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Numerical modeling and analysis of pullout tests of sheet and geogrid inclusions in sand Rousé, Pascale C. 2018

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NUMERICAL MODELING AND ANALYSIS OF PULLOUT TESTS OF SHEET AND GEOGRID INCLUSIONS IN SAND by PASCALE C. ROUSÉ B.Sc., Universidad de Chile, 2003M.A.Sc., The University of British Columbia, 2005A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Civil Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2018 © Pascale C. Rousé, 2018 ii The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled: Numerical modeling and analysis of pullout tests of sheet and geogrid inclusions in sand submitted by Pascale C. Rousé in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Civil Engineering Examining Committee: R. Jonathan Fannin, Civil EngineeringCo-supervisor Mahdi Taiebat, Civil Engineering Co-supervisor  John. A. Howie, Civil Engineering Supervisory Committee Member Ricardo Foschi, Civil Engineering University Examiner Gary Schajer, Mechanical Engineering University Examiner Additional Supervisory Committee Members: Erik Eberhardt, Earth, Ocean and Atmospheric Sciences Supervisory Committee Member Supervisory Committee Member iii Abstract One way of studying the soil-inclusion interaction in the pullout test is by numerical modeling. Several of the numerical studies available in the literature lack the integration of consistent material characterization as input for the numerical model, resulting in little phenomenological description of the soil-inclusion interface behavior. There is, therefore, a need for an improved evidence-based understanding of the factors influencing the pullout resistance of different inclusions. Accordingly, the main objective of this study was to capture the pullout response of different inclusions, for which extensive laboratory pullout test data existed, through a phenomenological numerical model that uses physically-based parameters. This numerical model is henceforth used in a parametric study to assess the adequacy of the laboratory test data in the literature and ASTM D6706-01 recommendations. The finite difference software FLAC was used to simulate the laboratory response of three sheet inclusions and three geogrids, embedded in a pullout box filled with a uniformly graded sand (Badger sand) and subjected to vertical stresses up to 17 kPa. In the numerical model, the inclusions were represented by an elastic continuum at the center of the pullout box. The sand was modeled using NorSand, a constitutive model that is able to capture the dilative behavior of dense sands. An alternative approach to the usual spring interface is proposed to model the soil-inclusion interaction, where a thin continuum layer following a NorSand behavior is used, and the friction angle changed according to the interface strength of each inclusion. The soil and interface parameters were obtained from a laboratory testing program on Badger sand including triaxial, direct shear and direct simple shear tests. The results of this dissertation yield three principal contributions: 1) plane strain conditions and a stress-dependency of the critical state friction angle prevail in the pullout box; 2) the use of a constitutive model that can simulate dilation to represent the soil-inclusion interface behavior is able to capture the complete pullout response of the different inclusions; and 3) different aspects of ASTM D6706-01 pullout recommendations deserve improvement for a correct interpretation of the soil-inclusion interaction factor.  iv Lay Summary The pullout test is used to determine the soil-inclusion interaction parameters that are considered in the design of reinforced earth structures. The available pullout test data have advanced our understanding of soil-inclusion interaction, but there are some aspects of the pullout behavior that we cannot measure experimentally. In consequence, there still exists some uncertainty related to the stress regime inside the pullout box, the influence of the relatively low stresses at which the pullout test is performed, and the influence of the boundary conditions of the box on the pullout response. In this study, a numerical model is proposed, then validated against laboratory pullout test data, and subsequently used in a parametric study. The objective is to better understand the influence of different attributes of the pullout test on the pullout response, and thus improve our interpretation of thesoil-inclusion interface behavior for design. v Preface This Ph.D research is a continuation of my M.A.Sc. thesis (2003 to 2005) at The University of British Columbia, which was co-supervised by Dr. Dawn Shuttle and Dr. Jonathan Fannin. In that study, a characterization of the properties and strength of Badger sand were established for purposes of numerical modeling of geosynthetic pullout tests. The general intent was to contrast properties of Badger sand with those of other sands and determine if the constitutive model NorSand was able to capture the behavior of well-rounded grains such as Badger sand. The M.A.Sc. findings raised some interesting questions that I elected to answer by pursuing a doctoral degree also under the co-supervision of Dr. D. Shuttle and Dr. J. Fannin (2005 to 2007). In 2007, Dr. D. Shuttle resigned her position at The University ofBritish Columbia and, since 2009, Dr. Mahdi Taiebat has co-supervised my research. From the findings of my M.A.Sc. thesis and the subsequent research pursued in my Ph.D. program I co-authored two publications with my co-supervisors. They are presented below along with the corresponding chapter (and objective) of the dissertation in which the work is reported.  TN1: Rousé, P.C., Fannin, R.J., and Shuttle, D.A. (2008) “Influence of roundness on the void ratio and strength of uniform sand”, Geotechnique 58, No 3, pp. 227-231 (Chapter 3 – Objective 1).  https://www.icevirtuallibrary.com/doi/full/10.1680/geot.2008.58.3.227 TN2: Rousé, P.C., Fannin, R.J. and Taiebat, M. (2014) “Sand strength for back-analysis of pullout tests at large displacement”, Geotechnique, 64, No 4, pp. 320-324 (Chapter 4 – Objectives 1 and 2).  https://www.icevirtuallibrary.com/doi/full/10.1680/geot.13.T.021 vi In 2008, I was offered an appointment to work as a full-time academic at Universidad Diego Portales in Chile. I took the offer for reasons of professional development and have worked on my Ph.D. dissertation from Chile since that time. During this ten-years research period, and while I was working at Universidad Diego Portales, I published two technical notes from independent studies that arose from my Ph.D. research. The two technical notes are presented below along with the Chapter to which they refer. TN3: Rousé, P.C. (2014) “Comparison of methods for the measurement of the angle of repose of granular materials”, Geotechnical Testing journal”, 37, No 1, pp 164-168 (Chapter 3). https://www.astm.org/DIGITAL_LIBRARY/JOURNALS/GEOTECH/PAGES/GTJ20120144.htmTN4: Rousé, P.C. (2018) “Influence of vertical stress in the critical state friction angle at very low stresses in sands”, accepted for publication on June 22, 2018, Soils and Foundations (Chapter 3). vii Table of contents Abstract……………………………………………………………………………...…….. iii Lay Summary……………………………………………………………………………… iv Preface……………………………………………………………………………….……... v Table of contents………………………………………………………………………….. vii List of Tables………………………………………………………………………..…..... xii List of Figures…………………………………………………………………………..... xiv List of symbols and acronyms……………………………………………………...…..... xix Acknowledgments…………………………………………………………………........ xxiii Chapter 1 Introduction……………….……………………………………………………... 1 1.1. Objectives and contributions…...………………………………………………........ 7 1.2. Structure of the thesis………………………………….………………………….…. 9 Chapter 2 Literature review………………………………………………….……….….... 14 2.1. Introduction………………………………………………………………………… 14 2.2. Soil-inclusion interaction…………………………………………………………... 15 2.2.1. Direct shear test.…………………………………………………………….. 16 2.2.2. Pullout test……......……………………………………………………...….. 16     2.2.2.1. Sheet inclusions……………………………………………………...…. 19     2.2.2.2. Geogrids………………………………………………………………... 19 2.3. Numerical modeling of pullout tests…………...………..……………………..….... 24 2.3.1. Analytical models……………………………………………………...…… 25 2.3.2. Finite element models…………………………………………………...….. 27 2.3.3. Discrete element models……………………………………………...…….. 31 2.3.4. Statistical analysis…………………………………………………………... 32 2.3.5. Mobilized strength of sands and selection of parameters for numerical modelling of pullout ……………………………………………………………… 34     2.3.5.1. Numerical relations……………………………………….…………… 35     2.3.5.2. Empirical data…………………………………….…………………… 39 2.4. Laboratory pullout tests…………………………………………………...….….… 40 2.4.1. Vertical stress and top boundary…………………………………………….. 41 2.4.2. Front wall effects…………………………………………………...……….. 41 viii 2.4.3. Height of the box…………………………………………………...……….. 43 2.4.4. Inclusion length…………………………………………………...……..….. 44     2.5. Summary of current knowledge and role of present research…………………..….. 45 Chapter 3 Description of material properties …………….…………………………...…… 61       3.1. Introduction……………...………………………………………….…………….. 61       3.2. Description of physical and mechanical properties of Badger sand……………… 61 3.2.1. Index properties…………………………………………………………….. 61 3.2.2. Roundness and sphericity……………………………………………………62     3.2.2.1. On the variation of extreme void ratio with roundness………...…….… 63 3.2.3. Badger sand strength…………………………………………………………65     3.2.3.1. Direct simple shear tests……………..……………………...….…….... 65     3.2.3.2. Direct shear tests………………...………………………….......…….... 66     3.2.3.3. Triaxial tests……………………………………….......................…..… 66     3.2.3.4. Angle of repose……………………………………….........................… 68     3.2.3.5. Influence of roundness on the strength of uniform sand………………... 69     3.2.3.6. Stress-dependency of the angle of friction at the critical state……….... 71      3.3. Description of physical and mechanical properties of inclusions tested………..… 73      3.4. Soil-inclusion interface properties……………………….….………...…………... 75 3.4.1. Sheet inclusions…………….......……………………………...…..………... 75 3.4.2. Geogrids……….………...……………………………………...…………... 76      3.5. Description of pullout box……………………………………………...…………..78      3.6. Summary…………………..…………………………………………...………..… 79 Chapter 4 Numerical simulation of pullout tests at large displacement.………...…...…… 97        4.1. Introduction………………………………….…………………………………… 97        4.2. Numerical simulation of planar inclusions at large displacement……………...… 98 4.2.1. Model parameters for Badger sand .………………………………………... 98 4.2.2. Model parameters for the inclusions ……………………..…………………100 4.2.3. Model parameters for the soil-inclusion interface.…………….……………100 4.2.4. Numerical model.……………………………….…………………………..101      4.2.5. Simulation results ………………………………………………….….……101     4.2.5.1. Pullout response at large displacement.………………………………..101 ix     4.2.5.2. Sensitivity analysis to input parameters…………………………..…... 103     4.2.5.3. Horizontal stresses at the front wall…………………………………... 103 4.3. Numerical simulation of geogrids at large displacement……………………..….. 104 4.3.1. Pullout response at large displacement……………………………………. 105 4.4. Numerical simulations using direct shear parameters..……………………....….. 106 4.5. Summary…………………………………..….………………….…………....… 108 Chapter 5 Numerical simulations of pullout tests in sand ………………………………. 124       5.1. Introduction ……………………………………………………..………...…..… 124       5.2. Constitutive model NorSand for modeling of pullout tests………....…………….126 5.2.1. Summary description of NorSand ……..…………………………………... 126 5.2.2. Calibration of NorSand to Badger sand……………………………………. 127     5.2.2.1. Elastic Parameters……………………………………………………. 128     5.2.2.2. Critical State Parameters……………………………………………... 128     5.2.2.3. Plastic Parameters……………………………………………………. 128     5.2.2.4. Strength of Badger sand at low stresses in the triaxial test...…..…..….. 129      5.3. Numerical simulation approaches of pullout tests using NorSand….…………….131 5.3.1. Numerical modeling using NorSand in combination with a beam element and Mohr-Coulomb springs…..………………………………….…………………… 131 5.3.2. Numerical modeling using a beam element “glued” to the NorSand mesh.... 132 5.3.3. Numerical modeling using NorSand in combination with an elastic continuum for the inclusions and Mohr-Coulomb springs…………………………………… 133 5.3.4. Numerical modeling using an elastic continuum for the inclusions attached to the NorSand mesh...……………………………………………………………….134 5.3.5. Numerical modeling using an elastic continuum for the inclusion and a NorSand continuum layer for the soil-inclusion interface………………...……….135 5.3.6.  Discussion about the elastic continuum combined with the “NorSand interface layer” approach……………………………………………...…………………… 137     5.4. Numerical simulations of the full pullout response of sheet inclusions …………. 139 5.4.1. Model description and characterization………………..…………………... 139 5.4.2. Model parameters………………..………………………………………… 140     5.4.2.1. Model parameters for inclusions……………………………………….140 x     5.4.2.2. Model parameters for Badger sand…………………………………… 140     5.4.2.3. Model parameters for soil-inclusion interface………………………... 140     5.4.2.4. Model parameters for soil-front wall interface……………..…….…… 141 5.4.3. Boundary and initial conditions…………………………………………… 141 5.4.4. Simulations results………………………………………………………… 142     5.4.4.1. Pullout resistance per unit width………………………………………142     5.4.4.2. Horizontal stresses at the front wall……………………………………144      5.5. Numerical simulations of the full pullout response of geogrids using NorSand…. 144 5.5.1. Treating the geogrids in the model………………………………………….144 5.5.2. Simulation results……………………………………………..…………… 146      5.6. Numerical simulations using a constant Mtc……………………………………... 147      5.7. Summary………………………...……………………………………...………... 148 Chapter 6 Sensitivity analysis…………………………………...………………….……. 168       6.1. Introduction……………………………………………………………………… 168       6.2. Definitions……………………...………………………….………………...….. 169 6.2.1. Definition of pullout …………….………………...………………………. 169 6.2.2. Definition of vertical stress ……………………………...………………… 169 6.2.3. Definition of base case……………..…………………...…………………  170       6.3. Tornado diagrams………………………………………...……………….…….. 171       6.4. Analysis of results……………………………………………………………….. 172 6.4.1. Influence of vertical stress on Pu at base case……………………….…….. 172 6.4.2. Influence of tan on Pu……………………………………...……….…….. 173 6.4.3. Influence of ff on Pu……………………………………...……….……….. 174 6.4.4. Influence of L on Pu……………………………………...……….……….. 175 6.4.5. Influence of E on Pu……………………………………...……….……….. 176 6.4.6. Influence of Hb on Pu……………………………………...……….…...….. 177 6.4.7. Summary of the analysis……………….………………...……….……….. 177      6.5. Discussion……………………….……………………………………………….. 178 6.5.1. Comparison of Tornado simulation results with laboratory test data …….. 178 6.5.2. Analysis of the laboratory test data in the literature ...…………….……….. 180     6.5.2.1. Inclusion characteristics………………………………………..…….. 180 xi     6.5.2.2. Boundary conditions………………………………………..…..…….. 184 6.5.3. Analysis of ASTM D6706-01 recommendations.………….….….……….. 189     6.5.3.1. Definition of ultimate pullout………………………………………..…189       6.5.3.2. Vertical stress……………………………………………………..……189     6.5.3.3. Length of the inclusion………………………………………………... 190     6.5.3.4. Height of the box…………………………………………...…………..191     6.5.3.5. Front wall………………………...…………………………………... 191      6.6. Summary……………………….…………………………………..…………….. 192 Chapter 7 Conclusions and recommendations…………………………………………… 215       7.1. Conclusions of the present research ………………………………………..……. 215 7.1.1. Badger sand characterization………………………………………………. 218 7.1.2. Plane strain conditions and stress-dependency in pullout..………………… 219 7.1.3. Simulation of pullout response ..………………………………………….. 220 7.1.4. Sensitivity analysis…………….………………………………………...… 221       7.2. Novel contributions of the present study……………………...…………………. 222 7.2.1. Objective 1………………………………………………………………….222 7.2.2. Objective 2………………………………………………………………….222 7.2.3. Objective 3………………………………………………………………….223 7.2.4. Objective 4………………………………………………………………….223       7.3. Future work and recommendations…………………….………………………... 225 References……………………………………………………………………………….. 227 Appendix A………………………………………………………………………………. 243 xii List of Tables Table 2.1: Failure modes and properties required for reinforced earth and GRS…..……... 48 Table 2.2: Pullout box characteristics from the literature………………………………..… 49 Table 2.3: Inclusion characteristics from the literature………………..………………..… 50 Table 3.1: Material properties for Cu < 2.5………………………………………………... 81 Table 3.2: Summary of properties of Badger sand……………………………………….... 84 Table 3.3: Material properties for inclusions…………………………………………….... 84 Table 3.4: Soil-inclusion interface friction angle for sheets from direct shear tests……….. 85 Table 3.5: Soil-inclusion interface properties for geogrids...…………………………….... 85 Table 4.1: Parameters for inclusions………………………………………………………110 Table 4.2: Values of cs,ps for the soil-inclusion interface using deduced plane strainparameters…………………………………………………………………..………….… 110 Table 4.3: Variation in pullout resistance (kN/m) with input parameters………………….110 Table 4.4: Values of friction angle and bearing stress ratio, for the soil-geogrid interface, considering plane strain parameters…………………………...…………………………..111 Table 4.5: Values of cs,ps (eq. 4.5; eq. 4.6) for the soil-geogrid interface using deduced plane strain parameters…………………………………………………………………….…….111 Table 4.6: Values of cs,ds for the soil-inclusion interface using measured direct shear parameters……………………………………………………..………………………….112 Table 4.7: Values of friction angle and bearing stress ratio, for the soil-geogrid interface, considering direct shear parameters…………………………………………………...…..112 Table 5.1: Different approaches for the numerical modeling of pullout tests……...……..150 Table 5.2: Summary of NorSand parameters……………………….………………..……151 Table 5.3: Values of Mtc (cs,tx)…………………………………………………...…....…. 151 Table 5.4: Values of friction angle and bearing stress ratio, for the soil-grid inclusion interface, considering triaxial parameters………………………………………...……… 152 Table 5.5: Values of cs,tx (Mtc), (eq. 5.5; eq. 5.6) for the soil-geogrid interface using triaxial parameters……………………………………………………………………...………… 152 xiii  Table 6.1: Variables to study and ranges.……………………………………...…...….… 196 Table 6.2a: Ranges of parameter values for which ultimate pullout occurs (dc=de)…….... 197 Table 6.2b: Ranges of parameter values for which only initial pullout occurs (dc >de)…... 197 Table A.1: Summary of equations for NorSand…………………………………..……… 248 Table A.2: Input parameters for plane strain element……………………………………. 249      xiv  List of Figures  Figure 1.1: Reinforced earth structure…………………………………………………….. 12 Figure 1.2: Schematic of a pullout test…………………………………………………..… 12 Figure 1.3: Components of the pullout test (Raju, 1995)……………………….………… 13  Figure 2.1: Horizontal stress distributions assumed for design………………….…..……..….. 51 Figure 2.2: Cross-sectional detail view for geosynthetic pullout setup……………….…….51 Figure 2.3: Bearing stress ratio as a function of friction angle of the soil …….…………... 52 Figure 2.4: Influence of particle size in bearing stress ratio ….................................……… 52 Figure 2.5: Predicted components of resistance to pullout force of geogrids A, B and C for geogrids lengths of 0.31 m …………………………………………………………….…... 53 Figure 2.6: Effect of transverse ribs on the pullout response of Conwed geogrid ………... 54 Figure 2.7: Pullout responses at low confining pressures……………………………..…... 54 Figure 2.8: Hyperbolic representation of stress-strain curve……………………………………….55 Figure 2.9: Predictions for pullout tests performed by Fannin and Raju (1993)………….. 55 Figure 2.10: Flow chart to obtain  cs,ps and  p,ps from triaxial and direct shear test data… 56 Figure 2.11: Maximum friction angle in triaxial and plane strain conditions on Chattahoochee sand Hussaini (1973)………………………………………………………………………. 57 Figure 2.12: Maximum friction angle in triaxial and plane strain conditions on Monterey sand Marachi et al. (1981)………………………………………………………………………. 57 Figure 2.13: Peak and large displacement friction angle in triaxial and plane strain conditions for Brasted sand (Cornforth 1964, 1973)…………………………………………………... 58 Figure 2.14: Lateral stresses at the front wall of the pullout box .……………………...…. 59 Figure 2.15: Influence of front wall roughness in the pullout resistance of sheets ………...59 Figure 2.16: Influence of front wall roughness in the pullout resistance of geogrids …….. 60 Figure 2.17: The effect of soil thickness on pullout response (Farragt et al., 1993)........…. 60 Figure 3.1: Grain size distribution of Badger sand ………………………………………... 86 Figure 3.2: Badger sand: sphericity and roundness ……………………………………..… 86 Figure 3.3: Influence of roundness on extreme void ratio…………………………………. 87 Figure 3.4: Direct simple shear tests on Badger sand……………………………………... 88 Figure 3.5: Direct shear tests on Badger sand………………………………………..…….. 89 xv  Figure 3.6: Triaxial test results from water pluviation……………………………………... 90 Figure 3.7: Dmin versus max for Badger sand……………………….……………………… 91 Figure 3.8: Triaxial test results from moist tamped …………………………………...…... 91 Figure 3.9: Unified relation between roundness, range of extreme void ratio and friction angles in uniform sands……………………………………………………………………. 92 Figure 3.10: Friction angles for Badger sand………………………………………………. 93 Figure 3.11: Critical state friction angle as a function of normal stress (from Lehane and Liu, 2013) …………………………………………………………...…………………………. 93 Figure 3.12: Stress-dependency of the critical state friction angle in the triaxial test (data from Boyle, 1995).…………………………………………………...…………………………. 94 Figure 3.13: Stress-dependency of the critical state friction angle of Skedsmo sand, in plane strain and triaxial test (data from Quinteros, 2014)..…………...…………………………. 94 Figure 3.14: Critical state friction angle in direct shear as a function of vertical stress (from Rousé, 2018)..…………...…………………………………………………...……………. 95 Figure 3.15: Difference of cv,ds between  = 4 kPa and  = 50 kPa as a function of roundness (from Rousé, 2018)……………………………………………...……………………...…. 95 Figure 3.16: Inclusions tested………………………………………………………..……. 96 Figure 3.17: Shear stress and normal stress relationship for various interfaces from direct shear tests ………………………………………………………………….…………….... 96 Figure 4.1: G in function of mean stress for Badger sand…………………………..……. 113 Figure 4.2: Calibration of Mohr-Coulomb to Badger sand……………………....……..… 113 Figure 4.3: Variation of Badger sand friction angle with applied stress …………………114 Figure 4.4: Numerical model using Mohr-Coulomb………………………………………114 Figure 4.5: Measured and simulated pullout response of a) APT, b) GMT and c) GMS…115 Figure 4.6: Measured and simulated horizontal stress on the front wall due to pullout, at peak pullout resistance, for the GMT sheet inclusion…………………..……………………….116 Figure 4.7: Measured and simulated pullout response of a) GGT, b) GGS, and c) GGM grids using lower (eq. 4.5) and upper (eq. 4.6) bounds for the bearing stress ratio………..…… 117 Figure 4.8: Comparison of measured and simulated pullout resistance using lower (eq. 4.5) and upper (eq. 4.6) bounds for the bearing stress ratio………………………….………… 118 Figure 4.9: Measured and simulated pullout response of a) GGT, b) GGS and c) GGM.. 119 xvi  Figure 4.10: Measured and simulated lateral stress on the front wall due to pullout, at peak pullout resistance, for the GGT, GGS and GGM geogrids……………………………..…120 Figure 4.11: Comparison of measured and simulated pullout resistance…………………121 Figure 4.12: Comparison of measured and simulated horizontal stress on the front wall due to pullout, at peak pullout resistance, of GMT and the three geogrids…………………….122 Figure 4.13: Variation of normalized horizontal stress ratio with depth for GMT………..123 Figure 4.14: Normalized friction angle in plane strain…………………………………….123  Figure 5.1: Determination of tc for Badger sand …………..…….…………………...… 153 Figure 5.2: Calibration of NorSand to Badger sand…………..……………..……..…...… 153 Figure 5.3: Determination of critical state parameters for Badger sand…………...….…..154 Figure 5.4: Independent check of NorSand parameters for Badger sand using triaxial tests on moist tamped samples……………………………………………………………………. 154 Figure 5.5: Variation of Badger sand critical state friction angle with applied stress……...155 Figure 5.6: Comparison of measured and simulated pullout response for APT08 using NorSand in combination with a beam element and an elastic continuum for the inclusion and springs for the interface …………………………………………………………………...155 Figure 5.7: Comparison of measured and simulated pullout response using NorSand glued to the beam element or attached to the elastic continuum for a) APT, b) GMT, c) GMS, at n = 8 kPa…………………………………………………………………………….……….. 156 Figure 5.8: Comparison of measured and simulated pullout response for GMS08 using the “NorSand interface layer” approach ……………………………………….……………..157 Figure 5.9: Pullout box and numerical model: a) schematic of the laboratory pullout box; b) FLAC numerical grid and boundary conditions; c) soil-front wall and soil-inclusion interface detail………………………………………………………………………………………158 Figure 5.10: Measured and simulated pullout results using a stress-dependent Mtc of a) APT; b) GMT; and c) GMS ……………………………...………………………………..…….159 Figure 5.11: Displacement of embedded end with displacement of clamped end for a) APT; b) GMT; and c) GMS……………………………………………………………………..160 Figure 5.12: Measured and simulated horizontal stresses at the front wall using a stress- dependent Mtc of a) APT; b) GMT; and c) GMS ……………………………..…………..161 xvii  Figure 5.13: Measured and simulated pullout response of a) GGT, b) GGM, and c) GGS grids using lower (eq. 5.5) and upper (eq. 5.6) bounds for the bearing stress ratio…...……..…. 162 Figure 5.14: Comparison of measured and simulated pullout resistance using lower (eq. 5.5) and upper (eq. 5.6) bounds for the bearing stress ratio…..…………………………...…… 163 Figure 5.15: Schematic of how each component of the pullout test is treated for numerical simulations……………………………………………………………………………..… 164 Figure 5.16: Measured and simulated pullout results using a constant Mtc of a) APT; b) GMT; c) GMS; d) GGT; e) GGM; and f) GGS……………………………………...…………… 165 Figure 5.17: Measured and simulated horizontal stresses at the front wall using a stress- dependent Mtc of a) APT; b) GMT; c) GMS; d) GGT; e) GGM; and f) GGS………….. 166 Figure 5.18: Comparison of measured and simulated values of a) pullout response and b) horizontal stresses, using Mtc = stress dependent and Mtc = 1.07…………….…………… 167 Figure 6.1: Definition of pullout…………………………………………………………..198 Figure 6.2: Definition of base case………………………………………………………..198 Figure 6.3: Tornado diagrams…………………………………………………………..…199 Figure 6.4: Influence of vertical stress on Pu at base case…………………………………200 Figure 6.5: Influence of tan on Pu………………………………………………………..201 Figure 6.6: Influence of ff on Pu………………………………………………………….202 Figure 6.7: Influence of ff , tan and n,u on Pu…………………………………......…....202 Figure 6.8: Influence of L on Pu………………………………………………………….. 203 Figure 6.9: Influence of E on Pu…………………………………………………..…...…..204 Figure 6.10: Influence of inclusion elongation on Pu at n,ASTM  = 12 kPa …..……………..204 Figure 6:11: Influence of Hb on Pu………………………………………………………...205 Figure 6.12: Ideal pullout test configuration………………………………………………205 Figure 6.13: Comparison of laboratory test data with Tornado simulations……………….206 Figure 6.14: Displacements distribution along the reinforcement (Farrag et al. 1993)…….206 Figure 6.15: Predicted and measured pullout load displacement relationships for geogrids A, B, C (Wilson-Fahmy et al., 1994)………………………………………………………....207 Figure 6.16: Pullout curves for LR = 1.13 m and LR = 0.4 m (Moraci and Recalcati, 2006)..208 Figure 6.17: Displacements measured along the specimen for LR = 0.4 m (Moraci and Recalcati, 2006)…………………………………………………………………………...209 xviii  Figure 6.18: Displacements measured along the specimen for LR = 1.15 m (Moraci and Recalcati, 2006)………………………………………………………………………….. 210 Figure 6.19: Displacement-time plots at different locations behind the front clamp: (a) displacements recorded at 500 mm behind front clamp; (b) displacements recorded at 1000 mm behind front clamp (Bathurst and Ezzein, 2015)……………………………………..211 Figure 6.20: Definition of region where ultimate pullout occurs………………………....212 Figure 6.21: Displacements of clamped versus embedded end (Raju, 1995)……………..212 Figure 6.22: Comparison of pullout experimental data with region where ultimate pullout occurs………………………………………………………………………...……..……. 213 Figure 6.23: Definition of ultimate pullout and initial pullout regions………………….. 213 Figure 6.24: Overestimation of pullout resistance due to box height……………………. 214 Figure 6.25: Influence of the front face roughness in the peak and large displacement apparent friction angle at the soil-inclusion interface from laboratory test data…………………….214 Figure A.1: Definition of state parameter () and infinity of NCL……………...……….. 250 Figure A.2: NorSand image condition on yield surface: a) very loose sand, b) very dense sand…………………………………………………………………………..……….….. 251 Figure A.3: Geometry, boundary and loading conditions for plane strain…………...….. 252 Figure A.4: Results of plane strain element………………………………………………. 252   xix  List of symbols and acronyms  A surface area of the embedded inclusion (m2) b intermediate effective principal stress factor B width of the inclusion (m) Be width along the edge of the inclusion influenced by restrained dilatancy effect (m) Bt  bearing element thickness (m) C reinforcement effective unit perimeter (C=2 for geogrids)  Cu  coefficient of uniformity Cc coefficient of curvature dc displacement of the clamped end of the inclusion (m) de displacement of the embedded end of the inclusion (m) D50 mean grain diameter (mm) DI degree of interference Dmin minimum dilatancy DR relative density e void ratio ec void ratio at the critical state emax  maximum void ratio emin   minimum void ratio E elastic modulus (kPa) fb pullout interaction coefficient fds coefficient of resistance to direct sliding F yield surface F* pullout resistance (or friction-bearing interaction) factor G elastic shear modulus (kPa) Gs specific gravity H plastic hardening modulus for loading for NorSand Hp soil height above the inclusion inside the pullout box (m) IC interaction coefficient kn  normal stiffness at interface (kPa/m) xx  ks  shear stiffness at interface (kPa/m) L  initial length of the inclusion (m) Le  effective/mobilized inclusion length (m) M stress ratio Mi stress ratio at image condition Mi,tc Mi at triaxial compression Mtc  stress ratio at critical state in triaxial compression  N volumetric coupling parameter for NorSand Nt number of transverse elements in a grid Np number of sand grains p mean effective stress (kPa) pi mean effective stress at image condition (kPa) P  pullout force per unit width (kN/m) Pi initial pullout (kN/m) Pm measured pullout resistance at large displacement (kN/m) Ps simulated pullout resistance at large displacement (kN/m) Pu ultimate pullout (kN/m) q deviatoric stress (kPa)  Q plastic potential surface ri radius of curvature of surface features (m) rmax radius of the largest inscribed circle (m) R roundness Rmin  radius of the smallest circumscribing sphere (m) Sb spacing between bearing elements (m) S sphericity (Sb/Bt)  spacing required just to give a fully rough case from bearing stresses                Tg  pullout load for a grid with N transverse members (kN/m) T1,g pullout load for a single isolated element (kN/m) 𝛼𝑏 fraction of the geogrid width available for bearing c  scale effect correction factor ds fraction of grid surface area that resists direct shear with soil xxi  𝛼𝑠  the fraction of solid surface area in a geogrid  altitude of the critical state line at 1 kPa   soil-inclusion interface apparent friction angle (°) cs,tx  soil-inclusion interface apparent friction angle in triaxial (°) cs,ps  soil-inclusion interface apparent friction angle in plane strain (°) cs,ds  soil-inclusion interface apparent friction angle in direct shear (°) ff interface friction angle between soil and front wall of the pullout box (°) s soil-solid area of geogrid interface friction angle (°) e difference of extreme void ratios h lateral stress on the front wall of the pullout box (kPa) hm measured lateral stress on the front wall of the box due to pullout, at large displacement (kPa) hs simulated lateral stress on the front wall of the box due to pullout, at large displacement (kPa) n increase in vertical stress at the soil-inclusion interface (kPa) •pq  plastic shear strain increment •pv  plastic volumetric strain increment max  maximum stress ratio  Lode angle (°)  slope of the critical state line s scalar   Poisson’s ratio   normal stress in direct or direct simple shear tests (kPa) 3  cell pressure in triaxial test (kPa) a  applied vertical stress at the soil level in the pullout test (kPa) s  vertical stress due to the soil above the inclusion in the pullout test (kPa) b  effective bearing stress on the grid (kPa) n  vertical stress at the soil-inclusion interface level in the pullout box (kPa) xxii  n,ASTM vertical stress at the soil-inclusion interface level in the pullout box as defined by  ASTM D6706-01 (kPa) n,u  vertical stress at the soil-inclusion interface level in the pullout box at ultimate pullout (kPa) av  average shear resistance acting along the soil-inclusion interface (kPa)  soil friction angle (°)  subscript p peak  cs constant volume or critical state tx triaxial ds direct shear ps plane strain dss direct simple shear MC maximum contraction rep repose tc relates minimum dilatancy to state parameter   state parameter i state parameter at image condition  angle of dilation (°) AASHTO  American Association of State Highway and Transportation Officials ASTM  American Society for Testing and Materials MSE  Mechanically Stabilized Earth   xxiii  Acknowledgements  This thesis has taken a long time, and it would not have been possible without incredible support from a number of sources. First, I thank my co-supervisors Dr. J. Fannin and Dr. M. Taiebat for their long-distance guidance, for challenging me and holding me always to the highest standards of research. Dr. J. Fannin’s supervision was on matters of laboratory pullout test data, and related material properties. Dr. M. Taiebat’s supervision was on the fundamental aspects of numerical methods, and his fruitful questions throughout the main program of research. I sincerely appreciate the ongoing constructive feedback that I have received from Dr. J. Howie and Dr. E. Eberhardt, members of my Supervisory Committee. I would also like to thank Dr. D. Shuttle for her insight to the fundamentals of NorSand during the initial stages of my Ph.D. research.  Aside from the numerical model developed in this study, I characterized the strength of Badger sand through different laboratory testing. I am particularly grateful to Mavi Sannin for her assistance with the triaxial equipment, Dr. D. Wijewickreme for facilitating the direct simple shear device and Antone Dabeet and Ainur Seidalinova for performing the direct simple shear tests on Badger sand.  I thank Universidad Diego Portales, Chile, for their institutional support as time to work on my Ph.D. while holding an appointment as an Assistant Professor. I am also thankful to Itasca Chile, and in particular to Patricio Gomez, for access to a FLAC license.  This research was initially funded by Natural Sciences and Engineering Research Council of Canada (NSERC). I am also extremely grateful for the support provided by Universidad Diego Portales as funding for travel expenses to Vancouver.   Finally, sincere thanks to my friends that have kept my spirits high during this period. Most heartfelt thanks to the four pillars of my life, my children Martina, Amelia and Vicente, and my mother. They gave me the strength to continue working on this dissertation when all my energy was depleted.   xxiv                 To Martina, Amelia and Vicente                      To my grandparents and Seba     1 Chapter 1 Introduction Soils have little to no tensile resistance. Accordingly, the inclusion of tension-resistant materials within the soil is an effective way of reinforcing them (McGown et al., 1985). Reinforced earth walls are widely used for the construction of vertical and near-vertical earth retaining structures. To ensure the internal stability of these geosynthetic-reinforced structures, the reinforcing element must extend beyond an assumed failure surface (typically a Rankine failure plane), in order to develop adequate pullout resistance (Figure 1.1). Current design methods are based on limit equilibrium and, as geotechnical design moves closer to a performance-based design, the use of ultimate limit state for predicting the behavior of these reinforced structures under operational conditions is questionable. However, the pullout test is still used to determine the soil-inclusion interaction parameters at displacements relevant to ultimate limit state and therefore, the condition of pullout is examined in this study.   In the pullout test, the inclusion is embedded in a body of soil inside the pullout box. While the inclusion is pulled out of the box, a pullout resistance develops at the soil-inclusion interface as shear stresses develop (see Figure 1.2). ASTM D6706-01 defines pullout as the “movement of a geosynthetic over its entire embedded length, with initial pullout occurring when the back of the specimen moves, and ultimate pullout occurring when the movement is uniform over the entire embedded length”. It suggests a minimum length of the inclusion tested of 0.6 m, a box height of 0.3 m, and vertical stresses up to 250 kPa at the soil-inclusion level. To minimize the influence of the rigid front wall of the box in the pullout resistance, and better represent the soil-inclusion interaction behind the failure surface in the reinforced earth structure, ASTM D6706-01 also suggests the use of an internal frontal sleeve in order to transfer the point of application of the pullout load behind the rigid front wall of the box into the soil.  2  Many pullout experimental studies report the pullout response of different inclusions by measuring the pullout resistance with displacement of the clamped end of the inclusion, horizontal stresses at the inside front wall of the box, and displacements or strains at different (and limited) locations of the inclusion. Some of these studies were performed before ASTM D6706-01 (see for example Palmeira and Milligan, 1989; Farrag et al., 1989; Raju, 1995) and others after ASTM D6706-01 (Sugimoto et al., 2001; Moraci and Recalcatti, 2006; Ezzein et al., 2014) and therefore, some inconsistency exists between the experimental studies and with ASTM D6706-01 that has not allowed for a uniform or unique interpretation of the pullout response of different inclusions. In addition, some of the shortcomings of pullout tests are that the soil and the inclusion are not visible during testing and that some aspects of the pullout behavior cannot be measured experimentally. Accordingly, numerical models are useful tools “to see” beyond the experimental test results.  Different authors have sought to study the different aspects that influence the pullout response of an inclusion, and hence the soil-inclusion interaction parameters, by means of numerical modeling. In numerical simulations of the pullout test, different constitutive models have been used to represent the behavior of the soil (linear elastic, Mohr Coulomb, hyperbolic, Ducker Prager, and Lade among others), the inclusion (linear elastic, hyperbolic relation, non-linear force strain relation, and rigid-perfectly plastic, among others), and the soil-inclusion interface (elastic, hyperbolic, joint elements, and hierarchical single surface, among others) as described in Yuan and Chua (1990), Yogarajah and Yeo (1994), Bergado and Chai (1994), Abramento and Whittle (1995), Sohbi and Wu (1996), Gurung and Iwao (1999), Gurung et al. (1999), Perkins and Cuelho (1999), Perkins and Edens (2003), Ling (2005), Aggarwal et al. (2008), Huang and Bathurst (2009).   3  The results of these numerical models in combination with the laboratory pullout test data, have helped to improve our understanding of the soil-inclusion interface behavior for sheet and geogrid inclusions. However, there still exist some “knowledge gaps” related to the stress regime inside the pullout box, the influence of the low stresses at which the pullout tests are performed and the influence of the boundary conditions of the box in the pullout response, that need to be addressed, in order to gain confidence in the numerical simulations and laboratory test results.   The driver for contributions of the current study is established in the Ph.D. thesis of Raju (1995). In his research, Raju (1995) developed a large-scale pullout apparatus at The University of British Columbia, that was 1.3 m long, 0.64 m wide and 0.6 m deep. The inclusions were approximately 1 m long and 0.5 m wide to provide a clearance of 7.5 cm between the inclusion and the side wall of the box. The upper boundary was composed of an acrylic bag that works as a flexible boundary, and the base plate was made of 13 mm thick aluminum plate that provides a rigid lower boundary. The study predated the ASTM D6706-01 and therefore, no sleeve was used to minimize front wall effects in the pullout response, and the inclusion was clamped outside the pullout box (see Figure 1.3). In consequence, the numerical simulations of this study will not consider a sleeve at the front boundary of the box. A uniformly-graded coarse sand, named Badger sand, was used for the tests, along with six different types of inclusions: a fully rough aluminum sheet, a textured geomembrane, a smooth geomembrane and three geogrids. The pullout tests were performed at low normal stresses at the inclusion level, in the range of 4 to 30 kPa. Independent measurements of pullout force per unit width, displacement of the clamped and embedded ends, and horizontal stresses on the front wall of the box were taken. This allowed for the “perfect dataset” to be available for numerical modeling purposes.   4  The first aspect with room for improvement in the numerical modeling of pullout tests is the characterization of the different constituents of the experiment and, in particular, the soil strength and the soil-inclusion interface friction angle. Reports of Palmeira and Milligan (1989) and Moraci and Montanelli (2000) have shown that the calculated value of the soil-inclusion interaction factor is largely influenced by the chosen value of soil friction angle. From examination of direct shear test data, and a companion theoretical analysis, Jewell and Wroth (1987) postulated the angle of friction obtained from a conventional interpretation of the shear box test underestimates the mobilized friction angle of sand by about 20%, yielding a hidden factor of safety of the order of 1.2 in design based on either the peak strength or a critical state strength. From laboratory testing in a plane strain unit cell device, Boyle (1995) advocated that plane strain soil properties be used for design of reinforced soil structures, noting the higher friction angle yields a smaller load per unit width in the reinforcement, a finding that is supported by Allen and Bathurst (2002), and acknowledged in BS 8006 (BSI, 2010). Moreover, using friction angles obtained from the direct shear and the triaxial test, Bathurst et al. (2002) suggest, from back-analysis of several geosynthetic reinforced soil structures, that design practice is overly-conservative.   An additional aspect to consider when referring to the magnitude of the value of friction angle to use for the numerical modeling and back-analysis of pullout tests, is the effect of stress level. It is usually accepted that the friction angle of a granular soil is stress-dependent. Similar observations have been made by different researchers of the pullout test regarding the soil-inclusion interaction factor. For example, Juran et al. (1988) showed that the soil-inclusion interaction factor increased with decreasing normal stress level. The authors suggest that this is due to the restrained dilatancy phenomenon occurring at the soil-inclusion interface and that this dilation is highly dependent on the stress level. Farrag et al. (1993) showed that the soil-inclusion interaction factor decreases with increasing vertical stress when determined by either the pullout test or the direct shear test. These results are consistent with the study of Moraci and Recalcati (2006) who also observed a stress-dependent behavior of the soil-inclusion interaction factor for stresses between 10 and 100 kPa. Huang and Bathurst (2009) showed through a statistical analysis, that the interaction factor decreases with increasing stress level up to stresses around 40 kPa and then remains 5  practically constant. By assuming that the compacted unit weight of the soil is 17.5 kN/m3, they suggest that the soil-inclusion interaction factor varies linearly with depth below the top of a reinforced wall at an equivalent depth of 2.3 m and remains constant thereafter.  As a consequence of the boundary conditions of the pullout test device and the relatively low magnitude of effective stress at which the test is performed, it is reasonable to believe that the plane strain condition, and a stress-dependency of mobilized frictional resistance, both prevail in the laboratory pullout test. However, several of the numerical studies found in the literature invoke a constant value of friction angle, determined from either the shear box or the triaxial test, without adjustment for the plane strain condition (e.g. Pal and Wathugala, 1999; Wang and Richwien, 2002; Sugimoto and Alagiyawanna, 2003; Perkins and Edens, 2003; Moraci and Gioffrè, 2006 and Bobet et al., 2006, amongst others).   Another aspect of the numerical modeling of pullout that needs to be improved is the capture of the complete pullout response of different inclusions. Besides the soil and soil-inclusion interface strength characterization, results of the numerical analyses found in the literature, depend on various factors such as type of finite element and constitutive relations to describe the soil, inclusion and soil-inclusion interface behavior. For example, in most of the finite element models, the soil is represented by a continuum mesh, the inclusion by a flexible beam element and the soil-inclusion interface by springs defined by a Mohr-Coulomb type of failure. Several of these numerical analyses have been able to capture the initial stiffness of the pullout response and the pullout resistance of different inclusions when there is a gradual increase in pullout resistance and no strain softening is observed. However, very few studies have been able to capture the complete pullout resistance from small to large displacements that includes the strain softening behavior found in some of the laboratory pullout test data, such as the data obtained by Raju (1995) for the rough aluminum sheet and the smooth geomembrane. Typical finite element analyses also consider the geogrids as sheet inclusions and, given the complexity of the three-dimensional structure of geogrids, parameters for the soil, inclusion and soil-inclusion interface used as input for these constitutive relations do not always have geotechnical or physical meaning, and adjustments to these parameters are made without clear explanation, 6  reaching sometimes unrealistic values in order to match the observed pullout response. For example, Jewell et al. (1985) suggest that the soil-inclusion interaction of geogrids can be captured through a soil-geogrid interaction factor that includes the geometry and surface roughness of the inclusion, which are in general known parameters, and a bearing stress ratio (as a function of the friction angle of the soil) that has a large range of values. As long as the shortcomings and limitations of each approach are known and understood, the results of these models have proved to be very useful for the back-analysis of pullout tests once the numerical model has been calibrated against laboratory pullout test results. However, when more phenomenological studies are required in order to have a better description of the soil-inclusion interaction, a good estimation of input parameters to the numerical model, with more geotechnical or physical meaning, is needed, with the strength of the soil being the most critical factor.  The final aspect that needs improvement, is the understanding of the influence that the different constituents of the laboratory pullout test exert in the pullout resistance of different inclusions, and hence, in the soil-inclusion interaction factor. For example, the study of Moraci and Recalcatti (2006) shows that the pullout response of grids depends on the inclusion length, the soil-inclusion interface (apparent) friction angle and the vertical stress applied. The roughness of the inside front wall has also shown to influence the pullout response of sheet and geogrid inclusions. The data of Palmeira (1987) reveals that for a metallic grid embedded in Leighton Buzzard sand, the soil-inclusion interaction factor increases with the roughness of the front wall. This observation agrees with the results of the laboratory testing of Raju (1995) for the two geomembranes and a grid, where the inside front wall of the pullout box was varied between an aluminum and an arborite surface. In order to decrease the influence of the front wall on the pullout resistance, ASTM D6706-01 suggests that “the box shall be fitted with a metal sleeve at the entrance of the box to transfer the force into the soil to a sufficient horizontal distance so as to significantly reduce the stress on the door of the box”. Another important aspect to consider is the size of the pullout box. ASTM D6706-01 suggests a minimum box height of 0.3 m. Laboratory test studies of Farrag et al. (1993) along with the numerical simulations of Dias (2003) suggest that the height of the soil sample above the inclusion will affect the pullout resistance. Their 7  results show that as the height of the box increases the pullout resistance decreases (all the other variables held constant) and that a minimum height of 0.6 m for the pullout box is to be used in order to minimize boundary effects. In his very comprehensive study, Palmeira (2009) concludes that “boundary conditions may influence test results”. However, limited evidence is found in the literature to further expand on this conclusion. On one side, it is an immense task to build boxes of different sizes with different front wall frictions to determine the influence of these conditions to the pullout resistance of different inclusions. On the other hand, few phenomenological numerical models are available in the literature that allow for a comprehensive sensitivity analysis of the major components that influence the pullout resistance.  1.1. Objectives and contributions   There is a need for an improved science-based understanding of the factors influencing the pullout resistance of different inclusions. This thesis aims to address the “knowledge gaps” that govern the soil-inclusion interaction under pullout conditions previously described using the existent database on material properties and laboratory pullout tests performed by Raju (1995) at The University of British Columbia (UBC), along with the results of numerical simulations of these tests. Accordingly, four objectives are identified:  1. Obtain a proper characterization of the materials used in the laboratory pullout tests performed by Raju (1995), namely the sand (Bagder sand), the inclusions and the soil-inclusion interface, giving consideration to:  - The strength characteristics of Badger sand in triaxial, direct shear and plane strain; and,  - Gain an understanding of the stress-dependent strength of granular soils at the low effective stresses used in pullout testing (typically lower than 50 kPa);  8   2. Put forward evidence to demonstrate that a stress-dependency of the soil-inclusion interface friction angle and plane strain conditions prevail in pullout testing, by comparison of measured and simulated values of pullout resistance, and lateral stresses at the inside front wall of the pullout box;  3. Examine different configurations of soil, inclusion and soil-inclusion interface options to propose a numerical model that can capture the complete pullout resistance of the different inclusions tested by Raju (1995), with emphasis on: - The use of a phenomenological constitutive model that allows for dilatancy and corrects for plane strain conditions;  - Capture the strain softening pullout response observed in some of the laboratory pullout tests on the sheet inclusions; - Test the validity of the Jewell et al. (1985) approach to represent the three- dimensional aspects of the geogrids as a sheet inclusion, giving special consideration to the choice on the friction angle used to calculate the bearing stress ratio; and,  4. Perform a numerical parametric study, to analyze the adequacy of the laboratory test data available in the literature and the ASTM D6706-01 pullout recommendations, identify possible weaknesses in the determination of the soil-inclusion interaction factor, and suggest an “ideal” pullout test that will ensure that ultimate pullout occurs.  9  1.2. Structure of the thesis   Thesis objectives are addressed in seven chapters, detailed as follows:  Chapter 1 includes a general introduction and motivation to the dissertation, and the four main objectives of the PhD research.  Chapter 2 summarizes the literature relevant to this research with emphasis on the pullout response of sheet and geogrid inclusions. The basic theories of pullout suggested by researchers are introduced and a number of laboratory pullout tests are also reported; specifically, a study of several factors that may affect pullout capacity is presented. A summary of the state of the art of numerical modelling of pullout followed by significant contributions to understanding the mobilization of sand strength and selection of parameters for numerical modelling is also presented. Finally, a summary is given of the research needs and intended novel contributions on this thesis that emerge from the literature review.  Chapter 3 describes the properties of the different constituents of the laboratory pullout tests performed by Raju (1995) at The University of British Columbia (UBC). Having the insight of the stress-dependency of the critical state friction angle in direct shear, for Badger sand at stresses lower than 150 kPa from Raju (1995), additional laboratory tests (direct shear, triaxial and direct simple shear) are done to characterize the strength of the sand. An analysis of the influence of particle shape on the friction angle of uniform sands is done to confirm a proper characterization of the strength properties of Badger sand, and a unified plot is proposed that relates sand strength and void ratio with the roundness of the sand particles. The stress-dependency of the critical state friction angle in direct shear is also shown and discussed for other sands in relation to the roundness of the particles. A description of the characteristics of the inclusions and the pullout box used by Raju (1995) is finally presented.  10  In Chapter 4, informed by the findings of the strength characterization in Chapter 3, deduced values of critical state friction angle in plane strain, also stress-dependent, are obtained and are used in the numerical modeling of the pullout response at large displacement of the three sheet inclusions tested by Raju (1995). The commercially available numerical software FLAC6.0 is used for the simulations. Badger sand is represented by the Mohr-Coulomb constitutive model, the inclusion by a structural beam element and the soil-inclusion interface using spring elements defined by a Mohr-Coulomb type of failure. A good match between laboratory tests and numerical model results is found at large displacement (for both, pullout response and horizontal stresses at the front wall of the box). Stress-dependent, plane strain parameters are used to calculate an equivalent soil-inclusion friction angle, using the Jewell et al. (1985) approach, for the characterization of the geogrids as sheet inclusions. A reasonably good agreement between measured and simulated values of pullout resistance at large displacement and horizontal stresses at the front wall of the box is also observed for the three geogrids.  In Chapter 5, in order to move beyond large displacement behavior, an improvement to the numerical model presented in Chapter 4 that allows for the capture of the complete pullout response of the inclusions tested by Raju (1995), is needed. The NorSand constitutive model is used to represent the soil and its calibration is based on the triaxial tests on Badger sand. Different approaches to represent the inclusion and the soil-inclusion interface are explored, while keeping constant the NorSand constitutive model to simulate the sand response, to define a model able to capture the complete pullout response of the sheet inclusions while using physically based parameters as input to the model. The chosen model uses a NorSand mesh to represent the behavior of Badger sand inside the pullout box, an elastic continuum to represent the inclusion and a continuum layer also with NorSand parameters for the soil-inclusion interface. This approach allows for the capture of two independent measurements of the laboratory pullout tests of the three sheet inclusions; i) the pullout response including the strain softening behavior found in some of the tests; and ii) the horizontal stresses at the front wall of the pullout box. The Jewell et al. (1985) approach for the calculation of the soil-geogrid interface friction angle is also used to verify 11  its potential to capture the complete pullout response of the different geogrids tested by Raju (1995).  Having a model able to capture the full pullout response of sheet inclusions, a parametric study is undertaken in Chapter 6, to gain insight to pullout testing. Different parameters related to the inclusion, the soil-inclusion interface, and the pullout box are varied to study their influence on the onset of pullout. The results of the parametric study are examined to assess the laboratory test data in the literature and ASTM D6706-01 recommendations. The results of this Chapter show that the current scientific literature and ASTM D6706-01 underestimate the magnitude of the soil-inclusion interaction factor and that there is a combination of factors for which pullout (as defined by ultimate pullout in ASTM D6706-01) does not occur, and the measured pullout response is an in-air stress-strain curve or a combination of in-air extension with elongation of the inclusion.  Chapter 7 presents a summary of the research findings, the novel contributions of this study and recommendations for future work. 12              Figure 1.1: Reinforced earth structure              Figure 1.2: Schematic of a pullout test   Length as a function of the soil-inclusion interaction factor Inclusion Potential failure surface Clamp Hydraulic actuator a) Surcharge a L  Sleeve n  P Hb  Lb  Inclusion Front wall 13              Figure 1.3: Components of the pullout test (Raju, 1995)               Surcharge (a) Clamp Hydraulic actuator Initial vertical stress (n,ASTM)  14  Chapter 2 Literature review  2.1. Introduction  Mechanically Stabilized Earth (MSE) is a technique where inclusions are embedded in a soil mass to increase its strength and stiffness. The inclusions may be of two basic types; inextensible materials such as steel strips and bars, and extensible materials such as geotextiles, geogrids, straps and other polymeric elements. The vast majority of MSE structures have been constructed using metallic reinforcements. Since the 1970’s, the use of polymeric materials in earth reinforced structures has increased due to their flexibility and lower cost. The use of geosynthetics, instead of metallic reinforcements with concrete facing panels, has proved to be 30 to 50% less expensive (Allen and Holtz, 1991).   As for unreinforced slopes and walls, the design of MSE structures has to be checked for internal and external stability. External stability is examined by limit equilibrium analysis - in the same way as conventional retaining structures namely, overturning, sliding and global failure - assuming that the soil-reinforcement composite acts as a mass. For internal stability check, a failure surface is assumed that determines a horizontal stress distribution (see Fig. 2.1) and the reinforcement length beyond that surface has to be verified to resist the pulling forces (see Table 2.1).   In order to have a better understanding of the behavior and performance of MSE structures, some of them have been monitored to obtain data about strains, loads and displacements that occurred in the reinforced layers. Bathurst et al. (2002) presented a study of 20 well-documented geosynthetic walls, covering a wide variety of geometries, heights, surcharges, facing type, reinforcement type and spacing. They showed that actual walls that exhibited good performance contain between one half and one eighth of the reinforcement typically required by the American Association of State Highway and Transportation Officials (AASHTO) specifications. On the other hand, walls that exhibited poor performance contained one tenth to one twelfth of the reinforcement typically required. So far, pullout failure has not been observed explicitly in an actual wall. Koerner and Koerner (2018)  15 reported a database of 320 failed walls of which 22% were attributed to internal instability with wide reinforcement spacings and short reinforcement lengths being the most common (however no pullout failure is reported). Holtz (2017) mentions that at a certain depth of burial of the reinforcement, it is more likely that the reinforcement will fail in tension than pullout due to the large overburden pressures. Hence, pullout will likely only occur in the very upper layers of the reinforcement structure.   As reinforced walls become taller, have more complex geometries, poorer backfill soil and are used in more challenging situations, such as seismic regions, one can place several longer layers to ensure the internal and external stability of the structure. As a result, this structure will be overdesigned, with more reinforcement than needed, more backfill soil, and hence more expensive. It is therefore important to better understand the principal factors that influence the pullout resistance of the soil-inclusion interface to ensure that these structures remain stable, while being economical.  This Chapter summarizes the state of knowledge on pullout tests preceding the current study. Section 2.2. has relevant definitions and descriptions of the soil-inclusion interaction for both sheet and grid inclusions, necessary for understanding the related work presented herein. The scope and limitations of past research are then summarized concerning the numerical modeling of pullout tests (Section 2.3.) and the laboratory pullout tests (Section 2.4.). Finally, the role and intended novel contributions of the present study are described in Section 2.5.  2.2. Soil-inclusion interaction  The soil-inclusion interaction is generally characterized by the apparent friction angle of the soil-inclusion interface and is given by (Jewell et al., 1985).   𝑃 = 2𝐿𝜎𝑛𝑡𝑎𝑛𝛿                   (eq. 2.1)      16 Where:   P: pullout force per unit width (kN/m)  L: initial length of the inclusion (m) n: vertical stress at the soil-inclusion interface level in the pullout box (kPa) : soil-inclusion interface apparent friction angle  Different tests can be used to study the soil-inclusion compound behavior, direct shear and pullout tests being the most commonly used to determine the interaction factor.   2.2.1. Direct shear test  The first soil-geosynthetic direct shear test standard was BS 6906: 1991, followed by ASTM D 5321-92. In this test, a soil mass is forced to slide along the inclusion surface, using a constant rate of displacement while a constant load is applied normal to the inclusion surface. The test is repeated for different normal stresses and the shear stress is recorded. Shear versus normal stresses are used to determine the Mohr-Coulomb envelope and the cohesion and friction angle of the soil-inclusion interface can be determined. The standard test method suggested that both square and rectangular shear boxes could be used. These boxes should have a minimum dimension greater than 300 mm and the shear force is normally applied at a rate of 1.0 mm/min.   2.2.2. Pullout test  In the pullout test, the inclusion is embedded in a body of soil inside the pullout box, where a vertical stress is applied at the top of the soil sample and the inclusion is subjected to a horizontal displacement. The force required to pull the inclusion out of the soil mass is recorded and a pullout resistance per unit width is specified at a particular condition of displacement. This test is intended to be a performance test to be conducted as closely as possible to represent design or construction conditions and is suitable for all inclusions and soils. ASTM D 6706-01 specifies that the box should be square or rectangular with minimum dimensions 610 mm long by 460 mm wide by 305 mm deep, if sidewall friction is minimized,  17 otherwise a minimum width of 760 mm is required (see Figure 2.2). Before the pullout test was standardized by ASTM D 6706-01, several experimentalists suggested that the front wall of the apparatus exerted a large influence on the measured pullout force (see for example, Johnston and Romstad, 1989; Palmeira and Milligan, 1989; Farrag et al., 1993; Sugimoto et al., 2001). To minimize front wall effects, the Standard specifies that “the box shall be fitted with a metal sleeve at the entrance of the box to transfer the force into the soil to a sufficient horizontal distance so as to significantly reduce the stress on the door of the box." A minimum inclusion embedded length of 610 mm beyond the frontal metal sleeve is required. ASTM D 6706-01 also specifies that the normal stress applied at the soil-inclusion interface level considers the normal stress due to the soil above the inclusion and the stress applied on top of the soil sample. Its magnitude depends on testing requirements, “however, stresses up to 250 kPa should be anticipated."   The principal difference between the direct shear and the pullout test is that in the first, there is no extension of the inclusion and hence, the peak frictional resistance is independent of the extensibility (or stiffness) and thickness of the inclusion (Mallick and Zhai, 1996).   The pullout response at failure of an inclusion embedded in a mass of soil depends on the properties of the geosynthetic material, the soil-inclusion interface apparent friction angle and the effective vertical stress applied. An early expression to describe pullout resistance was introduced by Jewell et al., (1985) and is given by eq. 2.1. Subsequently, additional expressions given by equations 2.2a, 2.2b and 2.2c, were proposed.   𝑃 = 2𝐿𝜎𝑛𝑓𝑏𝑡𝑎𝑛𝜙      (eq. 2.2a)   𝑃 = 2𝐿𝜎𝑛𝜇𝑆 𝐺𝑆𝑌⁄       (eq. 2.2b)   𝑃 = 2𝐿𝜎𝑛𝛼𝐹∗       (eq. 2.2c)     18 Where:   fb: the soil–inclusion pullout interaction coefficient  𝜙: the soil friction angle  𝜇𝑆 𝐺𝑆𝑌⁄ : the soil–inclusion interface apparent coefficient of friction  α: the scale effect correction factor account for a non-linear stress reduction over the embedded length of highly extensible inclusions (FHWA, 2009).  F*: the pullout resistance factor  In all these expressions, the pullout force appears as a function of the initial length of the inclusion, the vertical stress applied at the soil-inclusion interface level (as defined by ASTM D 6706-01) and the soil-inclusion interface friction angle. For inextensible inclusions, equation 2.3 is a valid way of assessing the soil-inclusion interaction factor or apparent coefficient of friction.  𝜏𝑎𝑣 𝜎𝑛⁄ = 𝑡𝑎𝑛𝛿 =  𝑓𝑏𝑡𝑎𝑛∅ =  𝜇𝑆 𝐺𝑆𝑌⁄           (eq. 2.3)  Where:  av: average shear resistance acting along the soil-inclusion interface (kPa)   As opposed to the direct shear tests, during pullout, extensible geosynthetic inclusions will elongate as pullout displacement progresses, and the friction along the soil-inclusion interface will develop progressively, with the front end of the inclusion attaining relatively large strains while the embedded end may not even feel the presence of the pullout effect. The overall pullout response is hence a reflection of the combined behavior of the properties of the inclusion material, and the soil-inclusion interface characteristics. Furthermore, the interface interaction between soil and geogrid is even more complex due to the existence of the transverse members that mobilize a passive resistance when the geogrid is pulled.      19 2.2.2.1. Sheet inclusions  The most common expression to estimate the pullout resistance per unit width of a sheet inclusion embedded in a granular material is given by (Jewell et al., 1985):   𝑃 = 2𝐿𝜎𝑛𝑡𝑎𝑛𝛿                   (eq. 2.4)  It transpires therefore, that the interaction of soil and sheet inclusions is rather simple and is mainly controlled by surface roughness between the soil and the inclusion. Contrary to direct shear, in the pullout test, the length and extensibility of the reinforcement have an influence on the pullout resistance. Larger extensible inclusions in a body of soil, have a more extensible behavior and hence, a progressive development of shear stresses at the interface level. Moraci and Recalcati (2006) showed that shorter inclusions result in larger peak pullout resistance and that extensibility effects were more evident in long inclusions and in high vertical stress resulting in a lack of peak pullout resistance.  2.2.2.2. Geogrids  Geogrids, mostly used in MSE structures as soil reinforcement, are geosynthetic materials that have an open grid-like appearance and both, the longitudinal and transverse elements contribute to the soil-inclusion interface strength. The interface friction or interaction factor between soil and geogrid in pullout is complex and comprises two main mechanisms; shearing between the soil and the solid area of the inclusion, and soil bearing against the transverse elements. The shearing interaction between soil and the solid area of the geogrid is similar to the surface friction that is measured conventionally for soils and sheet inclusions (eq. 2.4, Jewell et al., 1985) and only needs a small relative displacement to be fully mobilized (Bergado and Chai, 1994).   As the pullout test takes place and the inclusion moves horizontally, the soil particles inside the grid apertures move towards the transverse elements of the geogrid and a bearing resistance against the transverse elements develops. As the inclusion is pulled out of the soil  20 mass, the zone of soil surrounding the inclusion tends to dilate. However, the volume change is restrained by the surrounding non-dilating soil, inducing an increase in vertical stress at the soil-inclusion interface level. Additionally, for extensible geogrids, elongation of the longitudinal elements may occur during pullout, and hence, the shear distribution at each transversal element of the geogrid varies and it increases as it gets closer to the front wall of the pullout box (Bergado and Chai, 1994).  Jewell et al. (1985) proposed eq. 2.5 (or eq. 2.2a) for the estimation of the pullout resistance of grids considering the frictional and bearing components of the interface interaction, through the parameter fb:  𝑃 = 2𝐿𝜎𝑛𝑓𝑏𝑡𝑎𝑛∅                  (eq. 2.5)  Where fb is given by:           tan21tantan bbtnbssbSBf +=                  (eq. 2.6)  With:  s: the fraction of solid surface area in a geogrid s: soil-solid area of geogrid interface friction angle (°) Sb: spacing between bearing elements Bt: bearing element thickness  b/n: bearing stress ratio 𝛼𝑏: fraction of the geogrid width available for bearing  The first part of eq. 2.6 corresponds to the shearing component and the second to the bearing component of the soil-geogrid interaction. Generally, these two components are assumed independent of each other when, in reality, they interact to an extent not yet well understood or quantified (Cardile et al., 2016).    21 Inspection of eq. 2.6 shows that all the parameters are soil properties or geogrid geometry (that are in general available), except for the bearing stress ratio b/n. Jewell et al. (1985) proposed a lower boundary for b/n as a function of the friction angle of the soil by assuming a punching failure mode in the soil, given by equation 2.7. An upper estimate of b/n is found by considering the conventional stress characteristics of a footing rotated to the horizontal and considering a horizontal boundary stress in the soil equal to the vertical stress applied, given by equation 2.8 (see Figure 2.3).   (𝜎𝑏𝜎𝑛)= 𝑒(90+)𝑡𝑎𝑛tan (45 +  2⁄ )                (eq. 2.7)   (𝜎𝑏𝜎𝑛)= 𝑒𝑡𝑎𝑛tan2 (45 +  2⁄ )     (eq. 2.8)  Additionally, Palmeira and Milligan (1989) suggested that the relative size of the soil particles to the bearing members also influences the pullout resistance of geogrids. Results of pullout tests on geogrids with different transverse element sections show that the normalized bearing strength (b/ntan) decreases with increasing ratio of bearing element thickness (Bt) and the mean grain diameter, D50, of the soil and stabilizes for values of Bt/D50 larger than 10 (Figure 2.4). Also, the shape of the transverse elements shows an effect on the normalized bearing strength; square or rectangular elements result in a slightly larger value of b/ntan than circular transverse elements.  Subsequently, Alfaro et al. (1995) proposed an expression that contains the 2-D interaction mechanism developing over the middle section of the geogrid and the 3-D interaction mechanism developing at the transverse elements of the geogrid, given by:  DD PPP 32 +=                                (eq. 2.9)  tgLBtgBLP neesne += 42                (eq. 2.10)     22 Where: Le: effective/mobilized inclusion length  Be: width along the edge of the inclusion influenced by restrained dilatancy effect  n: increase in vertical stress at the soil-inclusion interface   Holtz et al. (1997) proposed a definition of the ultimate pullout resistance of the inclusion per unit width given by: LCFP nc*=                 (eq. 2.11)    Where:  C: inclusion effective unit perimeter (C = 2 for geogrids)  F*: pullout resistance (or friction-bearing interaction) factor c: scale effect correction factor  In this equation, F* and c are back-fitted to pullout tests using the specific backfill soil to be used in the project. Huang and Bathurst (2009) declare that in practice, no independent back-fitting to isolate F* and c has been attempted. Rather, F*c has been considered as a single value for each test. In the FHWA (2009) default values of c of 0.8 and 0.6 are suggested for geogrids and geotextiles, respectively, and F* = 2/3 tan.  More elaborate relations incorporate the extensibility of the longitudinal members of geogrids which translates in integrating to the model the non-linearity of the shear stress distribution. Sieria et al. (2009) describe an analytic model able to reproduce the stress transfer mechanism based on a rheological approximation suggested by Beech (1987) and Costalogna and Kuwajima (1995). This model considers the passive resistance generated in the transverse elements and the elongation of the longitudinal elements when the geogrid moves.     23 Some studies have focused on the relative magnitude of the frictional and bearing effect in the pullout resistance of grids. Figure 2.5 shows data from Wilson-Fahmy et al. (1994) where the influence of the different components on the pullout response of three different grids is separated. In all the cases, the principal effect on the pullout resistance at low mobilized load is due to friction of the soil along the longitudinal elements. This effect decreases as the pullout displacement increases. On the contrary, the effect of the bearing elements is a large displacement phenomenon; at small displacements, the passive resistance due to transverse elements is low and increases with pullout displacement reaching between 50% and 70% of the total pullout force at large displacements. Similarly, the data of Farrag et al. (1993) in Figure 2.6 show the influence of the transverse elements in the pullout resistance of a Conwed geogrid and reveals that the frictional resistance of longitudinal elements represents around 75% of the total pullout resistance and that the passive resistance of the transversal elements starts to develop at larger displacements. For displacements less than 5 mm, more than 80% - taking values up to 92% - of the pullout resistance is taken by the friction of longitudinal elements and 5% to 10% is taken by as friction by transverse ribs (see Wilson-Fahmy et al., 1994). A recent study by Bathurst and Ezzein (2016) on transparent granular soil shows that after about 3 mm of displacement of the clamped end of the inclusion, approximately 20% of the pullout load is taken by the transverse elements.   Interference between transverse elements is another factor influencing grid behavior. When a geogrid is pulled out of the soil mass two mechanisms appear at the transverse elements level: an increase in stress due to the passive resistance against the transverse element and an active zone behind the transverse component. If the transverse elements are closely spaced, the following members can enter this zone as pullout progresses affecting hence the pullout response of the geogrid. For example, the photo-elastic studies of Milligan et al. (1990), show that the interference effect reduced the friction between the soil and the inclusion. As the distance between the geogrid transverse elements increases, the interference between these members decreases and, in an extreme case, they will behave as a series of isolated transverse elements being pulled out of the soil mass.    24 Palmeira (2004) proposed a theoretical model for the back-analysis of the pullout behavior of geogrids, which considers the frictional component as a Mohr-coulomb criterion and the mobilized bearing force as a curve relating bearing resistance of the transverse element with element displacement. The effect of the transverse elements in a geogrid was considered as a degradation of bearing resistance due to interference between transverse members. He concluded that interference effect can reduce, by up to 22%, the bearing force in a geogrid compared to a single member of the inclusion. Observations of Palmeira (2004) of metal geogrids embedded in dense sand show that, for ratios of transverse elements spacing to transverse elements thickness of above 40, the grid transverse members behaved in isolation, under the experimental conditions adopted.   As previously described, the soil-inclusion interaction is a complex phenomenon, and various researchers have studied it by conducting pullout testing along with theoretical analysis and numerical simulation. The following Section summarizes the principal types of numerical models found in the literature and the different solutions adopted to describe the behavior of the three primary components of the pullout test, namely the soil, the inclusion and the soil-inclusion interface.  2.3. Numerical modeling of pullout tests  To have a better understanding of the soil-inclusion interface behavior, several researchers have sought to model the soil-inclusion interaction in the pullout test using different numerical models that allow the prediction of displacements, strains, and forces generated in the reinforcement during deformation and failure. However, the results of these analyses depend on various factors, such as types of finite elements, constitutive relations to describe the soil, inclusion and interface behavior and input parameters used for the constitutive relations. A summary of the principal analytical and numerical models found in the literature is presented below.     25 2.3.1. Analytical models  Most analytical or closed-form models for the soil-inclusion interface behavior use a linear or non-linear elastic model where the onset of sliding is defined by a Mohr-Coulomb criterion. For example, Abramento and Whittle (1995) used a shear-lag analysis for sheet inclusions where the shear movement of the soil was allowed. The proposed analysis assumed the soil and the inclusion to be linear isotropic elastic materials and the soil-inclusion interface was characterized by a constant friction angle. The analytical model was able to capture the results of pullout tests on steel and nylon sheet inclusions in dry, dense Ticino sand at vertical stresses between 30 and 98 kPa, especially at small displacement levels. One of the advantages of the proposed formulation is its simplicity, which allows an interpretation of the variation of the parameters of the model. They identified three principal parameters affecting the load-elongation response during a pullout test. They showed that higher soil-inclusion interface friction resulted in more significant inclusion elongations and larger pullout loads. Similarly, the results suggest that longer inclusions show larger elongations and result in greater pullout loads. The relative stiffness between the soil and the reinforcement also showed an influence in the load-elongation response but less important than the previously described parameters.  For the non-linear behavior, a hyperbolic relation is generally used to relate stresses and strains. Gurung and Iwao (1999) and Gurung et al. (1999) proposed a model that considers extensible sheet inclusions and incorporated a hyperbolic shear stress-displacement relationship for the soil-inclusion interface response calibrated from direct shear tests and characterized by an interface shear stiffness. They successfully captured the pullout test results of Raju (1995) for the textured geomembrane at vertical stresses of 4, 8 and 12 kPa to a displacement up to 20 cm (see Figure 2.8). They were also able to capture the results of Palmeira and Milligan (1996), at vertical stresses of 25, 50, 100, 150 kPa for displacements up to 25 cm. Similarly, Weerasekara and Wijewickreme (2010) proposed an analytical solution based on a non-linear hyperbolic stress–strain behavior (Fig. 2.9) of the inclusion and the soil-inclusion interface. The interface behavior was modeled considering the changes in vertical stress on the inclusion due to constrained dilation of the soil and assuming a strain  26 softening behavior. In the absence of reliable direct stress–strain data to describe the inclusion response, the hyperbolic parameters were determined through a calibration process to match a single pullout test of a series of tests and the remaining tests were predicted by changing the variables that have a physical meaning. They were able to capture the behavior of different pullout tests showing a strain hardening behavior including the results of the textured geomembrane tested by Raju (1995) for displacements ranging from 5 mm to 20 mm (see Figure 2.9). The authors stated that if soil dilation is not accounted for, it will lead to an underestimation of pullout resistance.  The analytical solution for the pullout resistance of geogrids is more complex given the two components acting at the soil-inclusion interface, namely the bearing resistance provided by the transverse elements and the shearing resistance given by the planar section of the geogrid. Bergado and Chai (1994) proposed an analytical solution to capture the behavior of an individual bearing member and a method to determine the pullout force/displacement curve for the complete geogrid. The model is based on the idea that the frictional component of the soil-inclusion interface response is similar to the friction resistance of an axially loaded pile and needs a small relative displacement to be mobilized. The friction resistance is represented by a shear stiffness and the maximum friction resistance is determined by a Mohr-Coulomb failure criterion. The bearing component is modeled by a hyperbolic function that is controlled by the backfill soil stiffness and bearing member rigidity. The strain softening of the pullout response can be incorporated by varying the backfill friction angle from peak to critical state values. They can also add the non-linear, strain rate and temperature dependent behavior of the inclusion by a successive iteration technique using a secant modulus for the corresponding load level, strain rate, and temperature. They suggested that if the soil particle size is very small compared to the bearing element thickness, the factors that most influence pullout resistance are the soil strength and the geogrid geometry. Their formulation is able to reasonably capture the behavior of two Tensar grids that do not show a strain softening behavior until pullout displacement of 12 and 60 mm. Similarly, Moraci and Gioffrè (2006) developed a theoretical method to determine the peak and residual pullout resistance of three HDPE extruded geogrids embedded in dense sand. The method considers the geogrid as an equivalent planar strip of uniform thickness and allows to evaluate the frictional and bearing  27 components of the pullout response by accounting for the reinforcement extensibility and geometry, along with a non-linear failure envelope. The frictional component was assumed as 1/3 of the backfill friction angle (or  / = 1/3) as suggested by the data of Fannin and Raju (1993). The bearing component was evaluated using peak friction angles at the different vertical stresses to consider the non-linearity of the failure envelope of the backfill soil. The comparison of the theoretical method and laboratory test data allowed for the evaluation of the bearing and frictional components of the pullout response. They concluded that surface friction represents less than 20% of the peak pullout resistance, and less than 6% of the residual pullout resistance.   2.3.2. Finite element models  The finite element method (FEM) has also been used to model pullout tests on sheet and geogrid inclusions. The advantage of this method is that it allows the modeling of the complete pullout box and hence to study other aspects of the pullout response, such as stresses at the front wall of the test device or the influence of boundary conditions. Generally, in FEM analyses, the soil is modeled using a continuum mesh with a constitutive model to represent its behavior. The inclusion is mostly modeled as a flexible beam element that requires elastic parameters, and the soil-inclusion interaction is usually modeled in two ways. The first approach considers the soil fully bonded to the inclusion and therefore no slippage is allowed until the shear stress between the soil and the inclusion reaches a critical value, often determined by a Mohr-Coulomb criterion. The second way attaches the soil and the inclusion through an interface model (usually springs) allowing relative movement between both materials. The springs are typically characterized by a shear stiffness (that controls the amount of movement until the shear stress is high enough to cause slippage) and a normal stiffness (that controls normal deformation).      28 For example, Chan et al. (1993) used a non-linear force-strain relation for the inclusion and a linear elastic model for the soil. The soil-inclusion interface shear behavior was modeled using an elastic-plastic model with a failure condition defined by a Mohr-Coulomb criterion. The shear stiffness characterizing the soil-inclusion interface was calculated from a pullout test and the normal stiffness was assigned a very high value to assure compatibility in the normal direction. The results of the numerical simulation using the calculated value of shear stiffness underestimated the pullout force observed from the laboratory test data. Additional inspection of the numerical results showed that the mobilized shear stresses at the soil-inclusion interface were highly non-uniform and decreased along the reinforcement. Back-analysis of the numerical simulations allowed the determination of the “true shear stiffness” of the soil-inclusion interface which was higher than the calculated one. The results of the numerical simulations using the true shear stiffness showed a better agreement.  Another characteristic of FEM analyses is that most of these finite element models consider the geogrid geometry as a two-dimensional structure or a continuous sheet in a three-dimensional analysis without accounting for the complex shape of transverse elements. For example, Wilson-Fahmy et al. (1994) suggested a method that transforms the two-dimensional structure of geogrids into an equivalent linear structure. They divided the geogrid in the longitudinal direction into a number of elements in which the nodal points were located at the transverse elements joint locations. The frictional resistance of the planar section of the grid was represented by springs. The soil-inclusion interface stress-displacement relation was determined by direct shear tests at the appropriate stress level to account for the stress-dependency of the response and is expressed as a hyperbolic relation. They focused their study on the flexibility of transverse ribs of the geogrid and its influence on the pullout resistance by using different models to describe the transverse element behavior. A flexible-rib model where the material is assumed to be linearly elastic, with an appropriate secant modulus depending on the expected stress level. A stiff-rib model where the transverse elements are assumed not to deflect under load and a beam model that considers the transverse element to behave as beams deflecting under loads. The method of analysis was used to study a hypothetical geogrid and was capable of showing differences in  29 pullout resistance, especially at low mobilized force in the inclusion, when the transverse geogrid members were modeled as either flexible or stiff ribs.   Yogarajah and Yeo (1994) used the finite element program CRISP to model the behavior of pullout tests. They developed a two-dimensional model where the soil was modeled with a Mohr-Coulomb failure criterion, the inclusion was modeled using bar elements and joint elements were used to model the interface between the soil and the inclusion. The properties of the backfill soil were obtained from triaxial tests and the apparent soil-inclusion friction angle was taken as 90% of the peak soil friction angle. The length of the inclusion was considered equal to the length of the pullout box and roller boundaries were taken for the front, back and lower boundaries. An elastic modulus was used to represent the inclusion and the soil-inclusion interface needed four parameters; a shear modulus, a residual shear modulus, a normal modulus and the soil-inclusion interface friction angle. Sugimoto and Alagiyawanna (2003) used an equivalent 2D model approximating the 3D structure of the geogrid, where the cross-sectional area of the truss element used to simulate the geogrid was equal to the total cross-sectional area of all longitudinal members in a meter width of the geogrid. They used a non-linear stress-strain model to describe the geogrid behavior. The sand was assumed as an elastic-perfectly plastic material obeying the Drucker Prager yield criterion with non-associated flow rule. The behavior of the sand-geogrid interface was modeled using a bond slip model, where the shear stress is a function of the relative displacement at the opposite side node of the interface element, and a Coulomb friction model. The finite element analysis was carried out using DIANA. They found that simulating the interface behavior using the bond-slip model can give reasonable results when the geogrid shows elongation but is not appropriate for strain levels less than 4%. The Coulomb friction model can reasonably simulate the geogrid behavior when the geogrid slips along the entire length.     30 Despite the best efforts to represent the non-linear behavior of the interface response, most of the methods previously described represented the soil-inclusion interface as having a linear elastic or a Mohr-Coulomb behavior. The shortcomings of these models are that they describe the shear behavior of the soil-inclusion interface in an approximate sense, but do not consider the coupling of shear and normal behavior and are therefore incapable of capturing dilation which is one of the essential factors of the pullout response.   To move forward in the representation of the soil-inclusion behavior, Pal and Wathugala (1999) used a disturbed state concept (DSC) to characterize the soil-inclusion interface behavior. The DSC treats the behavior of a material as a disturbance with respect to the behavior of the reference state. In their approach, the soil-inclusion interface was treated as an equivalent thin zone with a small finite thickness, responding to a DSC behavior. This proposed model is not only able to characterize dilation, but also hardening and softening responses of interfaces. Shear and normal stiffnesses for the soil-inclusion interface were determined from the initial slope of the shear-displacement curve and the stiffness of the soil, respectively. The results of the FEM analysis compared well to laboratory test data up to 20 mm of displacement when no strain softening was observed.   Similarly, Perkins and Edens (2003) used the commercial program ABAQUS for the three-dimensional numerical modeling of pullout tests. The inclusion was modeled using four-noded membrane elements, and considered both, the geotextile and the geogrid, as planar sheet inclusions. The inclusions were given a direction dependent isotropic hardening, elastic-plastic-creep model. The soil-inclusion interface was established by creating two contact surface pairs above and below the inclusion and was modeled as a Coulomb friction model dependent on direction and normal stress. The interface was described by a friction coefficient and a parameter Eslip that determines the slope of the shear displacement curve. Two constitutive models were considered for the soil, a linear elastic model and a bounding surface plasticity model. The latter needed 11 parameters, four of which describe conventional critical state soil mechanics properties and with the remainder describing shape and hardening parameters associated with the bounding surfaces. The boundary conditions were chosen so as to represent lubricated sides used in the pullout box. The interaction  31 parameters were adjusted such that a good match was found between the pullout results and the numerical simulations. The results of the finite element simulations were able to capture the initial stiffness of the pullout response of grids and predicted a peak and strain softening behavior at larger displacements than the laboratory test data. No strain softening was predicted for geotextiles. By removing some of the material model components for the soil and the inclusion, they found that eliminating the creep factor for the inclusion resulted in a slightly stiffer pullout response that shows a peak and strain softening behavior. And additional removal of the plasticity component of the inclusion increased the stiffness and also eliminated the strain softening behavior of the pullout response. A similar analysis was conducted to determine the influence of the constitutive model for the soil in the pullout response, where the inclusion was kept as an elastic-plastic-creep model and the soil was taken as a linear-elastic material. A virtually identical response was found to that when the bounding surface plasticity model for the soil was used and they concluded that the plasticity component for the inclusion is more critical than the creep component and that the soil model is not that important.   2.3.3. Discrete element models  Finite element models are widely used as numerical tools to model the soil-inclusion interaction for both, pullout tests and reinforced earth walls. This method, combined with laboratory tests, has helped to better understand the reinforcement mechanisms at macroscopic scales. As shown in the previous section, most of the finite element models, consider the three-dimensional structure of the geogrids as two-dimensional elements or planar continuous sheets. This simplification does not allow the separation of the contributions of the frictional and bearing resistance or, for example, the rolling of soil particles. Therefore, the geogrid reinforcement mechanisms cannot be satisfactorily investigated only with the FEM, especially at a microscopic scale.        32 Tran et al. (2013) used a combined discrete element (DE)-finite element (FE) model where the soil is modeled using the DE model, the geogrid using the FE model and interface elements were used to simulate the interaction between the two domains. A linear elastic material model was used for the geogrid and its properties were determined by matching the experimental load-displacement curve obtained from the conducted index tests at a medium strain of 2%. The sand used in the experiment was modeled using spherical particles with a mean diameter of 5.1 mm (15 times the real average diameter). Particle properties were determined by calibration against experimental triaxial tests. The local increase in joint thickness was not considered in the geogrid model to simplify the analysis. They studied the contributions of the frictional resistance of geogrid surfaces and the bearing resistance of transverse members on the total pullout resistance. The results suggest that friction in the longitudinal elements takes more than 75% of the total pullout resistance.  Wang et al. (2014) used PFC2D that is based on discrete elements. It uses rigid entities for particles and walls and soft contacts for the interaction. The physical properties of the sand and the geogrid were obtained by matching direct shear test and tensile test results, respectively. The sand was modeled as unbonded particles with the linear contact stiffness model. The geogrid was modeled as bonded particles with a length of 200 mm as in the laboratory test and twenty particles were generated in one row without any overlaps between the particles. The property of the geogrid was characterized by its tensile strength, and the tensile strength was determined by the parallel bond forces between the geogrid particles.  The model was able to show sand particle rotations within the specimen, especially in the vicinity of the geogrid, which also illustrated the load transfer behavior between the geogrid and the sand.  2.3.4. Statistical analysis  Huang and Bathurst (2009) used a statistical approach to study the accuracy of the current FHWA in the soil-geogrid model. In their study, no attempt was made to gain further insight into the soil-inclusion interaction, but to investigate the most commonly used pullout capacity model against results of tests in general conformity with ASTM D6706-01. The authors used 478 pullout test results from different sources and compared the predicted  33 pullout resistance to different deterministic models. The accuracy of the data interpretation and model type was quantified through the mean value of pullout resistance and the coefficient of variation. The model bias values were obtained using the ratio of measured to predicted pullout resistance. The first of these models expressed pullout as a linear function of normal stress (eq. 2.2c) and used a single value of F* obtained from different normal stresses. The second model fits a first order approximation of the F* data versus normal stress. The third model used default coefficient values for F* recommended in FHWA (2009). Model four (or general model) was a bi-linear stress dependent model in which the breakpoint is set at a normal stress 40 kPa. Model five used a non-linear stress dependent approximation. The results of the statistical analysis showed that the bi-linear and non-linear models resulted in greater model accuracy and that the use of the current recommendations for values of F* and  leads to an underestimation of pullout load 2.3 times less than the measured value. They suggest that the apparent soil-inclusion interface friction angle is stress dependent and decreases with depth below the top of the wall until 2.3 m (or close to a vertical stress of 50 kPa) and remains constant thereafter.   As shown in the previous sections, one of the most important parameters of the constitutive relations that has the major influence on the results of pullout resistance is the strength of the soil and the soil-inclusion interface. Some numerical models used friction angles obtained from direct shear or triaxial tests without adjustment for the plane strain condition. Given this, it appears therefore essential to have a proper characterization of this property for the numerical modeling and back-analysis of pullout tests.     34 2.3.5. Mobilized strength of sands and selection of parameters for numerical modelling of pullout   The most common and widely used tests to determine the strength of soils are the direct shear and the triaxial tests. Direct shear is the simplest test and the strength of sands is measured along a predefined plane of failure where a soil-soil interface is created by moving one half of the direct shear box in the horizontal direction. Deformation is only allowed in the vertical direction and the state of stress cannot be evaluated during the test, only at failure conditions. This test cannot be considered a true element test and the friction angle obtained should be avoided in advanced numerical modeling (although it is widely used). Compression triaxial tests are commonly used to characterize the strength of soils. A cylindrical specimen is typically subjected to a hydrostatic confining pressure and then sheared using an axial load. Axial displacement is measured as the specimen deforms; volumetric strain is measured if the test is performed under drained conditions, or pore water pressure if the test is undrained. Plane strain conditions differ from triaxial in terms of the stresses acting on a soil element. In axisymmetric conditions, movement is allowed in all the directions and the intermediate principal stress coincides with the minor principal stress for triaxial compression and the major principal stress for triaxial extension. In contrast, in plane strain, movement is restricted in the direction that coincides with the intermediate principal stress.   Several engineering cases such as embankments, earth dams and retaining walls approximate plane strain conditions, where movement in one direction is restricted. Design guidance for reinforced soil structures recommends the effective angle of friction of the backfill soil, and any effective cohesion, be determined by shear box or triaxial tests (BS 8006, FHWA). Yet there is widespread appreciation in practice for the potential, in a long embankment or retaining wall, to mobilize soil strength in a condition of plane strain. From laboratory testing in a plane strain unit cell device, Boyle (1995) advocated that plane strain soil properties be used for design of reinforced soil structures, noting the higher friction angle yields a lower magnitude of lateral stress and therefore a smaller load per unit width in the inclusion, a finding that is supported by Allen and Bathurst (2002). Moreover, using friction angles obtained from the direct shear and the triaxial test, Bathurst et al. (2002) suggest, from back- 35 analysis of several geosynthetic reinforced soil structures, that design practice is overly-conservative. This is consistent with the postulate of Jewell and Wroth (1987, p.60) that “the tangent of the plane strain angle of friction for typical sands is about 20-25% greater than the tangent of the direct shear angle of friction”, yielding a hidden factor of safety of the order of 1.2 in design based on either the peak strength or a critical state strength.  It is also believed that plane strain conditions prevail in the laboratory pullout test as a consequence of the boundary conditions of the test device. However, no compelling evidence has yet been observed, and numerical simulations previously shown used in back-calculation of pullout test results typically invoke a constant value of friction angle determined from either the shear box or the triaxial test, without adjustment for the plane strain condition.   In general practice, friction angles are obtained from triaxial tests; plane strain tests are mostly used by researchers to study and better understand the influence of the intermediate principal stress on the strength of soil. Hence, the availability of plane strain friction angles for design purposes is very limited. It is, therefore, necessary to find relations between the friction angles obtained from the general practice (triaxial and shear box) and plane strain conditions for back-calculation or back-analysis of long structures. The following Section describes a summary of the available numerical and empirical relationships between different friction angles that will be later used for the numerical simulations of the present study.  2.3.5.1. Numerical relations  The first attempt to relate the different friction angles was made by Rowe (1969), where he proposed an expression linking the peak friction angles for direct shear and plane strain ( dsp,  and psp , , respectively) and the critical state friction angle in plane strain (cs,ps) given by:                        tanp,ds = tanp,pscoscs,ps           (eq. 2.12)     36 The assumptions for this relation are: - The direction of principal stress and principal strain increments is the same - It is possible to define a same angle between the horizontal plane and principal stress and between the horizontal plane and principal strain increments on a simple shear apparatus  - Mohr circles of stress and strain increments are geometrically the same - A condition of zero strain in the horizontal direction in direct shear exists  Rowe (1969) noted that, for shearing at constant volume, eq. 2.12 may be simplified to:          sincs,ps = tancs,ds                    (eq. 2.13) Where: - cs,ds: critical state friction angle in direct shear  Lade and Lee (1976) proposed a relation between the peak friction angle in plane strain and triaxial conditions given by eq. 2.14a and 2.14b. Lee (2000) observed that these relations tend to overpredict the plane strain friction angle of Ottawa sand (rounded sand) and under-predict the plane strain friction angle of Rainier sand and RMC sand (angular sands) at low confining stresses (from 25 kPa to 100 kPa).   175.1 ,, −= txppsp   ( )34, txp              (eq. 2.14a) txppsp ,,  =   ( )34, txp               (eq. 2.14b)  Where: - txp, : peak friction angle in triaxial compression  Wroth (1984) proposed a relation between the friction angle in plane strain and triaxial given by eq. 2.15. Georgiadis et al. (2004) observed that this relation overestimates the plane strain friction angle for higher values of tx. It is important to mention that these equations do not include the effect of density, confining stress, mineralogy, fabric, etc.   37           txppsp ,,98  =                  (eq. 2.15)  Bolton (1986), from laboratory test data, proposed a relation between the peak friction angle and the constant volume friction angle for both, triaxial and plane strain conditions, that accounts for the effects of relative density and confining pressure in the value of the peak friction angle given by:                          p,tx - cs,tx = 3IR  (for triaxial)            (eq. 2.16)                                 p,ps - cs,ps = 5IR  (for plane strain)            (eq. 2.17)   Where: - IR = DR(10-lnp)-1 and 0 < IR < 4  - DR: relative density   He also proposed a relation between the peak friction angle, the critical state friction angle and the angle of dilation () in plane strain given by:                   p,ps = cs,ps + 0.8                (eq. 2.18)   Jewell and Wroth (1987), from Mohr’s circle of stress and strain increments, proposed a relation between the peak friction angle in plane strain and direct shear test and the dilation angle in direct shear given by:                              ( )dspdsppsp,,,tantan1costansin+=             (eq. 2.19)     38 Inspection shows eq. 2.13 to also follow logically from eq. 2.19, for the case of  = 0 at the critical state (Figure 2.10). This agreement was acknowledged by Lings and Dietz (2004) and Simoni and Houlsby (2006), and the relation is in general accordance with the postulate of Jewell and Wroth (1987) that cs,ps  1.2cs,ds.  Kulhawy and Mayne (1990) suggest that:   txppsp to ,, 2.112.1  =                     (eq. 2.20)  Schanz and Vermeer (1996), assuming that the friction angle at the critical state in triaxial and plane strain are equal, rearranged Bolton’s relation (eq. 2.16 and 2.17) and proposed an expression for the peak friction angle in plane strain given by:              p,ps = (5p,tx - 2cs,ps)/3                   (eq. 2.21)   Figure 2.10 shows a summary flow chart of the relations previously described that allow obtaining the large displacement friction angle in plane strain from friction angles obtained using triaxial and direct shear tests.  In addition, several authors have compared the magnitude of the friction angles measured with the different test devices (namely direct shear, triaxial and plane strain apparatus) and have shown that the peak friction angle obtained by using a plane strain loading apparatus is larger than the one obtained from triaxial tests (Rowe, 1969; Hussaini, 1973; Marachi et al., 1981; Kulhawy and Mayne, 1990). This is due to the movement constraint imposed in a plane strain condition, where the particles are restricted to move in one direction and hence develop higher stresses, resulting in consequence, in a larger associated friction angle. However, whether this difference in magnitude of the friction angle at the critical state is maintained, has been under discussion for over 50 years. Some authors have suggested that the difference between the peak friction angle found in triaxial and plane strain can differ by more than 5° in dense sands and that this difference is maintained at the critical state (Stroud, 1971). It has also been recognized that as the initial density of the sample decreases, i.e. the soil sample  39 becomes looser, the difference between the peak friction angle in plane strain and triaxial tests decreases. In the following section, a comparison of the magnitude of the friction angles in plane strain and triaxial conditions, measured by different experimentalists is shown in order to establish a relation between the strength of soils at large displacement.   2.3.5.2. Empirical data   Hussaini (1973) performed plane strain and triaxial tests on Chattahoochee sand at initial relative densities between 30% and 100% at a confining stress of 490 kPa (Figure 2.11). He observed that as the initial relative density decreased, i.e., the soil became looser, the difference of peak friction angle between plane strain and triaxial decreases (plane strain being always larger than triaxial). The difference was about 3 degrees for dense samples and less than 1° for loose samples, this being 31°-32°.   Marachi et al. (1981) performed plane strain and triaxial tests on Monterey sand at densities of 27%, 60% and 90% at confining stress of 70 kPa (Figure 2.12). They also found that as the sample gets looser, the difference between plane strain and triaxial friction angle decreases (plane strain being always larger). For dense samples, the difference is about 7° and less than 3° for loose samples (it is important to note that these tests were performed under very low confining stresses). A smaller difference between friction angles at a looser state in plane strain and triaxial was found for bigger values of cell pressure (3); about 2° for 3 = 560 kPa and less than 1° for 3 = 3300 kPa (at that stress level some particle crushing might occur).  Cornforth (1964, 1973) performed plane strain and triaxial tests on Brasted sand at initial porosities ranging from 34% to 44% at a confining pressure of 40 lb/sq in (about 275 kPa). As the samples became looser, the difference in peak friction angle between plane strain and triaxial decreases and approaches the value of critical state friction angle. He also reported data of the critical state friction angle obtained from triaxial and plane strain tests and found average values of 33° and 32.3°, respectively. Adding trend lines to the three sets of data they converge on an approximate unique value of about 32.4° (Figure 2.13). Rowe (1969) reported  40 data of peak friction angles on quartz sand from plane strain and triaxial tests at confining pressures between 4 and 5 lb/sq in (27.5 to 34.5 kPa). He found that as the initial relative density decreases (i.e., the sample becomes looser) the peak friction angle for plane strain and triaxial approach the same value of about 31º. This evidence suggests that the friction angle at the critical state for triaxial and plane strain differs in less than 3° depending on confining pressure hence, cs, ps   cs, tx (+ 1º to 3º).    2.4. Laboratory pullout tests  Many experimental researchers have sought to study the various factors that influence the pullout resistance of an inclusion, namely its type and characteristics, along with the characteristics of the soil and the testing device. A summary of the apparatus characteristics and testing procedures is shown in Tables 2.2 and 2.3 Important differences are observed between testing devices, in terms of box dimensions, load application system, methods to minimize boundary conditions effects and testing procedures. Most of the differences observed in the pullout response of similar soil-inclusion compounds are related to the difference in testing methods and boundary conditions of the pullout box. Moreover, several of the experimental data available was obtained before the publication of ASTM D 6706-01 and therefore, many test configurations do not comply with the minimum specifications described in Section 2.2.2, mostly factors related with the dimensions of the inclusion tested with respect to the box dimensions (see for example, Ingold, 1983; Palmeira and Milligan, 1989; Farrag et al., 1993; Wilson-Fahmy et al., 1994; Perkins and Cuelho, 1999; Sugimoto et al., 2001). Nevertheless, all the studies in Table 2.2. have helped to better understand the factors that influence the pullout resistance of different inclusions. The most important findings are therefore summarized in the following Sections concerning the parameters that will be used in the parametric study in Chapter 6. Although the pullout rate and side wall effects are beyond the scope of this study, their influence will be described for completeness.      41 2.4.1. Vertical stress and top boundary  Farrag et al. (1993) conducted tests at different vertical stresses to determine its influence on the pullout behavior. They found that the pullout resistance increases with vertical stress and that the shear stresses at the soil-inclusion interface are more uniform along the specimen at lower stresses. Higher vertical stresses result in an inclusion showing a more extensible behavior and therefore, a non-uniform shear stress distribution.   The way in which the surcharge is applied at the top of the soil sample, also influences the pullout resistance. Palmeira and Milligan (1989) used a rigid and a flexible top plate as load application device. The test results showed that using a rigid top plate, the maximum pullout force is larger than using a flexible one. This finding was later corroborated through a discrete element study of Wang et al. (2017), who also showed that at small clamp displacements both, a rigid and a flexible top plate, give similar pullout responses. Johnston and Romstad (1989) recommend that the vertical stress is applied with fluid-filled bags to distribute the applied vertical pressures during the pullout tests. Farrag et al. (1993) argue that a flexible membrane allows for a better and more uniform distribution of the vertical load at the soil-inclusion level.  2.4.2. Front wall effects  The front wall of the pullout box is also a major boundary problem that needs to be considered when analyzing the pullout resistance of different inclusions. Several authors have placed pressure transducers to study the horizontal stresses developed at the inside front wall of the pullout box as the test is carried out. Raju (1995) and Sugimoto et al. (2001) showed that a peak horizontal stress develops at the reinforcement level and decreases near the top and bottom of the box (see Figure 2.14). The shape of the horizontal stress profile at the inside front wall is not always symmetrical given the difference in boundary conditions on top and bottom of the box. They found that the horizontal stress is linearly proportional to the pullout resistance.    42  Some works in the literature have looked for ways to minimize the effect of the front wall in the pullout resistance, especially before the introduction of the use of sleeves by ASTM D 6706-01. The typical front wall conditions studied have been: using a flexible front wall, varying the inside front wall surface roughness from a lubricated one to a fully rough one, and adding a front sleeve. For example, Sugimoto et al. (2001) compared the pullout results of geogrids using a large pullout box with a rigid front wall and a flexible front wall. They showed that the mobilization of strains along the inclusion, and hence the interaction factor, varied with the mobility of the front wall. Larger deformations close to the front wall were observed for the rigid front wall due to a local increase of relative density of the soil. Raju (1995) compared the pullout response of three different inclusions (one grid and two membranes) using an aluminum front wall and an arborite front wall, with a surface friction angle (ff) equal to 15° and 12.5°, respectively. The laboratory test results show that the peak pullout resistance is slightly larger when using the front face made of aluminum. However, at large displacement, both pullout resistances seem to converge (see Figure 2.15). The same observation was made by Palmeira and Milligan (1989) using four different front wall configurations from a lubricated front wall (ff = 6°), a metallic front wall (ff = 16°), sand paper glued at the internal front wall surface (ff = 30°), and a fully rough wall to which sand particles were glued (ff = 45°). They observed that the rougher the front wall, the larger the peak pullout resistance or peak soil-inclusion interface apparent friction angle (see Figure 2.16). Compared to the lubricated front wall, the metallic front wall showed an increase in 1.6 times, the sand paper configuration in 1.8 times, and the fully rough front wall in 2.3 times. In terms of large displacement, the pullout resistance found using the plain metal and lubricated front wall tend to converge at around 10 mm. Similarly, the pullout resistance using the fully rough front wall and sand paper tends to decrease, suggesting that at larger displacements the pullout resistance might be similar.   An apparent way of reducing the soil pressures generated at the front wall of the pullout box is the use of sleeves. Sleeves transfer the point of application of the pullout load inside the soil mass far beyond the rigid frontal boundary (Farrag et al., 1993). Several authors have studied the influence of sleeve length on pullout test results. For example, Farrag et al. (1993)  43 conducted tests with no sleeve, and sleeve lengths of 20 cm and 30.5 cm. They found that as the sleeve length increases, the lateral pressure in the front wall and the pullout resistance decreases. Lopes and Ladeira (1996) found that for a smooth (but not lubricated) front wall with no sleeve the maximum pullout load was larger compared to a test with a sleeve 20 cm long. Chang et al. (2000) showed that the shorter the sleeve, the larger the pullout resistance. On the contrary, Dias (2003), by means of numerical models, compared the pullout resistance of a box with no sleeve and a lubricated front wall and with sleeves 15 cm and 30 cm long. The results showed that the presence of the sleeve yielded larger pullout loads than the lubricated front wall. Palmeira (2009) concluded, “these contrasting results suggest that a more comprehensive study on how to minimize the influence of the front box wall on pullout test results is required.”   2.4.3. Height of the box  Soil thickness (or generally the height of the box) also appears to influence the pullout response of different inclusions. Farrag et al. (1993) tested a 1 m long geogrid varying the height of the box and found that as the soil thickness decreases, the peak pullout resistance increases. They also observed that the difference between the soil thickness on top and below the inclusion does not influence the results (Figure 2.17). The authors suggest a minimum thickness of 30 cm above and below the inclusion to eliminate the effect of the top and bottom boundaries on pullout response. Similarly, Palmeira (2009) suggests that the soil sample should be greater than 0.6 m and that little influence of the box height in the pullout response exists for heights greater than the length of the reinforcement. In addition, for geogrids, Moraci and Recalcati (2006) suggest that the height of the box should be connected to the dimensions of the passive wedge that develops on the transversal elements of the geogrid. According to the authors, the size of the wedge can be taken as 40 times the thickness of the transversal element of the grid.      44 2.4.4. Inclusion length  After equation 2.1, it has generally been accepted that pullout resistance increases with the embedded length of the inclusion. A very comprehensive study was performed by Moraci and Recalcatti (2006) were the length of three different HDPE extruded geogrids and the applied vertical stress were varied. The length was varied between 0.4 and 1.15 m and the vertical stress between 10 to 100 kPa, resulting in 36 different pullout test combinations. They observed that the pullout response is a combination of inclusion length, soil-inclusion interface friction angle and vertical stress applied. Tests performed on “short” inclusions or “long” inclusions at low vertical stress (lower than 25 kPa) showed less elongation of the inclusion and a peak and strain softening pullout response due to a more uniform distribution of shear stresses along the soil-inclusion interface. Tests on “long” inclusions and confining stresses larger than 25 kPa resulted in a strain hardening pullout response because of the non-uniform shear stress distribution due to the elongation of the inclusion.   Recently, Ezzein et al. (2014) developed a pullout box with a transparent glass bottom and a combination of transparent soil and opaque particles at the soil-inclusion interface in order to visualize the movement of soil particles at the soil-geogrid interface. The length of the geogrid was 2.0 m. They were able to measure the relative horizontal displacement between the geogrid and the surrounding soil. They observed that 50% of the specimen length is mobilized at vertical stresses lower than 10 kPa and that this length dropped to 25% at vertical stresses of 50 kPa. Additionally, they were able to observe that when the vertical stress applied is large, there is less movement of the soil with respect to the geogrid. Subsequently, Bathurst and Ezzein (2015 and 2016) studied the effect of vertical stress on the mobilization of the active length in the geogrid. Those results allowed the proposal of an exponential function to approximate the longitudinal displacements and strains in the geogrid at vertical stresses higher than 12 kPa.     45 2.5. Summary of current knowledge and role of present research  The research described in this study is intended to contribute to a science-based understanding of the factors influencing the pullout resistance of different inclusions, by expanding on principles and concepts presented in the literature to date.  In the current Chapter, a summary of terminology and concepts concerning the soil-inclusion interaction is first presented. It is found that most of the relations proposed to describe soil-inclusion interaction in Section 2.2, in particular the Jewell et al. (1985) approach, make reference to a “soil’s” friction angle without specifying whether it is at peak or critical state, whether it should be obtained from the triaxial, direct shear or plane strain test or the confining pressure at which the friction angle should be considered. To help address this knowledge deficiency, in Chapter 3 of the present study, an extensive characterization of Badger sand is presented with specific emphasis on the strength characteristics under triaxial, direct shear and plane strain conditions at the low stresses at which pullout tests are mostly performed (less than 100 kPa). The obtained stress-dependent plane strain friction angles are used in Chapter 4 for the numerical modeling of pullout tests of sheet inclusions at large displacements, for which good physical properties are available to describe the soil-inclusion interface. Stress-dependent plane strain properties are also used in the Jewell et al. (1985) formulation to deduce a soil-inclusion interface friction angle between Badger sand and the geogrids to treat them as equivalent sheet inclusions. The results of the simulations provide evidence in support of the hypothesis that plane strain conditions and the stress-dependency of the friction angle at the critical state at stresses lower than 50 kPa must be considered for proper interpretation or back-analysis of pullout tests. This finding is further verified for the complete pullout response in Chapter 5.  The numerical models described in Section 2.3 have shown that, in addition to pullout box characteristics, one of the most important factors that influences the pullout response of an inclusion is its interaction with the surrounding soil, specifically the soil-inclusion interface. Furthermore, many of the models described in Section 2.3 have been able to capture the pullout resistance at small displacements, or at large displacement only when no strain softening exists. Consequently, the literature review identifies two aspects with scope for  46 improvement in the numerical modeling of pullout tests in sands:  - Many of the numerical models use input parameters that require calibration against laboratory pullout test data, and adjustments to these parameters are made until the simulation matches the measured pullout resistance, reaching sometimes unrealistic values. - Few models have been able to capture the complete pullout response at small and large displacement, including the peak and strain softening behavior found in some tests.   The present study aims to improve the numerical simulation of pullout tests in sands by integrating the consistent material characterization in Chapter 3 and the findings on Chapter 4 regarding plane strain conditions and the stress-dependency of the soil’s friction angle at the critical state. This, combined with a proposed representation of the soil-inclusion interface as a continuum layer responding to a constitutive model able to capture the dilatant behavior of dense sands, allows for the capture of the complete pullout response of the sheet inclusions, including the strain softening behavior observed in some of the pullout tests performed by Raju (1995), through a more phenomenological numerical model in Chapter 5. The Jewell et al. (1985) is also used to study its potential to capture the complete pullout response of the different geogrids tested by Raju (1995).  Section 2.4 confirms a well-accepted appreciation that test conditions and device configuration have an influence in the pullout resistance of different inclusions. Limited experimental data exist to increase the understanding of the influence of the different constituents in a laboratory pullout test. This is because it is either a tremendous and very time-consuming effort to vary every parameter in a large range to examine its influence or the available instrumentation is not able to capture the principal aspects of the soil-inclusion interaction. In order to address this knowledge gap, and with the benefit of the phenomenological numerical model developed in Chapter 5, a parametric study in Chapter 6 is undertaken to gain insight to the scientific literature. In particular, different parameters related to the inclusion and the boundary conditions of the pullout box are varied between the typical ranges found in the literature (Table 2.2 and Table 2.3) and the ASTM D6706-01 recommendations to better understand their influence in the pullout response. Based on the  47 results of the parametric study, a discussion regarding the soundness of the laboratory pullout tests found in the literature and the ASTM D6706-01 recommendations is undertaken and some flaws in the current pullout recommendations are identified.  An “ideal” pullout test is suggested that ensures that pullout occurs (as defined by ASTM D6706-01), and that addresses the shortcomings revealed from the parametric study. 48  Table 2.1: Failure modes and properties required for reinforced earth and GRS (Holtz, 2017) Reinforced earth Geosynthetics Properties required Ties break Tensile rupture Tensile strength Ties pullout Geosynthetic pullout Soil-reinforcement interface friction    49  Table 2.2: Pullout box characteristics from the literature Author (year) Box dimensions  L×W×H (m) Box front wall Soil thickness  (cm) Rate of loading  (mm/min) Ingold (1983) 0.5×0.285×0.3   1  Johnston and Romstad (1989) 1.3×0.91×0.51 steel    4.4  Palmeira and Milligan (1989) 0.5×0.25×0.15  1.0×1.0×1.0 fully rough  sand paper smooth steel lubricated 15 100  Farrag et al (1993) 1.52×0.92×0.76 sleeve length 0 – 20 – 30.5 cm 20, 40, 60, 70  4-20  Wilson-Fahmy et al (1994) 1.9×0.91×1.1 sleeve length 10 cm 55  1.5  Yogarajah and Yeo (1994) 2.0×0.5×1 steel 50  1  Raju and Fannin (1995) 1.3×0.64×0.63 aluminium arborite 30  0.5, 0.25, 1  Lopes and Ladeira (1996) 1.53×1.0×0.8 sleeve length 20 cm 30  1.8 to 22  Perkins and Cuelho (1999) 1.2×0.9×1.1 sleeve length 26 cm 55  1  Ta-teh et al (2000) 1.5×0.9×0.8 sleeve length 0, 7.5, 15, 20 cm  40  1, 5, 10, 20, 50  Sugimoto et al (2001) 0.68×0.3×0.625 sleeve length 25 cm   Moraci and Recalcati (2006) 1.7×0.6×0.68 sleeve length 25 cm  1  Lopes and Silvano (2010) 1.53×1.0×0.8 sleeve length 20 cm 30  2  Bathurst and Ezzein (2015) 3.7×0.8×0.3 sleeve length 20 cm 25 1    50  Table 2.3: Inclusion characteristics from the literature Author (year) Inclusion type Dimensions L×W (m) Tu  (kN/m) Vertical stress (kPa) Ingold (1983) Netlon 1168 geogrid FBM5 geogrid Steel grid 0.5×0.275  60 kN/m 288 kN/m 0 to 200  Johnston and Romstad (1989) Tensar SR2 geogrid   27 to 54  Palmeira and Milligan (1989) mild steel geogrid galvanized steel geogrid 0.75×0.25  0.5×1.0  25  Farrag et al (1993) Tensar SR2 geogrid, Conwed  9027 0.9×0.3 0.9×(0.15,0.3,0.45,0.75)  48.2 to 140  Wilson-Fahmy et al (1994) stiff PE geogrid (A) flexible geogrids (B-C) (0.31,0.92,1.7)×0.35 101.7 48.5 36.2 69  Yogarajah and Yeo (1994) Tensar SR80 geogrid (1.8,1.1)×  6.5  Raju and Fannin (1995) aluminum rough planar 2 geomembranes  3 geogrids 0.965×0.5 see Table 3.3 4 to 30  Lopes and Ladeira (1996) Tensar SR55 0.96×0.33 55  26 to 87.8  Perkins and Cuelho (1999) Geogrid Geotextile (0.3, 0.715)×0.3 20 31 5 to 35  Ta-teh et al (2000)  1×(0.1×0.2×0.25×0.3×0.4×0.5×0.6)  50  Sugimoto et al (2001) SR55 geogrid SS1 geogrid 0.5×0.3 54 11.8 5 to 93 Moraci and Recalcati (2006) GG1 geogrid GG2 geogrid GG3 geogrid (0.4, 0.9, 1.15)×0.58 73.06  98.99 118.29 10 to 100 Lopes and Silvano (2010) PP+PET polymer 0.9×0.3 115 kN/m 50 k Bathurst and Ezzein (2015) Biaxial PP Tensar geogrid 2.0× 12.5 kN/m   2 to 52    51           Figure 2.1: Horizontal stress distributions assumed for design: a) Rankine tieback wedge; b) distribution proposed by Broms; c) bilinear assumed by Reinforced Earth; d) coherent gravity (Holtz, 2017)             Figure 2.2: Cross-sectional detail view for geosynthetic pullout setup (ASTM D 6706-01)  HKa 0.65HKa HKo HKa Ko Ka 6.1 m 2.35 m a) b) c) d) Soil Soil Inclusion 52                Figure 2.3: Bearing stress ratio as a function of friction angle of the soil              Figure 2.4: Influence of particle size in bearing stress ratio Footing rotated to horizontal (eq. 2.8) Punching failure mode (eq. 2.7) Bearing stress ratio (b/n)  Soil friction angle,  F 1 Equation 3.12 Size ratio, Bt/D50 Data from: Jewell et al. (1984) Palmeira and Milligan (1989) 53                            Figure 2.5: Predicted components of resistance to pullout force of geogrids A, B and C for geogrids lengths of 0.31 m (Wilson-Fahmy et al., 1994) 54          Figure 2.6: Effect of transverse ribs on the pullout response of Conwed geogrid  (Farrag et al., 1993)               Figure 2.7: Pullout responses at low confining pressures (Gurung et al.,1999)  55             Figure 2.8: Hyperbolic representation of stress-strain curve            Figure 2.9: Predictions for pullout tests performed by Fannin and Raju (1993) (Weerasekara and Wijewickreme, 2010)    1 - 3  (1 - 3)ult (1 - 3)f 56                            Figure 2.10: Flow chart to obtain  cs,ps and  p,ps from triaxial and direct shear test data. (Circles represent input data from triaxial and direct shear test and diamonds represent the deduced friction angles in plane strain)  cs,ds  eq. 2.13 Rowe (1969)   eq. 2.14 Lade and Lee (1976)   eq. 2.18 Bolton (1986)  eq. 2.12 Rowe (1969)    p,tx  cs,ps  p,ds  eq. 2.19 Jewell and Wroth (1987)   eq. 2.15 Wroth (1984)    = 0 p,ps   57             Figure 2.11: Maximum friction angle in triaxial and plane strain conditions on Chattahoochee sand (Hussaini, 1973)               Figure 2.12: Maximum friction angle in triaxial and plane strain conditions on Monterey sand (Marachi et al., 1981)    27293133353739410 10 20 30 40 50 60 70 80 90 100Maximum angle of friction  ( )Initial relative density, DR (%)triaxialplane strain303540455055600 10 20 30 40 50 60 70 80 90 100Maximum angle of friction  ( )Initial relative density, DR (%)triaxialplane strain58             Figure 2.13: Peak and large displacement friction angle in triaxial and plane strain conditions for Brasted sand (Cornforth, 1964 and 1973)   303234363840424446480 20 40 60 80 100Friction angle ( ) Initial relative density, DR (%)peak triaxial 275 kPa peak plane strain 275 kPacritical state triaxialcritical state plane strain59             Figure 2.14: Lateral stresses at the front wall of the pullout box (Raju, 1995)                Figure 2.15: Influence of front wall roughness in the pullout resistance of sheets (Raju, 1995)  GMT GMS GMS08 GGT10 60              Figure 2.16: Influence of front wall roughness in the pullout resistance of geogrids  (Palmeria, 1989)              Figure: 2.17: The effect of soil thickness on pullout response (Farragt et al., 1993) 61  Chapter 3 Description of material properties1  3.1. Introduction  The research objectives of this thesis are investigated by means of numerical simulations of the laboratory pullout tests on sheet and geogrid inclusions performed by Raju (1995). To obtain a mechanically-based description of the pullout response of the inclusions, in the present study, the numerical models are based on physically-based material properties of the pullout test components. Accordingly, this Chapter addresses the characterization of the materials used in the laboratory pullout tests, with special emphasis on the strength of the sand, in agreement with research objective one. The properties of Badger sand, in particular the strength at stresses lower than 150 kPa, are determined through triaxial, direct simple shear and direct shear test data, with specific emphasis on the very low stress range used in the laboratory pullout tests (between 4 kPa and 30 kPa). Material properties and geometric characteristics of the inclusions and the soil-inclusion interface are identified for purposes of numerical simulations. Finally, a description of the pullout box and test conditions used by Raju (1995) is presented.  3.2. Description of physical and mechanical properties of Badger sand  3.2.1. Index properties  The sand used in the laboratory testing by Raju (1995) was obtained commercially from the Badger Mining Corporation, of Wisconsin, USA (for convenience, this sand is referred to as “Badger Sand”). It is derived from the Wonowoc sandstone formation and processed by a hydraulic washing operation. The mineralogy of the grains was almost entirely quartz, and                                                           1 Some of these results were published as a Technical Note: Rousé, P.C., Fannin, R.J., and Shuttle, D.A. (2008) “Influence of roundness on the void ratio and strength of uniform sand”, Geotechnique 58, No 3, pp. 227-231.  https://www.icevirtuallibrary.com/doi/full/10.1680/geot.2008.58.3.227  62  they had a specific gravity (Gs) of 2.65. Dry sieve analysis showed the grain size to vary between 0.6 mm and 2 mm, with a mean diameter D50 = 0.87 mm. The coefficient of uniformity, Cu, is 1.3 and the coefficient of curvature, Cc, is 1.1. Accordingly, it is described as uniformly-graded coarse sand with no fines, to which the symbol SP is assigned by the Unified Soil Classification System (Figure 3.1). The minimum and maximum void ratios, emin and emax, were determined according to ASTM D 4253-00 and ASTM D 4254-00 Method B, respectively, yielding values of emax = 0.69 and emin = 0.49 (Rousé, 2005).   3.2.2. Roundness and sphericity  Sphericity (S) is defined by the ratio of the surface area of a particle to its volume (Wadell, 1932). Krumbein (1941) sought to calculate it from measurement of three mutually perpendicular intercepts (termed the long, intermediate and short axes of the grain) and is well-suited to relatively large particles. In the case of small grains, a convenient form of the sphericity relation is that used by Santamarina and Cho (2004), and termed inscribed circle sphericity by Mitchell and Soga (2005), which requires analysis of a projected image of the grain profile to establish the ratio:       𝑆 =𝑟max𝑅min      (eq. 3.1)  where rmax is the radius of the largest inscribed circle and Rmin is the radius of the smallest circumscribing sphere (see Fig. 3.2).  Roundness (R) is defined as the ratio of the average radius of curvature of the corners and edges of the particle to the radius of the maximum sphere that can be inscribed (Wadell, 1932; Krumbein, 1941).  Again, it is convenient to calculate this from image analysis using the radius of curvature of surface features (ri) illustrated in Figure 3.2, and the radius of the largest inscribed circle (rmax), yielding the relation used by Santamarina and Cho (2004):        𝑅 =𝛴𝑟𝑖𝑁𝑝𝑟max      (eq. 3.2) 63  Where Np is the number of features examined.  In this study, the grain shape of Badger sand was quantified using the definitions of equations (3.1) and (3.2).  The first attempt to classify soil grains by means of roundness is attributed to Russell and Taylor (1937). From visual comparison with photographs, five classes were proposed for description of the shape, varying from angular to well-rounded. Powers (1953) subsequently modified the approach through further division at the lower end of the scale, thereby yielding six intervals that form the basis of current usage, namely, very angular (0.12 < R < 0.17), angular (0.17 < R < 0.25), subangular (0.25 < R < 0.35), subrounded (0.35 < R < 0.49), rounded (0.49 < R < 0.7), and well-rounded (0.70 < R < 1.00). Accordingly, measurements and comparisons of data in this study are made with reference to Powers (1953). Visual inspection of the Badger sand (see Figure 3.2) suggests the grain shape does not vary significantly. Recognising the law of strong numbers applies with a number of elements equal to 30 or more (Moore and McCabe, 2003), roundness and sphericity were calculated for 30 grains of sand that were selected at random. A mean value of R = 0.81 and S = 0.77 was obtained, respectively. Comparison with the Powers (1953) classification indicates the sand is well-rounded, a finding that holds for all of the grains examined since the lowest value of roundness exceeds 0.7. The largest values of roundness and sphericity, 0.98 and 0.95 respectively, imply some of the grains are almost perfectly rounded spheres.  Round spheres of equal diameter can be arranged in five idealised packings (Lade et al., 1998). The well-rounded grains of Badger sand yield a minimum void ratio of 0.49 that is larger than the theoretical lower bound of 0.35 for a pyramidal (each sphere of one layer rests in the opening between four spheres forming a square in the adjacent layer) or tetrahedral packing (each sphere of one layer rests in the opening between three spheres forming an equilateral triangle in the layer below). It is close to the theoretical value of 0.43 for a double-stagger arrangement. Badger sand exhibits a maximum void ratio of 0.69 that approximates the theoretical value of 0.65 expected of a cubic-tetrahedral packing, however the range emax - emin appears rather small for uniformly graded sand.    64  3.2.2.1. On the variation of extreme void ratios with roundness  Several authors report values of roundness and describe its influence on emax and emin. A summary of that data is given in Table 3.1: it comprises 66 values of emax and 46 values of emin that are compiled from six references on a total of 61 uniformly graded soils and 5 glass beads, together with values for the Badger sand examined in the current study. All of the materials are uniformly graded, given a Cu ≤ 4 for the gravels and Cu ≤ 2.5 for the sands and glass beads. The variation of maximum and minimum void ratio, with roundness, is plotted in Figure 3.3. Roundness varies from R = 0.1 for a crushed sand, to R = 1.0 for the glass beads.   Youd (1973) examined 21 sands of roundness 0.19 ≤ R ≤ 0.60, determined from Wadell’s method, and void ratios from ASTM D 2049-69. He postulated a curved relation between R and both emax and emin, wherein both values diminish with increasing roundness. Moroto and Ishii (1990) report similar data for 11 gravels of 0.27 ≤ R ≤ 0.58 but, given the different focus of the work, did not comment upon R and the values of void ratio. Shimobe and Moroto (1995) subsequently proposed a hyperbolic relation between roundness, determined from Wadell’s method, and emax which yielded a good correlation for 11 sands and gravels of 0.15 ≤ R ≤ 0.65 and 3 types of glass beads. Sukumaran (1996) sought to introduce two new parameters, a shape factor and angularity factor, from which Sukumaran (1996) provided values of roundness for 7 sands and 1 type of glass bead. An exponential function was found to describe the relation between these factors and emax and emin (Sukumaran and Ashmawy, 2001). Santamarina and Cho (2004) used the data of Youd (1973) to generate a hyperbolic relation between emax and emin and roundness. More recently, Cho et al. (2006) postulate the relation to be linear, based on 19 values of emax and 16 values of emin.  Data for the Badger sand are reported as a mean value, together with the minimum and maximum values of roundness obtained from image analysis of 30 grains (Table 3.1). The variation in measured roundness of these grains almost bounds the range (0.7 ≤ R ≤ 1.0) of the well-rounded class, while the variation in emax and emin is negligible. The data fits the general trend and is in excellent agreement with the little data available for well-rounded 65  sands. Greater scatter is evident in the data for very angular to subrounded grains, which is partly attributed to differences in measurement of void ratio, a consideration that was identified by Cubrinovski and Ishihara (2002), and partly to the quantification of a three-dimensional attribute by a two-dimensional image analysis (Bowman and Soga, 2005).   Analysis of the compiled database for soils with Cu ≤ 2.5 (Table 3.1), excluding the glass beads, yields a refinement to the relation between emax and emin and R (see Fig. 3.3), where:      𝑒max = 0.651+ 0.107𝑅−1               (eq. 3.3a)      𝑒min = 0.433+ 0.051𝑅−1    (eq. 3.3b)  The relations describe the strong non-linear trend of the angular grains, which becomes increasingly linear for rounded grains. They imply a value of emax = 0.76 and emin = 0.48 for a uniformly graded soil with perfectly rounded grains (R = 1), which shows reasonable agreement with the data for glass beads.  3.2.3. Badger sand strength  3.2.3.1. Direct simple shear tests   Three direct simple shear tests were performed at stresses of 50, 100 and 150 kPa by Dabeet (2013) on specimens reconstituted by air-pluviation to a relative density between 42% and 53%. The test data had been obtained using the Norwegian Geotechnical Institute (NGI) type DSS apparatus of The University of British Columbia, which allows the testing of specimens having a diameter of about 70 mm and a height of 20–25 mm. The specimen diameter is constrained against lateral strain using a steel-wire-reinforced rubber membrane, for constant volume tests, and the height constraint is obtained by clamping the top and bottom loading caps against vertical movement. Figure 3.4 shows a gradual increment of the stress ratio with shear strain and the three tests reach a critical state value of stress ratio of approximately 0.5.  66  3.2.3.2. Direct shear tests  A series of direct shear tests was performed by the author of this thesis, using a fully-automated shear box, 100 mm by 100 mm in plan area. Specimens were reconstituted by pouring the sand with a spoon close to the surface of the sand sample, to a medium loose state (DR between 12% and 38%) and 10 tests performed over a relatively large range of normal stress between 4 kPa and 150 kPa, with shearing at 1 mm/min. Figure 3.5 shows the variation of shear stress and vertical deformation with horizontal displacement for all the test specimens in this study. The shear resistance increases with normal stress and a small peak is observed for specimens tested at normal stresses larger than 20 kPa. The vertical displacement shows a slight contraction before the specimens start dilating; the samples tested at vertical stresses lower than 20 kPa showing a larger dilatancy.   3.2.3.3. Triaxial tests  In addition to obtain the strength characteristic of Badger sand through triaxial testing, the objective of the triaxial program was to establish parameters for Badger sand for the critical state soil model NorSand. Of the eight parameters that describe NorSand, three are critical state parameters, that are determined using triaxial compression tests on reconstituted samples of loose soil to ensure completely contractive behavior.  Triaxial tests were performed on Badger sand, reconstituted using a water pluviation technique that has been found to ensure replication of saturated uniform loose specimens (Chern, 1981; Sivathayalan, 2000). The length of individual specimens, which varied between 123.42 mm and 124.44 mm, was obtained by comparison to a dummy sample of known height. The average diameter of each specimen was constant at 63.38 mm.  The triaxial test program comprised a total of seven tests: three drained tests were performed at confining pressures of 50 kPa (void ratio, e = 0.623), 100 kPa (e = 0.624), and 150 kPa (e = 0.620) and two undrained tests at confining pressures of 100 kPa (e = 0.624) and 150 kPa (e = 0.620) at The University of British Columbia (Figure 3.6). The relative density of all 67  specimens, measured after completion of consolidation, was found in the narrow range of 33% to 35%. Accordingly, the method of reconstitution yields specimens that are deemed loose, yet close to that of a medium-dense state (DR ≥ 35%). Although looser samples were expected, this was the loosest condition that could be achieved with the water pluviation reconstitution method for Badger sand, and the specimens exhibited significant dilation. Studies have shown that even after 25% to 30% strain, dilative specimens have not reached the critical state (Sivathayalan, 2000). The same experience has been reported by Vaid and Chern (1985) for a rounded Ottawa sand.   Therefore, and given that the drained triaxial tests on Badger sand did not reach the critical state, a value for the critical state friction angle was obtained from interpretation of stress-dilatancy concepts. Specifically, the critical state ratio (Mtc) can be obtained by plotting maximum stress ratio max = (q/p)max (where q is the deviatoric stress and p the mean effective stress) against the corresponding minimum dilatancy (Dmin) in the drained triaxial test (Bishop, 1966). Dilatancy is given by Dmin because of the dilation-negative convention. By definition, Dmin = 0 at the critical state. Extrapolation of a best fit through the triaxial tests of this study yields a value of Mtc = 1.07 for max at Dmin = 0 (see Fig. 3.7).   Given the unusually low value of strength found for Badger sand, and to verify that the deduced magnitude of the critical state friction angle was correct, two more triaxial drained tests on Badger sand were performed by Golder Associates (2004) using the moist tamped reconstitution method at 3 = 100 kPa and 300 kPa and relative densities of 65% (e = 0.560) and 76% (e = 0.538), respectively (Figure 3.8). These tests were also plotted in Figure 3.7 and agree with the trend found for the water pluviated tests performed at UBC. Accordingly, a critical state friction angle under triaxial conditions was found to be cs,tx = 27º, in the range of confining pressures of 50 to 300 kPa.      68  3.2.3.4. Angle of repose  The angle of repose (rep) was determined according to ASTM C1444-00 and was found to be 30.9º, and to exhibit little variation in magnitude (Rousé, 2005). It is in excellent agreement with a value of 31º reported for the same Badger sand by Fannin et al. (2005), following the laboratory bench-test procedure described by Bolton (1986) involving excavation at the toe of a loose dry heap, which yields a value believed accurate to ± 0.5º. Comparison indicates this angle of repose for Badger sand, confirmed by two independent tests, is appreciably greater than the critical state angle of friction of 27º. A study of Rousé (2014)2 shows that the value of rep depends on the method used for the measurement and supports the results of Rousé et al. (2008) and Fannin et al. (2005), that ASTM C1444-00 and the Cornforth method give similar values of rep (a difference of 1.5° in average is observed). This study allows confirmation that, although the values of rep and cs,tx are unusually low for Badger sand, their relative magnitude is consistent with the values for other sands.   These results show that strength values of Badger sand are consistent with their expected relative value among tests but are lower than usual values for sands. Additionally, the difference of extreme void ratios is very low (emax - emin = 0.2) and the shape of the sand grains is very rounded. In order to verify that these unusual values are reliable, an analysis is done to study the influence of the shape of the sand particles on the strength of uniform sands.                                                              2 Rousé, P.C. (2014) “Comparison of methods for the measurement of the angle of repose of granular materials”, Geotechnical Testing journal”, 37, No 1, pp 164-168. https://www.astm.org/DIGITAL_LIBRARY/JOURNALS/GEOTECH/PAGES/GTJ20120144.htm  69  3.2.3.5. Influence of roundness on the strength of uniform sand  Figure 3.3 shows that the values of emax and emin for Badger sand fall within an acceptable range from data in Table 3.1. Different studies have also shown that the magnitude of the friction angle also depends on the shape of the sand particles.   Santamarina and Cho (2004) proposed a linear relation between constant volume friction angle and the roundness of soil grains. The constant volume friction angle was taken to equal the angle of repose, with the latter established in a laboratory bench test described by Santamarina and Cho (2001) from observation of a saturated deposit in a water-filled cylinder. The roundness was approximated by visual comparison to charts of Krumbein and Sloss (1963). A compilation of those data for 24 sands (extracted from Cho et al., 2006) is reproduced in Fig. 3.9. The friction angle diminishes with increasing roundness. Values of rep and cs,tx for Badger sand are included for purposes of comparison: they conform to the general trend observed by Santamarina and Cho (2004). A linear fit to the data for soils alone gives:  𝑟𝑒𝑝= 41.8 − 14.4𝑅      (eq. 3.4)  Also included in Fig. 3.9 are values of the angle of maximum contraction in drained tests (MC) from triaxial testing reported by Sukumaran (1996), together with the value of MC for Badger sand obtained from the drained triaxial tests. The maximum contraction defines the point where the volumetric deformation passes from contraction to dilation (Luong, 1980; Negussey et al., 1988). A linear fit to the soils data takes a similar form as that for the angle of repose:       MC = 34.3 − 9.6𝑅       (eq. 3.5)    70  Although the result for Badger sand is in excellent agreement with the trend of the data, the linear relation contrasts with an exponential function postulated by Sukumaran and Ashmawy (2001). Inspection confirms the two friction angles are different in magnitude and suggests the angle of repose is about 4º to 6º larger than that calculated at maximum contraction.  The magnitude of ∆e = emax - emin is also plotted in Figure 3.9 and exhibits a clear decrease with increasing roundness. Although unusually low for a sand, the range in extreme void ratios of 0.2 that was obtained for Badger sand appears entirely reasonable. Figure 3.9 shows the relation between void ratio range e (i.e. emax - emin) and roundness of the particle for 46 soils and glass beads from Table 3.1. It is given by   𝛥𝑒 = 0.07𝑅−1 + 0.138     (eq. 3.6)  In quantifying the grain size characteristics of cohesionless soils, Cubrinovski and Ishihara (1999) found e to be the most appropriate index parameter, allowing for the effects of grain size and grain size distribution. Using e in the relation with roundness yields a scatter (coefficient of determination R2 = 0.62) less than that obtained for emin (R2 = 0.35) and similar to that for emax (R2 = 0.65). Inspection of Figs. 3.3 and 3.9 suggests that greater scatter occurs in values of roundness for sub-angular and subrounded grains. This scatter is attributed largely to many values of roundness being estimated from visual inspection using charts, rather than being calculated from equation 3.2.  The unified plot of Fig. 3.9 depicts the generalized influence of roundness. In principle, the unified plot enables a correlation between void ratio and friction angle for uniformly graded soils. For purposes of clarity, the dataset of Table 3.1 is reported as three subsets in Fig. 3.9: R against e and rep (Subset 1: 8 data pairs); R against e and MC (Subset 2: 8 data pairs); and R against e only, or R against rep only (Subset 3: 28 data points for e and 13 data points for rep). Notwithstanding the scatter that is noted for very angular to sub-rounded grains, the data imply a utility of the unified plot for rounded and well-rounded sands. Substituting the measured value of R = 0.81 for Badger sand in equations 3.4, 3.5 and 3.6 71  yields values for, rep and MC and e of, 30.08, 26.58 and 0.22, respectively. These estimated values are very close to those obtained by means of index and laboratory tests and reported in Table 3.2.   3.2.3.6. Stress-dependency of the angle of friction at the critical state  Figure 3.10 shows a summary of the critical state friction angles found for Badger sand through direct shear (cs,ds), triaxial (cs,tx) and direct simple shear (cs,dss) tests at stress levels between 4 kPa and 150 kPa, along with the angle of repose. Two factors are shown to affect the magnitude of the value of the critical state friction angle; the type of test and the stress level. cs,dss obtained at stresses between 50 and 150 kPa, is slightly lower than cs,tx at the same stress level. These results are consistent with the findings of Vaid and Sivathayalan (1996) on Fraser River sand for stresses up to 200 kPa. Additionally, the critical state friction angles in direct shear obtained at stresses between 50 and 150 kPa are approximately 2° lower than the values of cs,tx at the same stress level. This observation is consistent with the results of Rowe (1969) on loose samples of glass ballotini tested at stresses between 68 and 413 kPa.  Regarding the values of friction angle at low stress levels, Figure 3.10 shows that for the same test condition, in this case direct shear tests, cs,ds is found to diminish with increasing normal effective stress for values of  below 50 kPa, yielding a difference of 2.9° for a change from 20 to 4 kPa. The values of cs,ds obtained by Raju (1995) using a box 76 mm × 76 mm in plan area are also included in Figure 3.10. Excellent agreement is found between both test series, with cs,ds ≈ 30º at 4 kPa, 26º at 30 kPa. A similar observation is reported by Srikongsri (2010), using the same fully-automated shear box, who found a difference of 6º in two uniform sands and two silt-sand mixes that were similarly tested at normal effective stresses up to 50 kPa.    72  Although there is concern for the reliability of data obtained at such low stresses in the shear box, the magnitude of the difference in the friction angles of Badger sand is in excellent agreement with that observed by Lehane and Liu (2013). In their study, the authors used a conventional shear box and a modified shear apparatus consisting of a low friction Teflon box, and tested three different uniformly graded sand samples with mean particle sizes ranging from 0.12 to 0.45 mm, at a stress range between 4 and 400 kPa. The values of the critical state friction angle in Figure 3.11 show a clear stress-dependency that decreases with increasing levels of . The largest decrease occurs between 4 and 50 kPa with a difference of 4º to 5º in the measured friction angle.   The stress-dependency of the critical state friction angle has also been observed in the triaxial test data of Ponce and Bell (1971). The authors used a uniformly graded rounded to sub-rounded sand and tested it at confining stresses from 241 to 1.4 kPa. The results of their study reveal an increase of nearly 10º in the critical state friction angle, with a reduction in confining pressure. Figure 3.12 shows the friction angles obtained by Boyle (1995) on a rounded Ottawa sand and a poorly graded sand using triaxial tests at stress levels between 20 and 100 kPa. A reduction of approximately 3º in the critical state friction angle is observed.  A similar observation was reported by Fukushima and Tatsuoka (1984) and Tatsuoka et al. (1986) on triaxial and plane strain tests, respectively, performed on sands at confining pressures between 5 and 400 kPa. Chu (1995) concluded that the critical state friction angle is independent of the initial void ratio but depends on the confinement pressure, especially at very low stresses.  A study of Quinteros (2014) compares the friction angles obtained using the triaxial, the shear box and the plane strain apparatus of a uniformly graded clean sand with trace of fine gravel, called Skedsmo sand. A series of tests was performed in each test device at almost the same value of effective stress ranging from 4 to 300 kPa and a relative density of 52%. Figure 3.13 shows the values of friction angle for the direct shear and plane strain tests. Inspection of the triaxial test results reveals that the tests were stopped before the critical state was reached and hence they are not reported in this study. A distinctive stress-dependency is observed at 73  a stress level lower than 50 kPa for both data sets. The critical state friction angle in plane strain shows a reduction of nearly 10º and cs,ds a reduction of 7º approximately.   A recent study of Rousé (2018)3, also shows a stress-dependent behaviour of cs,ds at values of stresses lower than 50 kPa for five sands (Figure 3.14) and reveals that the difference in the magnitude of cs,ds between  =  kPa and  =  kPa is influenced by the shape of the sand particles (Figure 3.15). As the roundness of the sand particles increases, the difference in cs,ds decreases. As a result, this analysis provides confidence in the strength values of Badger sand to be used in the numerical modeling of the pullout tests.  3.3. Description of physical and mechanical properties of inclusions tested  Three sheet inclusions and three geogrids (see Figure 3.16) were tested by Raju (1995):  - Planar rough inextensible aluminum sheet (APT): this is a 1.5 mm thick, 0.5 m wide and 0.95 m long aluminum sheet with particles of Badger sand glued on both sides (Fig. 3.16a). This resulted in a fully rough inextensible inclusion with a soil-inclusion interface that can be treated as a sand-sand interface that allows the checking of material properties.  - Planar rough extensible geomembrane (GMT): this is a textured high-density polyethylene geomembrane HDT, that has a soil-inclusion interface roughness very close to the APT sheet (Fig. 3.16b). It is 2 mm thick, with a density of 0.94 g/cc and a modulus of elasticity of 770 MPa. Tensile strength at yield and break (ASTM D638-14) are reported to be 29 and 6 kN/m, respectively.                                                               3 Rousé, P.C. (2018) “Influence of vertical stress in the critical state friction angle at very low stresses in sands”, accepted for publication on June 22, 2018, Soils and Foundations 74  - Planar smooth extensible geomembrane (GMS): this is a smooth HDPE membrane, 1.5 mm thick, with a density of 0.94 g/cc and a modulus of elasticity of 770 MPa (Fig. 3.16c). Tensile strength at yield and break (ASTM D638-14) are reported to be 20.7 and 29 kN/m, respectively.   - Tensar UX-1500 (GGT): this is a uniaxial geogrid formed by punching small holes in a solid sheet of HDPE and preferentially stretching it in one direction. Geometry and dimensions of GGT are shown in Figure 3.16d. The ultimate tensile strength (ASTM D4595-86) is reported to be 86 kN/m.  - Miragrid 15T (GGM): this is a biaxial grid made of polyester multifilament yarns interlocked by weaving such that the yarns retain their relative position which are covered with a black polymeric coating. Geometry and dimensions of GGM are shown in Figure 3.16e. The ultimate tensile strength (ASTM D4595-86) is reported to be 124 kN/m.  - Stratagrid 700 (GGS): This is a uniaxial grid made from a polyester yarn. Yarns are bonded and interwoven at the junctions to form a dimensionally stable structure and covered with a black polymeric coating. Geometry and dimensions of GGS are shown in Figure 3.16f. The ultimate tensile strength (ASTM D4595-86) is reported to be 146 kN/m.  The values of elastic modulus for the different inclusions were taken from Raju (1995) and manufacturer’s information (Table 3.3). These polymeric materials exhibit a load rate, and temperature dependent elastic-viscoplastic behavior and therefore, the value of the elastic modulus (which is of interest for the numerical simulations of this study) should be verified. EPA (1988) reports values of elastic modulus obtained from ASTM D638 for HDPE membranes. Letting aside the effect of temperature (assumed the same on both laboratories), they obtained a value of E = 750 MPa for an 80-mil geomembrane, which is very close to the value reported by Raju (1995) of 770 MPa. Assuming a 1 m long test specimen, a stiffness of 1540 kN/m and 1155 kN/m is deduced for the GMT and GMS inclusions, respectively. 75  3.4. Soil-inclusion interface properties  Different options exist for the numerical modeling of pullout tests of sheet inclusions and geogrids. As described in Section 2.2.2.1, the interaction between the soil and the sheet inclusion is rather simple and is mostly described by the soil-inclusion interface friction angle, . On the contrary, the interaction between the soil and the geogrid is more complex due to the interaction of the soil and the longitudinal elements of the geogrid (as shearing) and the soil and the transverse elements of the geogrid (as passive resistance). Two-dimensional numerical models usually simplify the soil-geogrid interaction by considering the geogrid as an equivalent rough sheet inclusion (see for example Wilson-Fahmy and Koerner, 1993 or Yogarajah and Yeo, 1994). Jewell et al. (1985) suggested that the frictional and bearing components of the soil-inclusion interface interaction can be captured through the parameter fbtan as described in Section 2.2.2.2.   In this Section, the parameters for the soil-inclusion interface are described. For the sheet inclusions, tan is experimentally measured. For the geogrids, the geometric parameters to populate the parameter fbtan are shown in Table 3.4, and are used in Chapters 4 and 5 to describe the soil-inclusion interface friction in the numerical simulations of pullout tests on geogrids, and assess the validity of Jewell et al. (1985) approach.  3.4.1. Sheet inclusions  The determination of the soil-inclusion interface friction angle for the sheet inclusions is done by analyzing the laboratory test data of Figure 3.10 along with the direct shear tests between Badger sand and the sheet inclusions performed by Raju (1995). For the fully rough aluminum inclusion with Badger sand particles glued on both sides (APT), the soil-inclusion interface friction angle is equivalent to a soil-soil interface. The rough nature of the textured membrane (GMT) surface, allows the presumption that both, the APT and GMT inclusions have a similar soil-inclusion interface roughness. The direct shear tests performed by Raju (1995) show that the APT and GMT have indeed, a soil-inclusion interface friction angle that is very similar and equivalent to a soil-soil interface (Figure 3.17). Therefore, for the 76  subsequent analysis of this study (Chapters 4 and 5), the strength of the soil-inclusion interface in direct shear for the APT and GMT sheet inclusions is taken as a stress-dependent critical state friction angle determined according to Figure 3.10.   For the smooth geomembrane, direct shear tests between Badger sand and GMS were also performed by Raju (1995) and show a slight stress-dependency of the soil-inclusion interface friction angle (Figure 3.17).. Accordingly, and with reference to the direct shear test data, the value of the critical state soil-inclusion friction angle for the GMS-Badger sand interface is 10, 8 and 6, for  = 4 kPa, 8 kPa and 12 kPa, respectively (Table 3.4).  3.4.2. Geogrids  To verify the goodness of the Jewell et al. (1985) approach to capture the complex interaction of the soil-geogrid interface through a two-dimensional numerical model, the “equivalent” soil-inclusion interface friction angle for geogrids is determined using the expressions proposed by Jewell et al. (1985) and later modified by Palmeira and Milligan (1989) as described in Chapter 2 (equations 2.5 to 2.8). In these formulations, the effect of the roughness of the solid area of the grid and the existence of the bearing elements is incorporated, along with the influence of the grain size of the soil and the shape of the bearing elements. The influence of the coating of the geogrids in the soil-inclusion interaction is not accounted for in this formulation and is a moot point for the numerical simulations of this study where the geogrids are treated as planar sheet inclusion.  The original relation proposed by Jewell et al. (1985) and later modified by Palmeira and Milligan (1989) is given by:  𝑃 = 2𝐿𝜎𝑛𝑓𝑏tan∅                 (eq. 3.7)  With:  𝑓𝑏𝑡𝑎𝑛 = 𝛼𝑠tan𝛿𝑠 +𝜎𝑏𝜎𝑛12𝐵𝑡𝑆𝑡𝛼𝑏                       (eq. 3.8)  77  With:  s: the fraction of solid surface area in a geogrid s: soil-solid area of geogrid interface friction angle (°) Sb: spacing between bearing elements Bt: bearing element thickness  b/n: bearing stress ratio 𝛼𝑏: fraction of the geogrid width available for bearing  The first element in eq. 3.8 represents the sliding component of the pullout resistance between the soil and the solid surface of the geogrid. The second accounts for the bearing component that develops against the transverse elements as the inclusion is pulled out of the box.   Examination of eq. 3.8 shows that s, Bt/St and b are geometric parameters, and are available for each geogrid as shown in Table 3.5. s is the friction angle between the soil and the solid surface of the geogrid and can be deduced through direct shear tests. Unfortunately, such data does not exist for this study and it is thus assumed. Studies of Potyondy (1961) have shown that the ratio of skin friction to soil friction for sands acting on concrete surfaces is approximately 0.8. Given the rough nature of the solid surface of the Miragrid and Stratagrid geogrids, values of 0.8 times the interface friction angle of the APT (and GMT) sheet inclusion were assumed for the GGM and the GGS geogrids, and a value of s equal to the soil-GMS interface friction angle for the Tensar grid was adopted.   The next parameter to be determined is the bearing stress ratio, 𝜎𝑏 𝜎𝑛⁄ . Data from Jewell et al. (1985) and Palmeira and Milligan (1989) suggest that 𝜎𝑏 𝜎𝑛⁄  is a function of  (as per (𝜎𝑏 𝜎𝑛⁄ )∅ from Figure 2.4), the particle size of the soil, F1, (specifically the mean grain size D50) and the shape of the transversal element, F2, and is given by:        𝜎𝑏 𝜎𝑛⁄ = (𝜎𝑏 𝜎𝑛⁄ )∅ × 𝐹1 × 𝐹2      (eq. 3.9)    78  Table 3.5 shows the values of F1 and F2 for the three geogrids. The values for F1 were obtained by replacing the D50 = 0.9 mm of Badger sand (see Figure 3.1) and the thickness of the transverse elements of each geogrid in eq. 3.10 and F2 was taken equal to 1.2 for the Tensar grid and 1.0 for the Miragrid and the Stratagrid (Palmeira and Milligan, 1989).  𝐹1 = (20−𝐵𝑡 𝐷50⁄10)         (eq. 3.10)  The last parameter needed for describing the Jewell et al. (1985) approach is (𝜎𝑏 𝜎𝑛⁄ )∅ that can be determined from Figure 2.4, which is a semi-log relation of the bearing stress ratio as a function of friction angle. This parameter is, however, the principal source of uncertainty given its variability by a factor of 2 to 3, for the same value of friction angle. The implications of this variability are discussed in Chapters 4 and 5, where the numerical simulations of the geogrids are presented.    3.5. Description of pullout box  The laboratory pullout box (Fig 1.3) was designed and constructed at the University of British Columbia by Raju (1995). It accommodates a soil sample 1.3 m long, 0.64 m wide and 0.6 m deep. A pullout test specimen 1 m long and 0.5 m wide was selected to provide a clearance of 7.5 cm between the inclusion and the side wall of the box. The upper boundary is composed of an acrylic bag that works as a flexible boundary (a stress-controlled boundary with no friction), and the base plate is made of 13 mm thick aluminum plate that provides a rigid lower boundary. The side walls of the box are formed by a 2.5 cm thick plexiglass sheet and a 0.3 cm thick glass sheet is glued to the inside wall to minimize wall friction effects. The front wall is made of two 13 mm thick aluminum plates (a strain controlled – zero strain boundary with little friction) and the back wall is made of one 13 mm thick aluminum plate. The inclusion was clamped outside the pullout box and no sleeve was used to minimize front wall effects in the pullout resistance. Instrumentation was used to measure pullout force, displacement of the inclusion at the clamped and embedded end, horizontal stresses at the inside front wall of the pullout box. The pullout force was measured using a load cell connected between the clamp and the hydraulic actuator. Displacements of the clamped and 79  the embedded end of the inclusion were measured using LVDTs. Horizontal stresses at the front wall of the pullout box were measured using six total pressure transducers. A hopper was used to place the sand inside the pullout box to a targeted density of 85 % to 90 % relative density. A regulator was used to maintain a constant pressure during testing, and tests were allowed to reach a displacement of 76 mm with a rate of 0.5 mm/min. A detailed description of the pullout device is found in Raju (1995) and Raju and Fannin (1998).  The experimental results obtained from the pullout laboratory tests are termed herein the UBC database that will be used for the numerical analysis of this thesis.  3.6. Summary  The characterization of Badger sand included basic properties such as the minimum and maximum void ratio and the shape of the particles, along with the strength in triaxial, direct shear, and direct simple shear. The principal findings are as follows:  1. A value of roundness (R) measured following Wadell’s (1932) method was found equal to 0.81, indicative of a well-rounded sand. Inspection of the unified plot (Figure 3.9) shows the range for e = 0.2 and the friction angle of Badger sand (cs,tx = 27°) to be consistent with the roundness of its particles, giving confidence to the unusual frictional strength of the Badger sand.  2. The critical state friction angle of Badger sand is found to be test-dependent. For the same stress range between 50 kPa and 150 kPa, cs,ds < cs,dss < cs,tx.    80  3. The critical state friction angle of Badger sand is found to be stress-dependent in direct shear tests, at values of normal effective stress less than 50 kPa. The largest decrease in the angle of shearing resistance occurs between 4 kPa and 30 kPa. More specifically, cs,ds ≈ 30º at 4 kPa, ≈ 26º at 30 kPa, and ≈ 25º at  between 50 and 150 kPa. This is consistent with the findings of Boyle (1995), Lehane and Liu (2013), Quinteros (2014) and Rousé (2018), amongst others, in the triaxial, direct shear and plane strain tests.  81  Table 3.1: Material properties for Cu < 2.5 Material Description  () emax emin Cu Roundness Reference Lapis Lustre sand  … 0.754 0.460 1.4 0.44 N11 Monterey sand 1  … 0.772 0.469 1.4 0.39 N1 Monterey sand 2  … 0.799 0.458 1.4 0.34 N1 Ottawa sand 1  … 0.704 0.408 1.4 0.60 N1 Ottawa sand 2  … 0.772 0.407 1.4 0.42 N1 Ottawa sand 3  … 0.830 0.460 1.4 0.38 N1 Del Monte white sand 1  … 0.971 0.503 1.4 0.27 N1 Del Monte white sand 2  … 1.082 0.550 1.4 0.23 N1 Del Monte white sand 3  … 1.203 0.636 1.4 0.21 N1 Crushed basalt 1  … 1.190 0.700 1.4 0.20 N1 Crushed basalt 2  … 1.260 0.722 1.4 0.19 N1 Crushed basalt 3  … 1.320 0.692 1.4 0.19 N1 Crushed basalt 4  … 1.350 0.747 1.4 0.18 N1 Crushed basalt 5  … 1.420 0.803 1.4 0.17 N1 MOL  … 0.799 0.458 1.4 0.34 N1 MOL  … 0.688 0.370 2.5 0.35 N1 CB  … 1.257 0.705 1.4 0.19 N1 CB  … 1.099 0.590 2.5 0.19 N1 Sand  … 1.156  … ≤ 2 0.165 N22 Sand  … 1.078  … ≤ 2  0.185 N2 Sand  … 1.047  … ≤ 2  0.240 N2 Sand  … 0.983  … ≤ 2  0.300 N2 Sand  … 1.000  … ≤ 2  0.325 N2 Sand  … 1.079  … ≤ 2  0.345 N2 Sand  … 1.094  … ≤ 2  0.356 N2 Sand  … 0.938  … ≤ 2  0.405 N2 Gravel  … 1.031  …  ≤ 2 0.315 N2                                                           1 N1: Youd (1973) 2 N2: Shimobe and Moroto (1995) 82  Material Description  () emax emin Cu Roundness Reference Gravel  … 0.983  …  ≤ 2 0.300 N2 Gravel  … 0.656  …  ≤ 2 0.655 N2 Glass beads  … 0.703  …  ≤ 2 1.000 N2 Glass beads  … 0.657  …  ≤ 2 1.000 N2 Glass beads  … 0.625  …  ≤ 2 1.000 N2 Syncrude Tailings sand 31.7 1.14 0.59 2.5 0.25 N33, N44 Ottawa 45 33.5 1.11 0.75 2.1 0.24 N3, N4 Fraser River sand 35.2 1.13 0.78 1.9 0.43 N3 Daytona Beach sand 32.4 1.00 0.64 1.4 0.30 N3, N4 Michigan Dune sand 29.6 0.80 0.56 1.5 0.32 N3 Ottawa 20/70 27.8 0.78 0.47 2.4 0.38 N3, N4 Ottawa 90 32.4 1.10 0.73 2.2 0.16 N3, N4 Glass Ballotini 23.3 0.91 0.34 1.4 ≈ 1.00 N4 Nevada sand 31 0.850 0.570 1.8 0.60 N55 Ticino sand 37 0.990 0.574 1.5 0.40 N5 Margaret river sand 33 0.870  … 1.9 0.70 N5 Ottawa sand 32 0.690  … 1.4 0.80 N5 Ponte Vedra sand 39 1.070  … 1.8 0.30 N5 9C1-crushed sand 39 0.910  … 2.3 0.25 N5 Jekyll island sand 40 1.040  … 1.7 0.30 N5 ASTM graded sand 30 0.820 0.500 1.7 0.80 N5 Blasting sand 34 1.025 0.698 1.9 0.30 N5 Glass beads 21 0.720 0.542 1.4 1.00 N5 Ottawa 20/30 sand  27 0.742 0.502 1.2 0.90 N5 Ottawa F-110 sand 31 0.848 0.535 1.7 0.70 N5 3P3-crushed sand  … 0.950 …  2.2 0.20 N5 Sandboil sand 33 0.790 0.510 2.4 0.55 N5                                                           3 N3: Sukumaran and Ashmawy (2001) 4 N4: Sukumaran (1996) 5 N5: Cho et al. (2006) 83  Material Description  () emax emin Cu Roundness Reference Gabbro  … 0.899 0.652 ≤ 1.6 0.27 N66 Greywacke  … 0.975 0.679  ≤ 1.6 0.30 N6 Slate  … 1.031 0.740  ≤ 1.6 0.31 N6 Greywacke  … 0.973 0.720  ≤ 1.6 0.31 N6 Gabbro+Greywacke  … 0.992 0.717  ≤ 1.6 0.32 N6 Rhyolite  … 1.020 0.740  ≤ 1.6 0.34 N6 Dolelite  … 0.961 0.685  ≤ 1.6 0.39 N6 River gravel  … 0.839 0.551  ≤ 1.6 0.43 N6 Beach gravel  … 0.744 0.525  ≤ 1.6 0.58 N6 Rounded C  … 0.922 0.654  ≤ 1.6 0.38 N6 Rounded A  … 0.846 0.605  ≤ 1.6 0.41 N6 Badger sand See Table 3.2 0.69 0.49 1.3 0.81 N77                                                              6 N6: Moroto and Ishii (1990) 7 N7: Rousé (2005)   84  Table 3.2: Summary of properties of Badger sand Property Average Maximum value Minimum value Number of tests Roundness 0.81 0.983 0.712 30 emax 0.69 0.697 0.691 6 emin 0.49 0.495 0.477 4 MC 25.8° 26.7° 25.4° 3 cs,tx 27.0° … … 3 rep 30.9 31.0 30.8 10   Table 3.3: Material properties for inclusions  Inclusion Elastic Modulus  (kPa) Ultimate strength  (kN/m) APT 7.7×107  4.8×104 GMT 7.7×105  29 GMS GGT GGS GGM 7.7×105  3.6×104 9.6×104 9.6×104 20.7 86 146 124 *Poisson’s ratio for the inclusion is 0.2.    85  Table 3.4: Soil-inclusion interface friction angle for sheets from direct shear tests Inclusion APT GMT GMS  = 4 kPa 29.4 29.4 10  = 8 kPa  28.2 28.2 8  = 12 kPa 27.8 27.8 6    Table 3.5: Soil-inclusion interface properties for geogrids  Geogrid s b Bt/St F1 F2 GGT 0.573 0.70 0.030 1.52 1.2 GGS 0.534 0.64 0.027 1.77 1.0 GGM 0.453 0.56 0.034 1.82 1.0    86             Figure 3.1: Grain size distribution of Badger sand               Figure 3.5: Badger sand: sphericity and roundness (Rousé et al, 2008)  Figure 3.2: Badger sand: sphericity and roundness (Rousé et al., 2008)    rmax Rmin rmax ri 1 mm 01020304050607080901000.01 0.1 1 10Percentage finer (%)Particle diameter (mm)87                Figure 3.3: Influence of roundness on extreme void ratio (Rousé et al., 2008)     1max 107.0651.0−+= Re  1min 051.0433.0−+= Re  88                               Figure 3.4: Direct simple shear tests on Badger sand at: a) 50 kPa, b) 100 kPa and        c) 150 kPa (Dabeet, 2013) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 a) 0.5 0.4 0.3 0.2 0.6 0.1 0.0 b) 0.5 0.4 0.3 0.2 0.6 0.1 0.0 c) 89                         Figure 3.5: Direct shear tests on Badger sand    90                   Figure 3.6: Triaxial test results from water pluviation: a) deviatoric stress drained tests, b) deviatoric stress undrained tests, c) volumetric strain drained tests, d) pore water pressure undrained tests    a) b) c) d) 3 = 50 kPa 3 = 150 kPa 3 = 100 kPa 3 = 50 kPa 3 = 100 kPa 3 = 150 kPa 3 = 150 kPa 3 = 100 kPa 3 = 150 kPa 3 = 100 kPa 91              Figure 3.7: Dmin versus max for Badger sand             Figure 3.8: Triaxial test results from moist tamped: a) deviatoric stress, b) volumetric strain     a) b) 3 = 100 kPa 3 = 300 kPa 3 = 100 kPa 3 = 300 kPa 1.051.101.151.201.251.301.351.40-0.40-0.35-0.30-0.25-0.20-0.15-0.10-0.050.00DminmaxCurrent studyGolder 2004CDP50CDP100CDP150max = 0.21Dmin+1.07Minimum dilatancy, Dmin Maximum stress ratio, max       Current study       Golder (2004) 3 = 100 kPa 3 = 150 kPa 3 = 50 kPa 3 = 300 kPa 92                             Figure 3.9: Unified relation between roundness, range of extreme void ratio and friction angles in uniform sands. Subset 1: data from Table 3.1, where values of R, e and rep exist. Subset 2: data from Table 3.1, where values of R, e and MC exist. Subset 3: data from Table 3.1, where values of R, and either e or  exist (Rousé et al., 2008) 0.00.20.40.60.81.0emax - eminRoundness, RSubset 1Subset 2Subset 3Badger sand0.00.10.20.30.40.50.60.70.80.91.00.010.020.030.040.050.0Friction angle, Subset 1Subset 2Subset 3Badger sandrep = 41.7-14.4RWell  RoundedRoundedSubroundedSubangularAngularVery angularemax-emin = 0.07R-1 + 0.138Cho et al.rep = 42-17RMC = 34.3-9.6R93                Figure 3.10: Friction angles for Badger sand     Figure 3.11: Critical state friction angle as a function of normal stress (from Lehane and Liu, 2013)  051015202530354045501 10 100 1000' cs(deg)Normal stress (kPa)SGBUWA-SandGB20222426283032340 20 40 60 80 100 120 140 160Fricition angle, ()Effective stress,  (kPa)AAAAAA AAAAAAA (current study) AAAAAAA (Raju 1995)AAAAAAA (current study) AAAAAAA (current study)cs,tx cs,dscs,ds cs,dssrep94          Figure 3.12: Stress-dependency of the critical state friction angle in the triaxial test (data from Boyle, 1995)       Figure 3.13: Stress-dependency of the critical state friction angle of Skedsmo sand, in plane strain and direct shear test (data from Quinteros, 2014)   30343842460 20 40 60 80 100 120Critical state friction angle ( )Effective stress (kPa)triaxial data soil Otriaxial data soil R20253035404550550 100 200 300 400Critical state friction angle ( )Effective stress (kPa)direct shearplane strain95              Figure 3.14: Critical state friction angle in direct shear as a function of vertical stress (from Rousé, 2018)           Figure 3.15: Difference of cs,ds between  = 4 kPa and  = 50 kPa as a function of roundness (from Rousé, 2018) 0246810120 0.2 0.4 0.6 0.8 1Difference of cs,dsbetween n= 4 kPa and n= 50 kPaRoundnessSand DSand ESand ASand CSand BSand FSkedsmo Sand20253035404550550 50 100 150Critical state friction angle in direct shear ( )Vertical effective stress (kPa)Sand ASand BSand CSand DSand ESand FSand F (Raju 1995)96              Figure 3.16: Inclusions tested: a) Rough aluminum sheet (APT), b) Rough membrane (GMT), c) Smooth membrane (GMS), d) Tensar grid (GGT), e) Miragrid 15T (GGM) and f) Stratagrid 700 (GGS), all dimensions in mm (Raju, 1995)           Figure 3.17: Shear stress and normal stress relationship for various interfaces from direct shear tests a) c) b) d) f) e) 02468100 5 10 15 20Shear stress (kPa)Normal stress (kPa)Sand-SandMS-SandMT-Sand97  Chapter 4 Numerical simulation of pullout tests at large displacement444.1. Introduction  The present Chapter aims to gain a better understanding of the stress and strength conditions affecting pullout resistance, through the numerical modeling of the laboratory pullout tests performed by Raju (1995). Numerical simulation, for an assumed plane strain condition in the vertical cross-section of the pullout box, is made using the two-dimensional non-linear finite-difference program FLAC5.0 (Itasca, 2005). Badger sand is modeled as a non-associated Mohr-Coulomb material and therefore, only large displacement conditions are considered in the analysis. The inclusions are modeled as an elastic-plastic structural beam and the soil-inclusion interface is represented by linear elastic springs with Mohr-Coulomb strength. The laboratory testing program presented in Chapter 3 allowed for the characterization of Badger sand and provides the input for the calibration of the Mohr-Coulomb constitutive model representing the behavior of Badger sand inside the pullout box.   As shown in Chapter 3, the parameters for the soil-sheet inclusion interface are experimentally derived from the direct shear tests performed by Raju (1995) and constitute, therefore, sound quality input parameters to the numerical model, giving confidence in the results of the numerical simulations. Accordingly, in this Chapter, numerical simulations of the pullout resistance at large displacement of sheet inclusions allow establishing evidence-based claims to address research objective two by: (1) putting forward evidence in support of the claim that plane strain conditions prevail in pullout and, (2) showing that it is important to account for the stress-dependency of the soil-inclusion interface friction angle in the back-analysis of pullout tests.                                                             4 Some of these results were published as a Technical note:  Rousé, P.C., Fannin, R.J. and Taiebat, M. (2014) “Sand strength for back-analysis of pullout tests at large displacement”, Geotechnique 64, No 4, pp 320 –324.  https://www.icevirtuallibrary.com/doi/full/10.1680/geot.13.T.021  98  In addition, and related to the last item of research objective three, the geogrids tested by Raju (1995) are likewise modeled using the numerical model described above. In this case, the geogrids are also represented by a structural beam simplifying, therefore, the complex three-dimensional aspects of the geogrids to sheet inclusions, and using the Jewell et al. (1985) approach for the determination of the soil-geogrid interface friction angle as per Section 3.4.2. The analysis of the results allows discussing the limitations of the simplification suggested by Jewell et al. (1985) for the two-dimensional numerical simulations of pullout tests on geogrids at large displacement.  4.2. Numerical simulation of sheet inclusions at large displacement   4.2.1. Model parameters for Badger sand   In this Chapter, the behavior of Badger sand inside the pullout box is modeled as a non-associated Mohr-Coulomb material, using four parameters, namely the shear modulus (G), Poisson’s ratio (), friction angle () and dilation angle (). Values for the shear modulus were taken from the additional triaxial tests on specimens of Badger sand reported by Golder Associates (2004) by attaching bender elements to the specimen and recording the shear wave velocities as the specimen was tested (Figure 4.1). The relation obtained between shear modulus and mean effective stress (p) is:                                  𝐺 = 𝑝𝑎9.35 (𝑝𝑝𝑎)0.47× 1000     (MPa)                  (eq. 4.1)   Where pa is the atmospheric pressure. It is recognized that the reconstitution method determines the fabric of a sand specimen. However, as elasticity parameters are needed in this study, this finding is reported, to establish a value for the same well-rounded sand rather than assuming a value believed appropriate to Badger sand.  Poisson’s ratio was assumed equal to 0.2 and a dilation angle of 3° was obtained from the volumetric strain-axial strain curve, as shown in Figure 4.2. Also shown in Figure 4.2 is the calibration of the Mohr-Coulomb constitutive model to the triaxial tests at confining stresses of 50 99  kPa, 100 kPa and 150 kPa, using a value of cs,tx = 27° deduced using Bolton’s method (Figure 3.7).  For the numerical simulations of this Chapter and to demonstrate that plane strain conditions and a stress-dependency of the soil-inclusion interaction factor prevail in pullout at large displacements, a plane strain friction angle at the critical state, cs,ps, is needed as input to the numerical model at different stress levels. Since no plane strain laboratory tests were performed on Badger sand, in this study, values of cs,ps, are deduced from the values of cs,ds reported in Figure 3.10 and using eq. 4.2 (see Section 2.3.5; Rowe, 1969):  sincs,ps = tancs,ds              (eq. 4.2)   The deduced values of cs,ps for Badger sand are shown in Figure 4.3. cs,ps exhibits the same stress-dependent behavior observed for cs,ds, which is consistent with the results of plane strain tests on Toyoura sand (Tatsuoka et al., 1986) and Skedsmo sand (Quinteros, 2014) that also infer a stress-dependency at confining stresses less than about 50 kPa. Additionally, an independent check can be done using equation 4.3a and 4.3b of Gutierrez and Wang (2009), which is a non-coaxial version of Rowe’s equation based on a relation proposed by Oda (1975) that needs the friction angle obtained from the direct simple shear test. From the measured values of cs,dss for Badger sand (Fig. 4.3), the use of equations 4.3a and 4.3b yields deduced values of 28.3 < cs,ps < 28.8 for 50 <  < 150 kPa.   𝑠𝑖𝑛𝑝𝑠=𝑡𝑎𝑛𝑑𝑠𝑠𝑠𝑖𝑛2𝛼−𝑐𝑜𝑠2𝛼𝑡𝑎𝑛𝑑𝑠𝑠               (eq. 4.3a)   𝑡𝑎𝑛𝑑𝑠𝑠= 𝑠𝑖𝑛𝑐𝑠.𝑑𝑠𝑠𝑡𝑎𝑛𝛼           (eq. 4.3b)  These results are in very good agreement with the values of 27.78 < cs,ps < 28.18 deduced from equation 4.2 over the same range of stress (Fig. 4.3). Accordingly, using equation 4.2 to deduce values of cs,ps  at   < 50 kPa is believed to be acceptable.    100  4.2.2. Model parameters for the inclusions  The inclusions are represented by a structural beam element in FLAC5.0, discretized in 42 elements. The beam is a two-dimensional element, with 3 degrees of freedom at each end node (x-translation, y-translation and rotation), that behaves as a linearly elastic material with both an axial tensile and compressive failure limit. Accordingly, the elastic model representing the inclusions requires the identification of the elastic properties, namely the Elastic modulus and Poisson’s ratio. Additional parameters related to the ultimate strength are also needed. In this study, for each inclusion, these properties were assigned according to manufacturer’s information (see Table 4.1 and Raju, 1995).  4.2.3. Model parameters for the soil-inclusion interface   The beam representing the inclusion and the mesh representing the soil are attached by an interface element characterized by linear elastic springs with Mohr-Coulomb strength, that need a normal and shear stiffness, kn and ks, respectively. In order to determine a value for the “springs” constants, a good rule-of-thumb is that kn and ks be set to ten times the equivalent stiffness of the stiffest neighboring zone (Itasca, 2017). Since the mesh of the numerical model is the same for the simulations of all the inclusions, the values for normal stiffness (kn = 540 MPa/m) and shear stiffness (ks = 54 MPa/m) were taken as equal for all the soil-inclusion interfaces. These “springs” are also defined by friction, cohesion and dilation angle. In this study, the cohesion was assumed equal to 0 kPa and the dilation angle equal to 3, as per Figure 4.2.   In order to test whether plane strain conditions and a stress-dependency of the soil-inclusion interaction factor prevail in pullout at large displacements, the correct soil-inclusion interface friction angle at the appropriate stress level is needed. In this study, for the APT and GMT sheet inclusions, that have a surface roughness equivalent to a soil-soil interface, the soil-inclusion interface friction angle is considered to be equal to that of Badger sand at the critical state (see Section 3.4.1), and taken as cs,ps at the different stress levels of 4, 8 and 12 kPa (Figure 4.3), namely 34, 32.3 and 31.5, respectively (see Table 4.2). For the smooth geomembrane, GMS, the soil-inclusion interface friction angle in plane strain was deduced using equation 4.2 and the 101  values in direct shear of 10, 8 and 6 for  = 4 kPa, 8 kPa and 12 kPa, respectively (see Section 3.4.1), resulting in values of cs,ps equal to 10.2, 8.1 and 6 (see Table 4.2).   4.2.4. Numerical model   Numerical simulation of the pullout tests at large displacement was done using the software FLAC5.0 (Itasca, 2017). The numerical model consisted of a mesh with 65 elements for the length of the pullout box and 30 elements for the height. Boundary conditions of the model are illustrated in Figure 4.4. The front and back boundaries of the pullout box are fixed in the horizontal direction which assumes frictionless walls. This is consistent with the data of Palmeira and Milligan (1989) and Raju (1995) that suggest that the pullout resistance obtained using a lubricated front wall and a metal front wall appear to converge at large displacement. Additionally, studies by Bayoumi (2000) and Bergado et al. (1992) use boundary conditions of the numerical model that are considered frictionless for the front and back walls (only the horizontal displacement was restrained). The upper boundary is a stress-controlled boundary, and the bottom is a fixed boundary in the horizontal and vertical directions.  A multi-stage simulation was performed, in which the soil was taken to geostatic equilibrium, a surcharge pressure applied to the top boundary to reach the desired vertical stress at the soil-inclusion interface level, and the inclusion then displaced in the horizontal direction at a velocity of 1x10-9 m/step.    4.2.5. Simulation results  4.2.5.1. Pullout response at large displacement  The pullout response of the sheet inclusions tested by Raju (1995) is shown in Figure 4.5. A total of five pullout tests were performed on the inclusions at a normal effective stress of n = 4 kPa 102  (APT04, APT04R2, APT04RR, GMT04 and GMS04), three tests at n = 8 kPa (APT08, GMT08 and GMS08) and three tests at n = 12 kPa (APT12, GMT12, GMS12). Mobilized pullout resistance increases with vertical stress. For the APT inclusion, in all cases except one, a peak value of pullout resistance develops that is followed by a strain softening behavior. The exception is a test at 4 kPa for which the response at small displacement is a consequence of an inadvertent “pre-shearing” of the inclusion at the time of clamping it to the control system (Raju, 1995). However, all three tests at 4 kPa yield a similar and constant value of pullout resistance at large displacement (with a difference of 4%). Similarly, the tests at 8 kPa and 12 kPa yield a constant value of pullout. The GMT inclusion does not show a peak and strain softening pullout response but a gradual increase in pullout force, and reaches similar large displacement values than the APT tests at the same n. The GMS inclusion also shows a peak and strain softening behavior of the pullout response for all the stress levels but reaches smaller values than the APT and GMT inclusions.  The simulations using the stress-dependent parameters in plane strain, are also shown in Figure 4.5 for the sheet inclusions. A comparison of the measured and simulated values of pullout force per unit width is only valid at large displacement for input, to the numerical model, of the stress-dependent angle of friction mobilized at the critical state. A very good agreement is observed between measured and simulated pullout resistance at large displacement for the APT, GMT and GMS sheet inclusions at all stress levels. The good agreement between measured and simulated values of pullout force per unit width using a stress-dependent cs,ps and cs,ps suggests that plane strain conditions and a stress-dependency of the soil-inclusion interaction factor prevail at large displacement in the pullout box.                                                                2 APT04, APT04R and APT04RR are three pullout tests on the aluminum rough inclusion at a vertical stress of 4 kPa. The APT04 test was repeated because it was inadvertently disturbed during campling of the inclusion (Raju, 1995). 103  4.2.5.2. Sensitivity analysis to input parameters  A sensitivity analysis was performed on the APT inclusion tested at a vertical stress of 8 Kpa, on four of the five parameters used to model the soil and the interface, while holding constant the parameters for the beam. The objective was to determine the relative influence of each significant parameter, with the exception of the friction angle of the sand. Choosing a nominal value of 34.3 for the friction angle of the sand, the declared values of G, , kn and ks were varied by a range of ± 10%, ± 50%, and + 100% to yield 12.5 MPa ≤ G ≤ 50 MPa, 0.1 ≤  ≤ 0.4, 27 MPa/m ≤ kn ≤ 108 MPa/m, and 270 MPa/m ≤ ks ≤ 1080 MPa/m. Results of the sensitivity analysis indicate the pullout resistance at large displacement to vary by less than ± 3% (see Table 4.3).  The variation is deemed acceptable, given the variation in measured pullout resistance at a normal effective stress of 4 kPa for the APT inclusion (see Fig. 4.5a).  4.2.5.3. Horizontal stresses at the front wall  Measurements of horizontal stress due to pullout on the front wall of the box, are available for tests on the GMT sheet inclusion (Raju, 1995). Here, the incremental horizontal stress, h, is defined as the difference between the horizontal stress at peak pullout resistance and that after surcharge loading. h increases with the vertical stress applied and the profile of this lateral stress is asymmetric (Fig. 4.6): the response is attributed to the different boundary condition imposed on the top and bottom of the sand specimen. A good agreement is found between the numerical simulations using stress-dependent plane strain parameters and the laboratory test results for the GMT sheet inclusion at all stress levels. This further suggests that plane strain conditions and a stress-dependency of the soil-inclusion interaction factor prevail in pullout testing.    104  4.3. Numerical simulation of geogrids at large displacement   As described in Section 2.2.2.2, the three-dimensional soil-geogrid interface interaction is a very complex mechanism and even more complex to represent in a two-dimensional numerical model. Hence, a typical way of modeling the soil-grid interaction has been to consider the geogrid as a sheet inclusion (see Section 2.3.2). Section 2.2.2.2 shows data from Wilson-Fahmy et al. (1994), Farrag et al. (1993) and Bathurst and Ezzein (2016) and suggests that the frictional resistance in a geogrid represents around 80% of the total pullout resistance and hence, the assumption of a two-dimensional rough sheet inclusion is not unreasonable. Jewell et al. (1985) suggested that the frictional and bearing components of geogrids can be captured using eq. 3.7 to eq. 3.10. Accordingly, in this study the suitability of the Jewell et al. (1985) approach to capture the large displacement pullout resistance of geogrids is studied, and the GGT, GGM and GGS geogrids are also considered as sheet inclusions and therefore, a value of the apparent soil-inclusion interface friction angle, cs,ps, is needed for each stress level of 4, 10 and 17 kPa.   In this Section, cs,ps is determined using the procedure proposed by Jewell et al. (1985), as given by equation 4.4.  𝑡𝑎𝑛𝑐𝑠,𝑝𝑠 = 𝑓𝑏𝑡𝑎𝑛𝑐𝑠,𝑝𝑠 = 𝛼𝑠tan𝛿𝑠 +12(𝜎𝑏𝜎𝑛)∅𝐹1𝐹2𝐵𝑡𝑆𝑡𝛼𝑏           (eq. 4.4)   The values of the geometrical parameters s, b, Bt/St, F1 and F2 are already shown in Table 3.4. s is the soil-solid are of the geogrid interface friction angle and is taken equal to the stress-dependent soil-GMS interface friction angle in plane strain for the GGT geogrid (that has a smooth surface) and for the GGS and GGM geogrids (that have a rough surface), as 0.8 times the stress-dependent soil-APT interface friction angle in plane strain, as per Section 4.2.3 (see Table 4.4).  As explained in Section 3.4.2. the principal source of uncertainty of the Jewell et al. (1985) approach is the adopted value of the bearing stress ratio (b/n), which is a function of the “soil’s friction angle” (see Figure 2.4). Inspection of eq. 4.4 shows that (b/n) has a direct influence on the value of the soil-geogrid interface friction angle and, therefore, on the value of the pullout 105  resistance at large displacement. As the objective of this Section is the numerical simulation of the pullout response at large displacements of geogrids, the “soil’s friction angle” used to obtain the value of (b/n) is taken equal to the stress-dependent cs,ps at  = 4 kPa, 10 kPa and 17 kPa (see Figure 4.3 and Table 4.4). The bearing stress ratio (b/n), is then evaluated considering eqs. 4.5 and 4.6 (see Table 4.4).   (𝜎𝑏𝜎𝑛)= 𝑒(90+𝑐𝑠,𝑝𝑠)𝑡𝑎𝑛𝑐𝑠,𝑝𝑠tan⁡(45 + 𝑐𝑠,𝑝𝑠2⁄ )           (eq. 4.5)   (𝜎𝑏𝜎𝑛)= 𝑒𝑡𝑎𝑛𝑐𝑠,𝑝𝑠tan2⁡(45 + 𝑐𝑠,𝑝𝑠2⁄ )             (eq. 4.6)  Replacing the values of the geometrical parameters (Table 3.4) along with the values of the upper and lower bounds of the bearing stress ratio in eq. 4.4, maximum and minimum values of cs,ps for the three geogrids tested at n = 4, 10 and 17 kPa are obtained (see Table 4.5).   4.3.1. Pullout response at large displacement  The pullout response of the three geogrids tested by Raju (1995) is shown in Figure 4.7. Three tests were performed at n = 4 kPa (GGT04, GGS04, GGM04), three tests at n = 10 kPa (GGT10, GGS10, GGM10) and three tests at n = 17 kPa (GGT17, GGS17, GGM17). Pullout resistance also increases with vertical stress and the three grids show similar large displacement pullout resistance at the same stress level, although the shape, surface texture and flexibility of bearing elements is very different (see Sections 3.3 and 3.4). The GGT geogrid shows mostly a strain hardening pullout response at all stress levels. The GGS geogrid shows a gradual increase of pullout resistance with displacement for n = 4 and 10 kPa and a slight strain softening response for n = 17 kPa without reaching a constant value at large displacement. A similar response is observed for the GGM geogrid however it appears that a constant value could have been reached at larger displacements for n = 17 kPa.  106  Also in Figure 4.7 are shown the results of the numerical simulations using the maximum and minimum values of cs,ps shown in Table 4.5, for each grid at each stress level. In general, the numerical simulations using the upper boundary of the bearing stress ratio (eq. 4.6) show a reasonable good agreement with the measured values of pullout resistance at large displacement. This suggests that considering the bearing elements as a footing rotated to the horizontal is a reasonable assumption, as suggested by Jewell et al. (1985). On the contrary, the simulations using the lower boundary of the bearing stress ratio underestimate the measured values of pullout resistance at large displacement.   Figure 4.8 shows a comparison of the measured values of pullout force per unit width at large displacement (Pm) and simulated (Ps) values using the maximum and the minimum values of cs,ps in all tests on geogrids. The use of eq. 4.6, which is the upper estimate of (b/n), shows, in general, a good fit to the measured values (Ps = 1.01 Pm) and shows that the use of eq. 4.5 underestimates the measured resistance by approximately 35% (Ps = 0.66 Pm).       4.4. Numerical simulations using direct shear parameters  Additional simulations are performed considering direct shear parameters. For the APT and GMT inclusion the values of cs,ds are taken as the stress-dependent cs,ds of Badger sand, and for the GMS inclusion cs,ds is obtained from the direct shear tests performed by Raju (1995) as described in Section 3.4.1. (see Table 4.6).   The good agreement between measured and simulated values for the geogrids using the approach proposed by Jewell et al. (1985) and eq. 4.6, allows for the repetition of the procedure but using direct shear parameters. The geometric parameters remain constant for each geogrid and the value of s is taken equal to the stress-dependent soil-GMS interface friction angle in direct shear for the GGT geogrid and for the GGS and GGM geogrids, as 0.8 times the stress-dependent soil-APT interface friction angle in direct shear (see Table 4.7). The “soil’s friction angle” used in eq. 4.6 to obtain a value of the bearing stress ratio is taken as the stress-dependent cs,ds. The values of (b/n) are shown in Table 4.7 and the calculated values of cs,ds for each geogrid at each stress level is shown in Table 4.6. 107  The numerical simulations using direct shear parameters underestimate the pullout resistance at large displacement for all the sheet inclusions (Figure 4.5) and the geogrids (Figure 4.9) at the different stress levels. Similarly, the profile of horizontal stresses for the simulations using direct shear parameters for the GMT sheet inclusion and the three geogrids are compared to the laboratory test results. Inspection shows the numerical simulations using direct shear parameters again underestimate the measured values (see Figures 4.6 and 4.10).  A comparison of the measured (Pm) and simulated (Ps) values of pullout force per unit width in all tests on sheet inclusions and geogrids (Figure 4.11), reveals simulation using plane strain parameters provides a good fit to the measured values (Ps = 0.98 Pm), and shows use of direct shear parameters to underestimate the measured response by approximately 25% (Ps = 0.77 Pm). The finding is consistent with the observation of Jewell and Wroth (1987) that use of the direct shear friction angle results in a hidden factor of safety of about 1.2 in design of a reinforced soil structure. Additionally, a direct comparison of the measured (hm) and simulated (hs) horizontal stresses at the front wall of the box (Figure 4.12), reveals simulation using plane strain parameters provides a good fit to the measured values (Δσhs = 0.98Δσhm), and shows that the use of direct shear friction angles also underestimates the measured response by approximately 25% (Δσhs = 0.74 Δσhm).   Figure 4.13 shows the variation of the horizontal stresses at peak pullout resistance normalized by the average shear resistance (h/av) deduced at peak pullout resistance for the GMT inclusion Raju (1995). The laboratory test results show that although the horizontal stresses increase with n, h/av appears to be constant at the different vertical stresses applied. This observation agrees with the numerical results of this Section that both, the pullout resistance and the horizontal stresses are underestimated in approximately 25% when using cs,ds.  A second analysis was performed to determine the influence of the magnitude of the friction angle used for back-analysis of pullout tests. In the literature, this friction angle has been usually chosen as constant, irrespective of the stress level in the pullout box; however, as described in Section 3.2.3.6, there is evidence that at low stresses the strength of the soil is stress-dependent due to dilatancy effects. Figure 4.14 shows the normalized values of cs,ps of Figure 4.3 by the values of cs,ps at n = 150 kPa and illustrates the importance of accounting for the stress-dependency of 108  mobilized friction angle: using a value of cs,ps obtained at   = 150 kPa would underestimate the pullout resistance, at values of n less than about 20 kPa, by approximately 10 to 30 %.  4.5. Summary   This Chapter shows the results of the numerical simulations in FLAC5.0 at large displacement of the laboratory pullout tests performed by Raju (1995) in three sheet inclusions and three geogrids. The six inclusions were modeled as sheet inclusions and were represented by a beam element and the soil by the Mohr-Coulomb constitutive model. The soil-inclusion interface was described by springs following a Mohr-Coulomb type of failure.  The following observations arose from the analysis on sheet inclusions (for which the confidence in material properties is strong):  1. The use of critical state stress-dependent plane strain parameters shows good agreement between measured and simulated values of pullout resistance at large displacements and horizontal stresses on the front wall of the box for the three sheet inclusions.  2. A back-calculation using direct shear instead of plane strain parameters underestimates the measured pullout resistance at large displacement, and the lateral stress on the front wall of the box due to pullout, by approximately 25%.  3. The stress-dependency of cs,ps and cs,ps must be considered in a back-calculation of pullout tests performed at n less than about 50 kPa. For example, the use of friction angles obtained in the stress range 50 to 150 kPa underestimates by 10% or more the measured pullout resistance at n less than 20 kPa.  From the numerical analysis of geogrids, the following observations can be made:  4. The use of the Jewell et al. (1985) approach to deduce an “equivalent” soil-geogrid interface friction angle to model the three-dimensional structure of a geogrid as a sheet inclusion shows: 109   a. The principal source of uncertainty of the Jewell et al. (1985) approach is the adopted value of the bearing stress ratio (b/n) and has a direct influence on the value of the soil-geogrid interface friction angle. b. The use of stress-dependent values in plane strain along with the upper boundary (eq. 4.6) for the calculation of the bearing stress ratio, shows good agreement between measured and simulated pullout resistance and horizontal stresses on the front wall of the pullout box at large displacement. c. The use of stress-dependent values in plane strain along with the lower boundary (eq. 4.5) for the calculation of the bearing stress ratio, underestimates the pullout resistance at large displacement by 35% for the geogrids of this study.  These results allow demonstrating that plane strain conditions prevail in pullout testing at large displacements for sheet inclusions and that the stress-dependency of the critical state friction angle of the soil and the soil-inclusion interface must be considered in a back-calculation of pullout tests performed at vertical stresses less than 50 kPa addressing, thus, objective two of this Dissertation. In addition, the results of the numerical simulations of the geogrids show that the use of the Jewell et al. (1985) approach, considering the stress-dependency of the critical state friction angle in plane strain, along with the upper boundary of the bearing stress ratio, allows for a reasonable capture of the pullout response of geogrids at large displacements when treated as equivalent sheet inclusions.  These findings suggest that to capture the pullout response of different inclusions, plane strain conditions and the stress-dependency of the critical state friction angle must be considered when modeling either sheet inclusions or using the Jewell et al. (1985) approach to establish a soil-geogrid interface friction angle. Accordingly, these aspects will be considered in Chapter 5 to capture the complete pullout response of the different inclusions tested by Raju (1995) in accordance to research objective 3. 110  Table 4.1: Parameters for inclusions   Inclusion Elastic Modulus  (kPa) Ultimate strength  (kN/m) APT 7.7×107  4.8×104 GMT 7.7×105  29 GMS GGT GGS GGM 7.7×105  3.6×104 9.6×104 9.6×104 20.7 86 146 124 *Poisson’s ratio for the inclusion is 0.2.    Table 4.2: Values of cs,ps for the soil-sheet inclusion interface using deduced plane strain parameters  Inclusion Surcharge, n (kPa) 4 8 12 APT 34.0° 32.3° 31.5° GMT 34.0° 32.3° 31.5° GMS 10.2° 8.1° 6°  Table 4.3. Variation in pullout resistance (kN/m) with input parameters  Change in X (%) Input Parameter (X) ν G kn  ks 100 10.9 10.9 10.7 10.9 50 10.9 10.7 10.7 10.7 10 10.8 10.8 10.7 10.7 0 10.7 10.7 10.7 10.7 -10 10.7 10.7 10.7 10.6 -50 10.7 10.8 10.8 10.6   111  Table 4.4: Values of friction angle and bearing stress ratio, for the soil-geogrid interface, considering plane strain parameters  Inclusion s° ; cs,ps° (b/n) (eq. 4.5) ; (b/n) (eq. 4.6)  n = 4 kPa n = 10 kPa n = 17 kPa n = 4 kPa n = 10 kPa n = 17 kPa GGT 10.2; 34.0 8.1; 31.3 6.0; 30.2 11.05; 29.4 8.98; 21.4 8.25; 18.8 GGS 27.2; 34.0 25.0; 31.3 24.2; 30.2 11.05; 29.4 8.98; 21.4 8.25; 18.8 GGM 27.2; 34.0 25.0; 31.3 24.2; 30.2 11.05; 29.4 8.98; 21.4 8.25; 18.8   Table 4.5: Values of cs,ps (eq. 4.5; eq. 4.6) for the soil-geogrid interface using deduced plane strain parameters  Inclusion Surcharge, n (kPa) 4 10 17 GGT 17.5; 33.7 14.2; 26.1 12.3; 22.8 GGS 23.9; 35.9 21.1; 29.9 20.1; 27.8 GGM 23.0; 36.6 20.1; 30.2 19.1; 27.9    112  Table 4.6: Values of cs,ds for the soil-inclusion interface using measured direct shear parameters  Zone Surcharge, n (kPa) 4 8 10 12 17 Soil-inclusion interface Planar APT  29.4°  28.2°   27.8°  GMT  29.4°  28.2°   27.8°  GMS  10.0°  8.0°   6°  Grid GGT 23.4°  20.0°  17.5° GGM 26.3°  24.0°  22.5° GGS 26.5°  23.8°  22.3°    Table 4.7: Values of friction angle and bearing stress ratio, for the soil-geogrid interface, considering direct shear parameters  Inclusion s°;cs,ds°  (b/n) (eq. 4.6)  n = 4 kPa n = 10 kPa n = 17 kPa n = 4 kPa n = 10 kPa n = 17 kPa GGT 10.0; 29.4   8.0; 28.0 6.0; 27.0  17.2 14.7 13.2 GGS 23.5; 29.4  22.4; 28.0  21.6; 27.0  17.2 14.7 13.2 GGM 23.5; 29.4  22.4; 28.0  21.6; 27.0  17.2 14.7 13.2     113                Figure 4.1: G in function of mean stress for Badger sand (Golder Associates, 2004)             Figure 4.2. Calibration of Mohr-Coulomb to Badger sand   )(100035.947.0MPapppGaa = = 3 114  202224262830323436381 10 100 1000Critical state friction angle, cs()Effective stress,  (kPa)AAAAAA (measured)AAAAAA (measured)AAAAAA (measured)AAAAAA (Rowe 1969, eq. 4.2)cs,dsscs,pscs,dscs,tx n (kPa) cs,ds cs,ps 4 29.4 34.0 8 28.2 32.3 12 27.8 31.5           Figure 4.3: Variation of Badger sand friction angle with applied stress ( is cell pressure in triaxial tests and normal stress in direct shear and direct simple shear tests) (Rousé et al., 2014)             Figure 4.4: Numerical model using Mohr-Coulomb 0.6 m – 30 elements  1.3 m – 65 elements  0.95  m – 42 elements  Inclusion Interface 115  05101520250 25 50 75Displacement of clamped end (mm)Measured (Raju 1995)Simulated (usingSimulated (using12 kPa8 kPa4 kPacs,ds)n =cs,ps)Pullout force(kN/m)05101520250 25 50 75Displacement of clamped end (mm)12 kPa8 kPa4 kPan =Pullout force(kN/m)0123450 25 50 75Displacement of clamped end (mm)12 kPa8 kPa4 kPan =Pullout force(kN/m)                             Figure 4.5: Measured and simulated pullout response of a) APT, b) GMT and c) GMS  a) b) c) 116  cv,ds-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.00 10 20 30 40 50Lateral stress due to pullout (kPa)test xx (Raju 1995)test xx (usingtest xx (usingn = 4 kPan = 8 kPan = 12 kPacs,ds)cs,ps)hshshmNormalized distance from inclusion, y/h              Figure 4.6: Measured and simulated horizontal stress on the front wall due to pullout, at peak pullout resistance, for the GMT sheet inclusion    117  05101520250 25 50 75Displacement of clamped end (mm)17 kPa10 kPa4 kPan =Pullout force(kN/m) Simulated using eq. 4.5Simulated using eq. 4.605101520250 25 50 75Displacement of clamped end (mm)17 kPa10 kPa4 kPan =Pullout force(kN/m)05101520250 25 50 75Displacement of clamped end (mm)17 kPa10 kPa4 kPan =Pullout force(kN/m)                           Figure 4.7: Measured and simulated pullout response of a) GGT, b) GGS, and c) GGM geogrids using lower (eq. 4.5) and upper (eq. 4.6) bounds for the bearing stress ratio  a) b) c) 118  024681012141618200 2 4 6 8 10 12 14 16 18 20Simulated pullout resistance, Ps (kN/m)Measured pullout resistance, Pm (kN/m)Simulated using eq. 4.5Simulated using eq. 4.6Ps = PmUsing eq. 4.6:Ps = 1.01PmR2 = 0.95Using eq. 4.5:Ps = 0.66PmR2 = 0.92                Figure 4.8: Comparison of measured and simulated pullout resistance using lower (eq. 4.5) and upper (eq. 4.6) bounds for the bearing stress ratio   119  05101520250 25 50 75Displacement of clamped end,  (mm)Measured (Raju 1995)Simulated (usingSimulated (using12 kPa8 kPa4 kPacs,ds)n =cs,ps)Pullout force(kN/m)051015250 25 50 75Displacement of clamped end (mm)17 kPa10 kPa4 kPan =Pullout force(kN/m)05101520250 25 50 75Displacement of clamped end (mm)17 kPa10 kPa4 kPan =Pullout force(kN/m)05101520250 25 50 75Displacement of clamped end (mm)17 kPa10 kPa4 kPan =Pullout force(kN/m)                          Figure 4.9: Measured and simulated pullout response of a) GGT, b) GGS and c) GGM    a) b) c) 120  cv,ps cv,dscv,ds-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.00 10 20 30 40 50 60 70 80Lateral stress due to pullout (kPa)test xx (Raju 1995)n = 4 kPan = 10 kPan = 17 kPa(Using cs,ps)hshmNormalized distance from inclusion, y/h (Using cs,ds)hscv,ds-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.00 10 20 30 40 50 60 70 80Lateral stress due to pullout (kPa)Normalized distance from inclusion, y/h cv,ds-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.00 10 20 30 40 50 60 70 80Lateral stress due to pullout (kPa)Normalized distance from inclusion, y/h                                           Figure 4.10: Measured and simulated lateral stress on the front wall due to pullout, at peak pullout resistance, for a) GGT, b) GGS and c) GGM  a) b) c) 121  024681012141618200 2 4 6 8 10 12 14 16 18 20Simulated pullout resistance, Ps(kN/m)Measured pullout resistance, Pm (kN/m)Simulated usingSimulated usingPlane strain:Ps = 0.98 PmR2 = 0.97 Direct shear:Ps = 0.77 PmR2 = 0.95Ps = Pmcs,dscs,ps                Figure 4.11: Comparison of measured and simulated pullout resistance   122  051015202530354045500 5 10 15 20 25 30 35 40 45 50Simulated horizontal stress due to pullout, hs(kPa)Measured horizontal stress due to pullout, hm (kPa)Simulated usingSimulated usinghs = hmPlane strain: hs = 0.98 hmR2 = 0.98 cs,dscs,psDirect shear: hs = 0.74 hmR2 = 0.91                   Figure 4.12: Comparison of measured and simulated horizontal stress on the front wall due to pullout, at peak pullout resistance, of GMT and the three geogrids           123  0.91.01.11.21.30 30 60 90 120 150Normal stress, n (kPa)32.3°31.5°29.8°28.8°28.1° 27.9°cs,ps34.0°27.7°(tancs,ps)/ (tancs,ps)150 kPan            Figure 4.13: Variation of normalized horizontal stress ratio with depth for GMT (Raju, 1995)               Figure 4.14: Normalized friction angle in plane strain  124  Chapter 5 Numerical simulations of pullout tests in sand  5.1. Introduction  The third aspect identified in the literature review regarding the numerical modeling of pullout tests with room for improvement is capturing the full pullout response of different inclusions. As described in Section 2.3, in the numerical simulations of pullout tests, different constitutive models have been used to represent the behavior of the soil, the inclusion and the soil-inclusion interface. Most of these studies have been able to capture the initial stiffness of the pullout resistance or the full pullout response when no strain softening is observed. However, very few studies have been able to capture the complete pullout behavior of different inclusions when a peak and strain softening behavior is observed, such as the data obtained by Raju (1995) for the APT and GMS sheet inclusions. Additionally, parameters for the different components of the numerical model, used as input for the constitutive relations, do not always have geotechnical or physical meaning, and adjustments to these parameters have been made without clear explanation, reaching sometimes unrealistic values in order to match the observed pullout response. Nevertheless, the results of these simulations have proved to be very useful for the back-analysis of pullout tests and to study the soil-inclusion interface behavior, once the numerical model has been calibrated against laboratory pullout test results. It appears, however that a more phenomenological approach in the numerical simulations is needed to have a more rational description of the soil-inclusion interaction.   The results of the numerical simulations in Chapter 4 show that to achieve a proper capture of the large displacement pullout resistance of different sheet inclusions, a numerical model must consider both, plane strain conditions and the stress-dependency of the sand strength at the critical state. However, to move beyond large displacement behavior, and capture the full pullout response of an inclusion, an improvement to the numerical model presented in Chapter 4 is needed.      125  Jewell et al. (1985) and Frost et al. (2004) state that most of the “pullout action” occurs at or near the soil-inclusion interface. Given this, and as explained in the literature review, it is, necessary to modify the numerical model in Chapter 4 to capture the dilatant behavior of the soil near the inclusion produced by shearing in the pullout test. Weerasekara and Wijewickreme (2010) stated that if soil dilation is not accounted for, it will lead to an underestimation of pullout resistance.   Accordingly, and in response to objective three of this thesis, in this Chapter the pullout test of the rough aluminum inclusion (APT) tested at a vertical stress of 8 kPa is used to study different alternatives for pullout modeling. This is an interesting test since it shows a strain softening behavior (see Figure 4.5a). Additionally, the tests on the textured (GMT) and smooth (GMS) geomembranes at n = 8 kPa are simulated in order to discuss the applicability of the proposed methods to different inclusions and soil-inclusion interfaces. In all the approaches presented below, the NorSand constitutive model is used for modeling the response of Badger sand inside the pullout box. The different methods followed in this Chapter to achieve a numerical model that can capture the complete pullout response of the sheet inclusions, including the strain softening behavior observed in the APT and GMS tests, are described below and summarized in Table 5.1:   1. Using NorSand to represent the behavior of Badger sand, a beam element for the inclusion in combination with a Mohr-Coulomb type of spring (as presented in Chapter 4) for the soil-inclusion interface. 2. Using NorSand to represent the behavior of Badger sand, a beam element for the inclusion and gluing the beam to the soil mesh through an interface characterized by normal and shear stiffness. 3. Using NorSand to represent the behavior of Badger sand and replace the beam by a continuum element with elastic properties to represent the sheet inclusion. This configuration simulates the elastic element being attached to the soil mesh (Badger sand) represented by NorSand. 4. Using NorSand to represent the behavior of Badger sand, an elastic continuum for the inclusion and a continuum layer with NorSand characteristics to mimic the response of the soil-inclusion interface.  126  Having selected a method for the successful numerical simulation of the three sheet inclusions, the approach proposed by Jewell et al. (1985) is explored to identify whether it is suitable to capture the complete pullout response of the three geogrids tested by Raju (1995).  5.2. Constitutive model NorSand for modeling of pullout tests  5.2.1. Summary description of NorSand   Many constitutive models must be calibrated at the in-situ void ratio. Hence, every time density changes, another set of parameters is needed and the model has to be re-calibrated. In Critical State Soil Mechanic (CSSM) models, the parameters are not associated to any particular value of initial void ratio. Consequently, the same set of parameters is able to capture the behavior of soils regardless of the initial void ratio. NorSand is a generalized CSSM model for soil able to capture the softening and dilatant behavior of sands through the state parameter () and by postulating an infinity of normal consolidation loci. The state parameter is simply defined as the difference between the void ratio in the current state and that at the critical state for the same mean stress. In essence,  represents whether the soil has a tendency to dilate or contract when sheared and by how much; a negative value indicates that the soil will dilate and a positive indicates that the soil will contract.   NorSand is a classic elasto-plastic model for soil, as such, and in common with other elasto-plastic models, it comprises four items: elasticity, a yield surface, a flow rule and a hardening law. NorSand dilates similarly to actual soil through the introduction of limited hardening. It requires eight soil properties, of which 7 are obtainable from conventional triaxial compression tests and 1 is assumed (Poisson’s ratio). Three parameters describe the critical state, the critical friction ratio Mtc (the subscript ‘tc’ denoting the reference triaxial compression condition) and Γ and λ which describe the critical state line in the in the e-logp space. Mtc is given by:         𝑀tc =6sincs,tx3−sincs,tx               (eq. 5.1)  127  NorSand takes triaxial compression as the reference condition and leaves the variation of M with proportion of the intermediate stress (or Lode angle, 𝜃) to the constitutive model, as described in equation 5.2 (Jefferies and Shuttle, 2011).       𝑀() = 𝑀𝑡𝑐 −𝑀𝑡𝑐23+𝑀𝑡𝑐𝑐𝑜𝑠(3 𝜃 2⁄ + 𝜋 4⁄ )                       (eq. 5.2)  Thus, Mtc becomes a soil property based on the critical state friction angle in the triaxial compression test and M(𝜃) is evaluated in terms of this property allowing therefore, the constitutive model NorSand, to correct internally the shear strength for plane strain conditions.   There are three parameters relating to the plastic behaviour: tc, N and H. The property tc is a constant for each sand and is defined as the slope of the maximum dilatancy (Dmin) to the initial state parameter taking triaxial compression as a reference and scales the maximum dilatancy to the state parameter. N can be obtained by plotting the maximum stress ratio versus Dmin for dense sands. The plastic hardening parameter, H, is obtained by fitting the results obtained by the drained triaxial tests and iterating to obtain the best match. A finite elastic shear modulus G is also adopted. A detailed description of NorSand is available in Appendix A.  5.2.2. Calibration of NorSand to Badger sand  The aim in calibrating NorSand to a particular soil is to obtain a consistent set of material parameters that adequately represents that soil’s behavior across a range of stresses and states. Hence the goodness of fit for any individual test is, by necessity, compromised to obtain a better representation of the overall soil response. For the calibration of NorSand, loose tests are usually required in order to reach the critical state within the limits of the triaxial equipment. As shown in Section 3.2.3.3., although the intent was to obtain loose samples of Badger sand by using the water pluviation reconstitution method (which gives the most homogeneous and loose samples), it was not possible to obtain specimens looser than 33% relative density after consolidation.     128  5.2.2.1. Elastic Parameters  For the calibration process of NorSand to the triaxial tests on Badger sand, the value of the Poisson’s ratio was assumed equal to 0.2 and the shear modulus was calculated from the additional triaxial tests on moist tamped samples of Badger sand tested by Golder Associates (2004) by attaching bender elements (see Figure 4.1).   5.2.2.2. Plastic Parameters  The volumetric coupling parameter, N, was deduced by plotting the maximum stress ratio versus Dmin (see Figure 3.7) and tc from the slope of minimum dilatancy versus state parameter (Figure 5.1), resulting in values of 0.21 and 2.0, respectively. The hardening parameter, H, was obtained by fitting the triaxial tests and taken constant and equal to 200 (Figure 5.2).  5.2.2.3. Critical State Parameters  The critical state parameters,  and , were obtained from the end of the three triaxial drained and the two undrained saturated water pluviated tests in the e-logp space (Figure 5.3). It is recognized that at 15% strain the samples have not reached the critical state and therefore, the critical state line has to be above the end of tests. A minor fitting process through the tests allowed to deduce values for  and  equal to 0.715 and 0.013, respectively. Mtc was obtained using Bishop (1966) method, by plotting the maximum stress ratio max = (q/p)max against the corresponding dilatancy (Dmin) obtained from the drained triaxial tests using the water pluviation (see Figure 3.7). This resulted in a value of Mtc = 1.07 for confining stresses between 50 kPa and 300 kPa.   A summary of the values obtained for Badger sand is shown in Table 5.2 along with typical values found in the literature. Despite the very rounded nature of the Badger sand grains, the narrow difference in extreme void ratios and the low values of friction angle, most of the parameters fall within the ranges generally accepted, except for . Nevertheless, this value of  ensures that the tests start and end at the left of the critical state line.   129  It is acknowledged that the reconstitution method (water pluviation), stress range (50 to 150 kPa) and sand densities (DR = 33% - 36%) used in the triaxial tests to obtain the NorSand parameters are different than the ones used in the pullout box (air pluviated, n = 4 to 17 kPa, DR = 85-95%). It is also important to recognize that the effects of fabric and anisotropy are very important in modeling the response of sands and that the version of NorSand used in the current study has no mechanism to capture these. In order to test the validity of the NorSand parameters already determined (Table 5.2), they were used to simulate the response of the two moist tamped triaxial tests performed by Golder Associates (2004). Figure 5.4 shows a very good match between simulated and measured values although the reconstitution method, the stress range (3  = 100 and 300 kPa) and densities (DR = 65% and 76%) are very different. Consequently, the same set of parameters can capture the behavior of Badger sand under two different reconstitution methods, at different densities and stress ranges. In this study, it is considered thus that the same parameters can be used for the numerical modeling of the pullout box where the sand was placed using the air pluviation method and a relative density between 85% and 90%. Even though the triaxial tests on Badger sand were performed at densities and stresses different from the ones of the pullout tests, NorSand is able to capture the fundamental behavior of Badger sand (Figure 5.4), especially dilatancy.   5.2.2.4. Strength of Badger sand at low stresses in the triaxial test  As shown in Chapter 3, the triaxial tests on Badger sand were performed at confining stresses between 50 kPa and 300 kPa, stresses that are higher than the ones used in the pullout tests performed by Raju (1995), that ranged between 4 kPa and 17 kPa. This was mainly due to equipment limitations and NorSand requirements for its calibration. The laboratory test data of Chapters 3 and the numerical simulations in Chapter 4 show that a stress-dependency of the critical state friction angle of the soil and the soil-inclusion interface friction angle exists at stresses lower than 50 kPa. Therefore, a stress-dependent value of cs,tx, as input for Mtc, is needed for NorSand at the very low stresses used in the pullout testing by Raju (1995).    130  As shown in Figure 4.3, the measured values of cs,ds and the deduced values of cs,ps for Badger sand are stress-dependent for   between 4 and 50 kPa. The studies of Fukushima and Tatsuoka (1984) and Boyle (1995), show that cs,tx is also stress-dependent at low stresses. It is thus expected that the strength of Badger sand in the triaxial test is likewise stress-dependent at the very low stresses used in the pullout box. Consequently, the value of cs,tx obtained from the triaxial tests at  = 50 – 150 kPa, is not representative of the strength of Badger sand at stresses between 4 and 50 kPa. Therefore, a method to deduce a value of cs,tx (and hence Mtc for NorSand) at low stresses (4 to 50 kPa) is required.   Figure 4.3 shows that the deduced values of cs,ps and the measured value of cs,tx for Badger sand at  = 50 – 150 kPa are very similar in magnitude (a difference of less than 1 is found for Badger sand), and allows the writing of eq. 5.3 as follows:  cs,tx = cs,ps -1                          (eq. 5.3)  This relation is consistent with the findings of Hussaini (1973), Marachi et al. (1981), Cornforth (1964, 1973), and Rowe (1969) as described in Section 2.3.5. Assuming that this difference between cs,ps and cs,tx for Badger sand holds true at very low stresses (between 4 and 50 kPa), the values of cs,tx as a function of  are deduced from eq. 5.3 and shown in Figure 5.5. By doing this, there is no intention to feed NorSand with a plane strain critical state friction angle (NorSand corrects internally for plane strain conditions through the Lode angle in equation 5.2). The use of eq. 5.3 to deduce a value of cs,tx from cs,ps, considers that, if values of cs,tx were obtained through laboratory testing at  from 4 to 12 kPa they would be very similar to cs,ps deduced using Rowe (1969) equation. Finally, this deduced value of cs,tx is used to determine a stress-dependent Mtc from eq. 5.1.      131  5.3. Numerical simulation approaches of pullout tests using NorSand   In this section, four different methods are tested for the numerical modeling of the three sheet inclusions tested by Raju (1995) at a vertical stress of 8 kPa. In all the approaches the behavior of Badger sand inside the pullout box is represented by NorSand, while the inclusion and the soil-inclusion interface are changed so the model can capture the pullout response of the different inclusions, including the strain softening behavior observed in the APT and GMS tests. It is important to recall here the most important characteristics of the soil-inclusion system that will define whether a numerical model is suitable to comply with objective 3 of the current study:  1) The APT inclusion is an aluminum inextensible sheet with Badger sand glued on both faces rendering the soil-inclusion interface as a soil-soil interaction. The pullout response shows a peak and a strain softening behavior. 2) The GMT inclusion is an extensible geomembrane with a rough surface with a soil-inclusion interface friction very similar to the soil-soil friction. Therefore, the main difference with the APT inclusion is the elasticity of the material. The pullout response shows a gradual increase in the pullout force per unit width, without strain softening. 3) The GMS inclusion has the same elastic properties than GMT but has a smooth surface. The pullout response shows a peak and strain softening behavior with lower magnitude than the GMT inclusion.  5.3.1. Numerical modeling using NorSand in combination with a beam element and Mohr-Coulomb springs  In this Section, the laboratory pullout data of the APT08 test is used to examine the NorSand-beam-spring combination. This approach consists on replacing Mohr-Coulomb from the numerical model in Chapter 4 (see Figure 4.4) with NorSand to represent the behavior of Badger sand inside the pullout box while keeping the beam element for the inclusion and the linear elastic springs with Mohr-Coulomb strength for the soil-inclusion interface.   132  The parameters used to describe the elastic behavior of the beam element are defined by FLAC. In this study, they were chosen so as to represent the real properties of the aluminum sheet used in the pullout test of the APT inclusion and are: an elastic Young’s modulus of 77×106 kPa and yield strength of 48×103 kPa (according to manufacturer’s information and Raju, 1995). The interaction between the soil and the sheet inclusion (beam element) is modeled using springs, with normal stiffness kn = 540×103 kPa/m and shear stiffness ks = 54×103 kPa/m (as per Section 4.2.3.), and a soil-inclusion friction angle, assumed equal to that of the sand at large displacement in plane strain, i.e., 32.3° for a vertical effective stress of 8 kPa (see Section 4.2.3.). The parameters for NorSand are defined in Table 5.2 with a value of Mtc of 1.26 corresponding to cs,tx = 31.3 as shown in Figure 5.5.  Figure 5.6 shows the results of the simulation using the NorSand-beam-spring approach. The pullout response is governed by the Mohr-Coulomb type of failure of the spring interface; when the soil strength reaches the soil-inclusion friction angle, the APT inclusion fails in pullout without being able to capture the strain softening behavior observed in the laboratory pullout test. As expected it reaches only the large displacement pullout resistance. These results suggest that, even if the soil on top and below the soil-inclusion interface is allowed to dilate through NorSand, it is the soil-inclusion interface that defines the response of the inclusion to pullout. It is, therefore necessary, to modify the numerical model so that the dilatancy occurring at the soil-inclusion interface level is captured.  5.3.2. Numerical modeling using a beam element “glued” to the NorSand mesh  This approach consists on “gluing” the beam element representing the inclusion to the NorSand mesh representing the soil. By doing this, no slip is allowed between the beam and the mesh and the pullout response should rely on the constitutive model NorSand assigned to the soil mesh. The parameters used to describe the elastic behavior of the beam element are chosen so as to represent the real properties of the inclusions used in the pullout tests performed by Raju (1995) and are defined in Table 4.1, according to manufacturer’s information and Raju (1995). The interaction between the soil and the beam element is defined by the normal stiffness kn = 540×103 kPa/m and the shear stiffness ks = 54×103 kPa/m (as per Section 4.2.3.). The parameters for NorSand are 133  defined in Table 5.2 with values of Mtc equal to 1.26 corresponding to cs,tx = 31.3 for the APT and GMT inclusions and equal to 0.26 corresponding to cs,tx = 7.1 for the GMS inclusion (as per eq. 5.3 and cs,ps = 8.1 in Table 4.2).  Figure 5.7 shows the comparison of the measured and simulated values of pullout resistance using the beam glued to the soil mesh for the three sheet inclusions. In general, a decent match is observed and the strain softening response is captured for the APT and GMS inclusions. For APT the simulated values are slightly lower at peak pullout resistance and slightly larger at large displacements. For the GMT inclusion, a slightly lower pullout response is observed and for the GMS inclusion slightly larger values of peak and large displacements are obtained.   5.3.3. Numerical modeling using NorSand in combination with an elastic continuum for the inclusions and Mohr-Coulomb springs  This approach consists on representing the inclusion by a continuum mesh 1 element high and 50 elements long to which the elastic constitutive model is assigned. This elastic continuum interacts with the NorSand mesh representing the behavior of Badger sand, through linear elastic springs with Mohr-Coulomb strength that correspond to the soil-inclusion interface.  The parameters used to describe the elastic behavior of the continuum element representing the APT inclusion tested at a vertical stress of 8 kPa are chosen so as to represent the real properties of the aluminum sheet used in the pullout test and are: an elastic Young’s modulus of 77×106 kPa and a poisson’s ratio of 0.2 (according to manufacturer’s information and Raju, 1995). The interaction between the soil and the sheet inclusion (elastic continuum) is modeled using a spring interface, with normal stiffness kn = 540×103 kPa/m and shear stiffness ks = 54×103 kPa/m (as per Section 4.2.3.), and a soil-inclusion friction angle, assumed equal to that of the sand at large displacement in plane strain, i.e., 32.3° for a vertical effective stress of 8 kPa (see Section 4.2.3.). The parameters for NorSand are defined in Table 5.2 with a value of Mtc of 1.26 corresponding to cs,tx = 31.3 as shown in Figure 5.5.  134  Figure 5.6 also shows the comparison of the measured and simulated pullout resistance for the APT08 sheet using the elastic continuum and the springs to represent the soil-inclusion interaction. The pullout response is governed by the Mohr-Coulomb type of failure of the spring interface, and the APT inclusion fails in pullout without being able to capture the strain softening behavior observed in the laboratory pullout test. As expected it reaches only the large displacement pullout resistance. Comparison of the simulated results using the beam element and the elastic continuum to represent the inclusion shows good agreement suggesting that both methods are equivalent and are governed by the strength of the soil-inclusion interface represented by the linear elastic Mohr-Coulomb springs.  5.3.4. Numerical modeling using an elastic continuum for the inclusions attached to the NorSand mesh  The “elastic continuum attached to the soil” approach is similar to the beam glued to the soil mesh (Section 5.3.2), but the inclusion is represented by a continuum mesh 1 element high and 50 elements long to which the elastic constitutive model is assigned. This approach can also be pictured as the method described in Section 5.3.3. but without the Mohr-Coulomb spring elements representing the soil-inclusion interface, relying thus on the soil to capture the pullout response of the sheet inclusions.   The soil is represented by the NorSand constitutive model (see Table 5.2 and Figure 5.5) and the properties of the elastic continuum are given the values shown in Table 4.1 for the three sheet inclusions. This approach keeps the inclusion attached to the soil mesh represented by NorSand, simulating, therefore, a fully rough soil-inclusion interface as is the case of APT and GMT, and no extra parameters (such as kn and ks in Section 5.3.2) are needed to define the soil-inclusion interface. The parameters for NorSand are the same than Section 5.3.2. and are defined in Table 5.2 with values of Mtc equal to 1.26 corresponding to cs,tx = 31.3 for the APT and GMT inclusions and equal to 0.26 corresponding to cs,tx = 7.1 for the GMS inclusion (as per eq. 5.3 and cs,ps = 8.1 in Table 4.2).  135  Figure 5.7a shows that the “elastic continuum attached to the soil” approach can also capture the peak and strain softening behavior of the APT inclusion tested at a vertical stress of 8 kPa, which is consistent with the results of the “beam glued to the soil” approach. A change in the elastic modulus of the inclusion from 7.7×107 kPa (for APT) to 7.7×105 kPa (for GMT) shows that the “elastic continuum attached to the soil” approach captures the pullout response of the GMT inclusion at n = 8 kPa (Figure 5.7b); the strain softening behavior is also suppressed and the large displacement pullout resistance is nicely captured. Keeping constant the elastic modulus of the inclusion but changing only the value of Mtc of the soil (from 1.26 to 0.26) to reflect on the low soil-inclusion interface friction angle, also captures the peak and strain softening pullout response of the GMS inclusion (Figure 5.7c). Comparison of the simulated pullout responses using the “beam glued to the soil” and the “elastic continuum attached to the soil” methods shows good agreement, indicative that both approaches are equivalent in terms of pullout response.   These results suggest that the NorSand constitutive model can capture the fundamental response of the soil-inclusion interface of sheet inclusions. This numerical approach seems acceptable when a soil-inclusion interface equivalent to a soil-soil interface is used in the pullout test, such as the APT and the GMT sheet inclusions. In that case, the strength of the soil and the strength of the soil-inclusion interface are the same and therefore, relying on the soil to capture the pullout response of these fully rough inclusions appears realistic. However, when pullout tests are performed on sheet inclusions with a soil-inclusion interface friction angle lower than the strength of the soil (such as GMS), modifying the strength of the complete soil mesh such that it matches the soil-inclusion interface friction angle, might not be representative of the behavior of the sand far from the inclusion. It appears therefore reasonable that a numerical approach that allows the interface to dilate while having a lower strength than the surrounding soil is needed.        136  5.3.5. Numerical modeling using an elastic continuum for the inclusion and a NorSand continuum layer for the soil-inclusion interface  Typical models for the soil-inclusion interface are based on springs characterized by a shear and normal stiffness as shown in Chapter 4 and Sections 5.3.1. and 5.3.3. of this study. This approach has allowed for the capture of the soil-inclusion interface behavior when a gradual increase in the pullout resistance is observed. However, to the best of the author’s knowledge, it has proven insufficient to capture the strain softening response observed in some pullout tests such as the APT (see Section 5.3.1.) and GMS tests of this study (Raju, 1995). Sections 5.3.2. and 5.3.3. showed that gluing or attaching the inclusion (beam or elastic continuum) to the soil mesh and allowing the NorSand constitutive model of the soil to represent the pullout response appears a good alternative when a fully rough soil-inclusion interface is simulated. However, when a smooth soil-inclusion interface is to be simulated, the use of a lower value of soil strength for the soil mesh is not representative of the experimental conditions and therefore, does not allow for analysis beyond the pullout response.   In the approach presented in the current Section, instead of attaching the soil mesh directly to the inclusion (elastic continuum element) a continuum layer, above and below the inclusion is added, also with the NorSand constitutive model, to mimic the interface response between the inclusion and the soil. The continuum layer is four elements high above and below the inclusion to allow for constant stress-strain to develop within each zone, with a total thickness of 15 mm in accordance with the observations of Roscoe (1970) and Frost et al. (2004), that shear occurs within a zone 10 to 20 times the mean grain size of the soil (D50 for Badger sand is close to 1 mm). This approach allows to keep constant the parameters for NorSand representing the soil, in particular the strength of the soil (Mtc as a function of cs,tx) and modify only the strength of the layer representing the soil-inclusion interface (while keeping constant the other NorSand parametes), for example for the simulation of the GMS inclusion. It is important to note that this approach is equivalent to the “elastic continuum attached to the soil” approach (Section 5.3.4.) when a fully rough soil-inclusion interface is simulated, such as the APT and GMT inclusions (i.e. the soil-inclusion interface friction angle equals the soil’s friction angle). In consequence, described below is the numerical 137  simulation of the GMS sheet inclusion for which the soil-inclusion interface friction angle is extremely lower than the friction angle of the soil.  The behavior of Badger sand inside the pullout box is represented by the NorSand constitutive model with the parameters shown in Table 5.2 and Figure 5.5 (Mtc equal to 1.26 corresponding to cs,tx = 31.3). The GMS inclusion is represented by an elastic continuum with the properties given in Table 4.1. The soil-inclusion interface for the GMS test is modeled as an equivalent continuum layer with a small finite thickness (15 mm), responding to a NorSand behavior. The soil-GMS interface is also given the NorSand parameters shown in Table 5.2, except for the value of the stress-dependent Mtc, which was given a value of 0.26 corresponding to a cs,tx of 7.1 for n = 8 kPa (as per eq. 5.3 and cs,ps = 8.1 in Table 4.2).   The results of the simulation of the pullout test on the GMS inclusion at n = 8 kPa is shown in Figure 5.8. The proposed approach allows the capturing of the pullout behavior of the GMS inclusion, including the peak, the strain softening, and the large displacement pullout resistance.   5.3.6.  Discussion about the elastic continuum combined with the “NorSand interface layer” approach  The results of the numerical simulations from the different approaches described in Section 5.3 reveal that, whether the surface roughness of the inclusion is large or not, the most important aspect to consider in the numerical modeling of pullout tests on sheet inclusions is the dilatancy of the soil-inclusion interface, as previously stated by Weerasekara and Wijewickreme (2010). The approach in Section 5.3.1., where the beam element is connected to the soil mesh through Mohr-Coulomb springs, is the most commonly used to represent the soil-inclusion interface. Although this approach allows for a discrete slip of the soil-inclusion interface, it has proven insufficient to capture the complete pullout response of different inclusions when a peak and strain softening pullout resistance is observed given that the soil-inclusion interface is driven by a Mohr-Coulomb type of failure. The approach in Section 5.3.5. replaces the beam element representing the inclusion by an elastic continuum and the Mohr-Coulomb springs by a continuum layer four elements high above and below the inclusion to allow for constant stress-strain to develop within each zone. This 138  “continuum layer” follows a NorSand behavior that allows for the soil-inclusion interface to dilate. This approach has proved successful to capture the pullout response of the different sheet inclusions tested at a vertical stress of 8 kPa, including the strain softening pullout response of the APT and GMS sheet inclusions.   Nevertheless, some potential limitations of this approach compared to the beam and Mohr-Coulomb springs approach can be identified as follows:  - The elastic continuum representing the inclusion does not resist flexural or torsional stresses (the structural beam does), however this is not of concern for this study. - The continuum interface layer does not allow separation between the structural element and the soil mesh in the vertical direction (NorSand does not resist tension), however this is not of concern for this study as the focus is on the displacement of the inclusion in the horizontal direction.  - There can be concern regarding to whether the continuum elements can sustain the level of strains imposed by pullout. The simulations in FLAC are done using small strains, meaning that gridpoints coordinates are not changed and the elements are not deformed. Therefore, the elements can sustain large levels of strains without compromising the connectivity. In any case, the pullout tests of the current study reach, in general, the peak pullout resistance at less than 0.5% strain and the large displacement pullout resistance at around 3% to 4% strain. Therefore, this is not an issue for this study. - The continuum interface layer represents a soil-inclusion interface with a finite thickness that is not representative of the experimental soil-inclusion interface. In order to restrict the thickness of this artificial interface layer, in this study, the height of the continuum interface layer was limited to 15 mm in accordance with the observations of Roscoe (1970) and Frost et al. (2004), that shear occurs within a zone 10 to 20 times the mean grain size of the soil, following two primary modes:   139  (1) If the continuum surface is smooth relative to the soil and the vertical stress remains below a critical stress that avoids particles embedding in the surface, the shearing mechanism consists of particles sliding along the continuum surface, which is the case of the smooth geomembrane (or GMS sheet inclusion);  (2) A change in the continuum surface roughness, the confining stress, the continuum hardness, or the particle angularity can result in internal shearing in the granular medium as it provides less resistance to shearing and hence, shearing within a finite zone in the adjacent granular material occurs, which is the case of the APT and GMT sheet inclusions.   As the third objective of the current study is to capture the complete pullout response of the different inclusions tested by Raju (1995), where fully rough and smooth surfaces exist and a strain softening pullout behavior is observed, the trade-off between the Mohr-Coulomb springs and the continuum layer appears reasonable and, therefore, the “NorSand continuum interface layer” approach is used to model the pullout response of the different inclusions tested by Raju (1995).  5.4. Numerical simulations of the full pullout response of sheet inclusions using NorSand  5.4.1. Model description and characterization  Section 5.3.5. showed that the “NorSand interface layer” approach, allows for the capture of the fundamental response of the soil-inclusion interface for the three sheet inclusions tested by Raju (1995) at a vertical stress of 8 kPa. In this Section, a refinement to the numerical model presented in Section 5.3.5. is done, to include the roughness of the front wall of the box of ff =15 (Raju, 1995). For that, the same approach used to represent the soil-inclusion interface is used; a continuum layer along the front wall of the box is attributed the same NorSand parameters than the soil and the soil-inclusion interface (see Table 5.2) and Mtc is given a value of 0.57 representative of a ff =15.    140  5.4.2. Model parameters  5.4.2.1. Model parameters for inclusions  In this Section, the sheet inclusions are treated as elastic elements and therefore, the elastic model representing the inclusions requires the identification of the elastic properties, namely the elastic modulus, E, and Poisson’s ratio, . In this study, for each inclusion, these properties were assigned according to manufacturer’s information (Raju, 1995) as shown in Table 4.1, as follows: APT = 7.7×107 kPa; GMT = 7.7×105 kPa; and GMS = 7.7×105 kPa. In all the cases the Poisson’s ratio was assumed as 0.2.  5.4.2.2. Model parameters for Badger sand  The behavior of Badger sand inside the pullout box is represented by the constitutive model NorSand. The model parameters are the ones obtained from the calibration of NorSand in Section 5.2.2. and presented in Table 5.2, with stress-dependent Mtc obtained from the stress-dependent values of cs,tx (as per Section 5.2.2.4.), i.e., Mtc = 1.33 (cs,tx = 33) for n = 4 kPa, Mtc = 1.26 (cs,tx = 31.3) for n = 8 kPa, Mtc = 1.22 (cs,tx = 30.5) for n = 12 kPa.  5.4.2.3. Model parameters for soil-inclusion interface  As described in Section 5.3.5. in the “NorSand interface layer” approach, a continuum layer with a small finite thickness (of 15 mm) responding to a NorSand behavior is used to mimic the response of the soil-inclusion interface above and below the inclusion. This thin continuum layer needs the same NorSand parameters than the soil (presented in Table 5.2), except for the value of Mtc that depends on the inclusion being modeled and the stress level of the test.   For the sheet inclusions, Mtc is taken from the element tests on Badger sand. As shown in Section 3.4, the soil-inclusion interface roughness for the APT and GMT inclusions is very similar and equivalent to a soil-soil interface (Raju, 1995). Therefore, the value of Mtc as a function of cs,tx = cs,tx for the soil-APT and soil-GMT interfaces is the same than the stress-dependent value of Mtc 141  for the soil at very low stresses (n = 4 to 12 kPa). Consequently, as per Section 5.2.2.4, the values of the stress-dependent Mtc for the APT and GMT are deduced from eq. 5.1 and Figure 5.5, considering that cs,tx = cs,ps -1 (eq. 5.3) and shown in Table 5.3.  For the smooth membrane, GMS, the stress-dependent values of Mtc as a function of cs,tx for the soil-GMS interface are deduced using eq. 5.3 from the plane strain soil-inclusion friction angles equal to 10.2, 8.1 and 6 (see Section 4.2.3.), for n equal to 4, 8 and 12 kPa, respectively. Using eq. 5.3, the values of cs,tx are 9.2, 7.1 and 5 for n = 4, 8 and 12 kPa, respectively. The corresponding values of Mtc for the soil-GMS interface using eq. 5.1 are shown in Table 5.3.  5.4.2.4. Model parameters for soil-front wall interface  Following the same approach than the soil-inclusion interface, the soil-front wall interface is also represented by a thin continuum layer with NorSand parameters. The laboratory test results between Badger sand and the aluminum resulted in an interface friction angle of 15 (Raju, 1995). Therefore, the parameters needed to describe the soil-front wall interface with NorSand are the same than for the soil and the soil-inclusion interface (Table 5.2) except for the value of Mtc = 0.57 corresponding to a soil-aluminum interface friction angle of 15  5.4.3. Boundary and initial conditions  Figure  5.9 shows the model geometry, the finite-difference mesh for the sand and the inclusion, the displacement and stress boundary conditions for the 2D model (Fig. 5.9b), and the interface element (Fig. 5.9c). The sample of Badger sand is represented by a 65 elements long and 30 elements high mesh. The front wall is fixed in the horizontal and the vertical directions and an interface with a friction angle of ff = 15, representative of the soil-aluminum interface friction angle (Raju, 1995), is used. No sleeve was used in the laboratory tests and therefore, the numerical model did not account for a frontal sleeve. The back wall of the pullout box is fixed in the horizontal direction assuming a frictionless boundary. The upper boundary is a stress-controlled boundary, and the bottom is a fixed boundary in the horizontal and vertical directions. The inclusion is modeled by a mesh 1 element high and 50 elements long in the center of the pullout 142  box. The nodes representing the upper and lower surfaces of the inclusion are not fixed in either direction to allow the specimen to move with respect to the front wall and deal with the possible concentration of strains in that zone. The soil-inclusion and soil-front wall interfaces are represented by a 15 mm thick NorSand mesh in accordance with the observations of Roscoe (1970) and Frost et al. (2004), that shear occurs within a zone 10 to 20 times the mean grain size of the soil (D50 for Badger sand is close to 1 mm). The model is brought into equilibrium under the self-weight of the sand and, if applied, any additional surcharge pressure. A velocity of 1×10-7 is then imposed at the free end of the inclusion through the two nodes representing the upper and lower surfaces of the inclusion element. The resulting pullout force per unit width, P, is determined by integrating the shear stresses in the thin layer representing the interface above and below the inclusion.  5.4.4. Simulation results  5.4.4.1. Pullout resistance per unit width  The pullout response of the sheet inclusions tested by Raju (1995) is shown in Figure 5.10. Mobilized pullout resistance increases with vertical stress for the three inclusions. For APT, in all cases except one, a peak value of pullout resistance develops that is followed by a strain softening behavior. The exception is a test at 4 kPa for which the response at small displacement is a consequence of an inadvertent “pre-shearing” of the inclusion at the time of clamping it to the control system (Raju, 1995). However, all three tests at 4 kPa yield a similar and constant value of pullout resistance at large displacement (with a difference of 4%). Similarly, the tests at 8 kPa and 12 kPa yield a constant value of pullout. The GMT inclusion does not show a peak and strain softening pullout response but a gradual increase in pullout force, and reaches similar large displacement values than the APT tests at the same n. The GMS inclusion also shows a peak and strain softening behavior of the pullout response for all the stress levels but reaches smaller values than the APT and GMT inclusions.  Figure 5.10 also shows the results of the simulations using the parameters presented in Tables 5.2 and 5.3 for the three sheet inclusions at the different vertical stresses, using a stress-dependent Mtc. 143  The numerical model can capture the full pullout response of the three inclusions. For APT (Fig. 5,10a), the numerical model simulates a slightly stiffer initial response, possibly due to the value of the plastic-hardening parameter, H, in the NorSand constitutive model which is believed to be fabric dependent (condition that the version of NorSand used in this study cannot respect), reaching the peak pullout resistance at a slightly smaller displacement than the laboratory test data. However, the model captures the magnitude of the peak and large displacement pullout resistance for the three vertical stresses. For the GMT simulations, all the parameters for the soil and the soil-inclusion interface were kept constant, and only the elastic modulus, E, of the inclusion was modified as shown in Table 4.1. The numerical model simulates a less stiff initial response for the three vertical stresses and a slight peak pullout resistance at n = 12 kPa. Overall, the numerical model can capture the lack of peak pullout response and the magnitude of the large displacement pullout resistance for the three vertical stresses applied (Figure 5.10b). Finally, the GMS simulations were obtained by keeping all the parameters of the GMT simulations constant but changing only the interface value of Mtc for the soil-smooth membrane as shown in Table 5.3. The numerical model is again able to capture the peak and the large displacement pullout resistance of the GMS inclusion at the three different vertical stresses (Figure 5.10c).   There might be some concern regarding the spikiness observed in the pullout resistance of the APT and GMS inclusions due to the lack of internal sleeve in the laboratory pullout box. Figure 5.11 shows the measured and simulated values of displacement of the embedded end (de) with displacements of the clamped end (dc) of the inclusion. The APT inclusion shows an inextensible behavior given by de = dc (Figure 5.11a) for the three vertical stresses applied. For the GMT inclusion (Fig. 5.11b), an extensible behavior is observed in the experimental data given by de < dc, that increases with the vertical stress applied. The numerical simulations of the GMT inclusion were done by keeping constant all the parameters of the APT simulations, except for the elastic modulus of the inclusion which was given a constant value of 7.7×105 kPa. The simulated displacements of the GMT inclusion also show an extensible behavior and capture the displacements of the embedded end for n = 4 kPa and 8 kPa. For the GMT inclusion tested at n = 12 kPa, smaller values of de are obtained with the numerical simulations, suggesting that a smaller value of E could have been used. For the numerical simulations of the GMS inclusion, all the parameters of the GMT simulations were kept constant and only the soil-inclusion interface 144  was changed. Inspection of the results (Fig. 5.11c) show an inextensible behavior of the GMS inclusion that is captured by the numerical simulations. These results suggest that the peak and strain softening behavior observed in the APT and GMS tests are apparently due to the inextensible behavior of the inclusions and that the peak pullout resistance is supressed by the elongation of the inclusion.  Unfortunately, no experimental tests were done with an internal sleeve to verify if this behavior would have also been observed however, the good comparison between measured and simulated values of displacement of the three sheet inclusions allow to suggest that the spikiness observed in the APT and GMS tests do not correspond to an anomalous behavior.   5.4.4.2. Horizontal stresses at the front wall  Horizontal stresses (h) along the front wall of the pullout box were also measured by Raju (1995) for all the tests except for the APT inclusion. Figures 5.12a and 5.12b show the values of h at maximum pullout resistance for the GMT and GMS inclusions. h is maximum at the inclusion level and decreases near the top and bottom boundaries of the box. It shows an asymmetric distribution due to the difference in the upper and lower boundary conditions. The maximum value of h increases with n and is larger for the GMT than the GMS sheet, which is consistent with the pullout response of both sheet inclusions (GMT has a larger pullout resistance than GMS at the same value of n). Figure 5.11 also shows the results of the lateral stresses at the front wall of the box of the numerical simulations for the sheet inclusions. They also show a stress-dependent behavior and an asymmetric distribution since the boundary conditions were selected in order to better represent the laboratory test conditions. Additionally, the distribution and the maximum values are captured by the numerical model. In summary, the numerical model can capture the distribution and magnitude of the lateral stresses for the tests on sheet inclusions at the different vertical stresses.     145  5.5. Numerical simulations of the full pullout response of geogrids using NorSand  5.5.1. Treating the geogrids in the model  Following the same idea developed in Chapter 4, in this Section, the approach proposed by Jewell et al. (1985) is tested to evaluate its applicability for the numerical simulation of geogrids as equivalent sheet inclusions and capture the complete pullout response obtained by Raju (1995). The objective is, therefore, to find an equivalent soil-geogrid interface friction angle given by 𝑡𝑎𝑛 = 𝑓𝑏𝑡𝑎𝑛 to feed the parameter Mtc of the soil-inclusion interface represented by the thin continuum layer responding to a NorSand behavior. It is important to recall that NorSand takes triaxial compression as the reference condition and therefore needs a critical state friction angle in triaxial to obtain the value of Mtc. As a consequence, the values of cs,ps calculated in Section 4.3 (see Table 4.5) are not valid for this analysis.    In this Section, cs,tx is determined following the same approach developed in Section 4.3. but using a stress-dependent cs,tx as input for the Jewell et al. (1985) method, as given in equation 5.4.  𝑡𝑎𝑛𝑐𝑠,𝑡𝑥 = 𝑓𝑏𝑡𝑎𝑛𝑐𝑠,𝑡𝑥 = 𝛼𝑠tan𝛿𝑠 +12(𝜎𝑏𝜎𝑛)∅𝐹1𝐹2𝐵𝑡𝑆𝑡𝛼𝑏          (eq. 5.4)  The values of the geometrical parameters s, b, Bt/St, F1 and F2 are already shown in Table 3.4. s is the soil-solid are of the geogrid interface friction angle and is taken equal to the stress-dependent soil-GMS interface friction angle in triaxial for the GGT geogrid (that has a smooth surface) and for the GGS and GGM geogrids (that have a rough surface), as 0.8 times the stress-dependent soil-APT interface friction angle in triaxial (deduced using eq. 5.3 and s in plane strain), as per Section 4.2.3 and Table 4.4 (see Table 5.4).  The values of the stress-dependent cs,tx at n = 4, 10 and 17 kPa are deduced using eq. 5.3. from cs,ps, obtaining therefore values of 33, 30.3 and 29.2, respectively. The bearing stress ratio (b/n), is then evaluated considering eqs. 5.5 and 5.6 (see Table 5.4).  146   (𝜎𝑏𝜎𝑛)= 𝑒(90+𝑐𝑠,𝑡𝑥)𝑡𝑎𝑛𝑐𝑠,𝑡𝑥tan⁡(45 + 𝑐𝑠,𝑡𝑥2⁄ )           (eq. 5.5)   (𝜎𝑏𝜎𝑛)= 𝑒𝑡𝑎𝑛𝑐𝑠,𝑡𝑥tan2⁡(45 + 𝑐𝑠,𝑡𝑥2⁄ )             (eq. 5.6)  Replacing the values of the geometrical parameters (Table 3.4) along with the values of the upper and lower bounds of the bearing stress ratio in eq. 5.4, maximum and minimum values of cs,ps for the three geogrids tested at n = 4, 10 and 17 kPa are obtained (see Table 5.5).   5.5.2. Simulation results  The values of the pullout resistance of the geogrids simulated as equivalent sheet inclusions using Jewell et al. (1985) approach (eqs. 5.5 and 5.6) are compared to the measured values obtained by Raju (199) in Figure 5.13. A decent match between measured and simulated values is found for the three geogrids when the bearing stress ratio is calculated using eq. 5.6. In general, the simulations using eq. 5.6 tend to slightly overpredict the pullout response at n = 4 kPa for the Tensar and the Miragrid and to slightly underpredict the value of P at n = 10 kPa. The simulations of the Stratagrid show a very good agreement at small and large displacement pullout with the laboratory test data at the three vertical stresses applied when using eq. 5.6, also capturing the evident strain softening observed at n = 17 kPa. The simulations using eq. 5.5 underpredict the pullout response at peak and large displacement for the three grids at all vertical stresses. Figure 5.14 shows a comparison of the measured values of pullout force per unit width at large displacement (Pm) and simulated values using the maximum and the minimum values of cs,tx in all tests on geogrids. The use of eq. 5.6, which is the upper estimate of (b/n), shows, in general, a good fit to the measured values (Ps = 0.98 Pm) and shows that the use of eq. 5.5 underestimates the measured resistance by approximately 35% (Ps = 0.63 Pm). The schematic drawing in Figure 5.15 shows how the different constituents of the numerical model should be treated.    147  As shown in Figure 5.13, although the use of the Jewell et al. (1985) and eq. 5.6 shows, in general, a decent match between measured and simulated values of pullout resistance, the method seems to do a better job capturing the pullout response of the Stratagrid (GGS) at the different vertical stresses. It is important to recall that the objective of the use of the Jewell et al. (1985) approach is to obtain a soil-geogrid interface friction angle to treat the three-dimensional structure of the geogrid as a sheet inclusion. This approach considers the geometrical characteristics and surface roughness of the geogrid, along with surrounding soil characteristics. However, the Jewell et al. (1985) method does not account for the degree of interference of the transverse elements as described by Palmeira (2009). Palmeira (2009) suggests that for a ratio of St/Bt larger than 40, the degree of interference is negligible. The values of St/Bt are 33 and 29 for the Tensar (GGT) and the Miragrid (GGM) inclusions, respectively, indicative of a non-negligible degree of interference. On the contrary, the value of St/Bt for the Stratagrid is 37, very close to the value of 40 needed for a negligible degree of interference.   5.6. Numerical simulations using a constant Mtc   In order to verify the importance of accounting for the stress-dependency of the critical state friction angle in the back-analysis of pullout tests, additional simulations are performed using a constant value of Mtc equal to 1.07 (equivalent to cs,tx = 27 for n between 50 and 150 kPa) for the two rough sheet inclusions (APT and GMT) and the three geogrids at all stress levels. Since no value of cs,tx is available for the soil-GMS sheet inclusion, no simulations are done for this case. For the sheet inclusions, the value of Mtc for the soil and the soil-inclusion interface are taken equal to 1.07, given the fully rough soil-inclusion interface as explained in Section 3.4.1. For the three geogrids, the value of Mtc for the soil is given a value of 1.07 (corresponding to cs,tx = 27) and for the soil-geogrid interface a constant value of friction angle equal to cs,tx = 27 was used in equation 5.6. Figure 5.16 shows that, even though the strain softening behavior is maintained, the numerical simulations using a constant Mtc deduced from a cs,tx at n between 50 and 150 kPa, underestimate the pullout resistance of the sheet inclusions and geogrids at n between 4 and 17 kPa. The same response is observed for the horizontal stresses at the front wall (Figure 5.18), an underestimation of h is obtained when using a constant value of Mtc obtained at n between 50 and 150 kPa. A comparison of measured and simulated values in all tests, (Fig. 5.18), reveals 148  simulation using a stress-dependent Mtc provides a good fit to the measured values, for both, the pullout resistance and the horizontal stresses (Ps = 0.99Pm and hs = 1.02hm, respectively). The use of a constant value of Mtc taken at effective stresses between 50 kPa and 150 kPa, underestimates the measured response by approximately 20% (Ps = 0.79Pm and hs = 0.79hm) at peak and large displacement. These results confirm that a stress-dependent soil strength must be used for the back-calculation of pullout tests at very low stresses.   5.7. Summary  This Chapter shows the results of different approaches to the numerical simulations of the laboratory pullout tests on sheet inclusions performed by Raju (1995). In all the cases, the soil inside the pullout box was represented by the NorSand constitutive model and different combinations of inclusion and soil-inclusion interfaces were tested. In all the cases the inclusion was treated as an elastic element (as a structural beam or an elastic continuum) and the strength of Badger sand and the soil-inclusion interface as a stress-dependent friction angle. The observations and conclusions from this analysis are as follows:  1. The use of a constitutive model for Badger sand that can capture the dilatant behavior of dense sands (such as NorSand), combined with a structural beam or an elastic continuum and Mohr-Coulomb type of springs to represent the behavior of the soil-inclusion interface, does not capture the complete pullout response of inclusions showing a strain softening pullout behavior.   2. The use of a structural beam glued to the soil mesh or an elastic continuum attached to the soil mesh is equivalent, and can capture the pullout response of the APT, GMT and GMS sheet inclusions. This approach, however, implies that the complete soil mesh has to be given the soil-inclusion strength characteristics. For the APT and GMT inclusions of this study, this simplification is valid given the fully rough nature of the soil-inclusion interface. For the GMS inclusion, that has a very smooth interface, this simplification is not representative of the soil conditions in the laboratory pullout test, as it underestimates the strength of the soil.  149   3. The representation of the soil-inclusion interface as a thin continuum layer obeying a NorSand behavior to which the value of Mtc can be modified depending on the soil-inclusion strength characteristics, captures the full pullout response (including the strain softening behavior when present) of the sheet inclusions, as well as the lateral stresses generated on the inside front wall of the pullout box.    4. The results of these simulations further suggest that a plane strain condition governs the pullout response of sheet inclusions at all displacements in the pullout box and that the stress-dependency of the soil-inclusion interface friction angle must be considered in a back-calculation of pullout tests performed at n less than about 50 kPa. For example, the use of Mtc obtained in the stress range 50 to 150 kPa underestimates by 20 % the measured pullout resistance and horizontal stress at n less than 20 kPa.  From the numerical analysis of geogrids, the following observations can be made:  1. The use of the Jewell et al. (1985) approach to deduce an “equivalent” soil-geogrid interface friction angle to capture the three-dimensional structure of geogrids as equivalent sheet inclusions shows: a. The use of stress-dependent values of friction angle in triaxial compression along with the upper boundary (eq. 5.6) for the calculation of the bearing stress ratio, shows good agreement between measured and simulated pullout resistance. b. The use of stress-dependent values of friction angle in triaxial compression along with the lower boundary (eq. 5.5) for the calculation of the bearing stress ratio, underestimates the pullout resistance at large displacement by approximately 35% for the geogrids of this study. c. The use of the Jewell et al. (1985) approach appears to do a better job in capturing the pullout behavior of geogrids, when the degree of interference as defined in Palmeira (2009) is negligible.   150  Table 5.1: Different approaches for the numerical modeling of pullout tests  Pullout response Section Soil model Inclusion type Interface type Large displacement Chapter 4 Mohr-Coulomb Beam element Springs Small and large displacement Section 5.3.1. NorSand Beam element Springs Small and large displacement Section 5.3.2. NorSand Beam element Glued Small and large displacement Section 5.3.3. NorSand Elastic continuum Springs Small and large displacement Section 5.3.4. NorSand Elastic continuum Attached Small and large displacement  Section 5.3.5. Section 5.4. NorSand Elastic continuum NorSand continuum    151  Table 5.2: Summary of NorSand parameters   Parameter Description Value Typical range* G  Shear modulus (MPa)  82   Ir = G/p = 200 - 800   Poisson’s ratio 0.2  0.1 – 0.3 N  Volumetric coupling parameter 0.2 0.2 – 0.45 tc  Relates minimum dilatancy to state parameter 2.0  2.0 – 4.5 H  Plastic hardening modulus for loading 200  50 – 500   Altitude of the critical state line in e-logp space defined p at 1 kPa 0.715  0.75 – 1.4   Slope of the critical state line in e-logp space 0.013  0.01 – 0.07 Mtc  Critical stress ratio in the p - q plane  See Table 5.3 1.0 – 1.5 *Values of typical range found in Shuttle (2006) and Ghafghazi and Shuttle (2008)  Table 5.3: Values of Mtc (cs,tx)    Inclusion Surcharge, n (kPa) 4 8 12 APT 1.33 (33.0°) 1.26 (31.3°) 1.26 (30.5°) GMT 1.33 (33.0°) 1.26 (31.3°) 1.26 (30.5°) GMS 0.34 (9.2°) 0.26 (7.1°) 0.18 (5.0°)    152  Table 5.4: Values of friction angle and bearing stress ratio, for the soil-geogrid interface, considering triaxial parameters  Inclusion s° ; cs,ps° (b/n) (eq. 5.5) ; (b/n) (eq. 5.6)  n = 4 kPa n = 10 kPa n = 17 kPa n = 4 kPa n = 10 kPa n = 17 kPa GGT 9.2; 33.0 7.1; 30.3 5.0; 29.2 10.23; 26.1 8.31; 19.0 7.63; 16.8 GGS 26.4; 33.0 24.2; 30.3 23.4; 29.2 10.23; 26.1 8.31; 19.0 7.63; 16.8 GGM 26.4; 33.0 24.2; 30.3 23.4; 29.2 10.23; 26.1 8.31; 19.0 7.63; 16.8  Table 5.5: Values of cs,tx (Mtc), (eq. 5.5; eq. 5.6) for the soil-geogrid interface using triaxial parameters  Inclusion Surcharge, n (kPa) 4 10 17 GGT 16.1 (0.61); 30.6 (1.23) 13.0 (0.49); 23.6 (0.92) 11.1 (0.41); 20.4 (0.79) GGS 22.9 (0.89); 33.6 (1.36) 20.2 (0.78); 28.0 (1.11) 19.2 (0.74); 26.0 (1.03) GGM 21.9 (0.85); 34.1 (1.38) 19.2 (0.74); 28.1 (1.12) 18.2 (0.70); 26.0 (1.03)    153           Figure 5.1: Determination of tc for Badger sand                     Figure 5.2: Calibration of NorSand to Badger sand 3= 50 kPa e = 0.623 DR = 34% 3= 50 kPa e = 0.623 DR = 34% 3 = 100 kPa e = 0.624 DR = 33% 3 = 100 kPa e = 0.624 DR = 33% 3 = 150 kPa e = 0.620 DR = 35% 3 = 150 kPa e = 0.620 DR = 35% 154            Figure 5.3: Determination of critical state parameters for Badger sand               Figure 5.4: Independent check of NorSand parameters for Badger sand using triaxial tests on moist tamped samples   3 = 100 kPa e = 0.56 DR = 65 % 3 = 300 kPa e = 0.538 DR = 76 % 155  202224262830323436381 10 100 1000Critical state friction angle, cs()Effective stress,  (kPa)AAAAAA (measured)AAAAAA (measured)AAAAAA (measured)AAAAAA (eq. 4.2)AAAAAA (eq. 5.3)cs,dsscs,txcv,pscs,dscs,tx         Figure 5.5: Variation of Badger sand critical state friction angle with applied stress             Figure 5.6: Comparison of measured and simulated pullout response for APT08 using NorSand in combination with a beam element and an elastic continuum for the inclusion and springs for the interface   024681012140 10 20 30 40 50Pullout resistance (kN/m)Displacement of clamped end (mm)MeasuredSimulated using beam elementSimulated using elastic continuum156  024681012140 10 20 30 40 50Pullout resistance (kN/m)Displacement of clamped end (mm)MeasuredSimulated using beam elementSimulated using elastic continuum024681012140 20 40 60 80 100Pullout resistance (kN/m)Displacement of clamped end (mm)012340 20 40 60 80 100Pullout resistance (kN/m)Displacement of clamped end (mm)                          Figure 5.7: Comparison of measured and simulated pullout response using NorSand glued to the beam element or attached to the elastic continuum for a) APT, b) GMT, c) GMS, at n = 8 kPa  a) b) c) 157             Figure 5.8: Comparison of measured and simulated pullout response for GMS08 using the “NorSand interface layer” approach   012340 10 20 30 40 50 60Pullout force per unit width (kN/m)Displacement of clamped end (mm)Measured Simulated 158                               Figure 5.9: Pullout box and numerical model: a) schematic of the laboratory pullout box; b) FLAC numerical grid and boundary conditions; c) soil-front wall and soil-inclusion interface detail 31 elements Elastic NorSand 65 elements Velocity, dc 50 elements Surcharge (n) a) Surcharge (a) Planar inclusion 1.3 m Badger sand P, dc Hb  = 0.6 m 0.95 m b) Not to scale Badger sand - NorSand Planar inclusion - Elastic Soil-Inclusion interface – NorSand (4 elements high = 15 mm) Surcharge (a) c) Soil-front wall interface – NorSand (4 elements wide = 15 mm) ff Velocity, dc 159  051015200 20 40 60 80 100Pullout resistance (kN/m)Displacement of clamped end (mm)Measured (Raju, 1995)Simulatedn = 12 kPan = 8 kPan = 4 kPa05101520250 20 40 60 80 100Pullout resistance (kN/m)Displacement of clamped end (mm)n = 12 kPan = 8 kPan = 4 kPa0123450 20 40 60 80 100Pullout resistance (kN/m)Displacement of clamped end (mm)n = 12 kPan = 8 kPan = 4 kPa                                  Figure 5.10: Measured and simulated pullout results using a stress-dependent Mtc of a) APT; b) GMT; and c) GMS a) b) c) 160  0204060801000 20 40 60 80 100Displacement of embedded end (mm)Displacement of clamped end (mm)Measured (Raju, 1995)Simulated0204060801000 20 40 60 80 100Displacement of embedded end (mm)Displacement of clamped end (mm)n = 12 kPan = 4 kPan = 8 kPa0204060801000 20 40 60 80 100Displacement of embedded end (mm)Displacement of clamped end (mm)                            Figure 5.11: Displacement of embedded end with displacement of clamped end for a) APT; b) GMT; and c) GMS  a) b) c) 161                      Figure 5.12: Measured and simulated horizontal stresses at the front wall using a stress-dependent Mtc of a) GMT; and b) GMS    Measured GMT GMS Simulated a) b) Incremental horizontal stress, h (kPa) Incremental horizontal stress, h (kPa) Normalized distance to inclusion Normalized distance to inclusion 162  05101520250 20 40 60 80 100Pullout resistance (kN/m)Displacement of clamped end (mm)n = 17 kPan = 10 kPan = 4 kPa05101520250 20 40 60 80 100Pullout resistance (kN/m)Displacement of clamped end (mm)n = 17 kPan = 10 kPan = 4 kPa              Figure 5.13: Measured and simulated pullout response of a) GGT, b) GGM, and c) GGS geogrids using lower (eq. 5.5) and upper (eq. 5.6) bounds for the bearing stress ratio  05101520250 20 40 60 80 100Pullout resistance (kN/m)Displacement of clamped end (mm)Measured (Raju, 1995)Simulated using eq. 5.5Simulated using eq. 5.6n = 17 kPan = 10 kPan = 4 kPaa) b) c) 163  024681012141618200 2 4 6 8 10 12 14 16 18 20Simulated pullout resistance, Ps (kN/m)Measured pullout resistance, Pm (kN/m)Ps = PmUsing eq. 5.6:Ps = 0.98PmR2 = 0.99Using eq. 5.5:Ps = 0.63PmR2 = 0.93                Figure 5.14: Comparison of measured and simulated pullout resistance using lower (eq. 5.5) and upper (eq. 5.6) bounds for the bearing stress ratio   164                    Figure 5.15: Schematic of how each component of the pullout test is treated for numerical simulations   Soil: Norsand Calibrated from triaxial tests, preferably at stress level used in pullout Inclusion: Elastic continuum Elastic Young modulus Poisson’s ratio Soil-front wall interface: NorSand  continuum Same parameters as soil except for Mtc Mtc = f (ff) Soil-inclusion interface: NorSand  continuum Same parameters as soil except for Mtc Sheet inclusions:  Mtc = f (cs,tx) = f () cs,tx = cs,ps - 1 sin cs,ps = tan cs,ds  Geogrids: modeled as sheet inclusions  Use tan cs,tx = fb tan cs,tx (Jewell et al., 1985) Mtc = f (cs,tx) = f ()  165                           Figure 5.16: Measured and simulated pullout results using a constant Mtc of a) APT; b) GMT; c) GMS; d) GGT; e) GGM; and f) GGS  Measured APT GMT GMS GGT GGM Simulated GGS a) d) b) e) c) f) 166                   Figure 5.17: Measured and simulated horizontal stresses at the front wall using a stress-dependent Mtc of a) APT; b) GMT; c) GMS; d) GGT; e) GGM; and f) GGS   Measured GMT GGT GGM Simulated GGS a) c) b) d) 167  Mtc - stress dependent GMT GMS GGT GGM GGS peak large displacement APT                        Figure 5.18: Comparison of measured and simulated values of a) pullout response and b) horizontal stresses, using Mtc = stress-dependent and Mtc = 1.07 a) b) Mtc = 1.07 168  Chapter 6 Sensitivity analysis  6.1. Introduction  The first three research objectives of the current study establish the basis for a sensitivity analysis in response to Objective 4 of this thesis. The characterization of the material properties comprising the pullout test (soil, inclusion and soil-inclusion interface) in Chapter 3 (Objective 1), were used as input parameters to the numerical models of Chapters 4 and 5. In particular, the parameters for the soil-sheet inclusion interface were experimentally derived from the direct shear tests performed by Raju (1995) and demonstrate, in Chapter 4 (Objective 2), that plane strain conditions prevail in pullout at large displacement and that the stress-dependency of the friction angle at the critical state has to be accounted for in a proper back-analysis of pullout test data. These findings, along with a soil-inclusion interface that is able to dilate following a NorSand behavior, were then considered in the numerical model of Chapter 5. This model allowed for the capture of two independent measurements of the laboratory pullout tests performed by Raju (1995) for the sheet inclusions, namely the pullout resistance (including the strain softening behavior found in the APT and GMS inclusions), and the horizontal stresses generated at the front wall of the pullout box, in accordance to Objective 3.  In this Chapter, a sensitivity analysis is done for a sheet inclusion, to study both the adequacy of the pullout laboratory data available in the literature, and the ASTM D6706-01 recommendations on the test and boundary conditions of the pullout box. Parameters related to the inclusion (length and elastic modulus), the soil-inclusion interface (apparent friction angle), the pullout box (height and front wall roughness) and test conditions (vertical stress applied) are varied within the range of typical values found in the literature. The results of these simulations identify the possible weaknesses in the determination of the soil-inclusion interaction factor, and allow testing of the hypothesis that there are combinations between the aforementioned parameters for which pullout does not occur and the measured response is primarily an in-air stress-strain curve, or a combination of in-air extension with elongation of the inclusion. Finally, an “ideal” pullout test is proposed that seems to ensure that pullout really occurs.  169  6.2. Definitions   6.2.1. Definition of pullout  ASTM D6706-01 defines pullout as the “movement of a geosynthetic over its entire embedded length, with initial pullout occurring when the back of the specimen moves, and ultimate pullout occurring when the movement is uniform over the entire embedded length”. In the current study, this definition is interpreted as follows (see Figure 6.1):  - Initial pullout (Pi) represents an extensible behavior of the inclusion and occurs when the displacement of the clamped end (dc) is larger than the displacement of the embedded end (de), irrespective of whether there is movement of the embedded end (dc > de = 0 or dc > de > 0); - Ultimate pullout (Pu) refers to an inextensible behavior of the inclusion where the displacement of the clamped end equals the displacement of the embedded end (or dc = de), with the onset of Pu occurring at the beginning of dc = de.   ASTM D6706-01 also defines ultimate pullout resistance as “the maximum pullout resistance measured during a pullout test”. This definition suggests that Pu and the maximum pullout resistance are the same.  6.2.2. Definition of vertical stress  ASTM D6706-01 defines the total vertical stress applied to the inclusion as the addition of the applied vertical stress (a) at the soil surface and the vertical stress due to soil above the inclusion (s), as shown in Figure 6.2. In this Chapter, the total vertical stress applied to the inclusion as per the ASTM D6706-01 definition is n,ASTM. In addition, n,u is used to define the vertical stress applied to the inclusion at the onset of ultimate pullout, Pu.    170  6.2.3. Definition of base case  Table 6.1 shows the variables that are changed in the sensitivity analysis; two are related to the inclusion (length, L, and elastic modulus, E), one is related to the soil-inclusion interface (apparent friction angle, ), two are related to pullout box conditions (soil-front wall friction angle, ff, and box height, Hb) and one is related to test conditions (vertical stress, n,ASTM).   Inspection of the literature review in Chapter 2 establishes the range in which each parameter has been studied (see Table 6.1), and determines that L varies between 0.25 m and 2.0 m, E varies between 4×103 kPa and 3×106 kPa, and Hb varies between 0.2 m to 1.1 m. The studies in which the roughness of the inside front wall is changed are very limited, and reveal the soil-front wall interface friction angle to be a minimum of ff = 0 for a frictionless front wall (essentially assumed in numerical simulations) and a maximum equivalent to the friction angle of the soil (as described in Palmeira, 1989). Intermediate values were found to be 5° to 6° for a lubricated front wall and 15° to 16° for an aluminum smooth front wall (Raju, 1995). The values of vertical stress ranged between 2 kPa and 200 kPa, with the majority of the tests performed at less than 100 kPa (Huang and Bathurst, 2009). ASTM D6706-01 suggests that vertical stresses up to 250 kPa must be anticipated. Also shown in Table 6.1, are the minimum dimensions for the length of the inclusion and the height of the box suggested by ASTM D6706-01, with values of 0.6 m and 0.3 m, respectively.   In this study, a “base case” for the sensitivity analysis is defined so as to ensure that ultimate pullout (or dc = de) occurs for a vertical stress up to 200 kPa. Accordingly, for the base case (see Figure 6.2), the length of the inclusion is the minimum suggested by ASTM D6706-01 (L = 0.6 m) and the elastic modulus is 1×106 kPa. The soil-inclusion interface friction angle is defined as 2/3cs,tx (with cs,tx as a stress-dependent parameter for n,ASTM lower than 50 kPa). The box height is chosen as 0.6 m, corresponding to the height used by Raju (1995) for the laboratory tests that are the basis of this study. The soil-front wall friction angle is 15 (equivalent to an aluminum front wall as Raju’s testing device). The base case is evaluated for different values of vertical stress (n,ASTM = 4 kPa, 12 kPa, 30 kPa, 50 kPa, 100 kPa and 200 kPa), and for each vertical stress, the 171  aforementioned parameters are changed within the ranges extracted from the literature review as reported in Table 6.1.  6.3. Tornado diagrams  The study of the influence of L, E, , ff and Hb on the onset of ultimate pullout is done using Tornado diagrams (one diagram for each value of n,ASTM). “A Tornado diagram is a special bar chart which is the graphical output of a comparative sensitivity analysis (Syncopation Software, Inc., 2015)”. The objective of the Tornado diagram is to establish which parameters most influence the desired output, in this case, Pu. Figure 6.3 shows the six Tornado diagrams obtained from the sensitivity analysis of this study. It is important to recall that only the simulations reaching a value of ultimate pullout (or dc = de) are included. The x-axis corresponds to the output of the sensitivity analysis, in this case, to the magnitude of Pu and the vertical line corresponds to the value of Pu of the base case (Fig. 6.2) for the current n,ASTM. The five parameters changed in the sensitivity analysis (L, E, , ff and Hb) for each value of n,ASTM are shown in the y-axis along with the input range for which ultimate pullout occurs. Each parameter is associated with a horizontal bar that ranges between the extreme values of Pu obtained from the simulations. Also, Pu obtained from each intermediate value within the range of study shown in the last column of Table 6.1, is added in the horizontal bar. The width of each bar shows how much impact that parameter has on the value of Pu. Finally, in order to quantify the influence of each variable on the onset of pullout, each value of Pu shown in the horizontal bars is normalized by the value of Pu at the base case and the extreme values are shown in brackets next to its bar; the larger the difference between the maximum and the minimum value, the larger the influence of the parameter on Pu.   The results of the sensitivity analysis presented in the Tornado diagrams (Figure 6.3) show that, for the ranges of the current study, and for n,ASTM = 4 kPa and 12 kPa (for which ultimate pullout always occurs), the inclusion characteristics (L and ) have the largest influence on the magnitude of Pu, followed by the boundary conditions of the pullout box (Hb and ff). The results suggest that the elastic modulus of the inclusion exerts the smallest influence on the magnitude of ultimate pullout, as evidenced by the narrower difference of extreme values of Pu. In all the cases, Pu increases with the parameter studied (L, , E and ff) except for the height of the pullout box, where 172  Pu decreases with increasing Hb as shown, for example in Figure 6.3b, where the values of Pu for Hb = 1.0 m and Hb = 0.3 m are 7 kPa and 11 kPa, respectively.  In the following Section, an evidence-based analysis of the Tornado diagrams results is performed in order to improve the understanding on the influence of each parameter on the onset of ultimate pullout. This will allow to make recommendations for the scientific literature and the ASTM D6706-01 Standard regarding the test conditions and boundary conditions of the pullout box.  6.4. Analysis of results  6.4.1. Influence of vertical stress on Pu at base case  Prior to the analysis of the influence of each parameter on the onset of pullout, the base case scenario is studied to gain more insight on the influence of the vertical stress on Pu. The first analysis is on the influence of n,ASTM on Pu, which is the typical vertical stress reported in the literature and assumed constant during the complete pullout test. Figure 6.4a (curve a) shows that for the base case with L = 0.6 m, E = 1×106 kPa, ff = 15, Hb = 0.6 m and a stress-dependent  = 2/3cs,tx for n,ASTM  ≤ 50 kPa, Pu increases with vertical stress, for values of n,ASTM up to 200 kPa. Further examination of the data for the base case shows a non-linear increase of Pu with vertical stress for n,ASTM between 4 kPa and 50 kPa (Figure 6.4b, curve a), that although small, it is not possible to ignore. A linear relation exists between Pu and n,ASTM, for a n,ASTM between 50 kPa and 200 kPa.   In order to determine the nature of the slight non-linear increase of Pu at the base case with vertical stress for n,ASTM between 4 kPa and 50 kPa, three additional simulations were made at n,ASTM = 4 kPa, 12 kPa and 30 kPa but using a constant value of cs,tx = 27 and a constant value of  = 2/3cs,tx = 18, corresponding to cs,tx at n,ASTM = 50 kPa (stress level at which no more stress-dependency was observed in the angle of friction of Badger sand, see Figure 3.10). The results in Figure 6.4b (curve b) show a reduction on Pu at n,ASTM = 4 kPa, 12 kPa and 30 kPa, but the non-linearity still exists. These results suggest that the slight (but non-negligible) non-linearity observed in the onset 173  of ultimate pullout at n,ASTM < 50 kPa (curve a) for the base case is explained in part by the stress-dependency of  and cs,tx at stresses lower than 50 kPa.   Examination of the distribution of vertical stresses at the inclusion level at Pu for curve a, shows that n,u is not constant along the length of the inclusion (Figure 6.4c); it is larger close to the front wall of the box and decreases with distance from the front wall. Inspection of the normalized value of n,u to n,ASTM shows that close to the front wall of the pullout box the vertical stress increases up to 4.5 times n,ASTM and decreases abruptly until 20% of the length of the inclusion. Between 0.2L and 0.9L, n,u is slightly larger than n,ASTM and between 0.9L and L, n,u is lower n,ASTM. The increase of n,u at the front wall is presumed to be due to the presence of the front wall with a friction angle of 15, and the decrease at the back end of the inclusion due to an arching effect against the back wall of the box.   Integration of the area below the curve of n,u plotted against the values of Pu (Figure 6.4d, curve c) shows a linear relation between n,u and Pu given by Pu = 0.44n,u. Back-calculation of  results in a value of 20 which is very close to the input value of  = 18 for n,ASTM  ≥  50 kPa. This analysis suggests that the non-linearity between Pu and n,ASTM observed at low stresses, even when constant values of cs,tx and  are used, disappears when Pu is plotted against the current value of n,u which is larger than n,ASTM.  6.4.2. Influence of tan on Pu  The results of the Tornado simulations show that the value of ultimate pullout increases with increasing soil-inclusion interface friction angle (Figure 6.5a). For n,ASTM  ≥ 100 kPa, high interface friction materials (or  = cs,tx) do not allow for ultimate pullout to occur. For 4 kPa ≤ n,ASTM  ≤ 100 kPa, tan exerts a non-linear influence on the onset of Pu that extrapolates to the origin. As per Figure 6.4a, this non-linearity between Pu and tan is expected and explained by the stress-dependency of  and cs,tx at stresses lower than 50 kPa.   174  Further examination of the vertical stresses at the soil-inclusion interface at Pu, and plotting Pu against n,u for  = cs,tx and  = 2/3cs,tx (Figure 6.5b) establishes a linear relation between Pu and n,u given by Pu = 0.65n,u. Back-calculation of  results in a value of 28.5, which is very close to the input of  = 27 for n,ASTM ≥ 50 kPa. As expected, this linear relation between Pu and n,u plots above the case in Figure 6.4d (curve c) where  = 2/3cs,tx.  6.4.3. Influence of ff on Pu  In Section 6.4.1. the increase in the distribution of vertical stresses close to the front wall of the box was presumed to be attributed to the presence of the front wall with a friction angle equal to 15 (equivalent to an aluminum front wall). In this Section, further analysis is presented on the influence of the front wall roughness. The results of the Tornado diagrams show that for n,ASTM  ≤ 50 kPa, the value of ultimate pullout increases with increasing roughness of the front wall, for 0 ≤ ff ≤  30 (Figure 6.6a). A fully rough front wall (ff = 30) does not allow for ultimate pullout at n,ASTM  ≥ 100 kPa. For n,ASTM  ≤ 50 kPa, a slight non-linear relation exists between Pu and ff .   Examination of vertical stresses at the soil-inclusion interface level at Pu for n,ASTM = 50 kPa (Figure 6.6b) shows that the distribution of n,u at the front wall of the box increases with ff . Even for a frictionless front wall (or ff = 0), n,u is 18% larger than n,ASTM suggesting that the sole presence of the rigid front wall results in an increase of vertical stresses at the soil-inclusion interface level.   The results of this analysis suggest that for a medium interface friction material (in this case  = 2/3cs,tx), a difference in front wall roughness from ff = 0 (corresponding to a frictionless front wall) to ff = 15 (corresponding to an aluminum front wall, which is the most common configuration after the presence of sleeves) does not make a significant difference to the value of Pu (less than 15%). In order to verify if this observation holds true for high interface friction materials, new simulations were performed using a stress-dependent value of  = cs,tx, for ff = 0 and ff = 15 for n,ASTM between 4 kPa and 50 kPa. Figure 6.7a shows that for a given stress level, Pu increases with ; for ff = 0, Pu is in average 14% larger for  = cs,tx than  = 2/3cs,tx and for 175  ff = 15, Pu is in average 40% larger for  = cv,tx than  = 2/3cs,tx. In addition, Figure 6.7a shows that the influence of an aluminum front wall (ff = 15) instead of a frictionless front wall yields an increase in around 40% in the value of Pu, when a rough soil-inclusion interface is used.   Further examination of the data allows for the analysis of the effect of a frictionless front wall (or ff  = 0) on Pu, which is a configuration not possible to obtain in the laboratory test. Figure 6.7b shows that there is a slight non-linear relation between Pu and n,ASTM. As per Section 6.4.1. this non-linearity is attributed in part to the stress-dependency of  and cs,tx at stresses lower than 50 kPa. Further inspection of the data for ff  = 15 (base case) and ff  = 30 reveals that at stresses lower than 50 kPa the non-linearity between Pu and n,ASTM increases with ff.   Inspection of the vertical stresses at the soil-inclusion interface level at Pu, and repeating the analysis of Sections 6.4.1. and 6.4.2., where the values of Pu are plotted against n,u for each front wall configuration, results in two main observations (Figure 6.7b). The first is that the non-linearity between Pu and the vertical stress disappears when plotted against n,u, for all values of ff. The second is that Pu vs n,u plots in a unique line irrespective of the value of ff. These results suggest that Pu is linearly related to n,u (more than n,ASTM) and is independent of the front wall roughness.  6.4.4. Influence of L on Pu  The Tornado diagrams show that Pu increases with L and that there is a critical inclusion length for which no ultimate pullout occurs, that depends on the magnitude of n,ASTM. For short inclusions (or L < 0.6 m), ultimate pullout occurs for n,ASTM ≤ 200 kPa. For an inclusion length between 0.6 m and 1.0 m ultimate pullout occurs for n,ASTM ≤  30 kPa and for long inclusions (or L > 1.0 m) ultimate pullout occurs for n,ASTM ≤  12 kPa. Inspection of the cases for which ultimate pullout occurs, shows a slight non-linear increase of Pu with L that extrapolates to the origin (Figure 6.8a). Further examination of the data for which Pu occurs for the complete range of normal stress (4 kPa ≤ n,ASTM ≤ 200 kPa) shows a non-linear relation between Pu and n,ASTM that is more evident for normal stresses lower than 50 kPa (Figure 6.8b). This non-linearity is expected due to the stress-dependency of  and cs,tx at n,ASTM  ≤ 50 kPa.  176   The same analysis is done in this Section were Pu is plotted against n,u (see Figure 6.8c). Again, the non-linearity at low stresses disappears, and linear relations between Pu and n,u are obtained given by Pu = 0.37n,u (for L = 0.5 m) and Pu = 0.21n,u (for L = 0.3 m).   6.4.5. Influence of E on Pu  The results of the Tornado diagram show that the lower the value of the elastic modulus of the inclusion, the lower the value of n,ASTM that allows for ultimate pullout to occur (Figure 6.9a). For E = 1×104 kPa, ultimate pullout occurs for n,ASTM up to 12 kPa, for E = 1×105 kPa, ultimate pullout occurs for n,ASTM up to 30 kPa, for E = 5×105 kPa, ultimate pullout occurs for n,ASTM up to 50 kPa and for E = 1×106 kPa (or the base case), ultimate pullout occurs for n,ASTM up to 200 kPa.   Inspection of the cases for which ultimate pullout occurs (Figure 6.9b) shows a non-linear increase of Pu with E. The non-linearity is more evident at lower values of E. In order to better understand the decrease of Pu with E, the results of the simulations for n,ASTM = 12 kPa for a range of elastic modulus between 1×106 kPa and 1×104 kPa are taken as an example and shown in Figure 6.10a. The results show that as E decreases the pullout resistance at small displacements also decreases; for the stiffer inclusions a peak and strain softening behavior is observed, and for the inclusion with a lower E a gradual increase of the pullout resistance is noticed. Figure 6.10b shows that Pu occurs before peak or maximum pullout resistance and therefore, for stiffer inclusions, where a peak and strain softening behavior is observed, Pu occurs at values higher than the large displacement pullout resistance. On the contrary, for more extensible inclusions, where only strain hardening behavior is observed, Pu occurs at values lower than the large displacement pullout resistance.       177  6.4.6. Influence of Hb on Pu  The results of the Tornado diagrams show that the height of the box exerts an influence on the onset of ultimate pullout (Figure 6.11a). As Hb increases, the value of Pu decreases non-linearly from Hb = 0.3 m (which is the minimum suggested by ASTM D6706-01) until a critical height of 0.6 m beyond which the reduction in the value of Pu is negligible (showing a difference of less than 5% which is a difference attributed to testing error, as per the APT sheet tested at a vertical stress of 4 kPa by Raju, 1995). This non-linear increase of Pu with n,ASTM is in part due to the stress-dependency of  and cs,tx at n,ASTM  ≤ 50 kPa.  To explain the non-linear inverse relation between Pu and Hb, the distribution of vertical stresses at the soil-inclusion interface at Pu for n,ASTM = 50 kPa, is shown as an example. Figure 6.11b shows an increase in the distribution of n,u along the length of the inclusion for values of Hb = 0.3 m and 0.4 m. As the distance between the top boundary and the soil-inclusion interface level increases, the value of n,u decreases and therefore so does Pu. These results suggest that a value of Hb = 0.6 m (or possibly Hb/L = 1), eliminates the influence of the top boundary and results, therefore, in an almost constant value of Pu for Hb ≥ 0.6 m.  6.4.7. Summary of the analysis  A compilation of the data obtained from the Tornado simulations results in Tables 6.2a and 6.2b, and establishes the combination of parameters for which ultimate pullout and initial pullout occur, respectively. Table 6.2a along with the aforementioned conclusions are used to suggest an ideal pullout box and test conditions, that appears to guarantee that ultimate pullout occurs with minimum boundary conditions effect (Figure 6.12).   - A front wall with a maximum roughness equivalent to an aluminum material, as per Section 6.4.3. (ff ≤ 15)  - A minimum box height of 0.6 m, as per Section 6.4.6. (Hb ≥ 0.6 m)  178  - A maximum inclusion length of 1.0 m according to design guidance of reinforced structures suggests a minimum length for the reinforcement of 1 m beyond the surface failure (L ≤ 1.0 m) - A maximum vertical stress applied of 30 kPa that allows for a 1.0 m long inclusion to reach ultimate pullout, as per Table 6.2a (n,ASTM ≤ 30 kPa) - A minimum elastic modulus of 1×105 kPa that allows for a 1.0 m long inclusion tested at n,ASTM = 30 kPa to reach ultimate pullout, as per Table 6.2a (E ≥ 1×105 kPa)  6.5. Discussion  The discussion of the implications of the analysis of the Tornado diagrams is made in three parts. First, the results of the Tornado simulations are compared to the results of Raju (1995), allowing for confidence in the results obtained in Section 6.4. The second part refers to the analysis of the available pullout data in the literature, and the last discussion is related to the adequacy of ASTM D6706-01 regarding the test and boundary conditions of the pullout box.  6.5.1. Comparison of Tornado simulation results with laboratory test data   In this Section, the results of the Tornado simulations are compared to the results of the laboratory pullout tests performed by Raju (1995). In particular, the closest combinations of the sensitivity analysis are compared to the data on the textured geomembrane tested at vertical stresses of 4 kPa and 12 kPa.   Although the range of values of the parametric study brackets the values of the laboratory data, none of the combinations of the input values for the Tornado simulations exactly match the values that Raju (1995) measured on the laboratory tests for the GMT tests at n,ASTM = 4 and 12 kPa (GMT04 and GMT12, respectively), namely L = 0.95 m, E = 7.7×105 kPa,  = cs,tx, Hb = 0.6 m and ff = 15. As shown in Figure 6.2, the base case for the Tornado simulations was given the following parameters, L = 0.6 m, E = 1.0×106 kPa,  = 2/3cs,tx, Hb = 0.6 m, ff = 15. Inspection of the combinations of the Tornado simulations shows that the closest combinations are:  179  o Combination 1: L = 0.6 m;  = cs,tx; E = 1.0×106 kPa; Hb = 0.6 m; ff = 15 o Combination 2: L = 1.0 m;  = 2/3cs,tx; E = 1.0×106 kPa; Hb = 0.6 m; ff  = 15  In combination 1 the length and the elastic modulus of the inclusion are different from the GMT parameters and in combination 2, the soil-inclusion interface friction angle and the elastic modulus are different from the GMT parameters. Therefore, none of the simulations is expected to replicate Raju’s (1995) laboratory data.   Figure 6.13 shows a comparison of the GMT04 and GMT12 tests and the combinations 1 and 2 of the Tornado simulations. As expected, none of the Tornado combinations plot on top of the laboratory test data. Both plotting positions are lower; combination 1 is lower than the laboratory test data because L is shorter and combination 2 is lower because the value of  is smaller. In both cases, the simulations show a slight peak pullout resistance followed by a strain softening behavior due to the larger elastic modulus of the inclusion in the simulation compared to the laboratory test. As observed in Figure 6.10a, stiffer inclusions present stiffer pullout responses and a peak and strain softening behavior, which is the case of the simulations.  In order to verify that the Tornado simulations can capture the behavior of the GMT04 and GMT12 laboratory test data, an additional simulation was carried out that falls in the range of the Tornado plots but matches closely the combination of the laboratory data, namely:   o Combination 3: L = 1.0 m;  = cs,tx; E = 1.0×106 kPa; Hb = 0.6 m; ff  = 15  The result of the simulations matches closely but not exactly the laboratory test data. The simulated curve is slightly higher than the laboratory test data (due to the peak pullout resistance) and slightly stiffer due to the higher value of the elastic modulus of the inclusion in the simulation.  The good relative comparison of the laboratory test data and the results of the Tornado simulations establishes that the numerical model works within the range of parameters established in Table 6.1. It is relevant to note that the sensitivity analysis did not allow for the comparison between laboratory test data and numerical simulations when only initial pullout occurs, given that all the 180  laboratory tests performed by Raju (1995) reached a condition of ultimate pullout. The analysis of the Tornado diagrams can, therefore, be used to elevate the understanding of the mechanisms occurring in the pullout box and to examine the validity of the laboratory test data available in the literature and the ASTM D6706-01 recommendations.  6.5.2. Analysis of the laboratory test data in the literature  Comparison of Table 6.1 with the recommended ideal pullout test (Figure 6.12) suggests that there is a number of laboratory pullout studies published in the literature that do not comply with the conditions for ultimate pullout to take place, implying that only initial pullout occurs. Most of these studies do not measure displacements at both ends of the inclusion to determine whether a condition of dc = de really happens. In particular, comparison to Tables 2.2 and 2.3 shows that some of the laboratory tests of Farrag et al. (1993), Wilson-Fahmy et al. (1994), Moraci and Recalcati (2006), Bathurst and Ezzein (2015), Ingold (1983), Palmeira and Milligan (1989), do not reach ultimate pullout. A detailed description of the pullout boxes and test characteristics used by the authors listed above is provided in Section 2.4 of the literature review. In this Section, an assessment of the pullout tests found in the literature is done considering two aspects namely, inclusion characteristics and boundary conditions of the testing device.  6.5.2.1. Inclusion characteristics  The testing program developed by Farrag et al. (1993) was intended to study the influence of geogrid type, specimen width, sleeve length, soil thickness, displacement rate, soil density and vertical stress on the pullout response of two different grids. Box dimensions were 1.52 m in length, 0.92 m in width and 0.76 m in height. The front wall was equipped with a 30 cm long sleeve. The length of the inclusions tested was 0.9 m and three different vertical stresses were applied of 48.2 kPa, 96 kPa and 140 kPa. Comparison of the inclusion length and vertical stress used in their laboratory pullout tests with the combinations suggested in Table 6.2a, implies that the tests performed at vertical stresses of 48.2 kPa might have attained ultimate pullout. Inspection of their results (Figure 6.14) suggests that, in effect, the front and rear node of the geogrid are reaching a nearly equal rate of displacement after 800 seconds of testing, indicative of a condition 181  of dc = de. On the contrary, the combinations summarized in Table 6.2a suggest that the tests performed at vertical stresses of 96 kPa and 140 kPa, did not achieve ultimate pullout. Fortunately, all the tests intended to study the influence of the different parameters were performed at a vertical stress of 48.2 kPa and therefore, an ultimate pullout condition was reached and therefore, their suggestions are reliable.  Wilson-Fahmy et al. (1994) conducted laboratory pullout tests as a benchmark for their finite element numerical model. The pullout box was 1.9 m long, 0.91 m, wide and 1.1 m high. A 10 cm long sleeve was used at the front wall of the testing device. Tests at a vertical stress of 69 kPa were performed on three different geogrids with different lengths ranging from 0.31 m and 1.7 m. Elastic modulus deduced at 2% strain resulted in E = 7.9×105 kPa for geogrid A, E = 2.5×105 kPa for geogrid B and E = 1.9×105 kPa for geogrid C. From pullout force versus displacement plots (Fig. 6.15), the authors state that geogrid A reached ultimate pullout for lengths of 0.31 m and 0.92 m and that geogrids B and C, at a length of 0.31 m, suffered pullout with some junction failures. In contrast, geogrid A at L = 1.7 m, geogrid B at L = 0.92 m and 1.7 m, and geogrid C at L = 0.92 m and 1.7 m are identified as showing pullout failure. These observations agree with the combinations in Table 6.2a except for geogrid A with a length of 0.92 m.  The work of Moraci and Recalcati (2006) suggests that the influence of the length of the inclusion cannot be assessed independently from the vertical stress applied. In their paper, they studied the influence of the inclusion stiffness and structure, embedded length and vertical effective stress applied on the pullout response of three different geogrids. The testing device was 1.7 m long, 0.6 m wide and 0.68 m high. The front wall of the box was equipped with a 25 cm long sleeve. They tested three different geogrids with lengths ranging from 0.4 m to 1.15 m at vertical stresses between 10 and 100 kPa. Deduced elastic modulus of the geogrids at 2% strain suggest E = 4.7×105 kPa for GG1, E = 6.77×105 kPa for GG2 and E = 4.7×105 kPa for GG3. In particular, they show that short inclusions (L < 0.4 m) and long inclusions at stresses lower than 25 kPa show a strain softening behavior whereas long inclusions (L > 1.15 m) tested at stresses larger than 25 kPa result in a strain hardening behavior (Fig. 6.16). This is due to the progressive development of the pullout interaction mechanism at the interface level as the inclusion elongates. These observations are consistent with the results of the Tornado simulations of this study, that show that for short 182  inclusions the strain softening behavior results in a value of Pu larger than the large displacement pullout resistance. For long inclusions the strain hardening behavior results in a value of Pu smaller than the large displacement pullout resistance. Additional inspection of their results shows that for the three 0.4 m long inclusions tested at 10 kPa and 100 kPa a condition of dc = de is reached (Fig. 6.17). Similarly, for the three 1.15 m long inclusions tested at 10 kPa a condition of dc = de is reached (Fig. 6.18). On the contrary, for the three inclusions 1.15 m long tested at a vertical stress of 100 kPa, ultimate pullout is not reached (Fig. 6.18). These observations are consistent with the ranges of inclusion length and stiffness resulting from the Tornado simulations as shown in Table 6.2a.  In their latest experimental program using a transparent soil and a transparent pullout box, 3.7 m in length by 0.8 m wide by 0.3 m high, Bathurst and Ezzein (2015) tested a 2 m long geogrid at vertical stresses ranging from 2 kPa to 52 kPa. A 0.2 m long sleeve was used at the front wall of the box. An elastic modulus of 3.4×104 kPa was deduced at 5% strain. Inspection of the results (Fig. 6.19) suggests that only the tests at 2 kPa and 7 kPa have reached ultimate pullout which is consistent with the combinations suggested in Table 6.2a from the Tornado simulations.  The comparison of the laboratory test data available in the literature with the ranges suggested in Table 6.2a for the length and elastic modulus of the inclusion and the range of vertical stress that allow for ultimate pullout to occur, show in general, good agreement. Table 6.2a appears therefore to be a useful tool to predict whether a certain configuration of inclusion characteristics and vertical stress applied allows for ultimate pullout to occur. These results also suggest that inclusion length and elastic modulus cannot be analyzed separately from vertical stress to determine whether ultimate pullout occurs or not, essentially because the combination of these three parameters plays a role in the amount of elongation of an inclusion. Additional inspection of Table 6.2a indicates that the soil-inclusion interface friction angle is also a parameter that influences the elongation of an inclusion. Keeping constant the length and elastic modulus of the inclusion and the vertical stress applied, the rougher the soil-inclusion interface, the more the inclusion elongates. It appears, therefore, natural to use the results of the Tornado simulations to find a range of inclusion characteristic that will allow for ultimate pullout to occur as a function of the vertical stress applied.   183  A useful way to plot the combination of inclusion length and elastic modulus, and soil-inclusion interface friction, versus vertical stress is shown in Figure 6.20. By doing this, no attempt is made to find a relation between these parameters. The intention is to find a potential region in which ultimate pullout occurs, that can be used for guidance to select inclusion characteristics. The results of the Tornado simulations in Figure 6.20 depict a distinct boundary that separates a region in which ultimate pullout occurs (open circles) from one where only initial pullout occurs (solid circles). With the benefit of the laboratory test data of Raju (1995) of displacements of the clamped and embedded ends of the inclusion, this boundary can be further refined. Figure 6.21 shows that all the tests on sheet membranes and geogrids (Raju, 1995) have reached ultimate pullout, and can thus be added in Figure 6.20. A function can be fitted using the last data points at the bottom of the ultimate pullout data set and another at the top of the initial pullout data set. An undefined region results between both boundaries due to a lack of available data in that zone.   In order to test the validity of the proposed regions, the data of Wilson-Fahmy et al. (1994), Moraci and Recalcatti (2006) and Bathurst and Ezzein (2015) are added and shown in Figure 6.22. Given that no information was available regarding the magnitude of the soil-inclusion interface friction angle, in all the cases, the value of  was assumed to be 2/3 of the reported friction angle of the soil. Figure 6.22 shows that all the data from Wilson-Fahmy et al. (1994) plot in the “initial pullout region” except for geogrid A tested with a length of 0.3 m, which is consistent with the observations of their study. The data of Moraci and Recalcatti (2006) for geogrids GG1, GG2 and GG3 tested at 10 kPa plot in the “ultimate pullout region” and all the data for L = 1.15 m in the “initial pullout region”, which is consistent with the observations of their study. Examination of the data of geogrids GG1 and GG2 tested at a length of 0.4 m indicates a possible refinement of the regions in Figure 6.20. The data of both tests suggest that ultimate pullout has in effect been reached which implies that the boundary of “ultimate pullout” can be collapsed into the boundary of “initial pullout”. Finally, the data of Bathurst and Ezzein (2015) at stresses of 2 kPa and 7 kPa fall in the “ultimate pullout region” and at stresses of 27 kPa and 52 kPa fall in the “initial pullout region” which is also consistent with the observations of their study. The good agreement between the proposed boundary between the zones of “ultimate pullout” and “initial pullout” and the laboratory test data of Wilson-Fahmy et al. (1994), Moraci and Recalcatti (2006) and Bathurst and Ezzein (2015), suggests the potential exists for a plot that can predict whether a combination of 184  inclusion characteristics tested at a certain stress level will result in ultimate pullout, as shown in Figure 6.23.  6.5.2.2. Boundary conditions  As shown in Section 2.4, it is well recognized that boundary conditions influence pullout test results. In particular, the height of the testing device and the front wall of the box, have been shown to have an important effect on the pullout response of different inclusions (Palmeira and Milligan, 1989; Farrag et al., 1993; Raju, 1995; Moraci and Recalcatti, 2006; Palmeria, 2009). Unfortunately, limited experimental data exist to have a better understanding of the magnitude of the influence of these boundary conditions. The results of the Tornado simulations of this study have provide some insight regarding the mechanism by which the pullout response is affected by the height of the pullout box and the front wall roughness.  a) Height of the box  The testing program of Farrag et al. (1993) described in Section 6.5.2.1., also included the variation of the box height to study the influence of the soil sample above and below the inclusion in the pullout response. The study was done using the Tensar SR2 geogrid, 0.9 m long and 0.3 m wide and tested at a vertical stress of 48.2 kPa, condition that, as per Section 6.5.2.1., allows for ultimate pullout to occur. Tests were performed with soil thicknesses on top and below the inclusion of 10 cm, 30 cm and 60 cm. One test was performed using a top soil thickness of 30 cm and a bottom soil thickness of 40 cm to study the influence of different soil thickness in the pullout response. The laboratory test results show that as the soil thickness decreases, the pullout response becomes stiffer and both the peak and large displacement pullout resistance increase. The results also suggest that a difference in soil thickness on top and below the inclusion does not influence the pullout response of the inclusion. The authors recommend that a minimum soil thickness of 30 cm on top and below the inclusion is used to minimize top and bottom boundary effects though, no further testing was done with soil thicknesses larger than 30 cm. Similarly, Palmeira (2009), based on the numerical model of Dias (2003), suggest that for typical reinforcement lengths (lower than 1 m) the height of the soil sample should not be smaller than 0.6 m. 185  The results of the Tornado simulations show that Pu increases in average 25% when Hb decreases from 0.6 m to 0.3 m and that the influence of the top boundary is negligible for a soil thickness larger than 0.6 m, confirming the observation of Farrag et al. (1993) and Palmeira (2009). The influence of the soil thickness is more evident at lower vertical stresses and can reach almost 50% difference in the magnitude of Pu at vertical stresses of 4 kPa.   Inspection of Tables 2.2 and 2.3 shows that the work of Ingold (1983), Johnston and Romstad (1989), Palmeira and Milligan (1989) and Bathurst and Ezzein (2015) might be influenced by boundary condition effects from the top and bottom boundaries. In effect, Palmeira and Milligan (1989) used two different box sizes in their study. The medium-sized box had dimensions of 0.5 m long, 0.25 m wide and 0.15 m high and the large box was a cube of 1.0 m long sides. The intention of their study was to investigate the effect of the distance of the inclusion to the front wall and most of the tests were performed in the large pullout box. One test was made in the medium-sized box and the authors concluded that “in trying to avoid one problem, another was created, for this test failure mechanism was different…. Failure developed from the ends of the reinforcement towards the sample surface” instead of at the soil-inclusion interface level. No further testing was performed in the medium-sized box.  An interesting study to analyze the effect of the overestimation of the pullout response for a soil height of 0.3 m (as the minimum suggested by ASTM D6706-01) is that of Ingold (1983). This analysis does not intend to discuss the results quantitatively but to show, in a qualitative way, that the use of a soil thickness of 0.3 m (along with a constant friction angle) results in a significant overestimation of the soil-inclusion apparent friction angle. The study of Ingold (1983) compared the soil strength and the soil-inclusion friction of three different grids namely, Netlon 1168, FBM5 and BRC-B503 by means of pullout tests. The box dimensions were 0.5 m long, by 0.285 m wide and 0.3 m high.  The reported results of the soil-inclusion apparent friction angle () compared to the soil strength is shown in Figure 6.24a. Inspection of the results allows for the following observations:   186  - A constant value of soil friction angle of 35 is considered for vertical stresses ranging from 4 kPa to 200 kPa. The results in Figure 4.12 of the current study suggest that the friction angle is higher at lower stresses attaining a value 20% larger at 4 kPa compared to 150 kPa. - The tests on Netlon 1168 were performed using a length of 0.5 m which resulted in tension failure reported by the authors. This is consistent with Table 6.2a of the present study. Tests on FBM5 and BRC-B503 were performed using a length of 0.325 m which ensure that ultimate pullout occurs. - Deduced values of soil-inclusion apparent friction angle greatly exceed the soil friction angle for the three geogrids, especially at stresses lower than 50 kPa. For Netlon 1168,  is between 2.9 and 0.6 times  = 35, for FBM5,  is between 8.1 and 1.3 times larger than  = 35 and for BRC-B503,  is between 16 and 2.1 times larger than  = 35.  Considering the stress-dependency of the friction angle as shown in Figure 4.12 along with the overestimation of Pu for Hb = 0.3 m reported in Section 6.4.6 a re-interpreted plot of the data reported by Ingold (1983) is presented in Figure 6.24b. An important decrease in the values of  transpires, that is even closer to the strength of the soil when its stress-dependency is considered. For Netlon 1168,  is between 1.3 and 0.5 times  = f(), for FBM5,  is between 3.6 and 1.1 times larger than  = f() and for BRC-B503,  is between 7.3 and 1.7 times larger than  = f().   These results suggest that using a box height of 0.3 m (as the minimum established by ASTM D6706-01) and a constant value of soil friction angle, can overestimate the efficiency factor (often expresses as tan / tan) by more than twice at very low stresses. This can result in an underestimation of the length required beyond the failure surface in a reinforced earth structure to ensure adequate pullout resistance.   187  b) Front wall  The front wall of the pullout box is another important boundary that influences the pullout response of different inclusions. Very limited data exist in the literature that allow for an understanding of the influence of the roughness of the front wall in the pullout response.   One study corresponds to Palmeira and Milligan (1989) in a cubic pullout box of 1.0 m long side. The length of the metallic grid tested was 0.75 m and was embedded in Leighton Buzzard sand 14/25 with a reported friction angle in direct shear of 51.3 at 25 kPa of vertical stress and a critical state friction angle of 35. A vertical stress of 25 kPa was used for testing that, according to Table 6.2a, allows for ultimate pullout to occur. Two limiting conditions for the front wall configuration were used, namely, a fully rough front wall for which particles of Leighton Buzzard sand were glued on the inside wall of the box (ff = 45), and a lubricated front wall (ff = 6). Two intermediate situations were also studied by using a smooth steel surface (ff = 16) and by gluing sand paper to the inside wall of the box (ff = 30). The second study was done by Raju (1995) in which two front walls were used, namely an arborite front wall (with ff = 12.5) and an aluminum front wall (with ff = 15) for the smooth and textured geomembranes tested at a vertical stress of 8 kPa (GMS08 and GMT08, respectively) and for the Tensar grid tested at a vertical stress of 10 kPa (GGT10).   Figure 6.25 shows the experimental values of tan = / for the GMS and GMT sheets tested at 8 kPa, and the GGT geogrid tested at 10 kPa by Raju (1995), along with the experimental values obtained by Palmeira (1987) on his pullout tests on a metallic grid at a vertical stress of 25 kPa. The data suggest that  at peak pullout resistance increases with the front wall roughness of the pullout box, and that the increase is larger for higher soil-inclusion interface friction, as shown in Figure 6.7. These laboratory test results along with the findings from the Tornado simulations (Figs. 6.6 and 6.7) reinforce the well-accepted idea that to minimize the effect of the front wall in the pullout resistance, a lubricated internal wall should be used, especially when inclusions with high surface roughness are tested.    188  Christopher (1976) used a sleeve around the pullout slot at the front wall of the pullout box in order to transfer the point of application of the pullout load behind the rigid front wall into the soil. Palmeira (2009) also recalls that to minimize front wall effects a sleeve must be used at the front wall. In effect, the experimental studies of Wilson-Fahmy et al. (1994), Lopes and Ladeira (1996), Moraci and Recalcati (2006), Lopes and Silvano (2010), and Bathurst and Ezzein (2015), have incorporated a sleeve at the front wall of the box. Inspection of the studies for which the sleeve length was varied to understand its influence on the pullout response shows contradicting results.  Farrag et al. (1993) used sleeve lengths of 20 cm and 30.5 cm and compared the pullout response to a sleeveless front wall. Their experimental results show that the pullout response is stiffer and larger values of pullout resistance are reached when no sleeve is used. As the length of the sleeve increases, measurements of stresses at the front wall of the box decrease and also does the pullout resistance. The authors suggest that the use of a 30 cm long sleeve minimizes the effect of the rigid front wall. Lopes and Ladeira (1996), experimentally showed that the (peak) pullout resistance decreases as the length of the sleeve increases. A similar result was obtained by Lopes and Ladeira (1996) where the pullout tests were performed using a smooth front wall without sleeve and a 20 cm long sleeve. The pullout resistance of the sleeveless test was around 10% greater than the one using the sleeve.   A different tendency is obtained from the numerical model of Dias (2003). In his study, he modeled a nominal pullout test for a short inclusion (around 0.6 m long) at a vertical stress of 25 kPa embedded in a soil sample 1 m high. As per Section 6.5.2.1 and the previous analysis regarding the influence of the soil height, this numerical test should allow for ultimate pullout to occur with no top and bottom boundary effects. The numerical simulations were done for three front wall conditions namely, a sleeveless lubricated front wall and sleeves 15 cm and 30 cm long. The results of his study show that pullout resistance obtained is larger when a sleeve is used rather than a lubricated front wall and that the size of the sleeve (15 and 30 cm) does not influence the pullout response. In the light of these contradicting results, Palmeira (2009) suggests that a more comprehensive study is needed on how to minimize front wall effects.   189  6.5.3. Analysis of ASTM D6706-01 recommendations  The findings of the Tornado simulations and the assessment of the experimental data found in the literature have built the basis for the last discussion of this study, related to the recommendations of ASTM D6706-01 on test and boundary conditions.  In the light of the results of this Chapter, the following recommendations arise for pullout testing on sheet inclusions.  6.5.3.1. Definition of ultimate pullout    ASTM D6706-01 defines ultimate pullout resistance as “the maximum pullout resistance measured during a pullout test”. The results of this study suggest that ultimate pullout defined as when the movement of the inclusion is uniform over the entire embedded end (as per ASTM D6706-01, see Figure 6.1) for sheet inclusions, happens before maximum pullout resistance, independently of whether a strain softening behavior is observed. It is recommended, therefore, for ASTM D6706-01, to make a distinction between maximum and ultimate pullout resistance given that they characterize different mechanisms of the soil-inclusion interaction.  6.5.3.2. Vertical stress  ASTM D6706-01 states that stresses up to 250 kPa should be anticipated for testing, which appears a large magnitude. In effect, none of the laboratory pullout tests found in the literature have been done at such high vertical stresses. The statistical study of Huang and Bathurst (2009) states that 65% of the tests available in the literature were performed at stresses lower than 40 kPa and 97% of the tests at stresses lower than 100 kPa, with the highest vertical stress being 192 kPa. In addition, and according to the results of this study, at a vertical stress of 200 kPa, ultimate pullout occurs only for stiff inclusions, shorter than 0.6 m and with a surface roughness lower than 2/3 of the friction angle of the soil. It is more likely that pullout will not occur at all at stresses of 250 kPa. It is recommended, therefore, that ASTM D6706-01 decreases the maximum vertical stress to 30 kPa for extensible sheet inclusions to a maximum of 50 kPa when stiffer inclusions are tested. This range of vertical stresses is in agreement with Holtz (2017), that suggests that pullout failure in a reinforced wall is more likely to occur in the upper layers of the structure where overburden 190  pressures are lower. Considering an average soil unit weight of 20 kN/m3, an overburden stress of 30 kPa to 50 kPa implies that pullout is likely to occur for inclusions in the uppermost 1.5 m to 2.5 m of the wall height.  In addition, and regarding the magnitude of the vertical stress applied at the soil-inclusion interface level, the results of this study show that the distribution of vertical stresses along the inclusion is not uniform and is larger than the one defined by ASTM D6706-01 as n,ASTM = a + s. This suggests that the current form of considering the vertical stress suggested by ASTM D6706-01 results in an overprediction of the value of the interaction factor, which can lead to an underdesign of reinforced structures. It is recommended, therefore, that to obtain a proper interpretation of the interaction factor of a sheet inclusion, ASTM D6706-01 considers the measurement of the vertical stress at the soil-inclusion level at the specified condition of displacement.   6.5.3.3. Length of the inclusion  ASTM D6706-01 suggests a minimum length of 0.6 m for the inclusion. According to the results of the Tornado simulations, this minimum length appears reasonable as it ensures that ultimate pullout occurs for applied stresses up to a 50 kPa for stiff and fully rough inclusions. It is important to note that design guidance of reinforced structures suggests a minimum length for the reinforcement of 1 m beyond the surface failure. One would be inclined consequently to test specimens 1 m long which is consistent with the median reported by Huang and Bathurst (2009). According to Table 6.2a, for an inclusion length of 1.0 m, pullout will occur for stresses up to 30 kPa. It is recommended, thus, that ASTM D6706-01 specifies that sheet inclusions larger than 1 m, a maximum vertical stress of 30 kPa is used to ensure that ultimate pullout really occurs.    191  6.5.3.4. Height of the box  The minimum box height suggested by ASTM D6706-01 is 0.3 m. The results of the Tornado simulations show that, for an inclusion length of 0.6 m, the minimum height suggested by ASTM D6706-01 overpredicts the pullout resistance in average 25%, reaching almost a 50% at stresses closer to 4 kPa. The analysis of the data of Ingold (1983) also shows the large overestimation of the soil-inclusion apparent friction angle when the use of a soil sample of 0.3 m instead of 0.6 m is used. Although additional data are needed to determine if the height of the soil sample should equal the length of the inclusion to avoid top and bottom boundary condition effects (as suggested in Palmeira, 2009), the results of this study, along with the previous observation of Farrag et al. (1993) and Palmeira (2009) strongly advise a change to the minimum height suggested by ASTM D6706-01 from 0.3 m to 0.6 m.  6.5.3.5. Front wall  The most critical and contradictory aspect of ASTM D6706-01 recommendations is the treatment of the front wall to minimize its effect on the pullout resistance of sheet inclusions. Section 6.4.3. reveals that the sole presence of the rigid front wall (with ff = 0) increases the vertical stress at the soil-inclusion interface by about 1.2 for a nominal vertical stress applied of n,ASTM = 50 kPa. In addition, the experimental results of Raju (1995) and Palmeira and Milligan (1989), along with the results of the sensitivity analysis of the current study, confirm that the lower the roughness of the front wall of the box, the lower the pullout resistance obtained.   On the other hand, ASTM D6706-01 recommends that a sleeve is added at the front wall of the box and that it should extend into the pullout box a minimum distance of 150 mm. The experimental results of Farrag et al. (1993) and Lopes and Ladeira (1996) suggest that the measured pullout resistance is lower with the presence of a sleeve compared to a sleeveless (and smooth or lubricated) front wall. These results are in contradiction with the numerical simulations of Dias (2003) that suggest that a lubricated front wall (without sleeve) results in a lower pullout resistance compared to a front wall with a 15 cm or a 30 cm long sleeve.   192  The objective of the sleeve is to transfer the point of application of the pullout force into the soil simulating, therefore, a virtual fully rough front wall. This configuration should increase the pullout resistance compared to a sleeveless lubricated or smooth front wall as per the numerical results of the current Chapter and Dias (2003), and the experimental data of Raju (1995) and Palmeira and Milligan (1989).  The contradicting observations previously described, suggest that there is no sufficient evidence to warrant that the presence of a sleeve at the front wall of the box will minimize front wall effects. The results of the Tornado simulations of this study clearly show a reduction in pullout resistance when a lubricated front wall is used. In the light of this opposing findings, it is advised that a more comprehensive study is done to understand the influence of the presence of a sleeve in the pullout response. In the meantime, it is suggested that ASTM D6706-01 modifies its recommendation regarding the treatment of the front wall by replacing the use of a sleeve by the use of a lubricated front wall.  6.6. Summary  This Chapter reports a sensitivity analysis where the length, elastic modulus and surface roughness of the inclusion were varied, along with the height of the box, and the inside front wall roughness of the test device. The analysis of the results allowed discussing the validity of the pullout experimental results found in the literature, and to study the adequacy of ASTM D6706-01 recommendations regarding pullout testing. An identification of the weaknesses of the current ASTM D6706-01 recommendations in the determination of the soil-inclusion interaction factor is made and an “ideal” pullout test that appears to guarantee the occurrence of ultimate pullout, as defined by ASTM D6706-01, is suggested.   The analysis of the results of the Tornado simulations allow for the following conclusions to be drawn, that contribute to a better understanding of the influence of L, E, , ff, Hb and vertical stress in the pullout response of sheet inclusions:  193  • The onset of ultimate pullout occurs before maximum pullout resistance. This suggests a modification is required to the definition of ultimate pullout in ASTM D6706-01;  • The distribution of vertical stresses at the soil-inclusion interface level at the onset of ultimate pullout (n,u) is non-uniform (it is larger close to the front wall) and is larger than the vertical stress defined by ASTM D6706-01(n,ASTM = a + s, see Figure 6.2);   • A non-linear relation exists between Pu and E due to the elongation of the inclusion at which Pu occurs. Stiffer inclusions suffer less elongation and show a peak and strain softening pullout response, and a value of Pu larger than the large displacement pullout resistance. Extensible inclusions show a strain hardening pullout response, and a value of Pu lower than the large displacement pullout resistance;  • A non-linear inverse relation exists between Pu and Hb due to the increase of n,u at lower values of Hb. For a box height equal or larger than 0.6 m (or, in this case, larger or equal to the length of the inclusion) an almost negligible decrease in n,u is observed (less than 5%) which results in a negligible difference in the value of Pu;  • A non-linear relation between Pu and ff exists due to the non-linear increase of n,u with ff. Soil-inclusion interfaces with a friction equal to 2/3 the strength of the soil are mildly influenced by a change in front wall friction from a frictionless front wall (or ff = 0) to an aluminum front wall (or ff = 15). A change from ff = 0 to ff = 15 results in a significant increase in the value of Pu (around 40%) for inclusions with high surface roughness.  • A non-linear relation exists between Pu and tan, Pu and L, and Pu and n,ASTM, due in part to the stress-dependency of  and cs,tx at n,ASTM  ≤ 50 kPa.  • The non-linear relation between Pu and n,ASTM disappears when Pu is plotted against n,u. The linear relation between Pu and n,u exists for all values of tan, L and ff of this study. In addition the relation between Pu and n,u is unique and independent of the value of ff. 194  The analysis of the studies available in the literature suggests, with the help of Table 6.2a, that some of the laboratory tests of Farrag et al. (1993), Wilson-Fahmy et al. (1994), Moraci and Recalcati (2006), Bathurst and Ezzein (2015), Ingold (1983), Palmeira and Milligan (1989), do not reach ultimate pullout. For example, the combination of inclusion length (L = 0.9 m) and vertical stress (48.2 kPa, 96 kPa and 140 kPa) used in the pullout tests of Farrag et al. (1993) suggests that only the test at 48.2 kPa reached ultimate pullout. Wilson-Fahmy et al. (1994) declare that the pullout test on geogrid A (n,ASTM = 69 kPa, L = 0.92 m and E = 7.9 × 105 kPa) reached ultimate pullout. Comparison to the combinations on Table 6.2a suggests that this test only reached initial pullout.  A boundary that separates the “initial pullout zone” to the “ultimate pullout zone” is proposed in Figure 6.23 based on a combination of inclusion characteristics such as length, elastic modulus and surface roughness for different vertical stresses. This plot can help to predict whether a combination of inclusion characteristics tested at a certain stress level will result in ultimate pullout.  The principal aspects deserving of improvement in the current ASTM D6706-01 recommendations are:  • The definition of ultimate pullout as per ASTM D6706-01 suggests that Pu and the maximum pullout resistance are the same, whereas the results of this study show that Pu occurs before maximum pullout resistance. A refinement of the ASTM D6706-01 definition is suggested to make a distinction between maximum and ultimate pullout.  • ASTM D6706-01 states that stresses up to 250 kPa should be anticipated for testing. The results of this study show that very few tests on sheet inclusions will reach ultimate pullout at a vertical stress of 200 kPa. It is recommended that ASTM D6706-01 decreases the maximum vertical stress to 30 kPa for extensible sheet inclusions to a maximum of 50 kPa when stiffer inclusions are tested, to ensure that ultimate pullout really occurs.  195  • The results of this study show that the vertical stresses at the soil-inclusion level are larger than considered by ASTM D6706-01 for the calculation of the soil-inclusion interaction factor. It is suggested that a systematic measurement of vertical stresses at the soil-inclusion interface level is done to assess the magnitude of the increment of n over n,ASTM at different pullout stages. The use of n,ASTM results in an underprediction of the soil-inclusion apparent friction angle (or interaction factor).  • The results of this study show that the use of a box height of 30 cm overpredicts the pullout resistance by an average of 25%, reaching nearly 50% at stresses closer to 4 kPa. It is strongly advised that ASTM D6706-01 changes the minimum height from 0.3 m to 0.6 m.  • ASTM D6706-01 recommends the use of a sleeve to transfer the point of application of the pullout force into the soil, in order to minimize the effect of the rigid front wall of the pullout box. This results in an inclusion interaction with a virtual fully rough front wall. The results of the current study, along with the experimental results of Raju (1995) and Milligan and Palmeira (1989) and the numerical results of Dias (2003) show that the pullout resistance increases with the roughness of the front wall. This is in contradiction with the experimental results of Farrag et al. (1993) and Lopes and Ladeira (1996), which show that the pullout resistance is lower with the presence of a sleeve compared to a sleeveless but smooth front wall. Given the uncertainty of the effect of the presence of a front sleeve, it is suggested that ASTM D6706-01 modifies its recommendation regarding the treatment of the front wall by replacing the use of a sleeve by the use of a lubricated front wall.  Finally, the ideal pullout test configuration that appears to guarantee that ultimate pullout occurs for sheet inclusions with a soil-inclusion interface friction angle equal or lower than the strength of the soil is shown in Figure 6.13 and is given by: a maximum inclusion length of 1.0 m, an elastic modulus of the inclusion larger than 1×105 kPa, a minimum box height of 0.6 m, a smooth front wall and a vertical stress applied at the inclusion level lower than 30 kPa.    196  Table 6.1: Variables to study and ranges Variable Literature ASTM D6706-01 Current study L (m) 0.25 – 2.0  0.6 (min) 0.3; 0.5; 0.6; 1.0; 1.5; 2.0  E (kPa) 4x103 - 3x106   1x104; 1x105; 5x105; 1x106  (°) 1/3 -   1/6; 1/3; 2/3;   with  = f() Hb (m) 0.15 – 1.1 m 0.3 m (min) 0.15; 0.3; 0.4; 0.6; 0.8; 1.0  ff (°) Sleeve Steel-aluminum (15-16°) Sand glued (soil) Lubricated (6°) Frictionless (numerical analysis) Sleeve (min length of 15 cm)  0; 5; 15; 30 n,ASTM (kPa) 2 – 200  250 (max) 4; 12; 30; 50; 100; 200      197  Table 6.2a: Ranges of parameter values for which ultimate pullout occurs (dc = de)  n,ASTM = 4 kPa n,ASTM = 12 kPa n,ASTM = 30 kPa n,ASTM = 50 kPa n,ASTM = 100 kPa n,ASTM = 200 kPa L (m) [0.3 - 2] [0.3 - 2] [0.3 - 1] [0.3 - 0.6] [0.3 - 0.6] [0.3 - 0.6]  () [1/3 - ] [1/3 - ] [1/3 - ] [1/3 - ] [1/3 - 2/3] [1/3 - 2/3] E (kPa) [1×104 - 1×106] [1×104 - 1×106] [1×105 - 1×106] [5×105 - 1×106] [1×106] [1×106] Hb (m) [0.15 - 1.0] [0.15 - 1.0] [0.15 - 1.0] [0.15 - 1.0] [0.15 - 1.0] [0.15- 1.0] ff () [0 - 30] [0 - 30] [0 - 30] [0 - 30] [0 - 15] [0 - 15]  Table 6.2b: Ranges of parameter values for which only initial pullout occurs (dc > de)  n,ASTM = 4 kPa n,ASTM = 12 kPa n,ASTM = 30 kPa n,ASTM = 50 kPa n,ASTM = 100 kPa n,ASTM = 200 kPa L (m) - - [1.5 ; 2.0] [1.0 – 2.0] [1.0 – 2.0] [1.0 – 2.0]  () - - - - [] [] E (kPa) - - [1×104] [1×104 ; 1×105] [1×104 ; 5×105] [1×104 ; 5×105] Hb (m) - - - - - - ff () - - - - [30] [30]    198                   Figure 6.1: Definition of pullout                 Figure 6.2: Definition of base case      Hb  = 0.6 m a  n,ASTM  Sheet inclusion: E = 1x106 kPa  = 2/3cs,tx = f(n,ASTM) L = 0.6 m ff = 15  P, dc  , Pi , Pu 0510152025300 5 10 15 20 25 30Displacement of embedded end, de (mm)Displacement of clamped end, dc (mm)Ultimate pulloutdc > de = 0 dc > de > 0Initial pulloutdc = de Onset of ultimate pullout dc > de = 0      dc > de > 0                     dc = de 199                             Figure 6.3: Tornado diagrams: a) n,ASTM = 4 kPa , b) n,ASTM = 12 kPa, c) n,ASTM = 30 kPa, d) n,ASTM = 50 kPa, e) n,ASTM = 100 kPa, f) n,ASTM = 200 kPa N.P. stands for “no ultimate pullout”   0 5 10 15 20 25 30Pu (kN/m)n,ASTM = 12 kPaL: 0.3 - 2.0 m: 1/6 - H: 1.0 - 0.3 mE : 1x104 - 1x106 kPaff: 0 - 30 [0.95 ; 1.52][0.46 ; 3.14][0.31 ; 1.41][0.79 ; 1.00][0.88 ; 1.14]Pu at base case0 1 2 3 4 5 6 7Pu (kN/m)n,ASTM = 4 kPaL: 0.3 - 2.0 m: 1/6 - H: 0.4 - 0.3 mPu at base caseE : 1x104 - 1x106 kPaff: 0 - 30 [1.00 ; 1.74][0.37 ; 2.18][0.28 ; 1.34][0.61 ; 1.00][0.73 ; 1.02]0 5 10 15 20 25 30 35Pu (kN/m)n,ASTM = 30 kPa Pu at base caseL: 0.3 - 1.0 mL: 1.5 - 2.0 m N.P.: 1/6 - H: 1.0 - 0.3 mE : 1x105 - 1x106 kPaE : 1x104 kPa N.P.ff: 0 - 30 [0.97 ; 1.25][0.46 ; 1.63][0.36 ; 1.40][0.84 ; 1.00][0.90 ; 1.09]0 10 20 30 40Pu (kN/m)n,ASTM = 50 kPaL: 0.3 - 0.6 mL: 1.0 - 2.0 m N.P.: 1/6 - H: 1.0 - 0.3 mE : 5x105 - 1x106 kPaE : 1x105 - 1x104 kPa ff: 0 - 30 [0.99 ; 1.22][0.48 ; 1.00][0.36 ; 1.40][0.96 ; 1.00][0.91 ; 1.07]Pu at base case0 25 50 75Pu (kN/m)n,ASTM = 100 kPaL: 0.3 - 0.6 mL: 1.0 - 2.0 m N.P.: 1/6 - 2/3:  N.P.H: 1.0 - 0.3 mE : 1x106 kPaE : 5x105 - 1x104 kPa ff: 0 - 15 ff: 30 N.P.[0.95 ; 1.33][0.62 ; 1.00][0.72 ; 1.00][1.00 ; 1.00][0.79 ; 1.00]Pu at base case0 20 40 60 80 100 120 140Pu (kN/m)n,ASTM = 200 kPaL: 0.3 - 0.6 mL: 1.0 - 2.0 m N.P.: 1/6 - 2/3:  N.P.H: 1.0 - 0.3 mE : 1x106 kPaE : 5x105 - 1x104 kPa ff: 0 - 15 ff: 30 N.P.[0.95 ; 1.38][0.63 ; 1.00][0.72 ; 1.00][1.00 ; 1.00][0.81 ; 1.00]Pu at base casea) b) c) d) e) f) Hb : Hb : Hb : Hb : Hb : Hb : 200  0204060801001200 50 100 150 200Base case, Pu(kN/m)n,ASTM (kPa)Curve a: cs,tx = f() ;  =   cs,tx = f() Curve a0510152025300 10 20 30 40 50Base case, Pu(kN/m)n,ASTM (kPa)Curve aCurve a: cs,tx = f() ;  =   cs,tx = f() Curve b: cs,tx = 27 ;  = 18Curve b0204060801001200 50 100 150 200 250Base case, Pu (kN/m)n,ASTM ; n,u  (kPa)Curve aCurve a: cs,tx = f() ;   =   cs,tx = f(); Pu = f(n,ASTM)Curve c: cs,tx = f() ;   =   cs,tx = f(); Pu = f(n,u)Curve cPu = 0.44n,u                 Figure 6.4: Influence of vertical stress on Pu at base case    0123450.00 0.20 0.40 0.60 0.80 1.00n,u/n,ASTMx/L4 kPa12 kPa30 kPa50 kPa100 kPa200 kPan,ASTMa) b) c) d) 201  0204060801001200.0 0.2 0.4 0.6 0.8Pu(kN/m)tan4123050100200= cs ,tx /6= cs ,tx /3= 2cs ,tx /3base case= cs,tx n,ASTM0102030400 10 20 30 40 50Pu(kN/m)n,ASTM ; n,u  (kPa)= cs,tx Pu = f(n,ASTM) = 2 cs,tx / 3Pu = f(n,ASTM) = cs,tx Pu = 0.65n,u = 2 cs,tx / 3Pu = 0.44n,ucs,tx = f (n,ASTM)                    Figure 6.5: Influence of tan on Pu     a) b) 202  010203040500 5 10 15 20Pu(kN/m)ff ()4123050 = cs,tx = 2cs,tx /3n,ASTM = 2cs,tx /3 = 2cs,tx /3 = 2cs,tx /3 = cs,tx = cs,tx = cs,tx      Figure 6.6: Influence of ff on Pu          Figure 6.7: Influence of ff, tan and n,u on Pu   0204060801001200 10 20 30 40Pu(kN/m)ff ()4123050100200n,ASTM01234560.0 0.2 0.4 0.6 0.8 1.0n,u/n,ASTMx/Ln,u = 61 kPa; ff = 5n,u = 68 kPa; ff = 30n,u = 63 kPa; ff = 15n,u = 59 kPa; ff = 0a) b) 01020304050600 10 20 30 40 50Pu(kN/m)n,ASTM ; n,u  (kPa)dff = 0dff = 15dff = 30ff ff = ff  Pu = f(n,ASTM )Pu = f(n,u )a) b) 203                   Figure 6.8: Influence of L on Pu   0204060801001200.0 0.5 1.0 1.5 2.0Pu(kN/m)L (m)41230501002000510152025300 10 20 30 40 50Pu(kN/m)n,ASTM (kPa)L = 0.6 mbase caseL = 0.5 mL = 0.3 m0204060801001200 50 100 150 200 250Pu(kN/m)n,u (kPa)L = 0.6 mPu = 0.44n,ubase caseL = 0.5 mPu = 0.37n,uL = 0.3 mPu = 0.21n,ua) b) c) 204  0204060801001200 20 40 60 80 100 120Pu(kN/m)E  104 (kPa)4123050n,ASTMn,ASTM = 100 kPan,ASTM = 200 kPa0102030400 20 40 60 80 100 120Pu(kN/m)E  104 (kPa)4123050n,ASTM       Figure 6.9: Influence of E on Pu               Figure 6.10: Influence of inclusion elongation on Pu at n,ASTM  = 12 kPa  01234567890 10 20 30 40 50Pullout resistance per unit width (kN/m)Displacement of clamped end (mm)E = 1x104E = 1x105E = 5x105PbaseE = 1×105 kPaE = 1×104 kPaE = 5 105 kPaE = 1 106 kPaPu or dc = de012345670 10 20 30 40 50Displacement of embedded end (mm)Displacement of clamped end (mm)a) b) a) b) 205          Figure 6:11: Influence of Hb on Pu               Figure 6.12: Ideal pullout test configuration     0204060801001201400.2 0.4 0.6 0.8 1.0 1.2Pu(kN/m)H (m)4123050100200n,ASTM012345670 0.2 0.4 0.6 0.8 1n,u/ n,ASTMx/LH=0.3 mH=0.4 mH=0.6 mH=0.8 mH=1.0 ma) b)     Hb  ≥ 0.6 m a  n,ASTM ≤ 30 kPa  Inclusion: E ≥ 1×106 kPa L ≤ 1.0 m ff ≥ 15  b  = 0.3 m b  = 0.4 m b  = 0.6 m b  = 0.8 m Hb  = 1.0 m Hb (m) 206           Figure 6.13: Comparison of laboratory test data with Tornado simulations Combination 1: L = 0.6 m;  = cs,tx; E = 1.0×106 kPa; Hb = 0.6 m; ff = 15 Combination 2: L = 1.0 m;  = 2/3cs,tx; E = 1.0×106 kPa; Hb = 0.6 m; ff  = 15 Combination 3: L = 1.0 m;  = cs,tx; E = 1.0×106 kPa; Hb = 0.6 m; ff  = 15             Figure 6.14: Displacements distribution along the reinforcement (Farrag et al. 1993) 0510152025300 10 20 30 40 50 60 70Pullout resistance per unit width (kN/m)Displacement of clamped end (mm)Laboratory test dataCombination 1Combination 2Combination 3n,ASTM = 12 kPan,ASTM = 4 kPa207  Figure 6.15: Predicted and measured pullout load displacement relationships for geogrids A, B, C (Wilson-Fahmy et al., 1994)      (a) Geogrid length = 0.31 m (a) Geogrid length = 0.31 m (a) Geogrid length = 0.31 m       (b) Geogrid length = 0.92 m (b) Geogrid length = 0.92 m (b) Geogrid length = 0.92 m       (c) Geogrid length = 1.70 m (c) Geogrid length = 1.70 m (c) Geogrid length = 1.70 m (i) GEOGRID A (ii) GEOGRID B (iii) GEOGRID C  0481216202428320 10 20 30 40 50 60 70 80PULLOUT FORCE [kN/m]DISPLACEMENT [mm]SHEET PULLOUT05101520253035400 5 10 15 20 25 30 35 40PULLOUT FORCE [kN/m]DISPLACEMENT [mm]SHEET PULLOUT(SOME JUNCTION FAILURES)0481216202428320 5 10 15 20 25 30 35 40 45 50PULLOUT FORCE [kN/m]DISPLACEMENT [mm]SHEET PULLOUT(SOME JUNCTION FAILURES)010203040506070800 10 20 30 40 50 60 70 80PULLOUT FORCE [kN/m]DISPLACEMENT [mm]SHEET PULLOUT01020304050600 5 10 15 20 25 30 35 40PULLOUT FORCE [kN/m]DISPLACEMENT [mm]TENSION FAILURE05101520253035400 5 10 15 20 25 30 35 40 45 50PULLOUT FORCE [kN/m]DISPLACEMENT [mm]SHEET PULLOUT(SOME JUNCTION FAILURES)0204060801001200 20 40 60 80PULLOUT FORCE [kN/m]DISPLACEMENT [mm]TENSION FAILURE01020304050600 5 10 15 20 25 30 35 40PULLOUT FORCE [kN/m]DISPLACEMENT [mm]TENSION FAILURE05101520253035400 5 10 15 20 25 30 35 40 45 50PULLOUT FORCE [kN/m]DISPLACEMENT [mm]SHEET PULLOUT(SOME JUNCTION FAILURES)208   Figure 6.16: Pullout curves for LR = 1.13 m and LR = 0.4 m (Moraci and Recalcati, 2006)   209                         Figure 6.17: Displacements measured along the specimen for LR = 0.4 m (Moraci and Recalcati, 2006)   210                     Figure 6.18: Displacements measured along the specimen for LR = 1.15 m (Moraci and Recalcati, 2006)         211                      Figure 6.19: Displacement-time plots at different locations behind the front clamp: (a) displacements recorded at 500 mm behind front clamp; (b) displacements recorded at 1000 mm behind front clamp (Bathurst and Ezzein, 2015)   212  100005010000100100001501000020010000250100000 50 100 150 200 250E/Ltan(kPa/m)n,ASTM (kPa)Initial pulloutUltimate pulloutRaju 1995          Figure 6.20: Definition of region where ultimate pullout occurs          Figure 6.21: Displacements of clamped versus embedded end (Raju, 1995)     01020304050607080901000 20 40 60 80 100Displacement of embedded end (mm)Displacement of clamped end (mm)GMS04GMS08GMS12GMT04GMT08GMT1201020304050607080901000 20 40 60 80 100Displacement of embedded end (mm)Displacement of clamped end (mm)GGT04GGT10GGT17GGS04GGS10GGS17GGM10GGM17a) b) Ultimate pullout boundary Initial pullout boundary 213            Figure 6.22: Comparison of pullout experimental data with region where ultimate pullout occurs           Figure 6.23: Definition of ultimate pullout and initial pullout regions   1000010100002010000301000040100005010000601000070100000 50 100 150 200 250E/Ltan(kPa/m)n,ASTM (kPa)Moraci and Recalcati (2006)Wilson-Fahmy et al (1994)Bathurst  and Ezzein (2015)Initial pulloutUltimate pulloutgeogrid A; L=0.3 mGG1, GG2, GG3; L=1.15 mGG1, L=0.4 m,  GG2, L=0.4 m,  1000010100002010000301000040100005010000601000070100000 50 100 150 200 250E/Ltan(kPa/m)n,ASTM (kPa)Moraci and Recalcati (2006)Wilson-Fahmy et al (1994)Bathurst  and Ezzein (2015)Initial pullout regionUltimate pullout regiongeogrid A; L=0.3 mGG1, GG2, GG3; L=1.15 mGG1, L=0.4 m,  GG2, L=0.4 m,  214         Figure 6.24: Overestimation of pullout resistance due to box height            Figure 6.25: Influence of the front wall roughness in the peak and large displacement apparent friction angle at the soil-inclusion interface from laboratory test data   0123450 10 20 30 40 50/n,ASTMff ( )GMS08GMT08GGT10Palmeira (1987)024681012140 50 100 150 200/n,ASTM = tann,ASTM (kPa)SandNetlon 1168FBM5BRC-B503024681012140 50 100 150 200/n,ASTM = tann,ASTM (kPa)Sand (corrected by stress-dependency)Netlon 1168Netlon 1168 (corrected by H)FBM5FBM5 (corrected by H)BRC-B503BRC-B503 (corrected by H)a) b)  215  Chapter 7 Conclusions and recommendations  7.1. Conclusions of the present research  This study identified four principal “knowledge gaps” regarding the pullout test and companion numerical analysis. First, a lack of proper material characterization, in particular soil and soil-inclusion interface strength, to use as input to the numerical models, which results in little phenomenological description of the soil-inclusion interface behavior at different stages of the pullout test. Second, little definitive evidence has been reported to-date on whether plane strain conditions prevail in pullout: in most cases, it is simply assumed that the plane strain condition exists. Third, few numerical models have been able to capture the complete pullout response of different inclusions, including the peak and strain softening behavior observed in some laboratory tests.  Finally, there is a fairly good understanding, in general, of the influence of size and boundary conditions of the pullout box; however, limited data exist to have an evidence-based understanding of the effect of inclusion type and box boundary conditions in the pullout response of different inclusions. Consequently, with the benefit of the laboratory pullout tests performed by Raju (1995) at The University of British Columbia on sheet and geogrid inclusions, four principal objectives were established in this study in order to gain a better understanding of the pullout response of different inclusions.   1. Obtain a proper characterization of the materials used in the laboratory pullout tests with emphasis on sand strength characterization at very low stresses;   2. Put forward evidence to demonstrate that a stress-dependency of the soil-inclusion interface friction angle and plane strain conditions prevail in pullout testing, by comparison of measured and simulated values of pullout resistance, and lateral stresses at the inside front wall of the pullout box;  3. Examine different configurations of soil, inclusion and soil-inclusion interface options to propose a numerical model that can capture the complete pullout resistance of the different inclusions tested by Raju (1995), with emphasis on:  216  - The use of a phenomenological constitutive model that allows for dilatancy and corrects for plane strain conditions;  - Capture the strain softening pullout response observed in some of the laboratory pullout tests on the sheet inclusions for which material properties are experimentally obtained; - Test the validity of the Jewell et al. (1985) approach to represent the three- dimensional aspects of the geogrids as a sheet inclusion, giving special consideration to the choice on the friction angle used to calculate the bearing stress ratio;  4. Perform a numerical parametric study, to analyze the adequacy of the laboratory test data available in the literature and the ASTM D6706-01 pullout recommendations, identify possible weaknesses in the determination of the soil-inclusion interaction factor, and suggest an “ideal” pullout test that will ensure that ultimate pullout occurs.  In order to achieve these objectives, the finite difference numerical program FLAC5.0 was used to simulate the laboratory pullout tests performed Raju (1995), on six different inclusions (three sheets and three geogrids) embedded in a dense Badger sand (relative density between 85% and 90%). Having the insight of the stress-dependency of the critical state friction angle in direct shear for Badger sand from Raju (1995), additional laboratory tests (direct shear, triaxial and direct simple shear) were done to characterize the strength of the sand at stresses lower than 150 kPa (Chapter 3).   Informed by the findings of this strength characterization, deduced values of the critical sate friction angle in plane strain, also believed stress-dependent, are obtained and are used to model the pullout response of the sheet inclusions at large displacement (Chapter 4). Badger sand is represented by the Mohr-Coulomb constitutive model, the inclusion by a beam structural element, and the soil-inclusion interface using Mohr-Coulomb type of springs. A good match between laboratory tests and numerical model results for the sheet inclusions is found at large displacement (for both the pullout resistance, and the horizontal stresses at the front wall of the box). The geogrids were also represented as an equivalent sheet inclusion  217  for which the stress-dependent soil-inclusion friction angle in plane strain was deduced using the Jewell et al. (1985). Comparison of laboratory test data and numerical simulation results, also show good agreement for the pullout response at large displacement and horizontal stresses at the front wall of the box.   In order to capture the complete pullout response of the sheet inclusions, in Chapter 5, the NorSand constitutive model was used to represent the behavior of the Badger sand, and the inclusion and soil-inclusion characteristics were changed following the subsequent steps:  1. Use NorSand to represent the behavior of Badger sand, using a beam element for the inclusion in combination with a Mohr-Coulomb type of spring (as presented in Chapter 4) for the soil-inclusion interface; 2. Use NorSand to represent the behavior of Badger sand, using a beam element for the inclusion and gluing the beam to the soil mesh through an interface characterized by normal and shear stiffness; 3. Use NorSand to represent the behavior of Badger sand and replace the beam by a continuum element with elastic properties to represent the sheet inclusion. This configuration was used with and without spring elements to represent the soil-inclusion interface interaction; 4. Use NorSand to represent the behavior of Badger sand, using an elastic continuum for the inclusion and a continuum layer with NorSand characteristics to mimic the response of the soil-inclusion interface.  5. Deduce an apparent soil-inclusion friction angle for the geogrids following the Jewell et al. (1985) approach to transform the three-dimensional properties of the soil-geogrid interaction into a soil-sheet inclusion friction angle.      218  Finally, in Chapter 6, with the benefit of a model that is able to capture the full pullout response of sheet inclusions, a parametric study is undertaken to gain insight into the current scientific literature. In particular, the boundary conditions of the pullout box are examined to reach conclusions regarding the adequacy of ASTM pullout test recommendations. Based on the results of the parametric study, the principal aspects deserving of improvement in the current practice and ASTM D6706-01 recommendations are identified and an “ideal” pullout test is suggested that appears to ensure that pullout, as defined by ASTM D6706-01, really occurs.  Specific findings of this research are summarized in this Chapter, focusing first on the outcomes of the sand characterization, then presenting evidence in support of the assumption that plane strain conditions and stress-dependency prevail in pullout at very low stresses, and finally studying the results of the numerical model and parametric study regarding boundary effects in the pullout response. Lastly, novel contributions of the present study are highlighted. It is important to notice that, given the unusually rounded shape of the Badger sand grains and in consequence, its unusual low frictional strength, the results of this study can be viewed as a lower bound to other soil types more commonly used.    7.1.1. Badger sand characterization  The characterization of Badger sand included basic properties such as the minimum and maximum void ratio and the shape of the particles, along with the strength in triaxial, direct shear, and direct simple shear. The findings are as follows:  • A value of roundness (R) measured following Wadell’s (1932) method was found equal to 0.81, indicative of a well-rounded sand. Inspection of the unified plot (Figure 3.9) shows the range for e = 0.2 and the friction angle of Badger sand (cs,tx = 27°) to be consistent with the roundness of its particles, giving confidence to the unusual frictional strength of the Badger sand.   219  • The critical state friction angle of Badger sand is found to be test-dependent. For the same stress range between 50 kPa and 150 kPa, cs,ds < cs,dss < cs,tx.   • The critical state friction angle of Badger sand is found to be stress-dependent in direct shear tests, at values of normal effective stress less than 50 kPa. The largest decrease in the angle of shearing resistance occurs between 4 kPa and 30 kPa. More specifically, cs,ds ≈ 30º at 4 kPa, ≈ 26º at 30 kPa, and ≈ 25º at  between 50 and 150 kPa. This is consistent with the findings of Boyle (1995), Lehane and Liu (2013), Quinteros (2014) and Rousé (2018), amongst others, in the triaxial, direct shear and plane strain tests.  7.1.2. Plane strain conditions and stress-dependency in pullout  • The analysis of the values of the friction angles obtained for Badger sand shows that the use of Rowe’s (1962) relation from direct shear tests, and Gutierrez and Wang (2009) from direct simple shear test, yield very similar values of critical state friction angle in plane strain for stresses between 50 and 150 kPa.   • The results of the two-dimensional numerical model of pullout tests in this study show that the use of stress-dependent values of cs,ps (deduced from the stress-dependent values of cs,ds and Rowe’s (1962) relation) in the Mohr-Coulomb constitutive model, agrees well with pullout laboratory test data and numerical results at large displacements for the three sheet inclusions. These results suggest that a stress-dependency at very low stresses and plane strain conditions are necessary for successful modeling of the soil-inclusion interaction.   • The results of the simulations using NorSand (that internally corrects for plane strain) and a stress-dependent value of Mtc, further suggest that that plane strain condition governs the pullout response of sheet inclusions at all displacements in the pullout box, and that the stress-dependency of the soil-inclusion interface friction angle must be considered in a back-calculation of pullout tests performed at vertical stresses less  220  than about 50 kPa. For example, the use of Mtc obtained in the stress range 50 to 150 kPa underestimates, by 20%, the measured pullout resistance and horizontal stress at vertical stresses less than 20 kPa.  • The use of a friction angle obtained from direct shear tests as an input to the numerical model underestimates the pullout resistance at small and large displacements by approximately 20%. This finding is consistent with the postulate of Jewell and Wroth (1987), who suggest the angle of friction obtained from a conventional analysis of the shear box test underestimates the mobilized frictional resistance of sand by about 20%, thereby yielding a “hidden” factor of safety of the order of 1.2 in design based on either the peak strength or a critical state strength.   7.1.3. Simulation of pullout response   • The use of a constitutive model for Badger sand that can capture the dilatant behavior of dense sands (such as NorSand), combined with the built-in beam element offered by a commercial software (such as FLAC5.0) and a typical Mohr-Coulomb spring configuration for the soil-inclusion interface, do not capture the complete pullout response of inclusions showing a strain softening pullout behavior. This is mostly due to the Mohr-Coulomb type of failure defined for the spring interface of the FLAC5.0 structural elements.  • The use of a structural beam glued to the soil mesh or an elastic continuum attached to the soil mesh is equivalent, and can capture the pullout response of the APT, GMT and GMS sheet inclusions. This approach, however, implies that the complete soil mesh has to be given the soil-inclusion strength characteristics. For the APT and GMT inclusions of this study, this simplification is valid given the fully rough nature of the soil-inclusion interface. For the GMS inclusion, that has a very smooth interface, this simplification is not representative of the soil conditions in the laboratory pullout test.    221  • The representation of the soil-inclusion interface as a thin continuum layer obeying a NorSand behavior to which the value of Mtc can be modified depending on the soil-inclusion strength characteristics, can capture the full pullout response (including the strain softening behavior when present) of the sheet inclusions, as well as the lateral stresses generated on the inside front wall of the pullout box.   • The use of stress-dependent values of cs,ps when using the Jewell et al. (1985) approach and eq. 4.6, corresponding to the upper boundary of the bearing stress ratio, to deduce an “equivalent” soil-geogrid friction angle to model the three-dimensional structure of a geogrid as a sheet inclusion, shows good agreement between measured and simulated pullout resistance and horizontal stresses on the inside front wall of the pullout box at large displacement.  • The use of stress-dependent critical state values of friction angles in triaxial compression, along with eq. 5.6, corresponding to the upper boundary of the bearing stress ratio in the method suggested by Jewell et al. (1985), to determine the NorSand parameter Mtc and therefore, characterize the strength of the soil-geogrid interface as a soil-sheet inclusion interface, can reasonably capture the full pullout response of geogrids.  7.1.4. Sensitivity analysis  • The onset of ultimate pullout (defined as the pullout resistance at dc = de) occurs before maximum pullout resistance.  • The distribution of vertical stresses at the soil-inclusion interface level at the onset of ultimate pullout (n,u) is non-uniform (it is larger close to the front wall and decreases along the inclusion with distance from the front wall) and is larger than the vertical stress defined by ASTM D6706-01.  • A minimum soil height above and below the inclusion of 30 cm is needed to minimize the effect of the top and bottom boundary of the pullout box.  222   • A lubricated front wall minimizes boundary effects on the pullout response, confirming the experimental observations of Palmeira (1989).  • A boundary that separates the “initial pullout zone” to the “ultimate pullout zone” is proposed in Figure 6.23 based on a combination of inclusion characteristics such as length, elastic modulus and surface roughness for different vertical stresses. This plot can help to predict whether a combination of inclusion characteristics tested at a certain stress level will result in ultimate pullout.  7.2. Novel contributions of the present study  The most important contributions of the present study are summarized below and listed according to the four principal objectives of this research:  7.2.1. Objective 1   • Consideration of the experimental results of this study, along with data reported by others in the literature, show the influence that grain shape exerts on the maximum and minimum void ratio of a grain assembly, and its angle of friction. The potential exists for a unified relation between roundness and these parameters, and the unified plot (Figure 3.14) enables a correlation between void ratio and friction angle for uniformly graded sands.  • A stress-dependency of the friction angle at large displacement is observed in triaxial, direct shear and plane strain conditions at stresses lower than 50 kPa.  7.2.2. Objective 2  • Plane strain conditions are demonstrated to prevail in pullout testing at both small and large displacements for sheet inclusions.   223   • The stress-dependency of the critical state friction angle of the soil and the soil-inclusion interface must be considered in a back-calculation of pullout tests performed at vertical stresses less than 50 kPa.   7.2.3. Objective 3  • The use of a constitutive model that can simulate dilation to represent the soil-inclusion interface behavior, that internally corrects for plane strain conditions, along with considering the stress-dependency of the soil-inclusion interface friction angle at stresses lower than 50 kPa, allows for the capture of the complete pullout response of sheet inclusions, including the strain softening behavior observed in some tests.  • The use of the Jewell et al. (1985) approach considering the stress-dependency of the critical state friction angle, along with the upper boundary of the bearing stress ratio, allows for a reasonable capture of the pullout response of geogrids when treated as equivalent sheet inclusions. The use of the lower boundary of the bearing stress ratio underestimates the complete pullout resistance of geogrids by approximately 35%. The use of the Jewell et al. (1985) approach appears to be more effective in capturing the pullout behavior of geogrids, when the degree of interference as defined in Palmeira (2009) is negligible.  7.2.4. Objective 4  • The most important aspects that need to be addressed in the ASTM D6706-01 recommendations for sheet inclusions are: o The following definitions are suggested to ASTM D6706-01 to make a distinction between maximum and ultimate pullout. - Maximum pullout resistance: the maximum pullout resistance measured during a pullout test  224  - Ultimate pullout resistance: pullout resistance measured when the displacement of the clamped end equals the displacement of the embedded end of the inclusion o A decrease in the vertical stress applied from 250 kPa to 30 kPa for extensible sheet inclusions to a maximum of 50 kPa when stiffer inclusions are tested, to ensure that a condition of ultimate pullout can be mobilized. o A measure of the vertical stresses at the soil-inclusion interface level to assess the magnitude of the increment of n over n,ASTM at different pullout stages, in order to make a correct interpretation of the soil-inclusion apparent friction angle (or interaction factor). o Increase the current minimum box height from 30 cm to 60 cm, in order to minimize the effect of the top and bottom boundaries of the pullout box. o The use of a lubricated front wall is recommended as a conservative approach until the studies have been undertaken to resolve the uncertainty with the use of a sleeve.  • The ideal pullout testing device and inclusion characteristics that guarantee that ultimate pullout occurs for a soil-inclusion interface roughness lower or equal to the strength of the soil, and minimizes top, bottom and front wall effects, should comply with the following characteristics: maximum inclusion length of 1.0 m, elastic modulus of the inclusion larger than 5×106 kPa, vertical stress applied at the inclusion level lower than 30 kPa, soil thickness above and below the inclusion of 30 cm and a lubricated front wall.          225  7.3. Future work and recommendations  Previous studies have suggested the use of plane strain strength for analysis of reinforced soil walls (see for example Boyle, 1995; Allen and Bathurst, 2002; BS 8006, 2010) or have assumed plane strain conditions to obtain closed-form solutions to pullout test data (Abramento and Whittle, 1995; Bobet et al., 2006). From a theoretical point of view, Jewell and Wroth (1987) postulated the angle of friction obtained from a conventional analysis of the shear box test underestimates the mobilized frictional resistance of sand by about 20%, yielding a hidden factor of safety of the order of 1.2 in design based on either the peak strength or a critical state strength. Hence, the main contribution of this dissertation is to put forward evidence in support of the assumption that plane strain conditions prevail in pullout testing of sheet and geogrid inclusions. Accordingly, this study has revealed the following opportunities for future work:  • The modeling approach used in this study considers the geogrids as equivalent sheet inclusions, where the effect of the transverse elements is accounted for in the soil-inclusion interface strength using Jewell et al. (1985) method. This approach was proved acceptable for the focus of this study. Although it is believed that no implication for the main conclusion of this thesis will occur, consideration of the effect of the bearing elements of the geogrids in a three-dimensional model is suggested, to verify that plane strain conditions and the stress-dependency of the friction angle at the critical state also prevails in pullout tests on geogrids.  • The validation approach of this study for the modeling of the pullout response of sheet and geogrid inclusions was done by focusing on the small-scale pullout test data obtained in the laboratory. A natural next step is evaluating the performance of the modeling approach for full-scale reinforced walls, by allowing the soil-inclusion interface to dilate using a constitutive model such as NorSand, considering plane strain conditions and the stress-dependency of the critical state friction angle at stresses lower than 50 kPa.   226  • The current study focused on a condition of ultimate pullout and therefore, no comparison was done between measured and simulated values of strains in the inclusion. The experimental study of Raju (1995) included the measurements of strains at five different places on the smooth geomembrane and two grids. 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Numerical Evaluation of the pullout box method for studying soil reinforcement interaction, Transportation research record 1278, Transportation research board, Washington, DC, 1990: 116-124.  243  Appendix A  NorSand was first introduced by Jefferies (1993) and its formulation is based on stress dilatancy. NorSand uses the plastic shear strains to control the hardening rule and is based on two axioms of Critical State Soil Mechanics (CSSM):  - Axiom 1: “A unique locus exists in q, p, e space such that soil can be deformed without limit at constant stress and constant void ratio; this locus is called the critical state locus (CSL)”.   - Axiom 2: “The CSL forms the ultimate condition of all distortional processes in soil, so that all monotonic distorsional stress state paths tend to this locus”.  NorSand is able to capture the softening and dilatant behavior of sands through the state parameter () and by postulating an infinity of normal consolidation loci (NCL) as shown in Figure A.1. The use of void ratio or relative density alone does not tell how the soil is going to behave, given that dilation or contraction depends not only on these parameters, but also on the stress level. Been and Jefferies (2004) showed that samples with similar relative densities and different stress levels behaved differently, while samples with same state parameter but different void ratios and stress level, had similar behavior. In NorSand, the state parameter tends to zero as shear strain accumulates and the minimum possible dilation rate (i.e. dilation at peak strength) is related to .  The idea of an Infinity of NCL was first introduced by Ishihara et al. (1975). They suggested that given that in a sand an infinity of void ratios can be obtained, an infinity of NCL exist. Jefferies and Been (2000) supported the idea with experimental data. Four samples of Erksak sand were tested at different void ratios. They performed load/unload tests for each sample and found that each isotropic loading line can be seen as a true NCL. Infinity of NCL can be viewed as an infinity of yield surfaces and each NCL can be viewed as a hardening law for an associated yield surface.   NorSand idealizes plastic work dissipation and hence, work conjugate stress and strain are needed. Accordingly, the shear strain increment is written as:  244  ( ) ( ) −+−+=••••321 cos3sinsin2cos3sin31 q  (eq. A.1)   Where  is the Lode angle (for triaxial compression and extension 𝜃 = 30𝑜and 𝜃 = −30𝑜, respectively), and •i i = 1,2,3 are the principal strain increments.  NorSand is an elasto-plastic model and hence it comprises four items. In this study, the version presented in Jefferies and Shuttle (2005) and Jefferies and Shuttle (2011) is used and the equations of the model are presented in Table A.1:  - Elasticity: In NorSand, isotropic elasticity is assumed and elastic parameters are defined by the shear modulus, G, and a constant Poisson’s ratio, . NorSand does not account for the influence of fabric in G, however it is possible to measure the elastic modulus in situ and use it as an input parameter.  - Yield surface: The NorSand yield surface has a bullet-like shape with a limit that avoids having unrealistically large dilations of dense soils (see Figure A.2). The yield surface expands and contracts as required by the hardening law, depending on the current state parameter and the direction of loading.  - Hardening rule: The hardening law describes whether the yield surface increases or decreases in size with plastic straining. In NorSand the size of the yield surface is controlled by pi’ which is the image stress (Figure A.2). The hardening or softening of the yield surface depends on the state parameter and the direction of loading. As shown in Figure A.2, the critical state line in NorSand does not always intersect with the yield surface (e.g. very loose sands) and hence, the hardening law moves the yield surface towards the critical state under the action of plastic shear strain.  245  - Flow rule: The flow rule is determined by the ratio of the plastic volumetric strain to the plastic shear strain increments as shown in Table A.1.  It has been found that a strong relation exists between the minimum dilatancy, 𝐷min𝑃  (ie, value close to the peak stress ratio) and i (the value of the state parameter, , at the image state), as shown in Figure A.2 and given by:                    𝐷𝑚𝑖𝑛𝑃 = 𝜒𝜓𝑖               (eq. A.2)  where  is a model property and it usually lies between 2.5 and 4.5 for triaxial compression (with a common value of 3.5).  In order to describe the NorSand constitutive model, 8 parameters are needed, of which 7 are obtainable from conventional triaxial compression tests and 1 is assumed (Poisson’s ratio). Most of these parameters are familiar (the same parameters needed for Mohr-Coulomb), and the unfamiliar are easily deduced.   - Elastic parameters: The elastic shear modulus is ideally measured using bender elements and Poisson’s ratio is usually assumed to be in a range between 0.1 and 0.3.  - Critical state parameters: The critical state parameters are defined by the altitude of the critical state line at 1 kPa ( ), and the slope of the critical state line or normally consolidated line in the e-logp’ space, ( ).  The critical state parameters in the e-logp space,  and  are obtained from drained and undrained tests on very loose reconstituted sand samples at different confining pressures. In that case, the samples will contract and reach the CSL in the e-logp space at strains within the limits of the triaxial device. Loose samples are preferred instead of dense samples, given that these later dilate and do not reach the critical state, sometimes at 25% strain and are prone to localize.    246  In the stress space, the CSL is defined as:  𝑞 = Mp′        (eq. A.3)   where, for triaxial compression condition, M is given in terms of the friction angle by equation A.4.     𝑀tc =6sincs,tx3−sincs,tx                                            (eq. A.4)  Triaxial compression is specified because the stress conditions in the critical state are a function of the magnitude of the intermediate principal stress. Because of this, NorSand takes triaxial compression as the reference condition and leaves the variation of M with proportion of the intermediate stress (or Lode angle, 𝜃) to the constitutive model. In doing this, triaxial compression conditions (Mtc) are taken as the reference case in which the soil properties are determined. Thus, Mtc becomes a soil property and M(𝜃) is evaluated in terms of this property. Plotting the Lode angle against the dilatancy for plane strain condition, it is found that  is not constant. Hence, there is no unique Mps (Jefferies and Shuttle, 2002). The question is how M varies with the Lode angle. Data on Brasted sand shows that for plane strain, M varies between the Mohr-Coulomb criterion and the Matsuoka-Nakai criterion. Hence, in NorSand, the friction ratio, M, is a function of the intermediate principal stress through the Lode angle (), and Mtc (Jefferies and Shuttle, 2011) as described in equation A.5.     𝑀 = 𝑀𝑡𝑐 −𝑀𝑡𝑐23+𝑀𝑡𝑐𝑐𝑜𝑠(3 𝜃 2⁄ + 𝜋 4⁄ )                       (eq. A.5)   For triaxial tests, there are different ways to find the value of Mtc. For loose sand specimens, Mtc is found as the value of the stress ratio at the end of the test. For dense sand specimens, it is obtained from the end of tests on very loose sand specimens or by the Bishop’s (1966) method (see Figure 3.7). By plotting the maximum stress ratio, max, against the minimum dilatancy, and extrapolating a straight line through these points,  = Mtc at the critical state when the dilatancy equal to zero. A couple of very dense drained tests are necessary to estimate the maximum dilatancy of the soil (Dmin).  247   - Plastic parameters: Three other parameters are needed to describe this constitutive model. N, which is defined in the same way as the parameter N found in Nova’s flow rule (Nova, 1982) and can be obtained by plotting the maximum stress ratio versus Dmin for dense sands (see Figure 3.7). However, the value of N does not vary greatly and if the plot of Figure 3.7 is not possible to obtain, a reasonable value for N is 0.3.  is a constant for each sand and is defined as the slope of the Dmin to the initial state parameter taking triaxial compression as a reference. The plastic hardening parameter, H, is obtained by fitting the results obtained by the drained triaxial tests and iterating to obtain the best match. It is in principle a function of soil fabric and .   In order to follow a computational geomechanics approach, where a phenomenologically constitutive model is used, the verification and validation approach is needed. After Tasiopoulou et al. (2015), verification is a process of determining that a model implementation accurately represents the developers’ conceptual description and specification. It is essentially a mathematics issue and it provides evidence that the model is solved correctly. For NorSand, the verification step is described in Shuttle and Jefferies (1998) by modelling triaxial compression tests and cavity expansion and comparing to the theoretical solutions.   The validation step is a process of determining the degree to which a model is accurate representation of the real world from the perspective of the intended uses of the model. It is a physics issue, and it provides evidence that the correct model is solved (Tasiopoulou et al., 2015). A first step to validate NorSand is to check the correction for plane strain through the Lode angle. The plane strain element in Figure A.3 is considered. This is a drained plane strain element test on a dense sand ( = -0.2) with a cell pressure of 200 kPa. The input parameters are shown in Table A.2. Figure A.4 shows the deviatoric stress, the volumetric strain, the value of M and the Lode angle. The simulated dense sample shows a strain softening behavior and reaches the critical state at a value different than Mtc for a Lode angle equal to 14°, that is different to 30° for triaxial compression. These results show the internal correction made by NorSand to account for plane strain conditions.   248  Table A.1: Summary of equations for NorSand (from Jefferies and Shuttle, 2011)  Aspect of NorSand Equations Internal model parameters ( )ppii ln +=  where cee −=  −=tciMNMM1  Critical state pec ln−=  ( )423cos32 ++−=tctctcMMMM  Yield surface −=ii ppMln1  Hardening rule ••−= pqiiitciiiippppppMMHppmax2, ( )tciitci Mpp,maxexp −= Stress dilatancy  −=••ipqpqP MD ||/    Elasticity G  and  Where:  -i: state parameter at image condition  - pi: mean effective stress at image condition (kPa)  - Mi: stress ratio at image condition  - Mi,tc: Mi at triaxial compression   -•pq : plastic shear strain increment   249  Table A.2: Input parameters for plane strain element Parameter Description Value H Plastic hardening modulus for loading 700  N Relates minimum dilatancy to state parameter Volumetric coupling parameter 4.0 0.35 G  Shear modulus (MPa)  2.5e5  Poisson’s ratio 0.2  Altitude of the critical state line in e-logp space defined at 1 kPa 0.816  Slope of the critical state line in e-logp space 0.014     Mtc  Critical stress ratio in the p - q plane  1.26    250                  Figure A.1: Definition of state parameter () and infinity of NCL (from Jefferies, 1993)    Critical State Line NCL NCL   Void ratio, e Mean effective stress, p 251                            Figure A.2: NorSand image condition on yield surface: a) very loose sand, b) very dense sand (from Jefferies and Shuttle, 2002)    b) a) Mean effective stress ratio, p/pi Deviator stress ratio, q/pi Critical state M Mi Image condition Yield surface Deviator stress ratio, q/pi Mean effective stress ratio, p/pi Critical state M Mi Image condition Yield surface 252               Figure A.3: Geometry, boundary and loading conditions for plane strain (from Jefferies and Shuttle, 2011)               Figure A.4: Results of plane strain element  

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