with 400 cycles and 15 cycles, e V  equal to 2.6.  /e *00  , is V  15  Therefore, using the e v values of 2 1 ' 2 to 3% obtained from  Fig. 7.22 the volumetric strains due to 400 cycles are 6.5% to 8.0%. Because N is expected to have been > 400 a value of e^ = 8% was considered in the analysis. To evaluate the settlement associated with the above volumetric srains, these strains were assigned as potential strains to the liquefied  0.5  VOLUMETRIC STRAIN, % 10 5 4 3 2  0.4 cy  cTq 0.3 ' / A R A H A M A ' /  0.2  0  ///  1  H 0 C H 1 N 0 H E , PI  ho H0CHIN0HE,'P6 9// I  / '  /  O ' // ' ^  0  N1IGATA,A  NII,GATA,C  0.  Vs / / /  „ // //  10  20  30  40  50  (N,)60  Figure 7.22 Volumetric strains induced by cyclic stresses and liquefaction (after Tokimatsu and Seed, 1987)  Figure 7.23 Relationship between (N^gg and depth  elements and finite element analysis were carried out to obtain a compatible distribution of the volume strain and the settlement.  The moduli and  load vector used in the above analysis are described in Appendix 7.5. The settlement results obtained following the above procedures are discussed below together with the field measurements. The settlement measurements of the top of the core surface along its E-W centreline are presented in Fig. 7.24 together with the settlement results computed from the analysis based on a volumetric strain value, e^ =  8%. It may be seen that both the trend and the magnitude of the predictions agree quite well with the field observations. Because the settlement areal extent is closely associated with the liquefied soil zone extent, it is concluded that the zone of liquefaction that occurred during the ice loading event of 12 April 1986 was adequately identified by the analysis procedure carried out herein. In addition, inclinometer observations were also carried out to assess the residual deformations of the sand fills of the Molikpaq structure. These observations were carried out in the inclinometer located as shown in Fig. 7.25. The displacement profile measured in the inclinometer located adjacent to the west wall of the caisson and the corresponding computed displacements from the analysis are presented in Fig. 7.26. It may be seen that the predictions underestimate the field measurements. This could be associated with the fact that a Poisson's ratio value of v = 0.0 was used in the analysis (see Appendix 7.5).  WEST WALL  Cenier Line •V\  EAST WALL \  \ • N \\  ' y^MEASUREMENTS PREDICTIONS A \\  \v \\ VV \\ N s \ V \  o  Distance, meters  Figure 7.24 Comparison between the settlement, measurements and predictions of the top of the core surface  TOP OF CORE E L . +1.8m  PLAN  Figure 7.25 Location of the inclinometer used to measure the residual horizontal displacements  DISPLACEMENT , m m •25  -20  -15  -10  -5  0  Legend PREDICTIONS. _  MEASUREMENTS  Figure 7.26 Comparison between the residual horizontal displacements, measured and computed at the location of the west side inclinometer  7.5 2-D Finite Element Analysis of the Dynamic Ice Load Event of April 12, 1986 To study the influence on the results of some key parameters, several 2-D analyses were carried out. A brief description of the key parameters studied is presented below. The details of the analysis are described in Appendix 7.6 and the main conclusions are presented afterwards,  7.5.1 Description of the Key Parameters Studied in the 2-D Analysis (a) Influence of the Interface Element Type and the Value of the Angle of Friction, 5 The interface between the steel structure and the sand fills was modelled in the 3-D analysis by a standard solid isoparametric brick element characterized by an angle of friction, 6 = 20°. To study the influence of the interface element type in the results, 2-D analyses were carried out using both the standard isoparametric element and the "thin" interface element that was described in Chapter A. For both cases a value of the angle of friction 6 = 20° was considered (see Apendix 7.6). To study the influence of 6 in the results, additional 2-D analyses were carried out using the "thin" element with values of 6 = 0 and 6 = <J>. The main conclusions from the above studies are presented later in section 7.5.2.  (b) Influence of the Stress Redistribution Method The procedures followed in the 3-D analysis to redistribute the shear stresses of the liquefied soil elements to the adjacent soil and structural elements was described earlier. This stress redistribution method is con-  sidered to be an adequate method, but not the only method. An alternative method to the above was also considered in the 2-D analysis and consisted briefly on the following. Once a soil element is identified as having liquefied its shear strength is assigned a low value corresponding with its residual strength, su . The amount of shear stress, AT that exceeds su is redistributed to the adjacent stiffer elements using the load shedding procedures developed by Byrne and Janzen (1984) and described earlier in Chapter 2. To study the influence of the two different methods of stressredistribution, in the results, 2-D analyses were carried out using both methods as is described in Appendix 7.6.  The main conclusions of this  study are presented later in section 7.5.2.  (c) Influence of the Constitutive Law The 3-D analysis of the Molikpaq were carried out following a combination of an elastic and hyperbolic stress-strain laws as described in section 7.3.3.  To study the influence of the stress-strain law in the  outcome of the results, 2-D analysis were also carried out in Appendix 7.6 using the modified SMP model which was developed and presented earlier in Chapter 2.  Ideally, the above items would have been studied with a 3-D analysis, but either due to the computer time required or due to the shortage of computer memory space, the influence of the above had to be studied with 2-D analysis. below:  The main conclusions from the 2-D analysis are presented  7.5.2 Conclusions from the 2-D Analysis a)  The same initial response of the Molikpaq loaded wall movement was computed by the analysis carried out using the "Thin" interface element and the "Standard" soil element for the same 5 = 20°. However at ice load levels of 400 MN the displacements obtained with the "Thin" element are shown to be about 85% of that computed by the "Standard" element.  b)  The same response of the Molikpaq loaded wall was computed by the analysis when using the "thin" interface element with 6 = 20° or 6 =  <(>.  For  the case 6 = 0° however, the displacements of the loaded wall  are shown to be about 10% larger than those computed by the above analysis. c)  The same initial response of the Molikpaq loaded wall was computed by the analysis carried out using the modified SMP model and the hyperbolic model. However, at ice load levels of 400 MN, the displacements computed by the modified SMP model are shown to be about 82% of that computed by the hyperbolic model. This difference is related with the Mohr-Coulomb failure criteria that is used in the hyperbolic model and also because this model does not compute increases in mean normal stress, Ao due to increases in shear stress AT m zx  d)  Similar responses of the Molikpaq back wall movements were computed from all analyses except for the movements computed by the analyses where the shear stresses of the liquefied soil elements were redistributed following the "load shedding" method.  This method computed  larger movements for the back wall than that computed from the stress redistribution method used in the 3-D analysis.  e)  Essentially the same 2-D liquefaction areal extent is computed from all analyses and coincides with the 3-D liquefaction assessment for the location of the E-W cross-section.  7.6 Summary and Conclusions An analysis procedure for a caisson-retained island type structure was described and used to predict the response of Gulf's Molikpaq structure to the ice load event of April 12, 1986. The liquefaction assessments from the 3-D analysis indicate that the computed liquefaction areal extent is in good agreement with the field liquefaction assessment based on porewater measurements and settlement observations. The results obtained from the 3-D analysis also indicate that the overall predictions of porewater pressure and accelerations agree well with the field measurements except for the time to liquefaction and the magnitude and time of occurrence of the maximum accelerations. These discrepancies are attributed to the idealized ice loading function used in the analysis and to the influence of the initial static bias, drainage and previous loading history which were not considered in the analysis. In addition, several 2-D analyses, which considered different types of interface elements ("thin" and "standard" elements) and constitutive models (modified SMP and hyperbolic models) indicate that the computed liquefaction areal extent is practically insensitive to the above parameters and in good agreement with that computed by the 3-D analysis. Based on the above it is concluded that both the procedures followed to obtain soil parameters and the analysis procedures followed in the 3-D analysis are adequate procedures for design purposes, and that the influence of the above 3 factors must be considered in future analysis.  CHAPTER 8 SUMMARY AND CONCLUSIONS  A procedure to analyze the response of large offshore drilling platforms to the high ice loading conditions of the Beaufort Sea has been presented in this thesis.  These platforms comprise a large steel box  infilled with a sand core for stability against high ice loading. One such structure was subjected to very severe ice loading, and being the response monitored, it allows a case study against which the proposed procedures were checked. To analyze the behaviour of these highly complex soil-structure interaction systems a 3-D F.E. computer program with soil, interface and structural elements was developed in this thesis.  The stress-strain  relations and the evaluation of the stress-strain parameters from in situ testing as well as interface elements were considered the key aspects of the analysis and were considered in detail.  These aspects together with  the analysis of the Molikpaq structure are summarized in this chapter.  8.1 3-D Constitutive Law for Sands The 2-D hyperbolic model developed by Duncan and Chang (1970) and Duncan et al. (1980) was expanded to 3-D and implemented into F.E. formulation. This model does not account for the dilation characteristics of the sand material and to account for this an additional dilatant parameter based on Byrne and Eldridge (1982) was expanded to 3-D and also implemented into F.E. formulation. However, preliminary analysis of simple shear tests on Ottawa sand indicate that the predictions obtained by the hyperbolic  model (with and without dilatant parameters) considerably underestimate the failure shear stresses measured in those tests. Based on the above, and because the simple shear stress path is the path most closely followed in the sand fills of the Molikpaq due to the horizontal ice loading on this structure, a review of the existing 3-D stress strain models was carried out in chapter 2 to select the most appropriate one. Special emphasis was focussed on the 3-D yield criterion, and stress-dilatancy theory of the models. From the review, presented in Chapter 2, the 3-D model developed by Matsuoka (1974,1983) following the concept of the Spatial Mobilized Plane (SMP) was selected.  This model was modified by the writer to make it more  practical, and, take into account the rotation of principal axes during the simple shear test.  The model so developed is called the modified SMP  model. Particular attention was addressed in this thesis to the development of procedures to evaluate soil parameters for use in the modified SMP. In all, eleven soil parameters (4 elastic and 7 plastic) are required. It was shown that these soil parameters can be obtained from the following sources: (a) Laboratory tests (Chapter 3); (b) Pressuremeter tests (Chapter 5); and (c) Laboratory and cone penetration tests (Chapter 6). The performance of the modified SMP model was evaluated by comparisons with laboratory measurements, pressuremeter chamber test measurements, and in situ measurements obtained from field tests. The laboratory data selected for the comparisons was obtained from simple shear test data on Leighton-Buzzard sand, simple shear, and true-  triaxial test data on Ottawa sand.  In addition comparisons were also made  against triaxial test data on Erksak 320/1 sand. From the comparisons between the predictions and the reported data the following is concluded: • Both the 3-D and 2-D plane strain formulations of the proposed model can reproduced well the reported simple shear test data on Leighton-Buzzard sand.  This indicates that: (i) the model takes into account he gradual  rotation of the axes of principal stresses and strains that occur during that test; and (ii) the 2-D formulation which is derived from the 3-D formulation by applying the appropriate boundary conditions give a good prediction of the intermediate principal stress o 2 . • The overall good predictions of the triaxial tests on Erksak 320/1 sand and both the simple shear and true-triaxial on Ottawa sand with the exception of the circular path test further indicate that the proposed model is able to predict the behaviour of sand with reasonable accuracy for the stress-paths of practical importance for the Molikpaq structure. • Because the circular path test is not representative of the stress paths that occur in the sand fills of the Molikpaq structure during either the construction or ice loading phases, the reasons of the poor predictions of this test by the modified SMP model were not investigated herein. To further validate the modified SMP model, F.E. predictions of pressuremeter chamber tests on Leighton-Buzzard sand were carried out as described in Chapter 5. The parameters for the model were determined from the simple shear test data on Leighton-Buzzard sand (Chapter 3) , the same sand used in the pressuremeter chamber tests. From the comparisons between the pressuremeter chamber test measurements and the F.E. predictions the following conclusions are made:  • The results indicate generally good agreement between computed and observed pressure-deflection relations at the face of the pressuremeter provided the actual boundary conditions of the chamber tests are modelled. The measured response is a little softer at the initial stages of loading. This may be due to disturbance. •  The computed displacement patterns in the chamber tests are  sensitive to the vertical stress oapplied at the base of the chamber and indicate that plane strain conditions did not prevail in the chamber tests.  8.2 Evaluation of Stress-Strain Parameters of Soil from Laboratory and/or In Situ Testing Particular attention was addressed in this thesis to the development of procedures to evaluate soil parameters for use in the modified SMP model and hyperbolic model.  The current methods of soil parameters evaluation  were reviewed, some expanded and applied in this thesis. It was shown that the soil parameters for use in these two models can be obtained from the following three sources: i)  Laboratory tests (Chapter 3);  ii) Pressuremeter tests (Chapter 5); and iii) Laboratory and cone penetration tests (Chapter 6).  8.2.1 Evaluation of Soil Parameters from Laboratory Tests It was shown in Chapter 3 that the laboratory data obtained from any test, including the standard triaxial test, that measures the three principal stresses and strains can be used to evaluate soil parameters for use in the modified SMP model.  The procedures used to evaluate the shear  (elastic and plastic) and failure parameters follow closely those developed by Duncan et al. (1980) to evaluate the E moduli and failure parameters used in the hyperbolic model.  The procedures used to evaluate the elastic  bulk parameters follow those developed by Byrne and Eldridge (1982) to evaluate the bulk moduli used in the hyperbolic model and the procedures used to evaluate the flow rule parameters of the model follow those developed by Matsuoka (1983).  8.2.2 Evaluation of Soil Parameters from the Pressuremeter Test The current methods to infer soil parameters from the unloading and first time loading pressuremeter test data were reviewed, expanded and applied in this thesis.  A summary of the work carried out is presented  below: •  A procedure  for analyzing  the unloading  pressuremeter was presented in Chapter 5.  response of the  The analysis considers the  effects of change in the average stress (o'+o')/2, the stress ratio o'/o', r b r o and shear induced volume change on the maximum modulus. Results of the analysis are presented in a chart which allows the in situ, Gmax,o to be J r computed from the equivalent elastic shear modulus G* taking into account both the level of pressuremeter loading and unloading. The predicted G r  max  values from pressuremeter chamber and field tests  using the proposed chart were compared with values obtained from resonant column and crosshole seismic test and are found to be in good agreement provided factors are included to account for disturbance and anisotropic effects. • The proposed chart was also used in Chapter 6 to evaluate G c c r max values from the SBP tests carried out at the Amauligak 1-65 site. These  values were shown to be in good agreement with the G m a x values obtained from the cone penetration test (CPT) using empirical correlations. • A procedure for analyzing the first time loading response of the pressuremeter has been developed by Manassero (1989).  His method was  applied to the finite element generated pressuremeter response for plane strain conditions with the outer boundary at infinity.  An excellent  agreement was obtained between the stress-strain and volume changes, predicted by Manassero's method, and computed by the modified SMP model. • Soil parameters for use in the modified SMP model can be determined from pressuremeter test data using Manassero's method provided that: (i) elastic parameters for the model are estimated first from the unloading response of the pressuremeter using the proposed G*/G chart; and (ii) max y o that Manassero's method is expanded to take into account the intermediate principal stress, a 3 . •  From a practical point of view  (i.e. to interpret in situ  self-boring pressuremeter tests) the method proposed by Manassero needs further validation to assess the influence of initial disturbance that might occur at the beginning of these tests. 8.2.3 Evaluation of Soil Parameters from Laboratory and In Situ Testing The stress-strain parameters for Erksak 320/1 sand used in the analysis of the Molikpaq structure at the Amauligak 1-65 site were evaluated following the procedures described in Chapter 6. The soil parameters were estimated on the basis of both in situ test and laboratory data.  The in situ test data consisted of cone penetration  test (CPT) data, self-boring pressuremeter (SBP) data, and direct shear measurements of shear wave velocity.  The laboratory data used in the  assessment consisted of monotonic drained triaxial tests.  The following three types of soil parameters were evaluated from the above data: a)  Moduli. These were subdivided as (1) repeated loading moduli, and, (2) first time loading moduli.  b)  Failure Parameters.  c)  Liquefaction resistance curves.  8.2.3.1 Summary of the Procedures Followed to Evaluate the Moduli Used in the Analysis One of the key parameters used in the procedures to define the different soil types and moduli used in the analysis was the in situ void ratio, e c >  This parameter was evaluated from the in situ state parameter,  , which was obtained from the CPT cone bearing, q c , following the procedures developed by Been et al. (1986).  Soil moduli such as the  Young's modulus, E, the shear modulus, G, and bulk modulus, B, and the plastic shear parameter, G^, are highly dependent on the consolidated void ratio, ec . Therefore once the in situ void ratio, ec was known, the in situ moduli and G^ could be estimated by combining the in situ void ratio, e c , with the existing laboratory data. Another key parameter used in the analysis was the in situ maximum shear modulus , G . This modulus was determined in Chapter 6 from r max empirical correlations between G and void ratio (Hardin and Drnevich, r max 1972), G and cone bearing q (Robertson and Campanella, 1984), and G IT13.X C JIleLX and K 2 r a a x ( S e e d a n d Idriss, 1970). The unloading SBP data obtained at the Amauligak 1-65 site was also used to evaluate G using the G*/G chart max max developed in Chapter 5. In addition the G max values determined by Golder r Associates (1986,1987) at the TARSIUT 1-45 site from SBP tests and shear  wave velocities measurements were also used to assess the in situ maximum shear modulus, G . The various method gave rise to considerable scatter 6 ' max obtained in the plot of G versus depth, and an average G was used in JT13.X IHdX the analysis. 8.2.3.2 Summary of the Procedures Followed Parameters Used in the Analysis  to Evaluate  the Failure  To evaluate the failure parameters <j)1 and A<f>, the relationship between failure friction angle, <p„, and state parameter, \p, developed by Golder r Associates  (1986) was used  together with the corresponding  in situ  effective mean normal stress, o^, and the procedures developed by Duncan et al., (1980). evaluated  The failure parameters (tSmp/'0SMP^ 1  in turn from the <p  1  a n d  and A<f> parameters  ^TSMP^°SMP^  W e r e  through linear  relationships developed based on laboratory data. 8.2.3.3 . Summary of the Procedures Followed to Evaluate the Liquefaction Resistance Curves or Erksak 320/1 Sand The  liquefaction  resistance  curves  for  Erksak  320/1  sand were  evaluated based on the cone penetration resistance, q c > of this sand fills and the chart developed by Seed and DeAlba (1986).  This chart shows the  relationship between the modified cone tip resistance, (q c ) 1 , and the cyclic stress ratio,  T  e q/°^ 0 »  causing liquefaction. This chart is valid  for earthquakes of magnitude 7.5 or 15 significant cycles of loading. Because during the April 12, 1986 ice load event the Molikpaq was subject to a substantially larger number of cycles than 15, it was necessary to develop a relationship between the shear stress, x^, to cause liquefaction in N cycles, and t 1 5 , which caused liquefaction in 15 cycles.  Such a  correlation was developed by Been (1988) based on both published data and data obtained by Golder Associates (1984).  Because test data was not available to determine the possible increase in liquefaction resistance of the sand core of the Molikpaq due to: (a) initial effects of the static bias developed by the ice loading; (b) past history of cyclic loading; and (c) drainage conditions developed by the water pumps, the liquefaction resistance curves used in the analysis did not consider the above factors and therefore represent a lower bound to liquefaction resistance.  8.3 Interface Elements To model the contact between the Molikpaq steel structure and its sand fills, an interface element following the concept of Desai's "thin" element was developed and implemented into F.E. formulation (Chapter A).  Both  formulations include the implementation of load' shedding techniques for interface elements that failed in tension or shear. Procedures to evaluate soil parameters for use with the "thin" element were also developed and follow those developed by Clough and Duncan (1971) combined with the procedures recommended by Desai. The performance of the "thin" element was assessed by comparing its F.E. results with the closed form solutions of a soil-pipe system developed by Burns and Richards (1964).  The. F.E. results show that an excellent  agreement with the closed form solutions was obtained when the "thin" element is used in both the 2-D and 3-D F.E. analysis. In addition, both the "thin" element and the "standard" soil element (using the hyperbolic and modified SMP models) predictions were compared with earth pressure measurements on a 10 m retaining wall field test. These F.E. studies were necessary to check the procedures followed in the construction analyses of the Molikpaq, since there were no earth pressure measurements during the core construction phase of this structure.  From the comparisons between the field measurements and the F.E. predictions the following conclusions are made: • All the combination of element types and constitutive model types give results that are in fair agreement with the field measurements for both the at-rest condition and active condition. • The field measurements carried out by Matsuo et al. (1978) and the F.E. predictions carried out by the writer, are in good agreement with the results of tests performed by Terzaghi (1934) and with the analytical work carried out by Clough and Duncan (1971). The field measurements show that the coefficient of earth pressure at rest, K 0 varies from a maximum K 0 = .74 at 1.0 m depth to a minimum K 0 = .28 at 5.0 m depth.  If the wall is  allowed to rotate away from the backfill than the coefficient of earth pressure, K decrases considerably to an average (IK) = .11 which flV corresponds to a movement of the top of the wall of 8.4 cm or .84% of the wall height. • From this particular case study it seems that the need for a "thin" interface type of element is not completely justified if "standard" solid elements with stress-strain models, such as the hyperbolic model (Duncan et al., 1980), or the modified SMP model, both expanded with load shedding capabilities, are used in the analysis. 8.4 Summary of the Analysis Procedure and the Results Obtained from the Analysis The following 3-D assessments of the Molikpaq response to different load conditions were carried out by: • Static assessment during the construction phase of the berm and core;  • Static assessment during the ice loading phase of the event of March 25, 1986; • Pseudo-static assessment during the high ice loading phases of the event of April 12, 1986; • Static assessment of the settlement phase after the ice loading event of April 12, 1986.  The above 3-D analyses were carried out using the hyperbolic model because the 3-D formulation of the modified SMP model, could not be used since the required computer memory exceeded the existing UBC computer capacity of 1 megaword. To study the influence on the results of some key parameters (interface element type, constitutive law, stress redistribution) several 2-D analyses were carried out using both the hyperbolic and the modified SMP models. A summary of the procedures followed in the analysis carried out and of the results obtained is presented next.  a)  Construction Phase of the Berm and Core  The construction of the berm and core was simulated in the 3-D analysis by placing the fill in one single layer. Although the ideal approach is to "analytically construct" the fill in layers, that procedure was not followed due to the large band width of the system of equations.  The  stresses so obtained were compared with the stresses obtained from 2-D plane strain construction analysis where the sand fills were built in 7 layers.  It was found that the stresses obtained from both 2-D and 3-D  analyses were in reasonable agreement.  b)  3-D Analysis of the Static Ice Load Event of March 25, 1986 Prior to the dynamic ice load event of April 12, 1986, the Molikpaq  structure was subject to several other ice load events.  Because the event  of March 25, 1986 was the best documented ice event, several 3-D analyses were carried out to predict the available field measurements and therefore calibrate the soil-structure parameters for use subsequently in the 3-D analysis of the dynamic ice load event of April 12, 1986.  From the  comparisons between the F.E. predictions and measurements, the following is concluded: • The 3-D deflected shape of the caisson is adequately modelled by the analysis except for the NE corner, where the results indicate movements in the opposite direction from the measurements. • The measured deformation profile, by the in-place inclinometer, in the core and berm is well modelled by the analysis and both show a remarkable difference between the shear behaviour of the core and the shear behaviour of the berm.  c)  3-D Analysis of the Dynamic Ice Load Event of April 12, 1986 On April 12, 1986 the Molikpaq structure was subject to severe dynamic  ice loads.  To analyse this event a 3-D finite element dynamic computer  program with an appropriate stress-strain law is required.  To date,  however, such a program does not exist. Adequate 2-D finite element dynamic computer programs do exist, such as the program RICEL developed by Yogendrakumar and Finn (1987).  This  program was used in 2-D dynamic and pseudo-dynamic analysis of the Molikpaq's response to the above ice load event by Finn et al. (1988), who showed that the Molikpaq's system damping was very large and consequently  no significant dynamic amplifications developed.  Hence, the response of  the structure can be studied from pseudo-dynamic or pseudo-static analysis which do not consider inertia forces. Based on the above, the 3-D dynamic response of the caisson to a series of cyclic ice loading pulses was determined by computing the static response to one-half cycle of load/unload.  The amplitude of displacement  and the cyclic stress ratios so computed were assumed to be the dynamic values corresponding to a dynamic amplification factor of 1. The 3-D analysis of the response of the Molikpaq to the ice loading event of April 12, 1986 were subdivided in Chapter 7 into four separate assessments: liquefaction, porewater pressure rise, accelerations, and settlement. • The liquefaction assessment was carried out by comparing the cyclic stress ratio, At^Jcorrespondent to one-half cycle of load/unload, mobilized in each soil element, with the cyclic resistance ratio, T  av^°vo' These comparisons were carried out at different stages of load-  ing  (i.e. for different numbers of  cycles) and the liquefaction  assessment updated until the end of the ice loading. • The pore pressure rise assessment  was carried out following the pore  pressure model developed by Seed et al. (1976).  • The acceleration assessment was carried out assuming that the response of the Molikpaq structure to cyclic loading was harmonic. • The settlement assessment was carried out based on the work developed by Tokimatsu and Seed (1987) and Byrne (1990).  From the comparisons between the F.E. predictions and the measured data at the Amauligak 1-65 site the following is concluded.  • The liquefaction assessments from the 3-D analysis indicate that the computed liquefaction areal extent is in good agreement with the field liquefaction assessment based on porewater measurements and settlement observations. • Both the trend and the magnitude of the predictions of settlement agree quite well with the field observations. • The residual displacement profile measured by the inplace inclinometer  located adjacent to the west wall of the caisson was  underestimated by the F.E. predictions.  This could be associated with the  fact that a Poisson's ratio value of v = 0.0 was used in the analysis of settlement. • The maximum residual excess porewater pressures computed from the analysis for the locations of the piezometers E x and E a (adjacent to the loaded wall) are in good agreement with the field measurements, which indicate that liquefaction was developed at these two locations.  However  the results indicate that the prediction of the time to liquefaction, at the location, of piezometer E l t is not correct.  This could be due to the  fact that the liquefaction resistance curves used in the analysis do not take into account the effects of the initial static bias, drainage, and previous dynamic ice loading events at the site. In addition, the analyses were carried out using an idealized ice loading function which is somewhat different from the actual ice loading of April 12, 1986. • The results also indicate that the predictions of the residual porewater pressure for the location of piezometer E 3 is substantially higher than the maximum measured value. This can also be explained by the above considerations.  • The predictions of the accelerations measured at the centre of the top of the core by the accelerometer ACC-841 underestimate the field measurements during the initial phase, but agree quite well with both the maximum acceleration value recorded and the time of its occurrence. On the other hand the predictions of the accelerations measured at the top of the loaded wall by the accelerometer TM-706 agree quite well with the field measurements during the initial phase but both the maximum acceleration recorded and its time of occurrence are not well predicted by the analysis. These differences are not unexpected because the computed values of the accelerations were based on the assumption that the response of the Molikpaq to cyclic loading was harmonic which is an extremely crude assumption.  In addition an idealized loading function was used in the  analysis. • The results obtained from the several 2-D analysis, which considered different types of interface elements ("thin" and "standard" elements) and constitutive models (modified SMP and hyperbolic models) indicate that the computed liquefaction areal extent is practically insensitive to the above parameters and in good agreement with both the 3-D analysis and field observations.  Based on the above, it is concluded that both the procedures followed to obtain soil parameters from laboratory and in situ testing and the procedures followed in the analysis have been validated.  Bellotti, R. , Ghionna, V., Jamiolkowswki, M. , Robertson, P.K. and Petterson, R.W. (1989) "Interpretation of Moduli from Self-Boring Pressuremeter Tests in Sand", Geotechnique, Vol. 39, No. 2, 269-292. Bruce, J. and Harrington, A.G. (1982) "Design Aspects of a Mobile Artie Caisson", Proc. 14th Annual Offshore Technology Conf., OTC 4333, Houston. Byrne, P.M. and Eldridge, T.L. (1982) "A Three Parameter Dilatant Elastic Stress-Strain Model for Sand", Int. Symposium on Numerical Models in Geomechanics, Switzerland, pp. 73-79. Desai, C.S., Zaman, M.M., Lightner, J.G. and Siriwardane, H.J. (1984) "Thin-Layer Element for Interfaces and Joints", Int. Journal for Numerical and Analytical Methods in Geomechanics, Vol. 8, pp. 19-43. Finn, W.D., Anderson, D.L., Byrne, P.M. and Stuckert, B. (1988) "Analysis of the Response of Molikpaq to the Ice Loading Event of April 12, 1986", Department of Civil Engineering, University of British Columbia, Vancouver, B.C. Fitzpatrick, J. and Stenning, D.G. (1983) "Design and Construction of Tarsiut Island in the Canadian Beaufort Sea", Proc. 15th Annual Offshore Technology Conf., OTC 4517, Houston, Vol. 2, pp. 51-60. Jefferies, M.G., Stewart, H.R., Thomson, R.A.A. and Rogers, B.T. (1985) "Molikpaq Deployment at Tarsiut P-45", Proc. ASCE Spec. Conf. on Artie Offshore, San Francisco. Jefferies, M.G. and Wright, W.H. (1988) "Dynamic Response of Molikpaq to Ice-Structure Interaction", Proc. 7th Int. Conf. on Offshore Mechanics and Artie Engineering. Jefferies, M.G., Rogers, B.T., Stewart, H.R., Shinde, S., James, D. and Williams-Fitzpatrick, S. (1988) "Island Construction in the Canadian Beaufort Sea". Proc. Conf. on Hydraulic Fill Structures, ASCE, Fort Collins, Colorado. Manassero, M. (1989) "Stress-Strain Relationship for Drained Self-Boring Pressuremeter Tests in Sand", Geotechnique, Vol. 39, No. 2, 293-307. Matsuoka, H. (1983) "Deformation and Strength of Granular Materials Based on the Theory of 'Compounded Mobilized Plane' (CMP) and 'Spatial Mobilized Plane' (SMP)", Advances in the Mechanics and the Flow of Granular Materials, Vol. II, Trans Tech Publications.  Matsuoka, H. (197A) "Stress-Strain Relationships of Sands Based on the Mobilized Plane", Soils and Foundations, Vol. 1A, No. 2. McCreath, D.R., Hodge, W.E. and Harrington, A. G. (1982) "Geotechnical Design Considerations for the Gulf Mobile Artie Caisson", 2nd Canadian Conference on Marine Geotechnical Engineering, Halifax. Seed, H.B. and De Alba (1986) "Use of SPT and CPT Tests for Evaluating the Liquefaction Resistance of Sands", Proc. In Situ 86, ASCE Specialty Conf. on "Use of In Situ Tests in Geotechnical Engineering", Blacksburg, Virginia. Stewart, H.R. and Brakel, J. (1986) "Molikpaq Berm Construction at Amauligak 1-65", World Dredging Conference, Brighton, U.K. Yogendrakumar, M. and Finn, W.D. (1987) "RICEL: Program for Determining Response to Ice Loading of Offshore Structures", Department of Civil Engineering, University of British Columbia, Vancouver, B.C.  Arthur, J.R.F., Dunstan, T., Al-Ani, Q.J.L. and Assadi, A. (1977) "Plastic Deformation and Failure in Granular Media", Geotechnique 27, No. 1, pp. 53-74. Bathe, K.J., Wilson, E.L. an Iding, R.H. (1974) "A Structural Analyis Program for Static and Dynamic Response of Nonlinear Systems", Report No. UC SESM 74-3, Structural Engineering Laboratory, University of California, Berkeley, California. Bishop, A.W. (1966) "The Strength of Soils as Engineering Materials", Geotechnique 16, No. 2, pp. 91-130. Byrne, P.M. and Janzen, W. (1984) "INCOIL: A Computer Program for Nonlinear Analysis of Stress and Deformations in Oil Sand Masses", Soil Mechanics Series No. 80, Dept. of Civil Engineering, University of British Columbia, Vancouver. Byrne, P.M. and Eldridge, T.L. (1982) "A Three Parameter Dilatant Elastic Stress-Strain Model for Sand", Int. Symposium on Numerical Models in Geomechanics, Switzerland, pp. 73-79. Duncan, J.M. and Chang, C.-Y. (1970) "Nonlinear Analysis of Stress and Strain in Soils", Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 96, No. SM5, September 1970, pp. 1629-1653. Duncan, J.M., Byrne, P.M., Wong, K.S. and Mabry, P. (1980) "Strength, Stress-Strain and Bulk Modulus Parameters for Finite-Element Analysis of Stresses and Movements in Soil Masses", Report No. UCB/GT/80-01, Dept. of Civil Engineering, University of California, Berkeley, California. Kondner, R.L. (1963) "Hyperbolic Stress-Strain Response: Cohesive Soils", Journal of the Soil Mechanics and Foundation Division, ASCE, Vol. 89, No. SMI, pp. 115-143. Kondner, R.L. and Zelasko, J.S. (1963) "A Hyperbolic Stress-Strain Formulation of Sands", Proc. of the 2nd Pan American Conference on Soil Mechanics and Foundation Engineering, Vol. 1, Brasil, p. 289. Lade, P.V. (1972) "The Stress-Strain and Strength Characteristics of Cohesionless Soils", Ph.D. Thesis, University of California, Berkeley. Lade, P.V. and Duncan, J.M. (1975) "Elastoplastic Stress-Strain Theory for Cohesionless Soil", Proc. ASCE, Vol. 101, No. GT10, pp. 1037-1053. Matsuoka, H. (1974) "Stress Strain Relationships of Sands Based on the Mobilized Plane", Soil and Found., Vol. 14, No. 2, 47-61.  Matsuoka, H. and Nakai, T. (1974) "Stress-Deformation and Strength Characteristics of Soil Under Three Different Principal Stresses", Proc. JSCE, No. 232, 59-70. Matsuoka, H. and Nakai, T. (1985) "Relationship Among Tresca, Mises, Mohr-Coulomb an Matsuoka-Nakai Failure Criteria", Soil and Found., Vol. 25, No. 4, 123-128. Matsuoka, H. (1983) "Deformation and Strength of Granular Materials Based on the Theory of 'Compounded Mobilized Plane' (CMP) and 'Spatial Mobilized Plane' (SMP)", Advances in the Mech. and the Flow of Granular Materials, Trans. Tech. Publications, 813-836. Matsuoka, H. and Sakakibara, K. (1987) "A Constitutive Model for Sands and Clays Evaluating Principal Stress Rotation", Soils and Foundations, Vol. 27, No. 4, pp. 73-88. Murayama, S. (1964) "A Theoretical Consideration on a Behaviour of Sand", Proc. IUTAM, Symposium on Rheology and Soil Mechanics, Grenoble, pp. 146-159. Murayama, S. and Matsuoka, H. (1973) "A Microscopic Study on Shearing Mechanism of Soils", Proc. 8th Int. Conf. SMFE, Vol 1, Part 2, 293-298. Roscoe, K.H. (1970) "The Influence of Strains in Soil Mechanics", Tenth Rankine Lecture, Geotechnique 20, No. 2, pp. 129-170. Rowe, P.W. (1962) "The Stress-Dilatancy Relation for Static Equilibrium of an Assembly of Particles in Contact", Proc. Royal Society, Series A, London, Vol. 269, pp. 500-527. Rowe, P.W. (1971) "Theoretical Meaning and Observed Values of Deformation Parameters for Soil, Stress-Strain Behaviour of Soils", Proc., Roscoe Memorial Symposium, Cambridge University, pp. 143-194. Saada, A. and Bianchani, G.S. (1987) Proc. of International Workshop on "Constitutive  Equations  for  Granular  Noncohesive  Soils",  Reserve University, Cleveland, Ohio (published by Balkema).  Case  Western  Sayao, A.S.F.J. (1989) "Behaviour of Sand Under General Stress Paths in the Hollow Cylinder Torsional Device", Ph.D. Thesis, University of British Columbia, Vancouver, B.C. Stroud, M.A. (1971) "The Behaviour of Sand at Low Stress Levels in the Simple Shear Apparatus", Ph.D. Thesis, Cambridge University. Symes, M.J., Hight, D.W. and Gens, A. (1982) "Investigating Anisotropy and the Effects of Principal Stress Rotation and of the Intermediate Principal Stress Using a Hollow Cylinder Apparatus", IUTAM Conf. on Deformation and Failure of Granular Materials. Symes, M.J., Gens, A. and Hight, D.W. (1984) "Undrained Anisotropy and Principal Stress Rotation in Saturated Sand", Geotechnique 34, No. 1, pp. 11-27.  Symes, M.J. and Hight, D.W. (1988) "Drained Principal Stress Rotation in Saturated Sand", Geotechnique 38, No. 1, pp. 59-81. Vaid, Y.P., Byrne, P.M. and Hughes, J.M.O. (1981) "Dilation Angle and Liquefaction Potential", Int. Conf. on Recent Avances in Geot. Earthquake Engineering and Soil Dynamics, St. Louis, pp. 161-166. Wood, D.M., Drescher, A. and Budhu, M. (1979) "On the Determination of Stress State in the Simple Shear Apparatus", Geotechnical Testing Journal, Vol. 2, No. 4. Wroth, C.P. (1984) "The Interpretation of In Situ Soil Tests", Rankine Lecture, Imperial College, London, OUEL Report No. 1541/84. SM 051/84. Yong, R.K. and Ko, H.Y. (1980), Proc. of the Workshop on "Limit Equilibrium, Plasticity, and Generalized Stress-strain in Geotechnical Engineering", McGill University, published by ASCE. Zienkiewicz, O.C., Valliappan, S. an King, I.P. (1968) "Stress Analysis of Rock as a 'No Tension Material'", Geotechnique 18, pp. 56-66.  Andrawes, K.Z. (1964) "The Behaviour of Particulate Materials in the 'At Rest' State", M.Sc. Thesis, University of Manchester. Bryne, P.M., Cheung, H. and Yan. L. (1987) "Soil Parameters for Deformation Analysis of Sand Masses", Can. Geot. J., Vol. 24, No. 3, pp. 366-376. Byrne, P.M. and Eldridge, T.L. (1982) "A Three Parameter Dilatant Elastic Stress-Strain Model for Sand", Int. Symposium on Numerical Models in Geomechanics, Switzerland, pp. 73-79. Carter, J.P. and Yeung, S.K. (1985) "Analysis of Cylindrical Cavity Expansion in a Strain Weakening Material", Research Report No. R505, The University of Sydney, Sydney, Australia. Duncan, J.M., Byrne, P.M., Wong, K.S. and Mabry, P. (1980) "Strength, Stress-Strain and Bulk Modulus Parameters for Finite-Element Analysis of Stresses and Movements in Soil Masses", Report No. UCB/GT/80-01, Dept. of Civil Engineering, University of California, Berkeley, California. Finn, W.D. Liam, and Vaid, Y.P. (1977) "Liquefaction Potential from Drained Constant Volume Cyclic Simple Shear Tests", Proceedings, 6th World Conference on Earthquake Engineering, New Delhi. Jambu, N. (1963) "Soil Compressibility as Determined by Oedometer and Triaxial Tests", European Conference on Soil Mechanics and Foundation Engineering, Wissbaden, Germany, Vol. 1, pp. 19-25. Kolbuszewski, J. (1965) "San Particles and Their Density", Lecture delivered at Materials Science Colub's Symp., London. Matsuoka, H. (1983) "Deformation and Strength of Granular Materials Based on the Theory of 'Compounded Mobilized Plane' (CMP) and 'Spatial Mobilized Plane' (SMP)", Advances in the Mech. and the Flow of Granular Materials, Trans. Tech. Publications, 813-836. Negussey, D. (1984) "An Experimental Study on the Small Strain Response of Sand", Ph.D. Thesis, University of British Columbia, Vancouver, B.C. Roscoe, K.H. "An Apparatus for the Application of Simple Shear to Soil Samples", Proceedings, 3rd Int. Conf. on Soil Mechanics, Zurich, Vol. 1, pp. 186-191. Stroud, M.A. (1971) "The Behaviour of Sand at Low Stress Levels in the Simple Shear Apparatus", Ph.D. Thesis, Cambridge University. Vaid, Y.P., Byrne, P.M. and Hughes, J.M.O. (1981) "Dilation Angle and Liquefaction Potential", Int. Conf. on Recent Avances in Geot. Earthquake Engineering and Soil Dynamics, St. Louis, pp. 161-166.  Yong, R.K. and Ko, H.Y. (1980), Proc. of the Workshop on "Limit Equilibrium, Plasticity, and Generalized Stress-strain in Geotechnical Engineering", McGill University, published by ASCE.  Anderson, H.W. and J.S. Dodd (1966). "Finite Element Method Applied to Rock Mechanics", Proc. 1st Congress, Int. Soc. for Rock Mechanics, Lisbon, Portugal. Burns, J.W. and Richards, R.M. (1964). Attenuation of Stresses for Buried Cylinders", Proc. Symposium on Soil-Structure Interaction, University of Arizona, Tucson, Arizona. Byrne, P.M., Cheung, H. and Yan, L. (1987). "Soil Parameters for Deformation Analysis of Sand Masses", Can. Geot. J. , Vol. 2A, No. 3, pp. 366-376. Byrne, P.M. and Janzen, W. (198A). "Incoil: A Computer Program for Nonlinear Analysis of Stress and Deformations in Oil Sand Masses", Soil Mechanics Series No. 80, Dept. of Civil Engineering, University of British Columbia. Clough, W.G. and Duncan, J.M. (1971). "Finite Element Analysis of Retaining Wall", Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 97, SM12, pp. 1657. Cook, R.D., Malkus, D.S. and Plesha, M.E. (1989). "Concepts and Applications of Finite Element Analysis". University of Wisconsin-Madison, Published by John Wiley & Sons, (3rd edition). Desai, C.S. (1981) "Behaviour of Interfaces Between Structural and Geological Media", Proc. Int. Conf. on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, University of Missouri-Rolla, Rolla, Missouri. Desai, C.S., Zaman, M.M., Lightner, J.G and Siriwardane, H.J. (198A). "Thin Layer Element for Interfaces and Joints", Int. Journal for Numerical and Analytical Methods in Geomechanics, Vol. 8, pp. 19-A3. Duncan, J.M., Byrne, P.M., Wong, K.S. and Mabry, P. (1980). "Strength, Stress-Strain and Bulk Modulus Parameters for Finite Element Analyses of Stresses and Movements in Soil Masses", Report No. UCB/GT/80-81, Department of Civil Engineering, University of California, Berkeley. Duncan, J.M. and Chang, C.-Y. (1970). "Nonlinear Analysis of Stress and Strain in Soils", Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 96, No. SM5. Goodman, R.E., Taylor, R.L. and Brekke, T.L. (1968) - "A Model for the Mechanics of Jointed Rock", J. Soil Mech. Found. Div., ASCE, 9A, SM3. Matsuo, M., Kenmochi, S. and Yagi, H. (1978). "Experimental Study on Earth Pressure of Retaining Wall by Field Tests", Soil and Foundations, Vol. 18, No. 3, pp. 27.  Nogo, D. and Scordelis, A.C. (1967). "Finite Element Analysis of Reinforced Concrete Beams", ACI Journal, 6A.3. Potyundy, J.G. (1961). "Skin Friction Between Various Construction Materials", Geotechnique, 11, A, pp. 339.  Soils and  Terzaghi, K. (193A). "Large Retaining Wall Tests. I. Pressure of Dry Sand", Engineering News Record, Vol, III, pp. 136-1AO. Yoshimi, Y. and Kishida, T. (1981). "A Ring Torsion Apparatus for Evaluating Friction Between Soil and Metal Surfaces", Geotechnical Testing Journal, GTJODT, Vol. A, No. A, pp. 1A5.  Bolton, M.D. (1986). "The Strength and Dilatancy of Sands", Geotechnique 36, No. 1, pp. 65-78. Bellotti, R., Ghionna, V., Jaraiolkowski, M., Robertson, P.K. and Petterson, R.W. (1989). "Interpretation of Moduli from Self-Boring Pressuremeter Tests in Sand", Geotechnique, Vol. 39, No. 2, 269-292. Byrne, P.M., Salgado, F.M. and Howie, J.A, (1990). "Relationship Between the Unload Shear Modulus from Pressuremeter Tests and the Maximum Shear Modulus for Sand", Proc. of the Third International Symposium on Pressuremeters, British Geotechnical Society, Oxford University. Byrne, P.M., Cheung, H. and Yan, L. (1987). "Soil Parmeters for Deformation Analysis of Sand Masses", Can. Geot. J., Vol. 24, No. 3, 366-376. Gibson, R.E. and Anderson, W.F. (1961) "In Situ Measurements of Soil Properties with the Pressuremeter", Civ. Eng. Pub, WKS Rev. 56, No. 658, May, 615-618. Hardin, B.O. and Black, W.L. (1966) "Sand Stiffness Under Various Triaxial Stresses", J. Soil Mech. and Found. Div., ASCE, Vol. 92, No. SM2, 27-42. Hardin, B.O. (1978) "The Nature of Stress-Strain Behaviour of Soils", Proc., ASCE, Geo. Eng. Div. Specialty Conf. Earth. Eng. and Soil Dyn., Pasadena, California, Vol. 1, 3-90. Hughes, J.M.O., Wroth, C.P. and Windle, D. (1977) "Pressuremeter Tests in Sands", Geotechnique, Vol. XXVII, No. 4, 455. Jewell, R.J., Fahey, M. and Wroth, C.P. (1980) "Laboratory Studies of the Pressuremeter Test in Sand", Geotechnique 30, No. 4, 507-531. Knox, D.P. (1982) "Effect of State of Stress on Velocity of Low Amplitude Shear Wave Propagating Along Principal Stress Direction in Dry Sand", Ph.D. Thesis, Texas University. Ladanyi, B. (1963) "Evaluation of Pressuremeter Tests in Granular Soils", 2nd Pan American Conf., Soil Mech. Found. Eng. 1, 3-20. Lee, S.H.H. (1986) "Investigation on Low Amplitude Shear Wave Velocity in Anisotropic Materials", Ph.D. Thesis, Texas Universty. Manassero, M. (1989) "Stress-Strain Relationship for Drained Self-Boring Pressuremeter Tests in Sands", Geotechnique 39, No. 2, 293-307. Negussey, D. (1984) "An Experimental Study of the Small Strain Response of Sand", Ph.D. Thesis, University of British Columbia.  Robertson, P.K. (1982) "In Situ Testing of Soil with Emphasis on its Application to Liquefaction Assessment", Ph.D. Thesis, University of British Columbia. Robertson, P.K. and Hughes, J.M.O. (1986) "Determination of Properties of Sand from Self-Boring pressuremeter Test", Proc. 2nd Int. Symp. on the Pressuremeter and its Marine Applications, Austin, Texas, American Society for Testing and Materials, 283-302. Rowe, P.W. (1971) "Theoretical Meaning and Observed Values of Deformation Parameters for Soils, Stress-strain Behaviour of Soils", Proc. Roscoe Memorial Symp., Cambridge University, 143-194. Stokoe, K.H. and Ni, F.H. (1985) "Effects of Stress State and Strain Amplitude on Shear Modulus of Dry Sand", Proc. 2nd Symp. Interaction of Non-nuclear Munitions with Structures, Panama City Beach, 407-412, Stroud, M.A. (1971) "Sand at Low Stress Levels in the Simple Shear Apparatus", Ph.D. Thesis, University of Cambridge. Vesic, A.S. (1972) "Expansion of Cavities in Infinite Soil Mass", J. Soil Mech. and Found. Div., ASCE, Vol. 98, No. SM3, 265-290. Yan, L. and Byrne, P.M. (1989) "Simulation of Downhole and Crosshole Seismic Tests on Sand Using the Hydraulic Gradient Similitude Method", Soil Mech. Series No. 132, Dept. of Civil Engineering, University of British Columbia, Vancouver, Canada, October. Yu, P. and Richart, F.E. (1984) "Stress Ratio Effects on Shear Modulus of Dry Sands", J. Geot. Eng., Vol. 110, No. 3, March, 331-345.  Been, K. and Jefferies, M.G. (1985). Geotechnique 35, No. 2, 99-112.  "A State Parraeter for Sands",  Been, K., Jefferies, M.G.,. Crooks, J.H.A. and Rothenburg, L. (1987). "The Cone Penetration Test in Sands. Part 2: General Inference of State". Geotechnique 37, No. 3, 285-299. Bjerrum, L. (1973). "Geotechnical Problems Involved in Foundations of Structures in the North Sea". Geotehcnique 23, No. 3, 319-358. Byrne, P.M. and Eldridge, T.L. (1982). "A Three Parameter Dilatant Elastic Stress-Strain Model for Sand". International Symposium on Numerical Models in Geomechanics, Switzerland, 73-79. Castro, G. (1969). "Liquefaction of Sand". Ph.D. Thesis, Division of Engineering and Applied Physics, Harvard University. Duncan, J.M., Byrne, P.M., Wong, K.S. and Mabry, P. (1980). "Strength, Stress-Strain and Bulk Modulus Parameters for Finite Element Analysis of Stresses and Movemnets in Soil Masses". Report No. UCB/GT/80-81, Department of Civil Engineering, University of California, Berkeley, California. Finn, W.D. Liam, Bransby, P.L., and Pickering, D.J. (1970). "Effect of Strain History on Liquefaction of Sand". Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 96, No. SM6, 1917-1934. Garga, V.K. and McKay, L.D. (1984). "Cyclic Triaxial Strength of Mine Tailings". Journal of Geotechnical Engineering Division, ASCE, Vol. 110, GT8, 1091-1105. Golder Associates (1986a). "Laboratory Testing on Sand Samples from Molikpaq Core at Amauligak 1-65 Delineation Well". Report submitted to Gulf Canada Resources Ltd. Golder Associates (1986b). "Fig. 4.23 in Contract Briefing Document. Supplied by Atlantic Geoscience Centre. Golder Associates (1987). "Laboratory Testing of Molikpaq Core Sand Under Repated Loading". Report submitted to Gulf Canada Resources Ltd. Hardin, B.O. and Drenvich, V.P. (1972). "Shear Modulus and Damping in Soils: Design Equations and Curves". Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 98, No. SM7, 667-692. Jefferies M.G. and Livinstone, W.R. (1985). "Molikpaq at Amauligak 1-65 Post-Construction Assessment", Gulf Canada Resources Ltd., Frontier Development Division.  Jefferies, M.G., Stewart, H.R., Thomson, R..A. and Rogers, B.T. (1985). "Molikpaq Deployment at Tarsiut P-45". Proc. ASCE Spec. Conf. on Artie Offshore, San Francisco. Jefferies, M.G. and Wright, W.H. (1988). "Dynamic Response of 'Molikpaq' to Ice-Structure Interaction". Proceedings of the 7th International Conference on Offshore Mechanics and Arctic Engineering. Jefferies, M.G., Rogers, B.T., Stewart, H.R., Shinde, S., James, D. and Williams-Fitzpatrick, S. (1988). "Island Construction in the Canadian Beaufort Sea". Proc, Conf. on Hydraulic Fill Structures, ASCE, Fort Collins Colorado. Liao, S.S.C. and Whittman, R.V. (1986). "Overburden Correction Factors for SPT in Sand". Journal of Geotechnical Engineering, ASCE, Vol. 112, No. 3, 373-377. MacKay, J.R. (1972). "Offshore Permafrost and Ground Ice, Southern Beaufort Sea, Canada". Canadian Journal of Earth Sciences, Vol. 9, No. 11, pp. 1550. Negussey, D. (1987). Private Communication. Robertson, P.K. and Campanella, R.G. (1983). "Interpretation of Cone Penetration Tests - Part I (Sand)". Canadian Geotechnical Journal, Vol. 20, No. 4. Seed, H.B. and Lee, K.L. (1966). "Liquefaction of Saturated Sands During Cyclic Loading". Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 92, No. SM6, 105-134. Seed, H.B. and Idriss, I.M. (1970). "Soil Moduli and Damping Factors for Dynamic Response Analyses". Report No. EERC 70-10, Earthquake Engineering Research Center, University of California, Berkeley, California. Seed, H.B., Martin, P.P. and Lysmer, J. (1976). "Pore-water Pressure Changes During Soil Liquefaction". Journal of Geotechnical Eng. Div., ASCE, Vol. 102, GT34, 323-346. Seed, H.B. (1979). "Soil Liquefaction and Cyclic Mobility Evaluation for Level Ground During Earthquake". Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 105, Nol. GT2, 201-255. Seed, H.B. , Idriss, I.M. and Arango, I. (1983) . "Evaluation of Liquefaction Potential Using Field Performance Data". Journal of Geotechnical Engineering Division, ASCE, Vol. 109, No. 3, 458-482. Seed, H.B. and De Alba, P. (1986). "Use of SPT and CPT Tests for Evaluating the Liquefaction Resistance of Sands". Proceedings, In Situ 86, ASCE Specialty Conference on 'Use of In Situ Tests in Geotechnical Engineering', Blacksburg, Virginia.  Seed, R.B. and Lee, S.R. (1988). "Penetration and Liquefaction Resistances: Prior Seismic History Effects". Journal of Geotechnical Engineering, ASCE, Vol. 114, No. 6, 691-697. Shibata, T., Yukitomo, H. and Miyoshi, M. (1972). "Liquefaction Process of Sand During Cyclic Loading". Soil and Foundations, Vol. 12, No. 1, 1-16. Tokimatsu, K.A.M. and Seed, H.B. (1987). "Evaluation of Settlement in Sands Due to Earthquake Shaking". Journal of Geotechnical Engineering, ASCE, Vol. 113, No. 8, 861-878. Thurber Consultants Ltd. (1985). "Self-Boring Pressuremeter Testing of Molikpaq Sand Core at Amauligak 1-65". Report submitted to Gulf Canada Resources Ltd. Vaid, Y.P. and Chern, J.C. (1983). "Effect of Static Shear on Resistance to Liquefaction". Soil and Foundations, Vol. 23, No. 1, 47-60. Vaid, Y.P. and Finn, W.D. Liam (1979). "Static Shear on Liquefaction Potential". Journal of the Geotechnical Engineering Division, ASCE, Vol. 105, GT10, 1233-1248.  Byrne, P.M. (1990). "Volume Change and Pore Pressure Rise in Sands Due to Cyclic Loading". Soil Mechanics Series No. , Dept. of Civil Engineering, University of British Columbia, Vancouver. Byrne, P.M. and Janzen, W. (1984). "INCOIL: A Computer Program for Nonlinear Analysis of Stress and Deformations in Oil Sand Masses". Soil Mechanics Series No. 80, Dept. of Civil Engineering, University of British Columbia, Vancouver. Duncan, J.M., Byrne, P.M., Wong, K.S. and Mabry, P. (1980). "Strength, Stress-Strain and Bulk Modulus Parameters for Finite-Element Analysis of Stresses and Movements in Soil Masses", Report No. UCB/GT/80-01, Dept. of Civil Engineering, University of California, Berkeley, California. Jefferies, M.G. (1987). Private Communication. Jefferies, M.G. and Wright, W.H. (1988). "Dynamic Response of 'Molikpaq' to Ice-Structure Interaction". Proceedings of the 7th International Conference on Offshore Mechanics and Arctic Engineering. Kulhawy, F.H., Duncan, J.M. and Seed, R.B. (1969). "Finite Element Analysis of Stresses and Movements in Embankments During Construction", Geotechnical Engineering Research Report No. TE-69-4, Department of Civil Engineering, University of California, Berkeley, November 1969. Seed, H.B. and DeAlba, P. (1986). "Use of SPT and CPT Tests for Evaluating the Liquefaction Resistance of Sands". Proceedings, In Situ 86, ASCE Specialty Conference on "Use of In Situ Tests in Geotechnical Engineering", Blacksburg, Virginia. Seed, H.B. and Idriss, I.M. (1982).  During Earthquakes".  California.  "Ground Motions and Soil Liquefaction  Earthquake Engineering Research Institute, Berkeley,  Seed, H.B., Martin, P.P. and Lysmer, J. (1976). "Porewater Pressure Changes During Soil Liquefaction". Journal of Geotechnical Engineering Division, ASCE, Vol. 102, GT4, 323-346. Seed, H.B., Seed, R.B., Harder, L.F. and Jong, H.L. (1988). "Re-evaluation of the Slide in the Lower San Fernando Dam in the Earthquake of February 9, 1971". Report No. UCB/EERC-88-04, Earthquake Engineering Research Center, University of California, Berkeley, California. Tokimatsu, K.A.M. and Seed, H.B. (1987). "Evaluation of Settlement in Sands Due to Earthquake Shaking". Journal of Geotechnical Engineering, ASCE, Vol. 113, No. 8. 861-878.  CHAPTER 2 APPENDIX INDEX  2.1 Evaluation of the friction angle <{>13 for b-values varying from 0.0 to 1.0. 2.2 Relationship between principal stresses and, the normal and shear stress on the SMP, Extrapolation of these relationships to the increments of plastic strain space. 2.3 Fundamental relationship between cartesian stresses and principal stresses. Extrapolation of these relations to the increments of plastic strain space. 2.A Development of A(T s m p /o s m p) in terms of AOj, Ao2, Ao3. 2.5 Relations between increments of principal stress and increments of cartesian stress in the 3-Dimensional stress space 2.6 Evaluation of the plastic constitutive matrix {C^} of the SMP model. 2.6.1 3-Dimensional 2.6.2 2-Dimensional 2.6.3 Axisymmetric 2.7 Load shedding formulation to use with the modified SMP model. 2.8 Discussion of the assumptions regarding the direction of the increments of principal strain, based on data derived from hollow cylinder tests.  CHAPTER 2 APPENDIX FIGURES INDEX Figure  Page  2.5.1  Sketch of a Principal Plane  350  2.7.1  Matsuoka-Nakai Failure Criterion  388  2.8.1  Initial Anisotropy Hollow Cylinder Drained Tests. Dense Ham River Sand (after Symes et al., 1982)  400  2.8.2  Initial Anisotropy Hollow Cylinder Undrained Tests. Medium Loose Ham River Sand (after Symes et al., 1984) .... 401  2.8.3  Initial Anisotropy Hollow Cylinder Drained Tests. Medium Loose Ham River Sand (after Symes et al., 1988) .... 402  2.8.4  Continuous Rotation Hollow Cylinder Drained Test. Medium Dense Ham River Sand (after Symes et al., 1982) .... 405  2.8.5  Continuous Rotation Hollow Cylinder Undrained Test. Test Rl: Medium Losse Ham River Sand (after Symes et al., 1984)  406  Continuous Rotation Hollow Cylinder Undrained Test. Test R2: Medium Loose Ham River Sand (after Symes et al., 1984)  408  "Continuous Rotation" Hollow Cylinder Drained Test. Test LR1: Medium Loose Ham River Sand (after Symes et al., 1988)  409  "Continuous Rotation" Hollow Cylinder Drained Test. Test LR3: Medium Loose Ham River Sand (after Symes et al., 1988)  410  "Continuous Rotation" Hollow Cylinder Drained Test. Test LR2: Medium Loose Ham River Sand (after Symes et al., 1988)  412  "Continuous Rotation" Hollow Cylinder Drained Test. Test LR4: Medium Loose Ham River Sand (after Symes et al., 1988)  413  2.8.6  2.8.7  2.8.8  2.8.9  2.8.10  2.8.11  "Shear After Rotation" Hollow Cylinder Undrained Test. Medium Loose Ham River Sand (after Symes et al., 1984) .... 414  APPENDIX 2 ~ LIST OF FIGURES (Cont'd) 2.8.12  "Shear After Rotation" Hollow Cylinder Drained Test. Medium Loose Ham River Sand (after Symes et al., 1988) .... A15  2.8.13  "Combined Rotation and Shear" Hollow Cylinder Drained Test. Medium Loose Ham River Sand (after Symes et al., 1988)  /  A17  EVALUATION OF FRICTION ANGLE <f>13 FOR VARYING b-VALUES  Evaluation of Friction Angle Matsuoka-Nakai  failure  for Varying b-Values  criterion  is expressed by the following  equation (see main text eq. (2.29),  =  I  (tan2  *ml2 +  tan  SMP  ^m23  + t a n  ' W  (2  -1-1)  This equation can also be written as:  T  SMP °SMP  W 2 (C0S ^12)2  2 , =  1  ( s i n  ,(  s i n  W 2 (C0S V23)J  ,(  8 i n  W V ^ (C0S ^13)2  since °l-°2  (sin  ^12  )J  =  =  "  ( 8 i n  W  and ( c 0 S  W  1  W  then (cost.J*  ml2  (Oj+a,)2 - (o r o 2 )2 = (Oj+OJ)2  or 4o  (cos  W2  i°2  „ . „  therefore ( T^)3 = cos*  (2.1.3)  1 2 1°  and Sin<J)  m23  T 1 ^ 1 ) 2 = -j  (  COS*  (2.1.4)  2°3  and (a -a ) 2 .  sin* 1 0 ( — =  (2.1.5)  13  Substituting eqs. (2.1.3) to (2.1.4) into eq. (2.1.2) the following eqution is obtained T  SHP  °SMP  2 = -r (— 3  ,  ^a-03)  + —: 4 2 3 ° °  C°x-°,)'  2  + —;  AOl  °3  X ,2  )  (2.1.6)  In order to perform a study of the influence of the b-value on the value of the friction angle <f> the above eq. (2.1.6) will be expressed as a function of the following quantities:  b =  o 2 -o 3  (2.1.7)  °i-°3  and 0,-03 a = ——— °l+°3  (2.1.8)  from (2.1.7) ° 2 = bcoj-o,) + o 3  (2.1.9)  substituting eq. (2.1.9) into eq. (2.1.6) the following is obtained: ^SMP  b^o^o,) 2  2  °SMP ~  3  +  ^  2 =- ( 3 ^ AOjOj  2  (1-b)2 (o1-o3)2 2 +  ^(b0103+03 -b03)  rci +  L U  bo 1 o 3 °1°3  2 = f (tan*2 [1 + 3 ml3  b+  +  b2 o3  )+ ( bo 3 2 ^bo,2 + °1°3  T = | tan*  (1-b)2 a1a3 b  °1°3  x 2 ' )) °!°3 °1°3  1,2 (1-b)2 — + )) o3 Oj b— b — +l-b  °1  or  ^(bo^+OjOj-bOjOj^  b2 o32 °1°3  1/a  °3  ) + o3(2-3b+b2) 7————rt ) bOj+Ojd-b)  0l (b+b  (  °SMP  2  l/a  (2.1.10)  from eq. 2.1.8. o, =  (1+a) (1-a)  (2.1.11)  substituting eq. (2.1.11) into eq. (2.1.10) the following is obtained:  T  SMP  2 =  ^  3  jilt} (b+b2) + o3(2~3b+b2) 1/a  , ml3  ( b  T  SMP = 2 oeMT> 3  ,  (Ui) Tl^Ij  } +  °3(1_b)  , (1+a)(b+b2)+(l-a)(2-3b+b2h ; ml3 b(l+a)+(l-a) (1-b)  (2.1.12)  Entering in the above equation values of b=0 (or b=l), which correspond to the triaxial stress path compression (or extension) then:  °SMP  =^  3  tan* 1 _ = K m 1 3  (2.1.13)  To note that the same value is obtained from eq. (2.1.1) by entering the triaxial stress path condition, i.e., <p . _ = <b , 0 and <p r ml2 mlJ m2J therefore eq. 2.1.12 is considered to be verified.  = 0 and  Now from eq. (2.1.12) the value of ta™!^^ i s obtained:  .  . ml3  =  ;  oSMp  ;  3 b(l+a)+(l-a)(1-b) 2 (1+a)(b+b2)+(l-a)(2-3b+b2)  T  SMP since the value of ( ) is assumed to be a constant, by Matsuoka-Nakai, °SMP when the sand reaches failure, then: t-T,* -V 1 I b(l+a) + (l-a) (1-b) tan* F13 f 2 (1+a)(b+b2)+(l-a)(2-3b+b2)  . U.i.i*;  where: K_ r  = constant value regardless of the stress path, and given by eq. (2.1.13).  Since the above eq. (2.1.14) is only valid when b=0 or b=l, for values of b^O and b^l, then this equation is rewritten as:  (2.1.15)  where: *  *F13  =  for b=0 or b=l *F13  for 0 < b < 1  ^F13 * *F13 *  is the failure friction angle, defined by o x and o 3 , for any r ii general stress path. i.e.,  RELATIONSHIP BETWEEN PRINCIPAL STRESSES AND THE NORMAL AND SHEAR STRESS ON THE SMP. EXTRAPOLATION OF THESE RELATIONS TO THE INCREMENTS OF PLASTIC STRAIN SPACE  Relationship Between Principal Stresses and the Normal and Shear Stress on the SMP. Extrapolation of These Relations to the Increments of Plastic Strain Space  The state of stress on a soil element .can be characterized by its principal stresses o^ (i = 1,2,3) or by the normal stress, Og^p and shear stress, "tg^p defined on the Spatial Mobilized Plane of the element. A relationship between these stresses, which is based on equilibrium of stresses in direction i, was developed by Matsuoka (1983) and is given by the following equation: (  °i " °SMP)ai " T SMP b i  =  (2  °  '2a)  where b. l a. l  = direction cosines of the shear stress direction = direction cosines of the normal stress direction (see main text eq. (2.24))  From eq. (2.2.1) b^ is obtained as follows:  °i"°SMP b. =  1  T  SMP  a.  (2.2.2)  1  To obtain Ae? as a function of A £ g M p  ^SMP'  M a t s u o  ^a  (1983)  assumes that the direction cosines of the normal component of the increment of plastic strain are the same as the direction cosines, a^ of the normal stress, °2j4p> i n the SMP, and that the direction cosines of the shear  component of the increment of plastic strain are equal to the direction cosines, b^ of the shear stress on the SMP. Based on these two assumptions Matsuoka developed the following equation:  A (Ae  R P  f -AeIMp)ai+ - F  b  i =0  (2  -2-3)  This equation is similar to eq. (2.2.1) developed for the stresses. From eq. (2.2.3), Ae? is obtained as follows:  Ae  i  = A£  SMP  +  W  (Ar  SMP/2)  (2  '2'A)  FUNDAMENTAL RELATIONSHIP BETWEEN CARTESIAN STRESSES AND PRINCIPAL STRESSES. EXTRAPOLATION OF THESE RELATIONS TO THE INCREMENTS OF PLASTIC STRAIN SPACE  Fundamental Relationship Between Cartesian Stresses and Principal Stresses. Extrapolation of these Relations to the Increments of Plastic Strain Space  • Fundamental relationship between Cartesian stresses and principal stresses: Extrapolation of these relations to the increments of plastic strain space The state of stress on a soil element can be defined using a Cartesian coordinate system (x,y,z) or a principal coordinate system (1,2,3) . The relationship between the stresses defined in these two coordinate systems is obtained from equilibrium of stresses in the x,y,z directions and are given by the following equation:  a  X T Xyx  zx  X oxy Xy  zy  X Xxz a yz z  ni  £  al  0  0  m2  m  0  °2 0  J2  m2  n2  n2  n  0  0  i3  m3  n3  o  £j mj (2.3.1)  where S i , m i , n i are the direction cosines of the principal stresses oi(i=l,2,3) i.e. iL = cos(i,x) nu = cos(i,y) and n^ = cos(i,z)  Developing the multiplication of matrices the following equations are obtained:  I  i=l,3  I  i=1.3  2  o.m.  ll  I o.n. i=l,3 l l  (2.3.2)  x = I o.S.m. xy i=it3 1 1 1  c  y  = z  1  i-i.3  o.m.n. 1  1  1  czx = . , l „ o.n.2. 111 i=l ,3  Assuming that the increments of plastic principal strain and the stresses have the same direction, than the relationships shown in Eq. (2.3.2) can be extrapolated for the increments of plastic strain as follows:  Ae*  Ae*  X i=l,3  Aep  A P m. Ae.  y  x  i=l ,3  Ae1  X i=l ,3  Ae? i n?I  2  (2.3.3) A/;  SL  A/; ZZ  =  X i=l,3  Ae? l 2.m. li  =  X i=l ,3  Ae. l m.n. li  =  X i=l,3  Ae? l n.2. 11  Af  n.  DEVELOPMENT OF A(o_Mp/o  ) IN TERMS OF Ao lf Ao2, AND Ao3  Development of A(T S M p /o S M p) i n terms of Aa lt Ao2, and Ao3  To evaluate A X  ^ SMP/'°SMP^ '  of the stress ratio on the SMP,  the increment t e r m s  increments of principal stress, Ao^ (i =  1,2,3), eq. (2.50) from the main text is differentiated as follows:  AT  , SMP  \  A T  SMP  =  T S M P SMP°SMP (OTT)^ SMP  A O  SMP  . ( 2  -  A  -  1 }  Therefore the terms Ax„wr. and Aanv/r. need to be evaluated. SMP  SMP  2.2.1 Evaluation of ATS0M„P n T  SMP  T  SMP  =  S i v e n by ( s e e main text, eq. (2.26))  [(°i-° 2 ) ,a i 2a 2 2 + (o2-o3)'a2Ja3* + (0,-0,)»a,» ai »](2.4.2)  Designating the term inside the square brackets by A then  T  SMP  =  '  [ A ] 1  A  ( 2  Differentiating eq. (2.4.3) the following is obtained  A T  SMP  =  \  A _ 1  '  J  A ( A )  -  4  *  3 )  or AT  A (A) SMP  2T  SMP  (2.A.A)  • Evaluation of A(A) As described,  A = (Oj-Oj) 2 a x 2 a 2 2 + (o 2 -o 3 ) 2 a 2 2 a 3 2 + (o 3 -o : ) 2 a 3 2 aj 2  (2.A.5)  differentiating with respect to o^ and a^ (i = 1,2,3) A(A) = 2(o 1 -o 2 )a 1 2 a 2 2 (Ao 2 -Ao 3 ) + (0,-0,)22axa22Aa, + (0,-0,)22at2a2Aa2 + 2(o 2 -o 3 )a 2 2 a 3 2 (Ao 2 -Ao 3 ) + (o 2 -o 3 ) 2 2a 2 a 3 2 Aa 2 + (o 2 -o 3 ) 2 2a 2 2 a 3 Aa 3 + 2(o3-o3)a32a12(Aoj-Ao,) + (o3-a\)22a3a12Aa3 + (o 3 -o x ) 2 2a 3 2 a 1 Aa 1  rearranging the above terras the following is obtained:  A(A) = Ao,(DSl) + Ao2(DS2) + Ao3(DS3) + Aa,(DAI) + Aa2(DA2) + Aa3(DA3)  where DSl = 2(o 1 -o 3 )a 1 2 a 2 2 - 2(o3-o,)a32a,2 DS2 = -2(o 1 -o 2 )a 1 2 a 2 2 + 2(o 2 -o 3 )a 2 2 a 3 2 DS3 = -2(o 2 -o 3 )a 2 2 a 3 2 + 2(o 3 -o x )a 3 2 a x 2  DAI = 2(o 1 -o 2 ) 2 a 1 a 2 2 + 2(o 3 -o t ) 2 a x a 3 2 DA2 = 2(o 1 -o 2 ) 2 a 1 2 a 2 + 2(o 2 ~o 3 ) 2 a 2 a 3 2 DA3 = 2(o 2 -o 3 ) 2 a 2 2 a 3 + 2(03-0,)2a12a3  (2.A.6)  in order to have A(A) as a function of Ao^ only, the terms Aa^ shown in eq. (2.A.6) will be evaluated as follows:  • Evaluation of Aa, From eq. (2.2A) of the main text  °1°2°3  l  »  =  (  ol  2a  1'2 2 +a i°2 0 3 +0 3°l 2 '  (2.A.7)  Designating the term inside the square brackets by A x , than  aa = [ A J ^  Differentiating (2.A.8)  Aa, = A(A1)/2a1  - Evaluation of A(A,) As described  °1°2°3 A,  1  =  °l  2  °2  + a  i  0  2  0  3  (2.A.9)  °3°1:  Designating the numerator by A X T and the denominator by A1B, than  A x = AjT/AJB  Differentiating A(AJT)AJB  =  - AJTACAJB)  (A^P  (2  '4-10)  - Evaluation of A(AtT) and A(A,B) As described  A  1  T  = °1°203  Differentiating  A(A,T) = Ao,o2o3 + A O j O j O j + Ao3o,o2  (2.4.11)  and A,B = o, 2 o 2 + o,o2o3 + o 3 o, 2  Differentiating  A(A,B) = 2o,o2Ao, + o,2Ao2 + A O , O 2 O 3 + Aa2o,o3 + Ao3o,o2 + A O 3 O , 2 + 2O3O,ACJ,  (2.4.12)  Substituting A(A,T) from eq. (2.4.11) and A(A1B) from eq. (2.4.12) into eq. (2.4.10) the following is obtained;  A(A,) = {Ao1(A1B)o2o3 + Ao 2 (A 1 B)o 1 o 3 + Ao3(A,B)o,o2) - [Ao,(A,T)(2o,o2+0203+2o,a3)) + AO2((A,T) (O, 2 +O,O 3 )) + AO3((A,T)(O,O2+O,2))]}/(A,B)2  rearranging in terms of Ao,, Ao 2 , Ao3  A(A,) = Ao,(A,,) + A o 2 ( A , 2 ) + A O 3 ( A , 3 )  (2.4.13)  where A1X  = [(A.BJOjOj - (AlT)(2o1oJ + o 2 o 3 + 2o1o3)]/(AjB)2  A  = [(AJBJOJOJ  12  (2.4.14)  + O1O3)]/(A1B)2  (2.4.15)  = [(A.Bjo.o, - (A 1 T)(O 1 O 2 + o 1 2 )]/(A 1 B)  (2.4.16)  - (AJT)(OJ 2  and A13  Substituting eq. (2.4.13) into eq. (2.4.8) the following is obtained:  (2.4.17)  Evaluation of Aa„ From eq. (2.24) of the main text  °1°2°3  1'2  (2.4.18)  Designating the term inside the square brackets by A 2 , than a 2 = [A 2 P' 2  Differentiating Aa2 = A(A2)/2a2  - Evaluation of A(A7) As described  (2.4.19)  A  2  O,O2  2  °1°203 + o22o3 +  0,0,0,  Designating the numerator by A 2 T and the denominator by A 2 B, than  A 2 = A 2 T/A 2 B  Differentiating  A(A2) -  A(A2T)A2B - A2TA(A2B) ^  (2.4.20  - Evaluation of A(A,T) and A(A,B) As described  A 2 T = 0,0,03  Differentiating  A(A 2T) = AO,O 2O3  + AO2O,O3  +  AA 3o,o2'  (2.4.21)  and A 2 B = o,o 2 2 + o 2 2 o 3 + o,o2o3  Differentiating  A(A2B) = Ao,o22 + 2Ao2o,o2 + Ao 3 o 2 2 + 2Ao 2 o 2 o 3 + Ao,o2o3 + Ao2o,o3 + Ao3o,o2 = Ao,(o 2 2 +o 2 o 3 ) + + Ao2(2o,o2+2o2o3+o,o3) + Aa 3 (o 2 2 +o,o 2 )  (2.4.22)  Substituting A(A2T) from eq. into eq.  (2.4.20)  and A(A2B) from eq.  (2.4.21)  (2.4.22)  the following is obtained:  A(A2) = AOj(Aal) + Ao 2 (A J 2 ) + Ao 3 (A 23 )  (2.4.23)  where:  A21  = [(AaB)(o2O3)-(A2T)(O22+O2O3)]/(A2B)  A22  = [(A2B)(O1O3) - (A2T)(20l02 + 2O2O3 + oxo3)]/(AaB)2  Aa3  = [(AaB)(o1oa) - (A2T)(o22 + o1oa)]/(AaB)  (2.4.24)  (2.4.25)  and  Substituting  eq.  (2.4.23)  into  eq.  (2.4.19)  (2.4.26)  the  following is  obtained:  (2.4.27)  Evaluation of Aa, From eq.  (2.24)  of the main text  a, = [3  1/ 2 a i°2°3  +  °2°32  +  °1°32"  Designating the term inside the square brackets by A 3 , than:  (2.4.28)  a 3 = [A 3 P' 2  Differentiating Aa3 = A(A3)/2a3  (2.4.29)  as described above _ A  0  1°203  3 - 0l0203 + o2o32 +  0l03*  Designating the numerator by A 3 T and the numerator by A3B, than  A 3 = A3T/A3B  Differentiating A(A,T)A3B - A3TA(A3B) A(AJ = rTTTt (A3B)  A(2.4.30)  Evaluation of A(A,T) and A(A,B)  A(A3T) = A(AXT) =  AOjOjOj  +  AO2OXO3  + Ao 3 o 1 o 2  (2.4.31)  and A 3 B = 0,0,0, + o 2 o 3 2 +  0l03  2  Differentiating A(A3B) = A a 1 a 2 a 3 + Ao 2 o x o 3 + A a 3 a 1 a 2 + A o 2 o 3 2 + 2A o 3 o 2 o 3 + Ao^, 2 + Ao x o 3 2 +  2AO3O1O3  = AO1(O2O3+O32)  + AO2(O1O3+O32) +  AO3(O1O2+2O2O3+2O1O3)  (2.4.32)  Substituting A(A3T) from eq. (2.4.31) and A(A3B) from eq. (2.4.32) into eq. (2.4.30) the following is obtained:  A(A3) = A O 1 ( A 3 1 ) + Ao 2 (A 3 2 ) + A O 3 ( A 3 3 )  (2.4.33)  where: [(A 3B)(o2o3) - A 3 T ( O 2 O 3 + O,»)]/(A 3 B)»  (2.4.34)  AS1  =  A32  = [(A 3 B)(O 1 O 3 ) -  A33  = [(A3B)(O1O2) - A 3 T(O 1 O 2 +1 2O 2 O 3 + 2O1O3)]/(A3B)*  A 3 T( 0 i 0 3 + O 3 2 ) ] / ( A 3 B ) 2  (2.4.35)  and  Substituting  eq.  (2.4.33)  into  eq.  (2.4.29)  the  (2.4.36)  following  is  obtained:  Aa3 = (A31/2a3)Aox + (A32/2a3)Ao2 + (A33/2a3)Ao3 (2.4.37)  Substituting the values of Aa, from eq. (2.4.27) and Aa3  (2.4.17), Aa2  from eq.  from eq. (2.4.37) into eq. (2.4.6) the following is  obtained:  A(A) = AoJDSj  +  AX1 I T DAi  +  A21 21"  +  A3 x 2i" D A 3 ]  A12 A22 A32 + AO2[DS2 + ^ DA, + DA2 + J - DA3] '1 2 A  13  A  23  A  33  + AO3[DS3 + — DA, + j - DA2 + j - DA3] '1 ""2  designating the terms in brackets by TM1, TM2 and TM3, respectively, the following is obtained: A(A) = AojTMl] + AO 2 [TH2] + Ao3[TM3]  (2.4.38)  Substituting the value of A(A) from eq. (2.4.38) into eq. (2.4.4) the value of AT™- is obtained:  (2.4.39)  2.1.2 Evaluation of Ao^p o M p is given by (see main text, eq, (2.25)) onr  °SMP = °i a i 2  +  °* a * 2  +  °3 a 3 2  (2.4.40)  Differentiating  A o S M p = Ao.a,2 + 2a 1 o l Aa 1 + Ao2a2> + 2a 2 o 2 Aa 2 + Ao 3 a 3 * + 2a 3 o 3 Aa 3 (2.4.41) Substituting the values of Aax  from eq. (2.4.17), Aa2  from eq.  (2.4.27) and Aa3 from eq. (2.4.37) into eq. (2.4.41), the following is obtained:  AO  = AOJ(SIGM01) + AO2(SIGM02) + Ao3(SIGM03)  (2.4.42)  where: SIGM01  (a, 2 + o,A,, + o 2 A 2 , +  SIGM02  (a22  + o,A, 2 + O 2 A  22  +  O3A32)  SIGM03  (a32  + o,A, 3 + O 2 A  23  +  O3A33)  Substituting the values of AT  O3A31)  from eq. (2.4.39) and A o ^ from eq.  (2.4.42) into eq. (2.4.1) the following is obtained:  / TM1 , ^ . , TM2 , A , TM3 ,, (Ao, (^ ) + Ao2 2 (— ) + Ao, )]o 3 (— 2T 2t SMPp v m ) SMP SMP A(^) = °SMP  - [Ao,(SIGM01) + AO2(SIGM02) + Ao3(SIGM03)] T SMP  l/(o SMp )  or rearranging  (2.4.43)  where: TM1  TSMOBI =  SMP T S M 0 B 2  =  TM?  SMP T S M 0 B 3  =  SMP  o S M p - (SIGM01)TSMp]/(oSMp)2  °SMP " CSIGM02)xSMp]/(oSMp)2  °SMP " (SIGM03)TSMp]/(oSMp)2  RELATIONS BETWEEN INCREMENTS OF PRINCIPAL STRESS AND INCREMENTS OF CARTESIAN STRESS IN 3-DIMENSIONAL STRESS SPACE  Relations Between Increments of Principal Stress and Increments of Cartesian Stress in 3-Dimensional Stress Space  2.5.1 Relation Between Principal Stresses and Cartesian Stresses In Fig. 2.5.1 it is shown the plane ABC defined in the 3-Dimensional cartesian stress system x,y,z. Assuming that the direction, i, perpendicular to the plane is a principal direction, than the normal stress to the plane, o^is a principal stress and the plane ABC, on which there is no shear stress, is a principal plane. A relationship between the principal stress, o^ and the cartesian stresses o , o , o , x , T and x can be developed based on force r x y z xy yz zx equilibrium in the x,y and z directions and given by the following equations:  (o -a.) cos(i,x) + T cos(i,y) + T cos(i,z) = 0 ,J x i ' xy xz T yx cos(i,x) + (oy-a.)I  cos(i,y) ' J + xyz cos(i,z) = 0  (2.5.1)  xzx cos(i,x) + xzy cos(i,y) + (oz-o.) x cos(i,z) = 0  where cos(i,x), cos(i,y) and cos(i,z) are the direction cosines of direction i (i = 1,2,3) in respect to directions x,y and z, respectively. From the known relation for direction cosines, a fourth equation is obtained  Figure 2.5.1  Sketch of a Principal Plane  cos2(i,x) + cos2(i,y) + cos2(i,z) = 1  (2.5.2)  2.5.2 Relations Between Increments of Principal Stress, Increments of the Rotation of Principal Directions and Increments of Cartesian Stress Differentiating eqs. (2.5.1) and (2.5.2) the following is obtained:  (Aox-Ao.)cos(i,x) - (ox-a.)sin(i,x)A(i,x) + Axxycos(i,y) i i - x sin(i,y)A(i,y) + Ax cos(i,z) - x sin(i,z)A(i,z) = 0 xy xz xz Ax cos(i,x) - x sin(i,x)A(i,x) + (Ao -Ao.)cos(i,y) yx yx y I - (Oy-a^)sin(i,y)A(i,y) + Ax^costi.z) - x^zsin(i,z)A(i,z) = 0 Ax cos(i,x) - x sin(i,x)A(i,x) + Ax cos(i,y) - x sin(i,y)A(i,y) + zx zx zy ^y + (Ao -Ao.)cos(i,z) - (o -o.)sin(i,z)A(i,z) = 0 z 1 z 1 cos(i,x)sin(i,x)A(i,x) + cos(i,y)sin(i,y)A(i,y) + + cos(i,z)sin(i,z)A(i,z) = 0 (2.5.3)  The above equations constitute a system of A equations with the A unknowns: Ao^, A(i,x), A(i,y) and A(i,z). Since i = 1,2,3 than 3 systems of A equations with A unknowns are obtained and those will be solved as follows:  2.5.2.1 Relations between increment of principal stress, Ao l t vith increments of cartesian stress Substituting i=l in eqs. (2.5.3) the following is obtained:  Aa cos(l,x) - AO.cos(l,x) - (o -o,)sin(l,x)A(1,x) + AT cos(l,y) x x xy - T  xy  AT  sin(l,x)A(l,y) + AT  xz  COS(1,Z) - T  xz  sin(l,Z)A(1,Z) = 0  sin(l,X)A(1,X) + Ao cos(l.y) - Ao.cos(l.y) yx y - (o^-o1)sin(l,y)A(l,y) + Ar^cosd.z) - T^sind,z)A(l,z) = 0  COS(1,X) - T  yx  AT cos(l,x) - R sin(l,x)A(l,x) + AT cos(l.y) - T sind,y)A(l,y) + xz zx zy zy + Ao cos(l,z) - Ao.cos(l,z) - (o -o1)sin(l,z)A(l,z) = 0 z z cos(l,x)sin(l,x)A(l,x) + cos (1 ,y) sind ,y)A(l ,y) + + cos(l,z)sin(l,z)A(l,z) = 0  To simplify the above equations take the following form:  All-AOjfij-Ad ,x)Bll+Cll-A(l ,y)Dll+Ell-A(l ,z)Fll = 0  (2.5.A)  A21-A(l,x)B21+C21-Ao1m1-A(l,y)D21+E21-A(l,z)F21 = 0  (2.5.5)  A31-A(l,x)B31+C31-A(l,y)D31+E31-Ao1n1-A(l,z)F31 = 0  (2.5.6)  (2.5.7)  AA1 A(l,x)+BA1 A(1,y)+CA1 A(l,z) = 0  where:  All = Aa cos(l.x) x  A21 = AT cos(l,x) yx  Bll = (ox-o.)sin(l.x) B21 = Tyxsin(l,x) 1  A31 = At cos(l.x) zx B31 = Tzxsin(l,x)  Cll = AT^cosd.y)  C21 = ACyCos(1,y)  Dll = T^sind.y)  D21 = (Oy-Oj)sin(l,y) D31 = Tzysin(l,y)  Ell = AT  E21 =  COS(1,Z) XZ  Fll = T sin(l,z) xz  AT  COS(1,Z)  yz  F21 = T sin(l,z) yz  C31 = ATzycos(l,y)  E31 = Ao cos(l,z) z F31 = (o -ojsind.z) z 1  AA1 = cos(l,x)sin(l,x)  = cos(l,x)  BA1 = cos(l,y)sin(l,y)  iDj = cos(l,y)  CA1 = cos(l,z)sin(l,z)  n a = cos(l,z)  Note: all the above equations are designated as (2.5.8)  - Evaluation of the System of Equations (2.5.A) to (2.5.7) From eq. (2.5.6) Aox is obtained:  . A31 Ao.1 = nx  B31 A M , C31 - — A(l,x) + nx nx  \  D31 A M » . E31 — A(l,y) + nx nt  F31 A n , — A(l,z) n.  and from eq. (2.5.7)  (2.5.10)  Substituting eqs. (2.5.9) and (2.5.10) into Eq. (2.5.5) the following is obtained: RAT Ch1 A^I ,J A21 + 77T + 777 A41 B21 A(l,y) A41 B21 A(l,z) v i ' + C21 - —n, m.1  B41 B31 . , C41 B31 , C31 ^ D31 . M — m, A(l.y) - m — m, A(l,z) - — m l + — ra, AA(l.y)  m  — m l + n— ra, A(l,z) - D21 A(l,y) + E21 - F21 A(l,z) = 0 n i i  collecting terms in the above equation: . rB41 _ 0 1 B41 B31 A D31 A(l.y) l m B21 - m — ra, + — ra, - D21] x AM + A(1  B31 ^ r C A 1 Bno 1 XF31 z) [ 21 1111 + « A4l " M l iTT" HT  mi  iron " F21]  + A21 + C21 - n, — m, — m. + E21 = 0 1 - — ra, n, 1 - n, Designating the terms inside the square brackets by [FM1] and [FJ1] the following is obtained  A(l,y) [FM1] + A(1 ,z) [FJl] + A21 + C21 - — n i  m, - — n i  m, - — n i  m, + E21 = 0  and solving for A(l,y)  am i = A U , y ;  _LZ1am nz ; A21 C21 E21 A31 m * C31 m i E31 m i A U FM1 ' FM1 FM1 FM1 FM1 nl FM1 n x FM1 n. (2.5.11)  Substituting now the values of Ao, from eq. (2.5.9) and A(l,x) from eq. (2.5.10) into eq. (2.5.A) the following is obtained:  All - A31 ^  +  B31 ^  (- M i  2  A(1  , y ) _ gLI A ( l f Z ) ) _ C 3 1 li  2  2  *1 *1 *1 RAl + D31 — A(l,y) - E31 — + F31 — A(l,z) + Bll A(l,y) C41 + Bll  A(l,z) + Cll - Dll A(l,y) + Ell - Fll A(l,z) = 0  Collecting terms the following is obtained:  A(l,y) [-B31  +  +  D31 ^  A(l,z) (-B31 2  +  ^  +  2  Bll  - Dll]  F31 ^  +  2  Bll j g - Fll)  + All - A31 n— - C31 n— - E31 — + Cll + Ell = 0 n i i i  designating the terra inside the square bracket by FP1 and substituting the value of A(l,y) from eq. (2.5.11) the following is obtained:  FT1  FPl  FP1  FPl  FP1  m  !  -FP1 f g A(l.z) - § g A21 - f § C21 - f g E21 + § g A31 +  FPl  m  i  CA1  FPl  +  m i  § g C31-  C41  E31 — + A(1,z) (-B31 777 — + F31 — + Bll 777 - Fll) FM1 nx AA1 n x nx A41 i  i  i  + All - A31 n— - C31 n— - E31 — + Cll + Ell = 0 n i i i collecting terms:  A(l.z) [-FP1  - B31  ^  +  F31 ^  FPl  +  Bll  FPl  - Fll]  FPl  FPl  m i  All + Cll + Ell - § g A21 - f g C21 - § g E21 + (§g - - - ) A31  +  FPl  +  rai  FPl  m i  ) C31 + FM1 n x  nx  FM1 n x  ) E31 nx  =0  Designating the term inside the square brackets by FL1 and solving for A(l,z) the following equation is obtained:  A(l,z) = All(Alz) + Cll(Clz) + Ell(Elz) + A21(A2z) + C21(C2z) + E21(E2z) + A31(A3z) + C31(C3z) + E31(E3z)  where:  Alz = Clz = Elz = -  A2z = C2z = E2z =  FL1  FPl FL1 FM1  and ra FPl i A3z - C3z - E3z - -<2L - - _ )  l _  Substituting the value of A(l,z) from eq. (2.5.12) into eq. (2.5.11) the value of A(l,y) is obtained:  A(l,y) = -All (Alz)  - C21(C2z)  _  . FJ1 tijiUjZ] FM1  - Cll (Clz) f g - Ell(Elz) f g - A21(A2z) f g  " E21 (E2z)  A21 FM1  C21 FM1  " A31(A3z)  E21 A31 FM1 -j.-, +FM1 17M1n, „  - C31(C3z)  +  C31 FM1 TTM1n, ~  E31 FM1 n,  rearranging terras:  A(l,y) = All(Aly) + Cll(Cly) + Ell(Ely) + A21(A2y) + C21(C2y) + E21(E2y) + A31(A3y) + C31(C3y) + E31(E3y) (2.5.13)  where:  Aly = _Alz  Ml  Cly = -Clz Ely = -Elz FJ1 1 A2y = -(A2z ^ + m ) C2y = -(C2z f g E2y = -CE2z  ^  +  +  and r^ m A3y = -(A3z  F J 1  -m  1  ,ri — M C3y = -(C3z  F J 1  -)  E 3 y  FTl  = -(E3z m  1  - Fiii  -^  1  _)  ^  Substituting A(lz) from eq. (2.5.12) and A(l,y) from eq. (2.5.13) into eq. (2.5.10) A(l,x) is obtained as follows:  A(l,x) = All(Alx) + Cll(Clx) + Ell(Elx) + A21(A2x) + C21(C2x) + E21(E2x) + A31(A3x) + C31(C3x) + E31(E3x)  where:  B41. , C41,., . Alx = - M l ( Aa1l y ) " M l ( A lCzl) x B41... . C41.. . A2x = - A 4 l ( A 2 y ) " A41  =  C 2 x =  C41, B41 ~ A 4 1 ( C l y ) A41 " ^ T ( C El lz x)  =  C41, ( E 1 z ) B41 ~ A 4 l ( E l y ) A41 " T^  C41, B41 B41 C41, 2 z ) = - ^ ( E 2 y ) - T7T(E2Z) " A 4 l ( C 2 y ) A41 " T T T ( CE2x A41  A3x = - Hj(A3y) - £ £ x ( A 3 z )C3x = - f^(C3y) - g^(C3z) E3x = - Mf(E3y) - ^(E3z)  Substituting the values of A(l,z) from eq. (2.5.12), A(l,y) from eq. (2.5.13) and A(l,x) from eq. (2.5.14) into eq. (2.5.9) the value of Ao^^ is obtained as follows:  Ao.1 = —n - —n All (Alx) - —n Cll (Clx) - —nE l l (Elx) - — n A21(A2x) a 2 x x a — C21 (C2x) - n— n i i ^ n i  E21 (E2x) -n— i  E31(E3x) + ^ n n ! !  }  A31(A3x) - n— i _ D31 c n i  n ( c l  C31(C3x)  }  — Ell(Ely) - — A21(A2y) - — C21(C2y) - —n E21(E2y) n n n i i i i ^ A31 (A3y) - 231 C31(C3y) -^ J J n, n, n,  E31(E3y) + H i _ 111 J n, n,  A U ( A l z )  — Cll(Clz) - — Ell(Elz) - — A21(A2z) - —n C21(C2z) n n n i i i i — — E31(E3z) n, E21(E2z) - — n. A31(A3z) - — n, C31(C3z) - n.  collecting terras  Ao, = All(QA11) + Cll(QCll) + Ell(QEll) + A21(QA21) + C21(QC21) + E2KQE21) + A31CQA31) + C31(QC31) + E31(QE31)  where:  QA11 = -  n  i  (Alx) _ D 3 1 ( A l y ) _ m n n i i  ( A l z )  QC11 = - ^(Clx) - ^(Cly) - ^(Clz) n, n, n, QE11 = - — (Elx) - —(Ely) - —n (Elz) n n i i i QA21 = - —(A2x) - —(A2y) - —(A2z) n n n i i i QC21 = - n—(C2x) - —(C2y) - n—(C2z) n i i i QE21 = - —(E2x) - —(E2y) - n—(E2z) n n i i i QA31 = - ^±(A3x) - ^(A3y) - |^(A3z) + n n n i i i  n  i  QC31 = - —(C3x) - —(C3y) - — — n, n, (C3z) + n, QE31 = - —(E3x) -— (E3y) - —(E3z) + n— n n n i i i i  Since from eqs. (2.5.8)  All =  2. AO  A21 =  JL  AT  ,  A31 =  £. AT  xy , ZX  1  1  1  , x  Cll =  , Ell = n. AT 1 yx xz C21 = m. Ao , E21 = n. AT 1 1 y yz C31 = m. AT zy , E31 = n, Ao„ 1 1 7.V' z M. AT 1  and assuming & that AT xy = AT yx, AT yz = AT zy and AT xz = AT zx, then  Aox = Aox(Qxl) + Aoy(Qyl) + Aoz(Qzl) + Ar^Qxyl) + AT  (Qyzl) + AT  (2.5.15)  (Qzxl)  where: Qxl = fi^QAll) Qyl = m1(QC21) Qzl = nx(QE31) Qxyl = £1(QA21) + m^QCll) Qyzl = mx(QC31) + n1(QE21) Qzxl = Jx(QA31) + nx(QE11)  2.5.2.2 Relations Between Increment of Principal Stress, Ao3 with Increments of Cartesian Stress The procedures followed to obtain Ao3  in terms of increments of  cartesian stress are the same as the procedures followed for Aox and consist on the following. Substituting i = 3 in eq. (2.5.3) the following equations are obtained:  A13 - A o 3 £ 3 - A(3,x)B13 + C13 - A(3,y)D13 + E13 - A(3,z)F13 = 0 (2.5.4a)  A23 - A(3,x)B23 + C23 - Ao 3 m 3 - A(3,y)D23 + E23 - A(3,z)F23 = 0  (2.5.5a)  A33 - A(3,x)B33 + C33 - A(3,y)D33 + E33 - Ao 3 n 3 - A(3,z)F23 = 0 (2.5.6a)  A43 A(3,x) + B43 A(3,y) + C43 A(3,z) = 0  (2.5.7a)  where: A13 = Ao cos(3,x) x  A23 = Ax cos(3,x) yx  A33 = Ax cos(3,x) zx  B13 = (o -o )sin(l,x) x  B23 = x sin(3,x) yx  B33 = x sin(3,x) zx  C13 = Ax cos(3,y) xy  C23 = Ao cos(3,y) y  C33 = Axz cos(3,y) y  D13 = t sin(3,y)  D23 = (oy-o3)sin(3,y) D33 = xzysin(3,y)  E13 = Ax cos(3,z) xz  E23 = Ax cos(3,z) yz  E33 = Ao cos(3,z) z  F13 = x sin(3,z) xz  F23 = x sin(3,z) yz  F33 = (o -o,)sin(3,z) z 3  A43 = cos(3,x)sin(3,x)  B43 = cos(3,y)sin(3,y)  C43 = cos(3,z)sin(3,z)  S 3 = cos(3,x)  ra3  = cos(3,y)  n 3 = cos(3,z)  Note: all the above equations are designated as (2.5.8a).  - Evaluation of the System of Equations (2.A.4a) to (2.4.7a) From Eq. (2.5.4a)  .  AO 3 =  A13 3  D13 A , 0 . E13 , . C13 j — A(3 ,x) + -r—  B13  -R— A(3,x) + j — 3  3  A3  *3  F13 A(3,z) (2.5.9a)  and from eq. (2.5.7a)  (2.5.10a)  Substituting  eqs.  (2.5.9a) and  (2.5.10a) into eq.  (2.5.5a) the  following is obtained:  A23 +  B23 B43 A ( 3  A43  + B13  ra, fi7  'y)  +B23  C43  A43  A ( 3 , z )  A43 A ( 3 ' y ) " M 3  A(3  +  C23  "  m,  A13  F  m3 m3 + 013 ' z ) ) ~ £0 1 3 ~ fi ~  m3 m3 - E13 -r- + F13 j- A(3,z) - D23 A(3,y) + E23 - F23 A(3,z) = 0 3  3  Collecting terms in the above equation:  A(3  'y)  B13 B43 .n i . m 3 , A ,, , rB23 C43 A43 17 + 0 1 3 27 " D 2 3 ] + A ( 3 ' Z ) A43  ... . rB23 B43 ,y A43 *"  ri3  C43  a/Q A43  m  3  m  + 7" 23  F13  3  ra3  T~ ~ F23] + A23 + C23 + E23 - A13 £3 7 x 3  m3 m3 - C13 TE13 j= 0 3 3 Designating the terras inside the square brackets by FM3 and FJ3 the following is obtained: m3 m3 A(3,y)[FM3] + A(3,z)[FJ3] + A23 + C23 + E23 - A13 j- ~ c 1 3 J  0  3  and solving for A(3,y):  AfU-3, y\; = _ 111 A(* ^ _ M l _ £23 _ E23 A13.^ C13 ^ E13 ^ FM3 FM3 FM3 FH3 FM3 £ 3 FM3 £ 3 FM3 £ 3 (2.5.11a)  Substituting the values of Ao3 from eq. (2.5.9a) and A(3,x) from eq. (2.5.10a) into eq. (2.5.6a) the following is obtained:  A33 + B 3 3 4 3 4 3 A(3,y) + ^ j ^ 4 3 A(3,z) + C33 - D33 A(3,y) + E33 n  - 1 7  3  n  -  B 1 3  3  RA3  1 7 HI  B13 C43 n 3 -^Mr117  n M3  z  - >  -  3  C 1 3  17  n n n 3 3 3 + D13 -iA(3,y) E13 -j+ F13 y*3 A(3,z) - F33 A(3,z) = 0 *3 y3  Collecting terms the following is obtained  . rB33 B43 B13 B43 . AO.y) A43 " 0 3 3 " A43 T t +  n 013  », . A,_ , . B33 C43 IT3 + A ( 3 ' Z ) ( "  W  n n B13 C43 n 3 3 3 F3 " A/o IT + F13 T ~ 3 ) + A33 + C33 + E33 - A13 -r~ A43 Jl 3 it 3 S3 3  n  - C13 r  * 3  n  3  - E13 r  * 3  = 0  Designating the term inside the square bracket by FP3 and substituting the value of A(3,y) from eq. (2.5.11a) the following is obtained:  FP3 FJ3 FP3 ^ i P A(3'z) - I i A +  FP3 m 3 FM3 JJ C 1 3  + F13  n  3  +  A 2 3  FP3 -Ii  FP3 A m 17 E 1 3  c 2 3  FP3 -Ii  , ,B33 A,_ + A ( 3 , Z )  F33) + A33 + C33 + E33 - A13  X3  FP3 m 3 + Ii 17A13  E 2 3  C43 B13 C43 A43 A43 "3  C13  *3  n  3  *3  Tt n  3  E13 -=— = 0 *3  Collecting terms:  Afi ^ r FP3 FJ3 . B33 C43 [- - p g r - +  B13 C43 n 3 X4T" 1 7 +  FP3 FP3 + C33 + E33 + A23 (- g g ) + C23 (F P 3  + A13  (  m  3  n  3  l i 17" £7>  + 013  n F13  F 3 3 ] +  A33  FP3 + E23 (- | g )  FP3 m 3 n 3 l i 1 7 ' £T}  (  3 17 "  + E13  n FP3 m 3 3 fe 17" 17) =  0  Designating the term inside the square bracket by FL3 and solving for A(3,z) the following is obtained:  A(3,z) = A13(A13z) + C13(Cl3z) + E13(El3z) + A23(A23z) + C23(C23z) + E23(E23z) + A33(A33z) + C33(C33z) + E33(E33z) (2.5.12a) where: A13z = C13z = E13z =  FP3 m 3 7 FM3 S 3  n  3 23  1 FL3  FP3 1 A23z = C23z = E23z = FM3 FL3 A33z + C33z + E33z = "  Substituting the value of A(3,z) from eq. (2.5.12a) into eq. (2.5.11a) the value of A(3,y) is obtained:  A(3,y) = -  A13(A13z) -  C13(C13z) - f g E13(E13z)  -  A23 (A23z) -  C23 (C23z) -  E23(E23z)  -  A33 (A33z) -  C33 (C33z) -  E33(E33z)  A23_C23_E23A13 ^ C 1 3 ^ E 1 3 ^ FM3 FH3 FM3 FM3 2 3 FM3 2 3 FM3 2 3  rearranging terms:  A(3,y) = A13(A13y) + C13(C13y) + E13(E13y) + A23(A23y) + C23(C23y) + E23(E23y) + A33(A33y) + C33(C33y) + E33(E33y) (2.5.13a)  where: p to  1  m  A13y = C13y = E13y = - fjg (A13z) + ^ •C" TO  A23y + C23y + E23y = - fjg (A23z) -  3  j1  ^  A33y = C33y = E33y = - f^f (A33z) Substituting A(3,z) from eq. (2.5.12a) and A(3,y) from eq. (2.5.13a) into eq. (2.5.10a), A(3,x) is obtained as follows:  A(3,x) = A13(A13x) + C13(C13x) + E13(C13x) + A23(A23x) + C23(C23x) + E23(E23x) + A33(A33x) + C33(C33x) + E33(E33x)  (2.5.14a)  where:  A13x =- ||f(A13y)- jj|f(A13z> C13x =- ||§(Cl3y)- ||f(C13z) E13x =- |||(E13y)- §£§(E13z)  A23x =- |||(A23y)- f£f(A23z) C23x =- fff(C23y)- ^||(C23z) E23x  f*f(E23y)- g|(E23z)  A33x =- |||(A33y)- JJ^U33z) C33x =- fff(C33y)- ^(C33z) E33x =- ||f(E33y)- jff(E33z)  Substituting the values of A(3,z) from eq. (2.5.12a), A(3,y) from eq. (2.5.13a) and A(3,x) from eq. (2.5.14a) into eq. (2.5.9a) the value of Ao3 is obtained as follows:  Ao3 = A13(QA13) + C13(QC13) + E13(QE13) + A23(QA23) + C23(QC23) + E23(QE23) + A33(QA33) + C33(QC33) + E33(QE33)  where: QA13 = - f^(A13x) - ^(A13y) - f^(A13z) + f3  3  3  3  QC13 = - f^(C13x) - ^(C13y) - y^(C13z) + f3  *3  *3  *3  QE13 = - f^(E13x) - j^(E13y) - f^(E13z) + }*3  *3  3  QA23 = - f^(A23x) - j^(A23y) - f^(A23z) *3  *3  *3  QC23 = - f^(C23x) - j^(C23y) - f^(C23z) A •  A n  A »  QE23 = - f^(E23x) - j^(E23y) - f^(E23z) *3  *3  *3  QA33 = - f^(A33x) - j^(A33y) - f^(A33z) x3 *3 x3 QC33 = - f^(C33x) - j^(C33y) - y^(C33z) *3  3  3  QE33 = - f^(E33x) - y^(E33y) - y^(E33z) A a  A a  A *  *3  Since from eqs. (2.5.8a)  A13 = J.Ao 3  A23 =  , C13 = m,Ax 3  x  , C23 = m.Ao  J,AT 3  3  xy  A33 = S,3 AT  zx  ,  '  C33 =  ,  E13 = n3Ax 3 zx  ,  E23 = n,Ax 3 yz  yx  y  M,Ax 3  zy ,  and assuming that A x ^ = A x ^ , A x y z = A x ^ and A T  E33  xz  = n.Ao 3 2 = AT^,  then  Ao3 = Aox (Qx3) + Aoy (Qy3) + Aoz(Qz3) + Axxy(Qxy3) + Axyz (Qyz3) + Axzx (Qzx3) (2.5.15a)  where: Qx3 = fi3(QA13) Qy3 = m3(QC23) Qz3 = n3(QE33) Qxy3 = J3(QA23) + m3(QC13) Qyz3 = m3(QC33) + n3(QE23) Qzx3 = 23(QA33) + n3(QE13)  2.5.2.3 Relations Between Increment of the Principal Stress, Ao? with Increments of Cartesian Stress Substituting i = 2 in eqs. (2.5.3) the following equations are obtained:  A12 - Ao 2 2 2 - A(2,x)B12 + C12 - A(2,y)D12 + E12 - A(2,z)F12 = 0 (2.5.4b)  A22 - A(2,x)B22 + C22 - Ao.m. A(2,y)D22 + E22 - A(2,z)F22 = 0  (2.5.5b)  A32 - A(2,x)B32 + C32 - A(2,y)D32 + E32 - Ao 2 n 2 - A(2,z)F32 = 0 (2.5.6b)  (2.5.7b)  A42 A(2,x) + B42 A(2,y) + C42 A(2,z) = 0  where: A12 = Ao 2. x'  A22 = Ax 2. yx -  A32 = Ax 2. zx '  B12 = (o -o,)sin(2,x) x  B22 = x sin(2,x) yx  B32 = x sin(2,x) zx  C12 = Ax ra, xy 2  C22 = Aa ra,  C32 = Ax ra, zy 3  D12 = x sin(2,y) xy  D22 = (o -o2)sin(2,y) D32 = x sin(2,y) zy y  E12 = Ax n, xz J  E22 = Ax n, yz 5  E32 = Aa n,5 z  F12 = x sin(2,z) xz  F22 = x sin(2,z) yz  F32 = (a  A42 = 2 2 sin(2,x)  2 2 = cos(2,x)  B42 = m, sin(2,y)  m 2 = cos(2,y)  C42 = n 2 sin(2,z)  n 2 = cos(2,z)  Note; all the above equations are designated as (2.5.8b)  z  -a. )sin(2,z)  - Evaluation of the System of equations (2.5.4b) to (2.5.7b) From eq. (2.5.5b) Ao2 is obtained:  . A22 Ao. 1 = m,  B22 ... . — A(2,x) m.  C22 D22 . . E22 F22 . ,0 x A(2,y) + A(2,z) + m, m m, m (2.5.9b)  and from eq. (2.5.7b)  (2.5.10b)  Substituting  eqs.  (2.5.9b) and  (2.5.10b) into eq.  (2.5.6b) the  following is obtained:  A32 + B 3 2 4 B A 2 A(2,y) +  A(2,z) + C32 - A(2,y)D32 + E32 - A22  n  * B22 B42 A,_ , B22 C42 A , 0 > . " 2 . A,0 A(2 y) A(2 ^ A42 ' " ^ ~A42 ' Z ) + 57 0 2 2 A ( 2 ' y ) n2 n2 — E22 + — F22 A(2,z) m2 m2  n2 C22 - F32 A(2,z) = 0 m2  collecting terms in the above equation:  , rB32 B42 n,_ B22 B42 A(2,y) AA2 " 0 3 2 " 57 A42  +  ^ ^  D22]  R9? CAO n 2 "a 1,0 + — F22 - F32] - A22 m2 A42 m2 m2  + A(2  . rB32 C42 ' z ) [ A42  "j C22 — - E22 — ra2 ra2  + A32 + C32 + E32 = 0  Designating the terras inside the square brackets by FM2 and FJ2 the following is obtained: n2 A(2,y)[FM2] + A(2,z)[FJ2] - A22 — - C22 — - E22 ra2 m2 ra2 + A32 + C32 + E32 = 0  solving for A(2,y)  . . _ , _ _ FJ2 A , 0  AU.y;  -  F M 2  A U . Z J  ^ ., A22 +  F M 2  ^  C22 +  F M 2  m  E22 ^  +  F M 2  m  A32 ^  FM2  C32 FM2  E32 FM2  (2.5.11b)  Substituting now the values of Ao2 from eq. (2.5.9b) and A(2,x) from eq. (2.5.10b) into eq. (2.5.4b) the following is obtained: fi £ £ A12 - A22 ji + B22 ^ A(2,x) + D22 ^ _ E 2 2 !i + m2  F22  m2  A(2  , z ) - C22 ^ m2  2 M2,z) + D22 ^ A(2,y)  +  siy« A42  + C12 - A(2,y)D12 + E12 - A(2,z)F12 = 0  ,+ m g i A42  M  collecting terras the following is obtained:  fij A(2  «  y)  B22 BA2 . ^  + D 2 2  57  2  -D 1 2 ]  +  A(2  'z)  2  B22  (- — m2  . B12 BA2 57 +  + F22 — + 6 1 2 ^ 4 2 - F12) + A12 + C12 + E12 m2 AA2  AA2  ^ 2  £ 2  - A22  ^2  C22 m2  E22 — = 0 m2  m3  Designating the term inside the square bracket by FP2 and substituting the value of A(2,y) from eq. (2.5.11b) the following is obtained:  Ar?  ^ r[  FP2 FP2 ^F M 2 - — ^ 5  n  FP? +  i f  a  57  A 2 2  2 2  C A 2  M 2  FP? +  i f  C 2 2  57  ++ F22 F22 —^ ++ 6  FP? + l i  E 2 2  1 2  FP2 57 - i f  2 2  wpo E32 + A12 + C12 + E12 - A22 FM2  C A 2  - F121  M 2  FP2 -  A 3 2  J 2  c 3 2  2 2 E22 — =0 m2  C22 m2  i f  m  Designating the term in brackets by FL2 and rearranging:  A(2,z) [FL2] + A12 + C12 + E12 + A22 cfgf £ FP2 n 2 +  E 2 2  (  i f  solving for A(2,z):  2  FP2  57 - 5 7 > " i f  A 3 2  "  FP2  i f  +  C22 (§|§ £  FP2  C 3 2  "  i f  E 3 2  =  ID,  0  A(2,z) = A12(A12z) + C12(C12z) + E12(E12z) + A22(A22z) + C22(C22z) + E22(E22z) + A32(A32z) + C32(C32z) + E32(E32z) (2.5.12b)  where: A12z = C12z = E12z = -  1 FL2 n  A32z = C32z = E32z =  £  2 FP2 2 1 ) FM2 m 3 ra/ FL2  A22z = C22z = E22z = -  FP2  FH2 FL2  Substituting the value of A(2,z) from above into eq. (2.5.11b) the value of A(2,y) is obtained as follows:  A(2,y) = - (fjjf (A12z))A12 -  1 +  (  +  (  (C12z))C12 -  FT9  FH2 57 " I f  1  (A22Z))A22  +  (El2z))E12  FT9  -  - M f (C22z))C22  Fi2 57 " M l CE22z»E22 - ( ^ + i f (A32z))A32  " (FH2 + l i (32z))C32- ( ^ + ] § (E32z))E32  or  A(2,y) + (A12y)A12 + (C12y)C12 + (E12y)E12 + (A22y)A22 + (C22y)C22 + (E22y)E22 + (A32y)A32 + (C32y)C32 + (E32y)E32 (2.5.13b) where: A12y = C12y = E12y = - A12z FJ2 FM2 A22y = C22y = E22y  1 n„ FM2 m,  A32y = C32y = E32y  1 FM2  FJ2 A22z FM2  FJ2 A32z FM2  Substituting A(2,z) from eq. (2.5.12b) and A(2.y) from eq. (2.5.13b) into eq. (2.5.10b) A(2,x) is obtained:  A(2,x) + A12(A12x) + C12(C12x) + E12(E12x) + A22(A22x) + C22(C22x) + E22(E22x) + A32(A32x) + C32(C32x) + E32(E32x) (2.5.14b) where:  A12x =- |||(A12y)- ^||(A12z) Cl2x =- |||(Cl2y)- fff(C12z) E12x =- ||F(El2y)- J£§(E12Z)  A22x =- |||(A22y)- |||(A22z) C22x =- |||(C22y)- ^|(C22z) E22x =- ||f(E22y)- ||f(E22z)  A32x  |||(A32y)- ^§(A32z) C32x =- |||(C32y)- |||(C32z) E32x =- |||(E32y)- |||(E32z)  Substituting the values of A(2,z) from eq. (2.5.12b), A(2,y) from eq., (2.5.13b) and A(2,x) from eq. (2.5.14b) into eq. (2.5.9b) the value of Ao2 is obtained as follows: '  A  A22 _ m2  2  B22JU2  m2  _ B22.C12 m2  . B22_M2(A22x) _ B2^22 m2 m2 _ B2^32 m2  _ B22JI32 m2  3  }  (c22x)  _ B22_E12 m2  _ 8^22 m2  _ D2^A12 m2  _ D22_E12 m2  (e12  j _ D22_A22 m2  . D21A32 m2  3  _ D22_C32 m2  _ D22_m m2  _ I21A12 m2  12  _ F22_C12 m2  _ 122^12 m2  . F22_C22 m,  J  _ F21E22 m,  _ D22_C22 m2  (E22z)  ( m x )  (E22x)  (A32x)  _ 022^12 m2  3  J  _ U22E22 m2  (  (E32 J  _ F22A32 m.  _^ m2  z)  , + E22 m2 _ F2M22 m2  (A32z)  m.  (A22z)  _ F22_C32  collecting terms  Ao2 = A12(QA12) + C12(QC12) + E12(QE12) + A22(QA22) + C22(QC22) + E22(QE22) + A32(QA32) + C32(QC32) + E32(QE32)  where:  (c32z)  QA12 = - —(A12x) - ^22(A12y) - ^(A12z) X m ra ma 2 2 QC12 = - —(C12x) - —(C12y) - ^(Cl2z) ra2 ra2 m2 QE12 = - —(E12x) - —(El2y) - —(E12z) m2 m2 m2 QA22 = - ^(A22x) - j=p(A22y) - ^(A22z) + ra, ra, ra. QC22 = - —(C22x) - ^(C22y) - ^(C22z) + 1 ra2 ra2 m2 m. QE22 = - —(E22x) - ^(E22y) J - ^(E22z) + ^ m2 ra ra2 ra2 2 QA32 = - —(A32x) - —(A32y) - —(A32z) m2 m2 m2 QC32 = - —(C32x) - —(C32y) - —(C32z) m2 m2 m2 QE32 = - —(E32x) - ^(E32y) - ^(E32z) m2 m2 m2  Since from eqs. (2.5.8b)  A12 = Ao 2. x'  C12 = Axyxm, 2  E12 =  AT  2 xzn,  A22 = AT 2. xy <  C22 = Ao m,  E22 =  AT  n, yz 2  A32 = AT 2. zx '  C32 = AT zyM,2  E32 = Aozn,2  and assuming 6 that AT xy = AT yx; AT yz = AT zy; and AT xz = AT zx then  Aoa = Ao (Qx2) + Ao (Qy2) + Ao (Qz2) + AT (Qxy2) x y z xy + AT  yz (Qyz2) +  AT  zx (QZX2) (2.5.15b)  where: Qx2 = Sa(QA12) Qy2 = m2(QC22) Qz2 = n2(QE32) Qxy2 = S2(QA22)+m2(QC12) Qyz2 = m 2 (QC32)+n2(QE22) Qzx2 = fi2(QA32)+n2(QE12)  EVALUATION OF THE PLASTIC CONSTITUTIVE MATRIX (CP) OF THE- SMP MODEL  2.6.1  3-DIMENSIONAL FORMULATION  2.6.2  2-DIMENSIONAL FORMULATION  2.6.3  AXISYMMETRIC FORMULATION  \  Evaluation of the Plastic Constitutive Matrix (CP) of the SMP Model  2.6.1 3-Dimensional Formulation Evaluation of the Plastic Constitutive Matrix {CP} of the SMP Model Substituting  eq.  (2.55) into eq.  (2.49)  (see main  text), the  relationship between increments of plastic Cartesian strain {AeP} and increment of Cartesian stress {Ao) are obtained and given by: AeP X AeP  CP(1,1) CP(1,2) CP(1,3) CP(1,4) CP(1,5) CP(1,6)  Ao  CP(2,1) CP(2,2) CP(2,3) CP(2,4) CP(2,5) CP(2,6)  Ao  AeyP z  CP(3,1) CP(3,2) CP(3,3) CP(3,4) CP(3,5) CP(3,6)  Ao  Arp 'xy  CP(4,1) CP(4,2) CP(4,3) CP(4,4) CP(4,5) CP(4,6)  AX  p Ar 'yz  CP(5 ,1) CP(5 ,2) CP(5 ,3) CP(5,4) CP(5,5) CP(5,6)  AT  Arp zx _  CP(6,1) CP(6,2) CP(6,3) CP(6,4) CP(6,5) CP(6,6)  AT  xy yz zx  (2.6.1) where: p  CP(1,4) = ( 2 M.J.) TMXY/G P i-1.3 1 1  CP(1,2) = ( I M.S. TMY/G p i=l,3 1 1  CP(1,5) = ( X M.J.) TMYZ/G P i=l,3 1 1  CP(1,3) = ( S M.J.) TMZ/G P i=l,3 1 1  CP(1,6) = ( I M.J.) TMZX/G P i=l,3 1 1 .  CP(1,1) = ( I M.S.) TMX/G i=l ,3 1 1  (2.6.2)  Since for rows 2 to 6 a similar structure to the above is obtained, only the first and last term of each row is given below CP(2,1) = ( Z M.m?) TMX/G 1 i=l,3 1 1  (2.6.3)  CP(2,6) = ( X M.m?) TMZX/G ] i=l,3 1 1 CP(3,1) = ( Z M.n?) TMX/G 1 i=l,3 1 1  (2.6.4)  CP(3,6) = ( Z M.n?) TMZX/G 1 i=l,3 1 1 CP(4,1) = 2( Z M.2.m.) TMX/G 1 i-1.3 1 1 1 CP(4,6) = 2( Z M.2.m.) TMZX/G . i-1,3 1 1 1  (2.6.5) 1  CP(5 ,1) = 2( Z M.m.n.) TMX/G 1 i-1.3 1 1 1  (2.6.6)  CP(5,6) = 2( Z M.m.n.) TMZY/G 1 i-1.3 1 1 1 CP(6,1) = 2( Z M.n.2.) TMX/G 1 i-1.3 1 1 1  (2.6.7)  CP(6,6) = 2( Z M.n.2.) TMZX/G 1 i-1.3 1 1 1 2.6.2 2-Dimensional Formulation 2-D FE analysis are, in general, carried out more often in practice that 3-D FE analysis because the 3-D analysis require more time to define and input the relevant data (nodes and soil elements) and also because a great deal more computer time is required to numerically solve the problem  at hand.  Therefore it is considered useful to adapt the 3-D formulation  that has been described in the previous sections to 2-D.  This simply  requires imposing the necessary plain strain boundary conditions, and this way the 3-D characteristics will not be lost because the 2-D formulation will still be able to consider the influence of the intermediate principal stress, o 2 , as will be described next. The stress-strain relationship given in Eq. (2.6.1) is rewritten below. Cll C12 C13 CIA C15 C16  Ao  C21 C22 C23 C2A C25 C26  Ao  AeP z  C31 C32 C33 C3A C35 C36  Ao z  P  CA1 CA2 CA3 CAA CA5 CA6  AT  A 'yz r  C51 C52 C53 C5A C55 C56  AT  ATP  C61 C62 C63 C6A C65 C66  AT  Ae?"  X  X  Ae  P  y  A  R 'xy  p  ' zx  y  xy yz zx  where the C.. terms were described in eq. (2.6.2) to (2.6.7). Assuming that the 2-D Cartesian coordinate system is defined by the x-axis (horizontal) and z-axis vertical, then all the terms associated with xy or yz, i.e.AR'xy' , Ar'yz' , AT xy and ATyz can be deleted since there is no J J i contribution from these terms in the 2-D plane strain analysis. Therefore, eq. (2.6.8) will take the following form:  ~C11 C12 C13 C16~  Ao  Ae  C21 C22 C23 C26  Ao  Ae z  C31 C32 C33 C36  Ao z  AR  C61 C62 C63 C66  AT  Ae X  y  ' zx  X  y  zx  By renumbering the above terms the following is obtained:  Ae  Cll C12 C13 CIA  Ao  Ae  C21 C22 C23 C2A  Ao  Ae z  C31 C32 C33 C3A  Ao  CA1 CA2 CA3 CAA  AT  Since in 2-D plain strain analysis, Ae  =0,  X  y  _A r z 3 i  (2.6.10)  zx  then from above, the  y  3  fo