Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Discrete element modeling of direct simple shear response of granular soils and model validation using… Dabeet, Antone 2014

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

Item Metadata

Download

Media
24-ubc_2014_september_dabeet_antone.pdf [ 5.92MB ]
Metadata
JSON: 24-1.0167563.json
JSON-LD: 24-1.0167563-ld.json
RDF/XML (Pretty): 24-1.0167563-rdf.xml
RDF/JSON: 24-1.0167563-rdf.json
Turtle: 24-1.0167563-turtle.txt
N-Triples: 24-1.0167563-rdf-ntriples.txt
Original Record: 24-1.0167563-source.json
Full Text
24-1.0167563-fulltext.txt
Citation
24-1.0167563.ris

Full Text

     DISCRETE ELEMENT MODELING OF DIRECT SIMPLE SHEAR RESPONSE OF GRANULAR SOILS AND MODEL VALIDATION USING LABORATORY TESTS   by   ANTONE DABEET   B.Sc. The American University in Cairo, 2005 MASc. University of British Columbia, 2009       A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQIURMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY    in    THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES  (CIVIL ENGINEERING)    THE UNIVERSITY OF BRITISH COLUMBIA  (VANCOUVER)   July 2014     © Antone Dabeet, 2014  ii  ABSTRACT  The direct simple shear (DSS) device is one of the most commonly used laboratory testing tools to characterize the shear behavior of soils. In the Norwegian Geotechnical Institute (NGI) version of the DSS test, where a cylindrical soil specimen is confined by a wire-reinforced membrane, only normal and shear stresses on the horizontal planes are measured. The knowledge of these stresses alone does not provide adequate information to calculate friction angles for use in geotechnical design. Further, the absence of complementary shear stresses at the soil-membrane interface causes stress non-uniformities within DSS specimens, which makes the task of interpreting DSS testing results even more difficult. With the recent advances in computers, it is now possible to model soil in a realistic manner as a collection of particles using the discrete element method (DEM). With this background, a DEM model of a cylindrical DSS specimen was developed to provide insight on the state of stress and strain in DSS specimens. A laboratory DSS testing program was undertaken on glass beads as part of this study. The results of the glass beads tests were used for comparison with the DEM model results. Further, free-form sensors (paper-thin flexible pressure sensors mounted on the reinforced part of the DSS membrane) were used to measure lateral stresses acting on reconstituted Fraser River silt specimens. It was shown that: i) the adopted DEM modeling approach is effective in capturing the salient characteristics of the DSS behavior of the tested glass beads; ii) during the shearing phase, the distribution of shear strains across the specimen is more uniform at lower shear strain levels; iii) significant stress non-uniformities during shearing are limited to a narrow zone of about two particles diameter near the lateral boundaries, while stresses at central specimen locations are relatively more uniform (i.e. most representative of “ideal” simple shear conditions); and iv) at large shear strains, the horizontal plane becomes the plane of maximum obliquity, and the friction angle calculated using the stress state on the horizontal plane is a good approximation to the mobilized friction angle at such strain levels.   iii  PREFACE  Chapter 5: Versions of Chapter 5 have been published as follows:    A. Dabeet, D. Wijewickreme, and P. Byrne. Application of discrete element modeling for simulation of cyclic direct simple shear response of granular materials. Accepted for publication in proceedings of the 10th U.S National Conference on Earthquakes Engineering, Anchorage, Alaska.  A. Dabeet, D. Wijewickreme, and P. Byrne. Simulation of cyclic direct simple shear loading response of soils using discrete element modeling. In 15th World Conference on Earthquakes Engineering, Lisboa, 2012.  A. Dabeet, D. Wijewickreme, and P. Byrne. Discrete element modeling of direct simple shear response of granular soils and model validation using laboratory element tests. In 14th Pan-Am. Conference and 64th Canadian Geotechnical conference, Toronto, 2011.  Chapter 6: Previous versions of Chapter 6 have been published or submitted for possible publication as follows:    A. Dabeet, D. Wijewickreme, and P. Byrne. Evaluation of stress strain uniformities in the laboratory direct simple shear test specimens using 3D discrete element analysis. Submitted for possible publication, 2014.  A. Dabeet, D. Wijewickreme, and P. Byrne. Evaluation of the stress-strain uniformities in the direct simple shear device using 3D discrete element modeling. In 63rd Canadian Geotechnical Conference, Calgary, 2010.  Chapter 7: A previous version of Chapter 7 has been published as follows:   D. Wijewickreme, A. Dabeet, and P. Byrne. Some observations on the state of stress in the direct simple shear test using 3D discrete element analysis. Geotechnical Testing Journal, 36(2):292–299, 2013.  I was the lead investigator, responsible for all major areas of concept formation, data collection and analysis, as well as manuscript composition for the above listed manuscripts. D. Wijewickreme was the supervisory author and P. Byrne was the co-supervisory author. They were involved in concept formation and in manuscript review. The remaining parts of the dissertation (i.e., Chapters 1,2,3,4, and 8) are original, unpublished, independent work of the author, A. Dabeet.    iv  TABLE OF CONTENTS  ABSTRACT................................................................................................................................................... II PREFACE .................................................................................................................................................... III TABLE OF CONTENTS .............................................................................................................................. IV LIST OF TABLES ....................................................................................................................................... VII LIST OF FIGURES .................................................................................................................................... VIII LIST OF SYMBOLS ................................................................................................................................... XV ACKNOWLEDGEMENTS ....................................................................................................................... XVIII 1 INTRODUCTION .................................................................................................................................. 1 1.1 Thesis objectives ........................................................................................................................... 3 1.2 Thesis organization ....................................................................................................................... 4 2 LITERATURE REVIEW ....................................................................................................................... 7 2.1 Typical soil response as observed from results of DSS testing ............................................... 8 2.2 Stress-strain uniformities in DSS specimens ............................................................................. 9 2.3 Interpretation of stress state in DSS specimens ...................................................................... 14 2.4 DEM analysis ................................................................................................................................ 16 2.5 Closure .......................................................................................................................................... 20 3 EXPERIMENTAL ASPECTS ............................................................................................................. 33 3.1 UBC-DSS device ........................................................................................................................... 33 3.2 Lateral stress measurement in DSS specimens ....................................................................... 34 3.2.1 Free form sensors system for lateral stress measurement ....................................................... 35 3.2.2 Calibration of free form sensors................................................................................................. 36 3.2.3 Accuracy of free form sensors ................................................................................................... 37 3.2.4 Evaluation of possible specimen disturbance due to the addition of free form sensors ........... 39   v  3.3 Selection of materials for DSS testing ....................................................................................... 40 3.4 DSS testing procedure ................................................................................................................ 41 3.4.1 Specimen setup ......................................................................................................................... 41 3.4.2 Specimen reconstitution method ............................................................................................... 42 3.4.3 Consolidation phase .................................................................................................................. 43 3.4.4 Shearing phase .......................................................................................................................... 43 3.5 Testing program ........................................................................................................................... 44 4 DSS TESTING RESULTS .................................................................................................................. 58 4.1 DSS testing of glass beads to generate data for validation of discrete element models (Series I) ..................................................................................................................................................... 58 4.1.1 Monotonic shearing ................................................................................................................... 58 4.1.2 Cyclic shearing .......................................................................................................................... 60 4.1.3 Discussion on DSS testing performed to generate data for validation of discrete element models 62 4.2 DSS testing for investigation of the development of lateral stresses during shearing (Series II) .................................................................................................................................................... 64 4.2.1 Consolidation phase .................................................................................................................. 64 4.2.2 Monotonic shearing ................................................................................................................... 65 4.2.3 Cyclic shearing .......................................................................................................................... 65 4.2.4 Discussion of results of DSS testing for the investigation of the development of lateral stresses during shearing ....................................................................................................................................... 68 4.3 Closure .......................................................................................................................................... 69 5 DEVELOPMENT OF A DISCRETE ELEMENT MODEL OF THE DSS TEST .................................. 93 5.1 Overview of the discrete element program PFC3D .................................................................... 94 5.2 Analysis methodology ................................................................................................................. 95 5.3 Input parameters and sensitivity analysis................................................................................. 97 5.4 Evaluation of the performance of the PFC3D model of the DSS test ..................................... 100 5.5 Observations on lateral stresses from simulations results .................................................. 102 5.6 Closure ........................................................................................................................................ 103 6 EVALUATION OF STRESS STRAIN UNIFORMITIES IN THE LABORATORY DIRECT SIMPLE SHEAR TEST SPECIMENS ..................................................................................................................... 120 6.1 Analysis methodology and numerical simulations ................................................................ 121 6.2 Results from numerical simulations ........................................................................................ 122 6.2.1 Stress uniformities at the end of consolidation ........................................................................ 122 6.2.2 Shear strain uniformities .......................................................................................................... 124 6.2.3 Stress uniformities during the shearing phase ........................................................................ 126   vi  6.3 Observations on stress uniformities from measured lateral stresses ................................. 132 6.4 Discussion on stress-strain uniformities ................................................................................ 133 6.5 Conclusions ................................................................................................................................ 137 7 EVALUATION OF THE STATE OF STRESS IN THE DIRECT SIMPLE SHEAR (DSS) TEST ..... 163 7.1 DEM analysis .............................................................................................................................. 163 7.2 Results of DEM Analysis ........................................................................................................... 165 7.2.1 DSS mobilized friction angle .................................................................................................... 165 7.2.2 Principal stresses and stress invariants .................................................................................. 167 7.3 Interpretation of DSS stress state using stresses measured at the boundaries ................ 169 7.4 Discussion and Conclusions .................................................................................................... 170 8 SUMMARY AND CONCLUSIONS .................................................................................................. 181 8.1 DSS testing using free form sensors ....................................................................................... 181 8.2 Evaluation of the performance of a DEM model of the DSS test .......................................... 182 8.3 Stress-strain uniformities in DSS specimens ......................................................................... 183 8.4 Interpretations of the state of stress in DSS specimens ....................................................... 184 8.5 Limitations of the current work and suggestions for future research ................................. 185 REFERENCES .......................................................................................................................................... 187 APPENDIX A: ADDITIONAL DEM MODEL RESULTS ........................................................................... 195    vii  LIST OF TABLES  Table 2.1. Summary of previous studies on stress-strain uniformities in DSS specimens. ........................ 21 Table 3.1. Summary of DSS testing program. ............................................................................................ 45 Table 4.1. Summary of the results of DSS testing performed on glass beads (Series I). .......................... 71 Table 4.2. Summary of the results of DSS testing performed to assess DSS lateral stresses on Fraser River silt (Series II). .............................................................................................................................. 72 Table 5.1. Summary of PFC3D model input parameters for glass beads. ................................................. 104 Table 5.2. Summary of simulations results at the end of the consolidation phase ................................... 105 Table 6.1. DEM model input parameters and boundary and loading conditions for the simulated cases. ........................................................................................................................................................... 139 Table 6.2. Stress uniformity parameters evaluated at the end of the consolidation phase. ..................... 140     viii  LIST OF FIGURES  Figure ‎1.1. Schematic showing a cross-section through a DSS specimen subjected to “ideal” simple shear conditions. .............................................................................................................................................. 6 Figure ‎2.1. Monotonic constant volume response of loose air-pluviated Fraser River sand (after Wijewickreme et al., 2005): a) stress path; b) stress-strain response. ................................................ 25 Figure ‎2.2. Cyclic response of loose air-pluviated Fraser River sand (after Wijewickreme et al., 2005): a) stress path; b) stress-strain response. ................................................................................................. 26 Figure ‎2.3. Cyclic response of Fraser River silt (after Sanin & Wijewickreme, 2006): a) stress path; b) stress-strain response.......................................................................................................................... 27 Figure ‎2.4. Normalized vertical effective stress distribution on the top and bottom horizontal specimen boundaries obtained from analysis and experimental results (after Budhu & Britto, 1987). ............... 28 Figure  2.5. Schematic representation of stress state in the center of a DSS element test. ....................... 29 Figure  2.6. Schematic showing Mohr circles and failure envelops representing two possible stress states for the DSS specimen as discussed in Section 2.3: a) horizontal plane corresponding to maximum shear stress; b) horizontal plane corresponding to maximum stress obliquity. ................................... 30 Figure ‎2.7. Schematic showing the projection of two balls in contact in a DEM simulation. ...................... 31 Figure ‎2.8. Flow diagram illustrating DEM calculation cycle (modified from O’Sullivan, 2011b). ............... 32 Figure  3.1. Schematic showing a cross-section through a DSS specimen parallel to the direction of shearing and stresses acting on the specimen: a) at the end of consolidation and before shearing; b) during the shearing phase. .................................................................................................................. 46 Figure  3.2. Schematic of the UBC-DSS device. ......................................................................................... 47 Figure  3.3. Photograph showing a free form sensor (D= 18 mm) connected to the data acquisition cable. ............................................................................................................................................................. 48 Figure  3.4. Schematic illustrating the arrangement used to calibrate free form sensors. .......................... 49 Figure  3.5. Typical results for a calibration of a free form sensor. .............................................................. 50 Figure  3.6. Results of a load-unload calibration performed on a free form sensor..................................... 51 Figure  3.7. Potential error due to hysteresis effect. .................................................................................... 52   ix  Figure  3.8. Shear stress-strain plots for tests with and without the addition of the free form sensors. ...... 53 Figure  3.9. Recorded equivalent shear stress for the membrane and sensor-arrangement plotted against shear strain. ......................................................................................................................................... 54 Figure  3.10. Shear stress strain plots for tests performed with and without the addition of free form sensors after applying the additional membrane stiffness correction. ................................................. 55 Figure  3.11. Photographs showing the free form sensors attached to the DSS membrane a) before specimen preparation; b) during specimen preparation; c) during the shearing phase. ..................... 56 Figure  3.12. Photograph of the tested glass beads placed in the DSS cavity taken at the end of specimen preparation phase. ............................................................................................................................... 57 Figure  4.1. Results of drained DSS tests performed on glass beads: a) shear stress versus shear strain; b) volumetric strain versus shear strain; c) stress ratio versus shear strain. ....................................... 73 Figure  4.2. Results of monotonic constant volume DSS tests performed on glass beads: a) shear stress strain plot; b) shear stress ratio c) stress path; d) normalized stress path. ......................................... 74 Figure  4.3. Results of a cyclic constant volume DSS test performed on glass beads ('vc = 100 kPa and CSR = 0.078): a) stress path; b) shear stress-strain response. .......................................................... 75 Figure  4.4. Results of a cyclic constant volume DSS test performed on glass beads ('vc = 100 kPa and CSR = 0.109): a) stress path; b) shear stress-strain response. .......................................................... 76 Figure  4.5. Results of a cyclic constant volume DSS test performed on glass beads ('vc = 200 kPa and CSR = 0.104): a) stress path; b) shear stress-strain response. .......................................................... 77 Figure  4.6. Cyclic stress ratio versus number of cycles to reach  of 3.75% (single amplitude) for the tested glass beads for 'vc values of 100 kPa, 150 kPa, and 200 kPa. ............................................... 78 Figure ‎4.7. Number of cycles to reach  of 3.75% versus normalized cyclic resistance ratio (CRR) for the tested glass beads. .............................................................................................................................. 79 Figure  4.8. Shear stress versus vertical effective stress relationship obtained from the results of DSS testing on glass beads calculated at end of test (i.e.  of 8%). ............................................................ 80 Figure  4.9. Coefficient of earth pressure at rest computed from the initial consolidation phase of DSS tests on Fraser River silt. ..................................................................................................................... 81   x  Figure  4.10. Response of Fraser River silt during monotonic constant volume DSS shearing: a) shear stress-strain response; b) vertical effective stress versus shear strain; c) horizontal effective stress versus shear strain. .............................................................................................................................. 82 Figure  4.11. Change in lateral earth pressure coefficient with the development of shear strain during monotonic loading. ............................................................................................................................... 83 Figure  4.12. Response of Fraser River silt during cyclic DSS shearing with lateral stress measurement ('vc = 200 kPa and CSR = 0.155): a) vertical effective stress path; b) horizontal effective stress path; c) shear stress-strain response. .......................................................................................................... 84 Figure  4.13. Response of Fraser River silt during cyclic DSS shearing with lateral stress measurement ('vc = 200 kPa and CSR = 0.121): a) vertical effective stress path; b) horizontal effective stress path; c) shear stress-strain response. .......................................................................................................... 85 Figure  4.14. Response of Fraser River silt during cyclic DSS shearing with lateral stress measurement ('vc = 300 kPa and CSR = 0.17): a) vertical effective stress path; b) horizontal effective stress path; c) shear stress-strain response. .......................................................................................................... 86 Figure  4.15. Development of lateral earth pressure coefficient with the increase in number of cycles for cyclic DSS shear testing of Fraser River silt conducted with 'vc = 200 kPa. ...................................... 87 Figure  4.16. Development of lateral earth pressure coefficient with the increase in number of cycles for cyclic DSS shear testing of Fraser River silt conducted with 'vc = 300 kPa. ...................................... 88 Figure  4.17. Typical results of a cyclic DSS test with lateral stress measurement performed on an over consolidated Fraser River silt specimen: a) vertical effective stress path; b) horizontal effective stress path; c) shear stress-strain. ................................................................................................................. 89 Figure  4.18. Development of lateral earth pressure coefficient with the increase in number of cycles for overconsolidated Fraser River silt. ...................................................................................................... 90 Figure  4.19. Cyclic resistance ratio versus number of cycles relation observed from DSS testing of Fraser River silt. .............................................................................................................................................. 91 Figure ‎4.20. Number of cycles to reach  of 3.75% versus normalized cyclic resistance ratio (CRR) for normally consolidated Fraser River specimens. .................................................................................. 92   xi  Figure  5.1. Side view illustration of the PFC specimen prepared by numerically simulated pluviation: a) particles just prior to pluviaton; b) particles after application of vertical stress. ................................. 106 Figure  5.2. Shear stress strain response for simulated specimens with  values of 5x10-7/cycle and 2.5x10-7/cycle. .................................................................................................................................... 107 Figure  5.3. Comparison between odometer test results and the results of a simulated odometer test performed using the non-linear Hertz-Mindlin model with shear modulus values of 1 GPa, 1.5 GPa, and 2.5 GPa. ...................................................................................................................................... 108 Figure  5.4. Comparison between odometer test results and the results of a simulated odometer test using a linear constitutive model with constant contact stiffness parameter, K, values of 100 kN/m, 250 kN/m, 500 kN/m, and 1000 kN/m. ...................................................................................................... 109 Figure  5.5. Effect of density scaling as observed from the results of two odometer test simulations. ..... 110 Figure  5.6. Effect of damping coefficient on the results of the simulated odometer test. ......................... 111 Figure  5.7. Results of three drained DSS simulations performed at 'vc values of 100 kPa, 150 kPa, and 200 kPa along with the corresponding DSS test results performed on glass beads a) shear stress strain response; b) volumetric strain vs. shear strain. ....................................................................... 112 Figure  5.8. Results of three monotonic constant volume DSS simulations performed at 'vc values of 100 kPa, 150 kPa, and 200 kPa along with the corresponding DSS test results performed on glass beads a) shear stress strain response; b) stress path plot. .......................................................................... 113 Figure  5.9. Results of a cyclic constant volume simulation with 'vc of 100 kPa and CSR of 0.08: a) stress path plot; b) shear stress strain response.......................................................................................... 114 Figure  5.10. Results of a cyclic constant volume simulation with 'vc 0f 100 kPa and CSR of 0.12: a) stress path plot; b) shear stress strain response. .............................................................................. 115 Figure  5.11. Results of a cyclic constant volume simulation with 'vc 0f 200 kPa and CSR of 0.1: a) stress path plot; b) shear stress strain response.......................................................................................... 116 Figure  5.12. CSR vs. number of cycles to reach  of 3.75% from results of simulations and laboratory DSS testing. ....................................................................................................................................... 117   xii  Figure  5.13. Variation in lateral earth pressure coefficient with the development of shear strain obtained from the results of PFC simulations: a) drained conditions; b) constant volume conditions. ............ 118 Figure  5.14. Change in lateral earth pressure coefficient during cyclic loading for simulated specimens. ........................................................................................................................................................... 119 Figure ‎6.1. Top view schematic of the DSS specimen showing the locations of the measurement spheres. ........................................................................................................................................................... 141 Figure ‎6.2. Schematic showing top view of the DSS specimen illustrating selected locations of segments used for radial stress calculation. ...................................................................................................... 142 Figure ‎6.3. The distribution of vertical effective stress at the end of consolidation, zzc, across the specimen for a specimen consolidated to boundary vertical effective stress of 100 kPa. ................ 143 Figure ‎6.4. Displacements in the x-axis direction of particles with centers located near x = -2.5 cm and y = 0 cm during the shearing phase. ....................................................................................................... 144 Figure ‎6.5. Calculated shear strains computed from particle locations (near y = 0 cm) at boundary shear strain of 20 %. .................................................................................................................................... 145 Figure ‎6.6. Internal shear strains calculated from particles movement versus boundary shear strain. .... 146 Figure ‎6.7. Side view of a simulated specimen: a) at the end of consolidation; b) at boundary shear strain of 20%. ............................................................................................................................................... 147 Figure ‎6.8. Stress ratio calculated at locations of MS 1- MS 5 on average and at the boundary versus boundary shear strain. ....................................................................................................................... 148 Figure ‎6.9. Stress uniformity coefficients c and b versus boundary shear strains for the simulation performed with F = 0.176 and 'vc of 100 kPa.................................................................................... 149 Figure ‎6.10. Results of drained simulations performed with F values of 0.176, 0.3, and 0.5: a) shear stress-strain response; b) volumetric strain versus shear strain. ...................................................... 150 Figure ‎6.11. Change in stress uniformity parameter, c, with the development of shear strain for simulations performed with F values of 0.176, 0.3, and 0.5. ............................................................. 151   xiii  Figure ‎6.12. Change in stress uniformity coefficient, c, with the development of shear strain for drained simulations performed on specimens consolidated to 'vc of 100 kPa, 150 kPa, 200 kPa, and 400 kPa. .................................................................................................................................................... 152 Figure ‎6.13. Change in stress uniformity coefficientc, with the development of shear strain for drained and constant volume monotonic simulations. .................................................................................... 153 Figure ‎6.14. Stress uniformity parameter, c, versus shear strain for simulations performed with lateral boundary friction coefficients of 0, 0.176, and 1. ............................................................................... 154 Figure ‎6.15. Drained cyclic response a simulated DSS specimen: a) shear stress versus shear strain; b) volumetric strain versus shear strain. ................................................................................................ 155 Figure ‎6.16. Stress uniformity coefficientc, versus number of loading cycles. ...................................... 156 Figure ‎6.17. Distributions of radial stresses along the specimen height for segments 1-4 calculated at the end of the consolidation phase. ......................................................................................................... 157 Figure ‎6.18. Distributions of radial stresses along the specimen height for segments 1-4 calculated at the instant of boundary shear strain of 20 %. .......................................................................................... 158 Figure ‎6.19. Distributions of normalized vertical stress ('v/'v(average)) acting on the top and bottom boundaries during shearing. .............................................................................................................. 159 Figure ‎6.20. Images showing simulated particles and force chains captured at the end of the consolidation phase: a) central cross-section; b) top specimen view. ............................................... 160 Figure ‎6.21. Images showing simulated particles and force chains captured at the instant of boundary shear strain of 20 %: a) central cross-section; b) top specimen view. .............................................. 161 Figure ‎6.22. Experimental results showing normalized horizontal stresses acting at the locations of sensors 1-4 with the development of shear strain. ............................................................................ 162 Figure  7.1. Computed stresses versus boundary shear strain () from DEM: a) stress ratio, 'xx b) lateral stress, xx. .............................................................................................................................. 173 Figure  7.2. Stress paths, Mohr circle, and stress state at maximum obliquity for the simulated DSS specimen at shear strains of 30% (F = 0.176). .................................................................................. 174   xiv  Figure  7.3. Angles , , and mob with the development of boundary shear strain,  , for simulations with Interparticle friction coefficients of: a) 0.176; b) 0.3; and c) 0.5, respectively.................................... 175 Figure ‎7.4. Principal stresses calculated for central specimen locations (MS 2 through MS 4) versus shear strain for the simulation performed with F value of 0.176. ................................................................. 176 Figure ‎7.5. Angle between the major principal stress, '1, and the vertical direction, , versus shear strain (F = 0.176). ........................................................................................................................................ 177 Figure ‎7.6. Stress ratio, h/'v, versus the tangent of  (F = 0.176). ....................................................... 178 Figure ‎7.7. Stress path in 'q 'm space for the simulated specimen with F of 0.176. ............................ 179 Figure ‎7.8. Angle  (calculated from boundary stresses) versus DSS mobilized friction angle, mob, for the simulation with F of 0.176. ................................................................................................................. 180    xv  LIST OF SYMBOLS   a, b   slope and intercept, respectively, as defined in Equation 3.1. CRR    cyclic resistance ratio equal to CSR required to reach  of 3.75% within 15 loading cycles.  CSR   cyclic stress ratio (cy/'vc).  D50   average particles diameter.  eo, ec   void ratios calculated prior to consolidation and at the end of the consolidation phase, respectively.   F    friction coefficient at contacts.Fn, Fs  contact forces in the normal and shear directions, respectively.  G    contact shear modulus used as an input parameter for the Hertz-Mindlin contact model. Gs   specific gravity.  Kn, Ks    normal and shear contacts stiffness parameters, respectively. k   coefficient of earth pressure.   ko   coefficient of earth pressure at rest.  k   correction factor for the effect of vertical effective stress as shown in Equation 4.1. MS   measurement sphere.OCR   overconsolidation ratio.  r  particles radius.  Stress ratio angles as defined in Equations 2.2 and 2.3, respectively.c, b   stress non-uniformity parameters evaluated for central measurement spheres and for near boundary measurement spheres, respectively.    angle between the direction of major principal stress and the vertical direction.  u   excess pore water pressure. ur excess pore water pressure calculated at the end of constant volume shearing. s   an increment of contact shear displacement. Fs   an increment of contact shear force.   xvi  t  time increment during a calculation cycles.     friction angle calculated from boundary stresses as defined in Equation 7.5. n, s    overlap distance at contacts in the normal and shear directions, respectively. mob  mobilized friction angle.    shear strain.   max   maximum shear strain recorded during constant volume cyclic loading.  x, y, z   strains in the x, y, and z directions, respectively. v   volumetric strain.     material constant as shown in Equation 7.4.  o, c initial water content prior to consolidation and water content at the end of consolidation, respectively.   contact Poisson’s ratio used as an input parameter for the Hertz-Mindlin contact model.   '1, '2, '3 major, intermediate, and minor principal effective stresses, respectively.  'xx, 'yy, 'zz normal effective stresses acting on the x, y, and z planes calculated from contact forces at locations within measurement spheres on average, respectively.   'zz(max), 'zz(min), 'zz(average)maximum, minimum, and average of 'zz values calculated at the locations of measurement spheres on average, respectively.   'm mean effective stress as defined in Equation 7.1.   q deviatoric stress as defined in Equation 7.2.  'h, 'hc average stresses in the horizontal direction acting on the lateral specimen boundaries during shearing and at the end of consolidation, respectively.  'L Average normal stress acting on the lateral specimen boundaries.  'v, 'vc average vertical stresses acting on the horizontal specimen boundaries during shearing and at the end of consolidation, respectively.  corrected  corrected average normal stress acting on free form sensor area as defined in Equation 3.1.   recorded  recorded average normal stress acting on free form sensor area.  loading, unloading average normal stress acting on free form sensor area during the loading and unloading phases, respectively.   shear stress xz calculated from stresses acting at contacts within the volume of a   xvii  measurement sphere.   h average shear stress in the horizontal direction equal to horizontal driving force divided by specimen cross-sectional area.  L Shear stress acting on the lateral specimen boundaries. v average shear stress in the vertical direction acting on the specimen lateral boundaries.  cy cyclic shear stress amplitude.  L Lode angle.     xviii  ACKNOWLEDGEMENTS I’d like to thank my supervisor Dr. Dharma Wijewickreme for all the advice, guidance and patience. Without his support throughout my PhD studies, this work would not have been possible. It is an honour to have had Dr. Peter Byrne as my co-supervisor. I am thankful for his support and encouragement.   I am thankful to my PhD committee members, Dr. Humberto Puebla and Dr. Mahdi Taiebat, for their constructive feedback and for reviewing my thesis. The time and effort that the university examiners, Dr. Carlos Ventura and Dr. Erik Eberhardt, put towards reviewing my thesis has greatly improved the dissertation. I am thankful to Dr. Dave Chan, the external examiner, for his critical review of my thesis and for his comments. Thanks to Dr. Douglas Scott for chairing my PhD oral examination committee.  I am grateful to the financial support provided by the University of British Columbia through the Four-Year Fellowship (FYF) and the funding provided by the National Research Council of Canada during my PhD studies.   I’d like to thank the geotechnical faculty members at UBC, Civil workshop technicians, and my colleagues and friends. Thanks to Bader, Yazan, Ainur, Prasanth, Lalinda, Awad, Miguel, M. Shamma, Achala, Mavi, Ruslan, Michael, Mallaz, Kaley, Mark, Ivan, Paul, Antonio, Santiago, Shamila, Sheri, Craig, Helia, Bronwyn, Amar, Housam, Marvin, Raj, Deep, Jhoti, and Pawan. Thanks to the wonderful Ash Harpaul Brar for all the support and patience. Last but not least, I’d like to thank my amazing mother Mona Tarazi, my aunt Nadia Tarazi, my grandmother Nahil Tarazi, my late grandfather Suad Tarazi and the rest of my family for not sparing any effort in helping and guiding me throughout my PhD journey.    1    1 INTRODUCTION   The direct simple shear (DSS) test has been increasingly used to characterize soil behavior when subjected to cyclic loading conditions (e.g. Wood & Budhu, 1980; Finn et al., 1982; Wijewickreme et al., 2005; Kammerer, 2006). This is mainly due to its simplicity compared to other laboratory tests such as the hollow cylindrical torsional shear test and due to its ability to effectively simulate field earthquake loading conditions including rotation of principal stresses.   Soil strength-deformation parameters such as friction angle, undrained shear strength, and shear stiffness can be typically obtained from the results of monotonic DSS testing performed to simulate static loading conditions that typically prevail in commonly encountered engineering problems (e.g., design to withstand the dead-weight of a building).  Cyclic loading is applied to DSS specimens to simulate earthquake loading. The results of cyclic DSS testing are typically used to assess the changes in effective stresses due to earthquake loading (as a result of development of excess pore water pressures) and the cyclic resistance of soils that is useful in liquefaction assessment.  A schematic of a soil element subjected to ideal simple shear conditions is presented in Figure ‎1.1. In an ideal simple shear test, a specimen is subjected to a state of “pure shear strain”. The specimen is prevented from straining in axial directions (i.e. x = y = z = 0). A DSS test performed with zero axial strains during shearing is known as a constant volume test (i.e. equivalent to undrained according to Dyvik et al., 1987). Alternatively, for drained DSS tests, the specimen is allowed to strain in the vertical direction (i.e. z  ≠ 0; x = y = 0).  Interpretation of DSS testing results has not been straightforward due to uncertainties with regard to the state of stress within DSS specimens as follows:    2  i) Due to the lack of complementary shear stresses acting on the lateral boundaries of DSS specimens, there is a potential for the development of a non-uniform stress state which implies that measured boundary stresses might not be representative of the ideal DSS behavior of the tested specimens. During the shearing phase, for ideal simple shear conditions, complimentary shear stresses acting on the horizontal planes and on planes parallel to the lateral boundaries, h and L, respectively, are developed. However, as nearly frictionless lateral boundaries are used for DSS testing, the value of L is nearly zero. The absence of complimentary shear stresses on the lateral boundaries has been suggested to cause stress non-uniformities within DSS specimens (Roscoe, 1953).  ii) In the most commonly used version of the DSS device, stresses on the lateral boundaries are not measured and only stresses acting on the horizontal boundaries on average are known. With the knowledge of only one stress point, it is not possible to plot the Mohr circle in a confident manner and consequently it is not possible to calculate mobilized DSS friction angle, as defined in Equation 1.1, on planes parallel to the direction of shearing accurately. This presents another difficulty in assessing DSS results even if stresses in the specimen were perfectly uniform.   31311''''sin mob         (1.1) where: mob = mobilized friction angle,  '1 = major effective principal stress, '3 = minor effective principal stress.  With the recent advances in computers, it is now possible to model soil as a collection of particles rather than a continuum, and more realistically simulate soil behavior using the discrete element method (DEM)   3  computer codes (e.g., Particle Flow Code, PFC, developed by Cundall & Strack,1979). DEM allows for local measurements of stresses and strains within the modeled volume, in addition to the average boundary measurements. Accordingly, it is an ideal analysis tool for simulating and investigating the distribution of stresses and strains within test soil specimens. Recent works on simulating laboratory testing conditions using DEM have demonstrated the potential of DEM in tracking the salient characteristics of soil behavior accurately and its suitability for modeling soil element testing (Thornton, 2000; Cui & O’Sullivan, 2006; Yan, 2009; Zhao et al., 2009; Fu & Dafalias, 2011; Härtl & Ooi, 2011; O’Sullivan, 2011a).  The main motivation for this work stems from the need to bridge the present knowledge gaps in regards to interpretation of the state of stress in DSS tests. It is noted that understanding the state of stress in DSS testing would help practicing engineers and researchers in obtaining better estimates of soil design parameters such as friction angle and developing better understanding of stress-strain behavior of DSS specimens. As such, a comprehensive research program was undertaken to assess the stress state using the DEM methodologies, and the research approach, findings, and conclusions are presented in this thesis.  1.1 Thesis objectives   The main aim of this study is to investigate the state of stress in cylindrical Norwegian Geotechnical Institute (NGI) type DSS specimens subjected to different boundary and loading conditions with emphasis on the following components:  - Evaluation of the extent and significance of possible stress non-uniformities within DSS specimens. - Evaluation of the potential for development of shear strain non-uniformities (i.e. deviation from imposed boundary shear strains) in DSS specimens.    4  - Interpretation of DSS data to calculate DSS mobilized friction angle, defined in Equation 1.1, for use in geotechnical design and stress invariants to facilitate the development of soil constitutive models.  To achieve these objectives, 3D DEM analysis along with fundamental experimental DSS programs were implemented. The DEM models soil as a collection of particles, which is a more realistic modeling approach compared to continuum modelling of soil. Simulated specimens were subjected to monotonic and cyclic shearing under constant volume (i.e. equivalent to undrained according to Dyvik et al., 1987) and drained (i.e. constant vertical effective stress) boundary conditions. In this thesis, all stresses are reported in terms of effective stresses. The effect of vertical effective stress level on the results was investigated. In this thesis, a laboratory DSS testing program was undertaken using glass beads as the test material to generate basic data for validation of the DEM model. The laboratory work was further supplemented by measurement of lateral stresses on DSS specimens of Fraser River silt using state-of the art free form pressure sensors (paper-thin flexible sensors). Measured lateral stresses were used for qualitative comparison with lateral stresses obtained from the DEM model.   1.2 Thesis organization  This thesis is organized into eight chapters. Chapter 2 presents and discusses previously published work on the state of stress in DSS specimens, along with relevant background information on discrete element analysis. Chapter 3 and Chapter 4 address experimental aspects and present the results of the DSS testing program undertaken as part of this study, respectively. Aspects related to DEM analysis, including selection of model input parameters and evaluation of the developed model by comparing simulations results with counterpart laboratory DSS testing results, are addressed in Chapter 5. In Chapter 6, the results of the DEM analysis are used to provide insight on stress-strain uniformities within DSS specimens. Calculated mobilized friction angles and stress invariants from model results are presented in Chapter 7. Chapter 8 presents a summary of the main contributions of this study along with conclusive   5  remarks and the limitations of the current analysis. Finally, relevant results from additional DEM modeling that was undertaken in support the study findings are included in Appendix A.     6   Figure ‎1.1. Schematic showing a cross-section through a DSS specimen subjected to “ideal” simple shear conditions.    xyzh'v'Lh'vL'LL  7  2 LITERATURE REVIEW   The DSS test was initially developed to overcome significant stress non-uniformities imposed by the direct shear (commonly referred to as “direct shear box”) test. Unlike the shear box test that mobilizes a shear zone at the boundary between the two halves of the box, the direct simple shear test engages the whole specimen in the shearing process. There are two commonly used types of direct simple shear tests: the cubical specimen Cambridge-type with rigid side boundaries initially developed by Roscoe (1953), and the cylindrical specimen NGI-type with the wire-reinforced membrane providing lateral confinement (Bjerrum & Landva, 1966). In another version of the cylindrical specimen type DSS test, steel rings that slide over one another placed around a typical rubber membrane are used to provide lateral confinement. The DSS test with cylindrical specimen has been more commonly used. The previous assessments on the performance and suitability of the DSS testing method have been made mainly with respect to the devices having rectangular specimens similar to that used in the Cambridge-type tests. For these reasons, the NGI-type DSS test was chosen as the subject of analysis in this study.  This chapter discusses relevant previously published works relating to numerical and experimental analysis of DSS testing and a brief background on DEM modeling. Typical observations obtained from DSS testing are initially presented. This is followed by a discussion on stress-strain uniformities in DSS specimens. Basic theory related to the interpretation of stress state in DSS specimens is also presented. Findings from previous works on interpretation of DSS stress state are then discussed. Finally, a brief description of the discrete element method (DEM) is presented along with a rationale that a methodology such as DEM, that represents the soil as a particulate material, may be suitable for simulating the response of a soil specimen in a DSS shear device.      8  2.1 Typical soil response as observed from results of DSS testing   The stress path and stress-strain response of loose air-pluviated Fraser River sand to monotonic constant volume DSS shearing is presented in Figure ‎2.1. Slight strain softening behavior is noted followed by strain hardening behavior.   Figure ‎2.2 and Figure ‎2.3 present the response of Fraser River sand and Fraser River silt specimens subjected to DSS cyclic loading, as reported by Wijewickreme et al. (2005) and Sanin & Wijewickreme (2006), respectively. The University of British Columbia (UBC) DSS device, which is an NGI-type DSS device, had been used by these researchers to perform the testing. The tests on both Fraser River sand and Fraser River silt had been conducted with specimens consolidated to vertical effective consolidation stresses, 'vc, of about 100 kPa. The consolidated Fraser River sand specimen that was loaded with a cyclic stress ratio CSR (cy/'vc) of 0.1 experienced an overall drop in vertical effective stress (or equivalent increase in pore water pressure) throughout the application of cyclic loading with a shear strain ( of 3.75% reached in about seven cycles of loading. This  = 3.75% condition in a DSS specimen is essentially equivalent to reaching a 2.5% single-amplitude axial strain in a triaxial soil specimen.  An identical definition has been previously used to assess the cyclic shear resistance of sands by the U.S. National Research Council (NRC, 1985), and it also has been adopted in many previous liquefaction studies at UBC. It is also noted that during the last loading cycle, the specimen experienced the transient condition of near zero vertical effective stress (or equivalent excess pore water pressure ratio, u/'vc, of about 100 %). The stress-strain response depicted in Figure ‎2.2b shows the development of very small strain prior to failure (i.e. throughout the first six loading cycles) with sudden increase in shear strain during the seventh cycle, which is a typically observed behavior for loose sands.   The response of a Fraser River silt specimen subject to DSS cyclic loading with CSR = 0.21 is shown in Figure ‎2.3. Contractive behavior (i.e. decrease in 'v or equivalent increase in pore water pressure) is   9  observed during the first cycle of loading. With the progression of cyclic loading, dilative behavior (i.e. increase in 'v or equivalent decrease in pore water pressure) is noted during the loading parts of the cycles followed by significant contractive behavior during unloading parts. The tested silt specimen reached a minimum 'v value of about 10 kPa which is equivalent to about 90% excess pore water pressure. Gradual development of shear strain with the progression of cyclic loading (i.e. “cyclic mobility type” strain development) is noted from the stress-strain response shown in Figure ‎2.3b which is a typically observed behavior for fine-grained soils.   2.2 Stress-strain uniformities in DSS specimens   The DSS testing process deviates from the ideal simple shear boundary conditions due to the absence of complementary shear stress on the lateral boundary; this has been suggested to cause stress non-uniformities within DSS specimens that can lead to progressive failure.   Table ‎2.1 presents a summary of previous analytical and experimental research on stress-strain uniformities in DSS tests. Roscoe (1953) presented the first work on the topic. Using basic elastic analysis, Roscoe (1953) showed that uniform shear stresses are anticipated at central specimen regions with highly non-uniform stresses near the boundaries. Lucks et al. (1972), based on the results from finite element (FE) analysis assuming linear elastic soil behavior, concluded that stress non-uniformities within the DSS specimen are local and that 70% of the specimen is under uniform stress conditions. Prevost & Hoeg (1976) performed linear elastic FE analysis of DSS specimens that allowed for slippage at the top and bottom caps. The results from their analysis indicated significant increase in non-uniformities extending to central specimen locations for the cases when slippage was allowed to occur compared to simulations performed with no slippage at the top and bottom caps. The results of Prevost & Hoeg (1976) are of particular significance as during DSS shearing of some soil types there is a potential for some slippage to occur at the interface between the horizontal boundaries and soil specimen. It is noted that possible slippage at the top and bottom boundaries was not accounted for in the studies of Roscoe   10  (1953) and Lucks et al. (1972). Saada & Townsend (1980), using the results of a photoelastic study conducted on an epoxy material, concluded that the DSS test does not yield either reliable stress-strain relations or absolute failure values, and that the method is only suitable for comparing descriptively similar soils. The conclusions of Saada & Townsend (1980) are based on observation of stresses in an isotropic elastic material. However, soils are highly plastic anisotropic materials. For example, Finn et al. (1982) showed that accounting for soil anisotropy in their analysis resulted in more uniform shear stresses.  The normalized vertical stresses calculated from experiments, elastic analysis, and numerical analysis reported by Budhu & Britto (1987) for a DSS specimen that is at shear strain of 14% are presented in Figure ‎2.4. These researchers noted that results of the elastic analysis overestimated the calculated normalized vertical stresses at locations where yield has occurred (near specimen ends), compared to stresses calculated from the Modified Cam-Clay model. Plotted experimental data was obtained by fitting a function to three measurement points as shown in Budhu & Britto (1987). The fitted function is in qualitative agreement with the results obtained from numerical analysis. Overall, the results obtained from Budhu & Britto (1987) indicate that stresses are more uniform at the specimen center compared to stresses at locations near the lateral boundaries.   Dounias & Potts (1993) conducted 2D FE analysis to investigate simple shear stress and strain non-uniformities using an elastic perfectly plastic material that obeys Mohr Coulomb's yield criterion. Their results indicated that the measured average stresses underestimate initial stiffness and peak strength by about 20% when compared to those from a case that represents ideal simple shear conditions. Furthermore, they noted that post-peak stresses are “highly non-uniform”. A more advanced 3D FE analysis using the Modified Cam-Clay model presented in Doherty & Fahey (2011) indicated that calculated simple shear friction angle was only 4% lower than that would be derived under ideal simple shear conditions. Moreover, the results of Doherty & Fahey (2011) showed that stresses were fairly uniform at the DSS specimen core.    11   The above reported work has assumed that stresses at the end of consolidation are perfectly uniform across a give DSS specimen. Analysis of DSS consolidation performed by Saada et al. (1983) has indicated that significant stress non-uniformities can also be prevalent at the end of the consolidation (i.e. inherent non-uniformities) phase. These observations, and the noted discrepancies thereof, suggest that more realistic analysis that takes into consideration the effect of inherent non-uniformities on stress non-uniformities during subsequent shearing is needed.   Analysis of a truly undrained DSS test and a constant volume DSS test was performed using a multi-yield constitutive model by Wang et al.(2004). Their results indicated that boundary conditions for simulated truly undrained simple shear test caused more stress non-uniformities at specimen core compared to boundary conditions for constant volume DSS tests. Finn et al. (1982) also noted that constant volume DSS tests are superior to truly undrained DSS tests as compliance effects are significant for the latter.  Fu & Dafalias (2011) conducted 2D DEM analysis simulating constant volume simple shear conditions using elliptical particles to study fabric evolution in shear bands. Their study focused on very large shear strains as high as 500% which are significantly larger than shear strains typically used in DSS testing (0-≈20%). The authors demonstrated the development of shear bands at large strains which is indicative of non-uniform stresses and strains within the simulated DSS specimen.  The experimental results presented in Wood & Budhu (1980) indicate that average stress ratios calculated from specimen boundaries are similar to those measured at the center of the DSS specimen, which in turn, implied that the effect of stress non-uniformities might not be significant during shear. Vucetic & Lacasse (1982) reported results on medium-stiff clay obtained from DSS testing conducted on specimens with different height to diameter ratios and using different membrane types. They concluded that neither the height to diameter ratio nor the membrane reinforcement type has any significant influence on the static stress-strain behavior of the tested clay. Budhu (1984) noted that the distribution of strains along the height of the specimen is fairly uniform for shear strains less than 5%. Amer et al.   12  (1987), showed that specimen size has a notable effect on the cyclic behavior of sands which seems to contradict the conclusions of the study performed by Vucetic & Lacasse (1982). Perhaps, the difference between the conclusions of Vucetic & Lacasse (1982) and those of Amer et al. (1987) is due to the difference between the tested soil materials used in the two studies. It was noted by Airey (1984) that DSS testing performed on clays had more uniform stresses compared to DSS testing performed on sands.   All previously discussed studies on DSS stress-strain uniformities acknowledge that stresses and strains can be non-uniform at locations within DSS specimens, particularly near the lateral boundaries. However, it seems that there is no consensus on the extent and significance of stress-strain uniformities in DSS specimens. The discrepancies between the conclusions of various numerical studies seem to be arising as an artefact of the difference in modeling approach used for the assessment with only few studies performed using: (i) realistic advanced constitutive models; (ii) model boundary conditions including the degree of slippage allowed at the top and bottom boundaries; (iii) whether a more realistic 3D analysis was performed as opposed to a relatively simpler plane strain 2D analysis; and finally, (iv) in most of the previous studies soil was treated as a continuum material. Experimental studies have indicated that DSS clay specimens tend to develop less non-uniformities compared to sandy DSS specimens. Further, stresses and strains in the rectangular Cambridge-type devices have been assessed to be more uniform than these in the NGI-type DSS devices (Budhu 1984). This finding is of particular significance as the NGI-type DSS device has been more commonly in use than the Cambridge-type device.   Overall, previous work as discussed above has one or more of the following limitations:   The use of a constitutive model that does not capture the main characteristic of soil behaviour accurately (e.g. linear elastic analysis).   13   An assumed perfectly uniform end of consolidation stress state prior to DSS shearing – i.e., non-uniformities in stress distribution at the end of consolidation can likely affect the degree of stress non-uniformities during subsequent shearing.   The use of a 2D representation in the assessment than performing a more realistic 3D analysis of the DSS device.   In experimental studies, load/deformation measurements can be made only at the specimen boundaries; this information would not necessarily allow understanding the degree of non-uniformities within the central parts of the specimen and, in turn, affecting the level of confidence of the assessed stress non-uniformities.   These considerations warrant additional investigation into stress-strain non-uniformities for the NGI-type DSS device. The Discrete Element Method (DEM), a methodology that represents the soil as a particulate material, is judged suitable for the analysis performed as part of this study due to the following reasons:   The capabilities of DEM to capture soil behavior has been previously demonstrated by several researchers (e.g. Thornton, 2000; Cui & O’Sullivan, 2006; Yan, 2009; Zhao et al., 2009; Fu & Dafalias, 2011; Härtl & Ooi, 2011; O’Sullivan, 2011a).  DEM can be used to capture stress non-uniformities arising from the specimen preparation and consolidation phases.  DEM allows for stress and strain computations at the boundaries and at locations within the specimen boundaries, making it an ideal tool to investigate stress-strain distribution within the specimen.   For these reasons, the use of DEM analysis was considered as a method that would be a prudent tool in providing a more realistic insight into stress-strain uniformities in DSS specimens.     14  2.3 Interpretation of stress state in DSS specimens  The stress state in a DSS specimen during the shearing process is schematically shown in Figure  2.5 considering an element located at the specimen core. In the DSS test, normal and shear stresses on the horizontal plane, 'zz and , are measured. However, the stress state on the vertical plane (i.e. 'xx and ) is usually not known. As such, assumptions with regard to the DSS specimen stress state must be made to construct the Mohr circle and compute the mobilized friction angle on the plane of maximum obliquity, mob:    31311''''sin mob,         (2.1) where: mob = mobilized friction angle,  '1 = major effective principal stress, '3 = minor effective principal stress.  For example, if maximum shear stress, max, is assumed to act on the horizontal plane, angle mob becomes equal to angle  as defined in Equation 2.2. This is illustrated in Figure  2.6a. The stress point representing the horizontal plane is at the lowest point of the Mohr circle and that of the vertical plane is on the opposite side. The stress point on the vertical plane in this case is ('xx , )1.   zz'sin 1           (2.2) where:  = stress ratio angle,   15  'zz = normal effective stress acting on the z-plane.  However, if the horizontal plane is assumed to be the plane of maximum obliquity instead, angle mob is then equal to angle β defined in Equation 2.3 (see Figure  2.6b). The stress point on the vertical plane for this case is ('xx , )2.    zz'tan 1           (2.3) where: = stress ratio angle, 'zz = normal effective stress acting on the z-plane.  As noted above, the interpreted value of the DSS mobilized friction angle (mob) clearly depends on the assumed specimen stress state. Atkinson et al. (1991) noted that the DSS critical state friction angle is not unique and that it depends on the ratio of horizontal to vertical stresses.    Wood et al. (1979) conducted a thorough analysis of the DSS specimen stress state using the results of an instrumented rectangular DSS device capable of measuring stresses at different locations at the DSS specimen boundaries. Their study presented a simplified approach that can guide in computing stress state of DSS specimens using data from commonly used DSS devices (i.e. with average boundary stress measurement on the horizontal plane only).  The orientation of the failure plane in a DSS specimen has been investigated by a number of researchers. Roscoe (1970) suggested that failure in the DSS specimen always occurs along the horizontal plane. However, Wroth (1987) pointed out the possibility of failure occurring along vertical planes of maximum   16  obliquity instead. De Josselin (1988) developed a model that acknowledges the possibility of failure on both horizontal and vertical planes of maximum obliquity depending on the initial stress state. The assumed stress state on the horizontal plane as illustrated above will critically determine how the Mohr circle of stress should be drawn and, in turn, the mobilized friction angle and the orientation of failure plane in a DSS test according to the Mohr-Coulomb criterion.   Stress non-uniformities in DSS specimens introduce further difficulties in regards to calculating the DSS mobilized friction angle. Due to non-uniform stresses particularly near the specimen boundaries (see Figure ‎2.4), the DSS is not an element test. As such, it is not possible to draw Mohr circle for DSS specimens undergoing shearing. However, as stresses are fairly uniform at central specimen locations, it is possible to draw a Mohr circle representing the stress states for central specimen locations and, in turn, calculate the mobilized friction angle.   2.4 DEM analysis   As noted above, DEM analysis is considered to present an alternate but a more suitable tool in the present study to model the performance of a DSS specimen containing soil particles in a more realistic manner. To provide the necessary background, some salient details related to DEM analysis are given below.   The Discrete Element Method (DEM), alternatively called the Distinct Element Method, was initially developed by Cundall & Strack (1979). In DEM, soil is modeled as a collection of particles, which is a more realistic representation of soils compared to continuum analyses. Since the initial work by Cundall & Strack (1979), interest in DEM has been increasing. However, as DEM simulations are computationally demanding, the use of DEM remained limited in the 1980s and 1990s. The recent dramatic increase in available computational power made it possible to model reasonable number of particles and to use more realistic 3D models, which prompted a widespread increase in the use of DEM. As noted by (O’Sullivan   17  2011a), in addition to the use of DEM in geomechanics, DEM has been extensively used by chemical and process engineers and by researchers in food technology, mining engineering, and pharmaceutical sciences. O’Sullivan (2011b) is the first dedicated book on discrete element modeling for geomechanics applications. DEM analysis has been shown to accurately simulate observed soil behavior (e.g. Thornton, 2000; Cui & O’Sullivan, 2006; Yan, 2009; Zhao et al., 2009; Fu & Dafalias, 2011; Härtl & Ooi, 2011; O’Sullivan, 2011a).  Particle Flow Code in three dimensions (PFC3D) developmed by Itasca Inc. (2005a) based on DEM has been extensively used to simulate soil behavior as observed from the results of laboratory tests (e.g. Chung & Ooi, 2007; Härtl & Ooi, 2011). In PFC3D, rigid spheres referred to as “balls” are used to represent soil particles. Rigid displacement controlled boundaries referred to as “walls” are used in PFC3D to represent model boundaries. Contacts between two entities (i.e. Ball-Ball or Ball-Wall) are simulated using the “soft contacts” approach which allows a very small overlap distance between the contacting entities as shown in Figure ‎2.7 for two balls in contact. It is noted that the use of the “soft contacts” approach violates the compatibility requirement. As such, DEM assumes that the overlapping volume between the entities in contact is very small and therefore is of negligible significance from a practical perspective.  A typical calculation cycle for a DEM simulation is illustrated in the diagram shown in Figure ‎2.8. Initially, ball locations and densities, model boundary locations, and contact model input parameters are defined. After this step, the calculation cycle starts by identifying contacts. This is followed by calculating contact forces from contact overlap distance using a force-displacement law as illustrated below. The resultant force acting on each particle is then calculated from contact forces while taking into account body forces and externally applied forces. Particle accelerations are calculated from forces using Newton’s law. The computed accelerations are then integrated twice to obtain particle displacements and rotations using a time integration scheme (typically central difference time integration is used in DEM codes). Particle positions are then updated and the cycle is repeated for another time increment (t).   18   As previously discussed, a force-displacement law is used to calculate forces acting at contacts. At any given contact there are two forces: normal force, Fn, and shear (tangential) force, Fs, as shown in Figure ‎2.7 for two balls in contact. For the simplest contact model, the linear elastic model, Fn is calculated as shown in Equation 2.4.         ,           (2.4)  where Fn is the normal force at the contact, Kn is normal contact stiffness, and n is the overlap distance (see Figure ‎2.7). It is noted that the overall behavior of a soil mass simulated using the linear elastic force-displacement model is non-linear as the number of contacts is altered with shearing and, in turn, the overall stiffness changes with the progression of shearing. Shear force, Fs, is calculated as follows:  - When a contact is first formed, shear force and tangential displacement are set to zero.  - For a given time increment, an increment of shear force, Fs, is equal to the product of contact shear stiffness, Ks, and an increment of displacement in the tangential direction, s. - Shear forces and cumulative displacements in the tangential direction are then updated.   Slippage between contacting entities occurs when the ratio of tangential force, Fs, to the normal force, Fn, exceeds the interparticle friction coefficient, which is a fundamental material property that is a function of the surface roughness of the two contacting entities.   The linear-elastic force-displacement model is an idealization of the actual behavior at contacts between soil particles. For example, Cavarretta et al. (2010) showed that force-displacement relation at particles contacts is non-linear. As such, a non-linear elastic Hertz-Mindlin contact model has been used in some   19  of the previous DEM studies to obtain better model performance. The Hertz-Mindlin contact model is described in Chapter 5.   As PFC3D uses an explicit solution scheme, the selection of an appropriate time increment (t) for analysis is necessary for obtaining meaningful results. According to the PFC3D user manual (Itasca Inc., 2005b), it is recommended to use time increments that are equal to 75% and 25% of the critical time increment (defined in Equation 2.5) for the cases of linear contact model and non-linear Hertz-Mindlin model, respectively, to ensure the stability of the solution. It is noted that the critical time step is a function of the stiffness at the contacts and particle densities. Critical time step increases (i.e. faster simulations) with the decrease and increase in contacts stiffness and particle densities, respectively:           √   ,           (2.5)  where tcrit is the critical time step used for PFC simulations, m is the mass of a particle, and k is contact stiffness.   In this section, relevant aspects of DEM and PFC3D for the current analysis were briefly discussed. A thorough description of DEM and PFC3D can be found in Itasca Inc. (2005b) and O’Sullivan (2011b). The ability of DEM to simulate soil behavior accurately makes it an ideal tool for use in geomechanics analysis to investigate fundamental aspects of soil behavior. The future of discrete element modeling is promising due to the ever increasing computational power and the development of new and more efficient DEM codes.         20   2.5 Closure  The DSS test has been commonly used to characterize soil behavior due to its simplicity and its ability to impose realistic loading conditions involving continuous rotation of principal stresses that are anticipated in the field during seismic loading. Interpretation of DSS test results has not been a simple task mainly due to uncertainties in regards to the degree of stress-strain non-uniformities in DSS specimens and because stresses acting on the lateral boundaries of DSS specimens are typically not measured. There is no consensus among previous researchers with regard to the degree and extent of stress-strain non-uniformities within DSS specimens. This is mainly due to the difference in constitutive models and input parameters used for numerical studies. Results of experimental studies have indicated that stresses in tested clayey specimens were more uniform compared to stresses within sandy DSS specimens.   It was shown that previous studies on stress-strain uniformities had several limitations. Furthermore, it was noted that interpreted DSS friction angle is not unique, as different assumptions with regard to the state of stress on the vertical plane can result in different calculated DSS friction angles. As such, more analysis of the stress state in the DSS device along with the stress non-uniformities during DSS testing are needed to provide further insight into the state of stress in DSS specimens and, in turn, enable researchers and engineers to better interpret DSS testing results. As DEM has been shown to provide accurate predictions of soil behavior, it was considered an ideal tool for use in the analysis of stresses during DSS testing.     21  Table ‎2.1. Summary of previous studies on stress-strain uniformities in DSS specimens. Type of study (experimental / numerical)  NGI-type / Cambridge-type Constitutive model / tested material Assumed end of consolidation stress distribution Major finding  Citation  Numerical analysis  2D analysis   Linear elastic  Perfectly uniform  Uniform shear stresses at the center of the specimen  Roscoe (1953) Numerical (3D FE analysis) NGI-type cylindrical specimen Linear elastic Perfectly uniform  70 % of the sample has fairly uniform stresses Lucks et al. (1972) Numerical analysis considering slippage at top and bottom boundaries 2D analysis  Linear elastic  Perfectly uniform  There is more potential for development of non-uniformities with the increase in slippage at the top and bottom specimen boundaries Prevost & Hoeg (1976) Experimental study Cambridge-type Leighton Buzzard sand - Average stress ratio calculated from specimen ends was similar to that calculated for specimen core Wood & Budhu (1980) Photoelastic experimental study  Rectangular-type and cylindrical-type Epoxy material Zero applied vertical stress  Shear stresses are not uniform even at specimen core  Saada & Townsend (1980)   22  Type of study (experimental / numerical)  NGI-type / Cambridge-type Constitutive model / tested material Assumed end of consolidation stress distribution Major finding  Citation  Experimental study  Cylindrical NGI-type Medium-stiff clay -  Height to diameter ratio and membrane type did not have any significant effect on observed stress-strain behavior Vucetic & Lacasse (1982) Numerical  analysis 2D analysis Linear anisotropic elastic material Perfectly uniform The results of anisotropic elastic analysis show increased stress uniformities compared to isotropic elastic analysis. Increased stress uniformity is noted with the increase in specimen width to height ratio.  Finn et al. (1982) Numerical analysis  2D analysis Linear elastic material Non-uniform stresses at the end of consolidation Non-uniformities of stresses at the end of consolidation can be considerable  Saada et al. (1983) Experimental study  Cambridge-type and NGI-type Lieghton Buzzard sand  - Distribution of shear strains along specimen height is fairly uniform for shear strains less than 5%. Soil behavior in Cambridge-type and NGI-type devices are different because of the difference in applied boundary conditions between the two devices Budhu (1984)   23  Type of study (experimental / numerical)  NGI-type / Cambridge-type Constitutive model / tested material Assumed end of consolidation stress distribution Major finding  Citation  Numercial analysis  2D FE analysis Modified Cam-Clay Perfectly uniform  Stresses calculated from the analysis using the Modified Cam-Clay model are more uniform compared to stresses calculated using elastic analysis Budhu & Britto (1987) Experimental study NGI-type Loose and dense specimens of Ottawa sand - Selected DSS specimen size has a notable effect on cyclic behavior of sands Amer et al. (1987) Numerical analysis 2D FE analysis Isotropic elastic-perfectly plastic material Perfectly uniform Interpretation of DSS results using stresses measured on average underestimates initial stiffness and peak simple shear strength by about 20% compared to the case of ideal simple shear conditions. As peak is reached, stresses become “highly non-uniform” Dounias & Potts (1993) Numerical analysis  2D FE analysis Multi-yield plasticity model Perfectly uniform Boundary conditions for simulated truly undrained simple shear test cause more stress non-uniformities at specimen core compared to boundary conditions for constant volume DSS tests Wang et al. (2004)   24  Type of study (experimental / numerical)  NGI-type / Cambridge-type Constitutive model / tested material Assumed end of consolidation stress distribution Major finding  Citation  Numerical analysis  Truly undrained cylindrical–type specimen  Modified Cam-Clay End of consolidation vertical stresses are within ±1% of average vertical stress Fairly uniform stresses at specimen core were noted. Calculated simple shear friction angle was only 4% lower than that for the case of ideal simple shear conditions Doherty & Fahey (2011) Numerical analysis 2D DEM  analysis using elliptical particles - Isotropic consolidation Calculated void ratios vary at locations within the specimen particularly at large strains. The authors demonstrated the development of shear bands at large strains which is indicative of non-uniform stresses and strains within the simulated DSS specimen.  Fu & Dafalias (2011)   25  a)  b)  Figure ‎2.1. Monotonic constant volume response of loose air-pluviated Fraser River sand (after Wijewickreme et al., 2005): a) stress path; b) stress-strain response.     0102030400 20 40 60 80 100 120Shear stress, (kPa)Vertical effective stress, 'v(kPa)'vc= 100 kPa; Drc= 40 %0102030400 5 10Shear stress(kPa)Shear strain (%)'vc= 100 kPa; Drc= 40 %  26  a)  b)  Figure ‎2.2. Cyclic response of loose air-pluviated Fraser River sand (after Wijewickreme et al., 2005): a) stress path; b) stress-strain response.    -20-10010200 20 40 60 80 100 120Shear stress, (kPa)Vertical effective stress, 'v(kPa)'vc= 100 kPa; Drc= 40 %; cyc/'vc= 0.1-20-1001020-10 -5 0 5 10Shear stress(kPa)Shear strain (%)'vc= 100 kPa; Drc= 4  %; cyc/'vc= 0.1  27  a)  b)  Figure ‎2.3. Cyclic response of Fraser River silt (after Sanin & Wijewickreme, 2006): a) stress path; b) stress-strain response.     'vc= 97.2 kPa; ec= 0.884; cyc/'vc= 0.21'vc= 97.2 kPa; ec= 0.884; cyc/'vc= 0.21  28    Figure ‎2.4. Normalized vertical effective stress distribution on the top and bottom horizontal specimen boundaries obtained from analysis and experimental results (after Budhu & Britto, 1987).   -7-6-5-4-3-2-10123456'v/'v(average)ExperimentalElasticModified Cam-ClaySand102345  29     Figure ‎2.5. Schematic representation of stress state in the center of a DSS element test.   x-axisz-axis'zz 'xx'xx'zz  30  a)  b)   Figure ‎2.6. Schematic showing Mohr circles and failure envelops representing two possible stress states for the DSS specimen as discussed in Section 2.3: a) horizontal plane corresponding to maximum shear stress; b) horizontal plane corresponding to maximum stress obliquity.    ('xx,)1('zz,)Normal stressShear stressmob = ('xx,)2('zz,)Normal stressShear stressmob =   31   Figure ‎2.7. Schematic showing the projection of two balls in contact in a DEM simulation.    Overlap disctance in the normal direction (n)Projection of contact planeVector between the centeriods of  the two balls in contactFnFs  32   Figure ‎2.8. Flow diagram illustrating DEM calculation cycle (modified from O’Sullivan, 2011b).   t = 0Define system geometry and contact modelt = to+ tIdentify contacting particles and calculte contact forces (Where tois the time at the end of the previous step and t is the time increment during the current step)t = to+ tCalculate resultant force acting on each particle including body forces and external forcest = to+ tCalculate particle accelerations and integrate to determine velocitiest = to+ tCalculate particle displacements and rotations in current time increment and update particle positionsMove forward one step (t) in time and reviseboundary positions as required  33  3 EXPERIMENTAL ASPECTS   In recognition of the importance of Direct Simple Shear (DSS) testing for use in industry and research (Bjerrum & Landva 1966; Wood & Budhu 1980; Finn et al. 1982; Wijewickreme et al. 2005; Kammerer 2006), a focused laboratory testing program that aims at evaluating DSS testing was undertaken. The experimental approach includes: (i) fundamental laboratory testing using glass beads as test material specifically selected for validation of discrete element numerical models of the DSS test; and (ii) testing undertaken using a newly developed instrumentation program to measure horizontal normal stresses on the DSS specimen during shear testing (i.e., the inner wall of DSS reinforced membrane was instrumented using state-of-the-art free form pressure sensors). This chapter describes the experimental approach, materials used for testing, testing procedure, and testing program.  3.1 UBC-DSS device   There are two commonly used types of direct simple shear tests: the rectangular specimen type with rigid side boundaries, initially developed by Roscoe (1953), and the cylindrical specimen NGI-type with a wire-reinforced membrane providing lateral confinement (Bjerrum & Landva, 1966). Alternatively, sometimes in the NGI-type DSS test, a stack of steel rings has also been used to provide lateral confinement instead of the wire used to reinforce the membrane. A schematic showing stresses acting on a DSS specimen is illustrated in Figure  3.1.  A version of the NGI-type of the DSS test device at the University of British Columbia (UBC) was used for the testing presented in this study. The UBC-DSS device allows the testing of a soil specimen having a diameter of around 70 mm and height of approximately 20 mm. In the DSS device, the specimen is constrained laterally using a steel-wire reinforced rubber membrane throughout the consolidation and shearing phases, which is believed to effectively simulate zero lateral strain conditions assumed under field direct simple shear conditions. The top specimen boundary is fixed in the vertical direction. The   34  bottom specimen boundary is allowed to move in the vertical direction during the consolidation phase and drained shearing phase. Drained conditions on soil specimens are imposed by maintaining a constant vertical effective stress on the top and bottom boundaries. Alternatively, constant volume conditions can be enforced by preventing the bottom specimen boundary from movement in the vertical direction during shearing phase. Constant volume testing is a simpler alternative to undrained testing in the direct simple shear device. It has been shown that the decrease (or increase) in vertical stress acting on the top and bottom boundaries is equivalent to increase (or decrease) in pore water pressure during an undrained test (Finn et al., 1978; Dyvik et al., 1987).  A detailed schematic of the UBC-DSS device is shown in Figure  3.2. A vertical loading arm, connected to a single acting piston, is capable of applying vertical stress up to about 800 kPa. The horizontal loading arm is connected to a double acting piston used to apply stress controlled cyclic loading (maximum cyclic horizontal stress is about ± 75 kPa) and a constant speed drive motor attached during strain controlled monotonic loading. An electro-pneumatic pressure regulator controlled by a computer, connected to one side of the double acting piston, enables applying the desired form of cyclic loading. The pressure on the other side of the double acting piston is kept constant.  Three linear variable displacement transducer (LVDTs) are used: one LVDT measures vertical displacement and the other two LDVTs are used to measure horizontal displacement. The use of two horizontal LVDTs provides higher resolution measurements at low strains. Two loads cells are used to measure vertical load and horizontal load.   A detailed description of the high speed data acquisition and control system that is connected to the UBC-DSS device is given in Sivathayalan (2000).  3.2 Lateral stress measurement in DSS specimens  In the most commonly used version of the DSS device lateral stresses are not measured. As a result, the complete stress state of the tested DSS specimen is not known. Important soil parameters such as friction   35  angle cannot be calculated without making assumptions in regards to the development of lateral stresses during shearing. Previous researchers reported measurement of average lateral stresses by utilizing the resistive properties of the metal wire used to provide lateral confinement for DSS specimens (Youd & Croven, 1975; Dyvik et al., 1981; Budhu, 1985). However, as noted by Budhu (1985), the usefulness of average measurements of lateral stresses is limited. Measurements of lateral stresses acting on the planes perpendicular to the direction of shearing and those parallel to the direction of shearing are required to characterize the constitutive behavior of tested specimens. In recognition of this, it was decided to use a paper-thin pressure sensor type glued to the inner wall of the DSS membrane to measure lateral stresses at selected measurement locations as described in the following section. The flexibility of these sensors made them ideal for the current application as they effectively conform to the shape of the DSS cylindrical cavity.   3.2.1 Free form sensors system for lateral stress measurement  Free form pressure sensors, available through several distributers (e.g. Sensor Products Inc. and Tekscan Inc.), come in different sizes and shapes. Tactilus Free Form Sensor System that consists of circular shaped sensors (D=18mm), cables connecting the sensors to a data acquisition unit, data acquisition unit, and data analysis software provided by Sensor Products Inc. was used for the current investigation. Figure  3.3 shows a photograph of a free form sensor and the cable connection to the data acquisition unit. A key feature of the used free form sensors is their small thickness of 0.3 mm.  The circular part of the sensor is the active part (i.e. the part that is sensitive to change in applied pressure). When pressure is applied to the active part of the sensor its resistance changes in proportion to the amount of applied pressure. The system is calibrated to establish a correlation between resistance and applied pressure. The used sensors are calibrated to measure pressures in the range of 0-100 PSI. Recorded pressures are average normal pressure over the area of a sensor. The data acquisition unit has the capacity to accommodate 32 channels. The system delivers high scan speed of 100Hz.     36  Free form sensor systems similar to the system used in the current study were previously used to measure contact pressure in several industries including for geotechnical applications. For example, Paikowsky & Hajduk (1997) demonstrated the calibration of a grid-based tactile pressure sensors system and its application to measure soil pressure distribution on the walls of a shear box. Recently, Palmer et al. (2009) have used free form sensors to measure the distribution of pressure acting on pipelines subjected to horizontal ground movements.  3.2.2 Calibration of free form sensors  The sensors were initially calibrated by the manufacturer (Sensor Products Inc.). Based on the information provided by Sensor Products Inc., the initial calibration was performed using an air bladder that was inflated between two fixed flat metal surfaces. During the calibration, individual sensors connected to the data acquisition unit and to the computer were placed between one of the metal surfaces and the air bladder. Accordingly, a correlation between applied air pressure and sensor resistance was established and used as basis to provide accurate pressure measurements displayed on the computer screen by the manufacturer.  In the current study, free form sensors were mounted on a curved rubber surface of the inside of the membrane supported by metal wires, which is different from the flat metal surface used for sensors calibration performed by the manufacturer. As such, it was required to calibrate the sensors while mounted on the curved inner wall of the wire-reinforced DSS membrane to obtain accurate pressure measurements.   A schematic illustrating the arrangement used to calibrate the sensors is shown in Figure  3.4. The ends of a typical rubber membrane were mounted around the top and bottom caps of the DSS device and were sealed using O-rings. The rubber membrane arrangement acts in a similar way as a pressure bladder. The sensors were attached to the reinforced DSS membrane and placed around the rubber membrane. A   37  metal mould was clamped around the reinforced membrane to provide support. Air pressure, applied through the bottom drainage line to the cylindrical cavity formed by the membrane, was increased in increments of 10 kPa. The free form pressure system reading displayed on the computer screen was recorded manually for each applied pressure increment. The top drainage line was closed at all times during the calibration. The readings of pressure gauges connected to the bottom and top drainage lines were identical, which indicated adequate sealing.   A typical calibration plot is shown in Figure  3.5 indicating the results of three subsequent calibrations done on a free form sensor. Air pressure (plotted on the y-axis) is increased in increments of 10 kPa in the range of 20 kPa – 100 kPa and corresponding sensor pressure readings is plotted on the x-axis. It is noted that recorded sensor pressure readings are about 20% higher than the applied air pressures. As indicated earlier, a difference between applied air pressure readings and corresponding sensor-pressure readings is expected because the latter is based on a calibration performed under conditions that are different from these for DSS testing (i.e. sensors mounted on flat metal surface compared to curved reinforced rubber DSS membrane surface). A linear trend line was used to fit calibration data as shown in Figure  3.5 and correction was made as shown in Equation 3.1.                                   (3.1)  Where,            is the corrected stress measurement,           is the recorded stress measurement, and, a and b are the slope and intercept of the linear trend line, respectively.   3.2.3 Accuracy of free form sensors  Accuracy of pressure readings obtained from free form sensors can be affected by a number of factors such as the number of uses for a particular sensor, hysteresis effects on the reinforced membrane, and creep (i.e. time dependant) effects. It was noted that sensor calibration would be affected with the number   38  of times used. As such, a calibration for all the sensors immediately after each DSS test was performed to overcome this concern.   It was noted that obtained pressure readings are not time dependent. For example, in response to an increase in air pressure, sensor pressure readings increase immediately in proportion to the increase in air pressure and essentially remain unchanged after that, provided that air pressure is kept constant. As such, no correction was necessary to account for possible sensor creep effects.   The response of the sensors during unloading was of  particular interest as DSS lateral stress can decrease during shearing. As such, unload-reload tests were performed to investigate possible hysteresis effect on the accuracy of the pressure measurements as shown in Figure  3.6. It can be seen that the unloading points run almost parallel to those obtained from the loading calibration. For the same applied pressure, the sensors indicate a higher pressure value upon unloading compared to that for loading. The percentage error due to hysteresis effects as defined in Equation 3.2 is used here to quantify the anticipated error due to hysteresis effects.                                             (3.2)  Where loading and unloading are recorded sensor pressures during loading and unloading phases, respectively, at the same applied air pressure and average is the average of loading and unloading.  Paikowsky & Hajduk (1997), for their Teckscan free form sensors, used the concept of sensor overconsolidation ratio (OCR) defined as the ratio between the maximum pressure a sensor is exposed to and the current pressure in a given load-unload cycle as basis for evaluating hysteresis effects. A similar approach is followed in this study. Figure  3.7 shows the percentage error plotted against the over consolidation ratio (OCR) for four sensors. Data similar to that presented in Figure  3.6 for four sensors   39  was used to calculate % error and OCR values. For example, at maximum applied pressure of 100 kPa shown in the load-unload plot in Figure  3.6, OCR is equal to 1 and percentage of error is zero as the loading and unloading parts of the plot share the same point. In an overall sense, the percentage error increases with the increase in OCR up to an average value of about 17% at OCR of 2.5. The calculated percentage error is less than that reported by Paikowsky & Hajduk (1997) for their Teckscan free form sensors.  Overall, the previous analysis indicates that the major source of error in reported pressures is due to hysteresis effects. The analysis was performed for unload stress paths that are typical of those anticipated during monotonic DSS tests on a strain softening material.   3.2.4 Evaluation of possible specimen disturbance due to the addition of free form sensors  As will be described in details in Section 3.4.1, typically, four free form sensors are glued on the inner wall of the DSS membrane during a test with lateral stress measurements. The effect of adding the sensors on the observed soil behavior is evaluated by comparing shear stress strain plots obtained from DSS tests performed without the addition of the sensors to those undertaken using the instrumented DSS membrane (i.e. with the sensors glued to the membrane inner walls) as shown in Figure  3.8. All tests were conducted using the same procedure on reconstituted specimens of Fraser River silt. Tests performed without the sensors by Sanín (2010) and as part of the current study resulted in lower peak shear stresses relative to the other two tests done using the instrumented DSS membrane. It was also suspected that the higher measured shear stress for tests done using the instrumented DSS membrane is due to additional membrane stiffness caused by the addition of the sensors. To test this hypothesis, the equivalent shear stress-strain behavior of the instrumented membrane was measured. This was done by mounting the instrumented membrane around the DSS top and bottom caps (without a soil specimen). The membrane was fixed in place using O-rings. A monotonic test was undertaken by manually increasing shear strain. The corresponding measured equivalent shear stress (i.e. recorded horizontal   40  force divided by DSS specimen cross-sectional area) was recorded. As shown in Figure  3.9, equivalent shear stress increases with the development of shear strain.   The equivalent shear stress versus shear strain data was fitted using two linear trend lines for ranges of strains of 0% - 1% and 1% - 20%. These linear fits were used as basis to correct the measured monotonic shear stresses for additional stiffness due to the sensors by deducting equivalent shear stresses in Figure  3.9 from the measured shear stress shown in Figure  3.8 (for tests done using the instrumented DSS membrane at the corresponding shear strains). Figure  3.10 shows the corrected shear stress vs. shear strain plots after applying the correction for the two tests performed using the instrumented membrane. Stress strain plots for the two tests performed without the sensors are shown in the same figure. The four plots are almost identical indicating the effectiveness of the applied correction.  As shown in Figure  3.9, the effect of additional membrane stiffness is more prominent at larger shear strains. As such, the effect of additional membrane stiffness is judged marginal for stress controlled cyclic tests with predominantly small shear strains. Accordingly, the reported results for cyclic tests as shown in Chapter 4 were not corrected for the addition of the membrane stiffness effect.  3.3 Selection of materials for DSS testing  DSS testing for evaluations of discrete element modeling approach was undertaken on spherical uniform glass beads with diameter of 2 mm and specific gravity (Gs) of 2.5. Glass beads are ideal for the testing intended for validation of discrete element models as shown by Härtl & Ooi (2008), Coetzee & Els (2009), and O’Sullivan (2011a). A typical specimen contains about 10,000 particles at the selected particle diameter of 2 mm yielding a reasonable computational time for the discrete element model. As such, the choice of particle diameter of 2 mm is judged appropriate.   It was required to perform additional DSS testing to measure lateral stresses in DSS specimens during shear. It was found that the measured lateral stresses on free form sensors were unusually low in DSS   41  tests conducted with Fraser River sand (i.e., a medium sand with D50 of about 0.25 mm); it appears that this is likely due to the arching of relatively coarse-grained sand around the free-form sensor location.  On the other hand, when the DSS tests were conducted with Fraser River silt, the response of the sensors were effective possibly due to the lack of arching in this fine-grained soil.  As such, it was decided to use Fraser River silt to investigate the development of lateral stresses.  Fraser River silt is obtained from a channel fill deposit located in the Fraser River Delta, British Columbia, Canada, and detailed description of this material is given in Sanin & Wijewickreme (2006). Fraser River silt has a specific gravity of 2.69 and plasticity index of 4.1.  3.4 DSS testing procedure  3.4.1 Specimen setup   In preparation for specimen setup, porous stones were boiled and placed on the top and bottom pedestal of the DSS. The bottom end of the reinforced DSS membrane is then mounted on the bottom DSS pedestal and the bottom O-ring is placed to fix the membrane in place and provide sealing against leakage. The split mould is then placed and clamped around the membrane. At this time, the upper end of the membrane is folded around the mould. The membrane is stretched to conform to the cylindrical wall of the split mould using vacuum. For DSS tests with lateral stress measurement, four free form sensors were glued to the membrane using water soluble glue as shown in Figure  3.11a. Four pieces of transparent tape, each is slightly larger than the area of a sensor, were then placed over the sensors for additional support. As shown in Figure  3.11a, two of the sensors run parallel to the direction of shearing while the other two sensors are perpendicular to the direction of shearing.   Figure  3.11b shows a photograph taken during soil placement in the DSS cavity. A sufficient amount of soil was placed, as per using the procedures given in Section 3.4.2, to achieve the desired initial specimen height. The top pedestal is then lowered into the DSS cavity and secured in place. This is   42  followed by the application of a seating load of about 10 kPa in small increments to avoid specimen squeezing. The top end of the membrane is then secured around the top pedestal using an O-ring and the mould is removed. At this time the specimen is ready for consolidation.   3.4.2 Specimen reconstitution method   3.4.2.1 Preparation of glass beads DSS specimens   Glass bead DSS specimens were prepared using the preparation method of air pluviation. The glass beads were rained into the specimen mould of the DSS device under gravity using a funnel with a bottom opening having a diameter equal to 1 cm. The raining process was performed such that the particles had a drop height of 17 cm above the top of the specimen. The final surface of the specimen was obtained by traversing a suction tube connected to a vacuum supply of about 30 kPa. The vacuum process allowed for the removal of excess particles and produced a final levelled surface with minimum disturbance to the particles below the top surface of the specimen. A photograph showing a typical specimen of glass beads at the end of specimen preparation phase is presented in Figure  3.12. The use of air pluviation is relevant in this context as it mimics the deposition process of spherical particles used in the discrete element model.  3.4.2.2 Preparation of Fraser River silt DSS specimens  Fraser River silt DSS specimens were reconstituted using the slurry deposition method. De-aired water was added to a batch of dry Fraser River silt. The batch was mixed thoroughly resulting in a homogenous paste. The paste was placed under vacuum for at least 24 hours to remove entrapped air bubbles that may have developed during mixing. The paste was then allowed to settle under its own weight and clear excess water emerging on the top surface was siphoned. The paste was carefully stirred before preparation of DSS specimens to ensure that it remain homogenous. A sample was taken at this time and used to measure the moisture content of the paste. An appropriate amount of the paste was then   43  spooned into the DSS cavity until the desired specimen height was achieved. The specimen was then secured within the DSS cavity using O-rings and a seating vertical pressure of about 10 kPa was applied in small increments. Similar specimen preparation procedure was followed by Sanín (2010). Control over end of consolidation moisture content is not possible when using the slurry deposition method which is believed to simulate the natural deposition and consolidation of the silt effectively.  3.4.3 Consolidation phase   Consolidation was performed by incrementally increasing the vertical stress to its desired value at the end of consolidation. The specimens were subjected to the consolidation stress for 1 hour before subsequent shearing; this allowed sufficient time for the completion of primary consolidation for the tested fine-grained specimens.   3.4.4 Shearing phase   For constant volume (i.e. equivalent to undrained) DSS tests, the vertical loading arm shown in Figure  3.2 is constrained against movement using a clamp and, in turn, forcing a total constant volume condition on the specimen as the specimen is constrained from lateral movement by the wire reinforced membrane.   For cyclic loading, stress-controlled cyclic shear stress variation having a sinusoidal form with a frequency of 0.1 Hz was applied to the tested DSS specimens. It is noted that the effect of applied cyclic shearing frequency on DSS behavior of tested specimens is not investigated in this study. The amplitude of the cyclic shear stress is controlled to achieve the desired cyclic stress ratio (cy/'vc). Monotonic shearing tests were undertaken at a slow fixed shear strain rate of 10 % per hour. Figure  3.11c shows a photograph of a specimen at the end of shearing tested using the instrumented DSS membrane.       44  3.5 Testing program  A testing program to assist evaluating the NGI-type DSS testing was undertaken as presented in Table  3.1. As indicated earlier, DSS testing on glass beads (Series I) was used as basis for validation of a discrete element model as described in Chapter 5, and, in turn, this testing provided insight on the use and interpretation of DSS testing results in characterizing the behavior of granular materials. The effect of vertical effective stress on monotonic and cyclic DSS response of the glass beads is investigated for 'vc values of 100, 150, and 200 kPa. Although DSS testing has been predominantly used for evaluation of the constant volume (i.e. equivalent to undrained) behavior of soils, drained DSS testing is of importance from a fundamental point of view for the development of constitutive models. As such, few monotonic drained DSS tests were also undertaken on the glass beads.   DSS testing on reconstituted Fraser River silt specimens (Series II) using the instrumented DSS membrane developed as part of this study was undertaken with emphasis on the measured lateral stresses. Limited number of tests were conducted to investigate the effect of vertical effective stress for 'vc values of 200 kPa, and 300 kPa and the effect of overconsolidation ratio for OCR values of 1, and 2 on measured lateral stresses for monotonic and cyclic constant volume tests.     45  Table ‎3.1. Summary of DSS testing program. Test Series  Material / Test 'vc          (kPa) OCR CSR (cy/'vc) Boundary condition  No. of Tests I a Glass beads: Effect of vertical effective stress  100, 150, and 200 1 Monotonic Drained and Constant Volume  6 b 0.08-0.11 Constant Volume  9 II a Fraser River silt: Effect of vertical effective stress on lateral stresses  200, and 300 1 Monotonic  Constant Volume  2 0.12-0.17 6 b Fraser River silt: Effect of OCR on lateral stresses 200 2 0.17-0.25 Constant Volume  3       Total = 26 tests      46  a)  b)  Figure ‎3.1. Schematic showing a cross-section through a DSS specimen parallel to the direction of shearing and stresses acting on the specimen: a) at the end of consolidation and before shearing; b) during the shearing phase.   xyz'hc= ko'vc'vcxyz+ h- h'v'h  47   Figure ‎3.2. Schematic of the UBC-DSS device.   Legend Load cellPressure regulatorElectro-Pneumatic pressure regulatorValveLCRE-PRR E-PLCLCLVDTLVDTSingle acting pistonDouble acting pistonLocating pinMotor driveSpecimenDrainage linesVertical loading armHorizontal loading armVertical displacement clamp  48   Figure ‎3.3. Photograph showing a free form sensor (D= 18 mm) connected to the data acquisition cable.      49   Figure ‎3.4. Schematic illustrating the arrangement used to calibrate free form sensors.    Air Pressure sourceO-ring for sealingRegular membrane Reinforced membrane Mould  Free-From Sensor Cable to Data acqusition systemValve Top drainage lineBottom drainage lineDSS top cap (Fixed in place during calibration)DSS bottom cap (Fixed in place during calibration)Cable to Data acqusition system  50   Figure ‎3.5. Typical results for a calibration of a free form sensor.    0204060801001200 20 40 60 80 100 120 140Applied air pressure (kPa)Sensors pressure reading (kPa)Calibration No. 1Calibration No. 2Calibration No. 3  51   Figure ‎3.6. Results of a load-unload calibration performed on a free form sensor.    0204060801001200 20 40 60 80 100 120 140Applied air pressure (kPa)Free-Form sensor reading (kPa)Loading Unloading  52   Figure ‎3.7. Potential error due to hysteresis effect.    051015202530354045501 1.5 2 2.5 3% error Sensor over-consolidation ratio (OCR) Sensor 1Sensor 2Sensor 3Sensor 4  53   Figure ‎3.8. Shear stress-strain plots for tests with and without the addition of the free form sensors.    0102030405060700 5 10 15Shear strain  (%)Shear stressh (kPa)CURRENT TESTING (with Free-Form sensors)SANIN (2010)CURRENT TESTING (without Free-Form sensors)  54   Figure ‎3.9. Recorded equivalent shear stress for the membrane and sensor-arrangement plotted against shear strain.      024681012141618200 5 10 15 20 25Recorded equivalent shear  stress (kPa) Shear strain (%) Membrane setup No. 1Membrane setup No. 2  55    Figure ‎3.10. Shear stress strain plots for tests performed with and without the addition of free form sensors after applying the additional membrane stiffness correction.    0102030405060700 5 10 15Shear stress, h(kPa)Shear strain (%)CURRENT TESTING (with Free-Form sensors)SANIN (2010)CURRENT TESTING (without Free-Form sensors)  56  a)        b)  c)               Figure ‎3.11. Photographs showing the free form sensors attached to the DSS membrane a) before specimen preparation; b) during specimen preparation; c) during the shearing phase.    57   Figure ‎3.12. Photograph of the tested glass beads placed in the DSS cavity taken at the end of specimen preparation phase.      58  4 DSS TESTING RESULTS  This chapter presents the results of the DSS experimental program previously described in Chapter 3. The chapter is organized into two main parts: i) results of DSS testing performed on glass beads to generate data for validation of discrete element models (Series I); and ii) the results of the testing performed on the Fraser River silt performed to investigate lateral stresses development during DSS shearing (Series II). Measured lateral stresses are used for qualitative comparison with lateral stresses obtained from the DEM model presented in Chapter 5. Each section includes a presentation of the testing results shown in plots and summary results for the testing program presented in a tabular format. This is followed by a discussion on the implications of the experimental findings. A summary of the results of the performed DSS tests for Series I and Series II is shown in Table  4.1 and Table  4.2, respectively.   In the following, drained conditions refer to tests performed under constant vertical effective stress, ’vc, throughout the shearing phase. As previously discussed, constant volume testing is a simpler alternative to undrained testing in the direct simple shear device. It has been shown that the decrease (or increase) in vertical stress acting on the top and bottom boundaries is equivalent to increase (or decrease) in pore water pressure during an undrained test (Finn et al., 1978; Dyvik et al., 1987). As such, equivalent pore water pressure values, u, presented in this chapter were considered equal to ’vc’v. It is noted that positive volumetric strains represent contraction.   4.1 DSS testing of glass beads to generate data for validation of discrete element models (Series I)  4.1.1 Monotonic shearing  The results of three monotonic drained tests performed with 'vc values of 100 kPa, 150 kPa, and 200 kPa are presented in Figure  4.1 (GB-D-100-M; GB-D-150-M; GB-D-200-M).The end of consolidation void ratios (ec) for the test are 0.605, 0.596, and 0.591, respectively. As typically observed for drained shearing   59  of granular soils, a rapid increase in the shear stress at relatively small shear strains is observed as shown in Figure  4.1a. This is followed by a gradual increase in shear stress with further development of shear strain. Volumetric strain vs. shear strain plot shown in Figure  4.1b suggests the following: i) contractive volumetric strains observed for the whole range of investigated shear strains of 0 % - 8 % which is typical of loose granular materials; ii) the three volumetric strain plots corresponding to 'vc values of 100 kPa, 150 kPa, and 200 kPa are very similar.  The results suggest that the effect of confining stress level on the volumetric shear response is not significant for the range of stress levels that were used in the testing. The normalized shear stress (i.e. h/'v) vs. shear strain plots as shown in Figure  4.1c are essentially the same for the three investigated 'v values.  The observations are in accord with drained shear behavior that is typically observed for granular soils.  The results of three monotonic constant volume tests conducted under the same 'vc values as for the drained tests are shown in Figure  4.2 (GB-CV-100-M; GB-CV-150-M; GB-CV-200-M). The tests were conducted after initially consolidating the specimens to vertical effective stresses of 100 kPa, 150 kPa, and 200 kPa after achieving consolidation void ratio (ec) values of 0.6, 0.588, and 0.59, respectively, under the three consolidation levels. A rapid increase in shear stress at small shear strains is noted. Shear stress increases at a smaller rate with subsequent increase in shear strain in a strain hardening manner (see Figure  4.2a). Prior to shearing, mobilized shear force at contacts between balls is nearly zero. With the progression of shear strain, shear force at contacts increases causing rapid increase in shear stress, as noted at small shear strains in Figure  4.2a. As shear strain further increases, slippage at some contacts occurs (i.e. maximum fiction at some contacts is mobilized). This results in the noted smaller rate of shear stress increase at larger shear strains. Shear stress ratio (h/'v) versus shear strain plots, shown in Figure  4.2b for tests performed with 'vc values of 100 kPa, 150 kPa, and 200 kPa, are very similar. This is in line with previous observation for the drained case as shown in Figure  4.1c. Stress paths shown in Figure  4.2c indicate an initially significant contractive behavior (i.e. decrease in vertical effective stress) followed by dilative behavior (i.e increase in vertical effective stress) which is the typically   60  observed behavior for loose to medium loose sands. It is noted that phase transformation for the three plots corresponding to tests with 'vc of 100 kPa, 150 kPa, and 200 kPa occurs at essentially the same stress ratio (h/'vc). A friction angle of 16 at phase transformation is estimated. The observations are in accord with uniqueness of mobilized friction angle at phase transformation as reported by Vaid & Sivathayalan (1996). Figure  4.2d shows the normalized stress path plot. The three plots fall within a relatively close range. The results suggest that the effect of confining stress level on the volumetric strain tendency is not significant for the range of stress levels that were used in the testing. This observation is in line with independence of measured volumetric strains from 'vc as previously discussed for drained tests. It is noted that additional testing covering a broader range of 'vc values is required to reach a more solid conclusion in regards to the effect of vertical effective stress.   4.1.2 Cyclic shearing   The results of constant volume cyclic shear tests performed on the glass beads as indicated in Table  4.1 are presented in this section.   4.1.2.1 Cyclic loading response  Figure  4.4 presents typical stress path and stress strain response for a test with a relatively low cyclic stress ratio (CSR=cy/'vc) of 0.078 and vertical effective stress ('vc) of 100 kPa (GB-CV-100-03). A gradual drop of vertical effective stress is observed from its initial value of 100 kPa with the application of cyclic shear until it momentarily reaches a'vc value of about 0 kPa during the last three cycles. The increase in effective stress observed in the loading parts of the last few cycles is associated with the dilative tendency of the tested glass beads. The momentary total loss of strength (i.e. reaching 'vc of about 0 kPa) is typically observed for cyclic testing on sands (Wijewickreme et al., 2005).The gradual   61  drop in effective stress is associated with gradually increasing shear strains with the progression of cyclic loading. The overall increase in shear strain is more dramatic during the last few cycles.   Typical results for a cyclic test with a higher CSR value of 0.109 are presented in Figure  4.3 (GB-CV-100-01). The plotted results indicate the following: i) A sudden drop in vertical effective stress from its initial value at end of consolidation of 100 kPa to a value of about 40 kPa is observed during the first cycle of loading; ii) Effective stress reaches a value close to 0 kPa during the last two cycles of loading followed by an increase in effective stress associated with dilative behavior during the loading parts of the cycles; and iii) a more rapid increase in shear strain compared to the test performed with the lower CSR value is observed. Similar observations were noted for the cyclic testing performed at a 'vc value of 200 kPa as shown in Figure  4.5.   4.1.2.2 Cyclic resistance  For comparison purposes, the cyclic shear resistance herein is defined as the number of load cycles required to reach a single-amplitude horizontal shear strain  = 3.75%, in a given constant volume DSS test under a given applied CSR. This  = 3.75% condition in a DSS specimen is essentially equivalent to reaching a 2.5% single amplitude axial strain in a triaxial soil specimen. An identical definition has been previously used to assess the cyclic shear resistance of sands by the U.S. National Research Council (NRC, 1985), and it also has been adopted in many previous liquefaction studies at the University of British Columbia.  The instance of reaching  = 3.75% for the above laboratory tests on glass beads are denoted by the dots on the stress-strain and stress path plots shown in Figure  4.3, Figure  4.4, and Figure  4.5, respectively.    62  Figure  4.6 presents the cyclic stress ratio (CSR) plotted against the number of cycles to reach  = 3.75% obtained from the results of nine cyclic tests performed with 'vc of 100 kPa, 150 kPa, and 200 kPa. A typical trend of decreasing cyclic stress ratio with the increase in the number of cycles to reach  = 3.75% is observed. The  points in Figure  4.6 corresponding to the selected 'vc values fall within a narrow range suggesting that the effect of 'vc on cyclic resistance is marginal within the investigated range of 'vc values. This is consistent with observations made on the effect of 'vc from the results of monotonic tests discussed in the previous section.   4.1.3 Discussion on DSS testing performed to generate data for validation of discrete element models  As per above, the results from the cyclic constant volume tests performed on the glass beads indicated that cyclic resistance is essentially the same for tests performed at 'vc values of 100 kPa, 150 kPa, and 200 kPa. The effect of 'vc on the cyclic resistance ratio (CRR) defined as the stress ratio at which the specimen reaches  of 3.75% in 15 cycles is typically expressed in terms of the correction factor k defined in Equation 4.1.                    (      )              (4.1)  Vaid & Sivathayalan (1996) and Vaid et al. (1985) demonstrated that k for tests performed on loose sands has a value of about 1 indicating that CRR values for loose sands are essentially not affected by stress level. It may be noted from the results of monotonic tests that the tested glass beads demonstrated loose contractive behavior. Hence, the observed, practically identical, CRR values for the cyclic tests performed as part of the current study at 'vc values in the range of 100 kPa – 200 kPa is judged   63  reasonable. Similar to cyclic tests, results of monotonic tests indicate very similar behavior for the selected 'vc values. Additional testing will be required if this observation is to be extended to a broader range of 'vc values.  Normalized cyclic resistance ratio (obtained by dividing cyclic resistance ratio for a given number of cycles by that  calculated for 15 cycles) versus number of cycles to reach  of 3.75% is showed in Figure ‎4.7. It can be seen that the relationship between normalized CRR and number of cycles to reach  of 3.75% follows the typical power relationship (as in Equation 4.2) for granular material with a “b” value of 0.18 (Idriss & Boulanger, 2008).              (   )           (4.2)  Figure  4.8 shows the end of test (i.e. at  = 8%) shear stresses plotted against the corresponding vertical effective stresses for constant volume and drained tests. The DSS effective friction angle (') calculated as the arctangent of the slope of a linear trend line is equal to 17o. Negussey et al. (1988) based on ring shear testing and triaxial testing results on glass beads with D50 of 0.4 mm reported constant volume friction angle of 24o. Fukuoka (1991) and Fukuoka & Sassa (1991) showed large strain friction angles in the range of 18o-23o. Santamarina & Cho (2001) indicated a critical state friction angle of 21o for their glass beads with D50 of 0.32 mm. Lately, Cavarretta et al. (2010) showed a critical state friction angle of 21o for glass beads based on triaxial compression testing. Their results indicated slight dependence of shear stress strain response on the roughness of individual glass beads particle surfaces. Milled glass beads demonstrated slightly higher strength compared to as delivered (i.e., with smoother surface) glass beads. It seems that the scatter in reported friction angles in literature is partly due to the difference in surface roughness between the glass beads used by the researches cited above in addition to the differences in particles sizes used in these studies.      64   In an overall sense, the above observations suggest that the results derived from DSS testing of glass beads are consistent and they would be suitable for use in calibrating or validating discrete element models as addressed in Chapter 5.   4.2 DSS testing for investigation of the development of lateral stresses during shearing (Series II)  4.2.1 Consolidation phase   The computed coefficient of lateral earth pressure at the end of consolidation (ko) with respect to the applied vertical effective stress ('vc) based on observations made during initial vertical consolidation of Fraser River silt specimens are presented in Figure  4.9. The value of ko was calculated by dividing the average of lateral effective stress (obtained from the four measurements made using free form sensors, as previously discussed in Chapter 3), by the applied vertical effective stress at the end of consolidation ('vc). A range of ko values of 0.4-0.5 was noted for the normally consolidated specimens. ko values were very similar for specimens consolidated to 'vc values of 200 kPa and 300 kPa. Using the relationship of ko = 1- sinby Jaky (1944) to obtain a rough estimate of ko of Fraser River silt yields a ko value of 0.42 (based on  of 35.5º obtained from triaxial test results reported by Wijewickreme & Sanin (2006)). The calculated ko of 0.42 falls within the range of measured ko values of 0.4-0.5 which suggests that the reported results in this regard are reasonable. Higher ko values in the range of 0.7-0.8 were noted for specimens with OCR of 2 compared to ko values for normally consolidated specimens.       65  4.2.2 Monotonic shearing  Figure  4.10 shows measured shear stress, vertical stress, and horizontal stress plotted against shear strain for a monotonic constant volume test (test FRS-200-M as shown in Table  4.2). The following is noted from the results shown in Figure  4.10: i) shear stress increases rapidly with the development of shear strain until it reaches an essentially constant value of about 40 kPa (Figure  4.10a); ii) a decrease in vertical effective stress with the increase in shear strain is observed (Figure  4.10b); and iii) very similar horizontal effective stresses were measured for the locations of sensor 1 – sensor 4 indicating uniform distribution of horizontal stresses acting on the inner wall of the DSS reinforced membrane (Figure  4.10c); iv) a decreasing trend is noted for average horizontal effective stress with the development of shear strain which is similar to the observed trend for the vertical effective stress.  The change in the coefficient of lateral earth pressure, k, with the development of shear strain for the two monotonic tests performed at 'vc of 200 kPa and 300 kPa are presented in Figure  4.11. An increase in k is noted from an initial (i.e., at  = 0%) value of about 0.5 to a value of about 1.1 at large shear strains. A similar trend of increasing k values with the development of shear strain to values higher than 1 at relatively large shear strains has been observed by Budhu (1985) from the results of DSS testing on Leighton Buzzard sand. It is noted that the potential for stress non-uniformities near the lateral boundaries may have affected the measured k values.   4.2.3 Cyclic shearing 4.2.3.1 Normally consolidated specimens   The results of the cyclic tests performed on Fraser River silt are summarized in this section with emphasis on the measured lateral stresses. The results of test FRS-200-01 (see Table  4.2) are presented in Figure  4.12. An overall cumulative contractive response is observed with the initial progression of cyclic   66  loading. A dilative response (i.e. increase of vertical effective stress) was observed during the loading parts in a given cycle. This is followed by a contractive response (i.e. decrease of vertical effective stress) during the unloading parts in a given cycle. The horizontal effective stress, calculated as average of stresses acting on the four sensors attached to the DSS reinforced membrane, is plotted in Figure  4.12b against shear stress. The trend observed for the horizontal effective stress is similar to that noted for the vertical effective stress. It is noted that the horizontal effective stress during the last loading cycle was not reported as the accuracy of the measurements at low stresses is judged inadequate (minimum stress that can be measured by the free form sensors with confidence is about 20 kPa). Shear stress strain behavior presented in Figure  4.12c indicates a gradual increase in maximum shear strain with the development of cyclic shearing (i.e., cyclic mobility type of behavior). As shown in Figure  4.13, similar observations were noted from the results of test specimen FRS-200-03 that was performed at a lower CSR value of 0.121 compared to CSR of 0.155 for test FRS-200-01. Cyclic tests performed on specimens consolidated to 'vc of 300 kPa displayed similar behavior as shown in Figure  4.14, for test FRS-300-01.   Figure  4.15 and Figure  4.16 show the change in lateral earth pressure coefficient with the increase in number of cycles for specimens consolidated to 'vc values of 200 kPa and 300 kPa, respectively. An overall increase in lateral earth pressure coefficient is noted with the progression of cyclic loading. During a given cycle, k value increases during the loading parts of the cycles followed by reduction in k value during the unloading parts of the cycle. The overall increase in k values is more gradual for the tests with lower CSR values relative to the tests performed with higher CSR values. The observed increasing trend of k values with the development of cyclic shearing is in qualitative agreement with the results of cyclic DSS testing performed on Ottawa sand by Youd & Croven (1975).       67  4.2.3.2 Effect of overconsolidation ratio  The effect of overconsolidation ratio (OCR) on the cyclic behavior was investigated for an OCR value of 2. Figure  4.17 presents the results from test FRS-200-OC-03 (see Table  4.2) representing a typical response. A more dilative response is observed associated with more dramatic increase in vertical effective stress during the loading parts of shearing cycles compared to normally consolidated specimens. Figure  4.18 shows the change in k value with the increase in number of cycles for three tests with CSR values of 0.173, 0.207, and 0.251. A smaller overall increase in k values is observed for the overconsolidated specimens compared to normally consolidated specimens as discussed in the previous section.  4.2.3.3 Cyclic resistance  Figure  4.19 shows cyclic resistance ratio (CRR) plotted against the number of cycles to reach  of 3.75% for the cyclic tests performed on Fraser River silt. The following is noted from the results: i) the CRR trends for the tests on NC silt performed at 'vc of 200 kPa and at 'vc of 300 kPa are very similar. This is in line with the findings of Sanin & Wijewickreme (2006) that indicated that cyclic resistance is fairly independent of the initial confining stress ('vc) for the results of testing performed on Fraser River silt; and ii) the cyclic resistance for specimens with OCR of 2 is considerably higher than that obtained for the normally consolidated specimens.   As previously noted in Section 4.1.3, the number of cycles to reach  of 3.75% versus normalized cyclic stress ratio (i.e. CRRN/ CRRN=15) plot for granular materials typically follows a power relation as in Equation 4.2 (Idriss & Boulanger, 2008). Number of cycles to reach  of 3.75% versus normalized cyclic resistance ratio for normally consolidated Fraser River silt is shown in Figure ‎4.20. It is noted that the plotted data follows the power relation as in Equation 4.2 with a “b” value of 0.15.    68   4.2.4 Discussion of results of DSS testing for the investigation of the development of lateral stresses during shearing  The following discussion addresses the results presented in the previous sections on DSS testing of Fraser River silt in relation to furthering our understanding of the lateral stresses in the DSS device during shear. A thorough discussion on aspects related to shear stress-strain response and cyclic resistance of Fraser River silt is presented in Sanín (2010).  4.2.4.1 Lateral stress development in DSS device during vertical consolidation  The coefficient of lateral earth pressure at rest, ko, typically reported for the NGI-type DSS testing with the wire reinforced membrane providing lateral confinement, are lower than those reported for Cambridge-type DSS testing or odometer tests with rigid boundaries. For example, Budhu (1985) reported ko values in the range of 0.2-0.32 for the NGI-type DSS test from experiments performed on Leighton Buzzard sand. However, ko values reported for the same sand, but using the Cambridge type test were in the range of 0.4-0.45. The lower trend of ko values obtained from the NGI-type test compared to those obtained from the Cambridge-type test is caused by the ability of the latter to more effectively provide lateral confinement. Youd & Croven (1975) also reported relatively low ko value of 0.34 for the NGI-type test calculated from results of testing performed on Ottawa sand. Higher ko values of 0.54 and 0.67 were obtained from NGI-type DSS testing for clays compared to these obtained for sands Dyvik et al. (1981). The reported ko values in the range of 0.4-0.5 for normally consolidated Fraser River silt fall in between ko for sands and clays which is typically expected for silts.   Northcutt (2010) showed a trend of increasing ko values with the increase in OCR for sands. The noted trend of higher ko values in the range of 0.7-0.8 from over-consolidated specimens (OCR=2) compared to those from normally consolidated specimens is in line with the findings of Northcutt (2010).    69  4.2.4.2 Lateral stress development in DSS device during shear loading  Budhu (1985) showed that measured average horizontal stress, h, in the NGI-type DSS device is neither equal to x nor equal to y measured in the Cambridge-type DSS device (coordinates were shown in Figure ‎1.1). As such, he concluded that measured h is not useful for interpretation of DSS testing results. The results presented in this study allow for opportunity to identify x and y independently, calculated as the average of measured stresses on the two sensors that run perpendicular and parallel to the direction of shearing, respectively. Further, measured lateral stresses obtained from the four sensors for a given test are used to assess the stress uniformities in the tested DSS specimens, as will be shown in Chapter 6.   Youd & Croven (1975), from the results of constant shear strain amplitude drained DSS tests, noted that the coefficient of lateral earth pressure increases with the increase in shear strain amplitude and number of cycles. It was not possible to evaluate the effect of shear strain amplitude and number of cycles on measured lateral earth pressure coefficient independently from the stress controlled constant volume cyclic DSS tests. However, the noted general trend of increasing k value with the increase in number of cycles is essentially in agreement with the findings of Youd & Croven (1975).   4.3 Closure  The results of DSS testing performed for use in validation of discrete element models and to investigate lateral stress development during DSS shearing were summarized. The previous analysis and discussion on the DSS testing performed on glass beads indicated the validity of the generated data for use to confirm discrete element models. This provides the rationale for using the generated data in Chapter 5 for the development and calibration of discrete element modeling of the DSS test. The results of the numerical model are used to provide insight on the state of stresses and strains in DSS specimens, along   70  with further analysis on the results of DSS testing performed with lateral stress measurements as shown in Chapter 6 and Chapter 7.      71  Table ‎4.1. Summary of the results of DSS testing performed on glass beads (Series I). Test ID Drainage condition e0 σ'vc (kPa) ec  Cyclic tests cy/σ'vc No. cyc. =3.75% (single amplitude) max (%) urσ'vcGB-D-100-M Drained 0.621 101.0 0.605 Monotonic GB-D-150-M 0.618 150.7 0.596 GB-D-200-M 0.618 200.4 0.591 GB-CV-100-M Constant volume  0.615 99.3 0.600 Monotonic GB-CV-150-M 0.610 151.3 0.588 GB-CV-200-M 0.615 187.6 0.590 GB-CV-100-01 0.612 99.3 0.596 0.109 1.8 6.2 1.00 GB-CV-100-02 0.614 100.2 0.598 0.092 4.7 6.4 0.99 GB-CV-100-03 0.614 100.0 0.600 0.078 12.8 6.0 1.00 GB-CV-150-01 0.608 150.1 0.589 0.097 5.7 5.2 0.98 GB-CV-150-02 0.611 149.4 0.587 0.081 15.7 5.3 0.99 GB-CV-150-03 0.611 149.1 0.588 0.061 68.7 5.1 1.00 GB-CV-200-01 0.609 197.0 0.586 0.104 3.7 6.4 0.98 GB-CV-200-02 0.603 198.7 0.579 0.084 13.7 6.2 1.00 GB-CV-200-03 0.612 191.2 0.588 0.065 25.7 6.0 1.00  eo void ratio after specimen preparation and application of seating load. ec void ratio at the end of consolidation. σ'vc vertical effective stress at the end of consolidation. cy single amplitude cyclic shear stress. max maximum shear strain during cyclic shearing. ur equivalent residual pore water pressure at the end of cyclic shearing.      72  Table ‎4.2. Summary of the results of DSS testing performed to assess DSS lateral stresses on Fraser River silt (Series II). Test ID o(%)σ'vc (kPa) c(%)OCR   Cyclic tests τcy/σ'vc No. cyc. to =3.75% γmax (%) urσ'vcFRS-200-M 48.0 198.3 29.4 1 Monotonic FRS-200-01 47.8 198.8 29.4 0.155 8.2 6.2 0.91 FRS-200-02 47.4 196.3 29.8 0.142 12.7 4.5 0.87 FRS-200-03 47.3 199.2 29.4 0.121 35.8 3.8 0.93 FRS-300-M 45.3 296.1 27.5 Monotonic FRS-300-01 45.7 293.5 28.3 0.170 4.7 6.8 0.86 FRS-300-02 45.7 295.0 28.3 0.145 13.7 5.7 0.89 FRS-300-03 46.7 298.1 27.9 0.123 40.7 4.9 0.91 FRS-200-OC-01 45.9 195.4 27.7 2 0.173 99.7 4.5 0.89 FRS-200-OC-02 46.7 195.7 27.8 0.207 29.7 5.1 0.86 FRS-200-OC-03 49.2 194.2 26.8 0.251 8.8 4.3 0.79  o initial specimen water content. c water content at the end of consolidation. σ'vc vertical effective stress at the end of consolidation. cy single amplitude cyclic shear stress. max maximum shear strain during cyclic shearing. ur equivalent residual pore water pressure at the end of cyclic shearing. OCR overconsolidation ratio.     73  a)  b)  c)   Figure ‎4.1. Results of drained DSS tests performed on glass beads: a) shear stress versus shear strain; b) volumetric strain versus shear strain; c) stress ratio versus shear strain.  0.00.20.40.60.81.00 2 4 6 8Volumetric strain, v(%)Shear strain (%)Contractive'vc= 100 kPa'vc= 150 kPa'vc= 200 kPa0.00.10.20.30.40 2 4 6 8Stress ratio, h/'vShear strain,  (%)'vc= 100 kPa'vc= 150 kPa'vc= 200 kPa  74     Figure ‎4.2. Results of monotonic constant volume DSS tests performed on glass beads: a) shear stress strain plot; b) shear stress ratio c) stress path; d) normalized stress path.   051015202530354045500 2 4 6 8Shear stress, h(kPa)Shear strain (%)'vc= 100 kPa'vc= 150 kPa'vc= 200 kPaa) b) c) d)   75  a)  b)  Figure ‎4.3. Results of a cyclic constant volume DSS test performed on glass beads ('vc = 100 kPa and CSR = 0.078): a) stress path; b) shear stress-strain response.      76  a)  b)    Figure ‎4.4. Results of a cyclic constant volume DSS test performed on glass beads ('vc = 100 kPa and CSR = 0.109): a) stress path; b) shear stress-strain response.      77  a)  b)  Figure ‎4.5. Results of a cyclic constant volume DSS test performed on glass beads ('vc = 200 kPa and CSR = 0.104): a) stress path; b) shear stress-strain response.      78     Figure ‎4.6. Cyclic stress ratio versus number of cycles to reach  of 3.75% (single amplitude) for the tested glass beads for 'vc values of 100 kPa, 150 kPa, and 200 kPa. 0.000.020.040.060.080.100.120.140.160.181 10 100Number of cycles to reach  of 3.75%Cyclic stress ratio (CSR) 'vc = 150 kPa'vc = 200 kPa'vc = 100 kPa  79   Figure ‎4.7. Number of cycles to reach  of 3.75% versus normalized cyclic resistance ratio (CRR) for the tested glass beads.     0.00.51.01.52.01 10 100CRRN/ CRRN=15Number of cycles to reach of 3.75%                 Lab data  80     Figure ‎4.8. Shear stress versus vertical effective stress relationship obtained from the results of DSS testing on glass beads calculated at end of test (i.e.  of 8%).   0204060800 50 100 150 200 250Shear stress, h(kPa)Vertical effective stress'v(kPa)DrainedConstant volume 17o  81    Figure ‎4.9. Coefficient of earth pressure at rest computed from the initial consolidation phase of DSS tests on Fraser River silt.   00.10.20.30.40.50.60.70.80.90 50 100 150 200 250 300 350 400Coefficient of lateral earth pressure at rest, koVertical effective stress, 'vc(kPa)OCR =2NC (OCR = 1)   82  a)  b)  c)   Figure ‎4.10. Response of Fraser River silt during monotonic constant volume DSS shearing: a) shear stress-strain response; b) vertical effective stress versus shear strain; c) horizontal effective stress versus shear strain.     83   Figure ‎4.11. Change in lateral earth pressure coefficient with the development of shear strain during monotonic loading.       84  a)   b)  c)  Figure ‎4.12. Response of Fraser River silt during cyclic DSS shearing with lateral stress measurement ('vc = 200 kPa and CSR = 0.155): a) vertical effective stress path; b) horizontal effective stress path; c) shear stress-strain response.     85  a)  b)  c)  Figure ‎4.13. Response of Fraser River silt during cyclic DSS shearing with lateral stress measurement ('vc = 200 kPa and CSR = 0.121): a) vertical effective stress path; b) horizontal effective stress path; c) shear stress-strain response.     86  a)  b)  c)  Figure ‎4.14. Response of Fraser River silt during cyclic DSS shearing with lateral stress measurement ('vc = 300 kPa and CSR = 0.17): a) vertical effective stress path; b) horizontal effective stress path; c) shear stress-strain response.     87   Figure ‎4.15. Development of lateral earth pressure coefficient with the increase in number of cycles for cyclic DSS shear testing of Fraser River silt conducted with 'vc = 200 kPa.      88   Figure ‎4.16. Development of lateral earth pressure coefficient with the increase in number of cycles for cyclic DSS shear testing of Fraser River silt conducted with 'vc = 300 kPa.       89  a)  b)  c)  Figure ‎4.17. Typical results of a cyclic DSS test with lateral stress measurement performed on an over consolidated Fraser River silt specimen: a) vertical effective stress path; b) horizontal effective stress path; c) shear stress-strain.     90   Figure ‎4.18. Development of lateral earth pressure coefficient with the increase in number of cycles for overconsolidated Fraser River silt.      91    Figure ‎4.19. Cyclic resistance ratio versus number of cycles relation observed from DSS testing of Fraser River silt.     0.000.040.080.120.160.200.240.281 10 100 1000Cyclic stress ratio (CSR) Number of cycles to reach  of 3.75 %'vc= 200 kPaOCR=2'vc= 300 kPaNC'vc= 200 kPaNC  92    Figure ‎4.20. Number of cycles to reach  of 3.75% versus normalized cyclic resistance ratio (CRR) for normally consolidated Fraser River specimens.    0.00.51.01.52.01 10 100CRRN/ CRRN=15Number of cycles to reach of 3.75%                 Lab data  93  5 DEVELOPMENT OF A DISCRETE ELEMENT MODEL OF THE DSS TEST  With the continued advances in computing power, it is now possible to model soil as a collection of particles rather than a continuum using the discrete element method (DEM) developed by Cundall & Strack (1979). DEM allows for determination of stresses and strains at specific internal zones in addition to their averages at the boundaries. Accordingly, DEM is an ideal analysis tool for simulating and investigating the distribution of stresses and strains within test soil specimens. Recent works on simulating laboratory testing conditions using DEM have demonstrated the potential of DEM in tracking the salient characteristics of soil behavior and its suitability for modeling soils element testing (Thornton, 2000; Cui & O’Sullivan, 2006; Yan, 2009; Zhao et al., 2009; Fu & Dafalias, 2011; Härtl & Ooi, 2011; O’Sullivan, 2011a). Dabeet et al. (2010) presented initial results of 3D DEM analysis of a cylindrical DSS specimen with emphasis on stress-strain uniformities. A 2D DEM study of the Cambridge-type DSS test was conducted by Shen et al. (2011). Wijewickreme et al. (2013) have discussed the results of a study on the state of stress in a simulated DSS specimen using 3D DEM analysis of spherical particles.   With this background, and in consideration of the extensive use of the DSS device in characterizing the seismic response of soils, analysis of DSS specimens subjected to drained and constant volume loading conditions was undertaken using discrete element modeling. Particle Flow Code in Three Dimensions (PFC3D) based on DEM, as discussed in detail in the following section, was used to perform the analysis. This chapter addresses the development of the DSS DEM model; the outcomes from the DEM simulation of laboratory DSS testing using the DEM model are then directly compared with data from laboratory DSS testing performed on glass beads. It is noted that the results of the DSS testing program performed on glass beads, as discussed in Chapter 4, are used for DEM model validation. The selection of spherical glass beads for validation of the DEM model is considered appropriate as soil is modeled as a collection of spherical particles in the current analysis.    94  5.1 Overview of the discrete element program PFC3D  Particle Flow Code in three dimensions (PFC3D) is based on the discrete element method (DEM) by Cundall & Strack (1979) and Itasca Inc. (2005b). Soil particles are modeled as rigid spheres (referred to as balls). The contacts between balls are modeled using the soft contacts approach that allows particles to virtually overlap (Itasca Inc. 2005a).  The magnitude of the overlap is related to the forces at the contacts through normal and shear stiffness values, Kn and Ks, respectively. For the linear contact model, Kn and Ks are constants. Alternatively, the non-linear Hertz-Mindlin contact model uses variable Kn and Ks values as defined in Equation 5.1 and Equation 5.2. The Hertz-Mindlin model is described in more detail in Itasca (2005b). The Hertz-Mindlin model has been in common use lately and is believed to simulate force-displacement relationship at particles contacts more effectively compared to the linear model. A comparison between the results of the linear model and the Hertz-Mindlin model is provided in Section 5.3.      (  √        )√            (5.1)     ( (          )      ) |  |           (5.2)  Where n is the overlap distance between the two contacting entities, |  | is the magnitude of normal force at the contact, G is the elastic shear modulus,   is Poisson’s ratio, and       [ ] [ ] [ ]  [ ]           (5.3)  Where r is sphere radius and the superscripts [ ] and [ ] refer to the two spheres in contact.    95   Kn and Ks have the units of force/displacement. Friction, F, defined as the ratio of shear to normal force at which slippage at contacts occurs can be specified. Boundaries are referred to as walls. The contacts between the balls and walls are modeled in a similar way to contacts between the balls. The code uses an explicit solution scheme. In PFC3D, continuum stresses can be calculated based on contact forces located within a spherical zone with a user-defined center and radius (see O’Sullivan 2011b; Itasca Inc. 2005b).  5.2 Analysis methodology  A DSS specimen comprising a diameter of 7 cm and a height 2.1 cm filled with “balls” was considered for the PFC simulations herein, leading to a height to diameter ratio of 0.3. The specimen size is similar to the actual specimen size used in NGI-type DSS devices.  Specimen preparation was simulated by “numerically raining” particles under gravity simulating the preparation of pluviated glass beads specimens that are used for the evaluation of the PFC model.   Specimens simulated using particle pluviation were initially formed in a very loose state in a cylindrical mould that had the same diameter as the specimen cavity. The locations of the balls were generated randomly inside the mould with the resulting uniform particle diameters of 2 mm which is the same particle size and distribution as that for the tested glass beads.   The soil placement simulation is graphically illustrated in Figure  5.1.  A mould containing particles is mounted on top of the specimen cavity as shown in Figure  5.1a. Gravity is then numerically activated and the process of balls falling under gravity and filling the specimen cavity is simulated. It is noted that the falling height of particles affects the specimen’s void ratio. The falling height of particles is selected by iterative trial and error process to achieve the desired void ratio for the glass beads (i.e. the void ratio that   96  is close to that used in the real experiment). At the end of the soil placement phase, particles with centers located above the top of the specimen are deleted and the top cap/boundary is placed (Figure  5.1b).  The consolidation of the specimen was simulated by moving the bottom boundary upwards (similar to consolidation in the UBC-DSS device) at a rate of 0.05 mm/second. The time step was set to 2x10-5 seconds /computation cycle (i.e., it takes one million computation cycles to displace each of the walls by 1 mm). The simulation can vary from several hours to days in terms of processing time depending on the number of balls in the model and the computation power available.    The specimen is bounded laterally by a cylindrical wall that consists of 15 rigid rings (each with height of 1.4 mm). The rings provide lateral confinement simulating boundary shear strain conditions for the cylindrical specimen DSS test. During shearing, the rings are moved independently at different rates to achieve uniform boundary shear strain, xz. The top boundary is displaced horizontally as a function of shear strain in the positive x-direction.  The bottom boundary is allowed to move in the z-direction during drained shearing to maintain the desired zz.   Average stresses were calculated at the boundaries for comparison with the results of the real DSS test.  Vertical effective stress, 'v, at the boundaries is equal to the sum of forces acting at the wall-ball contacts for the top and bottom boundaries divided by the total surface area of the top and bottom boundaries.  Horizontal effective stress, 'h, is equal to the summation of radial contact forces acting on lateral rings divided by inner surface area of the rings. The shear force driving the upper boundary and the lateral rings was calculated as the summation of boundary contact forces in the x-direction acting on the top cap and the rings located in the top half of the specimen. Shear stress, h, is equal to shear force divided by the area of the top boundary.    97  The boundary shear strain increment () used for the simulations reported herein was 2.5 x 10-7 /computation cycle and the time increment was set at 2 x 10-5 sec/computation cycle. The size of shear strain increment was selected based on the results of a sensitivity study that indicated non-changing response for shear strain increments larger than 2.5 x 10-7 /computation cycle. Figure  5.2 shows the shear stress strain plot for two drained simulations performed at  values of 2.5 x 10-7 /cycle and 5 x 10-7 /cycle. The stress-strain response for the two simulations is essentially identical indicating that the selection of shear strain increment of 2.5 x 10-7 /computation cycle for the current analysis is reasonable.  During drained shearing, vertical effective stress is kept constant at its end of consolidation value using a sub-routine that controls the movement of the bottom specimen boundary along the z-axis as needed to maintain constant vertical effective stress. Volumetric strain is calculated based on the change of the distance between the top and bottom boundaries during shearing. For constant volume simulations, the bottom boundary is fixed in the z-direction. It is noted that the top boundary remains fixed during all simulations (i.e. drained and constant volume) to reproduce the boundary condition imposed by the fixed top platen used in the UBC-DSS device.  5.3 Input parameters and sensitivity analysis  The selected model input parameters for glass beads are summarized in Table  5.1. The following explains the rationale behind the selection of model input parameters.   Cavarretta et al. (2010) measured the friction coefficient at the interface between two glass bead particles. They reported an interparticle friction coefficient of 0.176 for large (diameter = 2.4 mm to 3 mm) as supplied (i.e. not milled or crushed) glass beads. Their study is of relevance to the current analysis as the diameter of modelled glass bead particles (D = 2 mm) in this study is similar to the diameter of glass particles tested by Cavarretta et al. (2010). Although the material used in the current testing is not identical to that used by Cavarretta et al. (2010), interparticle friction of as-supplied glass beads seems to   98  fall within a narrow range of values. For example, Lorenz et al. (1997) reported similar F value of 0.180.02 for contacts between two glass beads. As such, the measured friction coefficient between as-supplied glass bead particles by Cavarretta et al. (2010) of 0.176 was adopted for the current analysis.  To simulate the effect of ripped plates used to minimize particle slippage in real DSS testing, a relatively high friction coefficient value of 10 was assumed at contacts between balls and the top and bottom boundaries. It is noted that while the use of such a high friction coefficient prevents slippage at the top and bottom boundaries, particles in contact with the top and bottom boundaries can displace horizontally relative to the top and bottom boundaries by rolling. Frictionless side boundaries were assumed in the PFC model to simulate the nearly frictionless DSS lateral boundaries.   Poisson’s ratio, , of 0.22 used for the Hertz-Mindlin contact model was selected based on Härtl & Ooi (2011). Similar Poisson’s ratio of 0.2 was used by Cavarretta et al. (2010) to fit the Hertz-Mindlin force vs. deformation curve to measured laboratory data.   The selected shear modulus, G, used for the Hertz-Mindlin contact model was derived based on comparison between PFC simulations results of an odometer test and laboratory odometer test performed on glass beads as part of this study. Figure  5.3 shows volumetric strains vs. vertical effective stress for three simulations performed with shear modulus, G, values of 1 GPa, 1.5 GPa, and 2.5 GPa along with the results of the laboratory odometer test. Overall, it is noted that the results of the simulation with G of 1.5 GPa are the closet to the experimental data relative to the results of the simulations performed with G values of 1 GPa and 2.5 GPa. As such, G of 1.5 GPa was selected for the simulations reported in this study. It is noted that the shear modulus parameter, G, controls the relative displacement between particles in contact. For example, the use of high G values results in lower compressibility of an assembly of particles in contact compared to using low G values. Hence, it was judged reasonable to determine the G parameter based on the compressibility of glass beads as measured from the results of an odometer test.   99   The linear contact model and the Hertz-Mindlin non-linear contact model have been in common use lately for DEM studies in the literature. Hence, it is of interest to compare the results of DEM analysis using the two models. Four odometer simulations were performed using the linear model with linear stiffness parameter, K, (where K= Kn= Ks) values of 100 kN/m, 250 kN/m, 500 kN/m, and 1000 kN/m. The results of the simulations for the linear model are shown in Figure  5.4 plotted along with the results of the laboratory odometer test results. By comparing the results of the Hertz-Mindlin model shown in Figure  5.3 to those for the linear model, it is clear that the Hertz-Mindlin model better captures the observed laboratory odometer behavior of the glass beads than the linear model.   The use of the non-linear Hertz-Mindlin model is more costly in terms of the time it takes to complete the simulations compared to the linear model. For this reason, it was decided to use density scaling to speed up the time it takes to perform the simulations. The density of the glass beads was multiplied by a factor of 104 as shown in Table  5.1. As a result the critical time step increases by a factor of 100 as critical time step is proportional to the square root of the mass and, in turn, dramatically reducing the required simulation time to perform the analysis.  This technique has been commonly used in DEM analysis for quasi-static solutions to reduce the simulation time (Thornton, 2000; Cui & O’Sullivan, 2006). The effect of density scaling on the results of a simulation of an odometer test was evaluatedby comparing the results of two simulations with scaled particle densities of 2500x102 kg/m3 and 2500x104 kg/m3 to a simulation that was performed by using actual particle density of 2500 kg/m3. as shown in Figure  5.5. The difference between the volumetric strain vs. vertical effective stress plots for the three simulations is relatively small. This suggests that the use of density scaling in this context is reasonable.  It is noted that gravity is set to zero during the consolidation and shearing phases.   Finally, viscous damping was used with an assumed damping ratio of 0.3. It was observed from running the code with different damping ratios of 0 and 0.3 that the results are essentially not sensitive to the selected damping ratio as shown in Figure  5.6 for the results of odometer simulations. It is noted that   100  simulated particles have small velocities and therefore the results are not sensitive to the viscous (i.e velocity dependent) damping ratio.  5.4 Evaluation of the performance of the PFC3D model of the DSS test  The performance of the developed PFC3D model of the DSS test as described earlier is evaluated in this section by comparing the simulation results to the results of the DSS laboratory tests performed on glass beads presented earlier in Chapter 4. It is also noted that  relevant results from additional DEM modeling that was undertaken in support of the study findings are included in Appendix A.   Table  5.2 shows the end of consolidation void ratios, ec, and the coefficient of lateral earth pressure at the end of consolidation, ko, for the three simulated specimens consolidated at 'vc values of 100 kPa, 150 kPa, and 200 kPa. The calculated ec values for the three simulations are similar to these obtained for the real DSS specimens shown in Table 4.1.   Figure  5.7 shows shear stress-strain response and volumetric strain development with the increase in shear strain for three drained simulations performed at 'vc of 100 kPa, 150 kPa, and 200 kPa along with those from the real DSS tests. It can be seen that the simulated specimens exhibit very similar shear stress-strain responses to those derived from the testing of real soil specimens. Contractive behavior is observed from the results of both the simulations and the physical tests indicating a response that is typical of loose soils. Similar volumetric strains were noted for the simulation performed with 'vc of 100 kPa, 150 kPa, and 200 kPa. The predicted volumetric strains are smaller than the measured volumetric strains. The former also shows some dilation for shear strains higher than 6%, which is noted for the measured volumetric strains. During a drained DSS test, lateral stress increases. This increase in lateral stress can possibly cause the DSS reinforced membrane to expand in the lateral direction and, in turn, results in increased recorded volumetric strain compared to the case of rigid boundaries used in the DEM   101  model. Analysis that accounts for the membrane stiffness effect is needed to investigate the effect of membrane stiffness on DSS testing results.   Figure  5.8 shows the predicted and measured stress-strain responses and the stress path plots for the monotonic simulations performed under constant volume conditions. A steep increase in shear stress is observed initially with the development of shear strain. This is followed by a more gradual increase in shear stress with further development of shear strain. The stress path plots show initially contractive behavior. Phase transformation is then reached followed by dilative behavior. Overall, the PFC model effectively captures the main characteristics of the observed monotonic constant volume response. It is noted that the maximum difference between model and DSS testing shear stresses at any given shear strain is 20%.   Figure  5.9 through Figure  5.11 present responses of simulated DSS specimens during cyclic constant volume shearing. The simulation depicted in Figure  5.9 was consolidated to 'vc of 100 kPa and loaded to CSR (cy/'vc) of 0.08. A significant overall drop in 'vc was noted with the increase in number of cycles. During the last loading cycle, a transient 'vc of about zero condition, which was similarly observed for the glass bead testing presented in Figure  4.3 through Figure  4.5, was reached. Simulation results for a specimen consolidated with 'vc of 100 kPa and loaded to CSR of 0.12 are presented in Figure  5.10. A similar overall trend of decreasing 'vc with the increase in number of cycles is noted with the transient near zero 'vc condition reached in about three loading cycles. The results of a simulation performed with 'vc of 200 kPa and CSR of 0.1 are presented in Figure  5.11 along with the results of a laboratory test performed with the same 'vc and with a slightly higher CSR of 0.104. The observed overall reduction in 'vc with the increase in number of cycles is also similar to trends noted for the simulations performed with 'vc of 100 kPa. Compared to the lab results, the DEM model predicts higher number of cycles to reach  of 3.75 %. This is partly because the laboratory specimen was sheared with a slightly higher CSR of 0.104 compared to the simulated specimen with CSR of 0.1. It is noted that the PFC model captures   102  stress paths and shear stress-strain response typically observed for loose granular soils including the dilative behavior observed during the last cycle and momentary loss of strength (i.e. near zero vertical effective stress) during the last loading cycle which was observed from the results of the glass beads presented in Chapter 4 (see Figure  4.3 through Figure  4.5 in Chapter 4 for direct comparison).  Figure  5.12 shows CSR versus number of cycles to reach  of 3.75% for 9 simulations performed at 'vc values of 100 kPa, 150 kPa, and 200 kPa. The results of the cyclic tests performed on the glass beads reported earlier in Chapter 4 are also included in the plot. Overall, the model predictions and the laboratory results fall within a narrow range indicating that developed PFC model of the DSS test has the ability to adequately simulate the cyclic response of the glass beads.   5.5 Observations on lateral stresses from simulations results  Figure  5.13 shows the change in lateral earth pressure coefficient, k, (calculated based on average horizontal stress acting on the rings providing lateral support) with the development of shear strain for the simulated cases for drained and constant volume conditions. A steep increase in k values is noted at low shear strains followed by a more gradual increase in k with further increase in shear strain. The observed behavior is qualitatively similar to that noted from DSS testing performed with lateral stress measurement reported in Chapter 4. The numerical modeling results shown in Figure  5.13 indicate that the relationship between the k value with shear strain is not substantially affected by 'vc for the simulated normal stress range between 100 kPa and 200 kPa. Similar trends were observed from the results of drained and constant volume simulations.   The evolution of lateral earth pressure coefficient, k, with the development of shear strain during constant volume cyclic DSS loading simulations are presented in Figure  5.14.  An overall increase in k values with the increase in number of cycles is also noted from the results of cyclic simulations as shown in   103  Figure  5.14, demonstrating qualitatively similar trends to these observed from the results of cyclic DSS testing performed on Fraser River silt shown in Chapter 4.   5.6 Closure  The development of a PFC3D model of the DSS test, selection of input parameters for the developed model, and performance evaluation of the model were presented in this chapter.  In general, the DEM model captured the DSS testing results as follows:   Drained monotonic tests: The maximum difference between model and DSS testing shear stresses at any given shear strain is 10%. Volumetric strain obtained from DSS testing is about twice as much as that obtained from the DEM model. It is noted that rigid lateral boundaries are used in the model compared to the reinforced membrane used in the DSS test to provide lateral confinement.  As the reinforced membrane is not infinitely rigid, small lateral displacements may occur during DSS shearing of the glass bead and, in turn, may cause the noted discrepancy between volumetric strain for the numerical and physical models.    Constant volume monotonic tests: The maximum difference between model and DSS testing shear stresses at any given shear strain is 20%.  Cyclic constant volume tests: cyclic stress ratio values obtained from the DEM model are about 20% higher than these obtained from the DSS test.      104  Table ‎5.1. Summary of PFC3D model input parameters for glass beads. Input parameter Used parameter value Comments Ball-ball friction coefficient (F) 0.176 After Cavarretta et al. 2010 for dry glass beads Ball-wall friction coefficient (top and bottom walls)  10 Very large value assumed to simulate rough top and bottom DSS caps Ball-wall friction coefficient (lateral walls) 0 Value assumed to simulate the nearly frictionless DSS lateral boundary Hertz-Mindlin shear modulus parameter (G) 1.5 GPa Based on comparison with results of an odometer test as part of this study (see Figure  5.3) Hertz-Mindlin Poisson’s ratio ( 0.22 After Härtl & Ooi (2011) Viscous Damping ratio  0.3 Assumed based on results of a sensitivity analysis that showed essentially marginal effect of the selected damping coefficient on model results (see Figure  5.6) Particles density  2500x104 kg/m3 Glass density multiplied by a factor of 10000 as discussed in the input parameters section     105  Table ‎5.2. Summary of simulations results at the end of the consolidation phase 'vc (kPa) ec* ko ('hc/ 'vc)** 100 0.607 0.65 150 0.604 0.64 200 0.601 0.63 * Calculated for locations at specimen core on average. ** Horizontal stress ('hc) was calculated as an average radial stress acting on the rings providing lateral support computed at the end of the consolidation phase.      106  a   Figure ‎5.1. Side view illustration of the PFC specimen prepared by numerically simulated pluviation: a) particles just prior to pluviaton; b) particles after application of vertical stress.   xyzTop boundary Bottom  boundary Lateral cylindrical boundary a)b)Balls falling under gravity  Specimen Cavity  Mould  b)   107    Figure  5.2. Shear stress strain response for simulated specimens with  values of 5x10-7/cycle and 2.5x10-7/cycle.    05101520253035400 2 4 6 8Shear stress, h(kPa)Shear strain,  (%) = 5x10-7/cycle = 2.5x10-7/cycle  108   Figure  5.3. Comparison between odometer test results and the results of a simulated odometer test performed using the non-linear Hertz-Mindlin model with shear modulus values of 1 GPa, 1.5 GPa, and 2.5 GPa.    -1.4-1.2-1-0.8-0.6-0.4-0.200 50 100 150 200Volumetric strain, v(%)Vertical effective stress, 'v(kPa)Odometer testG = 1 GPaG = 1.5 GPaG = 2.5 GPa  109   Figure  5.4. Comparison between odometer test results and the results of a simulated odometer test using a linear constitutive model with constant contact stiffness parameter, K, values of 100 kN/m, 250 kN/m, 500 kN/m, and 1000 kN/m.      110    Figure  5.5. Effect of density scaling as observed from the results of two odometer test simulations.    -1.4-1.2-1-0.8-0.6-0.4-0.200 50 100 150 200Volumetric strain, v(%)Vertical effective stress, 'v(kPa)Particles density multiplied by afactor of 10000Particles density multiplied by afactor of 100Real density of glass beads  111    Figure  5.6. Effect of damping coefficient on the results of the simulated odometer test.     -1.4-1.2-1-0.8-0.6-0.4-0.200 50 100 150 200Volumetric strain, v(%)Vertical effective stress, 'v(kPa)Damping coefficient = 0Damping coefficient = 0.3  112  a)  b)  Figure  5.7. Results of three drained DSS simulations performed at 'vc values of 100 kPa, 150 kPa, and 200 kPa along with the corresponding DSS test results performed on glass beads a) shear stress strain response; b) volumetric strain vs. shear strain.   0204060800 2 4 6 8Shear stress, h(kPa)Shear strain,  (%)PFC simulationsLaboratory tests'vc= 100 kPa'vc= 150 kPa'vc= 200 kPa-1.0-0.8-0.6-0.4-0.20.00 2 4 6 8Volumetric strain, v(%)Shear strain,  (%)Contraction  113  a)  b)   Figure  5.8. Results of three monotonic constant volume DSS simulations performed at 'vc values of 100 kPa, 150 kPa, and 200 kPa along with the corresponding DSS test results performed on glass beads a) shear stress strain response; b) stress path plot.   0204060800 2 4 6 8Shear stress, h(kPa)Shear strain,  (%)PFC simulationsLaboratory tests'vc= 100 kPa'vc= 150 kPa'vc= 200 kPa0204060800 50 100 150 200 250Shear stress, h(kPa)Vertical effective stress, 'vc(kPa)  114  a)  b)  Figure  5.9. Results of a cyclic constant volume simulation with 'vc of 100 kPa and CSR of 0.08: a) stress path plot; b) shear stress strain response.  -16-12-8-404812160 20 40 60 80 100 120Shear stress, h(kPa)Vertical effective stress, 'vc(kPa)'vc = 100 kPaCSR = 0.08-16-12-8-40481216-10 -5 0 5 10Shear stress, h(kPa)Shear strain,  (%)  115  a)  b)  Figure  5.10. Results of a cyclic constant volume simulation with 'vc 0f 100 kPa and CSR of 0.12: a) stress path plot; b) shear stress strain response.  -16-12-8-404812160 20 40 60 80 100 120Shear stress, h(kPa)Vertical effective stress, 'vc(kPa)'vc = 100 kPaCSR = 0.12-16-12-8-40481216- 0 -5 0 5 10Shear stress, h(kPa)Shear strain,  (%)  116  a)  b)  Figure  5.11. Results of a cyclic constant volume simulation with 'vc 0f 200 kPa and CSR of 0.1: a) stress path plot; b) shear stress strain response.     117   Figure  5.12. CSR vs. number of cycles to reach  of 3.75% from results of simulations and laboratory DSS testing.   0.000.040.080.120.160.200.241 10 100Cyclic stress ratio (CSR) Number of cycles to reach  of 3.75%Laboratory testsSeries3Series4Series1PFC with 'vc = 100 kPaPFC with 'vc = 150 kPaPFC with 'vc = 200 kPa  118  a)  b)  Figure  5.13. Variation in lateral earth pressure coefficient with the development of shear strain obtained from the results of PFC simulations: a) drained conditions; b) constant volume conditions.  0.40.60.81.00 2 4 6 8 10Lateral earth pressure coefficient, kShear strain,  (%)'vc= 100 kPa'vc= 150 kPa'vc= 200 kPaDrained PFC simulations0.40.6.81 00 2 4 6 8 10Lateral earth pressure coefficient, kShear strain,  (%)'vc= 100 kPa'vc= 150 kPa'vc= 200 kPaConstant volume PFC simulations  119   Figure  5.14. Change in lateral earth pressure coefficient during cyclic loading for simulated specimens.    0.40.60.81.00 5 10 15 20Lateral earth pressure coefficient, kNumber of cycles CSR = 0.12Cyclic PFC simulations with 'vc= 100 kPaCSR = 0.08CSR = 0.1  120  6 EVALUATION OF STRESS STRAIN UNIFORMITIES IN THE LABORATORY DIRECT SIMPLE SHEAR TEST SPECIMENS   The DSS test was initially developed to overcome significant stress non-uniformities imposed by the direct shear (commonly referred to as “direct shear box”) test. Unlike the shear box test that mobilizes a shear zone at the boundary between the two halves of the box, the direct simple shear test engages the whole specimen in the shearing process.   DSS testing process deviates from the ideal simple shear boundary conditions due to the absence of complementary shear acting on the lateral boundary; this has been suggested to cause stress non-uniformities within the DSS specimen that can lead to progressive failure. Background information and discussion, as presented in Chapter 2, concluded that there is a need to fill the knowledge gap with regards to stress-strain uniformities. There is no consensus on the degree and significance of non-uniformity of stresses and strains in DSS specimens.  It was shown in Chapter 5 that DEM and PFC3D in particular captured the observed behavior from DSS laboratory tests performed on glass beads. PFC3D provides means for tracking stresses at locations inside the specimen and on the boundaries. This makes it a potentially ideal tool for use in investigating the degree of uniformity of stresses across the specimen and along the boundaries at any stage during progress of the numerical test. Further, PFC3D can be used to provide insight on shear strains calculated based on the changes in particle locations at locations inside the specimen.  With this background and in consideration of the extensive use of the DSS device in characterizing the seismic response of soils, a study of stress and strain uniformities in a DSS specimen subjected to monotonic and cyclic loading conditions was undertaken using discrete element modeling (PFC3D 3.1 particle flow code in three dimensions developed by Itasca), and this chapter presents some key findings from this work.  The effect of shear strain level, selected interparticle friction coefficient, vertical effective   121  stress at the end of consolidation, loading end boundary conditions (i.e. constant volume or constant vertical effective stress) conditions , selected friction coefficient on the lateral boundaries, and loading type used for the simulated cases on stress-strain uniformities are also discussed. In the following, DSS tests performed with constant vertical effective stress are referred to as drained tests.   The use of free form sensors to measure lateral stresses at four locations on the vertical DSS specimen boundaries that was undertaken as a part of this study was discussed in Chapter 3. The results of DSS testing performed on Fraser River silt with lateral stress measurements were reported in Chapter 4. Further analysis of lateral stress measurements obtained from the free form sensors is presented in this Chapter to provide additional insight on stress uniformities in DSS specimens.   6.1 Analysis methodology and numerical simulations  PFC3D model details used for analysis of stress-strain uniformities are the same as those previously discussed in Chapter 5. In this chapter, the results of the PFC3D model are presented with particular emphasis on the distribution of stresses and strains within the DSS specimen and on the boundaries.   Average stresses were calculated from contact forces at internal specimen locations and at locations on the boundaries. Internal forces were used to calculate stresses at eleven locations inside the specimen. The Measurement Sphere (MS) routine in PFC3D is used to calculate local stresses. The center and radius of each measurement sphere is specified so that the stresses at the desired locations are computed.  The MS routine computes the stress tensor from forces at contacts averaged over the volume of the MS. Figure ‎6.1 shows a top view of the specimen illustrating the locations and sizes of the measurement spheres for the simulated DSS specimen. Stresses at the boundaries are equal to the sum of forces acting at the wall-ball contacts divided by surface area of the corresponding wall segment. The distribution of radial stresses along the specimen height was calculated at four segments on the perimeter   122  of the lateral boundary as shown in Figure ‎6.2. Segments 1 and 3 are perpendicular to the direction of shearing while segments 2 and 4 run almost parallel to the direction of shearing.  The locations of the centers of balls passing through vertical sections (i.e. parallel to the Z-axis) were recorded during shearing.  The computed data were used to assess the variation of shear strain along the x and z coordinates.  Shear strains were also monitored at the lateral boundaries.   Table ‎6.1 summarises the simulated DSS specimen cases presented in this chapter. The effect of interparticle friction on the uniformity of stresses and strains was investigated for F values of 0.176 which is the interparticle friction coefficient for glass beads as previously discussed in Chapter 5 and also for higher F values of 0.3 and 0.5. It will be shown in Section 6.2.3.2 that increasing F values results in a more dilative response similar to that typically observed for dense sands. The effect of vertical effective stress at the end of consolidation was investigated for 'vc values in the range of 100 kPa – 400 kPa. The effects of loading end conditions (i.e. drained or constant volume) and selected friction coefficient on the lateral rings on stress-strain uniformities are also discussed. A drained cyclic simulation was performed to evaluate the degree of stress-strain uniformity during cyclic shearing.   6.2 Results from numerical simulations  6.2.1 Stress uniformities at the end of consolidation  It was decided to evaluate the initial degree of stress uniformity across the DSS specimen in terms of the variation of vertical effective stresses across the simulated DSS specimen using the stress uniformity parameter at the end of consolidation, , defined as in Equation 6.1.   123  )((min)(max)'''averagezzzzzz    (6.1) Where,  (max)'zzand (min)'zz are the maximum and minimum vertical effective stresses evaluated at locations of measurement spheres on average, respectively.  )(' avergaezz is the average vertical effective stress at the locations of the measurement spheres.   Similarly defined parameters have been used by others for assessing stress uniformities in laboratory element testing specimens  (Hight et al., 1983; Wijewickreme & Vaid, 1991). An  value of zero represents a perfectly uniform vertical effective stress state at the end of consolidation at the locations of the measurement spheres. Stress non-uniformity increases with the increase of the value of .    Evaluation of stress uniformity was carried out over the central specimen locations represented by the locations of central measurement spheres MS 2 – MS 4 and over the locations closer to the lateral boundary represented by the remaining measurement spheres (MS 1, and MS 5 through MS11) shown in Figure ‎6.1. In the following evaluation, the stress uniformity parameter calculated for central locations is denoted by c and the stress uniformity parameter calculated for locations closer to the lateral boundary is denoted by b.   Table  6.2 presents the calculated c and b parameters at the end of consolidation for specimens consolidated to 'vc values of 100 kPa, 150 kPa, 200 kPa, and 400 kPa. The results shown in the Table  6.2 indicate the following: i) c values are significantly lower than b values which implies that stresses at the central zone of the specimen are initially (i.e. at the end of consolidation) more uniform than stresses at locations near the lateral boundaries; ii) both c and b are essentially not affected by   124  the vertical effective stress level and, in turn, the degree of stress uniformity at the end of consolidation is very similar for the investigated range of 'vc values of 100 kPa – 400 kPa.   The distribution of vertical normal stress (zzc) at the end of consolidation for the specimen consolidated to boundary vertical effective stress of 100 kPa was computed at the measurement spheres (MS 1 through MS 5) as shown in Figure ‎6.3. In other words, each point in the plot represents the average stress computed at the center of the corresponding measurement sphere.  The results presented in Figure ‎6.3 show a reasonably uniform stress distribution at the end of consolidation particularly for the central measurement spheres MS 2 through MS 4. Measurement sphere, MS 1 had the lowest recorded zzc value of 85 kPa. It is noted that locations represented by MS 1 are close to the lateral boundary. This is in line with the results shown in Table  6.2 as previously discussed.  6.2.2 Shear strain uniformities  The interpretation of particle movements in the x-direction during the shearing process provides an opportunity to assess strains along the specimen height.  The x-displacements of 9 randomly selected balls initially (pre-shearing state) aligned along the z-axis near x= -2.5 cm, during the simulation with F = 0.176 and 'vc of 100 kPa are presented in Figure ‎6.4. Each point on the figure represents the location of the center of a particle when the boundary shear strain is at 1%, 5%, 10%, and 20%, respectively. Although particle movements in the horizontal direction near the top and bottom boundaries show a non-linear trend, particle movements in the horizontal direction for particles within the middle two thirds of the specimen show only a slight variation from the ideal linear trend shown on the figure. Similar uniform particle movement trends in the horizontal direction within the middle two thirds of the specimen were observed for other x values. This noted approximate linear variation of the particle movements in the horizontal direction with respect to the vertical coordinate of the specimen suggests that, at a given x-coordinate value and within the middle two thirds of the specimen, shear strains are fairly uniform.    125   Figure ‎6.5 was produced by calculating the shear strains computed based on particles displacements at different x-locations along y = 0 for particles within the middle two thirds of the specimen (Note: y-axis is the horizontal axis perpendicular to the direction of the shear movement passing through the center of the specimen) when the boundary shear strain was at 20% for specimens with B-B friction of 0.176, 0.3, and 0.5. Lower shear strain is observed at locations near the center of the specimen relative to shear strain computed at locations that are closer to the lateral boundary. The overall distribution of shear strain along the x-coordinate is relatively more uniform (i.e. less deviation from boundary shear strain) for the specimen with B-B F=0.5. This can be considered reasonable since there is less opportunity for slippage at particle contacts for specimens with higher B-B friction during shearing, hence resulting in a more uniform shear strain distribution.   Figure  6.6 shows boundary shear strain (equal to the relative horizontal displacement between the top and bottom caps divided by the specimen height) plotted against shear strains calculated at x-coordinates of -2.5 cm, -1.25 cm, 0 cm, 1.25 cm, and 2.5 cm, respectively, for the simulation with F = 0.176 and 'vc of 100 kPa. Shear strain in DEM specimens at a given x-coordinate and y-coordinate values were considered equal to the reciprocal of the slope of a linear trend line passing through the particle displacement field (see Figure ‎6.4) at the corresponding x-coordinate and y-coordinate values. It is noted from the results presented in Figure  6.6 that internal strains (i.e. strains calculated at different x-locations within the specimen) are very similar at small boundary shear strains indicating a fairly uniform shear strain distribution at small shear strains. Strain non-uniformities increasingly develop with the progression of shearing as internal strains are more scattered at larger shear strains. As expected, internal shear strains at sections closer to the lateral boundary (i.e. at x-coordinates of -2.5 cm and 2.5 cm) are very similar to corresponding boundary shear strains.   Images showing a side view of the simulated specimen captured at the end of consolidation and at the instant of boundary shear strain of 20% are shown in Figure  6.7a and Figure  6.7b, respectively. Particles   126  in three zones within the specimen are colored in blue as shown in Figure  6.7a. At the end of consolidation, the vertical edges of the blues zones are approximately parallel to the lateral boundaries of the specimen. With the progression of shearing, some strain non-uniformities develop within the specimen. It can be seen that the edge of middle blue zone is no longer parallel to the specimen lateral boundary which is indicative of non-uniform strains at locations inside the simulated specimen (Figure  6.7b).   6.2.3 Stress uniformities during the shearing phase  It would be of direct relevance to compare the laboratory-testing-wise interpreted stresses with those numerically assessed for the internal measurement spheres. This can be accomplished by comparing the computed shear stress ratio [i.e., h/ ′v] for the individual measurement spheres with the computed average shear stress ratio, derived from the vertical and shear forces experienced by the DE balls at the boundaries of the specimen model as previously discussed in Chapter 5. The boundary shear strain plotted against shear stress ratio calculated over the volume of the five measurement spheres MS 1 – MS 5, are presented in Figure  6.8 (see Figure ‎6.1 for locations of the measurement spheres) along with average shear stress ratio. Average shear stress ratio calculated from boundary stresses falls within the range of shear stress ratios calculated for the measurement spheres. It is noted that at  of 20%, calculated stress ratio values are between 0.27 and 0.45. This indicates a relatively non-uniform shear stress ratio distribution across the simulated specimen. In light of this, an in depth analysis of stress uniformities in simulated DSS specimens is presented in this section. The effects of shear strain level, vertical effective stress level, selected interparticle friction coefficient, loading end conditions, and friction coefficient on the lateral boundary on the uniformity of stresses within the simulated DSS specimen are discussed. Further, the effect of the application of cyclic shearing on stress uniformity is investigated.   The degree of stress uniformity during shearing is evaluated using internal stresses averaged over the volume of measurement spheres and using stresses acting on the lateral specimen boundaries. The   127  former is performed using the stress uniformity parameter  as previously defined in Equation 6.1. The parameter  is evaluated for central measurement spheres (MS 2 – MS 4) and for near-boundary measurement spheres (MS 1 and MS 5 through- MS 11). As indicated earlier, the subscripts c and b are added to the parameter  to refer to central and near-boundary measurement spheres, respectively. Further, distribution of radial stresses acting on the lateral boundaries is used to provide insight on the degree of stress uniformity in DSS specimens.  6.2.3.1 Effect of shear strain level   The stress uniformity parameters c and b, calculated for a drained simulation with interparticle friction coefficient, F, of 0.176 and 'vc of 100 kPa, are plotted against boundary shear strain in Figure  6.9. As previously discussed, the end of consolidation (i.e. at  of 0%) value of b is higher than that for c. With the development of shear strain, c is noted to rapidly increase from near zero value (i.e., perfectly uniform stresses) to a steady value of about 0.25 at  of 6%. c remains essentially constant with further increase of shear strain. The plot of b does not seem to follow a particular trend. b values are higher than those for c for most shear strain values indicating a more uniform stress distribution in the central parts of the specimen relative to locations near the boundaries. Overall, based on the c versus shear strain plot, increased degree of stress non-uniformity is noted with the increase in shear strain. This implies that DSS results obtained at relatively small shear strains are more reliable than those obtained at large shear strains as stresses are more uniform at small shear strain levels.      128  6.2.3.2 Effect of interparticle friction  As per the previous discussion in Chapter 5, an interparticle friction coefficient, F, of 0.176 was used to simulate the DSS behavior for the glass beads. As interparticle friction for real soils can be higher than that for the glass beads, two additional simulations with F values of 0.3 and 0.5 were performed to assess the effect of interparticle friction coefficient on the degree of stress uniformity.   The shear stress strain response and volumetric strain versus shear strain response for the simulations performed with F values of 0.176, 0.3, and 0.5 are presented in Figure  6.10. Stiffer and more dilative responses are noted with the increase in interparticle friction coefficient. The shear strain versus volumetric strain plots for the simulations performed with F values of 0.3 and 0.5 show a response similar to those typically observed for very dense sands.   The change in stress uniformity parameter c, calculated from the results of the three simulations with F values of 0.176, 0.3, and 0.5, with the development of shear strain is shown in Figure  6.11. At most shear strain levels, the value of c generally seems to decrease with increasing interparticle friction coefficient.  In essence, a relatively more uniform stress distribution during shearing is noted at central specimen locations with the increase in interparticle friction.   6.2.3.3 Effect of vertical effective stress  The plots of shear strain versus c for drained simulations performed with 'vc values of 100 kPa, 150 kPa, 200 kPa, and 400 kPa are shown in Figure  6.12. Very similar c values are observed from the results of the four simulations for shear strains lower than about 8%. However, for shear strains in the   129  range of 8% - 20%, lower c values are noted for simulations performed with higher vertical effective stresses indicating increased stress uniformities for simulations performed with higher stress levels.   6.2.3.4 Effect of loading end conditions  Figure  6.13 shows the development of c values with the increase in shear strain for two simulations performed under constant volume and constant vertical effective stress (i.e. drained) shearing conditions. The results in Figure  6.13 show essentially the same c values for the two simulations. As such, although based on limited analytical work, the loading end condition does not seem to affect the degree of stress uniformity evaluated at central locations for the simulated specimens.   6.2.3.5 Effect of lateral boundary friction   Possible non-unifromities in DSS specimens have been typically referred to the lack of friction on the lateral boundaries (Roscoe, 1953). The degree of stress uniformity in simulations performed with lateral boundary friction coefficients of 0.176 (i.e., equal to ball-ball friction) and 1 during shearing is compared to that for the simulation performed with frictionless lateral boundaries. Figure  6.14 shows the c parameter for the three simulations with the development of shear strain. Similar trends were observed for the three simulations indicating essentially no significant effect of the selected boundary lateral friction on the degree of stress uniformity evaluated for central specimen locations.   6.2.3.6 Stress uniformities during the application of cyclic loading  Figure  6.15 presents the shear stress-strain response and volumetric strains vs. shear strains plots for a cyclic drained simulation performed with shear strain single amplitude of 2%. The observed overall   130  contractive behavior with the progression of cyclic loading is in accord with that typically observed for loose sands (Sriskandakumar, 2004). The stress uniformity parameter c plotted against the number of cycles is shown in Figure  6.16. The values of c do fluctuate within individual cycles; an overall trend of gently increasing c with the increase in number of cycles is noted.  6.2.3.7 Distribution of radial stresses acting on the lateral boundaries during shearing  Figure  6.17 shows the distribution of radial stresses along specimen height for the four segments (see Figure  6.2 for the locations of the segments) calculated at the end of the consolidation phase. Average radial stresses calculated for each of the segments are also indicated on the plots. It can be noted that radial stress distribution along the height of the specimen is fairly close to average radial stresses (in the order of ±15% of average radial stress).  Similar to Figure  6.17, Figure  6.18 shows the distribution of radial stresses along the specimen height for the four segments calculated at boundary shear strain of 20% obtained from the results of a monotonic simulation. Average radial stresses calculated for each of the segments are about 100 kPa as indicated on the plots in Figure  6.18. It can be noted that radial stress distribution along the height of the specimen is more uniform (i.e., closest to average radial stress values) for segments 2 and 4 that run parallel to the direction of shearing compared to segments 1 and 3 that are perpendicular to the direction of shearing. Segments 1 and 3 show increased radial stress non-uniformity along the specimen height with a maximum deviation from average radial stress of about ± 80%. This increased degree of stress non-uniformity was not observed from the analysis of internal stresses performed using average stresses over the locations of the measurement spheres.  This high level of non-uniformity, therefore, seems to be limited to volumetric zones adjacent to the boundaries of the specimen.     131  One of the outputs arising from the discrete element modeling are the forces at the contacts (called “force chains”); they can be viewed graphically at any stage of the simulation. Contact forces for the simulation performed with F of 0.176 and 'vc of 100 kPa are shown in Figure  6.20 and Figure  6.21 as black lines at shear strains of 0% and 20%, respectively.  The thickness of the black lines is proportional to the magnitude of contact forces. At 0% shear strain, a significant number of strong force chains can be seen to be aligned generally in the vertical direction (Figure  6.20a) which is the direction of the major principal stress after consolidation and before shear.  However, at 20% shear strain as shown in Figure  6.21a, most of the strong force chains are oriented in a direction that is inclined to the vertical direction. This is expected because with a shear stress applied on the specimen, the major principal stress will rotate and be inclined. The strong force chains will therefore rotate and be inclined in a direction consistent with the direction of major principal stress.  6.2.3.8 Distribution of vertical stresses on top and bottom specimen boundaries   Figure ‎6.19 shows calculated normalized vertical effective stress calculated by dividing vertical effective stress obtained from contact forces acting on the top and bottom boundaries by average vertical effective stress acting on the top and bottom boundaries. Normalized vertical effective stresses obtained from numerical and experimental studies performed by Budhu & Britto (1987) are also plotted in Figure ‎6.19 for qualitative comparison with the results of the current DEM analysis. Overall, the results indicate that normalized vertical effective stress distribution is more uniform at the specimen centre compared to that at locations near the lateral boundaries. It is noted that Budhu & Britto (1987) used a polynomial function to fit three experimental measurements. The portions of plynomial fuction indicating dramatic increase/decrease of stresses near the laterial boundaies were obtained by extrapolation. This dramatic  increase of normalized vertical effective stress near the left and right sides of the top and bottom boundaries, respectively, has not been observed from the results of the DEM model. It seems that results of the elastic and modified Cam-Clay analyses overestimate the degree of stress non-uniformities compared to the results of DEM analysis at locations near the boundaries.    132   Visual observations on density of force chains shown in Figure  6.20 and Figure  6.21 are used in this context to provide insight on observed stress uniformities. At 0 % shear strain, it seems that force chains are evenly distributed within the specimen which implies the absence of significant stress non-uniformities at this stage of the simulation. However, at 20% shear strain, contacts at locations near the upper right side and lower left side corners seem to have significantly lower forces compared to the rest of the specimen. These qualitative observations are consistent with the distribution of radial stresses on the lateral boundaries for segments 1 and 3 shown in Figure  6.18. It is also noted that these contacts with significantly lower forces than the rest of the specimen are limited to a relatively small zone of the specimen cross-section near the lateral boundary. A top view of the specimen shown in Figure  6.21b indicates that the extent of the significantly lower force contacts is about two particle diameters adjacent to lateral specimen boundary on the right side.    6.3 Observations on stress uniformities from measured lateral stresses  As previously discussed in Chapter 3 and Chapter 4, lateral stresses at four locations on the boundaries were measured using free-form sensors mounted on the reinforced DSS membrane for Fraser River silt specimens. Measured horizontal stresses for individual sensors, 'h(sensor), are used in this context to provide insight on the distribution of stresses around the circumference of the DSS membrane for a Fraser River silt specimen undergoing monotonic constant volume shearing. Figure  6.22 shows normalized horizontal stresses obtained from the sensors (i.e. 'h(sensor) /'h(ave.)), plotted against shear strain. For the case of perfectly uniform stresses, the ratio 'h(sensor) /'h(ave.) is equal to 1. The following is observed from the results shown in Figure  6.22: i) at the end of consolidation (i.e. at  = 0 %), measured stresses are within about ± 5 % from average stress which indicates the adequacy of the used slurry deposition method in producing fairly uniform specimens; ii) some increase in stress non-uniformities with   133  the increase in shear strain is noted (measured stresses are within about ± 10 % from average at  of 25 %).   6.4 Discussion on stress-strain uniformities   Analysis of internal stress acting in the vertical direction averaged over the locations of the measurement spheres indicated that stresses are uniform at central specimen locations with slight increase in non-uniformities at internal locations closer to the lateral boundaries. Further, analysis of radial stresses acting on four sections around the circumference of the lateral rings at the end of the consolidation phase indicated that calculated stresses are within ± 15% of average lateral stress (see Figure  6.17).  The majority of previous numerical studies on DSS testing do not account for possible non-uniformities arising during the consolidation processes, and a perfectly uniform stress distribution is typically assumed at the end of the consolidation phase (Lucks et al., 1972; Prevost & Hoeg, 1976; Budhu & Britto, 1987; Dounias & Potts, 1993). Saada et al. (1983), based on their research, concluded that potential non-uniformities arising during the consolidation phase in the DSS specimen can be as significant as those arising during DSS shearing. Their study was based on finite element analysis performed assuming an elastic material representing the DSS specimen. The results of the analysis suggested the possibility of significant non-uniformities near the specimen boundaries at the end of consolidation phase. DEM analysis presented in the previous sections suggests that the results of Saada et al. (1983) significantly over-estimate the degree of non-uniformities at the end of consolidation, likely due to using an elastic representation of soil. Recently, 3D finite element analysis of the consolidation phase for a DSS specimen performed using the Modified Cam-Clay model has shown that stresses across the specimen vary within about ± 1% of average stress (Doherty & Fahey 2011). It is believed that the results of the current DEM model are more realistic compared to the results of the continuum FE study performed by Doherty & Fahey (2011) as the latter does not account for irregularities at the soil specimen surface that likely are a major cause of stress non-uniformities.    134   Airey & Wood (1984) presented lab results showing the distribution of vertical stresses at the end of consolidation for loose and dense sand specimens and for a clay specimen. A more uniform stress distribution was obtained for the clay specimen compared to the sand specimens. Overall, their plots showed improved stress uniformity at central specimen locations which is in qualitative agreement to the results of the DEM model.   Computed shear strains at the center of the specimen were less than boundary shear strains for all the simulated cases herein. Slippage between particles results in reduced shear strains at the center of the specimen. It was noted that the degree of shear strain non-uniformity is a function of B-B friction between particles of the DEM model. Shear strain distributions, as shown in Figure  6.5, suggest that the non-uniformities are reduced for analysis with higher B-B friction of 0.5 compared to the cases where B-B friction is assumed to be 0.3 or 0.176. The latter resulted in the least uniform shear strain distribution.  This implies that shear strain non-uniformities could accentuate when there is more opportunity for particle slippage.   Reduction in calculated shear strain obtained from particle displacement patterns at central specimen locations is due to slippage at contacts and particles rolling. Shear displacement is transferred to the particles from the sides and top and bottom boundaries. As slippage and rolling at contacts occur, particle horizontal displacement is reduced at locations further away from the boundaries. At relatively small shear strains, there is less opportunity for slippage and rolling at contacts to occur compared to the case of larger shear strain. As such, calculated shear strain distribution is more uniform compared to that at large shear strain, which was noted from the results of DEM simulations.   Boundary shear strain typically obtained from DEM simulations may be corrected to obtain a more representative average shear strain using the results shown in Figure  6.6. For a given boundary shear   135  strain value, average shear strain may be calculated by averaging shear strains obtained at locations within the DSS specimen.   Budhu (1984), based on data from monitoring the change in positions of embedded lead shots during shearing, noted the presence of non-uniform strain distributions along the height of the specimen. The non-uniformities, in particular, were large for strains higher than 5%. In contrast, significant strain non-uniformities along the height (z-coordinate) were not observed from the current PFC modeling for the range of strains investigated. Note that rigid side boundaries were used in the PFC model which is different from the wire-reinforced membrane used in the lab test. It is possible that reduced stiffness of the membrane may contribute to non-uniform boundary strain conditions (Budhu, 1984).  Simple elastic analysis of a DSS specimen during shearing performed by Roscoe (1953) predicted uniform stress distribution at central specimen locations with non-uniformities developing at locations near the boundaries. This was later confirmed by experimental studies (e.g. Wood & Budhu, 1980; Budhu, 1984) and numerical studies (e.g. Lucks et al., 1972; J. Prevost & Hoeg, 1976; Budhu & Britto, 1987; Dounias & Potts, 1993; Wang et al., 2004; Doherty & Fahey, 2011).   The present study showed similar trends of less stress non-uniformities at central specimen locations compared to near boundary locations. It was shown that stress ratios calculated at central measurement spheres (MS 2-MS 4) are similar to each other while higher stress ratios were calculated for the near boundary MS 5. Comparison of stress uniformity coefficient c (calculated based on average stresses at central specimen locations as previously discussed) to stress uniformity coefficient b (calculated based on average stresses at near-boundary locations as previously discussed) indicated that, in an overall sense, c values are lower than b values. This suggests that stresses at central specimen locations are more uniform than those at locations closer to the lateral boundaries. Analysis of the distribution of radial stresses acting on the lateral boundaries further confirmed increased stress non-uniformities at locations adjacent to the boundaries. As such, it can be argued that central specimen locations are the most   136  representative of ideal simple shear conditions which is in agreement with the conclusions of previous studies as discussed above.   The effects of shear strain level, interparticle friction coefficient, vertical effective stress level, drainage conditions, friction coefficient on the lateral boundary, and number of loading cycles on the degree of stress uniformities at central specimen locations were previously discussed. Stress uniformities were shown to decrease with the increase of shear strain level. Slight improvements on stress uniformities were noted with the increase of vertical effective stress. The specimen with the highest interparticle friction coefficient of 0.5 showed a more uniform stress distribution compared to the specimens sheared with lower interparticle friction coefficients. It is noted that shear strains for the specimen with the highest interparticle friction are the most uniform compared to those for specimens with lower interparticle friction coefficients. It is possible that the observed more uniform shear strain distribution could have contributed to increased stress uniformities for the specimen performed with the highest interparticle friction coefficient of 0.5. Practically negligible effects of drainage conditions and friction coefficient on the lateral boundary on stress uniformities at central specimen locations were noted. This implies that central specimen locations are located “far away enough” from the boundaries that they are not influenced by boundary effects. Finally, the noted increasing overall trend in c with the increase in number of shearing cycles suggests that there is a potential for developing stress non-uniformities during the application of cyclic shearing.   Experimental results obtained from free form sensors at four locations around the circumference of the DSS reinforced membrane indicated similar measured stresses at large strains (within ±10 % of average horizontal stress). It is important to note that measurements obtained from free form sensors are average stresses over the area of the sensors and do not necessarily capture non-uniformities along the specimen height. The results of DEM analysis showed that stresses can be highly non-uniform at large shear strains along the height of the specimen at locations adjacent to the lateral boundaries (see Figure  6.18).    137  6.5 Conclusions   The laboratory direct simple shear (DSS) testing process was simulated using a 3D discrete element model that utilizes PFC3D to investigate non-uniformities of stresses and strains in a cylindrical specimen. Analytical specimens for the simulation were formed by “numerically pluviating” spherical particles under gravity. This was followed by numerically simulated consolidation and shearing phases. The results of the analysis indicated the following:   - Vertical stresses at the end of the consolidation phase are fairly uniform at central specimen locations with some non-uniformities noted at locations closer to the lateral boundaries. Radial stresses calculated at the end of consolidation at the lateral boundaries were within ± 15% of average radial stress.  - Lower shear strains were noted at locations inside the simulated DSS specimen compared to those at the specimen boundary. Strain non-uniformities were noted to accentuate when there is opportunity for particle slippage. - Essentially more uniform shear strains within the specimen were noted at lower boundary shear strain levels compared to shear strains within the specimen calculated at higher boundary shear strains.  - Fairly uniform normal stresses and stress ratios were observed at central specimen locations relative to those at locations closer to the lateral boundaries suggesting that central specimen locations are the most representative of ideal simple shear conditions. - It was shown that laboratory-testing-wise shear stress ratios calculated from boundary stresses fall within the range of stress ratios calculated for the measurement spheres (see Figure ‎6.8). -  Radial stresses acting on the lateral boundaries at large shear strains were highly non-uniform (within ± 80% of average radial stress). However, it was shown that these significant non-uniformities are limited to a narrow zone of about two particles diameter adjacent to the lateral boundaries.    138  - It was noted that the increase of shear strain level can result in an increase of stress non-uniformities. Higher vertical effective stress levels and interparticle friction coefficient values resulted in minor improvements in regards to stress uniformities (i.e. lower stress non-uniformities). Drainage conditions and friction coefficient on the lateral boundaries had practically negligible effects on the degree of stress uniformities in the central region of the simulated specimens.  - The results of a drained cyclic simulation indicated that non-uniformities can develop within the simulated specimen with the increase in the number of loading cycles.   From the above results, it can also be seen that the DEM, and PFC3D in particular, is a powerful tool that can be used in evaluating soil element tests which is important for the use and interpretation of the results.     139  Table ‎6.1. DEM model input parameters and boundary and loading conditions for the simulated cases.   Simulations series Interparticle friction, F  'vc (kPa) Loading end (i.e. drained or constant volume)  Lateral rings friction coefficient  Loading type a) Effect of interparticle friction, F 0.176,0.3, and 0.5 100 Drained 0 Monotonic  b) Effect of 'vc 0.176 100, 150, 200, and 400 Drained 0 Monotonic c) Effect of drainage condition  0.176 100 Drained and constant Volume 0 Monotonic d) Effect of  lateral rings friction coefficient 0.176 100 Constant Volume 0, 0.176, and 1 Monotonic e) Effect of loading type  0.176 100 Drained 0 Monotonic and cyclic   140  Table ‎6.2. Stress uniformity parameters evaluated at the end of the consolidation phase. 'vc (kPa) c b 100 0.03 0.30 150 0.03 0.30 200 0.03 0.28 400 0.04 0.25     141    Figure ‎6.1. Top view schematic of the DSS specimen showing the locations of the measurement spheres.    -3.5-2.5-1.5-0.50.51.52.53.5-3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5MS 1 MS 2MS 3MS 4 MS 5MS 6 MS 7MS 9MS 8MS 10MS 11X-coordinate (cm)Y -coordinate (cm)  142    Figure ‎6.2. Schematic showing top view of the DSS specimen illustrating selected locations of segments used for radial stress calculation.    Segment 1 (=330°- 30°)Segment 2 (=60°- 120°)Segment 3 (=150°- 210°)Segment 4 (=240°- 300°)  143   Figure ‎6.3. The distribution of vertical effective stress at the end of consolidation, zzc, across the specimen for a specimen consolidated to boundary vertical effective stress of 100 kPa.   020406080100120140160-3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5'zzc(kPa)X-coordinate (cm)MS 1MS 2 MS 3 MS 4MS 5  144    Figure ‎6.4. Displacements in the x-axis direction of particles with centers located near x = -2.5 cm and y = 0 cm during the shearing phase.      145    Figure ‎6.5. Calculated shear strains computed from particle locations (near y = 0 cm) at boundary shear strain of 20 %.   0510152025-3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5Shear strain  (%)  X-coordinate (cm) F = 0.176F = 0.3F = 0.5  146    Figure ‎6.6. Internal shear strains calculated from particles movement versus boundary shear strain.      147  a)   b)   Figure ‎6.7. Side view of a simulated specimen: a) at the end of consolidation; b) at boundary shear strain of 20%.      148   Figure ‎6.8. Stress ratio calculated at locations of MS 1- MS 5 on average and at the boundary versus boundary shear strain.       149    Figure ‎6.9. Stress uniformity coefficients c and b versus boundary shear strains for the simulation performed with F = 0.176 and 'vc of 100 kPa.   00.20.40.60.810 5 10 15 20Shear strain,  (%)F = 0.176 (Zone 1) F = 0.176 (Zone 2)cb  150  a)  b)  Figure ‎6.10. Results of drained simulations performed with F values of 0.176, 0.3, and 0.5: a) shear stress-strain response; b) volumetric strain versus shear strain.      151    Figure ‎6.11. Change in stress uniformity parameter, c, with the development of shear strain for simulations performed with F values of 0.176, 0.3, and 0.5.      152   Figure ‎6.12. Change in stress uniformity coefficient, c, with the development of shear strain for drained simulations performed on specimens consolidated to 'vc of 100 kPa, 150 kPa, 200 kPa, and 400 kPa.    00.20.40.60 5 10 15 20cShear strain,  (%)Sig'Vc = 100 kPa (Zone 1) Sig'Vc = 150 kPa (Zone 1)Sig'Vc = 200 kPa (Zone 1) Sig'Vc = 400 kPa (Zone 1)'vc= 100 kPa'vc= 400 kPa'vc= 200 kPa'vc= 150 kPa  153   Figure ‎6.13. Change in stress uniformity coefficientc, with the development of shear strain for drained and constant volume monotonic simulations.      154    Figure ‎6.14. Stress uniformity parameter, c, versus shear strain for simulations performed with lateral boundary friction coefficients of 0, 0.176, and 1.    00.20.40.60 5 10 15 20c Shear strain,  (%) Frictionless lateral boundariesLateral boundary friction of 0.176Lateral boundary friction of 1  155  a)  b)  Figure ‎6.15. Drained cyclic response a simulated DSS specimen: a) shear stress versus shear strain; b) volumetric strain versus shear strain.    -40-2002040-5 -4 -3 -2 -1 0 1 2 3 4 5Shear stress, h(kPa)Shear strain,  (%)00.511.52-5 -4 -3 -2 -1 0 1 2 3 4 5Volumetric strain, v(%)Shear strain,  (%)Contractive  156    Figure ‎6.16. Stress uniformity coefficientc, versus number of loading cycles.      157    Figure ‎6.17. Distributions of radial stresses along the specimen height for segments 1-4 calculated at the end of the consolidation phase.    0.00.51.01.52.00 25 50 75 100Specimen height (cm)Radial stress (kPa)Segment 1Average0.00.51.01.52.00 25 50 75 100Specimen height (cm)Radial stress (kPa)Segment 2Average0.00.51.01.52.00 25 50 75 100Specimen height (cm)Radial stress (kPa)Segment 3Average0.00.51.01.52.00 25 50 75 100Specimen height (cm)Radial stress (kPa)Segment 4Average  158    Figure ‎6.18. Distributions of radial stresses along the specimen height for segments 1-4 calculated at the instant of boundary shear strain of 20 %.   0.00.51.01.52.00 50 100 150 200Specimen height (cm)Radial stress (kPa)Segment 1Average0.00.51.01.52.00 50 100 150 200Specimen height (cm)Radial stress (kPa)Segment 2Average0.00.51.01.52.00 50 100 150 200Specimen height (cm)Radial stress (kPa)Segment 3Average0.00.51.01.52.00 50 100 150 200Specimen height (cm)Radial stress (kPa)Segment 4Average  159   Figure ‎6.19. Distributions of normalized vertical stress ('v/'v(average)) acting on the top and bottom boundaries during shearing.      160    a)  b)  Figure ‎6.20. Images showing simulated particles and force chains captured at the end of the consolidation phase: a) central cross-section; b) top specimen view.     161  a)  b)  Figure ‎6.21. Images showing simulated particles and force chains captured at the instant of boundary shear strain of 20 %: a) central cross-section; b) top specimen view.     162    Figure ‎6.22. Experimental results showing normalized horizontal stresses acting at the locations of sensors 1-4 with the development of shear strain.     163  7 EVALUATION OF THE STATE OF STRESS IN THE DIRECT SIMPLE SHEAR (DSS) TEST   The discussion presented in Section 2.3 showed that interpretation of DSS testing results is a challenging task as the stress state on the lateral boundaries is usually not known and due to the absence of complementary stresses on the lateral boundaries. Measurements of lateral stresses using free-form sensors at four locations on the lateral specimen boundaries throughout shearing were discussed earlier in Chapter 4. As shear stresses acting on the lateral boundaries are close to zero, it seems that measured lateral stresses can not be directly used to calculate the complete DSS specimen’s state of stress (i.e., it is not possible to draw the Mohr circle). As such, further analysis is performed in this Chapter with emphasis on the state of stress at central DSS specimen locations with uniform stresses (as discussed in Chapter 6) using the results of DEM modeling. In particular, the angles , and β, as defined in Equations 2.2 and 2.3 above are compared to the internal friction angle mob defined in Equation 2.1. Stress invariants and rotation of principal stresses during shearing are discussed. The DEM model results are used to provide insight on using boundary stresses to calculate the DSS specimen state of stress.   7.1 DEM analysis  The results of the DEM model of the DSS test previously discussed in Chapter 5 and Chapter 6 are used here to provide insight on the state of stress in DSS specimens. Drained monotonic simulations performed with a 'vc value of 100 kPa were considered for this analysis. As discussed in Chapter 5, it was shown that the use of interparticle friction coefficient of 0.176 is reasonable for simulating the behavior of glass beads. The state of stress was evaluated for simulations performed with two other interparticle friction coefficients of 0.3 and 0.5. A summary of model input parameters was previously shown in Table  5.1.   It was assumed that stresses are uniform at the specimen core and therefore stress conditions reflect those of true simple shear conditions. Hence, stresses evaluated over central specimen locations (i.e.,   164  MS 2- MS 4 as discussed in Chapter 6) on average were considered for the current investigation on DSS state of stress to calculate the values of angles , , and mob. Work undertaken in Chapter 6 as part of this research has shown that stresses at central specimen locations are fairly uniform compared to stresses at locations adjacent to the lateral boundaries. Accordingly, the computation of stresses for central specimen locations is considered reasonable.  Stress invariants expressed in terms of mean effective stress, 'm, deviator stress, q, and Lode angle, L, as defined in Equations 7.1, 7.2, and 7.3, respectively, were calculated from stresses obtained for central specimen locations. It is noted that stress invariants are important for use for constitutive model development and for comparison with stress conditions of other soil laboratory tests such as the triaxial test. The angle between the major principal stress and the vertical direction, , was computed throughout the shearing phase.                             (7.1)      √ [                                ]        (7.2)           √ [              ]         (7.3) Where Lode angle, L, ranges between -30º and 30º.       165  7.2 Results of DEM Analysis  7.2.1 DSS mobilized friction angle  The observed shear stress ratio, /'zz, versus boundary shear strain, , behavior during the simulations performed with F values of 0.176, 0.3, and 0.5 are shown in Figure  7.1a. It is noted that the plots shown in Figure  7.1 were developed based on stresses averaged over MS 2-MS 4 (see Figure ‎6.1 for the locations of measurement spheres). The observed response in terms of pattern is similar to the behavior expected from the shearing of a granular material – i.e., stress ratio increases with the development of boundary shear strain with a relatively higher initial stiffness; this is followed by lower rate of shear stress increase (i.e., lower tangent stiffness) until shear stress reaches steady values in the range of 0.35-0.45 at large boundary shear strains. Calculated stress ratios are noted to increase with the increase in interparticle friction coefficient, F, within the range of selected F values of 0.176-0.5 for a given shear strain level.   The computed change in the lateral stress (xx) during the shearing process is shown in Figure  7.1b. Lateral stresses increase with the development of shear strain and seem to somewhat plateau at around 30% shear strain. Computed xx values for the three simulations with interparticle friction coefficients of 0.176, 0.3, and 0.5 are very similar.  As the complete stress state of the DSS specimen can be assessed because of the ability to compute lateral stress (xx) from the DEM work, it is possible to plot the Mohr circle representing the complete stress state (i.e., stress state on all planes across the specimen). The development of changes to xx and zz during the shearing process (i.e., boundary shear strain changing from 0% to 30%) is depicted in Figure  7.2 with respect to a Mohr diagram (reported stresses are based on average values over the volume of central measurement spheres MS 2 – MS 4).  The initial stress points just after consolidation   166  and immediately prior to shearing correspond essentially to a ko state; at this state, the stresses on the horizontal and vertical planes can be expressed as (zz,) = (100 kPa, 0 kPa) and (xx,) = (64 kPa, 0 kPa), and they are identified by the Points B and A, respectively, on the Mohr diagram in Figure  7.2.  The Mohr circle shown in Figure  7.2 has been drawn to represent the stress state when the boundary shear strain was at 30%. It can be noted that, at 30% shear strain level, the lateral stress is higher than the vertical stress and stress state on the horizontal plane (xx,) seems to be approaching that for maximum obliquity (i.e. maximum (/zz) ratio). This implies that the mobilized friction angle, mob, at 30% boundary shear strain is approximately equal to angle , which is similar to the case presented earlier in Figure ‎2.6b (i.e., mob = 19.4º and  = 19.1º).   In an ideal drained DSS test with perfectly uniform stresses, vertical effective stress evaluated at any location within the specimen should be constant during shearing. The observed increase in vertical effective stress (i.e. 'zz as shown in Figure  7.2) with the increase in shear stress is due to the development of non-uniformities throughout the shearing phase. It is important to note that average vertical effective stress was maintained constant during the numerical simulation of shearing at a value of 100 kPa (at the bottom and top boundaries on average) while the computation of effective stress in the vertical direction shown in Figure  7.2 was made for central specimen locations on average.  The same information shown in Figure  7.2 can also be examined considering the variation of the computed angles , , and mob with increasing boundary shear strain, , from 0% to 30%, as shown in Figure  7.3a for average stresses computed at the central specimen locations (i.e. MS 2- MS 4) for the simulation performed with interparticle friction coefficient of 0.176. Careful observation of this figure indicates the following: i) the value of mob at  = 0% is essentially determined by stress conditions corresponding to the ko stress state; ii) the value of mob becomes approximately equal to angle  at a  value in the vicinity of about 20% (i.e., maximum shear stresses acts on nearly horizontal and vertical   167  planes); and iii) the value of mob becomes closer to angle ,  which is a result of maximum (/xx) ratio corresponding with the horizontal plane at larger shear strains.  In a similar manner, the variation of the computed angles , , and mob with increasing boundary shear strain, , from 0% to 30%, are shown in Figure  7.3b and Figure  7.3c based on average stress computations made at central specimen locations for the simulations performed with interparticle friction coefficients of 0.3 and 0.5, respectively. Similar to the case of F = 0.176, mob becomes approximately equal to angle  at larger  values.   7.2.2 Principal stresses and stress invariants   Figure ‎7.4 shows the calculated principal stresses for central specimen locations (MS 2-MS 4) on average. Ideally, intermediate principal stress, '2, and minor principal stress, '3, should have the same value at  of 0 %. However, likely due to some initial (i.e., at the end of consolidation) non-uniformities in the numerical model, the initial values of '2 and '3 are slightly different (71 kPa and 63 kPa for '2 and '3, respectively). A general trend of increasing principal stresses with the development of shear strain is noted. Only a slight increase in '3 from an initial value of 63 kPa to a value of 80 kPa at the end of shearing is observed.   Figure ‎7.5 shows the change of the angle between the direction of '1 and the vertical direction, , with the increase in shear strain. At the end of consolidation (i.e., at  = 0%), '1 is aligned with the vertical direction and is equal to zero. At relatively small shear strains (0%-2%), a rapid increase in  with shear strain is observed, followed by a more gradual increase in  with further increase in shear strain.   168  At shear strain of 30%,  reaches a value of about 50°. The observed trend is qualitatively similar to that shown in Roscoe et al. (1967).  The relation between stress ratio, h/'v, and tan () is shown in Figure ‎7.6. An approximately linear relation between h/'v, and tan () is observed. Oda (1975) and Wood et al. (1979) proposed Equation 7.4 to describe the relation between h/'v, and tan () for DSS testing.                          (7.4)  Where  is a material constant. Oda (1975) pointed out that the value of  is not affected by initial conditions and, in turn, knowledge of  for a certain material can provide insight on the DSS mobilized stress ratio. This was later adopted by Wood et al. (1979) to develop a simplified method to compute the DSS stress state. They calculated a  value for Lieghton Buzzard sand of 0.67. The reported results in Figure ‎7.6 for the simulated glass beads show a fairly linear trend betweenh/'v, and tan (); if assessed considering Equation 7.4, this results in a  value of 0.32. This is in line with the findings of Oda (1975) and Wood et al. (1979). The observed linearity of the relation betweenh/'v, and tan () suggests that the simulated glass beads have a qualitatively similar behavior, in this respect, to that of sands.  Stress invariants expressed in terms of deviatoric stress, 'q, mean effective stress, 'm, and Lode angle, L, are important for use in the development of generalized soil constitutive models. Figure ‎7.7 shows the stress path in 'q versus 'm for a DEM simulation of the DSS behavior of the glass beads. It seems that a linear trend line with a slope of 1 is a reasonable approximation to the slope of the DSS stress path as   169  noted from the results of the DEM simulation. The stress path of a conventional triaxial compression test is shown on the same plot for comparison. The calculated Lode angle at the end of shearing (i.e., at  = 30%) was 13°. Jefferies & Shuttle (2002) reported L values in the range of about 12°-20° calculated for plane strain tests performed on sands. Lode angle calculated form the results of the DSS test simulation falls within the range of Lode angles calculated for plane strain tests.   7.3 Interpretation of DSS stress state using stresses measured at the boundaries  Measured boundary stresses 'v, 'h, and h, as previously shown in Chapter 4 for results of DSS testing performed on Fraser River silt with lateral stress measurement, cannot be directly used to calculate the mobilized DSS friction angle, mob, of tested DSS specimens due to the absence of complimentary shear stresses acting on the lateral boundaries. To calculate friction angle using boundary stresses, it is necessary to assume that complimentary shear stress acting on the lateral boundaries, v, is equal to shear stress acting on the horizontal boundaries, h (i.e. h = v).The friction angle calculated from boundary stresses is denoted here as angle  as defined in Equation 7.5. Clearly, this computed angle  is different from the true mobilized friction angle, mob, as the former is calculated based on an unrealistic assumption of equal (complementary) shear stresses acting on the horizontal and vertical specimen boundaries. A relationship between angle  and the DSS mobilized friction angle, mob, is very useful as calculating the latter requires direct measurement of stresses at the specimen core, which is practically not possible.         [             ]         (7.5)    170  Figure ‎7.8 shows the change in friction angle  calculated from boundary stresses versus DSS mobilized friction angle, mob, calculated at central specimen locations for the simulated glass beads with F = 0.176. At large shear strains, angle  is larger than angle mob by about 20%.   The results of a monotonic DSS test performed on a Fraser River silt specimen consolidated to 'vc of 200 kPa with lateral stress measurements were previously discussed in Chapter 4. Lateral stresses were measured using four free form sensors that were mounted on the inner walls of the reinforced DSS membrane. As such, horizontal effective stress, 'h, (calculated based on average stresses acting on the four lateral free form sensors) in addition to vertical effective stress, 'v and shear stress, h, that are typically measured during DSS testing, are known. It is therefore possible to calculate angle  as previously defined in Equation 7.5 for the tested Fraser River silt specimen. Angle  calculated for the Fraser River silt specimen at large strain ( = 25%) is equal to 42º.   The results of the discrete element model showed that angle  is 20% higher than the true DSS mobilized friction angle, mob, calculated at the central specimen locations where stresses are fairly uniform. If this observation from the DEM model is extended to calculate the true DSS mobilized friction angle for Fraser River silt, it will result in mob of 34º (based on mob = 0.8 ).   7.4 Discussion and Conclusions  Discrete element modeling of a cylindrical DSS specimen was performed and the results were used to calculate the mobilized friction angle at planes of maximum obliquity.    171  The results from DEM analysis suggest that the planes of maximum obliquity in the DSS device would rotate with progression of shear strain. With the development of shear strain, planes of maximum obliquity rotate and the horizontal plane becomes a plane of maximum obliquity at large shear strains as specifically notable from the results of the simulation performed with interparticle friction coefficient of 0.176. Similar trends are inferred at much larger shear strains from the simulations performed with larger F values of 0.3 and 0.5 that showed a more dilative response compared to the simulation with F = 0.176. In essence, at large shear strains, angle  as defined in Equation 2.3 seems to be a reasonable approximation for the DSS mobilized friction angle, mob, with a margin of error of 10%.  It is clear that the mobilized friction angle in the DSS device cannot be determined using one interpretation for the stress state for the whole shear strain range investigated in this chapter. Rather, the planes of maximum obliquity seem to rotate continuously with the development of shear strain.  According to the Mohr-Coulomb failure criterion, it is assumed that failure occurs along planes of maximum obliquity. As such, the DEM model results suggest the possibility that failure would occur along horizontal planes at large shear strain. This is consistent with the horizontal failure zone observed by Roscoe (1970). The interpreted orientation of failure plane from the results of the DEM model also agrees with Wroth (1984) at large strains based on his interpretation of the results of a DSS test on normally consolidated Boston Blue clay. DSS testing results performed by Cole (1967) suggested continuous rotation of planes of maximum obliquity with the progression of shearing. His results also indicated that at large shear strains, the horizontal plan is nearly a plane of maximum obliquity for the case of testing performed on dense sands.   Principal stresses and angle  in the DSS device were calculated from stresses computed from the DEM simulations at central specimen locations. Continuous rotation of principal stresses was observed with the   172  development of shear strain. Angle  of about 50° was calculated at large shear strains ( = 30 %) which is in line with the findings of Roscoe et al. (1967).   Stress invariants are of particular interest for understanding the DSS generalized stress state and for the development of soil constitutive models. It was shown that the stress path for the simulated DSS specimen in 'q versus 'm space had a slope of unity. Lode angle, L, calculated at large strains for the DSS simulation was in the range of Lode angles calculated from the results of plane strain tests shown in Jefferies & Shuttle (2002). This finding is of particular significance as plane strain conditions are typically assumed for DSS testing.   The results of the DEM model were used to provide insight on the relation between angle , calculated using boundary stresses as defined in Equation 7.5 and DSS mobilized friction angle, mob, calculated for stress acting at central specimen locations. Analysis indicated that, at large shear strains, mob is equal to 80% of angle . This finding was applied to calculate friction angle from a DSS test performed on Fraser River silt with measured lateral stresses.      173  a)  b)  Figure ‎7.1. Computed stresses versus boundary shear strain () from DEM: a) stress ratio, 'xx b) lateral stress, xx.   00.10.20.30.40.50.60.70 5 10 15 20 25 30Stress ratio, ' zz  Shear strain,  (%) F = 0.176F = 0.3F = 0.50204060801001201400 5 10 15 20 25 30Horizontal stress, ' xx  (kPa) Shear strain,  (%)   174    Figure ‎7.2. Stress paths, Mohr circle, and stress state at maximum obliquity for the simulated DSS specimen at shear strains of 30% (F = 0.176).   -60-40-2002040600 20 40 60 80 100 120 140 160 180Shear stress, (kPa)Normal stress, ' (kPa)('zz,)('xx,)Stress path on the horizontal plane at central specimen locationsStress path on the vertical plane at central specimen locations Point of maximum 'mob≈ AB  175     Figure  7.3. Angles , , and mob with the development of boundary shear strain,  , for simulations with Interparticle friction coefficients of: a) 0.176; b) 0.3; and c) 0.5, respectively.   0510152025300 5 10 15 20 25 30Angle (Degrees)Shear strain,  (%)mob31311''''sinmobzz'sin1zz'tan1F = 0.176a)c)0510152025300 5 10 15 20 25 30Angle (Degrees)Shear strain,  (%)mob31311''''sinmobzz'sin1zz't n1F = 0.3b)c)0510152025300 5 10 15 20 25 30Angle (Degrees)Shear strain,  (%)mob31311''''sinmobzz'sin1zz'tan1F = 0.5c)  176   Figure ‎7.4. Principal stresses calculated for central specimen locations (MS 2 through MS 4) versus shear strain for the simulation performed with F value of 0.176.   40801201602000 5 10 15 20 25 30Principal stress (kPa)Shear strain (%)'1'2'3  177   Figure ‎7.5. Angle between the major principal stress, '1, and the vertical direction, , versus shear strain (F = 0.176).   01020304050600 5 10 15 20 25 30(deg.)Shear strain (%)'1'3  178   Figure ‎7.6. Stress ratio, h/'v, versus the tangent of  (F = 0.176).   00.10.20.30.40.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4h/ 'v(deg.)Tan    179   Figure ‎7.7. Stress path in 'q 'm space for the simulated specimen with F of 0.176.    040801201600 40 80 120Deviatoric stress, q(kPa)Mean effective stress, 'm(kPa)Typical conventional triaxial compressionstress pathSimple shear (DEM simulation)3111  180   Figure ‎7.8. Angle  (calculated from boundary stresses) versus DSS mobilized friction angle, mob, for the simulation with F of 0.176.    05101520250 5 10 15 20 25Angle (deg.)Mobilized friction angle, mob(deg.)10.8  181  8 SUMMARY AND CONCLUSIONS  The laboratory direct simple shear (DSS) testing process was simulated using 3D discrete element modeling that utilizes PFC3D (Particle Flow Code in Three Dimensions developed by Itasca Inc.). Analysis of simulation results was performed to fill the knowledge gaps in regards to the state of stress in DSS specimens. Focused DSS testing program was performed using glass beads particularly for use in evaluation of the DEM model performance. DSS testing program undertaken with an instrumented membrane using state-of-the-art free form sensors to measure DSS lateral stresses provided further insight into the state of stress in tested DSS specimens.  The conclusions and findings from this study are presented in this chapter and are organized as follows: DSS testing using free form sensors; evaluation of the performance of the developed DEM model of the DSS test; stress-strain uniformities in DSS specimens; and interpretation of the state of stress in laboratory DSS specimens. Finally, limitations of the current study and suggestions for future work are presented.   8.1 DSS testing using free form sensors  A DSS testing program was performed using free form (paper-thin) flexible pressure sensors mounted on the inner wall of the DSS membrane to measure lateral stresses. Reconstituted Fraser River specimens were used to perform the testing. The major findings are summarized as follow:    A range of ko values of 0.4-0.5 was noted for the normally consolidated specimens which is in agreement with the typically used relations of ko = 1- sinJaky, 1944)   Higher ko values in the range of 0.7-0.8 were noted for specimens with OCR of 2 compared to ko values for normally consolidated specimens.   182   An increase in k value was noted from an initial (i.e., at  = 0%) value of about 0.5 to a value of about 1.1 at large shear strains as observed from the results of monotonic constant volume tests.   An overall increase in lateral earth pressure coefficient is noted with the progression of cyclic loading. During a given cycle, the k value increases during the loading parts of the cycles followed by a reduction in the k value during the unloading parts of the cycle.  Overall, this study demonstrated the calibration and use of free form sensors for measurement of lateral stresses in DSS specimens. The results obtained suggest the adequacy of free form sensors for use in this regard.  8.2 Evaluation of the performance of a DEM model of the DSS test  The development of a PFC3D model of the DSS test, selection of input parameters for the developed model, and performance evaluation of the model were presented in Chapter 5. Evaluation of model performance was done by comparison with counterpart DSS laboratory testing results performed on 2mm diameter uniform glass beads as follows:   Drained monotonic stress-strain response obtained from simulation results is very similar to that obtained for the glass beads. Both simulations and laboratory testing results showed contractive responses with the latter demonstrating more shear induced contraction.  Initial contractive behavior followed by dilative response was observed from the laboratory results of monotonic constant volume tests which is in agreement with counterpart simulations results.   The overall reduction of vertical effective stress during cyclic loading including the transient ′v ≈ 0 condition during the last loading cycle observed from cyclic constant volume testing of glass   183  beads was accurately captured by the DEM model. The CSR versus number of cycles to reach  of 3.75% plots were very similar for simulations and laboratory tests.  Overall, the used DEM modeling approach, including the selected contact model and input parameters, seems to be effective in capturing the salient characteristics of the DSS behavior of the tested glass beads. This justifies the use of the developed model for use in evaluating the state of stress in DSS specimens.   8.3 Stress-strain uniformities in DSS specimens   The DEM model of the DSS test was used to evaluate stress-strain uniformities within DSS specimens. Stresses were calculated at locations within the specimen and at boundary segments. The effects of shear strain level, selected interparticle friction coefficient, consolidation vertical effective stress, drainage conditions and friction coefficient on the lateral boundaries were investigated. The main findings are presented here as follows:   End of consolidation stresses at central specimen locations are fairly uniform with increased non-uniformities at locations closer to the lateral boundaries. Calculated radial stresses at the end of consolidation are within ± 15% of average radial stress.  During the shearing phase, the distribution of shear strains across the specimen is more uniform at lower shear strain levels. Shear strains calculated at central specimen locations were about half of these calculated near the boundaries.   Significant stress non-uniformities during shearing are limited to a narrow zone of about two particles diameter near the lateral boundaries while stresses at central specimen locations are relatively more uniform (i.e. most representative of “ideal” simple shear conditions).   184   It was shown that laboratory-testing-wise shear stress ratios fall within the range of stress ratios calculated for the measurement spheres. This suggests that shear stress ratio calculated from DSS laboratory testing using boundary stresses is representative of that calculated for interior specimen locations.  It was noted that the increase of shear strain level can result in an increase of stress non-uniformities. Higher vertical effective stress levels and interparticle friction coefficient values resulted in minor improvements in regards to stress uniformities (i.e. lower stress non-uniformities). Drainage conditions and friction coefficient on the lateral boundaries had practically negligible effects on the degree of stress uniformities in the simulated specimens.   The results of a drained cyclic simulation indicated that some non-uniformities can develop within the simulated specimen with the increase in the number of loading cycles  8.4 Interpretations of the state of stress in DSS specimens   Simulation results were used to evaluate the state of stress for central specimen locations that are most representative of “ideal” simple shear conditions. The results indicate that planes of maximum obliquity rotate continuously with the progression of shearing. However, at large strains, the horizontal plane becomes a plane of maximum obliquity. As such, angle  defined in Equation 2.3 seems to be a reasonable approximation for the DSS mobilized friction angle, mob, with a margin of error of 10%.  Principal stresses, angle , and stress invariants, were computed from the results of DEM simulations. Angle increases with the development of shear strain to a value of about 50° at large shear strains. The slope of DSS stress path in 'q versus 'm space is equal to 1. Lode angle, L, calculated at large   185  strains is in the range of Lode angles calculated from the results of plane strain tests shown in Jefferies & Shuttle (2002).   A correlation between friction angle , calculated using boundary stresses as defined in Equation 7.5 and DSS mobilized friction angle, mob, calculated for stresses acting at central specimen locations was established (mob.   8.5 Limitations of the current work and suggestions for future research  The anticipated increase in computational power along with the development of more efficient DEM codes will make it more feasible to model realistic soil particle shapes. For example, DSS specimens consisting of perfect spheres were used to arrive at the conclusions in this study as using realistic particle shapes is currently computationally prohibitive. Modeling DSS behavior of angular soils would provide further insight on the DSS state of stress.   Infinitely rigid rings were used to represent the lateral DSS specimen boundaries. However, in the NGI DSS test, the used reinforced membrane has a finite stiffness. As such, the imposed conditions at the lateral boundaries in the DEM model deviate from the real boundary conditions in the DSS test. It is of interest to investigate the effect of membrane stiffness on the DSS state of stresses and strains.   A non-linear Hertz-Mindlin contact model was used to model the relation between forces and displacements at particle contacts. It is noted that the relationship between forces and displacements at real particle contacts is rather complex. As such, it would be of interest to investigate the potential benefits of using other more realistic contact models.      186  In this study the used DEM model was validated against the results of testing performed using an NGI-type DSS test. It is of interest to validate the model under different boundary conditions such as triaxial conditions and under general stress conditions by comparison with hollow cylindrical testing results.      187  REFERENCES  Airey, D., 1984. Clays in circular simple shear apparatus. University of Cambridge. Airey, D.W. & Wood, D.M., 1984. Discussion of specimen size effect in simple shear test by Vucetic, and M. Lacasse, S. Journal of the Geotechnical Engineering Division, 110, pp.439–442. Amer, M.I., Kovacs, W.D. & Aggour, M.S., 1987. Cyclic Simple Shear Size Effects. Journal of Geotechnical Engineering, 113(7), pp.693–707. Atkinson, J., Lau, W. & Powell, J., 1991. Measurement of soil strength in simple shear tests. Canadian Geotechnical Journal, 28, pp.255–262. Bjerrum, L. & Landva, A., 1966. Direct Simple-Shear Tests on a Norwegian Quick Clay. Géotechnique, 16(1), pp.1–20. Budhu, M., 1985. Lateral Stresses Observed in Two Simple Shear Apparatus. Journal of Geotechnical Engineering, 111(6), pp.698–711. Budhu, M., 1984. Nonuniformities imposed by simple shear apparatus. Canadian Geotechnical Journal, 20(2), pp.125–137. Budhu, M. & Britto, A., 1987. Numerical analysis of soils in simple shear devices. Soils and Foundations, 27(2), pp.31–41. Cavarretta, I., Coop, M. & O’Sullivan, C., 2010. The influence of particle characteristics on the behaviour of coarse grained soils. Géotechnique, 60(6), pp.413–423. Chung, Y.C. & Ooi, J., 2007. Influence of discrete element model parameters on bulk behavior of a granular solid under confined compression. Particulate Science and Technology, 26(1), pp.83–96.   188  Coetzee, C.J. & Els, D.N.J., 2009. Calibration of discrete element parameters and the modelling of silo discharge and bucket filling. Computers and Electronics in Agriculture, 65(2), pp.198–212. Cole, E., 1967. The behaviour of soils in the simple-shear apparatus. Cambridge University. Cui, L. & O’Sullivan, C., 2006. Exploring the macro- and micro-scale response of an idealised granular material in the direct shear apparatus. Géotechnique, 56(7), pp.455–468. Cundall, P.A. & Strack, O.D.L., 1979. A discrete numerical model for granular assemblies. Géotechnique, 29(1), pp.47–65. Dabeet, A., Wijewickreme, D. & Byrne, P., 2011. Discrete element modeling of direct simple shear response of granular soils and model validation using laboratory element tests. In 14th Pan-Am. Conference and 64th Canadian Geotechnical conference. Toronto. Dabeet, A., Wijewickreme, D. & Byrne, P., 2010. Evaluation of the stress-strain uniformities in the direct simple shear device using 3D discrete element modeling. In 63rd Canadian Geotechnical Conference. Calgary. Doherty, J. & Fahey, M., 2011. Three-dimensional finite element analysis of the direct simple shear test. Computers and Geotechnics, 38(7), pp.917–924. Dounias, G.T. & Potts, D.M., 1993. Numerical analysis of drained direct and simple shear tests. Journal of Geotechnical Engineering, 119(12), pp.1870–1891. Dyvik, R. et al., 1987. Comparison of truly undrained and constant volume direct simple shear tests. Géotechnique, 37(1), pp.3–10. Dyvik, R., Zimmie, T.F. & Floess, C.H.L., 1981. Lateral stress measurements in Direct Simple Shear Device. In Laboratory Shear Strength of Soils. ASTM International, pp. 191–206.   189  Finn, W.D.L., Bhatia, S.K. & Pickering, D.J., 1982. The cyclic simple shear test. In Soil Mechanics-Transient and Cyclic Loads. John Wiley & Sons Ltd., pp. 583–607. Finn, W.D.L., Vaid, Y.P. & Bhatia, S.K., 1978. Constant volume simple shear testing. In The second International Conference on Microzonation for Safer Construction-Research and Application. San Francisco, California, pp. 839–851. Fu, P. & Dafalias, Y.F., 2011. Fabric evolution within shear bands of granular materials and its relation to critical state theory. International Journal for Numerical and Analytical Methods in Geomechanics, 35(11), pp.1918–1948. Fukuoka, H., 1991. Variation of the friction angle of granular materials in the high-speed high-stress ring shear apparatus-influence of re-orientation, alignment and crushing of grains during shear. Bulletin of the Disaster Prevention Research Institute, Kyoto University, 41(4), pp.243–279. Fukuoka, H. & Sassa, K., 1991. High-speed high-stress ring shear tests on granular soils and clayey soils. In XIX World Congress of the International Union of Forestry Research Organizations. Montreal, Canada. Härtl, J. & Ooi, J., 2008. Experiments and simulations of direct shear tests: porosity, contact friction and bulk friction. Granular Matter, 10(4), pp.263–271. Härtl, J. & Ooi, J., 2011. Numerical investigation of particle shape and particle friction on limiting bulk friction in direct shear tests and comparison with experiments. Powder Technology, 212(1), pp.231–239. Hight, D.W., Gens, A. & Symes, M.J., 1983. The development of a new hollow cylinder apparatus for investigating the effects of principal stress rotation in soils. Géotechnique, 33(4), pp.355–383.   190  Idriss, I.M. & Boulanger, R., 2008. Soil liquefaction during earthquakes, Earthquakes Engineering Research Institute. Itasca Inc., 2005a. PFC3D (Particle Flow Code in Three Dimensions) version 3.1. Itasca Inc., 2005b. PFC3D user manual, Monneapolis, USA. Jaky, J., 1944. The coefficient of earth pressure at rest. J. Soc. Hung. Eng. Arch., pp.355–358. Jefferies, M. & Shuttle, D., 2002. Dilatancy in general Cambridge-type models. Géotechnique, 52(9), pp.625–638. De Josselin, D.J., 1988. Elasto-plastic version of the double sliding model in undrained simple shear tests. Géotechnique, 38(4), pp.533–555. Kammerer, A.M., 2006. Undrained response of Monterey 0/30 sand under multidirectional cyclic simple shear loading conditions. University of California, Berkeley. Lorenz, A., Tuozzolo, C. & Louge, M.Y., 1997. Measurements of Impact Properties of Small , Nearly Spherical Particles. Experimental Mechanics, 37(3), pp.292–298. Lucks, A. et al., 1972. Stress Conditions in NGI Simple Shear Test. Journal of the Soil Mechanics and Foundations Division, 98(1), pp.155–160. Negussey, D., Wijewickreme, D. & Vaid, Y.P., 1988. Constant-volume friction angle of granular materials. Canadian Geotechnical Journal, 25, pp.50–55. Northcutt, S.L., 2010. Effect of particle fabric on the one-dimensional compression response of Fraser River sand. University of British Columbia. NRC, 1985. Liquefaction of soils during earthquakes, Washington, D.C.   191  O’Sullivan, C., 2011a. Particle-based discrete element modeling : geomechanics perspective. International Journal of Geomechanics, 11(6), pp.449–464. O’Sullivan, C., 2011b. Particulate discrete element modeling: a geomechanics perspective, Taylor & Francis Group. Oda, M., 1975. On the relation tau/sigman=kappa.tan(phi) in the simple shear test. Soils and Foundations, 15(4), pp.35–41. Paikowsky, S. & Hajduk, E., 1997. Calibration and Use of Grid-Based Tactile Pressure Sensors in Granular Material. Geotechnical Testing Journal, 20(2), pp.218–241. Palmer, M. et al., 2009. Tactile Pressure Sensors for Soil-Structure Interaction Assessment. Journal of Geotechnical and Geoenvironmental Engineering, 135(11), pp.1638–1645. Prevost, J. & Hoeg, K., 1976. Reanalysis of simple shear soil testing. Canadian Geotechnical Journal, 13, pp.418–429. Roscoe, K., 1953. An apparatus for the application of simple shear to soil samples. In 3rd International Conference on Soil Mechanics. Zurich, pp. 186–191. Roscoe, K., 1970. The influence of strains in soil mechanics. Géotechnique, 20(2), pp.129–170. Roscoe, K., Bassett, R. & Cole, R., 1967. Principal axes observed during simple shear of sand. In The Geotechnical Conference on Shear Strength Properties of Natural Soils and Rocks. Oslo, pp. 231–237. Saada, A.S., Fries, G. & Ker, C., 1983. Stress induced in short cylinders subjected to axial deformation and lateral pressures. Soils and Foundations, 23(1), pp.114–118.   192  Saada, A.S. & Townsend, F.C., 1980. State of the Art: Laboratory Shear Strength of Soils. In Laboratory Shear Strength of Soils. ASTM International, pp. 7–77. Sanín, M.V., 2010. Cyclic shear loading response of Fraser River Silt. University of British Columbia. Sanin, M.V. & Wijewickreme, D., 2006. Cyclic shear response of channel-fill Fraser River Delta silt. Soil Dynamics and Earthquake Engineering, 26(9), pp.854–869. Santamarina, J. & Cho, G., 2001. Determination of critical state parameters in sandy soils—simple procedure. Geotechnical Testing Journal, 24(2), pp.185–192. Shen, C., O’Sullivan, C. & Jardine, R., 2011. tf. In International Symposium on Deformation Charateristics of Geomaterials. Seoul, pp. 314–321. Sivathayalan, S., 2000. Fabric, initial state and stress path effects on liquefaction susceptibility of sans. University of British Columbia. Sriskandakumar, S., 2004. Cyclic loading response of Fraser River sand for validation of numerical models simulating centrifuge tests. University of British Columbia. Thornton, C., 2000. Numerical simulations of deviatoric shear deformation of granular media. Géotechnique, 50(1), pp.43–53. Vaid, Y. & Sivathayalan, S., 1996. Static and cyclic liquefaction potential of Fraser Delta sand in simple shear and triaxial tests. Canadian Geotechnical Journal, 33, pp.281–289. Vaid, Y.P., Chern, J. & Tumi, H., 1985. Confining pressure, grain angularity, and liquefaction. Journal of Geotechnical Engineering, 111(10), pp.1229–1235. Vucetic, M. & Lacasse, S., 1982. Specimen size effect in simple shear test. Journal of the Geotechnical Engineering Division, 108(12), pp.1567–1585.   193  Wang, B., Popescu, R. & Prevost, J.H., 2004. Effects of boundary conditions and partial drainage on cyclic simple shear test results—a numerical study. International Journal for Numerical and Analytical Methods in Geomechanics, 28(10), pp.1057–1082. Wijewickreme, D., Dabeet, A. & Byrne, P., 2013. Some Observations on the State of Stress in the Direct Simple Shear Test Using 3D Discrete Element Analysis. Geotechnical Testing Journal, 36(2), pp.292–299. Wijewickreme, D. & Sanin, M., 2006. New sample holder for the preparation of undisturbed fine-grained soil specimens for laboratory element testing. Geotechnical Testing Journal, 29(3), pp.1–8. Wijewickreme, D., Sriskandakumar, S. & Byrne, P., 2005. Cyclic loading response of loose air-pluviated Fraser River sand for validation of numerical models simulating centrifuge tests. Canadian Geotechnical Journal, 42(2), pp.550–561. Wijewickreme, D. & Vaid, Y.P., 1991. Stress nonuniformities in hollow cylinder torsional specimens. Geotechnical Testing Journal, 14(4), pp.349–362. Wood, D. & Budhu, M., 1980. The behaviour of Leighton Buzzard sand in cyclic simple shear tests. In International Symposium on Soils under Cyclic and Transient Loading. Rotterdam: A. A. Balkema, pp. 9–21. Wood, D., Drescher, A. & Budhu, M., 1979. On the determination of stress state in the simple shear apparatus. Geotechnical Testing Journal, 2(4), pp.211–221. Wroth, C.P., 1987. The behaviour of normally consolidated clay as observed in undrained direct shear tests. Géotechnique, 37(1), pp.37–43. Wroth, C.P., 1984. The interpretation of in situ soil tests. Géotechnique, 34(4), pp.449–488.   194  Yan, W.M., 2009. Fabric evolution in a numerical direct shear test. Computers and Geotechnics, 36(4), pp.597–603. Youd, T.L. & Croven, T.N., 1975. Lateral stresses in sands during cyclic loading. Journal of Geotechnical and Geoenvironmental Engineering, 101(2), pp.217–221. Zhao, X. & Evans, T.M., 2009. Discrete simulations of laboratory loading conditions. International Journal of Geomechanics, 9(4), pp.169–178.     195  APPENDIX A: ADDITIONAL DEM MODEL RESULTS  Appendix A presents additional results obtained for the simulations shown in Chapter 5 and Chapter 6.   Figure A. 1. Monotonic constant volume shearing response for simulations with horizontal wall-ball friction of 0.5, 1, 2, and 10: a) shear stress strain; b) stress path.   a) b)   196    Figure A. 2. Monotonic constant volume shearing response for simulations with lateral wall-ball friction of 0, 0.1 and 0.176: a) shear stress strain; b) stress path.    a) b)   197    Figure A. 3. Monotonic constant volume shearing response for simulations with lateral ball-ball friction of 0.176, 0.3 and 0.5: a) shear stress strain; b) stress path.    a) b)   198     Figure A. 4. Monotonic drained shearing response for simulations with ’vc of 100 kPa, 150 kPa, 200 kPa and 400 kPa: a) shear stress strain; b) volumetric strain versus shear strain.    a) b)   199   Figure A. 5. Monotonic drained shearing response for a simulation with ball-ball friction of 0.5: a) shear stress strain; b) vertical effective stress versus shear strain; c) shear stress ratio versus shear strain.    a) b) c)   200    Figure A. 6. Monotonic constant volume shearing response for a simulation with ball-ball friction of 0.176: a) shear stress strain; b) vertical effective stress versus shear strain; c) shear stress ratio versus shear strain.    a) b) c)   201     Figure A. 7. Monotonic drained shearing response for a simulation with ’vc of 400 kPa: a) shear stress strain; b) vertical effective stress versus shear strain; c) shear stress ratio versus shear strain.  b) a) c) 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0167563/manifest

Comment

Related Items