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Improved characterization and analysis of bi-planar dip slope failures to limit model and parameter uncertainty.. 2009

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 IMPROVED CHARACTERIZATION AND ANALYSIS OF BI- PLANAR DIP SLOPE FAILURES TO LIMIT MODEL AND PARAMETER UNCERTAINTY IN THE DETERMINATION OF SETBACK DISTANCES  by  Brendan R. Fisher B.A. SUNY Potsdam, 1995 M.S. Radford University, 1997 M.S. Virginia Polytechnic University, 2000   A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY   in   The Faculty of Graduate Studies   (Geological Engineering)    THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)   July 2009     © Brendan R. Fisher, 2009      ii  ABSTRACT  A dip slope is a natural or man-made rock slope with a persistent discontinuity behind the slope face that is coincident with, or similar to the slope inclination.  Most dip slope failures occur in weak, orthogonally jointed sedimentary rock with toe breakout involving sliding along joints, plastic failure of intact blocks, and/or intense deformation of the slope that facilitates kinematic release.  Dip slope failures have been reported to extend more than 20 percent of the slope’s height behind the crest, making this rock slope failure mechanism relevant in the context of engineering projects.  Nonetheless, dip slope failure mechanisms and evaluation methods are not well understood because of the complexity of the toe breakout.  This thesis provides an overview of dip slope failure mechanisms where bi-planar failures may occur.  It provides specific guidance for evaluating a dip slope’s stability state predicated on an extensive literature review, numerical modeling, and parametric evaluations. The thesis provides methods for effectively planning and executing geotechnical investigations with the goal of establishing a dip slope’s stability state.  Finally the thesis uses a comprehensive case study where very detailed geotechnical information is available as data for dip slope design.  In summary, the results of this study and research suggest that: 1) typical failure mechanisms for dip slopes can be characterized and anticipated based on documented case histories and therefore site investigations should be customized towards evaluating the potential for those failure mechanisms, 2) typical geotechnical investigation and analysis methods (supplemented with numerical modeling where appropriate) may be used to evaluate a dip slope's stability state, 3) the influence of the rock mass shear strength at the toe and the failure mechanisms assumed for toe breakout is paramount while establishing a dip slope’s stability state where slopes are steeper than about 45 degrees, 4) the stability state of shallow dip slopes is dominated by the shear strength of the slope-coincident sliding surface, 5) statistical evaluations of other geotechnical parameters that dictate a dip slope’s stability suggest that at a scoping level, geotechnical investigation methods can be cost effectively planned to provide value to the geotechnical project, and 6) risk sharing dictates the current methods of dip slope evaluation and these methods can be improved based on the research contained herein. iii       TABLE OF CONTENTS Abstract ............................................................................................................................. ii  Table of Contents .............................................................................................................. iii  List of Tables .................................................................................................................... vii  List of Figures................................................................................................................... ix  Acknowledgements ......................................................................................................... xiv  Dedication ....................................................................................................................... xv  Co-Authorship Statement ................................................................................................ xvi  1.  Introduction ................................................................................................................ 1  1.1.  Problem Statement ............................................................................................. 1  1.2.  Thesis Objectives ............................................................................................... 2  1.3.  Structure of Thesis .............................................................................................. 4  1.4.  Literature Review – Relevant Research to Date ................................................. 6  1.4.1.  Geotechnical Uncertainty ............................................................................ 6  1.4.2.  Dip Slopes ................................................................................................... 7  1.4.3.  Bi-Planar .................................................................................................... 14  1.4.4.  Toe Failure Mechanisms for Bi-Planar Failures ........................................ 15  1.5.  References ....................................................................................................... 22  2.  Analysis of Toe Breakout Mechanisms to Develop an Improved Methodology for Assessing the Stability State of Dip Slopes 0F ................................................................... 29  2.1.  Introduction ....................................................................................................... 29  2.2.  Scope of the Problem ....................................................................................... 30  2.3.  Geological Environments .................................................................................. 33  2.3.1.  Orthogonally-Jointed Sedimentary Slopes ................................................ 33  2.3.2.  Faulted/Sheared Slopes ............................................................................ 34  2.4.  Toe Breakout/Internal Shearing ........................................................................ 35  2.4.1.  Active/Passive Controls ............................................................................. 35  2.4.2.  Persistent Structure ................................................................................... 36  2.4.3.  Step-Path Failures ..................................................................................... 36  2.4.4.  Sliding-Bending Model ............................................................................... 37  2.5.  Treatment of Toe Breakout and Internal Shearing ........................................... 37  2.5.1.  Limit Equilibrium Methods ......................................................................... 37  2.5.2.  Frictional Plasticity Theory ......................................................................... 38  iv  2.6.  Improved Dip Slope Stability State Assessment ............................................... 43  2.6.1.  Discontinuum Approach ............................................................................ 43  2.6.2.  Continuum Approach ................................................................................. 49  2.7.  Parametric Evaluation ....................................................................................... 52  2.8.  Discussion ........................................................................................................ 54  2.8.1.  Shear Strength Estimates of Heavily Jointed Rock Masses ...................... 55  2.8.2.  Shear Strength Estimates of Orthogonally-Bedded Dip Slopes ................ 57  2.9.  Practical Implementation – Case Histories ....................................................... 58  2.9.1.  Case History – Randomly Jointed Dip Slope at Aguas Claras Mine ......... 59  2.10.  Case History – Bedded Dip Slope at Smoky River Coal ............................... 61  2.10.1.  Numerical Modeling ................................................................................... 62  2.10.2.  Monitoring Data ......................................................................................... 63  2.11.  Conclusions ................................................................................................... 64  2.12.  References .................................................................................................... 68  3.  Assessment of Parameter Uncertainty Associated with Dip Slope Stability Analyses as a Means to Improve Site Investigations1F.................................................................... 73  3.1.  Introduction ....................................................................................................... 73  3.2.  Problem Statement – Parameter Uncertainty ................................................... 74  3.3.  Spearman Rank Correlations ........................................................................... 76  3.4.  Mohr-Coulomb Shear Strength Parameters ..................................................... 81  3.5.  Geotechnical Input Distributions ....................................................................... 84  3.6.  Parameter Uncertainty and Dip Slope Stability ................................................. 86  3.7.  Discussion and Practical Recommendations .................................................... 91  3.8.  Conclusions ...................................................................................................... 97  3.9.  References ....................................................................................................... 99  4.  Improved Design of Set-Back Distances for Dip Slopes in Bedded Rock2F ............ 102  4.1.  Introduction ..................................................................................................... 102  4.2.  Dip Slopes ...................................................................................................... 103  4.3.  Guidelines and Their Interpretation for Evaluating Dip Slopes ....................... 104  4.4.  Case History: Devil Canyon, Southern California ........................................... 106  4.4.1.  Regional and Site Geology ...................................................................... 107  4.4.2.  Seismicity ................................................................................................ 109  4.4.3.  Geotechnical Investigation ...................................................................... 109  4.4.4.  Geologic Structure ................................................................................... 109  v  4.4.5.  Weathering and Rock Grade ................................................................... 111  4.4.6.  Groundwater ............................................................................................ 111  4.4.7.  Laboratory Testing ................................................................................... 112  4.4.8.  Rock Mass Classification ......................................................................... 114  4.4.9.  Engineering Geologic Model ................................................................... 117  4.5.  Design of Dip Slope Setback Distances for the Devil Canyon Project ........... 118  4.5.1.  Bi-Planar Failure Mechanisms ................................................................. 118  4.5.2.  Analysis Methods .................................................................................... 121  4.5.3.  Case History Results and Discussion ...................................................... 122  4.6.  Case History Revisited: Path to an Improved Methodology ........................... 125  4.6.1.  Assessment of Dip Slope Failure Mechanisms ....................................... 125  4.6.2.  Assessment of Analysis Techniques ....................................................... 127  4.6.3.  Limiting Geotechnical Parameter Uncertainty ......................................... 129  4.6.4.  Reassessment of the Devil Canyon Project ............................................ 130  4.6.1.  Perceived Risk ......................................................................................... 133  4.6.2.  Reliability-Based Design .......................................................................... 136  4.6.3.  Probability of Failure ................................................................................ 137  4.7.  Discussion ...................................................................................................... 140  4.8.  Conclusions and Lessons Learned ................................................................. 142  4.9.  References ..................................................................................................... 144  5.  Thesis Discussion and Conclusions ...................................................................... 149  5.1.  Summary ........................................................................................................ 149  5.1.1.  Analysis of Toe Breakout Mechanisms ................................................... 149  5.1.2.  Analysis of Parameter Uncertainty .......................................................... 150  5.1.3.  Evaluation of Current Guidelines and Design of Setback Distances ....... 152  5.2.  Key Conclusions and Scientific Contributions ................................................ 153  5.2.1.  Methodology for Assessing Dip Slope Stability ....................................... 153  5.2.2.  Site Investigations ................................................................................... 154  5.2.3.  Design of Set-Back Distances ................................................................. 156  5.3.  Future Research ............................................................................................. 156  5.4.  References ..................................................................................................... 158  Appendix A.  International Database of Dip Slope Failures ...................................... 159  Appendix B.  Influence of Groundwater on D/H Ratios ............................................. 165  B.1     Introduction ..................................................................................................... 165  vi  B.2     Distribution of Water Pressures in Dip Slopes ................................................ 165  B.3     Apparent Friction Angles from Dip Slope Case Histories ................................ 168  B.4     References ...................................................................................................... 171  Appendix C.  Mohr-Coulomb Shear Strength Estimates for Dip Slopes .................... 172  C.1     Published Methodology ................................................................................... 172  C.2     Procedure to Estimate Mohr-Coulomb Parameters for Dip Slopes ................ 173  C.2.1  Step One – Establish the Hoek-Brown Failure Envelop .............................. 174  C.2.2  Step Two – First Approximation of Mohr-Coulomb Parameters .................. 174  C.2.3  Step Three – Estimate Normal Stresses at the Dip Slope Toe ................... 175  C.2.4  Step Four – Iterate Mohr-Coulomb Parameters Based on Normal Stresses  176  C.2.5  Step Five – Best Fit Mohr-Coulomb Parameters and Stability Check ......... 177  C.3     References ...................................................................................................... 181  Appendix D.  Udec Codes ......................................................................................... 182  Appendix E.  Dip Slope Failure Mechanisms ............................................................ 205  E.1   Seismic Loading ............................................................................................ 205  E.2   Increased Pore Water Pressure .................................................................... 206  E.3  Example UDEC Codes ................................................................................... 213  E.4  References ..................................................................................................... 230  Appendix F.  Chatsworth Project Laboratory Testing ................................................ 231  F.1     Uniaxial Compressive Strength Testing of Intact Rock ................................... 231  F.2     Direct Shear Tests on Saw Cut Samples ........................................................ 232  F.3     Direct Shear Tests on Natural Discontinuity Surfaces .................................... 233  F.4  Torsional Ring Shear Tests on Shale Bedding ............................................... 235  F.5  Direct Shear Tests on Intact Rock Samples ................................................... 235  F.6  Triaxial Testing of Intact Rock Samples ......................................................... 237  F.7  References ..................................................................................................... 239   vii       LIST OF TABLES Table 2.1.  Dip slope failure case histories reviewed for this paper. ............................... 31  Table 2.2.  Input parameters to initialize equilibrium conditions for the model shown in Fig. 2.8. ........................................................................................................................... 47  Table 2.3.  Input parameters for the model shown in Fig. 2.8. ........................................ 47  Table 2.4.  Input parameters to initialize conditions to equilibrium for the model shown in Fig. 2.9. ........................................................................................................................... 48  Table 2.5.  Input parameters for the model shown in Fig. 2.9. ........................................ 48  Table 2.6.  Input parameters for finite element model in Fig. 2.10. ................................. 52  Table 2.7.  Input parameters for finite element model in Fig. 2.11. ................................. 52  Table 2.8.  Geotechnical properties used for back analysis of Aquas Claras mine failure.  ........................................................................................................................................ 60  Table 2.9. Engineering geology of the East Limb No. 9 Mine. ........................................ 62  Table 2.10.  Input parameters for the Smoky River Coal back analysis. ........................ 63  Table 3.1.  Uncertainty associated with RMR89*, based on a detailed dataset for a dip slope in Southern California. ........................................................................................... 78  Table 3.2.  Uncertainty of Mohr-Coulomb Parameters. ................................................... 82  Table 3.3.  Typical relative cost of geotechnical investigations. ...................................... 92  Table 4.1.  Key findings and their interpretations from SCEC 2002. ............................. 105  Table 4.2.  Discontinuity spacing. ................................................................................. 111  Table 4.3.  Rock mass weathering and strength profiles. ............................................. 113  Table 4.4.  Distribution of results from laboratory testing. ............................................. 113  Table 4.5.  Range of measured GSI input values. ........................................................ 117  Table 4.6.  Input parameters to initialize conditions to equilibrium for the model shown in Fig. 4.11. ....................................................................................................................... 127  Table 4.7.  Input parameters for the model shown in Fig. 4.11. .................................... 127  Table 4.8.  Most likely values for the Grade R2 rock mass. .......................................... 131  Table 4.9.  Input parameters for finite element models in Fig. 2.10. ............................. 134  Table 4.10.  Change in factor of safety due to parameter variability under static loading conditions. ..................................................................................................................... 139  Table 4.11.  Probability of failure based on the results presented in Table 4.7. ........... 139   viii  Table B. 1a-c.  Dip slopes where groundwater was reported as a triggering mechanism.  ...................................................................................................................................... 169   Table C. 1.  Iterations to establish normal stresses at toe breakout and internal shear surface. ......................................................................................................................... 177  Table C. 2.  Input parameters for finite element model (Mohr-Coulomb best fit). ......... 180  Table C. 3.  Input parameters for finite element model (Hoek-Brown Failure Criterion).  ...................................................................................................................................... 180   ix  LIST OF FIGURES Fig. 1.1.  Dip slope along a highway (Photograph by E. Eberhardt). ................................ 7  Fig. 1.2.  Result of a buckling failure at the Fording-Greenhills coal mine (Photograph by E. Eberhardt). .................................................................................................................... 9  Fig. 1.3.  Buckling failure mechanisms. From Stead and Eberhardt (1997). ................... 10  Fig. 1.4.  Example of a ploughing failure from Stead and Eberhardt (1997). .................. 12  Fig. 1.5.  Photograph showing ploughing failure of thinly-bedded sedimentary rock (Photograph by E. Eberhardt). ........................................................................................ 13  Fig. 1.6.  Active-passive failures surface from Kvapil and Clews (1971) and after Stead et al., 2006. ..................................................................................................................... 15  Fig. 1.7.  Kinematic bi-planar slope failure analysis according to Nathanail, 1996. ........ 16  Fig. 1.8.  Complex slope failure presented by Hoek et al., 2000 showing rock mass failure in the Prandtl zone and step path failure that facilitates toe breakout. ................. 18  Fig. 1.9.  Rock slope with a discontinuous potential plane of weakness.  Jennings (1970) suggests using a weighted average of discontinuity and intact rock shear strength to estimate the overall “weighted” average shear strength of the plane. ............................ 19  Fig. 1.10.  Slope back analysis presented by Hoek, 2000.  The Hoek Brown Failure Criterion was proposed to estimate the shear strength of the weak rock mass within the active and passive portions of the sliding mass. ............................................................. 20  Fig. 2.1.  Simplified dip slope showing bi-planar failure mechanism. .............................. 30  Fig. 2.2.  Dip slope failure D/H ratios reported in the geotechnical literature. ................. 32  Fig. 2.3.  Active-passive rock slope failure model (after Kvapil & Clews 1979). ............. 35  Fig. 2.4.  Localization of shear surfaces according to Mohr-Coulomb criterion. ............. 39  Fig. 2.5.  Toe breakout and internal shear surface according to Mohr’s rupture theory. . 40  Fig. 2.6.  Bi-planar dip slope failure in sedimentary rock. ............................................... 42  Fig. 2.7.  Bi-planar dip slope failure in metamorphic rock. .............................................. 42  Fig. 2.8.  Failure mechanism for a 30-degree orthogonally jointed and bedded dip slope. 1) Model setup.  2) Bi-planar failure mechanism. ............................................................ 45  Fig. 2.9.  Failure mechanism for a 45-degree orthogonally jointed and bedded dip slope. 1) Model setup.  2) Bi-planar failure mechanism. ............................................................ 46  Fig. 2.10.  Failure mechanisms for a shallow dip slope treated as a continuum. ............ 50  Fig. 2.11.  Failure mechanisms for a steep dip slope treated as a continuum. ............... 51  x  Fig. 2.12.  Comparison of stability factors estimated using finite element, discrete element, the Mohr-Coulomb theory (to determine inclination of toe breakout and internal shear surfaces), and optimized Sarma methods. ............................................................ 56  Fig. 2.13.  Methodology for estimating the stability state of dip slopes. .......................... 58  Fig. 2.14.  Finite element analysis of Aquas Claras mine failure. ................................... 61  Fig. 2.15.  Smoky River Coal back analysis showing bi-planar failure mechanism. A.) Discrete element model set up.  B.) “Blow up” showing bi-planar failure mechanism.  C.) Comparison of measured displacement vectors with those predicted using discrete element. .......................................................................................................................... 67  Fig. 3.1.  Simplified dip slope showing bi-planar failure mechanism. .............................. 74  Fig. 3.2.  Spearman Rank Correlation Coefficient for a RMR89* rated “Fair” rock mass.  ........................................................................................................................................ 79  Fig. 3.3.  Influence of ci and RQD on Rock Mass Rating (10,000 data points). ............. 79  Fig. 3.4.  Influence of different rock mass rating inputs for different RMR89* classes. ... 81  Fig. 3.5.  Spearman Rank Correlations Coefficients for rm and crm. ............................... 84  Fig. 3.6.  Influence of Mohr-Coulomb shear strength parameters on dip slope stability. 88  Fig. 3.7.  Influence of Hoek Brown input parameters on dip slope stability. .................... 89  Fig. 3.8.  Relationship between geotechnical effort and uncertainty. .............................. 94  Fig. 3.9.  Geotechnical effort and perceived Pf for a proposed dip slope. ....................... 94  Fig. 3.10.  Typical geotechnical investigation for reliability-based geotechnical design for dip slopes. ....................................................................................................................... 96  Fig. 3.11.  Geotechnical tasks associated with reliability evaluations of dip slopes. ....... 97  Fig. 4.1.  Typical dip slope geometry showing the range of toe breakout surfaces (after SCEC 2002). ................................................................................................................. 104  Fig. 4.2.  Project area location map. ............................................................................. 107  Fig. 4.3.  Engineering geologic map showing the area proposed for residential development.................................................................................................................. 108  Fig. 4.4.  Stereonet showing the geologic structure of Devil Canyon. ........................... 110  Fig. 4.5.  Direct shear test results for Grade R1 intact rock samples ............................ 114  Fig. 4.6.  Geologic Strength Index showing combination of composition and structure and surface conditions of discontinuities mapped along outcrops in Devil Canyon (after Marinos and Hoek 2000). .............................................................................................. 115  xi  Fig. 4.7.  Outcrop of the Chatsworth Formation observed in the valley at the base of the canyon.  Note the orthogonal joints truncated at bedding.  Bedding is shown in blue. Truncated toe joints are shown in red. .......................................................................... 116  Fig. 4.8.  Dip slope “working” cross-section. ................................................................. 117  Fig. 4.9.  Geologic model used to establish structural setbacks based on a compromise on the part of the stakeholders. ..................................................................................... 124  Fig. 4.10.  Simplified dip slope showing bi-planar failure mechanism and internal shear surface. ......................................................................................................................... 125  Fig. 4.11.  Distinct element model showing bi-planar dip slope failure mechanism. ..... 126  Fig. 4.12.  Comparison of stability factors using numerical models and limit equilibrium. “Mohr-Coulomb” refers to the method used to establish the inclination of the toe breakout and internal shear surfaces. ........................................................................... 129  Fig. 4.13.  Stability state of the dip slope using finite element modeling with the most likely values (MLV) of geotechnical parameters.  This is the same slope as that shown in Fig. 4.8.  Rock grades (delineated by the green lines) and groundwater is included in the analysis. ........................................................................................................................ 135   Fig. B. 1.  Permeability ratio for a dip slope assuming anisotropy parallel to the slope face. .............................................................................................................................. 166  Fig. B. 2.  Water pressures within a bedded dip slope. ................................................. 167  Fig. B. 3.  Apparent friction angles and D/H ratios from reported dip slope failures. .... 170   Fig. C. 1.  Example dip slope with D/H ratio of 0.15. ..................................................... 173  Fig. C. 2.  RocLab estimate of Mohr-Coulomb parameters by fitting a straight line to the Hoek-Brown failure envelope.  This procedure is outlined in Hoek et. al., (2002). ....... 174  Fig. C. 3.  Screen capture of the Sarma program (Hoek, 1985).  Results show the normal stress at the toe breakout surface (base stresses for Side number 1) and internal shear surface (side stresses for Side number 1).  The coordinate xt is 0.00. ......................... 175  Fig. C. 4.  RocLab program is used to predict the instantaneous Mohr-Coulomb parameters by a tangential linear best-fit at the average normal stress. ...................... 176  Fig. C. 5.  Linear best fit to Hoek Brown failure envelope for range of normal stresses within the example dip slope. ........................................................................................ 178  Fig. C. 6.  Finite element evaluation showing maximum shear strain contours of the example dip slope using the Hoek-Brown failure criterion.  SSR factor is 1.23. ........... 179  xii  Fig. C. 7.  Finite element evaluation showing maximum shear strain contours of the example dip slope using the Mohr-Coulomb failure criterion.  SSR factor is 1.24. ....... 179   Fig. E. 1.  Seismic loading of a 30-degree dip slope with D/H ratio of 0.20.  1) Model setup and 2) failure mechanism predicted using distinct element modeling compared to that predicted using the Mohr-Coulomb theory to establish failure surface inclinations.  ...................................................................................................................................... 207  Fig. E. 2.  Seismic loading of a 45-degree dip slope with D/H ratio of 0.12.  1) Model setup and 2) failure mechanism predicted using distinct element modeling compared to that predicted using the Mohr-Coulomb theory (M-C) to establish failure surface inclinations. ................................................................................................................... 208  Fig. E. 3.  Seismic loading of a 60-degree dip slope with D/H ratio of 0.08.  1) Model setup and 2) failure mechanism predicted using distinct element modeling compared to that predicted using the Mohr-Coulomb (M-C) theory to establish the inclination of failure surfaces......................................................................................................................... 209  Fig. E. 4.  Increase in water pressure in a 30-degree dip slope with D/H ratio of 0.20.  1) Model setup, 2) failure mechanism predicted using distinct element modeling; steady state and partially saturated slopes, 3) failure mechanism predicted using distinct element modeling; steady state and fully saturated conditions. .................................... 210  Fig. E. 5.  Increase in water pressure in a 45-degree dip slope with D/H ratio of 0.12.  1) Model setup, 2) failure mechanism predicted using distinct element modeling; steady state and partially saturated slopes, 3) failure mechanism predicted using distinct element modeling; steady state and fully saturated conditions. .................................... 211  Fig. E. 6.  Increase in water pressure in a 60-degree dip slope with D/H ratio of 0.08.  1) Model setup, 2) failure mechanism predicted using distinct element modeling; steady state and partially saturated slopes, 3) failure mechanism predicted using distinct element modeling; steady state and fully saturated conditions. .................................... 212   Fig. F. 1.  Summary of uniaxial compressive strength of the Chatsworth Formation Sandstone. .................................................................................................................... 231  Fig. F. 2.  Summary of Chatsworth Formation Sandstone unit weight. ......................... 232  Fig. F. 3.  Direct shear testing on saw cut surfaces of Chatsworth Sandstone. ............ 233  Fig. F. 4.  Direct shear testing on natural discontinuity (clean) surfaces. ...................... 234  Fig. F. 5.  Direct shear testing results on clay-filled discontinuities. .............................. 235  xiii  Fig. F. 6.  Direct shear testing completed on Grade R1 intact rock samples. ............... 237  Fig. F. 7.  Principle stresses from triaxial tests on intact samples of the Chatsworth Sandstone – Grade R2 rock. ......................................................................................... 238   xiv  ACKNOWLEDGEMENTS  I offer sincere gratitude to my advisor; Professor Erik Eberhardt for his unwavering patience, optimism, and careful thought-provoking discussions while I completed this thesis work.  Special thanks are owed to Dr. Oldrich Hungr and Duncan Wyllie for their critical comments and suggestions while editing the thesis research chapters and to Dr. Hoek for taking the time to explain to me parts of the Hoek-Brown failure criterion.  Without the influence of Mr. Norm Norrish, I would have never had the motivation to attempt this venture.  Thanks also to Dr. Bill Gates for his mentoring; and to Kleinfelder Inc. for bearing with me during this process.   xv  DEDICATION  For Jane, Stone, and Summer  xvi  CO-AUTHORSHIP STATEMENT  This thesis is a “manuscript-based” thesis and as such; the research chapters (Chapters 2, 3 & 4) have been submitted to professional journals for review.  The professional papers were co-authored by the author, Dr. Erik Eberhardt, Dr. Oldrich Hungr, and Mr. Bruce Hilton.  Chapter 2 “Analysis of Toe Breakout Mechanisms to Develop an Improved Methodology for Assessing the Stability State of Dip Slopes” was coauthored by the author and Dr. Erik Eberhardt.  Mr. Brendan Fisher and Dr. Eberhardt identified and designed the research program.  Mr. Brendan Fisher performed the research, completed the data analyses, and prepared the manuscript.  Dr. Eberhardt reviewed the manuscript and provided editorial comments.  Chapter 3 “Assessment of Parameter Uncertainty Associated with Dip Slope Stability Analyses as a Means to Improve Site Investigations” was coauthored by the author, Dr. Erik Eberhardt, and Dr. Oldrich Hungr.  Mr. Brendan Fisher, Dr. Eberhardt, and Dr. Hungr identified and designed the research program.  Mr. Brendan Fisher performed the research, completed the data analyses, and prepared the manuscript.  Dr. Eberhardt and Dr. Hungr reviewed the manuscript and provided editorial comments.  Chapter 4 “Improved Design of Set-Back Distances for Dip Slopes in Bedded Rock” was coauthored by the author, Dr. Erik Eberhardt, and Mr. Bruce Hilton.  Mr. Brendan Fisher, Dr. Eberhardt, and Mr. Hilton identified and designed the research program.  Mr. Brendan Fisher performed the research, completed the data analyses, and prepared the manuscript.  Dr. Eberhardt and Mr. Hilton reviewed the manuscript and provided editorial comments.   1  1. INTRODUCTION 1.1. Problem Statement Between 2000 and 2005, the author was involved with a consulting project pertaining to the stability of a dip slope consisting of a weak sedimentary rock sequence.  The project site is located within the Simi Hills of Southern California.  A developer proposed building homes at the crest of the slope, for which the author was responsible for providing recommendations regarding the setback distance the developer had planned to use. Setback distance is defined here as the distance behind the crest of the slope where structures or other significant infrastructure can safely be placed.  The state of practice within Southern California dictates that a bi-planar failure mechanism is considered the primary mode of failure for a dip slope. Other modes of dip slope failure include buckling and ploughing, but these are not considered because the geology typically consists of very weak and thickly bedded sedimentary sequences. Buckling and ploughing failures are also generally shallow and therefore are not of great concern with respect to not affecting large areas of land behind the slope crest. During the course of the geotechnical investigation and design carried out by the author, it became clear that the existing procedures (contained in published guidelines) for evaluating bi-planar dip slope failures are either overly conservative or overly optimistic depending on the interpretation of the geologic conditions.  The geotechnical investigation carried out involved extensive geologic outcrop mapping, drilling, sampling, and laboratory testing which resulted in a sizeable amount of data and detailed characterization of the rock mass.  During the course of the project, the author realized that:  1. There was a general lack of published case histories providing precedent for this problem.  2. Geotechnical engineers involved in these types of projects in Southern California acknowledge the inadequacies of the analysis methods proposed by the regulatory agencies, but conform to the existing methods out of convenience. 2    3. The perceived risk associated with the project was thought to reside with the regulatory agency and therefore the author and developer were left with little recourse but to submit to the published guidelines even though the geologic conditions at the site did not conform to the assumptions upon which the guidelines are based.  This was an extremely frustrating situation for the regulatory agency, the developer, and the author. In the end, the regulatory agency reluctantly agreed to use an engineering geologic model modified from those typically used because of the abundance of geological information that was collected for the project. The model was a compromise between what the guidelines specified to be used and that which the author thought was more appropriate. Questions still remained as to how appropriate that geological model was, how to accurately establish the stability state of dip slopes, and how the notion of risk plays into decisions that are made regarding safe setback distances.  1.2. Thesis Objectives The primary objectives of this thesis are three fold. The first is to further our understanding of the bi-planar failure mechanism of dip slopes and the proper geologic models that should be used to establish dip slope stability.  Bi-planar failures are not well defined or understood. They also result in deeper failures than other modes of dip slope failure (i.e. ploughing and buckling) and are therefore much more important with respect to determining setback distances behind slope crests for residential developments and infrastructure in general.  The second objective is to evaluate geological and geotechnical uncertainty associated with bi-planar dip slope failures as a means to provide recommendations regarding optimization of geotechnical investigations for quantifying a dip slope’s stability state.  In addition to identifying the failure mechanisms, understanding the influence of geological and geotechnical inputs provide a means to constrain and optimize geotechnical investigations for dip slopes. 3   The third objective is to use the findings from the analyses carried out to evaluate existing guidelines in California (the consulting project area discussed in Section 1.1) for establishing the stability state of dip slopes and to evaluate how decisions related to structural setbacks behind dip slopes are made based on perceived risk and risk sharing. In doing so, specific recommendations were made meant to improve those specific guidelines currently used by practitioners in California.  As a means for completing these objectives, the following investigations and analyses were carried out and are presented in this thesis:   The geological conditions and geometries associated with bi-planar dip slope failures, as reported in the geotechnical literature, were investigated as a means for constraining the conditions and geologic regimes most common to dip slopes failures.   Distinct element and finite element analyses were carried out to better understand and explain the failure mechanisms associated with dip slopes.  The numerical models were then used as a means for predicting the bi-planar failure mechanism and relating that to practical limit equilibrium solutions, which were necessary for calculating the slope's stability state (i.e. factor of safety).   This integration of the two analysis techniques effectively provided a means to quantify model uncertainty associated with the bi-planar failure mechanism.   Methods for quantifying the uncertainty associated with the geological and geotechnical input parameters used to evaluate a dip slopes’ stability state were investigated. The statistical input required was evaluated using Spearman Rank correlation coefficients, which allow for direct comparison of the geological and geotechnical input and their influence on dip slope stability.  This effectively led to a methodology for evaluating geotechnical uncertainty and recommendations for optimizing geotechnical investigations for the purpose of predicting slope stability; providing a means to quantify geological and geotechnical parameter uncertainty associated with dip slopes.  4   The probability of failure and risk analyses associated with dip slopes was examined.  The detailed geotechnical characterization data set collected by the author for the Southern California project was used as a basis for evaluating existing regulatory guidelines for evaluating dip slope stability to those used in Southern California.  The Southern California case study was also used to evaluate perceived risk and the practical application of using model and geotechnical uncertainty to quantify risk associated with dip slopes.  This effectively provides an unbiased means for evaluating human uncertainty and decision making associated with residential development behind dip slopes.  1.3. Structure of Thesis This thesis is structured as a manuscript-based thesis.  The first chapter includes the motivation for choosing this thesis topic, the thesis objectives, the thesis structure, and a literature review that follows in the next sub-section.  The main body of the thesis (Chapters 2, 3, and 4) presents three manuscripts submitted to refereed journals for publication, each involving distinct yet related topics addressing the objectives of this thesis as previously outlined.   As such, they are formatted as manuscripts, and for consistent formatting, the rest of the thesis is formatted in a similar manner.  Chapter 2 contains a discussion on geologic environments in which dip slope failures are encountered, and uses case histories from the geotechnical literature to help provide insights into the controlling failure mechanisms. Numerical modeling and limit equilibrium analyses are then used to correctly treat these failure mechanisms and counter issues of model uncertainty.  This manuscript has been submitted to the American Society of Civil Engineers (ASCE) Journal of Geotechnical and Geoenvironmental Engineering.  Chapter 3 builds on Chapter 2.  It is a discussion about geotechnical uncertainty associated with dip slopes, specifically model and parameter uncertainty, and provides a methodology for optimizing geotechnical investigations of dip slopes based on inclination of the dip slope being investigated.  The concept of value engineering and methods of doing so are introduced and the author discusses practical applications for dealing with 5  parameter uncertainty.  This manuscript has also been sent to the ASCE Journal of Geotechnical and Geoenvironmental Engineering.  Chapter 4 is a culmination of the findings outlined in Chapters 2 and 3 but also discusses decision making based on risk and probability of failure of dip slopes.   This chapter focuses on the Southern California dip slope case study and the highly detailed geotechnical data set collected for it.  The case history illustrates how decisions are made regarding residential developments in Southern California where unstable slopes are expected and focuses on human uncertainty and its influence on how dip slopes are evaluated in practice.  It also discusses the current guidelines available to practicing geotechnical engineers, how they are interpreted, and how they can be improved.  This manuscript has been submitted to the Journal of the Association of Environmental and Engineering Geologists.  Chapter 5 summarizes the preceding chapters, outlines the main conclusions of the thesis, and provides guidance for future research.  These chapters are then followed by a series of Appendices that contain additional background material and details pertaining to Chapters 2, 3, and 4, which were not included for the purpose of brevity.  As previously noted, a manuscript-based structure was adopted for this thesis and this required that Chapters 2, 3, and 4 be written as concisely as possible.  Appendix A provides an international database of dip slope failure case histories as reported in the geotechnical literature.  This database was used to derive empirical relationships regarding the geologic conditions present and the depth of failure associated with bi-planar failures in dip slopes.  Appendix B discusses groundwater in dip slopes and its influence on a dip slope’s failure depth.  This appendix supplements the empirical evaluation of the international database discussed in Chapter 2.  Appendix C discusses the fitting of a linear Mohr-Coulomb failure envelope to a Hoek-Brown failure envelope specifically for describing the shear strength of the rock mass comprising the toe of a dip slope.  The methodology described is specific to dip slopes and has not been documented prior within the geotechnical literature.  Appendix D provides the UDEC distinct element input codes used in the analyses contained in this thesis and discusses how those UDEC codes were incorporated into the thesis work. Appendix E describes the results of numerical models completed to understand 6  triggering mechanisms such as increases in pore water pressure and earthquake loading and the influence of these triggers on the bi-planar dip slope failure mechanism. Appendix F provides supplemental information regarding the rock mass characterization investigations completed for the Southern California case study.  1.4. Literature Review – Relevant Research to Date 1.4.1. Geotechnical Uncertainty Uncertainty in geotechnical engineering is inherent.  Morgenstern (1995) broadly subdivided these uncertainties into three categories: 1) model uncertainty, 2) parameter uncertainty, and 3) human uncertainty.  Model uncertainty relates to the geologic model that is chosen and the analysis techniques that are employed. It arises from gaps in the scientific theory that is required to make predictions on the basis of causal inference. Parameter uncertainty concerns the spatial (distance) and temporal (time) variations of parameters such as shear strength and compressibility of soils.  In rock mechanics, the rock mass shear strength is one of the key sources of parameter uncertainty, and because  it is typically estimated based on empirical procedures, parameter uncertainty extends to the associated geological inputs such as uniaxial compressive strength, rock quality designation (RQD), discontinuity roughness, geological strength index, etc. (Bieniawski, 1989, Hoek, 1998, Hoek et al., 2002).   Human uncertainty involves subjectivity, human error, etc., which in some cases may overwhelm both model and parameter uncertainty.  An example of human uncertainty as it pertains to this study would be the ignoring of rational methods of analysis because of lack of experience with such methods.  Traditionally, when faced with uncertainty, geotechnical engineers have either: 1) ignored it, 2) relied on conservative assumptions, 3) relied on the observational method, or 4) quantified uncertainty to make rational decisions (Christian, 2004). Peck (1969) stressed that even with what might be considered a fundamental understanding of the geotechnical uncertainty involved and therefore an adequate prediction of engineering performance, this does not absolve the geotechnical engineer of using the observational method to confirm the performance of the engineered structure.  7   1.4.2. Dip Slopes A dip slope is a natural or man-made rock slope (i.e. cut) that has an inclination coincident with (or similar to) a prominent discontinuity or set of discontinuities (Fig. 1.1). The prominent discontinuity provides a plane of weakness upon which, sliding may occur.  Because the slope inclination is the same as the discontinuity inclination, the discontinuity does not ‘daylight’ the slope and is therefore precluded from a typical kinematic evaluation (Hoek and Bray, 1974).  In order for failure to occur, ‘toe-breakout’ is required and therefore, the key issue to understanding dip slope failures is to understand the toe-breakout mechanism.  Fig. 1.1.  Dip slope along a highway (Photograph by E. Eberhardt).  Dip slope failures occur in both natural and engineered cut slopes, for which there are three modes of failure described in the geotechnical literature: 1) buckling, 2) ploughing, and 3) bi-planar.  The key focus in this thesis is the bi-planar failure mode, although buckling and ploughing failures are also briefly reviewed here. 8   1.4.2.1. Buckling  Buckling is a phenomenon where a thin slab of rock, presumably under high stresses, deforms elastically, bends, yields and finally breaks.  Most reported cases involve mine slope failures in British Columbia and the United Kingdom, with those involving natural slopes receiving less attention.  Studies of buckling failures in coal mines suggest that most failures involve moderate to steeply dipping slopes (Brawner et al., 1971; Hoek and Bray, 1974; Hawley et al., 1986; Norrish and Wyllie, 1997). Brawner (1997) describes one such failure within the Line Creek Coal Mine in British Columbia where thinly bedded shale within a 35 to 45 degree; 125 m tall footwall buckled causing failure of the slope.  Serr de Renobales (1987) describes a similar failure in an open cast coal mine in Central Spain where an 80 m high; 45 degree thinly bedded sedimentary slope buckled at the base of a footwall.  Both case histories illustrate the fact that within coal mines, buckling is an important consideration because the footwalls may reach significant heights and consist of thin, weak sedimentary beds.  9   Fig. 1.2.  Result of a buckling failure at the Fording-Greenhills coal mine (Photograph by E. Eberhardt).  Buckling failures of thin rock slabs are less of a concern (or hazard) for most civil engineering applications given their limited depth of failure (Xu et al. 1991; Norrish and Wyllie, 1997; Stead and Eberhardt, 1997; and Fisher and Eberhardt, 2006).Examples of buckling failures reported for natural slopes include those by  Lembo Fazio et al. (1990) and Tommasi et al. (1999) of a 1200 m tall, shallow dipping (22) sedimentary rock slope in Italy,  where a roll in the bedding (signifying a change in dip of the beds) was observed as contributing  to the buckling. Stead and Eberhardt (1997) showed that the steepening of bedding in a dip slope is an important control in the initiation of a buckling failure.    Cruden (1985), Hu and Cruden (1993), and Cruden and Hu (1996) describe another buckling failure involving a thinly bedded natural slope in the Highwood Pass, Alberta, Canada.   Watters and Inghram (1983) and Watters and Roberts (1995) report one of the few cases of buckling failure not involving bedded sedimentary rock. They 10  report on a natural slope buckling failure involving very thin granite slabs formed through glacial unloading and exfoliation within the Sierra Nevada province of North America.  In the geotechnical literature, there are two buckling scenarios often described: Euler and three-hinged buckling.  These mechanisms are shown in Fig. 1.3 for planar and convex footwall slopes. Chen et al. (1991) state that Euler buckling is appropriate for one slab of rock but in the case of natural slopes, consideration needs to be given to multiple slab failure for practical concerns. Where dip slopes encompass orthogonal discontinuity sets, three hinged buckling may be relevant.  Cavers (1981), Cavers et al. (1986), and Wang et al. (2004) describe the conditions for three-hinged buckling.  First, the slope- coincident discontinuity needs to dip more steeply than the discontinuity friction angle resulting in a driving force parallel to slope. Second, a thin slab needs to be crosscut normal to the slope-coincident discontinuity forming the dip slope.  Third, in the case of a planar slope, there needs to be an external influence generating a force normal to the slope face.  Where the slope is “rolled”, no external force is required.  The external force may be locked in tectonic stresses (Beetham et al., 1992; Stead and Eberhardt, 1997; Eberhardt and Stead, 1998), or pore pressures (Cavers, 1981; Eberhardt and Stead, 1998; Wang et al., 2004).  Seismic loading has also been suggested as an external force required for three-hinged buckling although the discontinuities normal to the slope parallel sliding surface have been shown (numerically) to absorb much of the seismic energy and make the slope more ductile than slopes that do not have the orthogonal structure (Eberhardt and Stead, 1998). The fourth condition is that lateral release of the buckling blocks must be made feasible by the presence of highly persistent joints that strike perpendicular to the slope face (i.e. lateral release joints).    Fig. 1.3.  Buckling failure mechanisms. From Stead and Eberhardt (1997). 11   Three hinged buckling is a concern in deep open pit mines where there are thin slabs of rock with orthogonal jointing.  In the case of other civil engineering projects it appears that this failure mechanism may only be a concern where facilities are located at the base of the slope or immediately behind the slope crest.  This is because the failures are usually very shallow.  Analysis of buckling failures is typically carried out using closed-form solutions or solving for force and moment equilibrium. Limit equilibrium has proven adequate for the analysis of a single non-jointed thin buckling slab.  However, most analyses of Euler buckling have been carried out using simple closed-form solutions taken from classical buckling theory (i.e. Euler's Method; Cavers, 1981). Examples include those by Kutter (1974), Watters and Ingram (1983), Pei and Tianchi (1992), Hu and Cruden (1993), Froldi and Lunardi (1995), Ulusay et al. (1995), and Hu and Kempfert (1999).  In these cases, the analysis is conducted assuming both limit equilibrium and elastic theory with the upper portion of the rock slab providing a driving force that is resolved using force equilibrium. The ‘buckling’ portion of the slab is considered elastic and pinned at both ends.  The driving force is compared to the crucial buckling load in order to estimate a safety factor. The analysis typically shows that only very thin slabs are subject to this failure mechanism (Cavers, 1981).  The three-hinged buckling method proposed by Cavers (1981) is appropriate for assessing the stability of a single thin slab of rock with orthogonal jointing.  As with Euler buckling, this analysis may be completed using limit equilibrium. Stead and Eberhardt (1997) suggest that because Cavers’ model cannot account for intact rock breakage and deformation during rotation of the blocks, numerical methods (discontinuum modeling) is more appropriate.  1.4.2.2. Ploughing Ploughing failure occurs under similar conditions as three-hinged buckling, except that the cross-cutting discontinuity needs to dip at an angle greater than 90 degrees to bedding.  Failure occurs once the active force generated by the thin rock slab above the joint overcomes the tensile properties and the shear strength of the lower passive block and discontinuity.  The lower block moves up and outward, away from the slope face 12  while the upper block drives itself underneath the lower block.  Fig. 1.4 shows schematic of ploughing failure.  Fig. 1.5 is a photograph of ploughing failure.   Fig. 1.4.  Example of a ploughing failure from Stead and Eberhardt (1997).  Observations within the coal mines of British Columbia suggest that ploughing failures are limited to rock slabs less than 5 m thick and occur in the upper 40 to 60% of the slope height (Dawson et al., 1993). This failure type is unlikely to occur at the slope toe because the lower block is ‘locked’ into place by the confinement created by the pit floor and overburden (Stead and Eberhardt, 1997).  Limit equilibrium methods to evaluate ploughing failures were proposed by Hawley et al. (1986).  Stead and Eberhardt (1997) and Eberhardt and Stead (1998) suggest that discontinuum-based numerical methods like the distinct-element approach are a more appropriate tool to facilitate the analysis, and that this failure type may contribute to more complicated modes of failure involving sliding and bending. 13    Fig. 1.5.  Photograph showing ploughing failure of thinly-bedded sedimentary rock (Photograph by E. Eberhardt).  14   1.4.3. Bi-Planar Bi-planar failures have been reported widely within the geotechnical literature, although detailed case histories where forward or back analyses have been completed with certainty are rare.  This is because there is little precedence and lack of published and suitable analysis methods outside of simplified closed-form and limit-equilibrium solutions.  With advances in desktop computing and numerical modeling programs, this failure type may be more readily analyzed (Stead and Eberhardt, 1997).  Brawner et al. (1971), Walton and Atkinson (1978), Walton and Coates (1980), Hawley et al. (1986), Stead and Eberhardt (1997), Eberhardt and Stead (1998), and Wang et al. (2004), each provide detailed theoretical accounts of different types of bi-planar slope failure.  A bi-planar slope failure consists of an active and a passive block, and therefore, has been referred to as an active-passive slope failure.  This is shown in Fig. 1.6.  The active portion of the slide is bounded by a surface, for example bedding, that is coincident with the slope face.  The active block moves down slope creating a ‘bearing load’ on the more stable passive block, leading to rock mass damage, yield and failure between the active and passive blocks.  The bearing failure portion of the slide is termed the Prandtl zone and is the transition zone between the active and passive sliding blocks.  The Prandtl zone allows the force generated by the active block to be transmitted to the passive block below and is deformation controlled (Reik and Teutsch, 1976).  Prior to failure, where there is a joint dipping into the slope acting as an internal sliding surface, the passive block typically moves up and outward resulting in a unique slope morphology (Alfonsi et al., 2004). This is similar to that experienced for a ploughing mode of failure.  15   Fig. 1.6.  Active-passive failures surface from Kvapil and Clews (1971) and after Stead et al., 2006.  It appears that describing the shear strength of the Prandtl zone and toe breakout along the base of the passive zone is difficult for most slope geometries and geomechanical conditions (Hoek and Palmieri, 2000). As summarized by Stead and Eberhardt (1997), toe breakout at the lower portion of the bi-planar failure may be described as three major types.  These are low angle thrusts consisting of any type of discontinuity that ‘daylights’ at a downward angle out of the slope face, low angle discontinuities that dip away from the excavation, and rock mass failure at the toe. 1.4.4. Toe Failure Mechanisms for Bi-Planar Failures A review of case histories reported in the geotechnical literature suggest that the bi- planar failure mode can be divided based on the toe failure mechanism, which interestingly, often differs between mine and natural slope cases. Furthermore, it is evident that adverse structure at the toe (e.g. a fault) is responsible for more reported slope failures in mines than natural slopes.  These failures tend to occur rapidly when structure is daylighted. Step path failures and progressive failures occur much less rapidly and are more prevalent in natural slopes. The different toe failure mechanisms reported for bi-planar slope failures are listed below.  1.4.4.1. Structure Dipping Out of Slope  In order for this type of bi-planar slope failure to be kinematically feasible (as shown in Fig. 1.7), the following three conditions must be met (Nathanail, 1996).  First, the lower 16  edge of the active wedge must dip more steeply than the lower edge of the passive wedge.  Second, the lower edge of the passive wedge must daylight the slope but dip less steeply than the friction angle.  Thirdly, there needs to be a planar feature separating the active and passive wedge.  This feature must dip steeply back into the slope. Brawner et al. (1971) noted several cases of bi-planar slope failures where bedding or thrust planes dipped out of the slope at low angles within the coal mines of British Columbia.  He derived stability charts for footwall design utilizing limit equilibrium techniques with the assumption that the rock slabs involved act as rigid bodies and that sliding is translational.  Kvapil and Clews (1971) describe bi-planar failures and state that within the Prandtl (or transition zone) the rock mass is severely fractured (rock mass failure).   Fig. 1.7.  Kinematic bi-planar slope failure analysis according to Nathanail, 1996.  Where low angle joints are 15 or less to bedding, tensile failure and deformation between the active and passive block allows sliding to be kinematically feasible (Eberhardt and Stead, 1998).  In cases where the dip of the daylighting discontinuity is steep compared to bedding (approaching 30), more resistance is provided by the passive block and the failure surface develops more pronounced active, Prandtl, and passive zones (Eberhardt and Stead, 1998).  Where the daylighting discontinuity is Fault (Internal Shear Surface) 17  greater than 30 to bedding, the discontinuity may be actually dipping back into the slope.  Those cases are described below.  1.4.4.2. Structure Dipping Into Slope This type of bi-planar failure appears very unlikely, although case histories describing these failures were found in the geotechnical literature.  Stead and Eberhardt (1997) state that this failure mode is possible in high, steep slopes where the slope-coincident discontinuity is clay filled, thereby providing a substantial driving force on the lower active block.  Hughes and Clarke (2002) describe a bi-planar failure within the Durham Coalfield of Northeast England where sliding occurred along a normal fault coincident with the coal mine slope face.  The linear portion at the toe was inclined into the slope face so that movement occurred upward and out of the slope toe.  Calder and Blackwell (1980) describe a bi-planar failure that was facilitated by a low angle discontinuity dipping back into the slope.  The slope was monitored and movements of the passive block suggested that it was moving up and away from the toe.  Phienwej and Conviravong (1996) report four similar slope failures at the Mae Hoh Open Cast Mine in Northern Thailand between 1984 and 1993.  1.4.4.3. Step path and rock mass toe breakout Hungr and Evans (2004) describe toe breakout through intact rock as the most catastrophic failure mechanisms of massive natural rock slopes.  Steeply inclined and thinly bedded sedimentary rocks or foliated metamorphic rock slopes are most susceptible.  Eberhardt et al. (2005) provide a back analysis of a large natural dip slope failure within interbedded sedimentary units consisting of marl, sandstone, and conglomerate.  In this case, toe breakout was attributed to strength degradation of the marl through weathering and occurred in step-path fashion along different weathering horizons within the marl rock mass.  Ping and Zhang (1992) provide recommendations for a forward analysis of a dip slope failure where shearing was speculated to occur along clay-filled bedding and through intact limestone at the slope toe.  They suggest that the most critical failure surface though the intact rock would be at a downward inclination out of the slope.  Stacey et al. (1990) provide a back analysis of a slab failure at the Westar Greenhills Mine near 18  Elkford, British Columbia, where toe breakout occurred in a downward direction through weak intact rock.  Goodman and Kieffer (2000) note that rock mass failure through the toe of a dip slope may occur because of progressive failure of intact rock as the result of the initiation of fracturing and breakage of rock bridges allowing the rupture surface to daylight.  Hoek et al. (2000) describe a similar mechanism for toe breakout in dip slopes (see Fig. 1.8). Using model tests, Mueller and Hofmann (1970) also found that toe failure within a weak, bedded dip slope occurs as a combination of shearing along joints and tensile fracturing through intact rock.  The inclination of the toe breakout surface is similar to that of a passive wedge, and inter-block translational shearing is required between the active and passive wedge to facilitate movement. Similarly, Alfonsi et al. (2004) state that toe breakout may occur at the toe via a “curve linear” or “broken line” step-path surface nearly normal to the slope-coincident joint.  Phienwej and Conviravong (1996) report two such rock mass dip slope failures within the Mae Hoh Open Cast Lignite Mine in Northern Thailand between 1984 and 1993.   Fig. 1.8.  Complex slope failure presented by Hoek et al., 2000 showing rock mass failure in the Prandtl zone and step path failure that facilitates toe breakout.  A detailed and theoretically correct limit equilibrium solution for analyzing step-path failures was presented by Jennings (1970).  Jennings’ solution focuses on sliding that is 19  controlled by one or two non-continuous joint sets (Fig. 1.9).  He states that composite shear strength may be established by a weighted average of joint shear strength and intact rock tensile strength.  Failure occurs at an “average” angle based on the joint or joints’ dip.  Although this is a rigorous solution, its’ practicality has been questioned owing to the difficulty in properly estimating the extent of intact rock bridges within the rock mass (Hoek and Bray, 1974).  Fig. 1.9.  Rock slope with a discontinuous potential plane of weakness.  Jennings (1970) suggests using a weighted average of discontinuity and intact rock shear strength to estimate the overall “weighted” average shear strength of the plane.  Others have taken a less theoretically correct approach to describing step-path failure as a toe breakout mechanism; it is more convenient to use a “smeared” strength for this portion of the failure surface (Hoek et al., 2000).  For these cases, the Hoek-Brown Failure Criterion is sometimes employed.  Using the Hoek-Brown failure criterion to estimate the rock mass strength in the Prandtl zone appears appropriate, but it is unclear whether it is appropriate for step path failures at the toe.  Giani (1992) describes a back analysis of a dip slope failure within a gypsum mine where the slope was over steepened.  The constructed slope was 50 m high and coincident with the dip of a gypsum layer being mined.    There was a claystone layer beneath the gypsum also coincident with the slope. Movement was first initiated in 1982 20  triggered by heavy rains.    Failure occurred in a bi-planar fashion with sliding at the gypsum/claystone contact and the failure surface daylighted at the slope toe and extended through the gypsum layer.  In 1990, a second failure occurred also because of heavy rain.  Toe failure daylighted in an upward fashion approximately 25 m in front of the slope toe.  Therefore, the projection of the toe failure was down and into the mine floor.  Back analysis and stabilization design was facilitated using the Hoek-Brown failure criterion.  Hoek (2000) took a similar approach to back analyze a 100 m slope failure within a dip slope coincident with a fault; the active and passive failure surfaces were characterized using the Hoek-Brown Failure Criterion (Fig. 1.10).  Fig. 1.10.  Slope back analysis presented by Hoek, 2000.  The Hoek Brown Failure Criterion was proposed to estimate the shear strength of the weak rock mass within the active and passive portions of the sliding mass.  1.4.4.4. Progressive failure mechanisms Many of the slope failures reported in the literature include toe break failures where the first signs of failure consisted of deformation, and/or buckling of the slope at the toe. Rock mass yield, buckling and shear failure appears to accommodate kinematic movement where prominent and highly persistent daylighting structures are not present.  Wang et al. (1992) discuss what they refer to as a “sliding bending” type dip slope case history where failure is attributed to sliding along a slope coincident bedding plane, with toe breakout occurring across the bedding at the slope toe.  At the time the authors were reporting their case history, located along the banks of the Yalong River in the Jinlong 21  Mountains, China, the slide was showing signs of slow deformation through buckling. They suggest that because there is considerable stress buildup at the toe of the slide, buckling deformations may occur for an extended period with stress release that is rapid (i.e. brittle) once released.  This appears to be a case of rock mass failure brought about by initial bending or buckling of the rock where failure is clearly deformation controlled. Wang et al. (1992) also discusses dip slope failures where toe breakout is across bedding and preceded by displacement and bending at the slope toe.  They propose that a thin plate bending theory may be used to explain toe buckling but do not provide guidance on assessing the rock mass strength.  Cruden and Masoumzadeh (1987) describe slope monitoring and failure prediction at the Luscar Mine in Alberta, Canada.  They describe a slope failure where the primary sliding took place along a siltstone/shale contact.  It appears that buckling at a roll in the slope may have occurred prior to rock mass failure across bedding.  The roll likely initiated a buckling failure that in turn provided a release point and subsequent rock mass failure back in the slope.  Xu et al. (1991) and Li et al. (1992) describe base friction model testing of dip slope failures over a 50-month period.  The model was constructed to represent bedded strata that were inclined past the residual bedding friction angle.  They observed progressive buckling and then interbed slippage.  The deformations advanced as the outer layers of the model began to buckle and plough.  This was followed by the development of a shear band across bedding and finally rapid failure.    Chen et al. (1992) studied numerous landslides along the Yangtze River in China and found that progressive failure is prominent within dip slopes composed of weak sedimentary units that dip greater than 20 degrees and range in height from ten to 80 m.  Beetham et al. (1992) describe progressive toe bending of steep slopes in Central Otago, New Zealand which appear to present similar mechanisms as those described by Chen et al. (1992).  Dawson et al. (1993) describes a bi-planar failure within the Upper East Pit footwall of the Smoky River Coal Mine, in Grande Cache, Alberta.  He completed the back analysis of the slope failure and rock anchor design using limit equilibrium.  He noted that the failure was likely much more complex than modeled and was due to a combination of ploughing and rock mass failure at the slope toe. 22   1.5. References Alfonsi, P, J Durville, and P. Potherat (2004) "The morphology of deep-seated slope deformations: simple explanations or sophisticated interpretations?" In: Lacerda, Ehrlich, Fontoura, and Sayao, Eds., Landslides: Evaluation and Stabilization: Taylor & Francis Group, London, pp 1111 - 1117.  Beetham, R. D., K. E. Moody, D. A. Fergusson, D. N. Jennings, and P. J. Waugh (1992) "Landslide development in schist by toe buckling; Landslides; proceedings of the sixth international symposium," Sixth international symposium on Landslides, Christchurch, New Zealand, Feb. 10-14, 1992, Vol. 6, pp 17-23.  Bieniawski, Z. T. (1989) Engineering rock mass classifications: a complete manual for engineers and geologists in mining, civil, and petroleum engineering, New York: Wiley. Brawner, C.O. (1997) "Rock mechanics engineering in Canadian surface mining," Geotechnical News, Vol. 15, No. 4, pp 110 - 116.  Brawner, C.O., D.L. Pentz, and J.C. Sharp (1971) "Stability studies of a footwall slope in layered coal deposits." Proceedings of the 13th US Symposium on Rock Mechanics, p 329 to 365.  Calder, P. N., and G. H. Blackwell (1980) "Investigation of a complex rock slope displacement at Brenda Mines," CIM Bulletin, Vol. 73, No. 820, pp 73-82. Cavers, D.S. 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(1970) "A Mathematical Theory for the Calculation of the Stability of slopes in Open Cast mines," In: Anonymous, Ed., Planning open pit mines. South Africa (ZAF): A. A. Balkema, Cape Town, pp 87-102.  Mencl, V. (1966) "Mechanics of landslides with non-circular slip surfaces with special reference to the Vaiont slide." Geotechnique, Vol. 16, No. 4, pp 329-337.  Kvapil, R., and K.M. Clews (1971) "An examination of the Prantl mechanism in large- dimension slope failures," Transactions of the Institution of Mining & Metallurgy, Section A, pp A1-A5.  Kvapl, R. Clews, M. (1979) "An examination of the Prandtl mechanism in large slope failures," Transactions of the Institution of Mining & Metallurgy, Section A, Vol. 88, pp 1 - 5.  Lembo Fazio, A., R. Ribacchi, and P. Tommasi (1990) "Sheared bedding joints in rock engineering; two case histories in Italy; Rock joints," In: Nick Barton, and Ove Stephansson, Eds., International symposium on Rock joints, Loen. Netherlands (NLD): A. A. Balkema, Rotterdam, Netherlands (NLD).  26  Li, Q., and Z.Y. Zhang (1990) "Mechanism of buckling and creep-buckling failure of the bedded rock mass on the consequent slopes," 6th International Association of Engineering Geologists Congress. Amsterdam, pp 2229 - 2233.  Li, T. B., J. Xu, and L. S. Wang (1992) "Ways and methods for the physical simulation of landslide; Landslides; proceedings of the sixth international symposium," Sixth international symposium on Landslides, Christchurch, New Zealand, Feb. 10-14, 1992, Vol. 6, pp 487-491.  Morgenstern, N. R. (1995) "Managing risk in geotechnical engineering," 10th Pan- American Conference on Soil Mechanics and Foundations Engineering. Guadalajar, pp 102-126.  Mueller, Leopold, and H. Hofmann (1970) "Selection, Compilation and Assessment of Geological Data for the Slope Problem," In: Anonymous, Ed., Planning open pit mines. South Africa (ZAF): A. A. Balkema, Cape Town.  Nathanail, C. P. (1996) "Kinematic analysis of active/passive wedge failure using stereographic projection," International Journal of Rock Mechanics and Mining Sciences, Vol. 33, No. 4, pp 405-407.  Norrish, N. I., and Duncan C. Wyllie (1997) "Rock Slope Stability Analysis," In: K. Turner, and R. Schluster, Eds., Landslides Investigation and Mitigation. Washington D.C.: Transportation Research Board, pp 391-425.  Peck, Ralph B. (1969) "Advantages and limitations of the observational method in applied soil mechanics," Geotechnique, Vol. 19, No. 2, pp 171-187.  Pei, L., and L Tianchi (1992) "The sigma-3 effect in the formation of rockslides in Erosion, Debris Flows, and Environment in Mountain Regions," The Chengdn Symposium: IAHS, pp 253 - 261.  27  Phienwej, Noppadol, and Sajjad Anwar (2005) "Rock mass characterization for the underground cavern design of Khiritharn pumped storage scheme," Geotechnical and Geological Engineering, Vol. 23, No. 2, pp 175-197.  Ping, T, and D. Zhang (1992) "Stability research and treatment of high and steep open- pit slope," Sixth international symposium on Landslides. Christchurch, New Zealand, pp 1631-1634.  Reik, G., and Chr Teutsch (1976) "Use of equivalent models in slope stability investigation” International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, Vol. 13, No. 12, pp 321-330.  Serra de Renobales, T. (1987) "Strata buckling in footwall slopes in coal mining," Congress of International Society of Rock Mechanics. Montreal, pp 527 - 531.  Stacey, P.F., B.  Bjornson, and D. Parsons (1990) "Mining implications of footwall movement at the Westar Greenhills operation," Mine Planning and Equipment Selection. Calgary, pp 411 - 412.  Stead, D., E. Eberhardt, and J. S. Coggan (2006) "Developments in the characterization of complex rock slope deformation and failure using numerical modeling techniques," Engineering Geology, Vol. 83, No. 1-3, pp 217-235.  Stead, Douglas, and Erik Eberhardt (1997) "Developments in the analysis of footwall slopes in surface coal mining," Engineering Geology, Vol. 46, No. 1, pp 41-61.  Tommasi, P., E. Pettinelli, R. Ribacchi, A. Campedel, and L. Veronese (1999) "Instability phenomena on a high dip slope in layered limestones with clayey-marly interbeds, Lavini di Marco, Italy," In: G. Vonille, and P. Berest, Eds., 9th International Congress on Rock Mechanics. Paris: Balkema, pp 139 - 144.  Ulusay, R., O. Aydan, M. Karaca, and A. Ersen (1995) "Buckling failure at an open-pit coal mine and its back analysis." Tokyo, Japan: A.A. Balkema, pp 451 - 454.  28  Walton, G., and T. Atkinson (1978) "Some geotechnical considerations in the planning of surface coal mines," Transactions of the Institution of Mining & Metallurgy, Section A, Vol. 87, pp 147 - 171.  Walton, G., and H. Coates (1980) "Some footwall failure modes in South Wales opencast workings," 2nd International Conference on Ground Movements and Structures. Cardiff, pp 435-435-449.  Wang, Fa-Wu, Ye-Ming Zhang, Zhi-Tao Huo, Tatsunori Matsumoto, and B-Lin Huang (2004) "The July 14, 2003 Qianjiangping landslide, Three Gorges Reservoir, China," Landslides, Vol. 1, pp 157-162.  Wang, Lansheng, Zhuoyuan Zhang, Mindong Cheng, Jin Xu, Tianbin Li, and Xiaobi Dong (1992) "Suggestion on the systematical classification for slope deformation and failure; Landslides--Glissements de terrain," Sixth international symposium on Landslides, Christchurch, New Zealand, Feb. 10-14, 1992, Vol. 6, Vol. 3, pp 1869-1877.  Watters, Robert J., and Brent J. Inghram (1983) "Buckling failure of granite slabs in natural rock slopes as an indications of high residual stresses ". Boise, Idaho, USA: Idaho Transportation Dept., Div of Highways, Boise, Idaho, USA, pp 83-96.  Watters, Robert J., and K. Roberts (1995) "The Kaiser effect and it's applications to slope instability," 5th Conference on Acoustic Emission/microseismic Activity in Geologic Structures and Materials. Pennsylvania State University: Pennsylvania State University Trans Tech. Publications, pp 233 -242.  Xu, Jin, Mingdong Chen, Tianbin Li, and Langsheng Wang (1991) "Geomechanical simulation of rockmass deformation and failure on a high dip slope." Christchurch, NZ: Published by A.A. Balkema, Rotterdam, Netherlands, p 601.      29  2. ANALYSIS OF TOE BREAKOUT MECHANISMS TO DEVELOP AN IMPROVED METHODOLOGY FOR ASSESSING THE STABILITY STATE OF DIP SLOPES0F1  2.1. Introduction A dip slope is a natural or engineered/cut rock slope that has an inclination coincident with a prominent discontinuity or set of parallel persistent discontinuities, for example bedding planes in sedimentary rock. The prominent discontinuity provides a plane of weakness upon which sliding may occur. Because these discontinuities are sloping parallel and do not ‘daylight’ in the slope face, they are precluded from a straight-forward planar/wedge kinematic evaluation (e.g. Hoek and Bray 1974).  In order for failure to occur, ‘toe breakout’ is required and therefore, the key issue to understanding dip slope failures is to understand the toe-breakout mechanism.  In practice, though, the geologic conditions and toe breakout mechanism are greatly simplified to enable the use of limit equilibrium techniques. Toe breakout is assumed to occur along a daylighting, fully-persistent, cross-cutting joint or fault, with failure occurring in a bi-planar mode (Fig. 2.1). Through the limit equilibrium analysis, the location and inclination of the slope parallel sliding surface and toe breakout surface are varied until reaching a lower bound estimate of the stability state. Issues may arise, however, because geologic conditions seldom lend themselves to this simplified model as toe breakout occurs in a complex manner involving plastic yielding of intact rock (in weak rock), fracturing of rock bridges, sliding along joints, and deformation of the rock mass to facilitate kinematic release.   1 A version of this chapter has been submitted for publication.  Fisher, B.R., and Eberhardt, E. (2009) Analysis of toe breakout mechanisms to develop an improved methodology for assessing the stability state of dip slopes. In review. 30    Fig. 2.1.  Simplified dip slope showing bi-planar failure mechanism.  Therefore, more robust avenues including finite element and discrete element numerical models may be required to better represent the geologic structure, rock mass strain softening, and corresponding toe breakout mechanisms that exist for natural, civil, and mining slope failures. 2.2. Scope of the Problem A review of the geotechnical literature suggests that dip slope failures are most prevalent in weaker, bedded sedimentary rock masses (also referred to as flysch or molasse sequences).  Table 2.1 shows the dip slope failures that were reviewed for this paper. The failure depths to slope height ratios are reported in the table (D/H ratio) and suggest that dip slope failure depths are generally less than one-third the slope height measured normal to the slope face.  The D/H ratios range from about 0.10 to 0.35 with a general increase in the failure depth with a decrease in the slope angle.   For most of the case histories reported in Table 2.1, the slope parallel sliding surface is nearly planar but the slope faces undulate.  Therefore, for a given slope or cut there is a range in D/H ratios. As shown in Table 2.1, there is a nearly equal representation of natural and mining slopes presented in the geologic literature and most of the case histories are within sedimentary lithologies. This is not to say that the stability of dip slopes is not a concern in civil applications.  Design of slopes for civil projects generally involves high safety factors and therefore, dip slope failures are not common. 31  Table 2.1.  Dip slope failure case histories reviewed for this paper. Slope Number & Location (Numbers coincide with Fig. 2.2) Geology Slope Type Angle, deg Height, m Min. Depth, m Max. Depth, m Min. D/H ratio Max. D/H ratio Trigger Reference 1 - Nichanga Open Pit, Zambia Sedimentary Mine 30 184 10 20 0.05 0.11 1,2 Terbugge and Hanif, 1981 British Columbia Sedimentary Mine 40      1 Hawley et al., 1986 2 - Luscar Mine Alberta, Canada Sedimentary Mine 33 100 26 35 0.26 0.35 3 Cruden and Masoumzadeh, 1987 Sinking Creek Mtn, VA; USA Sedimentary Natural       4 Shultz and Southworth, 1989 Liujiaxia Damsite Schist Natural 45      3 Li and Zhang, 1990 3 - Bawanshan Landslide Carbonate Rock Natural 40 940 10 10 0.01 0.01 3 Li and Zhang, 1990 4 - Taipei City Sedimentary Natural 30 26 6 6 0.23 0.23 2 Chen, 1992 5 - Smokey River Coal*** Sedimentary Mine 60 100 8 15 0.08 0.15 1, 2 Dawson, 1991 Jipazi Landslide, China  Natural       3 Jin et. Al., 1992 Yangtze River, China Sedimentary Natural 55      3 Jin et. Al., 1992 Xintan, Yangtze River, China Sedimentary Natural       2 Li et al., 1992 6 - Jinlong Mountain*** Sedimentary Natural 22.5 600 100 200 0.17 0.33 3 Wang, et al., 1992 Greece Evaporate, Mine  50     2 Giani, 1992 7 - Aquas Claras Mine, Brazil*** Iron Ore Mine 44 240 30 30 0.13 0.13 1 Behrens da Franca, 1997 8 - Fuxin, China Sedimentary Mine 19 150 35 40 0.23 0.27 2,3 Yang et. al., 2004 9 - South Australia Sedimentary Mine 30 60 8 10 0.13 0.17 1 Lucas, 2006 10 - NE Switzerland Sedimentary Natural 20 56 5 5 0.09 0.09 2, 5 Eberhardt et al., 2005 11 - Portilla, Chile Volcanic Natural 50 930 100 150 0.11 0.09 4 Welkner, 2008 Notes: ***Back Analysis Completed for this Paper 1. Mining 2. Increase in water pressure 3. Long-term creep and buckling at toe 4. Seismic acceleration 5. Weathering  32  Natural slope failures have been reported to develop as deep as 100 to 200 m behind the crest of the slope (Wang et al. 1992) making these failure types relevant in the context of residential or infrastructure development above naturally over-steepened slopes.  The current methods for describing the stability of dip slopes are not well- developed and failures of these types may carry large economic impacts (Hoek et al. 2000).  Therefore, improving the analysis methods available so that practitioners may arrive at more accurate predictions regarding slope stability and setback distances appears well merited. (The setback distance is the distance behind the crest of a slope for which development is deemed safe).  Fig. 2.2 graphically shows the D/H ratios of those case histories included in Table 2.1. There are three lines drawn on Fig. 2.2 to represent the general trend in the data showing a decrease in the depth of dip slope failures with an increase in the slope angle. The two “Best-Fit” lines are regression curves that were assigned to the data based on spreadsheet algorithms.  The best fit lines show the “average” fit of the data but do not effectively bracket the maximum failure depths.   Fig. 2.2.  Dip slope failure D/H ratios reported in the geotechnical literature.   33  Undoubtedly, the case histories reported in the literature do not represent all of the slope failures that have occurred, and therefore, based on the authors judgment, the author has included a “Proposed Upper Bound” line to bracket what is believed to be a reasonable upper limit for the D/H ratios for dip slopes.  Bracketing the upper bound failure depth has practical implications that will be discussed further in subsequent sections.  In addition to the compilation of D/H results presented in Fig. 2.2, an attempt was made to correlate D/H ratios with water pressure.  This attempt was largely unsuccessful because of the lack of adequate information regarding water pressures in the case histories reported in Table 2.1.  See Appendix B for a summary of D/H ratios related to water pressures in dip slopes.  This paper presents an improved methodology for analyzing and assessing the hazard posed by a dip slope, which incorporates finite element and discrete element, with traditional limit equilibrium analyses.  The proposed methodology is then demonstrated by means of two case histories involving well documented dip slope failures.  2.3. Geological Environments 2.3.1. Orthogonally-Jointed Sedimentary Slopes The most common dip slope failures reported in the geotechnical literature involve slopes composed of sedimentary sequences sometimes referred to as flysch or molasse (Table 2.1).  Wyllie and Mah (2004) provide a detailed account of the consolidation and jointing of sedimentary rock, which is useful for understating the geologic conditions (e.g. development of jointing) that provide avenues for the failure mechanisms associated with dip slopes.  In summary, because clastic sedimentary units are deposited in horizontal layers, the initial major principal stress (1) felt by the rock is in the vertical direction while the minor principal stress (3) develops horizontally. During consolidation and induration of the rock mass, slip occurs along the depositional contacts (i.e. bedding) and because the principal stresses are perpendicular, two sets of orthogonal joints form perpendicular to bedding.  Tensile stresses are not transmitted across 34  bedding during the jointing process and therefore, many sedimentary rocks have three orthogonal joint sets: bedding accompanied by two sets of joints truncated at bedding.  Where sedimentary rocks compose dip slopes, the critical slopes are those with bedding parallel to the slope face because bedding is the most persistent discontinuity set in the rock mass.   During uplift and erosion, the rock may be folded or faulted because of tectonic stresses during mountain building and dip slopes may form on the distal edges of anticlines, synclines, or homoclines where the rock becomes tilted such that bedding is parallel to the slope face. 2.3.2. Faulted/Sheared Slopes Although less common, dip slopes may also form where faults, shears, or lithologic boundaries occur such that they form the slope parallel sliding surface.  Conceivably, these slopes may occur in numerous geologic regimes and there may be many different explanations regarding the slope genesis.  If faults behind slopes are pseudo randomly oriented, then it is more likely that a fault would daylight the slope face; for instance during excavation of a pit slope. Where this occurs, the problem reduces to one that can be treated more simply.  In comparison, it would seem to be less probable to encounter a fault, or shear whose strike and dip happens to be parallel or sub-parallel to the slope face. More commonly, dip slopes may be excavated in situations where a mining plan requires that pit slopes be parallel to lithologic boundaries.  A good example of this is presented by Behrens da Franca, P. R. (1997).  He completed a detailed back analysis of a mine slope that was excavated exposing a soft hematite layer with a deposit of itabirite located approximately 30m behind and parallel to the slope face.  At the contact of the hematite and itabirite, there was a thin layer of soft itabirite or leached iron formation which acted as the slope parallel sliding surface during the slope failure.  35  2.4. Toe Breakout/Internal Shearing 2.4.1. Active/Passive Controls Kvapil and Clews (1979) describe dip slope bi-planar failures, where the presence of a Prandtl wedge (or transition zone) develops between the active and passive zones of the slide mass. This transition zone is characterized by severe fracturing and secondary shearing of the rock mass as the forces are transmitted from the active to passive block. The rock mass is literally “squeezed” between the active and passive sections, with large transverse displacements (or internal shearing) and bulging of the rock mass being observed within the transition zone.  In contrast, very little rock mass deformation occurs in the active zone; the majority of the deformation is concentrated along the slope- parallel sliding surface that serves as the release surface in the upper part of the slope. Likewise, there is minimal disturbance of the rock mass within the passive zone, although more than in the active zone.   Fig. 2.3 shows the complex process required for bi-planar slope failure.  Fig. 2.3.  Active-passive rock slope failure model (after Kvapil & Clews 1979). Prandtl Wedge 36  Clearly, the rock mass strength within the Prandtl wedge, and at the slope toe, largely dictates the amount of movement and ultimately the stability of the rock slope. The strength along the rupture surface in the upper slope is also important, as it controls the amount of driving force transmitted to the Prandtl wedge and passive zone below.  2.4.2. Persistent Structure The simplest toe breakout and internal shearing mechanism would involve sliding along the slope parallel sliding surface with a persistent joint dipping out of the slope and a joint (or other persistent discontinuity) dipping steeply into the slope separating the active/passive wedges and providing kinematic release of the slope (Nathanail 1996). The slope and structural geology model required for this type of failure is shown Fig. 2.1 where it could be envisioned that the internal shearing and toe breakout surface both consist of through-going discontinuities.  This model illustrates that in addition to the slope parallel sliding surface, internal shearing is required for slope release.  2.4.3. Step-Path Failures A step-path (or rock mass) failure is one where toe break out develops through the combined sliding along non-persistent joints and through intact rock bridges. Jennings (1970) presented a detailed and theoretically correct model for estimating the stability of a slope where step-path failure may occur.  The model effectively uses a weighted average of the shear strength of the joints and the shear strength (or tensile strength) of the intact rock through which the sliding surface develops.  More recently, the Hoek- Brown failure criterion presents a means to weight the combined influence of intact rock and non-persistent joint strength as an equivalent continuum rock mass strength (Hoek and Brown 1997; Carvalho et al. 2007). Giani (1992) provides a detailed account of a dip slope back analysis where the toe breakout was modeled using the Hoek-Brown failure criterion.  It should be noted though, that the treatment of the step-path problem as an equivalent continuum neglects the important kinematic and directional controls that exist where the discontinuities are of medium persistence. In such cases, the equivalent continuum 37  approach may only apply to the rock mass strength of the rock bridges to account for smaller-scale discontinuities that serve to weaken the rock bridge.  2.4.4. Sliding-Bending Model A sliding-bending dip slope failure is characterized by “buckling” of the outermost layers that is present at the toe of the slope as the toe breakout surface and internal shearing develops.  Buckling prior to kinematic release in layered sedimentary rock has been reported along the Yalong River, China (Wang et al. 1992), Yangtze River, China (Li et al. 1992), and within various coal mines in Canada and the UK (e.g. Scoble 1981; Cruden and Masoumzadeh 1987; Stead and Eberhardt 1997).  Jin et al. (1992) and Li et al. (1992) both provide details of a friction model testing of these bi-planar slope failures over a 50-month period.  The model was constructed to represent bedded strata in which “bedding” was inclined past the residual friction angle.  They observed progressive failure of the model slope with the first indications of failure being buckling of the outermost “layers”.  The deformations advanced further behind the slope face followed by bi-planar shearing across bedding.    This model suggests that one of the first indications of bi-planar slope failure may be bulging of the slope toe.  2.5. Treatment of Toe Breakout and Internal Shearing 2.5.1. Limit Equilibrium Methods Limit Equilibrium represents the most common method for estimating the stability of dip slopes where bi-planar failures are considered.  Brawner et al. (1971) and Stimpson and Robinson (1982) provide methods where the toe breakout surface is varied while the internal shearing is assumed normal to the slope face.  This failure mechanism is the same as shown in Fig. 2.1 with the internal shear assumed normal to bedding.  No consideration is given to the strength of the internal shear.  Others, such as Hawley (1983), Olauson (1984), and Hawley et al. (1986), provide similar solutions where the internal shear is normal to the slope face and shear strength parameters are assigned. Clearly the assumptions used to generate these solutions apply to very specific geological conditions.  38  In some cases, acceptable practice guidelines (e.g. SCEC 2002) also allow limit equilibrium “method of slices” approaches derived for circular or curvilinear shear surfaces to be used for estimating the stability of dip slopes (e.g. Bishop Modified, Janbu Generalized, Spencer, etc.; see Wyllie and Mah 2004 for descriptions).  However, Fisher and Eberhardt (2007) have shown that these methods are not reliable because they do not properly incorporate the influence of the location, orientation, and shear strength of the interslice shear surface when estimating dip slope stability. A suitable alternative here is Sarma’s (1979) method, as it allows for inclined slices to be considered in the “method of slices” solution. Still, the force balance based approach adopted in the form of a limit equilibrium analysis inherently neglects the complex interaction of rock mass deformation, yielding and shearing in the development of a Prandtl zone between the active and passive blocks as required for kinematic release.  2.5.2. Frictional Plasticity Theory In cases where there are no clear planes of weakness forming either the toe breakout or internal shear surfaces (or both), shearing must occur through the rock mass.  As noted above, limit equilibrium methods adopt a holistic approach, which require generalizing assumptions with regards to the nature of the rupture surface and failure kinematics. To counter these deficiencies, frictional plasticity theory was examined to develop a better mechanistic understanding regarding the development of toe release and internal shear surfaces.  The term frictional plasticity theory is defined by Terzaghi (1943) as “the theory on which the computation of stresses in a state of plastic equilibrium is based”. This theory, as it pertains to geologic materials, is typically based on Mohr’s theory of rupture.  In practice, the Coulomb theory is used in conjunction with Mohr’s rupture theory (i.e. Mohr-Coulomb diagrams and strength parameters).  For ease of reference, and because geotechnical engineers are more familiar with the Mohr-Coulomb theory, this term will be used in place of the more theoretically-correct “plasticity theory” throughout the remainder of this paper.  Quick scoping calculations immediately suggested that the location of the internal shear may be described by the Mohr-Coulomb theory where expected failures are relatively shallow and the major principal stress (1) is parallel to the slope face. These 39  calculations are based on the Mohr-Coulomb failure hypothesis (Fig. 2.4) relating localization and the shear failure surface that develops () to the plane of the major principal stress (1) as a function of the friction angle of the material ().   Fig. 2.4.  Localization of shear surfaces according to Mohr-Coulomb criterion.  The practical importance of this is that because 1 within dip slopes is oriented parallel to the slope face, and therefore bedding, it becomes possible to calculate the orientation  of the shear failure surface at the toe along which toe breakout should develop. The angle  also describes the orientation of the internal shear surface that develops to 40  accommodate slippage between the active and passive blocks and facilitate sliding along the toe breakout surface.                                                                                               Eq. 2.1  It then follows that the inclination of the toe breakout surface may be described by Eq. 2.2.                                                             Eq. 2.2  Where  is the toe breakout angle and  is the slope angle, both measured from the horizontal.  Fig. 2.5 shows an orthogonally bedded dip slope with the failure surfaces that would be predicted using Mohr’s theory. It should be noted that this relationship is not affected by the presence of orthogonal cross-jointing as the slope/bedding parallel orientation of 1, i.e. perpendicular to the cross joints, means that no shear stresses develop along the cross-joints; the normal stress acting on the cross joints is equal to 1.  Fig. 2.5.  Toe breakout and internal shear surface according to Mohr’s rupture theory.  245 '  o452 '   41  Fig. 2.6 and Fig. 2.7 show examples of bi-planar rock slope failures where the toe breakout surface and internal shear correspond to those predicted by the Mohr-Coulomb theory.  The sedimentary rock mass in Fig. 2.6 is orthogonally jointed and a tension fracture can be seen that formed behind the internal shear normal to the slope coincident discontinuity.  The orthogonal joint opened in tension as the internal shear initiated and propagated through the rock.  The slope in Fig. 2.7 is a rock slope composed of metamorphic rock and similarly, it can be seen that the internal shear developed through the rock mass as a rock mass failure.  Sliding at the toe is up and outward from the slope face.  42    Fig. 2.6.  Bi-planar dip slope failure in sedimentary rock. Fig. 2.7.  Bi-planar dip slope failure in metamorphic rock. 43  2.6. Improved Dip Slope Stability State Assessment The remainder of this paper discusses detailed analyses using limit equilibrium, and numerical modeling to arrive at an improved methodology for predicting the stability state of dip slopes.  In summary, the author proposes that the key to understanding the toe breakout mechanism and predicting the stability state of dip slopes is contingent on the geological/geotechnical model, especially whether the rock mass should be treated as a continuum or discontinuum in the analysis carried out.  This decision regarding whether to use a continuum or discontinuum analysis is based on whether the analyses being performed to better understand the failure mechanism (and therefore the factors controlling failure) or can be performed quickly for a large number of parameter variations.  In other words, this decision is based on a balance between the integrated use of numerical methods and limit equilibrium. In deciding between different numerical methods to use, distinction is required between problems in which geological structure will play an important controlling role, in which case the distinct-element method should be used, or those where the rock mass is assumed to behave as an equivalent continuum, in which case a finite-element or finite-difference continuum analysis may be suitable. The details of these methods, including their different advantages and limitations, as applied to rock slope stability investigations is summarized in Stead et al. (2006).  2.6.1. Discontinuum Approach As suggested through the review of dip slope failures reported in the geotechnical literature, the most common dip slope failures occur in weak sedimentary sequences containing orthogonal joints perpendicular to and truncated at bedding.  The empirical assessment presented in Fig. 2.2 suggests that the depth to height ratio (D/H) of failed slopes ranges from about 0.1 to 0.35 with the deepest failures occurring in relatively shallow, natural dip slopes (D/H of 0.35).  The most critical slopes are those where the slope parallel sliding surface forms along continuous bedding.  Discrete- and distinct-element codes are ideal for modeling the influence of geologic structures on dip slope failure development without implicitly specifying a failure 44  mechanism a priori. The model must incorporate a sufficient (representative) joint network density to realistically simulate the kinematic controls the actual geological structure and conditions will have. This does not necessarily require a one-to-one representation between model and reality as the joint frequency may be scaled and factored into the rock mass strength used to define the block behavior in the model.  In this sense it’s important to note that distinct-element codes like UDEC (Itasca 2007) can simulate both the complex interaction of block yielding, localization and plastic shear required for toe breakout and internal shearing, as well as sliding along discontinuities.  Fig. 2.8 and Fig. 2.9 show the failure mechanism for 30 and 45 degree, orthogonally- jointed dip slopes.  Geomechanical properties used during the numerical analyses are provided in Tables 2.2 through 2.5.  The toe breakout surfaces are clearly linear (as opposed to a curvilinear or log spiral surface) and develop according to those predicted by Eg. 2.1 and Eq. 2.2.  Internal shearing develops at an angle approaching that suggested in Eq. 2.1.  Fig. 2.8 and Fig. 2.9 also show the principal stress orientations within the slopes and provide a comparison between the toe breakout mechanisms predicted using Eq. 2.1 and 2.2 assuming that 1 is parallel to the slope face and the toe breakout mechanism predicted using the discontinuum model.   In this case, because the orthogonal joints do not influence the orientation of the toe breakout or internal shear surfaces, the rock mass may be subsequently treated as a continuum. Other results may show that a step-path failure mechanism develops, in which case the continuum assumption may not be valid although the failure mechanism may be represented assuming an “equivalent continuum” (Hoek et. al, 2000).  Here though, the confirmation that the continuum assumption can be applied means that the predicted inclinations of toe breakout and internal shear derived from the Mohr- Coulomb theory can be used directly as input to control the setup of a Sarma limit equilibrium analysis. Sarma’s method (Sarma 1979; Hoek 1987) is advantageous for estimating the slope’s stability in this case because it considers the shear strength and inclination of the internal shear. 45         a) D/H ratio of 0.125.             b) D/H ratio of 0.25.  Fig. 2.8.  Failure mechanism for a 30-degree orthogonally jointed and bedded dip slope. 1) Model setup.  2) Bi-planar failure mechanism. 46           a) D/H ratio of 0.10.             b) D/H ratio of 0.20.  Fig. 2.9.  Failure mechanism for a 45-degree orthogonally jointed and bedded dip slope. 1) Model setup.  2) Bi-planar failure mechanism. 47  Table 2.2.  Input parameters to initialize equilibrium conditions for the model shown in Fig. 2.8. Unit Field Stress r1, kN/m3 Bulk2, Pa Shear3, Pa Dil4, deg. 5, deg. c6, Pa Rock Gravity 22.73 672e6 350e6 0 45 12e6  Unit kn7, Pa/m2 ks8, Pa/m2 Ten, Pa , deg. c, Pa Joints 5570e6 557e6 0 35 12e6 Slope Parallel Sliding Surface 5570e6 557e6 0 50 12e6 Notes: 1. r is the rock unit weight. 2. Bulk is the bulk modulus based on rock mass deformation modulus and Poisson’s ratio. 3. Shear is the shear modulus based on rock mass deformation modulus and Poisson’s ratio. 4. Dil is the dilation of the rock during shearing. 5.  is friction angle. 6. c is cohesion intercept. 7. kn is the joint normal stiffness based on rock mass deformation modulus and joint spacing. 8. ks is joint shear stiffness.  This is taken as 1/10 of kn. 9. Ten is the tensile strength of the discontinuities.  Table 2.3.  Input parameters for the model shown in Fig. 2.8. Unit Field Stress r1, kN/m3 Bulk2, Pa Shear3, Pa Dil4, deg. 5, deg. c6, Pa Rock Gravity 22.73 672e6 350e6 0 34 0.7e6  Unit kn7, Pa/m2 ks8, Pa/m2 Ten, Pa , deg. c, Pa Joints 5570e6 557e6 0 35 0 Slope Parallel Sliding Surface 5570e6 557e6 0 12 0 Notes: 1. r is the rock unit weight. 2. Bulk is the bulk modulus based on rock mass deformation modulus and Poisson’s ratio. 3. Shear is the shear modulus based on rock mass deformation modulus and Poisson’s ratio. 4. Dil is the dilation of the rock during shearing. 5.  is friction angle. 6. c is cohesion intercept. 7. kn is the joint normal stiffness based on rock mass deformation modulus and joint spacing. 8. ks is joint shear stiffness.  This is taken as 1/10 of kn. 9. Ten is the tensile strength of the discontinuities.        48  Table 2.4.  Input parameters to initialize conditions to equilibrium for the model shown in Fig. 2.9. Unit Field Stress r1, kN/m3 Bulk2, Pa Shear3, Pa Dil4, deg. 5, deg. c6, Pa Rock Gravity 22.73 672e6 350e6 0 45 12e6  Unit kn7, Pa/m2 ks8, Pa/m2 Ten, Pa , deg. c, Pa Joints 5570e6 557e6 0 35 12e6 Slope Parallel Sliding Surface 5570e6 557e6 0 50 12e6 Notes: 10. r is the rock unit weight. 11. Bulk is the bulk modulus based on rock mass deformation modulus and Poisson’s ratio. 12. Shear is the shear modulus based on rock mass deformation modulus and Poisson’s ratio. 13. Dil is the dilation of the rock during shearing. 14.  is friction angle. 15. c is cohesion intercept. 16. kn is the joint normal stiffness based on rock mass deformation modulus and joint spacing. 17. ks is joint shear stiffness.  This is taken as 1/10 of kn. 18. Ten is the tensile strength of the discontinuities.  Table 2.5.  Input parameters for the model shown in Fig. 2.9. Unit Field Stress r1, kN/m3 Bulk2, Pa Shear3, Pa Dil4, deg. 5, deg. c6, Pa Rock Gravity 22.73 672e6 350e6 0 34 0.7e6  Unit kn7, Pa/m2 ks8, Pa/m2 Ten, Pa , deg. c, Pa Joints 5570e6 557e6 0 35 0 Slope Parallel Sliding Surface 5570e6 557e6 0 24.5 0 Notes: 10. r is the rock unit weight. 11. Bulk is the bulk modulus based on rock mass deformation modulus and Poisson’s ratio. 12. Shear is the shear modulus based on rock mass deformation modulus and Poisson’s ratio. 13. Dil is the dilation of the rock during shearing. 14.  is friction angle. 15. c is cohesion intercept. 16. kn is the joint normal stiffness based on rock mass deformation modulus and joint spacing. 17. ks is joint shear stiffness.  This is taken as 1/10 of kn. 18. Ten is the tensile strength of the discontinuities.  This may be counter intuitive because the slope has a well-defined joint pattern, however for the reasons previously noted, in this case the geologic structure does not need to be explicitly accounted for in the stability assessment for practical purposes. 49  2.6.2. Continuum Approach Where the rock blocks within a slope are small compared to the slope’s height and there are no preferred jointing patterns, the rock mass may be treated as a continuum (Hoek and Brown 1997).  A typical dip slope in this case might involve a large excavation in an open pit mine where a fault or lithologic contact is located behind and coincident with the slope face.  As with the discontinuum approach, the continuum approach requires that the failure mechanism be confirmed first. Here, a finite element or finite difference code is ideally suited for the task.  Fig. 2.10 and Fig. 2.11 show failure mechanisms for 30 and 45 degree dip slopes, respectively.  (Tables 2.6 and 2.7 provide the input parameters used in the finite element analyses performed for Fig. 2.10 and Fig. 2.11.)  In both cases, the toe breakout surfaces are log spiral; unlike bedded dip slopes where the toe breakout surface is linear; a result of rotation of 1 in the slope.  Fig. 2.10 and 2.11 also show the failure surfaces predicted using finite element compared to those predicted using Eq. 2.1 and 2.2 with 1 assumed parallel to the slope face.  Clearly, the linear toe breakout surface assumed for the 30 degree slope does not show a good match with the actual toe breakout surface shown in the finite element code.  A linear failure surface may be appropriate for the steeper, 45 degree slope. 50  Fig. 2.10.  Failure mechanisms for a shallow dip slope treated as a continuum. 51   Fig. 2.11.  Failure mechanisms for a steep dip slope treated as a continuum. 52  Table 2.6.  Input parameters for finite element model in Fig. 2.10. Unit Field Stress r1, kN/m3 Erm2, GPa  t4, kPa 5, deg. c6, kPa Rock Gravity 20 5e4 0.4 125 25 125  Unit kn7, kPa/m ks8, kPa/m t, kPa , deg. Slope Parallel Sliding Surface 1e5 1e4 0 15 Notes: 1. r is the rock unit weight. 2. Erm is the deformation or Young’s modulus. 3. is Poisson’s ratio. 4. t is the tensile strength. 5.  is friction angle. 6. c is cohesion intercept. 7. kn is the joint normal stiffness. 8. ks is joint shear stiffness.  Table 2.7.  Input parameters for finite element model in Fig. 2.11. Unit Field Stress r1, kN/m3 Erm2, GPa  t4, kPa 5, deg. c6, kPa Rock Gravity 20 8e5 0.3 700 34 700  Unit kn7, kPa/m ks8, kPa/m t, kPa , deg. Slope Parallel Sliding Surface 1e5 1e4 0 24.5 Notes: 1. r is the rock unit weight. 2. Erm is the deformation or Young’s modulus. 3. is Poisson’s ratio. 4. t is the tensile strength. 5.  is friction angle. 6. c is cohesion intercept. 7. kn is the joint normal stiffness. 8. ks is joint shear stiffness. 2.7. Parametric Evaluation Numerical codes relate shear stresses in the models to the shear strength of the materials to arrive at an estimate of the slope’s stability state.  The ratio of the shear strength of the materials and the applied stresses is referred to as shear strength reduction (SSR) in finite element codes or factor of safety (FOS) in discrete element codes.  The shear strength of the model materials are divided by the factor.  When the ratio is less than unity, the shear strength values of the materials result in an unstable system.  When the factor is greater than unity, then the slope is deemed stable. The 53  finite element code Phase2 by RocScience and the discrete element code UDEC by Itasca both use a bracketing procedure proposed by Dawson, et al., (1999) to vary the material properties and predict the stability state of the model.  The critical stability factor in Phase2 is based on displacement of the slope, which is the primary indication of non-convergence.  UDEC uses the ratio of unbalanced forces in the system.  The remainder of this paper will refer to shear strength reduction, factor of safety, and limit equilibrium safety factor as “stability factors”.  The stability factors from limit equilibrium, finite element, and discrete element provide a basis for comparison of model results.  The author completed a parametric evaluation using 1) Sarma’s limit equilibrium approach coupled with the Mohr-Coulomb theory (for predicting the linear failure surfaces of dip slope failure mechanism), 2) Sarma’s limit equilibrium approach varying the orientations of the toe breakout surface and internal shear to “minimize” the stability factor, 3) finite element analysis and 4) discrete element analysis to compare stability factors predicted for these three common modeling techniques.  Three slope inclinations were chosen.  Those include a 30 degree, 45 degree, and 60 degree dip slopes.  The author calculated stability factors while toggling the distance of the slope parallel sliding surface from the slope face.  As stated above, the main differences in the models are that the toe breakout mechanism is specified in the limit equilibrium analysis and not the finite element or discrete element models and the stability factors for all three methods are calculated differently.  Fig. 2.12 provides a normalized summary of the parametric evaluation based on the height of the slope (H) and the depth of the slope parallel sliding surface (D; measured normal to the slope face) for 30, 45, and 60 degree slopes.  The stability factors estimated for the 30 degree slope while optimizing Sarma’s method are conservative for slopes where the failure surface to slope height (D/H) ratio is less than about 0.3.  The assumption of linear toe breakout and internal shear surface as predicted using Eq. 2.1 and 2.2 yield conservative stability factors for D/H ratios to about 0.2.  These values increase for shallower slopes.  The results for the 45 and 60 degree 54  slopes show that the limit equilibrium methods are conservative at D/H ratios below about 0.25 and 0.2, respectively.   The discrete element results are not shown for the 60 degree slope because the results were highly ambiguous.  Based on the results of the 30 and 45 degree slopes, it is reasonable to assume that the discrete element stability factors should be similar to the finite element stability factors because both models were set up to treat the rock mass as a continuum. 2.8. Discussion The toe failure mechanisms predicted using numerical models are more rigerous than those predicted with the assumption that the 1 acts parallel to the slope face at depth. Fig. 2.12 shows that it is conservative to use linear failure surfaces up to certain D/H ratios and these ratios decrease with an increase in the slope angle.  At larger D/H ratios, the principal stresses within the slope begin to rotate from parallel to the slope face and assuming linear toe breakout and internal shear surfaces will result in optimistic assessments of the slopes’ stability state.  Fig. 2.2 shows that it is unlikely that D/H ratios in “real-world” dip slopes will exceed those where limit equilibrium provides conservative estimates of dip slope stability state.  The failure mechanism in shallow slopes mimics a log spiral passive earth pressure failure.  Therefore, the finite element models predict the dip slope’s stability state more accurately than assuming linear toe breakout surfaces.  The same log-spiral surfaces were observed in steeper slopes although once again, for shallow failures, the assumption that the toe breakout surface and internal shear may be predicted using Eq. 2.1 and Eq. 2.2 is sufficiently conservative for estimating the slope’s stability where the D/H ratios are within practical limits as defined by Fig. 2.2.  In addition to the modeling presented in this chapter, additional modeling results that consider water pressure and seismic loading are presented in Appendix E.  The addition of water and seismic loading does not appear to change the dip slope failure mechanism.  The practical implications of the parametric analysis are as follows.  From published accounts of dip slope failures (Table 2.1 and Fig. 2.2) we know that D/H ratios of failed 55  slopes increase with a decrease in slope angle.  The parametric evaluation showed that limit equilibrium methods are conservative within the D/H ratios expected for 30, 45, and 60 degree slope angles in the “real-world”.  Therefore, it is appropriate to use limit equilibrium as a first approximation and for completing parametric evaluations prior to utilizing more complicated and time consuming methods such as finite element.  While completing stability assessment of dip slopes where the rock mass is treated as a continuum, it is conservative to assume that the principal stresses act parallel to the slope face and that a first estimate of the slopes stability state may be ascertained using limit equilibrium methods.  Where more precise stability evaluations and merited, finite element analysis is recommended.  2.8.1. Shear Strength Estimates of Heavily Jointed Rock Masses Procedures for estimating the shear strength of heavily jointed rock masses have been well-documented in the geotechnical literature.  Some recent and comprehensive papers include Hoek and Brown, 1997; Hoek et al., 2002 & Carvalho et al., 2007.  These papers describe the Hoek-Brown failure criterion and how it may be applied to shear strength estimates for the purpose of performing slope stability analyses.  56   Fig. 2.12.  Comparison of stability factors estimated using finite element, discrete element, the Mohr-Coulomb theory (to determine inclination of toe breakout and internal shear surfaces), and optimized Sarma methods. 57  2.8.2. Shear Strength Estimates of Orthogonally-Bedded Dip Slopes  The rock mass shear strength of a bedded dip slope is clearly anisotropic and therefore, taking into the consideration the influence of joints and bedding is difficult.  Simplifying assumptions are required to establish a logical means for establishing the shear strength of an orthogonally jointed, bedded dip slope.  First, we assume that there is a slope parallel sliding surface that has been identified behind the slope crest with a shear strength that is less than that of the typical bedding.  In this case, there are two joint sets to consider; those that consist of bedding parallel to the slope face and orthogonal joints that are assumed to truncate at bedding.  The slope parallel sliding surface is dealt with separately and it is typical to assign Mohr-Coulomb or Barton-Bandis (Barton and Bandis, 1991) shear strengths for discontinuities.  Hoek and Brown (1980), in their derivation of the Hoek-Brown failure criterion, rely heavily on Eq. 2.1 and physical testing of an orthogonally-jointed “rock mass” simulated using bricks (Ladanyi and Archambault, 1970).  In summary, Hoek (1983) states that according to the Mohr rupture theory, when the inclination of primary joints (or bedding in this case) is parallel to 1 while the secondary discontinuities (in this case, orthogonal joints truncated at bedding) are normal to 1, there is theoretically no decrease in the rock mass shear strength from that of the intact rock.  In reality, and based on actual scaled testing, the measured shear strengths are less than intact rock.  This may be attributed to rotation and crushing of individual blocks within the rock mass during the scaled tests (Hoek, 1983).  Intuitively, a decrease in the shear strength may also be attributed to defects in the intact rock itself and it appears appropriate to down-grade the rock mass strength from the intact rock strength for practical applications because the scale of the intact rock blocks existing in the slope are generally larger than those tested in the lab.  Marinos and Hoek (2001) provide recommendations regarding assigning Geological Strength Index (GSI) to orthogonally-jointed sedimentary (flysch) sequences based on the distribution of sedimentary lithologies in the rock mass and the tectonic disturbance that the rock mass has been subjected to in the past.  This coupled with the Hoek-Brown 58  Failure criterion and recommendations from Carvalho et al., 2007 provides a rational means for down grading the strength of the individual blocks of a discrete element model where shearing is expected in a direction other than along bedding or orthogonal joints truncated at bedding.  2.9. Practical Implementation – Case Histories Fig. 2.13 is a flow chart that illustrates the procedure recommended for estimating the stability state of dip slopes.  It is broken down into bedded slopes where principal stresses are parallel to the slope face at depth and randomly-jointed slopes where principal stresses are not parallel to the slope face at depth.  In both cases, the steps involve developing a geologic and geotechnical model and also estimating the shear strength of the rock mass.  The process diverges after the shear strength estimate where the bedded dip slope’s stability factor is predicted using linear failure surfaces and limit equilibrium but then checked using distinct element codes.  In the case of the randomly jointed dip slope, limit equilibrium using Sarma’s method is appropriate for parametric evaluations. Finite element analysis is required for a more accurate estimate of the toe breakout and internal shear surface and to estimate the stability factor of the slope.  These steps are illustrated in two case histories.   Fig. 2.13.  Methodology for estimating the stability state of dip slopes. 59   2.9.1. Case History – Randomly Jointed Dip Slope at Aguas Claras Mine The Aquas Claras Mine is located in Brazil at Curral Mountain southwest of Belo Horizonte.  The mine produced iron ore and there was a major dip slope failure that occurred in 1992.  Details of the mining activities, geotechnical aspects of the mine, and the failure may be found in Behrens da Franca, 1997.  Behrens da Franca completed a detailed back analysis of the failure using finite element analysis.  This is therefore a re- evaluation of the back analysis previously performed with emphasis on the failure mechanism and the application of the Mohr-Coulomb theory to predicting the failure mechanism coupled with limit equilibrium to estimate the slopes’ stability state.  The benched slope was approximately 240m high with an overall slope angle of 44 degrees.  The surface of the slope consisted of “soft” hematite with a body of itabirite located approximately 30m behind and parallel to the slope face (D/H ratio of 0.125).  At the contact of the hematite and itabirite, there was a thin layer of soft itabirite or leached iron formation.  This thin leached iron formation is believed to be the slope parallel sliding surface of the dip slope failure.  The geotechnical properties used in the re- evaluation are presented below in Table 2.8 and are based on the work completed by Behrens da Franca (1997).             60    Table 2.8.  Geotechnical properties used for back analysis of Aquas Claras mine failure. Unit Field Stress r1, kN/m3 Erm2, kPa  t4, kPa 5, deg. c6, kPa Hematite Gravity 36 7e6 0.3 70 43 70 Itabirite Gravity 30 5e6 0.25 50 36 50  Unit kn7, kPa/m ks8, kPa/m t, kPa , deg. c, kPa Leached Iron Contact 1e5 1e4 0 40 35.5 Notes: 1. r is the rock unit weight. 2. Erm is the deformation or Young’s modulus. 3. is Poisson’s ratio. 4. t is the tensile strength. 5.  is friction angle. 6. c is cohesion intercept. 7. kn is the joint normal stiffness. 8. ks is joint shear stiffness.  The finite element approach predicts a stability factor of 0.97 while Sarma’s solution coupled (using linear toe breakout and internal shear surfaces predicted Eq. 2.1 and 2.2) predicts a stability factor of 0.99.  Fig. 2.14 shows the slope geometry and the failure surfaces predicted by finite element.  This well-documented case history shows that there is minimal error in using Eq. 2.1 and Eq. 2.2 to estimate the toe breakout and internal shear inclination while utilizing Sarma’s method to estimate dip slope stability where the rock mass is treated as a continuum with D/H ratio of about 0.125. 61   Fig. 2.14.  Finite element analysis of Aquas Claras mine failure. 2.10. Case History – Bedded Dip Slope at Smoky River Coal The Smoky River Coal Company Limited operated a coal mine within the foothills of the Rock Mountains near Grand Cache Alberta.  On June 20, 1987, a 250 m long section of the upper East limb No. 9 Mine failed.   A detailed account of the mining activities and back analysis of the slope failure was presented by Smoky River Coal Limited and Piteau Associates Engineering Ltd., 1987.  Similarly, Dawson et al., 1999 provides a summary and back analysis of the slope failure.  The failed slope was about 97 m high and inclined at about 60 to 65 degrees.  The upper two-thirds of the slope was reinforced with rock dowels to prevent translational sliding of daylighted geologic structure. Drainage was provided by horizontal drains drilled back to the No. 3 coal seam which acted as the slope parallel sliding surface for the slope failure.  The No. 3 coal seam was located eight to 15 m behind the slope face and therefore the D/H ratio for the slope is 0.08 to 0.15.  A summary of the rock lithologies within the slope based on published reports by Smoky River Coal and Piteau Associates Engineering Ltd., 1987 and Dawson and Barron, 1999 is presented in Table 2.9. 62   Table 2.9. Engineering geology of the East Limb No. 9 Mine. Lithology Thickness , m ci, MPa Bedding Spacing, m Bedding ’ Cross Joint Spacing, m Joint ’ GSI Silty Sandstone 3.05 55 0.5 30 0.76 35 45-50 Mudstone 3.05 20 0.03 25 0.38 30 40-45 Bench Sandstone 6.1 60 0.3 to 0.6 30 0.68 35 50-55 Silty Shale 1.5 7 0.03 to 0.09 25 0.38 30 40-45 No. 3 Coal Seam 0.9 0.5  25  30  2.10.1. Numerical Modeling The author utilized the computer code UDEC by Itasca as a check on the failure mechanism and then computed the stability factor of the slope using the UDEC code and Sarma’s limit equilibrium approach with the toe breakout surface estimated using Eqs. 2.1 and 2.2.  The UDEC model predicted a stability factor of 1.01.  The limit equilibrium stability factor is 0.99.  Table 2.10 provides the input parameters used in the UDEC evaluation.  The Mohr-Coulomb parameters used for rock mass while completing the Sarma analysis were  of 45 degrees and a cohesion intercept of 350 kPa.  (See Appendix C for a detailed methodology for establishing Mohr-Coulomb parameters from Hoek-Brown parameters for dip slopes.)               63  Table 2.10.  Input parameters for the Smoky River Coal back analysis. Unit Field Stress r1, kN/m3 Bulk2, Pa Shear3, Pa mb 4 s5 ci, Pa Rock Mass Gravity 25 1e9 0.7e9 0.341 0.0003 60e6  Unit kn7, Pa/m ks8, Pa/m Ten9, Pa , deg. c11, Pa No. 3 Coal Seam 12.5e9 12.5e8 0 25 0 Shear 12.5e9 12.5e8 0 25 0 Notes: 1. r is the rock unit weight. 2. Bulk is the bulk modulus based on rock mass deformation modulus and Poisson’s ratio. 3. Shear is the shear modulus based on rock mass deformation modulus and Poisson’s ratio. 4. mb is an empirical parameter used with the Hoek-Brown Failure Criterion (Hoek et. al., 2002). 5. s is an empirical parameter used with the Hoek-Brown Failure Criterion (Hoek et. al., 2002). 6. ci is the uniaxial compressive strength of intact rock. 7. kn is the joint normal stiffness based on rock mass deformation modulus and joint spacing. 8. ks is joint shear stiffness.  This is taken as 1/10 of kn. 9. Ten is the tensile strength of the discontinuities. 10.  is friction angle. 11. c is cohesion intercept.   2.10.2. Monitoring Data To establish the plausibility of the treating the rock mass as a continuum, the author compared the slope deformations estimated by the UDEC model to the deformations recorded during the mining activities.  A series of five monitoring prisms were installed in the area where the cross-section was generated.  Those points were “monitored” during the numerical modeling and a comparison of the measured data to that predicted by the model is shown in Fig. 2.15. The measured displacements show three stages of movement (Dawson and Barron, 1991).  Those include 1) initial outwards movement of the slope because of slope relaxation, 2) slope dilation prior to failure, and 3) post peak shear strength movements and failure of the slope.  Inspection of Fig. 2.15  reveals that the numerical model predicted the slope movements recorded during stages two and stage three; slope dilation prior to failure and post peak shear strength movements and failure the slope failure.  This validates the bi-planar failure mechanism predicted by the UDEC model. 64   The back analysis of the dip slope failure at Smoky River Coal confirms that because the principal stresses within a bedded; orthogonally-jointed rock slope are parallel to the slope face, limit equilibrium analysis may be confidently used to estimate the stability state of the slope. 2.11. Conclusions The most common dip slope failures occur in weak; orthogonally-jointed sedimentary rock (sometimes referred to as flysch or molasse sequences).  Within orthogonally- jointed sedimentary rock, toe breakout involves sliding along joints, plastic failure of intact blocks, and intense deformation of the slope to allow may kinematic release.  Dip slope failures have been reported to exceed 200 meters in depth making this failure mechanism relevant in the context of infrastructure development behind steep slopes. Where the slope is not manmade, failure progress over many years (sliding bending model) and because the slope is bedded, the major principal stress may be assumed parallel to the slope face.  Therefore, the Mohr-Coulomb theory is a practical means to estimate the location and inclination of the internal shear and toe breakout surface.  The failure mechanisms associated with bi-planar failures where the toe breakout surface is not constrained by a prominent discontinuity that dips out of the slope are complex.   But even so, there are practical methods for assessing the slope stability. Published case histories suggest that the depth of observed dip slope failures increases with a decrease in the slope inclination.  In all cases, the failure mechanisms involve sliding that occurs along a slope parallel sliding surface, and a toe breakout surface facilitated by internal shearing. Without the existence of these three surfaces, bi-planar sliding cannot occur.   Where persistent structure does not bound the toe breakout or the internal shear, the failure mechanisms may be analyzed using Eqs. 2.1 and 2.2 assuming that the major principal stress acts parallel to the slope face.  Where the sliding mass is deep within the slope, or the slope is shallow, and the rock mass is treated as a continuum, the toe breakout surface and internal shear are not linear and using Eqs. 2.1 and 2.2 to predict the location and orientation of the toe breakout surface and internal shear is less accurate than using finite element. 65   Published case histories of dip slope failures provide an indication of the maximum failure depths expected for a range of slope inclinations.  This paper successfully related the “D/H” ratio of these slopes (D is the depth of failure normal to the slope face and H is the slope height) to the slope inclination and has shown a general decrease in D/H ratios with an increase in slope inclination.  The detailed literature review performed for this paper suggests that during bi-planar slope failure of dip slopes, deformation of the near surface layers of rock occurs during progressive failure in natural slopes.  Many times, the deformation is buckling.  Buckling of the outer layers of rock releases some of the shear strain that develops deeper in the slope and may therefore is a good indicator of impending deep-seated bi-planar failure in bedded slopes.  This paper provides a practical methodology for estimating the stability of dip slopes based on the slope angle, the geological conditions, and the anticipated depth to the slope parallel sliding surface.  In summary, Sarma’s limit equilibrium method may be used to estimate dip slope stability with certainty where the slopes are steep and the depth to the slope parallel sliding surface is shallow.  As the depth to this surface increases or the slope angle decreases, using this approach becomes less accurate because 1) 1 rotates from parallel to the slope face in the case of shallow slopes and 2) the toe breakout surface is log spiral and the assumption that it is linear provides less accurate results. 66     67        Fig. 2.15.  Smoky River Coal back analysis showing bi-planar failure mechanism. A.) Discrete element model set up.  B.) “Blow up” showing bi-planar failure mechanism.  C.) Comparison of measured displacement vectors with those predicted using discrete element.   68  2.12. References Behrens da Franca, P. R. (1997) "Analysis of slope stability using limit equilibrium and numerical methods with case examples from the Aguas Claras mine, Brazil," Department of Mining Engineering. Ontario, Canada: Queen's University, p 204.  Carvalho, J. L., T. G. Carter, and M. S. Diederichs (2007) "An approach for predictions of strength and post yield behavior for rock masses of low intact rock strength," In: E. Eberhardt, D. Stead, and T. Morrison, Eds., Rock Mechanics: Meeting Society's Challenges and Demands, Proceedings of the 1st Canada-US Rock Mechanics Symposium. Vancouver.  Cavers, D.S. (1981) "Simple methods to analyze buckling of rock slopes," Rock Mechanics and Rock Engineering, Vol. 14, pp 87 - 104.  Chen, Hongery (1992) "Appropriate model for hazard analysis in slope engineering; Landslides; proceedings of the sixth international symposium," Sixth international symposium on Landslides, Christchurch, New Zealand, Feb. 10-14, 1992, Vol. 6, pp 349-354.  Cruden, D.M., and S. Masoumzadeh (1987) "Accelerating creep of the slopes of a coal mine," Rock Mechanics and Rock Engineering, Vol. 20, pp 123 - 135.  Dawson and Barron. (1991)  "Design guidelines for bedded footwall slopes - final report." Department of mining, Meturgical and Petroleum Engineering; University of Alberta, Edmonton, Alberta.  Dawson, E., Motamed, F., Nesarajah, S., and Roth, W. (2000) "Geotechnical stability analysis by strength reduction; Slope stability 2000." Geo-Denver 2000, Denver, CO, United States, Aug. 5-8, 2000, Vol. 101, pp 99-113.  Dawson, E. M., Roth, W. H., and Drescher, A. (1999). "Slope stability analysis by strength reduction." Geotechnique, 49(6), 835-840. 69    Dawson, Ethan, Farid Motamed, Saddanathapillai Nesarajah, and Wolfgang Roth (2000) "Geotechnical stability analysis by strength reduction; Slope stability 2000," Geo-Denver 2000, Denver, CO, United States, Aug. 5-8, 2000, Vol. 101, pp 99-113.  Eberhardt, E., K. Thuro, and M. Luginbuehl (2005) "Slope instability mechanisms in dipping interbedded conglomerates and weathered marls--the 1999 Rufi landslide, Switzerland," Engineering Geology, Vol. 77, No. 1-2, pp 35-56.  Fisher, B. R., and Erik Eberhardt (2007) "Dip slope analysis and parameter uncertainty - Case history and practical recommendations," In: E. Eberhardt, D. Stead, and T. Morrison, Eds., Rock Mechanics: Meeting Society's Challenges and Demands, Proceedings of the 1st Canada-US Rock Mechanics Symposium. Vancouver: Taylor & Francis, London, pp 871-878.  Giani, G. P. (1992) Rock Slope Stability Analysis, Rotterdam: A. A. Balkema. Hawley, P. M., D. C. Martin, and C. P. Acott (1986) "Failure mechanics and design considerations for footwall slopes " CIM Bulletin, Vol. 79, No. 896, pp 47-53.  Hoek, E. (1983) "Twenty-third Rankine lecture; Strength of jointed rock masses," Geotechnique, Vol. 33, No. 3, pp 185-223.  Hoek, E. (1987) "General two-dimensional slope stability analyses," In: E. T. Brown, Ed., Analytical Computational method in Engineering Rock Mechanics: Allen & Unwin; London, pp 95-128.  Hoek, E., and E. T. Brown (1997) "Practical estimates of rock mass strength," International Journal of Rock Mechanics and Mining Sciences, Vol. 34, No. 8, pp 1165- 1186.  Hoek, E., C. Carranza-Torres, and B. Corkum (2002) "Hoek-Brown criterion - 2002 edition," NARMS-TAC Conference. Toronto, pp 267-273.  70  Hoek, E., J. Read, A. Karzulovic, and Z. Chen, Y., (2000) "Rock Slopes in Civil and Mining Engineering," In: International Society for Rock Mechanics., Ed., Proceedings of the GeoEng2000 at the international conference on geotechnical and geological engineering. Melbourne: Technomic.  Hoek, Evert, and John Bray (1974) Rock slope engineering, London: Institution of Mining and Metallurgy.  Hoek, Evert, and E. T. Brown (1980) Underground excavations in rock, London: The Institution of Mining and Metallurgy.  Jennings, J. E. (1970) "A Mathematical Theory for the Calculation of the Stability of slopes in Open Cast mines," In: Anonymous, Ed., Planning open pit mines. South Africa (ZAF): A. A. Balkema, Cape Town, pp 87-102.  Jin, X., Mingdong, C., Tianbin, L., and Lansheng, W. (1992) "Geomechanical simulation of rockmass deformation and failure on a high dip slope." Christchurch, NZ, pp 601.  Kvapl, R. Clews, M. (1979) "An examination of the Prandtl mechanism in large slope failures," Transactions of the Institution of Mining & Metallurgy, Section A, Vol. 88, pp 1 - 5.  Ladanyi, B., and G. Archambault (1970) "Simulation of shear behavior of a jointed rock mass, Chapter 7; Rock mechanics, theory and practice; 11th symposium on Rock mechanics, proceedings," 11th symposium on Rock mechanics, Berkeley, CA, United States, 1969, pp 105-125.  Li, Q., and Z.Y. Zhang (1990) "Mechanism of buckling and creep-buckling failure of the bedded rock mass on the consequent slopes," 6th International Association of Engineering Geologists Congress. Amsterdam, pp 2229 - 2233.  Li, T. B., J. Xu, and L. S. Wang (1992) "Ways and methods for the physical simulation of landslide; Landslides; proceedings of the sixth international symposium," Sixth 71  international symposium on Landslides, Christchurch, New Zealand, Feb. 10-14, 1992, Vol. 6, pp 487-491.  Lucas, D. (2006). "Stress failure of a shallow open cut mine." Australian Centre for Geomechanics - December 2006 Newsletter, pp 4-6.  Mencl, V. (1966) "Mechanics of landslides with non-circular slip surfaces with special reference to the Vaiont slide." Geotechnique, Vol. 16, No. 4, pp 329-337.  Nathanail, C. P. (1996) "Kinematic analysis of active/passive wedge failure using stereographic projection," International Journal of Rock Mechanics and Mining Sciences, Vol. 33, No. 4, pp 405-407.  Olausen, R. C. (1984) "Development of a computer program to assess the bi-linear failure mechanisms in footwall slopes," University of British Columbia, Vancouver. Sarma, S.K. (1979) "Stability analysis of embankments and slopes," Geotechnique, Vol. 23, pp 423 - 433.  Scoble, M.J. (1981) "Studies of ground deformation in British surface coal mines," Mining Engineering. Nottingham: University of Nottingham.  SCEC (2002) Recommended Procedures for Implementation of DMG Special Publication 117 Guidelines for Analyzing and Mitigating Landslide Hazards in California. Southern California Earthquake Center, 63 pp.  Smoky River Coal Limited, and Piteau Associates Engineering Limited (1987) "Upper east limb open pit rock anchoring and slope monitoring project."  Stead, D., E. Eberhardt, and J. S. Coggan (2006) "Developments in the characterization of complex rock slope deformation and failure using numerical modeling techniques," Engineering Geology, Vol. 83, No. 1-3, pp 217-235.  Terzaghi, K (1943) Theoretical soil mechanics, New York, London, Sydney: John Wiley and Sons, Inc. 72   Terbrugge, P. J., and M. Hanif (1981) "Discussion of a large failure on the footwall of the Nichanga open Pit, Zimbia," International Symposium on Weak Rock. Tokyo, pp 1499- 1501.  Wang, Lansheng, Zhuoyuan Zhang, Mindong Cheng, Jin Xu, Tianbin Li, and Xiaobi Dong (1992) "Suggestion on the systematical classification for slope deformation and failure; Landslides--Glissements de terrain," Sixth international symposium on Landslides, Christchurch, New Zealand, Feb. 10-14, 1992, Vol. 6, Vol. 3, pp 1869-1877.  Welkner, D. (2008) "Integrated field investigation, numerical analysis and hazard assessment of the Portillo Rock Avalanche site, Central Andes, Chile," MSc Thesis; Geological Engineering. Vancouver: University of British Columbia, p 202.  Wyllie, Duncan C., and Christopher W. Mah (2004) Rock slope engineering: civil and mining, New York, NY: Spon Press.  Yang, T. H., T. Xu, R. Q. Rui, and C. A. Tand (2004) "The deformation mechanism of a layered creeping coal mine slope and the associated stability assessments," International Journal of Rock Mechanics and Mining Science & Geomechanics Abstracts, Vol. 41 (n SUPPL. 1), No. 3B 10, pp 1-6.   73  3. ASSESSMENT OF PARAMETER UNCERTAINTY ASSOCIATED WITH DIP SLOPE STABILITY ANALYSES AS A MEANS TO IMPROVE SITE INVESTIGATIONS1F2  3.1. Introduction Uncertainty in rock slope engineering is inherent. Often, field data (e.g. geological structure, rock mass properties, groundwater, etc.) are restricted to surface observations or limited by inaccessibility and can never be known completely. This leads to two forms of uncertainty: model uncertainty, which arises from gaps in understanding required to make predictions on the basis of causal inference, and parameter uncertainty, where geological heterogeneity contributes to spatial variations in rock mass properties (Morgenstern 1995). For most rock slope stability problems, these two forms of uncertainty are interconnected. Without a proper understanding of the acting failure mechanism, it is not possible to select the correct set of equations to carry out a stability analysis. Once the failure mechanism is understood, the issue then becomes selecting a specific set of values or range of values to be used in the analysis.  This is certainly true in the case of bi-planar dip slope problems, where in the absence of daylighting bedding, the geologic conditions dictate the mechanism by which failure may occur. Fisher et al. (2008a) show that the dip slope failure mechanism consists of (Fig. 3.1):   Sliding along a slope parallel sliding surface.  This is most commonly bedding in sedimentary sequences but may also consist of a fault, shear, or lithologic contact that is sub parallel to the slope face.   Toe breakout that may occur along a persistent discontinuity but more often occurs by means of a step path developing along discontinuities of limited persistence and/or shearing through weak rock at the toe of the slope.   2 A version of this chapter has been submitted for publication.  Fisher, B.R., Eberhardt, E. and Hungr, O. (2009) Geotechnical uncertainty associated with bi-planar dip slope failures. In review. 74   Kinematic release by shearing through the rock mass near the toe of the slope.  Based on these findings, Fisher et al. (2008a) address the issue of model uncertainty by providing a set of procedures for carrying out a dip slope stability assessment. These include empirical constraints that relate the slope height to failure depth (D/H) ratio (as observed through a number of case histories), together with the application of different analytical tools (e.g. limit equilibrium, finite element, finite difference, and discrete element), the combination of which depends on the geological conditions present. Fisher et al. (2008a) used the Sarma (1979) limit equilibrium method to analyze a range of scenarios and to provide a parametric study based on rational and somewhat conservative factors of safety.   Fig. 3.1.  Simplified dip slope showing bi-planar failure mechanism.  3.2. Problem Statement – Parameter Uncertainty Managing parameter uncertainty is one of the key aspects to understanding the reliability of a slope design (Duncan et al., 2003).  Authors that have focused on evaluating circular slope failures suggest that a slopes’ stability state is contingent on the averaged shear strength of the slope lithologies as opposed to the presence of local heterogeneities. Therefore the spatial distribution of the soil shear strength can be represented by the “averaged” strength along the slip surface (El Ramly et al. 2002). This is not the case with the stability state of dip slopes in rock since the failure mechanism consists of sliding along a slope-parallel persistent discontinuity and toe 75  breakout surface (Fig. 3.1), the latter often developing through the rock mass (Fisher et al. 2008a). Therefore, unlike other slope types documented in the geotechnical literature, a dip slope must be treated as both a continuum and discontinuum.  Another consideration is that a thorough understanding of the influence of the geotechnical input on the outcome of the geotechnical evaluation (e.g. parameter sensitivity) can greatly assist in planning the geotechnical investigation. This is especially important where access as well as budget may be limited. A typical geotechnical budget for a project may range between one-half and three percent of the total project cost, with up to eight percent being reported for large tunneling projects (Parker 2004).  Therefore, it is paramount that the geotechnical investigation (e.g. background research, field investigation, and laboratory testing) be planned as efficiently as possible.  Of course this applies to all geotechnical projects, not only dip slopes.  Planning of the geotechnical investigation is usually based on the judgment of the engineer who is in charge of the project.  Although experience and judgment cannot be replaced, there are methods available for project planning that aid in the decision making process with simple up front scoping calculations that provide a means for justifying the site investigation tasks undertaken.  Uncertainties specific to dip slopes can be quantified through a sensitivity analysis which effectively brackets the influence of the geotechnical input parameters on the outcome of the slope stability calculations. These can be carried out relatively quickly when performing a limit equilibrium analysis, although in the case of numerical slope stability analyses, much time and effort can be expended to test the sensitivity of the model results to parameter uncertainty.  Recently, other authors have proposed using Monte Carlo simulations and first order second moment (FOSM) calculations to predict the reliability of geotechnical engineering designs (El Ramly et al. 2002, Harr 1987, Hoek 1997, 2007, Duncan 2000).  El-Ramly et al. (2002) discuss Spearman Rank Correlation Coefficients (Spearman, 1904) that relate the uncertainty and statistical distribution of geotechnical input parameters to the outcome of geotechnical calculations.  Given an understanding of the simple methodology associated with calculating the Spearman Rank Correlation Coefficients, 76  and the efficiency with which these calculations can be accomplished, it becomes clear that this procedure is suitable for prioritization of limited budget funds during planning of the geotechnical investigations. 3.3. Spearman Rank Correlations Spearman Coefficients are calculated using rankings of the input values and not the actual values themselves (as done with a typical linear correlation).  The correlation is a value between negative one and one and provides an indication of the influence of one input parameter on the result of another. The closer the correlation is to negative one or one, the better the fit.  A positive correlation suggests that a high value of the input results in a high value of the output value.  A negative correlation suggests that a high input value results in a low output value. Spearman Rank Correlation Coefficients can be calculated using Eq. 3.1.  nn d R   3 2 2 61          Eq. 3.1  R2 is the Spearman Rank Correlation Coefficient, d is the difference in the ranks between the input and output variables, and n is the number of samples.  The Spearman Rank is calculated by ranking the input parameters based on their numerical values from highest to lowest.  For instance, if the sample population consists of five input values and the numerical values ranged from one to five, five would be ranked first (as “one”) and one would be ranked last (as “five”).  Similarly, if the input value of five corresponded with an output value of -10 while an input of one corresponded with an output of negative one; the negative one would be ranked as “one” and the negative ten would be ranked as “five”.  The difference in the rankings is the ranking of the input minus the ranking of the associated output.  A simple spreadsheet can be written to perform a Spearman Rank Correlation calculation, although there are also spreadsheet add-ons such as @RISK (Palisade Corporation) that performs the calculations for numerous inputs very efficiently.  The 77  @RISK add-on was used in the following example showing the influence of different input parameters on the results of a Rock Mass Rating (RMR) estimate.  The Rock Mass Rating System by Bieniawski (1989), initially developed to empirically aid in tunnel support design, has since become a standard rock mass mapping index (the same could be said for Barton’s Q-system, for which the following analysis could have been similarly performed). There are five geotechnical parameters required to estimate the basic Rock Mass Rating (RMR89):  1. Intact rock strength (ci).  2. Drill core quality (e.g. RQD).  3. Discontinuity spacing.  4. Discontinuity condition (e.g. roughness).  5. Groundwater condition.  Although the groundwater conditions are an important consideration when using the RMR89 system directly for empirical design or classification purposes, for the purpose of characterization and establishing rock mass properties, it is often not included as being a characteristic of the rock mass (e.g., Hoek and Brown 1997). Instead, the maximum rating value is assigned (e.g. 15) for the groundwater parameter. This modified system is referred to as RMR89*.  A rating is assigned to each of the input parameters and those ratings are summed to arrive at the RMR89* value.  Each one of the input parameters has an influence on the outcome of the RMR89* although depending on the rock mass quality, the influence of the input is not equal.  For example, consider the following input parameters and distributions presented in Table 3.1. These distributions are based on a data set collected for a dip slope located in Southern California (Fisher et al. 2008c).  78    Table 3.1.  Uncertainty associated with RMR89*, based on a detailed dataset for a dip slope in Southern California. Geotechnical Property Distribution Mean  Min. Max. Input Parameters ci (MPa) Normal 15 6 7 25 RQD (%) Triangular 63 26 0 100 Discontinuity Spacing (mm) Log Normal 450 1,260 10 3,000 Discontinuity Condition Rating Log Normal 17 3.5 0 30 Groundwater condition None 15 0 15 15 Output Values RMR89* Log Normal 54 6 34 76  The mean RMR89* generated suggests that the rock mass may be described as “Fair Rock” and follows a log normal statistical distribution with a standard deviation () of  about 6.  A similar mean value and standard deviation were obtained using a normal distribution.  A Spearman rank correlation coefficient chart is presented in Fig. 3.2.  It is clear from Fig. 3.2 that for the rock mass considered the influences of the input parameters are not equal.  The most influential rating parameters are the RQD, discontinuity spacing (directly related to RQD) and the condition of the discontinuities.   The intact rock strength (ci) has little influence on the RMR89* estimated as shown by the low ranking in Fig. 3.2.   79   Fig. 3.2.  Spearman Rank Correlation Coefficient for a RMR89* rated “Fair” rock mass.   Fig. 3.3.  Influence of ci and RQD on Rock Mass Rating (10,000 data points).  80  An alternative presentation of the correlation coefficient is provided in Fig. 3.3, which shows the correlation between the calculated RMR89* and the inputs RQD and ci. Inspection of Fig. 3.3 suggests that there is a much better correlation between RQD and the calculated RMR89* than that with ci.   Clearly, if the goal of a geotechnical investigation is to calculate RMR89*, and there is a preliminary indication that the rock mass consists of “Fair” rock, more resources should be focused on constraining the distribution of RQD than that of the uniaxial compressive strength. An approximation of the latter would be sufficient.  Fig. 3.4 expands on the previous example by now considering each of the different rating classes as defined by Bieniawski (1989), from Very Poor to Very Good quality rock masses. The relative influence of each geotechnical input parameter on the calculated RMR89* is shown across the different classes.  The input distributions used to generate this chart are based on the author’s experience and judgment.  In the case of the “Good” quality rock mass, the parameters are equally weighted, corresponding to the majority of the case histories that Bieniawski used to develop the RMR system (Bieniawski 1989).  This chart was generated by “normalizing” the correlation coefficients for each of the rock qualities.  The influence of the geotechnical input parameters on the outcome of the RMR89* are directly related to the recommended numerical values assigned by Bieniawski (1989) to describe the individual rock mass properties.  For example, Poor and Very Poor Rock masses are sensitive to the roughness of the discontinuities.  Bieniawski (1989) suggests that range of numerical ratings for discontinuities typically found in Poor to Very Poor rock masses (gouge infilled to being slightly rough) is 20.  Better quality rock (where the discontinuities are typically slightly rough to very rough) have a numerical rating range of 10.  Therefore RMR89* for the poorer quality rock masses is more sensitive to the discontinuity characteristics than better quality rock masses.  RQD and discontinuity spacing are directly related until the spacing of the joints reaches about 600mm according to Bieniawski (1989).  At this point, the rating for spacing increases but RQD is about 100 percent and there is no increase in the RQD numerical value assigned.  This trend can be seen in Fig. 3.4 by inspection of the RQD and Spacing “lines”.  For the poorer quality rock, there is a general increase in the influence 81  of RQD and Spacing.  Where the rock quality is Fair or better, the influence of RQD begins to diminish while an increase in discontinuity spacing continues to impact on the RMR89* estimated.  Uniaxial compressive strength of the intact rock does not play a large part on the influence of the RMR89* because narrow ratings are assigned to large ranges of ci and ranges of ci can be easily constrained using simple field tests (ISRM, 1981).   Fig. 3.4.  Influence of different rock mass rating inputs for different RMR89* classes.  3.4. Mohr-Coulomb Shear Strength Parameters In many situations, the goal of a geotechnical investigation is to establish the rock mass shear strength properties required to carry out a stability analysis.  The Mohr-Coulomb failure criterion is widely employed for this purpose, although issues arise with respect to appropriately scaling laboratory based values to those that are more representative at the rock mass scale. For this, Hoek et al. (2002) provide a methodology for establishing the Mohr-Coulomb “rock mass” shear strength parameters based on the geological and 0% 10% 20% 30% 40% 50% 60% 70% 80% V. Poor Rock Poor Rock Fair Rock Good Rock V. Good Rock Rock Quality Pe rc en t I nf lu en ce RQD Roughness Rating Spacing Uniaxial Compressive Strength Input Equally-Weighted 82  geotechnical site conditions.  This procedure, encoded in the Rocscience program RocLab (2007), uses a non-linear Hoek-Brown failure envelope to define the laboratory- based intact rock strength and the Geological Strength Index (GSI) to scale to the rock mass properties. The GSI can be assessed directly in the field or estimated from the RMR (Hoek and Brown 1997), and is based on the blockiness of the rock mass and the surface conditions of the discontinuities (Hoek et al. 1995). An estimate of the maximum value of the minimum principal stress (3max) is then used to superimpose a linear Mohr- Coulomb shear strength failure envelope from which the rock mass cohesion and friction angle values are obtained. This procedure is sufficiently accurate for the purposes of establishing Mohr-Coulomb parameters for sensitivity analyses.  Appendix B provides a more rigorous approach for establishing Mohr-Coulomb parameters from Hoek-Brown shear strength envelopes for dip slopes based on the actual normal stresses at the toe breakout and internal shear surfaces.  A Spearman Rank Correlation simulation was carried out for the purpose of testing the sensitivity of the Mohr-Coulomb shear strength parameters calculated for the “Fair” rated rock mass characterized earlier in Table 3.1.  Table 3.2 lists the Hoek-Brown geotechnical input parameters used and estimates of their statistical distributions. These are taken from the same dataset used for Table 3.1.  Table 3.2.  Uncertainty of Mohr-Coulomb Parameters. Geotechnical Property Distribution Mean  Min. Max Input Parameters GSI Log Normal 49 6 29 71 ci (MPa) Normal 15 6 7 25 mi Normal 17 0.67 15 19  (kN/m3) Normal 22.5 1.67 17.5 27.5 Disturbance (D) Normal 0.2 0.07 0.2 0.6 Slope Height (m) NA 100 0 100 100 Output Values crm (kPa) Log Normal 190 40 90 490 rm (deg) Normal 46 3.5 31 56  Fig. 3.5 shows the Spearman Rank Correlation Coefficients for the “Fair” rock mass friction angle (rm) and cohesion (crm), generated for a 100m slope. The height of the slope in this case is used to calculate the 3max value required to fit the linear Mohr- 83  Coulomb envelope to the non-linear Hoek-Brown envelope. As can be seen, the intact compressive strength (ci) has the greatest influence on the rm and crm values.  GSI is ranked second, slope height, unit weight (, and the Disturbance Factor (D) are ranked next, with mi having the least influence on the calculated Mohr-Coulomb shear strength parameters.  Given the curved nature of the Hoek-Brown failure envelope, as either slope height or  increases, 3max the result is the flattening of the superimposed linear Mohr-Coulomb envelope and thus an increase in crm but decrease in rm.  Based on Fig. 3.5, it is clear that if completing a geotechnical investigation directed towards estimating rock mass shear strength parameters for the “Fair” rock mass in question with properties as shown in Table 3.1 and 3.2, the limited resources available for site investigation would be better spent on quantifying the distribution and uncertainty of ci and GSI as opposed to mi.  a) Rock Mass Friction        -0 .4 -0 .2 0 0. 2 0. 4 0. 6 0. 8 qu, Mpa GSI Slope Height mi D Rock Unit Weight Spearman Rank Correlation Coefficient ci 84  b) Rock Mass Cohesion  Fig. 3.5.  Spearman Rank Correlations Coefficients for rm and crm. 3.5. Geotechnical Input Distributions For the previous calculations, it may appear that a detailed data set is required to determine to which input parameters limited site investigation resources should be directed. However, Spearman Rank Correlations are independent of the statistical distribution of the input parameters.  The expected minimum and maximum values simply provide a means for constraining the correlation.  For reliability based calculations, input distributions are paramount and the goal of establishing the correlation coefficients is to give an indication regarding which input parameter distributions have the most influence on the outcome of a deterministic or reliability-based engineering calculation.   Therefore, there is merit in discussing typical distributions for the geotechnical input parameters required to establish output variables such as RMR89* or the Mohr-Coulomb rock mass shear strength parameters generated from the Hoek-Brown/GSI procedure.  Hoek (2007) suggests that a normal distribution is the most common distribution used in geotechnical engineering and that when the actual distribution is unknown, a normal -0 .2 -0 .1 0 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 sci GSI Slope Heigth Rock Unit Weight D mi Spearman Rank Correlation Coefficient ci 85  distribution should be chosen and that the distribution should in many cases be truncated so that numerical stability using statistical sampling techniques such as Monte- Carlo (Harr 1987) or Latin Hypercube (Imam et al. 1980, Startzmann and Watterbarger 1985) can be maintained.  Besides normal distributions, other distributions such as beta, exponential, log normal or triangular have been used in geotechnical engineering (Wyllie and Mah 2004).  Hudson and Harrison (2000) suggest that discontinuity spacing data (length measurement) is best represented by a negative exponential distribution.  According to Wyllie and Mah (2004) triangular distributions are common for data sets where the minimum, maximum and most likely value can be estimated (such as RQD).  Normal and log normal distributions are common for describing joint roughness.  Hoek (1989, 2007) suggests that variables such as GSI, mi, and ci, and Mohr-Coulomb values such as rm and crm can be adequately described using normal distributions.  Sensitivity analyses completed as part of this study suggests that the distribution of RMR89* varies according to the average or mean rock quality with all but Poor and Very Poor rock masses having a normal distribution.  Poor and Very Poor rock masses showed a log normal distribution but could be described using a normal distribution with minimal error. If a correlation between RMR89* and GSI is used (e.g. Hoek and Brown 1997), then it follows that similar distributions would be expected for GSI. Hoek (2007) states that normal distributions are appropriate for GSI, if GSI is determined using qualitative descriptions of the rock mass.  The authors have also found that the distribution of the safety factor calculated using Sarma’s (1979) method for bi-planar failures in dip slopes is log normal.  This alleviates concerns regarding inaccuracies associated with estimating probability of failure (Pf) using simplified methods such as Rosenblueth’s (1981) point estimate (also referred to as First Order Second Moment, or FOSM, calculations).  The FOSM method has been suggested by Harr (1987), Hoek (1989), and Duncan (2000) for estimating probability of failure for geotechnical applications.  86  3.6. Parameter Uncertainty and Dip Slope Stability The influence of geotechnical input parameters on the calculated bi-planar dip slope factor of safety was evaluated by modifying a limit equilibrium solution published by Sarma (1979), Hoek (1987), and Watson (2000).  The solution allows non-vertical slices and explicitly accounts for internal shear forces and shear strength.  The solution is iterative and solves for force equilibrium only.  The evaluation was carried out using an EXCEL spreadsheet (Microsoft 2003) with the following inputs:   Slope geometry including height, slope angle, and dip slope bedding/slab thickness.   Rock mass properties including unit weight and Mohr-Coulomb shear strength parameters.  These were derived by first estimating RMR89*, and then using correlations provided by Hoek and Brown (1997) to establish GSI and Mohr- Coulomb values based on the input for RMR89*.   Mohr-Coulomb shear strength parameters for the slope parallel sliding surface, toe breakout surface, and internal shear (see Fig. 3.1).  Statistical input parameters were used for each of the input variables.  Slope geometries are the same as that shown in Fig. 3.1 with the depth of the slope parallel sliding surface based on empirical relationships for a large number of case histories reported by Fisher and Eberhardt (2008a).  The shear strength of the slope parallel sliding surface was estimated assuming the discontinuity in question would be a bedding plane within a siltstone layer.  Based on the relatively low plasticity index of the fine-grained siltstone, a fully-softened friction angle was assumed having a normal distribution with a mean value of 25° and standard deviation of 1°.  The toe breakout and release surface was assumed to develop and dip out of the slope at the most critical angle predicted by the Mohr rupture theory.  Two different shear strength scenarios for this feature were considered: toe breakout through shearing of the rock mass, thus requiring use of the rock mass shear strength parameters provided in Table 3.2; or, toe breakout along a persistent daylighting 87  discontinuity at the slope toe.  The later scenario provides a means for comparing the effect of using a “friction only” shear strength model for the toe breakout surface.  Fig. 3.6 provides a summary of the sensitivity of the dip slope stability analysis to the slope parallel sliding surface and toe breakout and internal shear strengths.   Fig 3.7 provides a further breakdown of this analysis to include the different input parameters used in the Hoek-Brown procedure to derive the Mohr-Coulomb rock mass shear strength parameters.  For this, a negative correlation coefficient was assigned to crm and rm so that during the Monte Carlo Simulation, if a high value for crm was chosen, a low value of rm was calculated.  The correlation coefficient reflects the curvature of the Hoek-Brown failure envelope.  The input parameters shown in Fig. 3.6 and Fig 3.7 that have not yet been described include ’bedding, and ’toe joint which is the friction angle used for the slope parallel sliding surface and persistent discontinuity forming the toe breakout surface, respectively.  The first key finding that can be drawn from these results is that the influence of the geotechnical input on the calculated factor of safety is directly related to the inclination of the slope and the resulting distribution of shear stresses that develop in the toe of the slope. Fig 3.6 and Fig 3.7 illustrate this change in parameter influence.  In effect, as the slope inclination increases, the normal stresses acting across the sliding surface decreases and thus the frictional component of shear strength decreases (frictional strength is a function of the acting normal stress). This reduction in shear strength means that more active load is transferred to the toe of the slope so that stability of the slope becomes more contingent on the shear strength on the internal shear and toe breakout surface.  Thus, the influence of shear strength along the slope parallel sliding surface diminishes with an increase in the slope angle while the influence of the toe breakout surface and internal shear increases with an increase in slope inclination.  88   Fig. 3.6.  Influence of Mohr-Coulomb shear strength parameters on dip slope stability.    89   Fig. 3.7.  Influence of Hoek Brown input parameters on dip slope stability.   90   In the case of toe breakout along a continuous, adversely dipping, daylighting discontinuity (Fig 3.6 & Fig 3.7 d, e, & f), the importance of the frictional strength increases with an increase in the slope angle.  Once again this is due to the increase in stress at the toe of the slope as the slope inclination increases. At low slope angles, the influence of the toe joint is negligible, and the shear strength of the bedding dictates the slopes stability state.  Furthermore, where the slope is shallow, internal shear plays a more important role in the dip slope stability state, thus increasing the importance of rm and crm. This suggests that considering an infinite slope (Duncan, 1996) would allow for a very conservative result to be obtained because the infinite slope solution does not account for the increase in the slopes’ stability state because of internal shearing.  The influence of the internal shear strength can be observed by comparing Fig. 3.6 & Fig. 3.7 b, d, and f.  At shallow angles, the toe joint strength (and thus the rock mass shear strength at the toe in Fig 3.6 & Fig 3.7 a, c, and e) does not have much influence on the slope’s stability state. As the slope inclination increases, more influence is placed on shear strength at the toe breakout surface as well as the internal shear.  Another finding is that crm influences the factor of safety calculation much more than rm. This is demonstrated by comparing both the rankings of the crm and the influence of the unit weight () relative to rm.  Where the slope is steeper than 30 degrees, crm receives the highest ranking and  has a large negative influence on the factor of safety.  While carrying out the same analysis, but for weak, poor quality rock masses (RMR<20) with low ci values, the results suggest that although the shear strength of the slope parallel sliding surface remains influential for shallow dipping angles, rm becomes more influential than the crm as the slope angle increases.  Whether represented as rm with or without crm, it is clear that the rock mass shear strength at the toe is the most important input parameter to constrain for a dip slope stability analysis when dealing with steep dip slopes whereas the bedding shear strength becomes the more important parameter to constrain for shallow dipping dip slopes. 91  3.7. Discussion and Practical Recommendations The preceding sections illustrate the use of Spearman Rank Correlation Coefficients as a means to quantify the influence of geotechnical input to the calculated outcome. RMR89* is sensitive to different input depending on the quality of the rock mass, but in general, it would seem unnecessary to expend considerable efforts on determining the intact rock strength, ci when estimating RMR. When calculating Mohr-Coulomb rock mass shear strength parameters for “Fair” rock according to the recommendations in Hoek and Brown (1997), GSI and ci hold the greatest influence in the output of the results.  Therefore, establishing reliable values during the geotechnical investigation should be a primary focus while expending resources to establish mi values is not well merited.  In the case of bi-planar failures of dip slopes, the influence of the input parameters can be directly related to the slope inclination and the transference of the active driving forces (e.g. distribution of stresses) in the slope.  For shallow-dipping dip slopes, the shear strength of the slope parallel sliding surface is most critical and considerable effort should be spent quantifying its shear strength.  For steeper dip slopes, more load is transferred from the upper slope to the slope toe, resulting in the shear strength of the internal shear and toe breakout surface having more influence.  If shearing is expected through the rock mass, then the focus will be on quantifying the strength of the rock mass.  If a persistent joint or fault daylights the slope toe, then establishing its shear strength holds the greatest value.  Thus there appears to be great value in performing quick scoping calculations to establish the influence of parameter uncertainty on the outcome of design calculations (e.g. factor of safety) before planning the geotechnical data collection campaign.  This of course moves towards the objective of increasing the efficiency and value return of the field and laboratory investigations. Value of the return can be measured by evaluating the decrease of excess cost that may be incurred if the uncertainty associated with the dip slopes’ stability state is decreased to a minimal or “residual” value.  Table 3.3 lists the site investigation steps taken and geotechnical input used to establish the stability state of a dip slope in Southern California for the purpose of designing a 92  setback distance (a setback distance is the distance behind the crest of the slope where stability can be assured and land commercially developed).  Although based on the California dip slope investigation, these costs represent relative costs of typical rock mechanics geotechnical investigations.  In the case of the California dip slope, field mapping, core drilling, and laboratory testing were considered integral here to carrying out an effective site investigation and reliability-based geotechnical design. However, each parameter was collected assuming a more or less equal weight in importance.  Table 3.3.  Typical relative cost of geotechnical investigations. At the onset of a project involving a dip slope such as that for residential development, a developer has an idea of where they would like to situate a structure behind the slope crest to maximize profit.  In the case of an open pit mine, the mine planner may propose a certain depth to maximize profit.  It is the geotechnical engineer’s task to determine whether placing the residences where proposed is feasible from a geotechnical standpoint and in the case of the open pit mine, how deep the dip slope can be constructed while maintaining an adequate margin of safety. Study Phase Site Investigation Item Cost Desk Study  1. Typical mi values 2. Disturbance factor (D) 3. Typical discontinuity friction angles $ Field Mapping 4. Outcrop mapping (geology, structure, etc.) 5. ci estimate using simple field tests 6. Outcrop RMR & GSI estimates $$ Core Drilling & Index Properties 7. Index testing 7a. Rock unit weight       7b. Atterberg Limits for correlation with slope parallel sliding surface strength cci from laboratory testing 8. GSI from RMR estimates on core $$$ Laboratory Testing & In Situ Measurements 9. Direct shear on saw cut sample (for potential toe joints) 9a. Estimates of JRC to complement saw cut direct shear tests 10. Direct shear testing of natural joints 11. Torsional ring shear testing for bedding 12. Structural mapping from down hole televiewer 13. mi estimate from triaxial testing $$$$ 93   The amount of geotechnical data available will determine the level of accuracy of the predictions regarding whether the slope will perform as intended.  This is analogous with estimating the probability of failure (Pf) of the slope.  Because a proposed slope will either perform as intended or will not perform as intended (regardless of the amount of geotechnical information collected), the Pf established by the geotechnical engineer is really a perceived Pf based on the geotechnical information available.  Once the geotechnical engineer makes his/her prediction, that information is shared with the stakeholders so that an informed decision may be made weighing the risks and benefits associated with the project.  There is a direct relationship between the geotechnical effort and the cost of slope mitigation costs; such as the cost of stabilization measures beyond what is necessary, excessive geotechnical conservatism, and setback distances for infrastructure.  This is illustrated in Fig. 3.8 which shows a generalized relationship between geotechnical effort and the excess cost of slope mitigation.  Intuitively, as the amount of geotechnical information available increases, geotechnical uncertainty decreases which leads to a more appropriate geotechnical design and decrease in (in this case) excess setback distance for infrastructure. 94   Fig. 3.8.  Relationship between geotechnical effort and uncertainty.   Fig. 3.9.  Geotechnical effort and perceived Pf for a proposed dip slope. 95   The concept outlined in Fig. 3.9 can be expanded to other typical geotechnical tasks for various geotechnical projects and Fig. 3.10 graphically shows the general process associated with data collection required for a reliability-based geotechnical design.  Thus, at the onset of an investigation (far left-hand side of the graph), a dip slope problem has been identified but no data has been collected yet. As a result, the reliability of the proposed design, and slope itself in terms of stability, is unknown as is the probability of failure. The accuracy of any geotechnical predictions at this point would be very low.  The geotechnical investigation then begins with the desk study (background research) and field mapping. The costs for this portion of the investigation are relatively small. Core drilling, lab testing and in situ measurements may be carried out next.  After performing the drilling, sampling, and lab testing program, there is usually enough information to make a judgment regarding the applicability of the proposed design. Either it is decided that the proposed configuration will not be reliable (line ‘A’, Pf unacceptable) or that the proposed slope is considered reliable for the specific design conditions (line ‘B’, Pf is acceptable).  If the latter is the case, the geotechnical engineer may be justified in proposing additional investigation and design to minimize the uncertainty associated with the geotechnical input while maximizing the design and value for the client.  If the former is the case, either the geotechnical engineer or the project team will derive another proposed scenario to be evaluated.  This process is of course, iterative.  The tasks (Table 3.3) associated with maximizing the accuracy of geotechnical predictions for both shallow and steep dipping dip slope designs are shown in Fig. 3.11. Where the dip slope is shallow, the minimal geotechnical investigation might consist of performing a field mapping regime, followed by a coring and testing program that concentrates on establishing the bedding shear strength.  This is because the shear strength of the bedding dictates the slope stability for shallow dip slopes.  Note that Atterberg Limits testing of material forming the slope parallel sliding surface is included because there are excellent correlations between these and the frictional strength of fine grained geotechnical materials Stark and Eid (1997) & Mesri et al, (2003).  Within 96  sedimentary sequences, the slope parallel sliding surface usually consists of a siltstone or shale.  Unnecessary tasks may include laboratory testing to establish the Hoek-Brown empirical coefficient mi for the rock mass, especially given the high costs associated with completing consolidated undrained triaxial testing.  To a lesser degree, the shear strength of the rock mass or discontinuities at the toe of the slope may be sufficiently estimated using mapping data and typical values from the literature. More expensive testing is not necessary because the value associated with further constraining the rock mass shear strength at the toe is minimal and existing methods for doing so are empirical (e.g. Hoek et. al, 2002).  For steep dip slopes, a detailed understanding of the rock mass shear strength becomes the primary focus given the change in failure mechanism for which toe breakout and internal shearing dictate the slope’s stability state.  Once again laboratory testing to establish the mi value is not necessary, but in addition, expensive torsional ring shear tests are now not justified. Instead, empirical correlations regarding bedding shear strength based on the sedimentary lithology are sufficient.  Fig. 3.10.  Typical geotechnical investigation for reliability-based geotechnical design for dip slopes. 97     Fig. 3.11.  Geotechnical tasks associated with reliability evaluations of dip slopes.  3.8. Conclusions Uncertainty is inherent in geotechnical design.  In regards to estimating the stability state of dip slopes, most of the uncertainty lies in the geologic model assumed, and the geotechnical parameters used in the evaluation.  Fisher et al. (2008a) show bi-planar sliding in dip slopes occurs along a slope parallel sliding surface with toe breakout occurring at the base of the failure.  Internal shearing is required to facilitate kinematic release.  All three of these surfaces work together for the slope to fail, but with different degrees of importance depending on the dip of the slope.  With this in mind, increased efficiency and value with respect to the site investigation budget can be gained by working towards minimizing the uncertainty of those parameters that have the greatest bearing on the outcome of the slope stability analysis. This can be done quickly and inexpensively by performing scoping calculations facilitated by the use of Spearman Rank Correlation Coefficients.  These calculations Task numbers refer to those outlined in Table 3.3. 98  can be completed through programming of computer-based spreadsheets, or more conveniently, by using statistical add-on packages (in this study the Excel spreadsheet add-on @RISK was used).  The results presented demonstrate that for shallow dipping dip slopes, stability is primarily dictated by the shear strength of the slope parallel sliding surface and therefore, efforts should be focused on constraining the shear strength of this surface. For steep dip slopes, the shear strength related to the toe breakout and internal shear release surfaces becomes dominant and therefore, the rock mass shear strength as well as that for any adversely dipping persistent discontinuities should be the focus of the geotechnical investigation.  Once the minimum information is collected, there may be an opportunity to provide value engineering to the project.  With regards to steep dip slopes, it may be justifiable to constrain the rock mass characteristics and shear strength properties based on detailed rock core logging, downhole televiewing, and additional testing.  For shallow slopes, value engineering may be added by supplementing the outcrop mapping with core logging and downhole televiewing to analyze bedding spacing and shear strength, as well as performing Atterberg limits testing to better constrain the shear strength of the slope parallel discontinuity.  In either case, establishing mi values for the purpose of determining the rock mass shear strength is unnecessary, despite the fact that it is often used as the basis for carrying out laboratory triaxial testing. Based on the parameter uncertainty analysis carried out here, simple estimates of the intact rock properties based on typical values taken from the literature are sufficient.  This paper shows that a better understanding of the influence of the geotechnical input coupled with engineering judgment can help to focus the geotechnical investigation so that it is as efficient as possible.  99  3.9. References Bieniawski, Z. T. (1989) Engineering rock mass classifications: a complete manual for engineers and geologists in mining, civil, and petroleum engineering, New York: Wiley. Microsoft Corporation (2003) "Microsoft Office EXCEL 2003." Redmond, WA.  Duncan, J. M. (1996) “Soil slope stability analysis” in Landslides Investigation and Mitigation.  Special Report 247.  Transportation Research Board. 673 pp.  Duncan, J. M., M. Navin, and T. F. Wolff (2003) "Discussion of "Probabilistic slope stability analysis for practice"," Canadian Geotechnical Journal, Vol. 40, pp 848-850.  Duncan, J. Michael (2000) "Factors of safety and reliability in geotechnical engineering," Journal of Geotechnical and Geoenvironmental Engineering, Vol. 126, No. 4, pp 307- 316.  El-Ramly, H., N. R. Morgenstern, and D. M. Cruden (2002) "Probabilistic slope stability analysis for practice," Canadian Geotechnical Journal, Vol. 39, No. 3, pp 665-683.  Fisher, B. R., and Eberhardt, E. (2008) "Analysis of Toe Breakout Mechanisms to Develop an Improved Methodology for Assessing the Stability State of Dip Slopes." in press, 19pp.  Harr, M.E. (1987) Reliability-based design in civil engineering, New York: McGraw-Hill.  Hoek, E. (1987) "General two-dimensional slope stability analyses," In: E. T. Brown, Ed., Analytical Computational method in Engineering Rock Mechanics: Allen & Unwin; London, pp 95-128.  Hoek, E. (1989) "A limit equilibrium analysis of surface crown pillar stability," Surface crown pillar evaluation for active and abandoned metal mines. Ottawa: Department of Energy, Mines, and Resources Canada, pp 3-13.  100  Hoek, E. (1994) "Strength of Rock and Rock Masses," ISRM News Journal, Vol. 2, No. 2, pp 4-16.  Hoek, E. (2007). "Practical Rock Engineering (2007 eds)."  Hoek, E., and E. T. Brown (1997) "Practical estimates of rock mass strength," International Journal of Rock Mechanics and Mining Sciences, Vol. 34, No. 8, pp 1165- 1186.  Hoek, E., C. Carranza-Torres, and B. Corkum (2002) "Hoek-Brown criterion - 2002 edition," NARMS-TAC Conference. Toronto, pp 267-273.  Hoek E., Kaiser P. K., and Bawden W. F. (1995) Support of Underground Excavations in Hard Rock, Balkema, Rotterdam.  Hudson, J. A., and Harrison, J. P. (2000) Engineering Rock Mechanics an Introduction to the Principles, Elsevier Science Ltd, Oxford.  Iman, R. LO., J. M. Davenport, and D. K. Zeigler (1980) "Latin Hypercube sampling (A program user's guide).” Technical Report SAND79-1473. Albuquerque, New Mexico: Sandia Laboratories.  Mesri, G., and M. Shahien (2003) "Residual shear strength mobilized in first-time slope failures," Journal of Geotechnical and Geoenvironmental Engineering, Vol. 129, No. 1, pp 12-31.  Morgenstern, N. R. (1995) "Managing risk in geotechnical engineering," 10th Pan- American Conference on Soil Mechanics and Foundations Engineering. Guadalajar, pp 102-126.  Parker, H.W. (2004) “Planning and site investigation in tunneling.” 1º Congresso Brasileiro de Túneis e Estructuras Subterrâneas. Seminário Internacional South American Tunneling, 6 pp.  101  Rocscience (2007). RocLab - Rock mass strength analysis using the Hoek-Brown failure criterion, ver. 1.0. Rocscience, Toronto.  Rosenbleuth, E. (1981) "Two-point estimates in probabilities," Journal of Applied Mathematical Modeling, Vol. 5, pp 329-335.  Sarma, S.K. (1979) "Stability analysis of embankments and slopes," Geotechnique, Vol. 23, pp 423 - 433.  Spearman, C. (1904) "The proof and measurement of association between two things" Amer. J. Psychol. 15 pp. 72–101.  Stark, Timothy D., and Hisham T. Eid (1997) "Slope stability analyses in stiff fissured clays," Journal of Geotechnical and Geoenvironmental Engineering, Vol. 123, No. 4, pp 335-343.  Startzmann, R. A., and R. A. Wattenbarger (1985) "An improved computation procedure for risk analysis problems with unusual probability functions," Proceedings of the Symposium of the Society of Petroleum Engineers Hydrocarbon Economics and Evaluation. Dallas.  Watson, A. (2000) "Parametric analysis of the bi-linear failure mechanism using method of slices and shear strength reduction technique in the finite difference code UDEC," Geological Engineering. Vancouver, BC: university of British Columbia, p 38.  Wyllie, Duncan C., and Christopher W. Mah (2004) Rock slope engineering: civil and mining, New York, NY: Spon Press.   102  4. IMPROVED DESIGN OF SET-BACK DISTANCES FOR DIP SLOPES IN BEDDED ROCK 2F3  4.1. Introduction High-density residential development within Southern California is becoming increasingly difficult for a number of reasons.  The geology of the area is tectonically disturbed (folded and faulted), and seismically, is very active.  Reviewing agencies are becoming more and more stringent in their requirements for grading permits because of the public’s willingness to bring legal action.  Yet the public’s demand for residential development motivates developers to maximize the amount of developed land above steep slopes.  There are published guidelines that provide practical recommendations for mitigating landslide hazards in Southern California.  Two of the more widely used publications include “Special Publication 117” by California Divisions of Mines and Geology (1997) and “Recommended Procedures for Implementation of Special Publication 117” by Southern California Earthquake Center (2002).  The intent of these publications is to assist engineering geologists and geotechnical engineers with geological investigation planning and geotechnical design related to seismic hazards and landslides and also to provide practical tools for regulators while evaluating site-investigation reports (SCEC, 2002).  The documents provide a reference as to the “state-of-practice” in Southern California for assessing and mitigating landslide hazards, although the interpretation and implementation of these guidelines is left to the consultant, regulator, and other stakeholders such as developers.  Consequently, they were never meant to provide a “one size fits all” methodology for landslide mitigation and SCEC (2002) is currently being updated to reflect new developments in landslide risk management and mitigation (R.A. Hollingsworth, personal communication).  Moreover,  3 A version of this chapter has been submitted for publication.  Fisher, B.R., Eberhardt, E. and Hilton, B. (2009) Improved Design of Set-Back Distances for Dip Slopes in Bedded Rock.  In review. 103  although SCEC (2002) provides a broad brush overview of the methodologies available for analyzing landslide hazards, the committee assembled to author SCEC (2002) encourages the use of other “more sophisticated” solutions when evaluating slope hazards where those procedures can be shown to provide reasonable solutions in line with the current state of knowledge.  This paper discusses a case history involving a residential development above a dip slope consisting of extremely weak to moderately weak rock in which geologic conditions differed from those typically encountered in Southern California. Therefore, the typical procedures for evaluating landslide hazards were supplemented with more sophisticated methods to arrive at a more reasonable design solution.  Unfortunately, the regulators tasked with review of the site-investigation report(s) were unfamiliar with the methodologies that were proposed, requiring a highly detailed geological investigation and design to validate the approach taken.  The results from these investigations are presented, highlighting the assumptions made, how risk was perceived, and the analysis methods that were agreed upon by the stakeholders. Recommendations are then provided based on how the project could have benefited from a more thorough understanding of dip slope failure mechanisms and uncertainty associated with geotechnical data.  4.2. Dip Slopes A dip slope is a natural or engineered/cut rock slope that has an inclination coincident with a prominent discontinuity or set of parallel persistent discontinuities, for instance, bedding planes in sedimentary rock. An example of a famous “dip slope” landslide within Southern California is the October 2, 1978 Bluebird Canyon Landslide that occurred in a residential section of Laguna Beach (Miller and Tan 1979).  The prominent discontinuity (slope parallel sliding surface) provides a plane of weakness upon which, sliding may occur. Because these discontinuities are slope-parallel and do not ‘daylight’ in the slope face, they are precluded from a straight-forward planar or wedge kinematic evaluation (e.g. Hoek and Bray 1974).  In order for failure to occur, ‘toe breakout’ is required and therefore, the key issue to understanding dip slope failures is to understand the toe- 104  breakout mechanism (Fisher and Eberhardt 2007a).  A typical dip slope showing a range of potential toe breakout surfaces is provided in Fig. 4.1.   Fig. 4.1.  Typical dip slope geometry showing the range of toe breakout surfaces (after SCEC 2002). 4.3. Guidelines and Their Interpretation for Evaluating Dip Slopes SCEC (2002) provides clear guidelines based on committee consensus regarding the implementation of Special Publication 117 (DMG 1997) and a summary of their key recommendations related to dip slopes consisting of weak sedimentary bedrock materials where the slope parallel sliding surface consists of bedding.  In their concluding remarks, the committee suggests that at current, geotechnical practice is prone to “gross conservatism” and “almost purely judgment-based design” and that although newer and more accurate methodologies may exist, this will be the case until there is clear recognition of the benefits of thorough geotechnical practice by all stakeholders as well as the geotechnical consultants themselves.  Based on experience and discussions with other engineering geologists and geotechnical engineers that practice in the area, the state-of-practice in Southern California for evaluating the stability of dips slopes is quite different from those published.  The motivation for this appears to be that: 1) conservative designs are more easily approved by regulatory agencies, and 2) the public is very willing to bring legal action against contractors and engineers; especially in situations involving residential 105  development.  Table 4.1 is a comparison of SCEC key findings and how those key findings are interpreted and used in practice based on the author’s experience.  Table 4.1.  Key findings and their interpretations from SCEC 2002. SCEC (2002) Key Finding Practical Implementation Geologic structure is paramount to a dip slopes stability state and therefore, a geologist should evaluate the potential failure mechanisms and the geotechnical engineer should assist the geologist in identifying and assessing that which is most critical. Geologic structure is considered paramount to the dip slopes stability state and considerable effort is expended identifying the orientation of bedding planes in sedimentary units that compose dip slopes. Two sets of shear strength values are required, that “along-bedding” and that “across-bedding”.  Shear strength is determined by direct shear testing along bedding (using residual strengths), and by either direct shear testing or triaxial testing on intact rock samples for the across-bedding strength.  Over a range of normal stresses, effective stress shear strengths are not linear and therefore, using non- linear strength criteria is appropriate. Two sets of shear strength values are used in the evaluation:  that along bedding using lab-based residual strengths and that across bedding, where because most of the geologic units encountered are very weak and poorly indurated, direct shear testing of intact rock samples are used. When evaluating the results, the ultimate shear strength is chosen to account for strain incompatibility.  The Mohr- Coulomb criterion is almost exclusively used to describe the shear strength of the along-bedding and across-bedding strength. A safety factor of 1.5 is required for establishing static slope stability.  However, if peak shear strengths are used for the across-bedding strength, the engineer is cautioned to use a safety factor of 2.0.  Alternatively, the engineer may choose to “down-grade” the across-bedding shear strength because of varying cementation or the presence of fractures in the rock. A safety factor of 1.5 is used to evaluate static slope stability regardless of the extent and reliability of the geotechnical data available. Slope stability calculations should be completed using Spencer’s method (1973) because it accommodates non-circular surfaces. Slope stability calculations are completed using a number of different limit equilibrium methods and the method that provides the most conservative answer is assumed to be correct. Failure geometries with a near 90-degree angle between bedding and the toe breakout (Fig. 4.1) should be avoided because these geometries lead to normal stress concentrations that are unrealistic. Moreover, these failure geometries may not be The failure geometry that provides the most conservative safety factor is assumed to be correct regardless of whether that geometry is kinematically feasible. 106  SCEC (2002) Key Finding Practical Implementation kinematically feasible. Finite element/difference codes may be more accurate for determining failure mechanisms than limit equilibrium.  When elasto-plastic models are used in finite element, strength reduction factors are comparable to safety factors calculated by limit equilibrium for the same failure mechanism (Griffins and Lane 1999). Limit equilibrium analyses are preferred over finite- element/-difference codes and shear strength reduction techniques. Seismic stability may be established using pseudo- static evaluations, displacement analyses, or a combination of both. Seismic stability is established using pseudo-static evaluations.  Displacement analyses are considered in that they are implicit in the recommended (by SCEC) pseudo-static evaluations.  The above interpretations and practiced methodologies and assumptions can have a profound impact on the estimate of a dip slopes’ stability state. The factors that have the greatest impact are the uncertainty associated with the geotechnical parameters and their treatment, the toe breakout mechanism chosen, the across-bedding shear strength, and the limit equilibrium method chosen.  The case history presented in the next section provides a detailed accounting of the geotechnical investigation and analysis carried out for a dip slope involving a proposed residential development.  Highlighted are the different assumptions that were made, how risk was perceived, and the analysis methods that were agreed upon by the stakeholders. 4.4. Case History: Devil Canyon, Southern California A high density real-estate development consisting of 375 single-family residential lots was proposed on a 230-acre site above Devil Canyon, north of the City of Chatsworth, California.   The general location of the proposed development is shown in Fig. 4.2.  Developments such as these are common. Given the value of the land, countered by the geologic conditions, the most challenging aspect of the geotechnical design was providing recommendations for the minimum setback distance; i.e. how close to the slope crest could development proceed, while balancing issues of public safety and maximizing land use. 107           Fig. 4.2.  Project area location map. 4.4.1. Regional and Site Geology The project area is located in the western portion of Los Anglees County along the southern foothills of the Santa Susana Mountains and within the Transverse Ranges geomorphic provinces.  The Transverse Ranges province is characterized by west- northwest trending structure primarily the result of faulting and folding related to the compression of the region along the San Andreas Fault system.  The provinces are comprised of complex crystalline and sedimentary rock including Pre-Cambrian intrusions, Mesozoic plutonic rock, various metamorphic facies, and Cretaceous sedimentary rock.  Within the Devil Canyon area, the geology consists of slightly folded Cretaceous bedrock of the Chatsworth Formation (Colburn et al. 1981). The Chatsworth Formation is a turbidite or flysch series, which in a very broad sense consists of thickly-bedded 108  sandstones and thinly bedded fine-grained sandstone and siltstone. The type section is located along Woolsey Canyon road, a few kilometers south of the project site.  Thicknesses for the formation exceed 1,800 m, with Johnson and Ogawa (1981) reporting ratios, in the proximity of the project area, of 80 percent bedded sandstone and 20 percent thin-bedded, fine-grained sandstone and siltstone.  The thick-bedded sandstone series averages about 7 m with a sandstone to siltstone ratio of about 20:1. The thinly bedded fine-grained sandstone and siltstone series averages 3 m with sandstone to siltstone ratios of 1:1.  A engineering geologic map of Devil Canyon is presented in Fig. 4.3.  The bedded nature of these rocks is reflected in the geomorphology of Devil Canyon, with the southern slopes reaching heights of 60 m and dipping gently coincident with bedding (i.e. shallow-dipping dip slopes).  The north side of the canyon is steeper, with bedding dipping into the slope and with heights approaching 85 m.  At the base of the canyon is an intermittent stream. Fig. 4.3.  Engineering geologic map showing the area proposed for residential development. 109  4.4.2. Seismicity The project area is located in the seismically active Southern California region. Major historic earthquakes felt in the vicinity of Chatsworth have usually originated from faults located outside the area.  These include the 1857 Fort Tejon, 1925 Santa Barbara, 1933 Long Beach, 1952 Arvin-Tehachapi, 1971 San Fernando, 1987 Whittier Narrows, 1992 Landers, and 1994 Northridge earthquakes.  The peak horizontal ground acceleration (PGA) for the site was first evaluated deterministically assuming a 6.6 moment magnitude event (similar to that of the 1994 Northridge event) along the Santa Susana fault at a distance of 2.4 km from the site. Based upon the mean rock value using Abrahamson and Silva’s (1997) attenuation model, an acceleration of 0.87g is estimated for strong rock, 0.59g for soil, and an average value of 0.73g was computed for the site-specific geologic conditions.  A probabilistic acceleration greater than 0.70g was estimated from the Probabilistic Seismic Hazard Map (Peterson et al. 1999) for a 10% probability of exceedance in 50 years at “soft” rock sites.  A value of 0.73g accommodates both methodologies.  4.4.3. Geotechnical Investigation In addition to aerial photograph interpretation, regional geologic mapping, and outcrop structural mapping, the site investigation included the drilling of thirteen rock cores behind the crest of the north slope and within the south slope of Devil Canyon.  The cores ranged in depth from 18 to 95m below the surface.  The boreholes were video logged using a down-hole camera to measure the orientation of geologic structures and discontinuity traces intersecting the borehole walls and to confirm water levels after drilling.   The cost associated with the drilling program approached about $500,000 by the end of the project.  4.4.4. Geologic Structure Thousands of discontinuity orientations were measured during outcrop mapping, core logging, and through borehole televiewer logs.  In general, the geological structure is 110  similar throughout Devil Canyon: homoclinal.  The general relationships of the orientation of the discontinuities within the rock mass are represented in Fig. 4.4 which was generated from the outcrop mapping on the dip slope. There are three primary discontinuity sets within the rock mass.  Fig. 4.4.  Stereonet showing the geologic structure of Devil Canyon.  The first discontinuity set consists of bedding which dips to the northeast at about 35 degrees.  Stereonets generated from data collected at other locations along Devil Canyon suggest that this dip varies from 20 to 40 degrees and is coincident with that of the northeast facing slopes.  Bedding is persistent and individual beds can be correlated between borings that were drilled along strike or dip of the beds. The second joint set acts as a “lateral” joint that dips as steep as 90 degrees towards the southeast.  Data from other borings show that the dip direction can vary from northwest to southeast and the dip of this joint set may range from 60 to 90 degrees.  The lateral joints are thought to be very persistent.  The third discontinuity set consists of joints that dip approximately 45 degrees southwest (ranging from 30 to 70 degrees).  This joint set is generally normal to bedding and considered the “cross-bed” joint.  Observations at outcrops suggest that the cross-bed joints are truncated at the intersection with weaker bedding planes.  These three discontinuity sets intersect to form a series of orthogonal blocks.  Bedding bounds the top and bottom of the blocks, with the two joint sets bounding the sides of the blocks.  Because of past folding activity in the region, the actual orientations of the 111  discontinuity sets vary slightly along Devil Canyon although the spatial relationship between the discontinuity sets remains generally consistent throughout the study area.  True spacing of the discontinuity sets were estimated using the borehole televiewer data, specifically the core orientation, dip of the discontinuities, and apparent spacing measured within the borehole (Wyllie and Mah 2004).  Table 4.2 shows the distributions and means of the discontinuity spacing for the three primary discontinuity sets.  Table 4.2.  Discontinuity spacing. Discontinuity Distribution Type Mean Spacing Bedding Log Normal 0.4m Lateral Joint Log Normal 0.6m Toe Joint Neg. Exponential 1.5m  4.4.5. Weathering and Rock Grade A site-specific weathering criteria was developed for this project which was later correlated to the intact strength (ci) of the Chatsworth Formation rock types.  For the purposes of estimating the slope stability, the rock mass engineering properties were assigned based on Rock Grade as discussed by Brown (1981).  Weathering is directly related to the amount of oxidation of the rock mass and generally decreases with depth (with Rock Grade increasing).  The weathering criteria used for the project is presented in Table 4.3. 4.4.6. Groundwater Surface water was observed to flow intermittently along Devil Canyon, with shallow and perched groundwater tables occurring onsite as a seasonal condition.  Springs were observed about 6 m above the canyon floor on the north side of the canyon and are associated with phreatophytic vegetation.  Groundwater was encountered at depth in several of the core borings.  On the dip slope side of Devil Canyon, groundwater is generally located at or slightly above the elevation of the intermittent stream flowing along the canyon bottom.  Groundwater increases in 112  elevation to the south and occurs with “slightly weathered” or “fresh” bedrock where oxidation of the rock mass is minimal and the color of the rock is gray. 4.4.7. Laboratory Testing Laboratory testing was performed to characterize the rock mass and derive rock mass properties.  The focus of the testing was on the decomposed through moderately weathered rock grades because it was decided by all stakeholders that for the slope stability evaluation it would be assumed that the rock strength does not exceed that of the Grade R2 rock mass.  In summary, the laboratory testing program consisted of:  1. Direct shear tests on saw cut samples of rock core.  2. Direct shear tests on natural joints within the rock mass.  3. Direct shear tests on clay-filled joints within the rock mass.  4. Ring shear tests on siltstone bedding.  5. Direct shear tests on intact rock core (R0 and R1 intact strength).  6. Triaxial tests on intact rock core (mainly R2 intact rock).  7. Uniaxial compressive strength testing on Grade R2 rock.  Table 4.4 summarizes the results of the laboratory testing campaign.  Appendix F contains the results of individual laboratory tests.             113   Table 4.3.  Rock mass weathering and strength profiles. Weathering  Grade Description Fresh R3 to R4 Gray rock showing no discoloration. This rock is typically encountered below the water table where oxidation is not occurring. Slightly Weathered R2 to R3 Slight oxidation.  95 percent of the rock material is not oxidized.  This rock is typically gray with reddish brown staining along discontinuities or individual beds. Moderately Weathered R2 to R3 Rock is moderately oxidized and is reddish brown.  Less than half of the material is fully oxidized or decomposed. Highly Weathered R1 to R2 Light brown, or gray brown rock that is almost completed oxidized. Decomposed R0 to R1 Light brown, or gray brown rock is completely oxidized.  This rock typically crumbles easily to sand-sized particles and/or is easily deformed by finger pressure. Original rock fabric is observable. Residual Soil R0 Decomposed with no evidence of original rock structure or bedding observable.  Table 4.4.  Distribution of results from laboratory testing. Parameter Units No. Tests Mean Std. Dev Min. Max Distribution Compressive strength, ci (2) MPa 26 14.9 5.7 6.8 25 Normal Basic friction, ’b (3) deg. 35 36.4 3.0 30 44.7 Log normal Joint friction, ’joint (4), deg.  39 38.3 5.0 27.7 47.7 Normal Bedding friction, ’bed (5) deg.  9 24.5 4.4 17.6 29.5 Normal Unit weight, ’rock (2) kN/m3 37 23.2 1.6 19.9 26.2 Normal Notes: (1)  Normal stress applied during test. (2)  Testing on Grade R2 samples. (3)  Measured using direct shear tests on saw cut core samples. (4)  Peak value measured using direct shear tests on natural joints. (5)  Residual value measured using direct shear and ring shear tests.  Although the results from direct shear testing of intact rock core were ultimately not used while determining the structural setbacks for residential development, it is useful to illustrate that these tests were performed according to the standard of practice per SCEC (2002) in Southern California.  Fig. 4.5 shows the results of direct shear testing on 114  highly weathered (Grade R1) intact rock samples.  “Ultimate” shear strength is chosen at the inflection of the stress-strain curve between the peak and residual strength values.   Fig. 4.5.  Direct shear test results for Grade R1 intact rock samples  Direct shear tests were not performed on the Grade R2 rock because of their higher strengths.  Instead, consolidated undrained triaxial testing was performed.  All 15 tests were completed at confining stresses that ranged between zero and about one-half the UCS of the Grade R2 rock.  The test results suggest a Mohr-Coulomb shear strength envelope with a cohesion intercept of 2.7 MPa and a ’ of 43 degrees.  4.4.8. Rock Mass Classification Rock Mass Rating (Bieniawski 1989) and Geologic Strength Index (Marinos and Hoek 2001) values were recorded at 35 outcrops at the base of Devil Canyon.  Fig. 4.6 115  highlights the range of GSI values mapped superimposed on Marinos and Hoek’s (2001) GSI chart for heterogeneous rock masses and turbidite sequences.  Because of erosion by the creek, the rock exposed as outcrops was predominantly the more resistant Grade R2 rock consisting of non-tectonically deformed sandstone with minor interbeds of siltstone (Fig. 4.7).  The GSI for these rocks varied from about 40 to 55.   Fig. 4.6.  Geologic Strength Index showing combination of composition and structure and surface conditions of discontinuities mapped along outcrops in Devil Canyon (after Marinos and Hoek 2000).   116    Fig. 4.7.  Outcrop of the Chatsworth Formation observed in the valley at the base of the canyon. Note the orthogonal joints truncated at bedding.  Bedding is shown in blue.  Truncated toe joints are shown in red.  A more detailed evaluation of GSI for the Grade R2 rock was carried out statistically using information from the borings facilitated by the EXCEL add-on @RISK (Palisade Corporation 2007).  Hoek and Brown (1997) suggest that GSI may be estimated from Bieniawski's (1989) RMR with the following linear adjustment:  RMR89 – 5. In this conversion, the rating for groundwater is set to 15 and the adjustment for discontinuity orientation is set to zero.  This relationship is considered valid where the GSI rating is greater than 30.  Table 4.5 provides the input data required to estimate GSI as well as the resulting GSI distribution for the 35 outcrops mapped. 117  4.4.9. Engineering Geologic Model Based on the geotechnical investigation and laboratory testing, a working geologic model was established for the dip slope south of Devil Canyon.  Fig. 4.8 shows a working cross-section taken through this model.  The geologic section provided in Fig. 4.8 is considered accurate although over simplified because it does not show individual siltstone layers that were encountered.  This is appropriate because given the distribution of thin layers of siltstone encountered in the borings; it is advisable to assume that siltstone along bedding may be encountered at any depth.  Table 4.5.  Range of measured GSI input values. Parameter Units Mean  Std. Dev. Min. Max Distribution Intact strength, ci MPa 14.9 5.7 6.8 25 Normal Discontinuity Condition - 17.1 3.4 0 30 Log normal Discontinuity Spacing mm 448 1260 9 3000 Log Normal RQD - 63 25.5 0 100 Triangular GSI - 49 6 34 76 Log Normal    Fig. 4.8.  Dip slope “working” cross-section. 118  4.5. Design of Dip Slope Setback Distances for the Devil Canyon Project A series of slope stability analyses were undertaken to determine the ‘setback’ distance behind the crest of the dip slope from which construction of residential housing could safely proceed. Fig. 4.3 shows the outline of one such large slope failure that had occurred nearby in the past.  Geologic mapping and aerial photograph interpretation suggested that this landslide occurred because of undercutting at the toe of the slope, most likely in response to periods of increased water flow along the bottom of the canyon.  Projecting a similar failure mechanism, however, results in a minimal setback distance. First, undercutting would result in the daylighting of only one or two bedding planes, and this over a relatively long timeframe. Secondly, although these beds would be highly unstable given the kinematically simple planar failure mode that would develop, the failure would not be very deep (approximately equal to the thickness of the beds).  Instead, it is typical practice to assume a bi-planar failure mode (Fig. 4.1).  Translation sliding of the bedding following toe breakout was thus considered, as well as other more complex modes such as buckling and ploughing (Stead and Eberhardt 1997). From this it was found that the bi-planar failure mode provided the more conservative setback distance for design.  4.5.1. Bi-Planar Failure Mechanisms The breakout mechanism at the toe of the dip slope may be represented in a number of ways depending on assumptions regarding the geology and nature of the rock mass. Within Southern California, the typical practice is to assume one of two potential failure mechanisms.  These are:   Failure at the toe through upward and outward movement along a highly persistent joint intersecting the bedding at an acute angle; and   Failure at the toe through shearing of intact rock material (i.e. at an angle across bedding).  119  4.5.1.1. Persistent Joint Model Although a flat dipping persistent daylighting joint could easily enable a deep bi-planar failure, the stereonet presented in Fig. 4.4 shows that for the geological conditions encountered at the project site, the candidate joint sets that could serve as cross-cutting toe release joints are for the most part normal to the bedding (i.e. the failure surface at the toe would form a 90 degree angle with the bedding as shown in Fig. 4.1).  In this case, only the shear strength of the persistent joint is considered with regards to failure at the toe.  The regulatory agency preferred this toe breakout model and suggested its use together with the lowest measured shear strength of the joints.  As such, the results of direct shear tests performed on saw cut cores would be used to represent the shear strength of joints at the toe of the slope.  However, the project engineer (the author) believed that this model was overly conservative as observations at the toe of the dip slope suggested that the cross-cutting “toe” joints were significantly limited in persistence, truncating at their intersection with the bedding planes (Fig. 4.7).  Moreover, using the saw cut samples to represent the shear strength of the joints discounted the strength afforded by the required dilation of the joints and the increase in frictional strength created by the natural surface roughness and waviness of the joints.  4.5.1.2. Intact Rock Model Toe breakout enabled through intact rock failure and shearing across bedding is a more often chosen alternative in Southern California over the persistent toe joint model, because within Southern California, typical rock units are young, poorly indurated, thickly bedded, and very weak.  Observations of landslides within sedimentary lithologies on dip slopes appear to confirm this mode of failure (B. Hilton, personal communication) and therefore, this was a viable option in the view of the regulatory agency.  With this model, the estimate of the intact rock strength is critical and the strength reducing influence of smaller joints within the rock mass (i.e. rock mass strength) is neglected.  Furthermore, based on the SCEC (2002) guidelines, a safety factor of 2.0 is recommended. 120   The developer obviously preferred this toe breakout model as it provided him with a significantly reduced setback distance. However, due diligence would hold that the use of intact rock strengths in this case would be overly optimistic because the influence of jointing within the rock mass would act to decrease the overall strength of the rock at the toe of the dip slopes. Although the intact rock strength assumption may not be too gross of an oversimplification for R0 and R1 grade rocks, it would be for the R2 grade rocks encountered for the Chatsworth project.  4.5.1.3. Step-Path Rock Mass Model Accordingly, an alternative model was proposed that would account for the presence of non-persistent jointing combined with intact rock failure at the toe of the dip slopes (i.e. “rock mass” failure).  The shear strength for the upper portion of the failure would be that of bedding while the shear strength at the toe of the slope would be based on rock mass properties derived from procedures involving the Hoek-Brown Failure Criterion and GSI (Hoek and Brown 1997).  The geology at the base of Devil Canyon is conducive to this type of strength estimation for the following reasons:   Shearing at the toe would likely have to take place through both intact rock and joints (e.g. step-paths),   “Cross-bedding” joints are truncated at bedding,   The individual blocks within the rock mass are much smaller than the slope, and   The actual shearing method at the toe would be difficult to evaluate without assuming an equivalent continuum model.  Therefore, in the absence of a more accurate methodology to analyze step-path failure (i.e. sliding along joints and shearing through intact rock bridges); the methodology proposed by Hoek and Brown (1997) provides a practical solution that appears to fit the geologic conditions encountered at the project site. 121   The regulatory agency was at first reluctant, but relented and insisted that only the most conservative input parameters be used in developing any empirical-based shear strength envelope.  They were more comfortable with the Mohr-Coulomb than the Hoek- Brown failure criterion, despite the SCEC (2002) guidelines designating non-linear strength criteria as being appropriate (Mohr-Coulomb is linear and Hoek-Brown non- linear). They also insisted that any shear strength parameters determined using the Hoek-Brown criterion be converted to Mohr-Coulomb parameters, so that they could be compared to independent shear strength estimates from other related projects that they were either currently being reviewed or had been reviewed in the past.  Lastly, the regulatory agency insisted on including a persistent joint that was 10m long, oriented at the “worst-case” inclination and that daylighted at any location along the slope.  The worst case inclination would be determined by the stereonet generated closest to the cross-section.  Ultimately, this is the toe breakout model that was used.  4.5.2. Analysis Methods The typical practice for analyzing dip slopes is to perform limit equilibrium analyses for static and pseudo-static conditions.  The models include anisotropic shear strengths: one for along-bedding and one for cross-bedding.  Although other methodologies such as numerical modeling are available, limit equilibrium methods coupled with prescribed safety factors have provided satisfactory results for many years (SCEC 2002). Thoroughness (best practice) requires using various limit equilibrium methods (regardless of their limitations) so that the safety factors generated can be compared, with the most conservative of these being the one adopted.  Although the actual peak ground acceleration (PGA) anticipated at this location was estimated to be about 0.73g, pseudo-static evaluations were completed using a seismic coefficient of 0.15g (the prescribed coefficient in this area) and a safety factor criterion of 1.1.  Therefore, the pseudo-static coefficient chosen was approximately 20 percent of the PGA. This procedure has been adopted by Los Anglees County as a “screening” analysis for slopes within hazard zones (SCEC 2002).  The pseudo static value of 0.15g 122  coincides with an 8.25 magnitude earthquake and based on the Makdisi and Seed (1978) displacement method, if a pseudo-static slope stability calculation shows that a particular slope has a safety factor of 1.15 using the 0.15g coefficient, it is expected that that slope will experience less than one meter displacement during the magnitude 8.25 design earthquake.   Los Angeles County has modified the 1.15 safety factor requirement and recommends using a safety factor criterion of 1.1 based on their local experience.  Thus, for design purposes, the setback distance was defined as the farthest location behind the slope crest at which a slip circle with a calculated safety static factor of 1.5 develops, or in the pseudo-static case, a safety factor of 1.1. 4.5.3. Case History Results and Discussion Fig. 4.9 shows the engineering geologic model that was decided upon for the project, the failure mechanism that was considered, and an example of the results of the setback distances that were predicted using Janbu’s simplified procedure (Janbu, 1973).  In many cases, Janbu’s procedure provided the most conservative setback distance.  The model incorporates a ‘jointed zone’ that extends 10m below the foot of the slope.  In this zone, it was assumed that continuous cross-cutting joints daylight at surface and extend at depth to intersect bedding.  The critical bedding plane may be located anywhere within the slope and is considered to contain a thin interbed of siltstone with shear strength properties less than the sandstone. At a depth of greater than 10 m, the shear strength used was determined using the Hoek-Brown derived rock mass properties for the Grade R2 rock.  Because the intact compressive strength (ci) of Grade R2 rock is defined as ranging between five and 25 MPa (Brown 1981), the regulatory agency insisted on using 5 MPa when establishing the rock mass shear strength.  The GSI value chosen was 40 because that was the lowest of the GSI values recorded at the outcrops mapped at the base of the canyon or calculated from the borings.  In most cases, the failure mechanism that was most critical was one that followed bedding and then extended up along the persistent joint to the surface.  Occasionally, 123  the most critical surfaces extended along bedding and exited the toe at a different orientation than along the persistent joints.  This typically occurred where the lower R0 and R1 rock was thicker at the slope surface.  Although it was believed that this failure mechanism was kinematically unlikely, it gave the most conservative answer, and was therefore used to provide setback recommendations that would meet with the least resistance from the regulatory agency.  Suffice to say, the setback distances that were used for the project were deemed excessive by the project team  and the team went through a few iterations of discussion with the regulatory agency trying to convince them of such.  This was extremely frustrating for the developer who was initially patient with the process, but who expended considerable resources and incurred significant time delays during the review process. Ultimately, the developer decided that the costs involved far outweighed the benefits and sold the project.  Based on this experience, and discussions with other engineering geologists and geotechnical engineers working on similar projects, it became clear that a better understanding of the failure mechanisms associated with dip slope failures is much needed, and that the uncertainty associated with geotechnical parameters and methods of analysis used for dip slope evaluations should be more closely evaluated.  124   Fig. 4.9.  Geologic model used to establish structural setbacks based on a compromise on the part of the stakeholders. 125  4.6. Case History Revisited: Path to an Improved Methodology 4.6.1. Assessment of Dip Slope Failure Mechanisms A comprehensive database of case histories involving dip slope failures was compiled and examined by Fisher and Eberhardt (2009a).  Based on this review, it was found that dip slopes tend to fail in a bi-planar manner with sliding occurring along a slope parallel discontinuity and toe breakout occurring either along a persistent joint or through the rock mass at the toe. This is largely in agreement with the assumptions made in the SCEC (2002) guidelines.  However, in order for a dip slope failure to be kinematically-feasible, internal shearing must occur above the toe breakout.  The geologic conditions dictate how the toe breakout and internal shear occurs. A typical bi-planar dip slope failure geometry is shown in Fig. 4.10.  Fig. 4.10.  Simplified dip slope showing bi-planar failure mechanism and internal shear surface.  The depth (measured normal to the slope face) of reported failures can be related to the slope inclination and height.  This depth to height ratio (D/H) increases with a decrease in the slope angle, with upper bound estimates of D/H equaling 0.30, 0.23, and 0.15, for slope inclinations of 30, 45, and 60 degrees, respectively.  This effectively constrains the depth of dip slope failures and provides an estimate of the distance behind the slope crest at which failure would extend.  Fisher and Eberhardt (2009a) also investigated the effect of orthogonal jointing on the failure mechanism for bedded dip slopes, performing back analyses of failed dip slopes reported in the geotechnical literature. Parametric evaluations were provided using limit equilibrium and numerical methods (finite- and distinct element).  A typical distinct element model result is presented in Fig. 4.11 for a shallow dip slope. The input parameters used for the distinct 126  element model are presented in Table 4.6 and Table 4.7. The failure mechanism observed was seen to be consistent across a range of water pressures, “locked-in” in situ stresses, and earthquake loads, for both shallow and moderately steep slopes.  Steeper slopes provided some exceptions, experiencing buckling when external forces such as high groundwater pressures or seismic loads were applied (Fisher and Eberhardt 2009a).  Otherwise, without these large external forces, only normal stresses develop across the joints oriented normal to bedding; i.e. no shear stresses are resolved and therefore no shear occurs along the cross joints (Fisher and Eberhardt 2007a).  Fig. 4.11.  Distinct element model showing bi-planar dip slope failure mechanism. 127  Table 4.6.  Input parameters to initialize conditions to equilibrium for the model shown in Fig. 4.11. Unit Field Stress r1, kN/m3 Bulk2, Pa Shear3, Pa Dil4, deg. 5, deg. c6, Pa Rock Gravity 22.73 672e6 350e6 0 45 12e6  Unit kn7, Pa/m ks8, Pa/m Ten, Pa , deg. c, Pa Joints 5570e6 557e6 0 35 12e6 Slope Parallel Sliding Surface 5570e6 557e6 0 50 12e6 Notes: 19. r is the rock unit weight. 20. Bulk is the bulk modulus based on rock mass deformation modulus and Poisson’s ratio. 21. Shear is the shear modulus based on rock mass deformation modulus and Poisson’s ratio. 22. Dil is the dilation of the rock during shearing. 23.  is friction angle. 24. c is cohesion intercept. 25. kn is the joint normal stiffness based on rock mass deformation modulus and joint spacing. 26. ks is joint shear stiffness.  This is taken as 1/10 of kn. 27. Ten is the tensile strength of the discontinuities.  Table 4.7.  Input parameters for the model shown in Fig. 4.11. Unit Field Stress r1, kN/m3 Bulk2, Pa Shear3, Pa Dil4, deg. 5, deg. c6, Pa Rock Gravity 22.73 672e6 350e6 0 45 12e6  Unit kn7, Pa/m ks8, Pa/m Ten, Pa , deg. c, Pa Joints 5570e6 557e6 0 35 0 Slope Parallel Sliding Surface 5570e6 557e6 0 12 0 Notes: 19. r is the rock unit weight. 20. Bulk is the bulk modulus based on rock mass deformation modulus and Poisson’s ratio. 21. Shear is the shear modulus based on rock mass deformation modulus and Poisson’s ratio. 22. Dil is the dilation of the rock during shearing. 23.  is friction angle. 24. c is cohesion intercept. 25. kn is the joint normal stiffness based on rock mass deformation modulus and joint spacing. 26. ks is joint shear stiffness.  This is taken as 1/10 of kn. 27. Ten is the tensile strength of the discontinuities.  4.6.2. Assessment of Analysis Techniques Because the major principal stress (1) for an orthogonally jointed dip slope remains parallel to the slope face with depth, the toe breakout and internal shear surfaces can be constrained 128  based on the Mohr rupture theory using simple equations (Fisher and Eberhardt 2007a).  This assumes that no additional persistent structure is present (e.g. an isolated fault). Once these angles are known, the shape of the failure surface can be defined and a limit equilibrium method of slices analysis can be carried out.  For the range of failure surfaces corresponding to the D/H ratios reported in the literature, Fisher and Eberhardt (2007b) found that the Sarma (1979) and Hoek (1987) solutions provided a correct, estimate of the safety factor for dip slopes, with Spencer’s (1967) method providing a not as accurate but reasonable estimate of dip slope stability. In contrast, other methods such as the Modified Bishop (Bishop 1957) and Janbu Generalized (Janbu 1973) methods were found to be inadequate, as these methods do not consider the contribution to the total shear strength made available from internal shearing. For dip slope failures, this is an extremely important consideration. Sarma’s (1979) method accounts directly for the shear strength of the internal shear boundary during the computations, and therefore was found to be the most appropriate limit equilibrium method for evaluating bi-planar failures in dip slopes (Fisher and Eberhardt 2007b).  This was confirmed through a number of numerical simulations, which enabled the toe breakout and internal shears to be modeled directly as opposed to being approximated beforehand using the Mohr-Coulomb theory to describe the inclination of the toe breakout and the internal shear surfaces (as was required for the Sarma analysis).  The use of shear strength reduction techniques (Dawson, 1999), built into the finite-element program Phase2 (RocScience 2007) and distinct-element program UDEC (Itasca 2007), further enabled comparisons to be made with respect to the factors of safety calculated by the different techniques.  Fisher and Eberhardt (2009a) showed that the stability factors calculated using distinct element, finite element, and Sarma limit equilibrium methods are similar for the range of D/H ratios of failed dip slopes reported in the geotechnical literature (Fig 4.12).      129    Fig. 4.12.  Comparison of stability factors using numerical models and limit equilibrium.  “Mohr-Coulomb” refers to the method used to establish the inclination of the toe breakout and internal shear surfaces. 4.6.3. Limiting Geotechnical Parameter Uncertainty The rock mass shear strength of a bedded dip slope is clearly anisotropic and therefore simplifying assumptions are required to limit the parameter uncertainty associated with the input values needed for the stability analyses.  First, the shear strength of the slope parallel sliding surface is assumed to be that of the weakest bedding plane (i.e. the siltstone or clay shale in the turbidite sequences, as opposed to the sandstones).  Second, the toe breakout and internal shear must develop through the rock mass consisting of the slope parallel bedding and orthogonal cross-joints that truncate at bedding. This necessitates the use of rock mass shear strength values. As previously noted, Marinos and Hoek (2001) provide recommendations regarding assigning GSI values to orthogonally-jointed and tectonically disturbed sedimentary sequences. Carvalho et al. (2007) provide a rational means for scaling rock mass shear strength values for weak rock.  130  The relative influence of parameter uncertainty on the outcome of dip slope stability evaluations was also seen to be directly related to the distribution of stresses within the slope, and therefore, the slope inclination (Fisher et al. 2009b).  Understanding this influence provides a starting point for scoping the geotechnical investigation before any analysis is undertaken.  For shallow dipping slopes (those up to about 45 degrees) the most critical shear strength to constrain is that for the slope parallel sliding surface and considerable effort should be spent to quantify it.  This is because a greater component of the gravitational stresses is acting normal to the bedding, and therefore the frictional strength parallel to bedding will dictate the percentage of the active forces transferred to the passive resisting blocks at the toe of the slope. In the case of sedimentary units, the most critical materials are fine grained units such as siltstones and shales.  Shear strengths can be established through laboratory testing (e.g. drained direct shear, torsional ring shear, etc.), or correlations between liquid limit, plasticity index and/or clay content and residual shear strengths of clay and clay shales (e.g. Stark and Eid 1994, 1997; Mesri et al. 2003).  When the dip slope is steep (greater than 45 degrees) the stresses in the slope are concentrated at the slope toe and therefore, the shear strength of the internal shear and toe breakout surface have more influence than the slope parallel sliding surface (Fisher et al. 2009b).  It is justifiable to therefore expend considerable effort quantifying the rock mass shear strength at the toe of the slope.  If shearing is expected through the rock mass, then the focus will be on estimating GSI values and the ci of the rock to empirically derive rock mass strengths (as previously described).  If a persistent joint set aligns with either the internal shear or the theoretically-critical toe breakout surface, then establishing the shear strength of those discontinuities holds greater value.  4.6.4. Reassessment of the Devil Canyon Project Given the benefits of a better understanding of dip slope failure mechanisms, from improving the stability analysis undertaken to reducing parameter uncertainty through a more targeted site investigation, the Devil Canyon project was reassessed looking back at the original assumptions made and restrictions imposed by regulators.  131  The revised design procedure would first downgrade the importance of direct shear testing of Grade R0 and R1 intact rock samples. A reduced number of less expensive (and more appropriate) triaxial tests would suffice.  For the given slope, it is clear that the shear strength of these materials has little practical bearing on the setback distance calculated, as the toe breakout and internal shears extend primarily though the Grade R2 rock. Thus the stability state would only be marginally influenced by the weaker materials.  The shear strength of the Grade R2 rock should be established using the GSI and Hoek-Brown failure criterion relationships for rock mass properties. Given the detailed information available to base the GSI and compressive strengths (ci) on, the adoption of a narrow range of most likely values introduces enough conservatism into the calculation of the rock mass properties, as well as all subsequent analyses. Further downgrading of these values is unnecessary, especially considering the safety factor criteria being designed to in this jurisdiction.  Table 4.8 shows the range of most likely values (i.e. geotechnical input) for the Grade R2 rock, based on a distribution fitting of the input data and Monte-Carlo simulation to establish output results of the data set collected during the site investigation.  Table 4.8.  Most likely values for the Grade R2 rock mass. Parameter Distribution Mean Std. Dev. Min. Max ci, MPa  Normal 14.9 5.7 6.8 25 GSI Log Normal 45(1) 6 34 76 ’RM, deg (2) Normal 50 3.4 37 60 c’RM, kPa (2) Log Normal 170 42 10 490 Notes: 1. GSI was downgraded from 49 based on core logging to 45 based on observations at rock outcrops within the Canyon. 2. Rock mass properties calculated using the procedures outlined in Hoek et al. (2002)   The next revision would be to recognize that the required assumption of an orthogonal persistent (>10 m) toe joint is not merited given that the cross joints were mapped in the field as truncating at bedding, which in turn has an average spacing of about 0.4m (Table 4.2 and Fig. 4.2).  In addition to the geologic conditions not supporting this failure model, the principal stresses are parallel to the slope face and therefore the dip slope failure mechanism is not heavily influenced by orthogonal joints. 132   Groundwater conditions were measured insitu during the summer months.  The piezometer levels coincided with the slightly weathered rock, which was not oxidized suggesting that the levels measured were representative of the groundwater table in the slope.  That being said, it is likely that the levels do fluctuate with the seasons and that this should be accounted for in the analysis.  It is appropriate to vary the phreatic surface to better understand the influence of groundwater on the stability state of the slope.  Earthquake loading in Southern California is a real concern and there are well established methods for evaluating slope stability under seismic loads.  The methodology in this jurisdiction requires assigning a pseudo-static coefficient of 0.15 with a safety factor criterion of 1.1.  With this method, the slope deformation during the design-based earthquake (return period of 475 years) should be less than 1 m (SCEC, 2002). However, hillside construction is sensitive to slope deformation and a more appropriate criteria might be a displacement limit on the order of 5 cm (SCEC, 2002).  Therefore, the setback established by the pseudo-static evaluation should be used to ensure that displacements will be minimal for the design earthquake with a return interval of 475 years.  Thus, for the reevaluation of the seismic slope stability, a screening analysis based on a prescribed displacement should be conducted.  If the slope “fails” the screening analysis, a more detailed displacement analysis would be required. The screening analysis requires establishing a pseudo-static coefficient based on a limiting displacement and in this case, that limiting displacement is taken as 5 cm.  Per the procedures outlined in SCEC (2002) and given a maximum horizontal acceleration of 0.73; an earthquake moment magnitude of 6.6 and an earthquake distance of 2.4 km, the coefficient used in the evaluation is 0.29g (double the prescribed 0.15g for less than 1 m limiting displacement typically used in Los Anglees County). The safety factor required for the limiting displacement is 1.0.  Fig. 4.13 shows the results of a finite element evaluation assuming a siltstone interbed that acts a weak plane that intersects the upper slope surface at the slope crest.  This surface was chosen because 1) rock coring suggested that siltstone interbeds could be encountered within the sandstone units, 2) presumably, the developer would want to maximize the area of buildable land up to the crest and 3) if a safety factor of greater than 1.5 was achieved at the crest of the slope, then according to the prescribed guidelines, the setback distance could be set at the 133  slope crest which would result in the greatest possible amount of buildable space.  The shear strength reduction (SSR) value for the static case, based on the scaled rock mass properties given in Table 4.5, is 1.92 and the SSR value for the pseudo-static “screening” case is 1.20. The actual input parameters used in the finite element analysis are presented in Table 4.9.  If instead laboratory intact rock strength properties were used directly, as would be allowed by the guidelines, the resulting factor of safety would be more than three times these values (static S.F. of 7.1).  The pseudo-static screening evaluation (for less than 5 cm of movement) suggests that movement in front of the crest will be minimal given the design earthquake loading.  Based solely on this evaluation, it would be allowable to construct residences at the crest of the slope. In reality, the setback would be either modified to account for shallow sloughing of the slope in front of the crest because of heavy precipitation or shallow reinforcement of the near surface materials might be specified to ensure erosion would not impact the structures.  The SSR values estimated are lower where it is assumed that the siltstone bedding is closer to the slope face.  As stated above, it is assumed that the developer would like to develop only to the slope crest and therefore, finding the lowest SSR or the failure surface where the SSR’s are equal to 1.5 and 1.0 for the static and pseudo-static conditions is not necessary in this case.  4.6.1. Perceived Risk Given the overriding influence that regulatory agencies have on the overall outcome of residential development projects, it would be logical to assume that part of the legal risk resides with the regulatory agency.  This is not the case.  In fact, towns do not have the responsibility of protecting private property or to ensure the quality of residential structures.  In addition, the town has no duty to properly enforce building codes and legal liability lies solely with the contractor, developer, or seller (Levin-Epstein and Borghesani 2004; also Storm v. Town of Ponce Inlet, 5D02-3555, Court of Appeals of Florida, Fifth District, 2004; Kaisner v. Kolb, 543 So. 2d 732, 1989; and Trianon Park Condominium Association v. City of Hialeah, 468 So. 2d 912, 1985).  134    Table 4.9.  Input parameters for finite element models in Fig. 2.10. Unit Field Stress r1, kN/m3 Erm2, kPa  t4, kPa 5, deg. c6, kPa Grade R0 Rock Gravity 18 5e4 0.4 0 23 44 Grade R1 Rock Gravity 19 5e4 0.4 0 41 50 Grade R2 Rock Gravity 22.8 9.2e5 0.3 14 50 170  Unit kn7, kPa/m ks8, kPa/m t, kPa , deg. c, kPa Leached Iron Contact 1e5 1e4 0 25 0 Notes: 1. r is the rock unit weight. 2. Erm is the deformation or Young’s modulus. 3. is Poisson’s ratio. 4. t is the tensile strength. 5.  is friction angle. 6. c is cohesion intercept. 7. kn is the joint normal stiffness. 8. ks is joint shear stiffness. 135  Fig. 4.13.  Stability state of the dip slope using finite element modeling with the most likely values (MLV) of geotechnical parameters.  This is the same slope as that shown in Fig. 4.8.  Rock grades (delineated by the green lines) and groundwater is included in the analysis.  Regardless of the legal liability, there are other consequences associated with improper setbacks above dip slopes and these could include loss of property and/or loss of life, as well as unwanted political repercussions such as a loss of confidence in the regulatory agency and geotechnical profession.  Obviously, it is imperative to involve all stakeholders when assessing the consequences of failure given that the geotechnical engineer is not qualified to identify all of the consequences involved (Fell et al. 2005).  It follows that regulations established by the regulatory agency based on best practices are required to ensure that minimum building codes are met.  Internal Shear 136  From the developer’s perspective, their risk is the loss of buildable land and the cost of additional investigation and design if a setback needs to be reestablished because the regulatory agency decides that the geotechnical evaluation was too liberal.  In the case of the Devil Canyon dip slope evaluation, it is estimated that the cost of reestablishing the building setback was approximately $125,000 per redesign, as a result of the additional efforts required by the design team.  The original site layout included structures that were planned at the crest of the slope (whether practical or not), and for every lot that lost approximately 10m at the perimeter, the estimated loss of revenue to the developer was an additional $150,000.  Thus, given the setback distance of 20m that was established following the guidelines imposed by the regulatory agency, shown in Fig. 4.9, this would affect ten or more lots, costing the developer in excess of $1.5million.  Alternatively, if the setback distance is too liberal and a slope failure occurs, the developer would be liable for the cost of the legal action that would surely be brought because of loss of property.  Given the above, the developer was highly motivated to ensure that a proper geotechnical investigation and design was completed, for which fees exceeded $2.0USD million.  4.6.2. Reliability-Based Design During the project, the developer’s accountants suggested performing risk based calculations so that they could weigh their costs according to their risk.  This evaluation was not completed because it was perceived that the outcome would have no impact on the regulatory agency that clearly had the final say on the setbacks that were required.  This was a rational request because uncertainty in geotechnical engineering is inherent and the fact that slopes have performed poorly after geotechnical evaluation and design shows that being conservative is not always sufficient (Duncan 2000).  In recent years, there has been a shift to reliability-based slope stability computation, although most regulatory agencies and owners have yet to embrace these more logical approaches. Apprehension to adopting these procedures appears to be rooted in: 1) most engineers having only elementary training in statistics and probability theory, 2) it is thought by many that probability analysis requires more detailed information than deterministic analysis, 3) there are few published case histories that show the ease of using probability theory, and 4) the acceptable probabilities of unsatisfactory performance are not defined in the geotechnical 137  literature (El-Ramly et al. 2002).  This is unfortunate because factor of safety coupled with a simple reliability analysis adds great value to the geotechnical design (Duncan 2000). 4.6.3. Probability of Failure Probability and reliability procedures used for geotechnical evaluations vary in their assumptions, limitations, mathematical complexity and capacity to handle different failure mechanisms (El-Ramly et al. 2002).  There are two preferred methodologies to consider probability and reliability in geotechnical engineering: Monte-Carlo simulation and the First- Order Second-Moment (FOSM) procedure.  There is an excellent discussion of Monte-Carlo sampling and its application to reliability based civil design in Harr (1987).  Harr (1987), Hoek (1989, 2007) and Duncan (2000) describe FOSM and its application to probability of failure (Pf).  In reassessing the Devil Canyon design, the FOSM procedure was applied to evaluate Pf because of its simplicity and its capacity to be completed using limit equilibrium and finite element analyses.  In addition, there is some precedence for its use in Southern California for slope stability applications (Moriwaki and Barneich 2001).  According to Duncan (2000), the FOSM process first involves estimating the mean and standard deviation of the input parameters required for the stability analysis.  For the Devil Canyon case, these include the Hoek-Brown shear strength parameters for the Grade R2 rock mass, the shear strength of the siltstone bedding, and the fluctuation of the water table in the slope. Secondly, a Taylor series (based on Rosenblueth’s 1981 point estimate method) is used to estimate the coefficient of variation (VF) and standard deviation () of the output, i.e. the factor of safety (F):                         Eq. 4.1            Eq. 4.2  Where F1 = (F1+ - F1-).  F1+ is the safety factor estimated by adding one  to the mean value of one of the input parameters.  F1- is the safety factor estimated by subtracting one standard deviation from the mean value of that input parameter.  FMLV is the most likely (or mean) value of         2222 2 4 2 3 2 2 2 1 FFFF F  MLV F F F V  138  the safety factor.  While calculating F1 all other input parameters are kept at their most likely values.  With F and VF known, the distribution of the output must be assumed (usually either log normal or normal distribution) so that the Reliability Index () associated with the probability distribution, VF and the FMLV, can be estimated.  Fisher et al. (2009b) found that the safety factor associated with dip slope failures is log normal.  LN, assuming a log-normal distribution, can be computed as:     Eq. 4.3   At this point, Pf can be estimated using standard statistics look-up tables or using built-in spreadsheet functions like NORMSDIST in Microsoft Excel (Duncan 2000).  The geotechnical input and distributions used to establish the Pf for the Devil Canyon setback design (static case) are summarized in Table 4.8.  Seismic loading is not considered because there are prescribed guidelines (and also desired design criteria) for performing deterministic as well as probabilistic slope stability evaluations (SCEC, 2002).  These methods have been adopted by the geotechnical community and there is no need to attempt to improve on those procedures here.  Table 4.10 shows the changes in rock mass shear strength estimates, as a function of varying the geotechnical input values, and the resulting safety factors using Rosenblueth’s (1981) point estimate.  It is not appropriate in this case to use the distributions of ’RM and c’RM directly in the Pf calculations because they are co-dependent.  An increase in ’RM coincides with a decrease in c’RM because of the curvature of the non-linear Hoek-Brown failure criterion used to scale and fit the linear Mohr-Coulomb envelope to.     2 2 1ln 1 ln F F MLV LN V V F       139  Table 4.10.  Change in factor of safety due to parameter variability under static loading conditions. Variable Values c’RM (kPa)  c'RM (kPa) ’RM (deg) 'RM (deg) F F GSI 51.0 196 52.0 2.04  39.0 148 48 48.3 3.7 1.80 0.24 ci 20.6 195  52.4  2.05  9.2 138 57 46.8 5.6 1.73 0.32 mi 18.3 172 50.8 1.94  15.7 166 6 49.5 1.3 1.88 0.06 D 0.27 164 49.6 1.88  0.13 175 -11 50.8 -1.2 1.95 -0.07  24.8 176  49.7  1.92  21.6 163 13 50.7 -1.0 1.91 0.01 GWT high 170 50 1.82  low 170 0 50 0 2.03 -0.21 Definitions: c’RM is the cohesion intercept for the rock mass.  c'RM is the difference in cohesion intercept for the rock mass while varying the input values. ’RM is the friction angle for the rock mass. 'RM is the difference in rock mass friction angle while varying the input values. F is the safety factor. F is the change in safety factor for the dip slope while varying the input values.   Table 4.11.  Probability of failure based on the results presented in Table 4.7. FMLV 1.92 VF 14% LN 4.73 Pf 1.1E-06   The Pf for the dip slope failure mechanism assumed (i.e. bi-planar), with a given failure surface that coincides with a setback distance at the slope crest, assuming static conditions and seasonable variation in the groundwater surface is about 1.1x10-6.  This Pf is very low and coupled with a minimum safety factor of 1.92 for the slip surface assumed, the developer (and regulatory agency) would likely conclude that the risk associated with constructing houses close to the crest of the slope is acceptable.  Of course, issues arise when trying to explain to potential homeowners that there is any Pf associated with their residence to begin with. Therefore, it is not difficult to understand why this practice has not become common place.  140  It is the author’s opinion that routinely calculating Pf holds benefits in terms of loss-prevention for the geotechnical consultant and can help to set a baseline criterion for the design safety factor and amount of data required to carry out the design in conjunction with building codes and regulatory requirements.  The author regularly uses this type of analysis internally to check the reliability of recommendations provided to clients.  This analysis can be extended to consider frequency analysis related to triggering mechanisms such as earthquakes or an increase in pore water pressures so that a more formal risk assessment may be performed.  For detailed information regarding the state of practice of landslide risk analysis, refer to Hungr et al. (2005). 4.7. Discussion A general consensus for the Devil Canyon Project was achieved between all stakeholders regarding the geologic model that was used to evaluate the stability state of the dip slope, as presented in the case history here.  The model was overly conservative, however, as the regulatory agency was unwilling to accept any arguments to the contrary. In the process, the developer expended considerable resources to have the project approved.  Ultimately, because of the cost and time involved, the developer decided the costs outweighed the benefits and they sold the project before it was built.  This disappointing outcome prompted the author to research dip slope problems further, examining the failure mechanisms involved, appropriate means for evaluating dip slope stability, and the uncertainty associated with the engineering geologic and geotechnical parameters required to complete dip slope evaluations.  The goal of the research was to develop a better mechanistic understanding of the problem and to use this to suggest practical recommendations that could be used to update guidelines, as well as to provide a framework for other practitioners facing similar design issues.  There are excellent guidelines available to assist engineering geologists and geotechnical engineers with evaluating the stability state of dip slopes (DMG 1997; SCEC 2002). Unfortunately, these guidelines are not always followed and the result is an unreasonable estimate of the bedrock shear strength and failure mechanism.  At Devil Canyon, assuming that toe breakout occurs along a highly persistent toe joint even though the geologic conditions 141  clearly indicate that this is unlikely, coupled with an imposed acceptance of a kinematically- infeasible failure mechanism, resulted in a dip slope design that was overly conservative.  On the other hand, choosing the alternative set of assumptions allowed by the regulators, where intact rock strength could be used to calculate the stability against toe breakout and shear even though the geology suggested that rock mass strength would be more appropriate, would (in other situations) have resulted in an overly-optimistic, and potentially unsafe, design.  (In this case, accounting for the actual rock mass shear strength, geological conditions, and failure mechanism suggests that structures can be safely constructed at the slope crest.)  Of course, there are other site conditions where either of these situations (i.e. persistent joint model of intact rock model) might be applicable and therefore, a thorough understanding of the geologic conditions is paramount.  In situations such as the Devil Canyon site conditions, where the cross-bed joints are truncated at bedding and the size of the blocks is small compared to the overall slope, the toe breakout mechanism occurs through rock mass failure and therefore it is the rock mass shear strength that is required for analysis.  In this case, state-of-the-practice dictates that GSI and the non- linear Hoek-Brown failure criterion are used to scale the linear Mohr-Coulomb shear strength parameters required.  This paper shows that numerical modeling is an appropriate tool to be used together with limit equilibrium for determining the stability state of dip slopes.  If numerical methods are not feasible due to certain project constraints, Sarma’s (1979) approach should be used to provide a conservative estimate of the dip slope stability state (Fisher and Eberhardt 2009a).  Hoek (1987) provides an algorithm and code based on a variation of Sarma’s method.  Geotechnical uncertainty related to dip slopes can be minimized by scoping the geotechnical investigation so that resources are spent on identifying the geotechnical input that is most influential to the dip slopes stability estimate (Fisher et. al., 2009b).  For shallow dip slopes, because the majority of stresses in the slope are distributed along the slope parallel sliding surface, it is advisable to characterize the shear strength of the bedding planes.  For steeper slopes, stresses are more distributed at the toe and therefore, a clear understanding of the rock mass quality at the toe of the slope is required.  142  Reliability based geotechnical design is becoming more prevalent.  Regulatory agencies, and in some cases the clients, may not be interested in these rational methods.  A client’s apprehension is not an excuse for not using these methods, however, given the clear value they provide in terms of loss-prevention and as a means for geotechnical engineers to evaluate the need for more detailed information and where to focus the limited resources available to them. 4.8. Conclusions and Lessons Learned Through the course of this study and based on a better understanding of dip slopes, the following suggestions can be forwarded:   Minimum requirements for geological and geotechnical investigation reports are beneficial but can be misinterpreted leading to unrealistic accounts of slope stability when the proper geological site conditions, rock mass shear strength, and failure mechanisms are not taken into consideration.   Dip slopes fail by a combination of sliding along a slope parallel discontinuity and through the development of a toe breakout surface.  Without internal shearing, this failure mechanism is not possible and therefore, the influence of this internal shear has to be taken into account when establishing the stability state of dip slopes.   Typical method of slices limit equilibrium solutions, i.e. those that were meant for circular slip surfaces, are not appropriate for estimating the stability state of dip slopes mainly because of their treatment of internal shear forces.  Finite and distinct element codes are preferred, although Sarma’s (1979) and Hoek’s (1987) solutions are the most appropriate limit equilibrium methods for evaluating dip slope stability and are recommended for practical purposes.   The Mohr rupture theory provides a means for determining the inclination of the toe breakout surface and internal shear and should be used as a first approximation for identifying these for input using Sarma’s (1979) or Hoek’s (1987) limit equilibrium solutions.   Near 90 degree failure surfaces (similar to that shown in Fig. 4.9) at the dip slope toe (without internal shear) are not kinematically-feasible and should be disregarded. 143    There are well-published methods of determining the shear strength of rock masses in bedded sedimentary sequences (e.g. Marinos and Hoek 2001; Hoek et al. 2002; Carvalho et al. 2007).  Therefore, using laboratory estimates of intact rock strength coupled with higher safety factors is not necessary or recommended.  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Schwartz (1996) "Probabilistic Seismic Hazard Assessment for the State of California," In: California Division of Mines and Geology, Ed., Open-File report 96-08, p 59.  RocScience (2007) "Phase2 Finite Element Program."  Rosenbleuth, E. (1981) "Two-point estimates in probabilities," Journal of Applied Mathematical Modelling, Vol. 5, pp 329-335. 148   Sarma, S.K. (1979) "Stability analysis of embankments and slopes," Geotechnique, Vol. 23, pp 423 - 433.  Southern California Earthquake Center (SCEC). 2002. Recommended Proceedings for Implementation of DMG Special Publication 117 Guidelines for Analyzing and Mitigating Landslide Hazards in California. In: T.F. Blake, R.A. Hollingsworth and J.P. Stewart (Editors). Southern California Earthquake Center, University of Southern California, Los Anglees, pp. 110 plus appendices.  Spencer, E. (1967) "A method of analysis of the stability of embankments assuming parallel inter-slice forces," Geotechnique, Vol. 17, pp 11-26.  Stark, Timothy D., and Hisham T. Eid (1994) "Drained residual strength of cohesive soils," Journal of Geotechnical Engineering, Vol. 120, No. 5, pp 856-871.  Stark, Timothy D., and Hisham T. Eid (1997) "Slope stability analyses in stiff fissured clays," Journal of Geotechnical and Geoenvironmental Engineering, Vol. 123, No. 4, pp 335-343.  Wyllie, Duncan C., and Christopher W. Mah (2004) Rock slope engineering: civil and mining, New York, NY: Spon Press.         149  5. THESIS DISCUSSION AND CONCLUSIONS This thesis presents a thorough investigation of the geologic environments, structural control, failure mechanisms, model and parameter uncertainty, and risk associated with bi-planar dip slope failures.  This study was predicated on a difficult consulting project with which, the author was involved between 2000 and 2005.  At the conclusion of the project, a number of questions remained regarding dip slopes and how to properly evaluate their stability state and the setback distance behind which residential development may safely proceed.  In addition, it was clear that these limitations in understanding have provided obstacles to others working on similar problems  and that there would be great merit in studying this issue further in the hopes of improving the engineering practice of dip slope geotechnical evaluations.  The purposes of this thesis were to 1) further our understanding of the bi-planar failure mechanism of dip slopes and the proper geologic models that should be used to establish dip slope stability, 2) evaluate geological and geotechnical uncertainty associated with bi-planar dip slope failures as a means to optimize geotechnical investigations, and to evaluate the existing guidelines within California for evaluating dip slopes in the context of the knowledge gained from this thesis research.  Following are the key findings and scientific contributions made from this research.  5.1. Summary 5.1.1. Analysis of Toe Breakout Mechanisms Our understanding of the bi-planar failure mechanisms was furthered by a thorough review of the literature on published case histories of dip slope failures that was carried out showing that most dip slope failures occur in weak, orthogonally jointed sedimentary rock and toe breakout involves sliding along joints, plastic failure of intact blocks, and intense deformation of the slope to enable kinematic release. Dip slope failures have been reported to exceed 200 meters in depth making this failure mechanism relevant in the context of planned development behind steep dip slopes.  Failure may develop over many years (sliding bending model), especially with natural slopes, and because the slope is bedded, the major principal stress may be assumed parallel to the slope face.  150  Analyses were carried out to investigate toe breakout mechanisms in dip slopes using a discontinuum-based numerical approach (distinct-element method). This approach represents the state-of-the-art practice for evaluating rock slope stability in discontinuous rock, although prior to this thesis, it had yet to be fully applied to dip slope problems.  The results indicate that the orientations of the principal stresses within the slope dictate the failure mechanism, but in all cases, internal shearing is required to facilitate slope release.  Bedded dip slopes fail in a manner that is predictable using classical the Mohr-Coulomb theory, but a transition exists for rock slopes where the dip parallel structure is much deeper into the slope (e.g. a slope parallel fault several 10's of metres behind the face), the slope may be treated as a continuum and toe breakout follows a log-spiral failure surface.  Because the depth of dip slope failures are generally shallow, assuming a linear toe breakout surface combined with internal shearing provides an accurate estimation of the actual failure surface.  This understanding provides guidance in selecting the correct limit equilibrium solution to apply in calculating a factor of safety for a dip slope by stipulating that the solution must account for internal shearing, for example as is done in Sarma's method.  A more accurate evaluation of the dip slope stability state can be achieved through the additional use of numerical modelling, but for practical application, limit equilibrium is sufficiently accurate.  However, as the depth to the basal shear surface increases or the slope angle decreases, limit equilibrium solutions that assume a linear toe breakout release surface becomes less accurate. This is because: 1) 1 rotates from the slope-parallel orientation seen for cases involving thinner bi-planar slabs; and 2) the toe breakout surface is more circular in shape, invalidating many of the assumptions that apply to the linear toe breakout case. 5.1.2. Analysis of Parameter Uncertainty Uncertainty associated with the geotechnical parameters required to properly carry out a dip slope stability analysis is similar to that encountered in other rock engineering situations. Results from the analyses carried out show that for bi-planar failures of dip slopes, the influence of the geotechnical input parameters is directly related to the slope inclination and the transference of the active driving forces (e.g. distribution of stresses) in the slope.  For shallow dipping dip slopes, the shear strength of the slope parallel sliding surface is most critical and considerable effort should be spent quantifying its shear strength.  For steeper dip slopes, more 151  stress is transferred from the upper slope to the slope toe, resulting in the shear strength of the internal shear and toe breakout surface having a more significant influence.  Thus for steeper dip slopes, where shearing is expected through the rock mass, focus must also be extended to the quantification of the rock mass properties.  If a persistent joint or fault daylights the slope toe, then establishing its shear strength holds the greatest value.  Based on these results, guidelines were developed to provide a basis for planning geotechnical field investigations and laboratory testing for the purpose of rock mass characterization of dip slopes.  This moves towards the objective of increasing the efficiency and value return of field and laboratory investigations.  Overall, because of the relative influence of different rock mass parameters on the calculated dip slope stability state, value engineering should focus on those input parameters that have the greatest influence on the dip slope stability.  Thus, where the dip slope is shallow Less than 45 degrees, the minimum geotechnical investigation required for due diligence is one that consists of performing a field mapping regime, followed by a testing program that concentrates on establishing the bedding shear strength.  To a lesser degree, characterization of the rock mass and/or discontinuity shear strength at the toe of the slope may be included as part of the program, although it may be sufficient to estimate this using mapping data and typical values from the literature. More expensive testing is not necessary because the value associated with further constraining the rock mass shear strength at the toe is minimal; in most cases, establishing the rock mass shear strength using existing empirical techniques is sufficient.  One unnecessary task, which was believed to be important prior to this study, was laboratory testing to establish the Hoek-Brown empirical coefficient mi for the rock mass, especially given the high costs associated with completing consolidated undrained triaxial testing.  For steep dip slopes (greater than about 45 degrees), in contrast to shallow dip slopes, a detailed understanding of the rock mass shear strength becomes the more critical set of parameters to constrain, given the change in failure mechanism to one where toe breakout and internal shearing dictate the slope’s stability state.  Once again laboratory testing to establish the mi value is not necessary, and expensive torsional ring shear tests to establish the bedding shear strength are not justified. Instead, empirical correlations to derive bedding shear strength based on the sedimentary lithology are sufficient. 152  5.1.3. Evaluation of Current Guidelines and Design of Setback Distances A detailed case history involving a dip slope stability assessment and setback design project in Southern California was carried out. This project provided much of the motivation for the research carried out in this thesis.  On this project, despite the availability of professional guidelines specifically developed and intended for the problem at hand (i.e., the stability analysis of a dip slope), published guidelines and the current state-of-practice were found to be largely inadequate. Details regarding the geology and likely failure mechanism were not properly accounted for in the guidelines forcing the use of unreasonable estimates of rock mass shear strength and assumptions regarding the failure mechanism.  In summary, the review provides an account of how the rock mass characterization of the dip slope in Southern California and subsequent analyses, completed according to the existing guidelines, did not provide a correct result for the in situ conditions encountered.  Yet because the risk for the project was thought to reside with the regulatory agency, they in turn dictated the geologic model to be used for the project.  A re-evaluation of the dip slope stability assessment was performed to illustrate how a firm understanding of the geological conditions at the site coupled with an understanding of dip slope failure mechanisms, would have resulted in a more sound design of the dip slope setback distance.  The results suggest that the regulatory agency's required assumption of toe breakout along a highly persistent toe joint was not supported by the geologic conditions mapped.  This, coupled with an imposed acceptance of a kinematically-improbable failure mechanism, resulted in a dip slope design that was unrealistic.  On the other hand, the alternative set of assumptions allowed by the regulators, where intact rock strength could be used to calculate the stability against toe breakout and shear, likewise was not supported by the geology and would have resulted in an overly-optimistic, and potentially unsafe, design.  Based on the comparative analysis presented, a set of improved recommendations were developed as follows:   Reliance on the actual geologic conditions to establish the most likely dip slope failure mode as opposed to predetermining the failure mechanism as is commonly done,  153   Accounting for internal shearing and its influence on calculated safety factors,   Using limit equilibrium analysis tools to establish the dip slopes’ stability state and then checking that estimate against more elaborate numerical codes.   Using “plasticity theory” and Mohr rupture criterion to estimate the inclination of both the toe breakout surface and internal shear surface where toe breakout is expected through the rock mass and not along pre-existing discontinuities such as persistent joints or faults, and   Understanding that kinematically-infeasible failure surfaces are not rational even though some (inadequately applied) limit equilibrium models may show theses surfaces produce the lowest safety factors. 5.2. Key Conclusions and Scientific Contributions This thesis provides an important update to the state-of-practice for evaluating dip slopes and designing setback distances for infrastructure.  It is based on a detailed literature review, evaluation of published case histories, numerical modeling, and statistical analysis to verify failure mechanisms and analysis techniques.  Furthermore, many of the findings presented, especially those related to means developed to address issues of geotechnical uncertainty, are also applicable to rock mass characterization and geotechnical risk evaluations in general.  The analysis techniques presented are practical and can easily be incorporated by practitioners which was one of the main goals of the thesis research.  The main conclusions and key contributions made through the different components of the research carried out in this thesis study are listed below in the following subsections.  5.2.1. Methodology for Assessing Dip Slope Stability  The most common dip slope failures occur in weak; orthogonally jointed sedimentary rock (sometimes referred to as flysch sequences).  Within orthogonally-jointed sedimentary rock, toe breakout involves sliding along joints, plastic failure of intact blocks, and intense deformation of the slope to allow kinematic release. 154    A first approximation of the depth of dip slope failures can be related to the slope height and inclination using D/H ratios as described in Chapter 2.   Typical dip slopes are bedded, and the major principal stress may be assumed parallel to the slope face.  Therefore, the Mohr-Coulomb theory is a practical means to estimate the location and inclination of the internal shear and toe breakout surface.   Where the sliding mass is deep within the slope, or the slope is shallow, and the rock mass is treated as a continuum, the toe breakout surface and internal shear are not linear and using the Mohr-Coulomb theory to predict the location and orientation of the toe breakout surface and internal shear is less accurate  than using finite element.   The failure mechanisms associated with bi-planar failures where the toe breakout surface is not constrained by a prominent discontinuity that dips out of the slope, are complex but even so, there are practical methods (such as assuming that the rock mass is an equivalent continuum) for assessing the slope stability.   This thesis provides a practical methodology for estimating the stability of dip slopes based on the slope angle, the geological conditions, and the anticipated depth to the slope parallel sliding surface.  In summary, Sarma’s limit equilibrium method may be used to estimate dip slope stability with certainty where the slopes are steep and the depth to the basal shear surface is shallow.  As the depth to the basal shear surface increases or the slope angle decreases, using this approach becomes less accurate because 1) 1 rotates from parallel to the slope face in the case of shallow slopes and 2) the toe breakout surface is log spiral and the assumption that it is linear provides less accurate results. 5.2.2. Site Investigations  With regards to calculating the stability state of dip slopes, most of the uncertainty lies in the geologic model assumed, and the geotechnical parameters used in the evaluation. Bi-planar sliding in dip slopes occurs along a slope parallel sliding surface with toe breakout occurring at the base of the failure.  Internal shearing is required to facilitate 155  kinematic release.  All three of these surfaces work together for the slope to fail, but with different degrees of importance depending on the dip of the slope.   Increased efficiency and value with respect to the site investigation budget can be gained by working towards minimizing the uncertainty of those parameters that have the greatest bearing on the outcome of the slope stability analysis.  This can be done quickly and inexpensively by performing scoping calculations facilitated by the use of Spearman Rank Correlation Coefficients.  These calculations can be completed through programming of computer-based spreadsheets, or more conveniently, by using statistical add-on packages (in this study the Excel spreadsheet add-on @RISK was used).   For shallow dipping dip slopes (less than about 45 degrees), stability is primarily dictated by the shear strength of the slope parallel sliding surface and therefore, efforts should be focused on constraining the shear strength of this surface.   For steep dip slopes (greater than about 45 degrees), the shear strength related to the toe breakout and internal shear release surfaces becomes dominant and therefore, the rock mass shear strength as well as that for any adversely dipping persistent discontinuities should be the focus of the geotechnical investigation.   Once the minimum information is collected, there may be an opportunity to provide value engineering to the project.  With regards to steep dip slopes, it may be justifiable to constrain the rock mass characteristics at the toe and shear strength properties based on detailed rock core logging, downhole televiewing, and additional testing.  For shallow slopes, value engineering may be added by supplementing the outcrop mapping with core logging and downhole televiewing to analyze bedding spacing and shear strength, as well as performing Atterberg limits testing to better constrain the shear strength of the slope parallel discontinuity.  In either case, establishing mi values for the purpose of determining the rock mass shear strength is unnecessary, despite the fact that it is often used as the basis for carrying out laboratory triaxial testing. Based on the parameter uncertainty analysis carried out, simple estimates of the intact rock properties based on typical values taken from the literature are sufficient.  156   This thesis shows that a better understanding of the influence of the geotechnical input coupled with engineering judgment can help to focus the geotechnical investigation so that it is as efficient as possible. 5.2.3. Design of Set-Back Distances  Minimum requirements for geological and geotechnical investigation reports are beneficial but can be misinterpreted leading to either overly optimistic or conservative accounts of slope stability when the proper geological site conditions, rock mass shear strength, and failure mechanisms are not taken into consideration.   Typical method of slices limit equilibrium solutions, i.e. those that were meant for circular slip surfaces, are not appropriate for estimating the stability state of dip slopes mainly because of their treatment of internal forces.  Finite and distinct element codes are preferred, although Sarma’s (1979) and Hoek’s (1987) solutions are the most appropriate limit equilibrium methods for evaluating dip slope stability and are recommended for practical purposes.   Kinematically-infeasible failure surfaces at the dip slope toe (without internal shear) are not rational and should be disregarded.   There are well-published methods of determining the shear strength of rock masses in bedded sedimentary sequences.  Therefore, following the prescribed methodology of using laboratory estimates of intact rock strength coupled with higher safety factors is not necessary or recommended. 5.3. Future Research Although the research presented herein is comprehensive, there are several areas where further research would be well merited.  These include:   Confirming dip slope failure mechanisms using more sophisticated modeling software such as ELFEN, a finite/discrete element code that incorporates brittle fracturing of intact rock.  The toe breakout failure mechanism modelled using UDEC was implied based on localized shear strain and plasticity indicators. Brittle fracture codes like ELFEN would 157  enable failure to be modelled explicitly in terms of the initiation and propagation of brittle fractures across a system of non-persistent joints and intact rock at the toe of the dip slope.   Establishing how fracture damage, energy, fatigue, and time (i.e. progressive failure) contribute to dip slope failures in natural and engineered rock slopes, scenarios that involve different time frames in terms of how the instability develops (i.e. one over periods in line with natural geological processes, and one over the time frame of an operating open pit mine) .   Refining methods for establishing rock mass shear strength where toe breakout is expected through step-path failure at the toe of dip slopes. This could involve a combination of empirical and laboratory data analysis supported by numerical modeling.   Performance engineering of natural or constructed dip slopes as a predictive tool for dip slope stability.  158  5.4. References Hoek, E. (1987) "General two-dimensional slope stability analyses," In: E. T. Brown, Ed., Analytical Computational method in Engineering Rock Mechanics: Allen & Unwin; London, pp 95-128.  Sarma, S.K. (1979) "Stability analysis of embankments and slopes," Geotechnique, Vol. 23, pp 423 - 433.     159  APPENDIX A. INTERNATIONAL DATABASE OF DIP SLOPE FAILURES  1. Nchanga, Zambia:  Terbrugge, P. J., and Hanif, M. (1981) "Discussion of a large failure on the footwall of the Nchanga open Pit, Zambia." International Symposium on Weak Rock, Tokyo, pp 1499- 1501.  Terbrugge and Hanif (1981) describe the failure of a soft rock mass within a benched footwall of the Nchanga open pit Mine in Chingola, Zambia.  The slope height at failure was 184 m and inclined at about 30 degrees with a failure depth (normal to the slope) ranging from about ten to 20 m.  The geology consisted of feldspathic quartzite ore, banded sandstone, banded shale, and quartzite. The failure was attributed to undercutting of the feldspathic quartzite ore and high water pressures within the slope.  The location of the water table at the time of failure is unknown although moisture contents of 30 percent were reported for the banded sandstone.  Judging from the cross-section provided, the top of the banded sandstone was about 10m above the failure plane.  2. Luscar Coal Mine, Canada:  Cruden, D. M., and Masoumzadeh, S. (1987) "Accelerating creep of the slopes of a coal mine." Rock Mechanics and Rock Engineering, Vol. 20, pp 123 - 135.  Cruden and Masoumzadeh (1987) describe the process of testing four rock creep models for prediction of a dip slope failure within the Luscar coal mine in Alberta, Canada.  The slope in question consisted of Lower Cretaceous coal-bearing rocks consisting of interbedded sandstones and siltstones dipping at about 33 degrees and coincident with the mined slope. The toe breakout surface was located at a bend (kink) in the slope although the design cross- section clearly depicts a bi-planar failure and not a buckling failure which would be more common in a slope that has a bend in the geology.  The slope height measured from the toe breakout surface to the slope crest is approximately 100m while the thickness of the failed mass ranged from about 26 to 35 or more meters.  The 26 m section was located near the toe 160  indicating the thickness of the mass through which sliding occurred.  Failure was attributed to slope creep.  3. Bawnshan, China:  Li, Q., and Zhang, Z. Y. (1990) “Mechanism of buckling and creep-buckling failure of the bedded rock mass on the consequent slopes." 6th International Association of Engineering Geologists Congress, Amsterdam, pp 2229 - 2233.  This paper discusses the sliding/bending type creep failure of dip slopes that eventually evolves into a bi-planar failure with the toe breakout occurring at the base of the slope.  Li and Zhang (1990) provide a solution for this type of bi-planar sliding based on an energy method of the stability of thin plates.  They state that although the solution is highly theoretical, it may be applicable to this failure mechanism.  Within the paper they provide a description and cross- section of the Bawnshan landslide located along the Yalong River in China depicting the evolution of the failure mechanism.  The slope consists of what is drawn as orthogonally-jointed limestone and marl units with a height of 940 m and a failure depth of 10m.  The slope is inclined at 40 degrees.  Li and Zhang use this slope as a case history to show that pure elastic buckling should not occur and that the sliding bending model is more appropriate.  4. Taipei City, Taiwan  Chen, H. (1992)  "Appropriate model for hazard analysis in slope engineering" Sixth international symposium on Landslides, Christchurch, New Zealand, Feb. 10-14, 1992, No.  6, pp 349-354.  Chen (1992) discusses the back analysis of a slope located below a residential development. This is a natural slope although a roadway was excavated into the slope at about half the slope height.  The slope consists of sandstone and shale dipping approximately 30 degrees and coincident with the slope face.  Tension cracks are located at two locations, one about 16 m above the slope toe and one set located at 26 m above the toe.  The larger mass is considered a planar slide along bedding while Chen suggests that the toe area may be modeled as a circular failure.  The larger planar slide appears to be about 6 m thick.  161  5. Smoky River Coal, Canada:  Dawson and Barron. (1991)  "Design guidelines for bedded footwall slopes - final report." Department of mining, Meturgical and Petroleum Engineering; University of Alberta, Edmonton, Alberta.  Dawson and Barron (1991) provide a detailed slope stability evaluation of a footwall failure at Smokey River Coal in Alberta, Canada.  The slope was approximately 100m tall and inclined at about 60 degrees.  The geology consisted of orthogonally-jointed sedimentary units consisting of interbedded sandstone, siltstone, mudstone, and coal.  Back analysis of the slope suggest that the failure was bi-linear with the toe breakout occurring through the sedimentary rock mass and the internal shear consisting of a “shear” zone located above the toe.  The depth of the failure was estimated at 12 m; ranging from eight to 15 m.  Failure is attributed to daylighting of the shear zone and an increase in water pressure within the rock slope.  Back analysis of the rock slope completed for this thesis suggests that the groundwater was located approximately 30m above the toe of the slope.  6.Jinlong Mountain at the Yalong River, China:  Wang, L., Zhang, Z., Cheng, M., Xu, J., Li, T., and Dong, X. (1992)  "Suggestion on the systematical classification for slope deformation and failure; Landslides--Glissements de terrain." Sixth international symposium on Landslides, Christchurch, New Zealand, Feb. 10-14, 1992, No. 6, Vol. 3, pp 1869-1877.  Wang et al. (1992) provide a classification system for slope failures.  They describe progressive failure of dip slopes as a process where creep of the slope causes disturbed stresses at the slope toe.  Over time the slope fails by bending and cracking with eventual toe breakout where bending is observed at the base of the slope.  They provide a cross-section of a deforming rock mass along the bank of Jinlong mountain, Yalong River, China.  The section suggests that the slope consists of sedimentary units that are orthogonally-jointed with a dip about 20 to 25 degrees.  The slope height is 600 m with a thickness (measured normal to bedding) of about 100 to 200m.  Failure is attributed to creep deformation of the slope.  162   7. Aguas Claras Mine, Brazil:  Behrens da Franca, P. R. (1997) “Analysis of slope stability using limit equilibrium and numerical methods with case examples from the Aguas Claras Mine, Brazil." MSc Thesis, Queen's University, Ontario, Canada.  The Aquas Claras Mine is located in Brazil at Curral Mountain southwest of Belo Horizonte and was the location of a large dip slope failure that occurred in 1992.  The benched slope was approximately 240 m high with an overall slope angle of 44 degrees.  The surface of the slope consisted of “soft” hematite with a layer of itabirite located approximately 30m behind and parallel to the slope.  At the contact of the hematite and itabirite, there was a thin layer of soft itabirite or leached iron formation.  This thin leached iron formation is believed to be the slope parallel sliding surface of the dip slope failure.  Failure of the slope is attributed to mining activities and oversteepening.  8. Fuxin Haizhou Coal Mine, China:  Yang, T. H., Xu, T., Rui, R. Q., and Tand, C. A. (2004) "The deformation mechanism of a layered creeping coal mine slope and the associated stability assessments." International Journal of Rock Mechanics and Mining Science & Geomechanics Abstracts, 41 (n SUPPL. 1) (3B 10), pp 1-6.  Yang et al. (2004) provide an analysis of long term creep of a dip slope within the Fuxin Haizhou open pit coal mine which is located in Fuxin City, China.  The slope is inclined at about 18 to 20 degrees with a fault located near the slope toe.  On the hanging wall of the fault, located along the lower portion of the slope, the geology consists of coal, coarse sandstone and gravel.  The footwall consists of sandy shale, sandstone and carbonaceous shale.   Sliding is occurring on the bedding within the footwall of the slope (upper portions) while shearing is occurring through the rock mass of the hanging wall.  The authors state that the shearing through the rock mass is occurring at an angle (measured from bedding) of 45 – ’/2 where ’ is the friction angle of the rock mass.  The fault enables kinematic release of the slope (internal shear).  The slope is approximately 150 m high with a failure depth of 35 to 40 m. Although creep is considered the 163  main failure mechanism, high water pressures occur in the slope and drainage (as well as rock bolting through the slope toe) has been employed to arrest movement.  The authors provide a cross-section that shows the piezometric head at an elevation above the ground surface prior to the onset of the slope failure.  Drainage adits targeted the slope parallel sliding surface and intersection of the toe breakout surface with the slope parallel sliding surface.  9. Leigh Creek Coal Mine, Australia:  Lucas, D. (2006) “Stress failure of a shallow open cut mine." Australian Centre for Geomechanics December 2006 Newsletter, pp 4-6.  A 60m tall pit slope failure occurred in the Leigh Creek coal mine in South Australia in 2001. The pit slope was excavated coincident with the layering of the coal, thereby creating a dip slope situation.  Failure occurred at a depth of eight to 10m behind the face along a “bedding” shear surface.  Failure was attributed to locked-in stresses in the slope.  The failure was modeled using UDEC and appeared to show a toe breakout surface that is consistent with that predicted using the Mohr-Coulomb theory assuming that the principal stresses are parallel to the slope face after excavation.  10. Rufi, Switzerland:   Eberhardt, E., Thuro, K., and Luginbuehl, M. (2005) “Slope instability mechanisms in dipping interbedded conglomerates and weathered marls – The 1999 Rufi landslide, Switzerland." Engineering Geology, Vol. 77, No. 1-2, pp 35-56.  Eberhardt et al. (2005) describe the back analysis of a natural dip slope using numerical modeling where the toe of the slope was modified by construction of a roadway.  The slope failure is located in the northeastern part of Switzerland.  The 20 degree slope consists of a series of interbedded marls, conglomerates, and sandstones (subalpine Molasse).  Photographs of the geology presented in the paper indicate that the sedimentary units are orthogonally- jointed.  Failure occurred beneath the roadway with the toe breakout surface daylighting about approximately five to 10 m down slope. Failure occurred through a marl bed following a 164  weathering profile, triggered by snow melt and heavy rain.  The slope failure height is approximately 56 m with an average depth of failure of about 5.3m.  Groundwater conditions are not explicitly reported although the authors completed a series of back analysis to establish the shear strength of bedding within the slope.  Groundwater conditions modeled included (i) a dry slope, (ii) a water table located 1.25 m below surface, (iii) a water table at surface and (iv) artesian conditions.  11. Portillo, Chile:  Welkner, D. (2008) "Intergrated field investigation, numerical analysis and hazard assessment of the Portillo Rock Avalanche site, Central Andes, Chile," MSc Thesis, University of British Columbia, Vancouver.  As part of her Master’s thesis, Welkner (2008) investigated a large prehistoric rock avalanche that failed along a dip slope consisting of interbedded andesitic lava flows, volcaniclastics, and sedimentary sequences.  Through runout volume calculations and numerical modeling, Welkner was able to reconstruct the geometry and kinematics of the failed volume.  The slope was approximately 930m tall and the failure surface was estimated at 100 to 150m deep.  The failure mechanism was most likely related to undercutting of the toe during glacial advance and retreat, and was potentially triggered by a very large earthquake.  165  APPENDIX B. INFLUENCE OF GROUNDWATER ON D/H RATIOS  B.1     Introduction Chapter 2 presents a summary of 11 case histories reported in the geotechnical literature that were used as a basis to relate maximum depth of failure (D; measured normal to the slope face) to height of slope (H) for failed dip slopes.  Table 2.1 tabulates this data while Fig. 2.2 provides a graphical representation.  In addition, Appendix A provides a detailed account of the case histories referenced for this thesis.  Fig. 2.2 suggests that with an increase in dip slope inclination, D/H ratios decrease.  Upper bound D/H ratios are bracketed to establish a practical limit and first approximation of the depth of dip slope failures expected based on slope inclination.  Intuitively, there should be a relationship between water pressure in dip slopes and the depth at which failure occurs.  To test this hypothesis, four of the 11 case histories were reevaluated (those where increased groundwater pressures were attributed to failure) with the emphasis placed on groundwater pressures; their influence on a decrease in effective shear strength along the slope parallel sliding surface, and D/H ratios.  Groundwater within dip slopes decreases the stability state of the slope because of an increase in water pressure that develops along the slope parallel sliding surface, internal shear, and toe breakout surfaces.    In some cases, the presence of water also decreases the strength of a rock mass if that rock mass encompasses degradable rock materials. B.2     Distribution of Water Pressures in Dip Slopes Hoek and Bray, 1984 demonstrate the influence of slope anisotropy on the distribution of equipotential lines (from flow nets) and groundwater pressure distributions.  Dip slopes are typically bedded and therefore, the permeability of the slope is greatest parallel to the slope face.  Fig. B.1 is taken from Hoek and Bray, 1984. 166   Fig. B. 1.  Permeability ratio for a dip slope assuming anisotropy parallel to the slope face.  Inspection of Fig B.1 suggests that assuming the water pressure along a slope parallel sliding surface or toe breakout surface is hydostatic and can be taken as the height of the water above the failure plane is not conservative.  Consider Fig B.2 taken from Dawson and Barron, 1991.  167   Fig. B. 2.  Water pressures within a bedded dip slope.  The hydrostatic water pressure at Point B may be estimated as the height of water (dh) times the unit weight of water.  A piezometer installed below the toe of the dip slope would most likely read dP2 and in this case, the actual water pressure would be calculated as dP2 times the unit weight of water (see Eq. B.1).  vB= dp2 x w x unit slope        Eq. B.1  Where vB is the uplift pressure at Point B, dP2 is the estimated pressure head at Point B, and w is the unit weight of water.  168  In practice, the actual height of dP2 would be difficult to estimate without piezometer readings.  dh may be related to the slope inclination and depth (D) to slope parallel sliding surface measured normal to the slope face as follows:  bh = D/cos           Eq. B.2   is the slope angle.  The total stress at Point B (Fig. B. 2) is calculated as follows:    = r x dh          Eq. B.3  Where r is the rock mass unit weight and dh is the height of the rock above Point B.  The pore water pressure coefficient (ru) is simply the ratio between the total stress and uplift pressure.  ru = vb/          Eq. B.4  The apparent friction angle at Point B may be calculated as:  app = tan-1[(1-ru) tan        Eq. B.5  Typically, the slope parallel sliding surface consists of a fine-grained material with a friction angle of 25 degrees or less. B.3     Apparent Friction Angles from Dip Slope Case Histories Of the 11 case histories reported in Chapter 2, four attribute dip slope failure to an increase in water pressure. Table B. 1, below provides a summary of the slope geometries, anticipated friction angles along the slope parallel sliding surfaces, the authors interpretation of groundwater pressures within the dip slopes, and apparent friction angles (app). 169   Table B. 1a-c.  Dip slopes where groundwater was reported as a triggering mechanism. a) Case histories as reported with groundwater location inferred. No.  Case History  Angle, deg Height, m Ave. Depth, m Ave. D/H r, kN/m3  w, kN,m3 '  1  Nchanga Open Pit  30 184 15 0.08 25 9.81 25 5  Smoky River Coal  60 97 12 0.12 25 9.81 25 8  Fuxin, China  19 150 38 0.25 25 9.81 25 10  NE Switzerland  20 56 5 0.09 25 9.81 20  b) Case histories as reported with inferred shear strength parameters. No.  Case History dh, m dP2 kPa vb, kPa ru app, deg.  1  Nchanga Open Pit 17 15 433  147 0.34 18 5  Smoky River Coal 24 6 600  59 0.10 24 8  Fuxin, China 40 80 1004  789 0.78 6 10  NE Switzerland 5 5 133  52 0.39 13  c) Case histories as reported with notes regarding groundwater conditions. No.  Case History  Notes Regarding Groundwater Conditions 1  Nchanga Open Pit  Groundwater height not reported; 30 % moisture in a "Banded Sandstone" located 10 m above the failure surface.   5  Smoky River Coal  Groundwater assumed to be that which resulted in safety factor of 1.0 during back analysis for this thesis. 8  Fuxin, China  Groundwater prior to failure located at the ground surface.   10  NE Switzerland  Groundwater reported to range from dry to artesian conditions. Failure through intact rock.    170  Fig. B.3 relates the apparent friction angles calculated and presented in to D/H ratios reported in Table B. 1.  Fig. B. 3.  Apparent friction angles and D/H ratios from reported dip slope failures.  The following conclusions may be drawn from Table B.1 and Fig. B.3: 1. There is a general decrease in the frictional strength available at slope failure as the slope inclination decreases i.e. app ranges from 24 degrees for a 60 degree slope to 6 degrees for a 19 degree slope (Table B.1). 2. Fig. B.3 suggests that as the effective frictional strength of the slope parallel sliding surface decreases, (i.e. app) there is an increase in the depth of failure.  This is intuitive. 3. The groundwater elevations stated in Table B.1 (dP2) are largely unknown and based on the judgment of the author.  Therefore the validity of Fig. B.3 is questionable. 4. Given the uncertainty associated with Fig. B.3, more reliance should be placed on estimating D/H ratios from Fig. 2.2 (Chapter 2).  Groundwater is implicitly included in that figure.    171  B.4     References Dawson and Barron. (1991)  "Design guidelines for bedded footwall slopes - final report." Department of mining, Meturgical and Petroleum Engineering; University of Alberta, Edmonton, Alberta.  Hoek, Evert, and John Bray (1974) Rock slope engineering, London: Institution of Mining and Metallurgy.  172  APPENDIX C. MOHR-COULOMB SHEAR STRENGTH ESTIMATES FOR DIP SLOPES C.1     Published Methodology As discussed in Chapter 3, in many situations, the goal of a geotechnical investigation is to establish the rock mass shear strength properties required to carry out a stability analysis.  The Mohr-Coulomb failure criterion is widely employed for this purpose, although issues arise with respect to appropriately scaling laboratory-based values to those that are more representative at the rock mass scale. For this, Hoek et al. (2002) provide a methodology for establishing the Mohr-Coulomb “rock mass” shear strength parameters based on the geological and geotechnical site conditions.  This procedure, encoded in the Rocscience program RocLab (2007), uses a non-linear Hoek-Brown failure envelope to define the laboratory-based intact rock strength and the Geological Strength Index (GSI) to scale to the rock mass properties. The GSI can be accessed directly in the field or estimated from the RMR (Hoek and Brown 1997), and is based on the blockiness of the rock mass and the surface conditions of the discontinuities (Hoek et al. 1995). An estimate of the maximum value of the minimum principal stress (3max) is then used to superimpose a linear Mohr-Coulomb shear strength failure envelope from which the rock mass cohesion and friction angle values are obtained.  The procedure outlined in Hoek et al., (2002) is based on the assumption that a circular failure occurs through a rock slope considered homogeneous and isotropic without prominent discontinuities influencing the location of curvature of that failure surface.  Hoek et al., (2002) rely on closed form estimates of 3 (minor principal stress) for these slope types to generate a characteristic (empirical) curve relating slope height, rock unit weight, and the rock mass uniaxial compressive strength to maximum 3 within the rock mass at failure.  Although appropriate for a first estimate of the rock mass shear strength, this published methodology may not be appropriate for fitting a linear Mohr-Coulomb envelope to a Hoek-Brown failure envelop for dip slopes where the rock at the toe of the slope may be considered a homogeneous rock mass.  This is because the assumptions required to develop the characteristic curves of Hoek et al. (2002) are not applicable for dip slopes where the slope parallel sliding surface greatly influences the location of sliding and distribution of stresses at the slopes’ toe.  173  This Appendix outlines a more appropriate method for establishing a linear Mohr-Coulomb fit to a Hoek-Brown failure envelope using an iterative process to evaluate normal stresses acting on the toe breakout and internal shear surfaces of dip slopes. C.2     Procedure to Estimate Mohr-Coulomb Parameters for Dip Slopes Fig. C. 1 is a cross-section of a dip slope where the stability state of the slope is in question. The slope is composed primarily of sandstone. One shale layer has been identified at a distance of 21.2 m behind the crest of the slope.  This shale layer is weaker than the surrounding sandstone (’ of 25 degrees and c’ of zero kPa) and therefore is a candidate for the slope parallel sliding surface.  The slope height is 100 m and the D/H ratio of the slope (described in Chapter 2) is about 0.15.  Field mapping at the toe of the dip slope suggests that the sandstone is heavily jointed and can be assumed homogeneous and isotropic.  Therefore, the shear strength of the sandstone rock mass at the toe may be described using the Hoek-Brown Failure Criterion.  The uniaxial compressive strength of the intact sandstone is about 7 Mpa, the unit weight of the sandstone is 22 kN/m3, and the GSI is about 40.  From Hoek et al., (2002), the mi value for sandstone is 17 and because this is a natural slope, the Disturbance Factor (D) is assumed zero.  Fig. C. 1.  Example dip slope with D/H ratio of 0.15.   174  C.2.1  Step One – Establish the Hoek-Brown Failure Envelop The Hoek-Brown failure envelop for the sandstone rock mass may be established using the procedures in Hoek et al., (2002).  The algorithms for establishing the envelope are encoded in the RocLab program from RocScience (2007).  C.2.2  Step Two – First Approximation of Mohr-Coulomb Parameters A first approximation of the Mohr-Coulomb parameters for the anticipated stresses within the dip slope is achieved by inputting the unit weight of the rock mass and the height of the slope in RocLab.  The computer program estimates the stresses within the dip slope and the Mohr- Coulomb parameters for the anticipated range of stresses are  of 33 degrees and c of 320 kPa (Fig. C. 2)  Fig. C. 2.  RocLab estimate of Mohr-Coulomb parameters by fitting a straight line to the Hoek-Brown failure envelope.  This procedure is outlined in Hoek et. al., (2002).  175  C.2.3  Step Three – Estimate Normal Stresses at the Dip Slope Toe Given  of 33 degrees, the inclinations of the toe breakout and internal shear surfaces may be estimated using Eq. 2.1 and 2.2.  A first approximation of the dip slope stability and the normal stresses generated along the toe breakout and internal shear surfaces may be determined using the “Sarma” computer program by Hoek (1985).  The input parameters required include the geometry of the slope, the inclination of the toe breakout and internal shear surfaces, and the engineering properties of the rock mass and shale unit.  A screen capture showing the results of the computer simulation is presented in Fig. C. 3.  Screen capture of the Sarma program (Hoek, 1985).  Results show the normal stress at the toe breakout surface (base stresses for Side number 1) and internal shear surface (side stresses for Side number 1).  The coordinate xt is 0.00..  The normal stress on the toe breakout surface is 475.42 kPa and the normal stress on the internal shear surface is 259.02 kPa.  The average normal stress is about 367 kPa.  The safety factor for the slope is about 1.21.  Fig. C. 3.  Screen capture of the Sarma program (Hoek, 1985).  Results show the normal stress at the toe breakout surface (base stresses for Side number 1) and internal shear surface (side stresses for Side number 1).  The coordinate xt is 0.00. 176   C.2.4  Step Four – Iterate Mohr-Coulomb Parameters Based on Normal Stresses Fig. C. 4 is a printout from the RocLab program showing a line drawn tangent to the Hoek- Brown failure envelope at the average normal stress of about 370 kPa.  The Mohr-Coulomb parameters for this normal stress are  of 44 degrees and c of 150 kPa.    Fig. C. 4.  RocLab program is used to predict the instantaneous Mohr-Coulomb parameters by a tangential linear best-fit at the average normal stress.  Because the anticipated friction angle of the rock mass has changed from 33 to 44 degrees, the inclinations of the toe breakout and internal shear surfaces have also changed (from Eq. 2.1 and Eq. 2.2).  Based on the new failure surfaces and Mohr-Coulomb parameters, the Sarma computer program predicts a safety factor of 1.15 with normal stresses on the toe breakout and internal shear surfaces of about 355 kPa and 170 kPa, respectively.  177  With an average normal stress of about 270 kPa, the RocLab program predicts  of about 47 degrees and c of 120 kPa.  The third iteration of the Sarma program shows a safety factor of 1.16 with a normal stresses ranging from 159 to 336 kPa; an average of 248 kPa.  The change in the average normal stress is about 8 percent between the second and third iterations which suggests that another iteration to establish normal stresses and Mohr-Coulomb parameters is not necessary.  Table C.1 is a summary of the iterations required to arrive at the Mohr-Coulomb parameters  of 47 degrees and c of 150 kPa.   Table C. 1.  Iterations to establish normal stresses at toe breakout and internal shear surface. Iteration Toe Breakout n, kPa Internal Shear n, kPa Ave. n, kPa , degrees c, kPa Safety Factor 1 475 259 376 33 320 1.21 2 360 175 270 44 150 1.15 3 336 159 248 47 120 1.16  C.2.5  Step Five – Best Fit Mohr-Coulomb Parameters and Stability Check Steps one through four above provide an estimate of the Mohr-Coulomb parameters based on a tangent linear fit to the Hoek-Brown failure envelop for the “average” stresses at the toe breakout and internal shear surfaces of the dip slope.  A tangential fit to the non-linear failure envelop is only appropriate for the normal stress considered (i.e. final average stress of 248 kPa).  The actual stresses range from about 159 to 336 kPa.  It is necessary to refine the Mohr- Coulomb linear fit because using the linear fit at the “average” stress overestimates the actual shear strength of the rock mass at the actual stress range at the toe of the dip slope.  Fig. C. 5 shows the non-linear Hoek-Brown envelope with a linear Mohr-Coulomb envelope superimposed to account for the range of normal stresses at the toe breakout and internal shear surfaces of the dip slope.  The parameters to use for a more detailed estimate of the dip slope stability state are of 49 degrees and c of 100 kPa (from the author’s judgment).  178   Fig. C. 5.  Linear best fit to Hoek Brown failure envelope for range of normal stresses within the example dip slope.  A “check” on the Mohr-Coulomb parameters is carried out using the finite element program Phase2 by RocScience (2007) which is used to compare the shear strength reduction (SSR) factor of the dip slope where the rock mass using the non-linear Hoek-Brown failure envelope and the linear Mohr-Coulomb approximation.  Fig. C. 6 and Fig. C. 7 show the SSR factors for the Hoek-Brown strength parameters and Mohr-Coulomb parameters, respectively.  Table C.2 and Table C.3 provide the input parameters used in the finite element evaluation.  The difference in SSR factors is minimal which suggests that linear approximation of the Hoek-Brown failure criterion, using the methodology provided above, is sufficiently accurate for practical purposes.  179   Fig. C. 6.  Finite element evaluation showing maximum shear strain contours of the example dip slope using the Hoek-Brown failure criterion.  SSR factor is 1.23.   Fig. C. 7.  Finite element evaluation showing maximum shear strain contours of the example dip slope using the Mohr-Coulomb failure criterion.  SSR factor is 1.24. 180  Table C. 2.  Input parameters for finite element model (Mohr-Coulomb best fit). Unit Field Stress r1, kN/m3 Erm2, GPa  t4, kPa 5, deg. c6, kPa Sandstone Gravity 20 3.2e6 0.4 0 49 100  Unit kn7, kPa/m ks8, kPa/m t, kPa , deg. c, kPa Joint 1e5 1e4 0 25 0 Notes: 9. r is the rock unit weight. 10. Erm is the deformation or Young’s modulus. 11. is Poisson’s ratio. 12. t is the tensile strength. 13.  is friction angle. 14. c is cohesion intercept. 15. kn is the joint normal stiffness. 16. ks is joint shear stiffness.  Table C. 3.  Input parameters for finite element model (Hoek-Brown Failure Criterion). Unit Field Stress r1, kN/m3 Erm2, GPa  GSI4 mi5 ci, MPa D7 Sandstone Gravity 20 3.2e6 0.4 40 17 7 0  Unit kn8, kPa/m ks9, kPa/m t, kPa , deg. c, kPa Joint 1e5 1e4 0 25 0 Notes: 1. r is the rock unit weight. 2. Erm is the deformation or Young’s modulus. 3. is Poisson’s ratio. 4. t is the tensile strength. 5.  is friction angle. 6. c is cohesion intercept. 7. D is Disturbance Factor 8. kn is the joint normal stiffness. 9. ks is joint shear stiffness.       181  C.3     References Evert Hoek (1985) “Sarma Dos-Based Program.”  Hoek, E., and E. T. Brown (1997) "Practical estimates of rock mass strength," International Journal of Rock Mechanics and Mining Sciences, Vol. 34, No. 8, pp 1165-1186.  Hoek, E., C. Carranza-Torres, and B. Corkum (2002) "Hoek-Brown criterion - 2002 edition," NARMS-TAC Conference. Toronto, pp 267-273.  Hoek E., Kaiser P. K., and Bawden W. F. (1995) Support of Underground Excavations in Hard Rock, Balkema, Rotterdam.  RocScience (2007) "Phase2 Finite Element Program."  RocScience (2007) "RocLab Program."  182   APPENDIX D. UDEC CODES  1. The following UDEC input code allows the user to set up a dip slope problem with a specified bedding thickness, number of beds, and “cross-bed” inclination.  ------------------------------------------------------------ new def setup  ;GEOMETRY PARAMETERS slope_height = s_height Slope_angle = s_angle Height_of_region_one = r1_height Height_of_region_two = r2_height  ;DEFINE X AND Y COORDINATES x_one = x1 x_two = x2 x_three = x3 x_four = x4 x_five = x5 x_six = x6 x_seven = x7 x_eight = x8 x_nine = x9 x_ten = x10 x_eleven = x11 x_twelve = x12 x_thirteen = x13 y_one = y1 y_two = y2 y_three = y3 y_four = y4 y_five = y5 y_six = y6 y_seven = y7 y_eight = y8 y_nine = y9 y_ten = y10 y_eleven = y11 y_twelve = y12 y_thirteen = y13  ;BEDDING GEOMETRY PARAMETERS bedding_dip = bd 183  bedding_trace = btr bedding_gap = bg bedding_spacing = bs number_of_bedding_layers = numlyr bedding_x = bx bedding_y = by beddinga_spacing = bsa  ;JOINT GEOMETRY PARAMETERS joint_1_dip = j1d joint_1_trace = j1t joint_1_gap = j1g joint_1_spacing = j1s joint_1_x = j1x joint_1_y = j1y joint_1a_gap = j1ag joint_1a_spacing = j1as joint_1a_trace = j1at joint_1a_gap = j1ag end  ;ALL INPUT PARAMETERS GO IN THE FOLLOWING ARGUMENTS   set s_height = 60 ;Input the slope height in meters   set s_angle = 35 ;Input the slope angle in degrees   set bd = s_angle ;Dip angle coincident with slope angle   set btr = 300   ;Input the trace length of bedding   set bs = 2  ;Input the spacing normal to bedding   set numlyr = 10 ;Input the number of bedding layers   set j1d = -35  ;Input dip of cross joints  ;DONE WITH INPUT PARAMETERS   set bg = 0.0  ;Input the gap length of bedding (0)   set x1 = 0  ;Model starts at the x,y, origin   set y1 = 0  ;Model starts at the x,y origin   set x2 = 0  ;x2 is zero   set y2 = s_height ;Distance from origin to slope toe is assume equal to slope height   set x3 = s_height ;Distance in front of toe is assumed to be equal to the slope height   set y3 = s_height ;Flat slope in front of slope  def y4 y4 = y3 + s_height end  def x4 x4 = (y4 - y3)/(TAN(s_angle*PI/180)) + x3 end  def x5 x5 = x4 + s_height ;Distance behind the crest is equal to the slope height  184  end   set y5 = y4   set x6 = x5   set y6 = y1  ;CREATE OUTER AREA OF THE SLOPE ro 0.1 bl x1,y1 x2,y2 x3,y3 x4,y4 x5,y5 x6,y6  ;SET UP TO DIVIDE SLOPE INTO JOINT SPACING ZONATION   set x7 = x1  def y7 y7 = y2 - ((bs*numlyr)/(cos(s_angle*PI/180))) ;Makes the joint region equal to bedding thickness x number layers end  set x8 = x3 set y8 = y7  def x9 x9 = x4 + ((bs*numlyr)/(sin(s_angle*PI/180))) end  def y9 y9 = y4 end  set x10 = x5 set y10 = y8  ; ZONE THE MODEL INTO THREE REGIONS crack x7,y7 x8,y8 crack x8,y8 x9,y9 crack x8,y8 x10,y10  ;ASSIGN JOINT REGIONS (2 TOTAL) jregion id 1 x7,y7 x2,y2 x3,y3 x8,y8 jregion id 2 x8,y8 x3,y3 x4,y4 x9,y9  ;ASSIGN Material REGIONS (2 TOTAL) Only assigning 2 surficial regions ;All else is default mat 1 and elastic change mat=2 range region x1,y1 x7,y7 x10,y10 x6,y6 change mat=2 range region x8,y8 x9,y9 x5,y5 x10,y10  ;JOINT SET GENERATION def j1t j1t = bs/(sin((bd-j1d)*PI/180)) end  def j1s 185  j1s = bs/(sin((90-(bd-j1d))*(PI/180))) end  set j1g = j1t  ;Joint gap is equal joint trace length  ;ASSIGN JOINT PROPERTIES PER REGION ;joint sets joint region 1 jset bd,0 btr,0 bg,0 bs,0 x3,y3 range jregion 1 jset j1d,0 j1t,0 j1g,0 j1s,0 x3,y3 range jregion 1  ;joint sets joint region 2 jset bd,0 btr,0 bg,0 bs,0 x3,y3 range jregion 2 jset j1d,0 j1t,0 j1g,0 j1s,0 x3,y3 range jregion 2  ; meshing gen edge 10 range mat=2 gen edge 4 range mat=1  ; material properties prop mat 1 density=2000.0 bulk=1.0E9 shear=3.0E8 prop jmat 1 jkn 1e9 jks 1e8 jfric 40  ; material properties prop mat 2 density=2000.0 bulk=1.0E9 shear=3.0E8 prop jmat 2 jkn 1e9 jks 1e8 jfric 40  def setup boundary_x1 = bx1 boundary_x2 = bx2 boundary_x3 = bx3 boundary_x4 = bx4 boundary_y1 = by1 boundary_y2 = by2 boundary_y3 = by3 boundary_y4 = by4 boundary_y5 = by5 end  def bx1 bx1 = x1-1 end  def bx2 bx2 = x1+1 end  def bx3 bx3 = x5+1 end  def bx4 186  bx4 = x5-1 end  def by1 by1 = y1-1 end  def by2 by2 = y1+1 end  def by3 by3 = y5-1 end  def by4 by4 = y5+1 end  def by5 by5 = y2+1 end  ; boundary conditions boun bx1 bx3 by1 by2 yvel 0 xvel 0 boun bx4 bx3 by1 by4 xvel 0 boun bx1 bx2 by1 by5 xvel 0  gravity 0 -10.0 solve force 10+3  ------------------------------------------------------------ 187  2. The following UDEC input code allows the user to set up a dip slope with a specified bedding thickness, number of beds, with persistent “cross-bed” inclination.  ------------------------------------------------------------ new def setup  ;GEOMETRY PARAMETERS slope_height = s_height Slope_angle = s_angle Height_of_region_one = r1_height Height_of_region_two = r2_height  ;DEFINE X AND Y COORDINATES x_one = x1 x_two = x2 x_three = x3 x_four = x4 x_five = x5 x_six = x6 x_seven = x7 x_eight = x8 x_nine = x9 x_ten = x10 x_eleven = x11 x_twelve = x12 x_thirteen = x13 y_one = y1 y_two = y2 y_three = y3 y_four = y4 y_five = y5 y_six = y6 y_seven = y7 y_eight = y8 y_nine = y9 y_ten = y10 y_eleven = y11 y_twelve = y12 y_thirteen = y13  ;BEDDING GEOMETRY PARAMETERS bedding_dip = bd bedding_trace = btr bedding_gap = bg bedding_spacing = bs number_of_bedding_layers = numlyr bedding_x = bx bedding_y = by 188   beddinga_spacing = bsa  ;JOINT GEOMETRY PARAMETERS Joint_1_dip = J1d joint_1_trace = j1t joint_1_gap = j1g joint_1_spacing = j1s joint_1_x = j1x joint_1_y = j1y joint_2_x = j2x joint_2_y = j2y joint_1a_gap = j1ag joint_1a_spacing = j1as joint_1a_trace = j1at joint_1a_gap = j1ag end  ;ALL INPUT PARAMETERS GO IN THE FOLLOWING ARGUMENTS   set s_height = 60 ;Input the slope height in meters   set s_angle = 35 ;Input the slope angle in degrees   set bd = s_angle ;Dip angle coincident with slope angle   set btr = 300   ;Input the trace length of bedding   set bs = 0.4  ;Input the spacing normal to bedding   set numlyr = 5 ;Input the number of bedding layers   set j1s = 1.6  ;Input joint spacing normal to joints  ;DONE WITH INPUT PARAMETERS   set bg = 0.0  ;Input the gap length of bedding (0)   set x1 = 0  ;Model starts at the x,y, origin   set y1 = 0  ;Model starts at the x,y origin   set x2 = 0  ;x2 is zero   set y2 = s_height ;Distance from origin to slope toe is assume equal to slope height   set x3 = s_height ;Distance infront of toe is assumed to be equal to the slope height   set y3 = s_height ;Flat slope in front of slope  def y4 y4 = y3 + s_height end  def x4 x4 = (y4 - y3)/(TAN(s_angle*PI/180)) + x3 end  def x5 x5 = x4 + s_height ;Distance behind the crest is equal to the slope height end    set y5 = y4   set x6 = x5   set y6 = y1 189   ;CREATE OUTER AREA OF THE SLOPE   ro 0.1 bl x1,y1 x2,y2 x3,y3 x4,y4 x5,y5 x6,y6  ;SET UP TO DIVIDE SLOPE INTO JOINT SPACING ZONATION   set x7 = x1  def y7 y7 = y2 - ((bs*numlyr)/(cos(s_angle*PI/180))) ;Makes the joint region equal to bedding thickness x number layers end  set x8 = x3 set y8 = y7  def x9 x9 = x4 + ((bs*numlyr)/(sin(s_angle*PI/180))) end  def y9 y9 = y4 end  set x10 = x5 set y10 = y8  ; ZONE THE MODEL INTO THREE REGIONS crack x7,y7 x8,y8 crack x8,y8 x9,y9 crack x8,y8 x10,y10  ;ASSIGN JOINT REGIONS (2 TOTAL) jregion id 1 x7,y7 x2,y2 x3,y3 x8,y8 jregion id 2 x8,y8 x3,y3 x4,y4 x9,y9  ;ASSIGN Material REGIONS (2 TOTAL) Only assigning 2 surficial regions ;All else is default mat 1 and elastic change mat=2 range region x1,y1 x7,y7 x10,y10 x6,y6 change mat=2 range region x8,y8 x9,y9 x5,y5 x10,y10  ;ORTHOGONAL JOINT SET GENERATION   set bx = x3  ;Input the x reference point on the slope (toe)   set by = y3  ;Input the y reference point on the slope (toe)  def j1d j1d = -1*(90 - bd) ;Sets the joint normal to bedding end    set j1t = bs  ;Joint trace is equal to the bedding spacing   set j1g = bs*numlyr  ;Joint gap is equal to the bedding spacing 190    set j1x = x3  ;x reference point to x3   set j1y = y3  ;y reference point to y3  def j2x   ;The location of the x coordinate is defined based on the joint set geometry  j2x = bx-(1.5 * j1s * cos(bd*(PI/180)) + bs*(sin(bd*PI/180))) end  def j2y   ;The location of the x coordinate is defined based on the joint set geometry j2y = by-(1.5 * j1s * sin(bd*(PI/180)) + bs*(sin(j1d*PI/180))) end  ;ASSIGN JOINT PROPERTIES PER REGION ;Orthogonal joint sets joint region 1 jset bd,0 btr,0 bg,0 bs,0 bx,by range jregion 1 jset j1d,0 j1t,0 j1g,0 j1s,0 j1x,j1y range jregion 1  ;Orthogonal joint sets joint region 2 jset bd,0 btr,0 bg,0 bs,0 bx,by range jregion 2 jset j1d,0 j1t,0 j1g,0 j1s,0 j1x,j1y range jregion 2  ; meshing gen edge 10 range mat=2 gen edge 4 range mat=1  ; material properties prop mat 1 density=2000.0 bulk=1.0E9 shear=3.0E8 prop jmat 1 jkn 1e10 jks 1e9 jfric 45  ; material properties prop mat 2 density=2000.0 bulk=1.0E9 shear=3.0E8 prop jmat 2 jkn 1e10 jks 1e9 jfric 45  def setup boundary_x1 = bx1 boundary_x2 = bx2 boundary_x3 = bx3 boundary_x4 = bx4 boundary_y1 = by1 boundary_y2 = by2 boundary_y3 = by3 boundary_y4 = by4 boundary_y5 = by5 end  def bx1 bx1 = x1-1 end  191  def bx2 bx2 = x1+1 end  def bx3 bx3 = x5+1 end  def bx4 bx4 = x5-1 end  def by1 by1 = y1-1 end  def by2 by2 = y1+1 end  def by3 by3 = y5-1 end  def by4 by4 = y5+1 end  def by5 by5 = y2+1 end  ; boundary conditions boun bx1 bx3 by1 by2 yvel 0 xvel 0 boun bx4 bx3 by1 by4 xvel 0 boun bx1 bx2 by1 by5 xvel 0  gravity 0 -10.0 solve  ------------------------------------------------------------       192  3. The following UDEC input code allows the user to set up a dip slope with a specified bedding thickness, number of beds, and “cross-bed” inclination.  ------------------------------------------------------------ new def setup  ;GEOMETRY PARAMETERS slope_height = s_height Slope_angle = s_angle Height_of_region_one = r1_height Height_of_region_two = r2_height  ;DEFINE X AND Y COORDINATES x_one = x1 x_two = x2 x_three = x3 x_four = x4 x_five = x5 x_six = x6 x_seven = x7 x_eight = x8 x_eightplusone = x8plusone x_eightminusone = x8minusone x_nine = x9 x_nineplusone = x9plus0ne x_nineminusone = x9minusone x_ten = x10 x_eleven = x11 x_twelve = x12 x_thirteen = x13 y_one = y1 y_two = y2 y_three = y3 y_four = y4 y_five = y5 y_six = y6 y_seven = y7 y_eight = y8 y_nine = y9 y_ten = y10 y_eleven = y11 y_twelve = y12 y_thirteen = y13  ;BEDDING GEOMETRY PARAMETERS bedding_dip = bd bedding_trace = btr bedding_gap = bg 193  bedding_spacing = bs number_of_bedding_layers = numlyr bedding_x = bx bedding_y = by beddinga_spacing = bsa  ;JOINT GEOMETRY PARAMETERS Joint_1_dip = J1d joint_1_trace = j1t joint_1_gap = j1g joint_1_spacing = j1s joint_1_x = j1x joint_1_y = j1y joint_2_x = j2x joint_2_y = j2y joint_1a_gap = j1ag joint_1a_spacing = j1as joint_1a_trace = j1at joint_1a_gap = j1ag end  ;DEFINE THE RANGE OF ANGLES FOR ASSIGNING JOINT PROPERTIES def bdplus1 bdplus1 = bd + 1 end  def bdminus1 bdminus1 = bd - 1 end  def j1plus1 j1plus1 = j1d + 25 end  def j1minus1 j1minus1 = j1d - 25 end  ;ALL INPUT PARAMETERS GO IN THE FOLLOWING ARGUMENTS   set s_height = 285 ;Input the slope height in meters   set s_angle = 45 ;Input the slope angle in degrees   set bd = s_angle ;Dip angle coincident with slope angle   set btr = 500   ;Input the trace length of bedding   set bs = 32.5  ;Input the spacing normal to bedding   set numlyr = 1 ;Input the number of bedding layers ****  def j1s   j1s = bs*3 ;Input joint spacing normal to joints end  ;DONE WITH INPUT PARAMETERS 194    set bg = 0.0  ;Input the gap length of bedding (0)   set x1 = 0  ;Model starts at the x,y, origin   set y1 = 0  ;Model starts at the x,y origin   set x2 = 0  ;x2 is zero   set y2 = s_height ;Distance from origin to slope toe is assume equal to slope height   set x3 = s_height ;Distance infront of toe is assumed to be equal to the slope height   set y3 = s_height ;Flat slope in front of slope  def y4 y4 = y3 + s_height end  def x4 x4 = (y4 - y3)/(TAN(s_angle*PI/180)) + x3 end  def x5 x5 = x4 + s_height ;Distance behind the crest is equal to the slope height end    set y5 = y4   set x6 = x5   set y6 = y1  ;CREATE OUTER AREA OF THE SLOPE   ro 0.1 bl x1,y1 x2,y2 x3,y3 x4,y4 x5,y5 x6,y6  ;SET UP TO DIVIDE SLOPE INTO JOINT SPACING ZONATION set y7 = y1  def x7 x7 = x3 + ((y7-y3)*(tan((90-s_angle)*PI/180))) ;Makes the joint region equal to bedding thickness x number layers end  def x9 x9 = x4 + ((bs*numlyr)/(sin(s_angle*PI/180))) end  def x8plusone x8plusone = x8 +1 end  def x8minusone x8minusone = x8 - 1 end  def y9 y9 = y4 end 195   def x9plusone x9plusone = x9 + 1 end  def x9minusone x9minusone = x9 - 1 end  def x8 x8 = x9 - ((y9-y1)*(tan((90-s_angle)*PI/180))) ;Joint region equal to bedding thickness x number layers end  set y8 = y1  ; ZONE THE MODEL INTO THREE REGIONS crack x7,y7 x3,y3 crack x8,y8 x9,y9 id = 10   ;ASSIGN JOINT REGIONS (2 TOTAL) jregion id 1 x7,y7 x4,y4 x9,y9 x8,y8 jregion id 2 x1,y1 x2,y2 x3,y3 x7,y7 jregion id 2 x8,y8 x9,y9 x5,y5 x6,y6  ;ASSIGN Material REGIONS only to jointed area ;Material 1 will be elastic and is hard to define change mat=2 range region x7,y7 x4,y4 x9,y9 x8,y8  ;ORTHOGONAL JOINT SET GENERATION   set bx = x3  ;Input the x reference point on the slope (toe)   set by = y3  ;Input the y reference point on the slope (toe)  def j1d j1d = -1*(90 - bd) ;Sets the joint normal to bedding end    set j1t = bs  ;Joint trace is equal to the bedding spacing   set j1g = bs  ;Joint gap is equal to the bedding spacing   set j1x = x3  ;x reference point to x3   set j1y = y3  ;y reference point to y3  def j2x   ;The location of the x coordinate is defined based on the joint set geometry j2x = bx-(1.5 * j1s * cos(bd*(PI/180)) + bs*(sin(bd*PI/180))) end  def j2y   ;The location of the x coordinate is defined based on the joint set geometry j2y = by-(1.5 * j1s * sin(bd*(PI/180)) + bs*(sin(j1d*PI/180))) 196  end  ;ASSIGN JOINT PROPERTIES PER REGION  ;joint sets joint region 1 jset bd,0 btr,0 bg,0 bs,0 x3,y3 range jregion 1 ;jset j1d,0 j1t,0 j1g,0 j1s,0 j1x,j1y range jregion 1 ;jset j1d,0 j1t,0 j1g,0 j1s,0 j2x,j2y range jregion 1  ;joint sets joint region 2 ;jset bd,0 btr,0 bg,0 bs,0 x3,y3 range jregion 1 ;jset j1d,0 j1t,0 j1g,0 j1s,0 x3,y3 range jregion 1  ; meshing gen edge 5 range mat=2 gen edge 30 range mat=1  change cons=3 range mat=2 change cons=3 range mat=1 change jmat=1 range ang bdminus1 bdplus1 change jmat=2 range ang j1minus1 j1plus1      ;CREATES THE LOW STRENGTH SLOPE change jmat=10 range id=10 ;COINCIDENT JOINT!  ; m-c properties prop mat 2 bulk=672e6 shear=350e6 prop mat 2 coh=12e6 fric=45 dil=0 prop mat 2 den=2273  prop mat 1 coh=12e6 fric=50 dil=0 prop mat 1 bulk=672e6 shear=350e6 den=2273  ;m-c discontinuity properties for jmat1 (Bedding) and jmat2 (joints) prop jmat=1 jfric 45 jcoh 0 ;KEEP THESE HIGH TO INITIALIZE MODEL prop jmat=1 jkn 5570e6 jks 557e6;KEEP THESE HIGH TO INITIALIZE MODEL prop jmat=2 jfric 35 jcoh 0 ;KEEP THESE HIGH TO INITIALIZE MODEL prop jmat=2 jkn 1e10 jks 1e9 ;KEEP THESE HIGH TO INITIALIZE MODEL  prop jmat=10 jfric 50 jcoh 0 ;KEEP THESE HIGH TO INITIALIZE MODEL prop jmat=10 jkn 1e10 jks 1e9 ;KEEP THESE HIGH TO INITIALIZE MODEL  def setup boundary_x1 = bx1 boundary_x2 = bx2 boundary_x3 = bx3 boundary_x4 = bx4 boundary_y1 = by1 boundary_y2 = by2 boundary_y3 = by3 boundary_y4 = by4 197  boundary_y5 = by5 end  def bx1 bx1 = x1-1 end  def bx2 bx2 = x1+1 end  def bx3 bx3 = x5+1 end  def bx4 bx4 = x5-1 end  def by1 by1 = y1-1 end  def by2 by2 = y1+1 end  def by3 by3 = y5-1 end  def by4 by4 = y5+1 end  def by5 by5 = y2+1 end  gravity 0 -9.81  ; boundary conditions boun bx1 bx3 by1 by2 yvel 0 xvel 0 boun bx4 bx3 by1 by4 xvel 0 boun bx1 bx2 by1 by5 xvel 0  hist unbal solve ------------------------------------------------------------  198  4. The following UDEC input code allows the user to set up a dip slope with a specified bedding thickness, number of beds, with orthogonal “cross-beds” with a set spacing truncated at bedding.  In addition, there is a slope-coincident joint at the base of the bedding layers that can be assigned properties different than the bedding layers.  ------------------------------------------------------------  new def setup  ;GEOMETRY PARAMETERS  slope_height = s_height Slope_angle = s_angle Height_of_region_one = r1_height Height_of_region_two = r2_height  ;DEFINE X AND Y COORDINATES x_one = x1 x_two = x2 x_three = x3 x_four = x4 x_five = x5 x_six = x6 x_seven = x7 x_eight = x8 x_eightplusone = x8plusone x_eightminusone = x8minusone x_nine = x9 x_nineplusone = x9plus0ne x_nineminusone = x9minusone x_ten = x10 x_eleven = x11 x_twelve = x12 x_thirteen = x13 y_one = y1 y_two = y2 y_three = y3 y_four = y4 y_five = y5 y_six = y6 y_seven = y7 y_eight = y8 y_nine = y9 y_ten = y10 y_eleven = y11 y_twelve = y12 y_thirteen = y13 199   ;BEDDING GEOMETRY PARAMETERS bedding_dip = bd bedding_trace = btr bedding_gap = bg bedding_spacing = bs number_of_bedding_layers = numlyr bedding_x = bx bedding_y = by beddinga_spacing = bsa  ;JOINT GEOMETRY PARAMETERS Joint_1_dip = J1d joint_1_trace = j1t joint_1_gap = j1g joint_1_spacing = j1s joint_1_x = j1x joint_1_y = j1y joint_2_x = j2x joint_2_y = j2y joint_1a_gap = j1ag joint_1a_spacing = j1as joint_1a_trace = j1at joint_1a_gap = j1ag end  ;DEFINE THE RANGE OF ANGLES FOR ASSIGNING JOINT PROPERTIES def bdplus1 bdplus1 = bd + 1 end  def bdminus1 bdminus1 = bd - 1 end  def j1plus1 j1plus1 = j1d + 25 end  def j1minus1 j1minus1 = j1d - 25 end  ;ALL INPUT PARAMTERS GO IN THE FOLLOWING ARGUMENTS   set s_height = 285 ;Input the slope height in meters   set s_angle = 45 ;Input the slope angle in degrees   set bd = s_angle ;Dip angle coincident with slope angle   set btr = 500   ;Input the trace length of bedding   set bs = 3  ;Input the spacing normal to bedding   set numlyr = 10 ;Input the number of bedding layers ****  200  def j1s   j1s = bs*3 ;Input joint spacing normal to joints end  ;DONE WITH INPUT PARAMETERS   set bg = 0.0  ;Input the gap length of bedding (0)   set x1 = 0  ;Model starts at the x,y, origin   set y1 = 0  ;Model starts at the x,y origin   set x2 = 0  ;x2 is zero   set y2 = s_height ;Distance from origin to slope toe is assume equal to slope height   set x3 = s_height ;Distance infront of toe is assumed to be equal to the slope height   set y3 = s_height ;Flat slope in front of slope  def y4 y4 = y3 + s_height end  def x4 x4 = (y4 - y3)/(TAN(s_angle*PI/180)) + x3 end  def x5 x5 = x4 + s_height ;Distance behind the crest is equal to the slope height end    set y5 = y4   set x6 = x5   set y6 = y1   ;CREATE OUTER AREA OF THE SLOPE ro 0.1 bl x1,y1 x2,y2 x3,y3 x4,y4 x5,y5 x6,y6  ;SET UP TO DIVIDE SLOPE INTO JOINT SPACING ZONATION set y7 = y1 def x7 x7 = x3 + ((y7-y3)*(tan((90-s_angle)*PI/180))) ;Makes the joint region equal to bedding thickness x number layers end  def x9 x9 = x4 + ((bs*numlyr)/(sin(s_angle*PI/180))) end  def x8plusone x8plusone = x8 +1 end  def x8minusone x8minusone = x8 - 1 201  end  def y9 y9 = y4 end  def x9plusone x9plusone = x9 + 1 end  def x9minusone x9minusone = x9 - 1 end  def x8 x8 = x9 - ((y9-y1)*(tan((90-s_angle)*PI/180))) ;Joint region equal to bedding thickness x number layers end  set y8 = y1   ; ZONE THE MODEL INTO THREE REGIONS crack x7,y7 x3,y3 crack x8,y8 x9,y9 id = 10   ;ASSIGN JOINT REGIONS (2 TOTAL) jregion id 1 x7,y7 x4,y4 x9,y9 x8,y8 jregion id 2 x1,y1 x2,y2 x3,y3 x7,y7 jregion id 2 x8,y8 x9,y9 x5,y5 x6,y6  ;ASSIGN Material REGIONS only to jointed area ;Material 1 will be elastic and is hard to define change mat=2 range region x7,y7 x4,y4 x9,y9 x8,y8  ;ORTHOGONAL JOINT SET GENERATION   set bx = x3  ;Input the x reference point on the slope (toe)   set by = y3  ;Input the y reference point on the slope (toe)  def j1d j1d = -1*(90 - bd) ;Sets the joint normal to bedding end    set j1t = bs  ;Joint trace is equal to the bedding spacing   set j1g = bs  ;Joint gap is equal to the bedding spacing   set j1x = x3  ;x reference point to x3   set j1y = y3  ;y reference point to y3  def j2x   ;The location of the x coordinate is defined based on the joint set geometry 202  j2x = bx-(1.5 * j1s * cos(bd*(PI/180)) + bs*(sin(bd*PI/180))) end  def j2y   ;The location of the x coordinate is defined based on the joint set geometry j2y = by-(1.5 * j1s * sin(bd*(PI/180)) + bs*(sin(j1d*PI/180))) end  ;ASSIGN JOINT PROPERTIES PER REGION ;joint sets joint region 1 jset bd,0 btr,0 bg,0 bs,0 x3,y3 range jregion 1 jset j1d,0 j1t,0 j1g,0 j1s,0 j1x,j1y range jregion 1 jset j1d,0 j1t,0 j1g,0 j1s,0 j2x,j2y range jregion 1  ; meshing gen edge 5 range mat=2 gen edge 30 range mat=1  change cons=3 range mat=2 change cons=3 range mat=1 change jmat=1 range ang bdminus1 bdplus1 change jmat=2 range ang j1minus1 j1plus1      ;CREATES THE LOW STRENGTH SLOPE change jmat=10 range id=10 ;COINCIDENT JOINT!  ; m-c properties prop mat 2 bulk=672e6 shear=350e6 prop mat 2 coh=12e6 fric=45 dil=0 prop mat 2 den=2273  prop mat 1 coh=12e6 fric=50 dil=0 prop mat 1 bulk=672e6 shear=350e6 den=2273  ;m-c discontinuity properties for jmat1 (Bedding) and jmat2 (joints) prop jmat=1 jfric 45 jcoh 0 ;KEEP THESE HIGH TO INITIALIZE MODEL prop jmat=1 jkn 5570e6 jks 557e6;KEEP THESE HIGH TO INITIALIZE MODEL prop jmat=2 jfric 35 jcoh 0 ;KEEP THESE HIGH TO INITIALIZE MODEL prop jmat=2 jkn 1e10 jks 1e9 ;KEEP THESE HIGH TO INITIALIZE MODEL  prop jmat=10 jfric 50 jcoh 0 ;KEEP THESE HIGH TO INITIALIZE MODEL prop jmat=10 jkn 1e10 jks 1e9 ;KEEP THESE HIGH TO INITIALIZE MODEL  def setup boundary_x1 = bx1 boundary_x2 = bx2 boundary_x3 = bx3 boundary_x4 = bx4 boundary_y1 = by1 boundary_y2 = by2 boundary_y3 = by3 203  boundary_y4 = by4 boundary_y5 = by5 end  def bx1 bx1 = x1-1 end  def bx2 bx2 = x1+1 end  def bx3 bx3 = x5+1 end  def bx4 bx4 = x5-1 end  def by1 by1 = y1-1 end  def by2 by2 = y1+1 end  def by3 by3 = y5-1 end  def by4 by4 = y5+1 end  def by5 by5 = y2+1 end  gravity 0 -9.81  ; boundary conditions boun bx1 bx3 by1 by2 yvel 0 xvel 0 boun bx4 bx3 by1 by4 xvel 0 boun bx1 bx2 by1 by5 xvel 0  solve  ------------------------------------------------------------ 204  5. The following UDEC input code is used to change the intact rock properties and joint properties of the previous “set-up” files (1-4 above).  ------------------------------------------------------------  reset disp jdisp rot  ; m-c intact rock block material  Prop mat 1 coh=1e6 fric=45 dil=0 prop mat 2 coh=0.700e6 fric=34 dil=0  ;m-c discontinuity properties for jmat1 (Bedding) prop jmat=1 jfric 24.5 jcoh 0  ;m-c discontinuity properties for jmat2 (Cross-beds) prop jmat=1 jfric 35 jcoh 0  ;m-c discontinuity properties for jmat10 (Slope-Coincident Discontinuity) prop jmat=1 jfric 10 jcoh 0                    205  APPENDIX E. DIP SLOPE FAILURE MECHANISMS  The author completed static analyses of dip slopes using finite element and distinct element codes to determine the failure mechanisms associated with bi-planar failures in dip slopes.  To verify that the failure mechanism does not change because of triggering mechanisms such as earthquake loading, and increases in pore water pressures, similar analyses were carried out using distinct element codes.  This appendix provides a summary of the modeling results.  E.1  Seismic Loading Seismic loading was performed using the methodology suggested by Itasca (2004).  A free-field boundary is used to account for the free-field motion that would exist if the seismic source was generated at a great distance from the slope.  The boundaries are placed at the edge of the model so that the boundary of the model retains its non-reflecting properties.  UDEC executes this by performing a one-dimensional free-field calculation in parallel with the main analysis. The lateral boundaries are assumed to be quiet boundaries and the UDEC code simulates this by applying viscous dashpots so that the unbalanced forces from the free-field are transmitted to the deformable-block boundary at the boundary grid points (Itasca 2004).  By using this method, the seismic waves (in this case modeled as stress waves) that propagate upwards are not affected by the boundary and are therefore identical to a model with an “infinite” boundary.  UDEC applies mechanical damping to the system so that the system does not oscillate indefinitely.  In nature, damping is attributed to internal friction and slippage along joints (Wyllie and Mah 2004).  Rayleigh Damping is commonly used to simulate natural damping in UDEC (Itasca 2004).  The external dynamic loading is applied as a stress or a velocity at the model boundary or within the model at internal blocks.  In this case, a velocity history was represented by a harmonic sine function and the velocity history was transformed to a stress because a velocity history cannot be applied to the quite boundary.  The purpose of performing the dynamic analysis was to observe the failure mechanism, and not testing the effect of a particular design-based earthquake load. Three slopes were modeled: a 206  30-degree, 45-degree, and 60-degree slope.  The D/H ratios were constrained by the literature review performed for Chapter 2.  The slopes were initially brought to equilibrium to simulate gravity loading.  The second step was to set up the free-field conditions (e.g. quiet boundaries). An earthquake load was then applied and the shear strength values of the rock mass was decreased until, based on the authors judgment, localization of shear bands, shear and tensile failure (e.g. plasticity indicators), and the time histories of the unbalanced forces showed the slope system failing.  Fig. E. 1 through Fig. E. 3  show the results of the earthquake modeling.  The model results suggest that the failure mechanisms under earthquake loading are similar to those under static loading for shallow slopes.  As the slope angle increases, there is the possibility of buckling of the outer beds in the orthogonally bedded slopes, followed by collapse of the outer layers.  This suggests that buckling is a valid failure mechanism to investigate for steep slopes subjected to external dynamic loading.  In the case of the 60 degree slope, the failure occurs deeper along a slope parallel sliding surface suggesting that buckling may result in failure depths of the same magnitude as bi-planar failures in steep slopes.  E.2  Increased Pore Water Pressure The failure mechanisms associated with an increase in pore water pressure was investigated using a steady state flow algorithm in UDEC (Itasca 2004).  The algorithm models groundwater flow as a coupled hydro-mechanical process which means that an increase in water pressure results in an increase in aperture which causes the joint hydraulic conductivity to increase.  The joint hydraulic conductivity is controlled by a cubic law relationship. Steady state is achieved once the model equilibrates and the area underneath a specified water table is fully saturated and the water is flowing at a steady state within the joints. The formulation assumes that the blocks are impermeable, restricting flow to the joint network.  Models were performed for 30, 45, and 60 degree slopes to investigate changes in failure mechanisms. The results are shown in Fig. E. 4 through Fig. E. 6.  In summary, the models show that for the most part the failure mechanism does not change from that observed for the unsaturated situation. As the slope approaches 60 degrees, the fully saturated slope case does show buckling of the outer bedded slope, which occurs through plastic failure behind the 207  buckling bed.  Therefore, steep slopes subjected to internal water pressures acting normal to the slope face should be investigated for buckling as well as bi-planar failures.   Fig. E. 1.  Seismic loading of a 30-degree dip slope with D/H ratio of 0.20.  1) Model setup and 2) failure mechanism predicted using distinct element modeling compared to that predicted using the Mohr-Coulomb theory to establish failure surface inclinations. . 50m 40m 30 Slope height: 200m D/H ratio: 0.20 Toe Breakout Surface (Mohr-Coulomb Theory) Internal Shear (Mohr-Coulomb Theory) Slope Parallel Sliding 1 2 *  Plastic Yield O Tensile Failure  208   Fig. E. 2.  Seismic loading of a 45-degree dip slope with D/H ratio of 0.12.  1) Model setup and 2) failure mechanism predicted using distinct element modeling compared to that predicted using the Mohr-Coulomb theory (M-C) to establish failure surface inclinations.   100m Slope height: 200m D/H ratio: 0.12 4550m Toe Breakout Surface (M-C Theory) Slope Parallel Sliding 1 2 *  Plastic Yield O Tensile Failure  *  Plastic Yield O Tensile Failure  Internal Shear (M-C Theory) 209   Fig. E. 3.  Seismic loading of a 60-degree dip slope with D/H ratio of 0.08.  1) Model setup and 2) failure mechanism predicted using distinct element modeling compared to that predicted using the Mohr-Coulomb (M-C) theory to establish the inclination of failure surfaces.  100m Slope height: 200m D/H ratio: 0.08 40m 60 Toe Breakout Surface (M-C Theory) Internal Shear (M-C Theory) Slope Parallel Sliding 1 2 *  Plastic Yield O Tensile  210   Fig. E. 4.  Increase in water pressure in a 30-degree dip slope with D/H ratio of 0.20.  1) Model setup, 2) failure mechanism predicted using distinct element modeling; steady state and partially saturated slopes, 3) failure mechanism predicted using distinct element modeling; steady state and fully saturated conditions.  50m 30 Slope height: 200m D/H ratio: 0.20 Slope Parallel Sliding Surface 1 50m Toe Breakout Surface (M-C Theory) Internal Shear (M-C Theory) 2 40m 3 Toe Breakout Surface (M-C Theory) Internal Shear (M-C Theory) *  Plastic Yield O Tensile Failure  *  Plastic Yield O Tensile Failure  211   Fig. E. 5.  Increase in water pressure in a 45-degree dip slope with D/H ratio of 0.12.  1) Model setup, 2) failure mechanism predicted using distinct element modeling; steady state and partially saturated slopes, 3) failure mechanism predicted using distinct element modeling; steady state and fully saturated conditions. 100 Slope height: 200m D/H ratio: 0.12 45 Slope Parallel Sliding Surface 1 Toe Breakout Surface (M-C Theory) 40 3 40 2 Toe Breakout Surface (M-C Theory) Internal Shear (M-C Theory) *  Plastic Yield O Tensile Failure  *  Plastic Yield O Tensile Failure  Internal Shear (M-C Theory) 212   Fig. E. 6.  Increase in water pressure in a 60-degree dip slope with D/H ratio of 0.08.  1) Model setup, 2) failure mechanism predicted using distinct element modeling; steady state and partially saturated slopes, 3) failure mechanism predicted using distinct element modeling; steady state and fully saturated conditions.  100m Slope height: 200m D/H ratio: 0.08 60 Slope Parallel Sliding Surface 1 Toe Breakout Surface (M-C Theory) Internal Shear (M-C Theory) 80m 2 3 80m *  Plastic Yield O Tensile Failure  *  Plastic Yield O Tensile Failure  213  E.3 Example UDEC Codes 1. The following UDEC input code allows the user to set up a dip slope with a specified bedding thickness, number of beds, with orthogonal “cross-beds” with a set spacing truncated at bedding.  In addition, there is a slope-coincident joint at the base of the bedding layers that can be assigned properties different than the bedding layers.  This model considers steady state flow and a partially saturated slope and demonstrates the failure mechanisms associated with dip slopes. ---------------------------------------------------------- new def setup  ;GEMETRY PARAMTERS  slope_height = s_height Slope_angle = s_angle Height_of_region_one = r1_height Height_of_region_two = r2_height  ;DEFINE X AND Y COORDINATES  x_one = x1 x_two = x2 x_three = x3 x_four = x4 x_five = x5 x_six = x6 x_seven = x7 x_eight = x8 x_eightplusone = x8plusone x_eightminusone = x8minusone x_nine = x9 x_nineplusone = x9plus0ne x_nineminusone = x9minusone x_ten = x10 x_eleven = x11 x_twelve = x12 x_thirteen = x13 y_one = y1 y_two = y2 y_three = y3 y_four = y4 y_five = y5 y_six = y6 y_seven = y7 y_eight = y8 y_nine = y9 y_ten = y10 214  y_eleven = y11 y_twelve = y12 y_thirteen = y13  ;BEDDING GEOMETRY PARAMTERS  bedding_dip = bd bedding_trace = btr bedding_gap = bg bedding_spacing = bs number_of_bedding_layers = numlyr bedding_x = bx bedding_y = by  beddinga_spacing = bsa  ;JOINT GEOMETRY PARAMTERS  Joint_1_dip = J1d joint_1_trace = j1t joint_1_gap = j1g joint_1_spacing = j1s joint_1_x = j1x joint_1_y = j1y  joint_2_x = j2x joint_2_y = j2y  joint_1a_gap = j1ag joint_1a_spacing = j1as joint_1a_trace = j1at joint_1a_gap = j1ag   end  ;DEFINE THE RANGE OF ANGLES FOR ASSIGNING JOINT PROPERTIES  def bdplus1 bdplus1 = bd + 1 end  def bdminus1 bdminus1 = bd - 1 end  def j1plus1 j1plus1 = j1d + 25 end  def j1minus1 j1minus1 = j1d - 25 end  215  ;ALL INPUT PARAMTERS GO IN THE FOLLOWING ARGUMENTS    set s_height = 200 ;Input the slope height in meters   set s_angle = 30 ;Input the slope angle in degrees   set bd = s_angle ;Dip angle coincident with slope angle   set btr = 500   ;Input the trace length of bedding   set bs = 5  ;Input the spacing normal to bedding   set numlyr = 8 ;Input the number of bedding layers ****  def j1s   j1s = bs*3 ;Input joint spacing normal to joints end  ;DONE WITH INPUT PARAMATERS   set bg = 0.0  ;Input the gap length of bedding (0)   set x1 = 0  ;Model starts at the x,y, origin   set y1 = 0  ;Model starts at the x,y origin   set x2 = 0  ;x2 is zero   set y2 = s_height ;Distance from origin to slope toe is assume equal to slope height   set x3 = s_height ;Distance infront of toe is assumed to be equal to the slope height   set y3 = s_height ;Flat slope in front of slope  def y4  y4 = y3 + s_height  end  def x4  x4 = (y4 - y3)/(TAN(s_angle*PI/180)) + x3  end  def x5  x5 = x4 + s_height ;Distance behind the crest is equal to the slope height  end    set y5 = y4   set x6 = x5   set y6 = y1   ;CREATE OUTTER AREA OF THE SLOPE  ro 0.1 bl x1,y1 x2,y2 x3,y3 x4,y4 x5,y5 x6,y6  ;SET UP TO DIVIDE SLOPE INTO JOINT SPACING ZONATION 216   set y7 = y1  def x7 x7 = x3 + ((y7-y3)*(tan((90-s_angle)*PI/180))) ;Makes the joint region equal to bedding thickness x number layers end  def x9 x9 = x4 + ((bs*numlyr)/(sin(s_angle*PI/180))) end  def x8plusone x8plusone = x8 +1 end  def x8minusone x8minusone = x8 - 1 end  def y9 y9 = y4 end  def x9plusone x9plusone = x9 + 1 end  def x9minusone x9minusone = x9 - 1 end  def x8 x8 = x9 - ((y9-y1)*(tan((90-s_angle)*PI/180))) ;Joint region equal to bedding thickness x number layers end  set y8 = y1   ; ZONE THE MODEL INTO THREE REGIONS crack x7,y7 x3,y3 crack x8minusone,y8 x9minusone,y9  id = 10   ;ASSIGN JOINT REGIONS (3 TOTAL) jregion id 1 x7,y7 x4,y4 x9,y9 x8,y8 jregion id 2 x1,y1 x2,y2 x3,y3 200, 131 jregion id 3 x8,y8 x9,y9 x5,y5 x6,y6  ;ASSIGN Material REGIONS only to jointed area ;Material 1 will be elastic and is hard to define change mat=2 range region x7,y7 x4,y4 x9,y9 x8,y8  217  ;ORTHOGONAL JOINT SET GENERATION   set bx = x3  ;Input the x reference point on the slope (toe)   set by = y3  ;Input the y reference point on the slope (toe)  def j1d  j1d = -1*(90 - bd) ;Sets the joint normal to bedding  end    set j1t = bs  ;Joint trace is equal to the bedding spacing   set j1g = bs  ;Joint gap is equal to the bedding spacing   set j1x = x3  ;x reference point to x3   set j1y = y3  ;y reference point to y3    def j2x   ;The location of the x coordinate is defined based on the joint set geometry  j2x = bx-(1.5 * j1s * cos(bd*(PI/180)) + bs*(sin(bd*PI/180)))  end  def j2y   ;The location of the x coordinate is defined based on the joint set geometry  j2y = by-(1.5 * j1s * sin(bd*(PI/180)) + bs*(sin(j1d*PI/180)))  end  def bsa   bsa = 4 * bs  ;Input the spacing normal to bedding outside region of interest end    set j1at = bsa ;Joint trace is equal to the bedding spacing   set j1ag = bsa ;Joint gap is equal to the bedding spacing  def j1as   j1as = 4 * j1s ;This is the spacing of joints outside the region of interest end  def j2ax  ;The location of the x coordinate is defined based on the joint set geometry  j2ax = x8-(1.5 * j1as * cos(bd*(PI/180)) + bsa*(sin(bd*PI/180)))  end  def j2ay  ;The location of the x coordinate is defined based on the joint set geometry  218  j2ay = y8-(1.5 * j1as * sin(bd*(PI/180)) + bsa*(sin(j1d*PI/180)))  end  ;ASSIGN JOINT PROPERTIES PER REGION  ;joint sets joint region 1 jset bd,0 btr,0 bg,0 bs,0 x3,y3 range jregion 1 jset j1d,0 j1t,0 j1g,0 j1s,0 j1x,j1y range jregion 1 jset j1d,0 j1t,0 j1g,0 j1s,0 j2x,j2y range jregion 1  ;joint sets joint region 2 jset bd,0 btr,0 bg,0 bsa,0 bx,by range jregion 2 jset j1d,0 j1at,0 j1ag,0 j1as,0 x8,y8 range jregion 2 jset j1d,0 j1at,0 j1ag,0 j1as,0 j2ax,j2ay range jregion 2  ;joint sets joint region 3 jset bd,0 btr,0 bg,0 bsa,0 bx,by range jregion 3 jset j1d,0 j1at,0 j1ag,0 j1as,0 x8,y8 range jregion 3 jset j1d,0 j1at,0 j1ag,0 j1as,0 j2ax,j2ay range jregion 3  ; meshing gen edge 5 range mat=2 gen edge 30 range mat=1   change cons=3 range mat=2 change cons=3 range mat=1 change jmat=1 range ang bdminus1 bdplus1 change jmat=2 range ang j1minus1 j1plus1      ;CREATES THE LOW STRENGTH SLOPE change jmat=10 range id=10 ;COINCIDENT JOINT!  ; m-c properties prop mat 2 bulk=672e6 shear=350e6 prop mat 2 coh=12e6 fric=45 dil=0 prop mat 2 den=2273  prop mat 1 coh=12e6 fric=50 dil=0 prop mat 1 bulk=672e6 shear=350e6 den=2273   ;m-c discontinuity properties for jmat1 (Bedding) and jmat2 (joints) prop jmat=1 jfric 45 jcoh 0 ;KEEP THESE HIGH TO INITIALIZE MODEL prop jmat=1 jkn 5570e6 jks 557e6;KEEP THESE HIGH TO INITIALIZE MODEL prop jmat=2 jfric 35 jcoh 0 ;KEEP THESE HIGH TO INITIALIZE MODEL prop jmat=2 jkn 1e10 jks 1e9 ;KEEP THESE HIGH TO INITIALIZE MODEL  prop jmat=10 jfric 50 jcoh 0 ;KEEP THESE HIGH TO INITIALIZE MODEL prop jmat=10 jkn 1e10 jks 1e9 ;KEEP THESE HIGH TO INITIALIZE MODEL  def setup boundary_x1 = bx1 219  boundary_x2 = bx2 boundary_x3 = bx3 boundary_x4 = bx4 boundary_y1 = by1 boundary_y2 = by2 boundary_y3 = by3 boundary_y4 = by4 boundary_y5 = by5 end  def bx1 bx1 = x1-1 end  def bx2 bx2 = x1+1 end  def bx3 bx3 = x5+1 end  def bx4 bx4 = x5-1 end  def by1 by1 = y1-1 end  def by2 by2 = y1+1 end  def by3 by3 = y5-1 end  def by4 by4 = y5+1 end  def by5 by5 = y2+1 end  gravity 0 -9.81  ; boundary conditions boun bx1 bx3 by1 by2 yvel 0 xvel 0 boun bx4 bx3 by1 by4 xvel 0 boun bx1 bx2 by1 by5 xvel 0   220  hist unbal  solve  save case30_ini.sav  reset disp jdisp rot  set delc off  fluid den 1000   prop mat 2 coh=0.100e6 fric=30 dil=0  prop jmat=1 jfric 30 jcoh 0 ;bedding  prop jmat=2 jfric 35 jcoh 0 ;joints  prop jmat=10 jfric 15 jcoh 0 ;slope coincident joint  reset disp jdisp rot  ;generate free-field both lateral boundaries & fixed bottom ffield gen left y by1, by5 np=200;  ff on left side yl and yu np is height ffield gen right y by1, by4 np=600; ff on right side yl and yu np is height ffield change mat=1 cons=3 ffield change mat=2 cons=3  ;fix bottom ffield base xvel=0 ffield base yvel=0   reset time hist hist ffyd x6,y6 hist ffsxx x6,y6  step 100000  reset disp jdisp rot  prop mat 2 coh=0.100e6 fric=30 dil=0; fails under static conditions  bound mat=1 bound mat=2 bound ff range bx1,bx2 y1,y2 bound ff range bx4,bx3 y1,y5 bound xvisc range bx1,bx3 by1,by2  ;amplitude of shear wave: 0.5MPa, freq: 3 Hz, T=3 sec bound stress 0 0.5e6 0 hist sine (3,2) range bx1,bx3 by1,by2 ;sine (freq,time)  ;fix yvel at bottom bound yvel=0 range bx1,bx3 by1,by2 221   ;apply free-field boundary conditions ff base sxy=0.5e6 hist sine(3,2) ffield base yvel=0 ffield base xvisc  reset hist time disp hist xvel x4,y4 yvel x4,y4 hist xdis x4,y4 ydis x4,y4  ;damping conditions damp 0.02 10 cycle time 10   222  2. The following UDEC input code allows the user to set up a dip slope with a specified bedding thickness, number of beds, with orthogonal “cross-beds” with a set spacing truncated at bedding.  In addition, there is a slope-coincident joint at the base of the bedding layers that can be assigned properties different than the bedding layers.  This model considers earthquake loading and demonstrates the failure mechanisms associated with dip slopes. --------------------------------------------- new def setup  ;GEMETRY PARAMTERS  slope_height = s_height Slope_angle = s_angle Height_of_region_one = r1_height Height_of_region_two = r2_height  ;DEFINE X AND Y COORDINATES  x_one = x1 x_two = x2 x_three = x3 x_four = x4 x_five = x5 x_six = x6 x_seven = x7 x_eight = x8 x_eightplusone = x8plusone x_eightminusone = x8minusone x_nine = x9 x_nineplusone = x9plus0ne x_nineminusone = x9minusone x_ten = x10 x_eleven = x11 x_twelve = x12 x_thirteen = x13 y_one = y1 y_two = y2 y_three = y3 y_four = y4 y_five = y5 y_six = y6 y_seven = y7 y_eight = y8 y_nine = y9 y_ten = y10 y_eleven = y11 y_twelve = y12 y_thirteen = y13  223  ;BEDDING GEOMETRY PARAMTERS  bedding_dip = bd bedding_trace = btr bedding_gap = bg bedding_spacing = bs number_of_bedding_layers = numlyr bedding_x = bx bedding_y = by  beddinga_spacing = bsa  ;JOINT GEOMETRY PARAMTERS  Joint_1_dip = J1d joint_1_trace = j1t joint_1_gap = j1g joint_1_spacing = j1s joint_1_x = j1x joint_1_y = j1y  joint_2_x = j2x joint_2_y = j2y  joint_1a_gap = j1ag joint_1a_spacing = j1as joint_1a_trace = j1at joint_1a_gap = j1ag   end  ;DEFINE THE RANGE OF ANGLES FOR ASSIGNING JOINT PROPERTIES  def bdplus1 bdplus1 = bd + 1 end  def bdminus1 bdminus1 = bd - 1 end  def j1plus1 j1plus1 = j1d + 25 end  def j1minus1 j1minus1 = j1d - 25 end  ;ALL INPUT PARAMTERS GO IN THE FOLLOWING ARGUMENTS    set s_height = 200 ;Input the slope height in meters   set s_angle = 30 ;Input the slope angle in degrees 224    set bd = s_angle ;Dip angle coincident with slope angle   set btr = 500   ;Input the trace length of bedding   set bs = 5  ;Input the spacing normal to bedding   set numlyr = 8 ;Input the number of bedding layers ****  def j1s   j1s = bs*3 ;Input joint spacing normal to joints end  ;DONE WITH INPUT PARAMATERS   set bg = 0.0  ;Input the gap length of bedding (0)   set x1 = 0  ;Model starts at the x,y, origin   set y1 = 0  ;Model starts at the x,y origin   set x2 = 0  ;x2 is zero   set y2 = s_height ;Distance from origin to slope toe is assume equal to slope height   set x3 = s_height ;Distance infront of toe is assumed to be equal to the slope height   set y3 = s_height ;Flat slope in front of slope  def y4  y4 = y3 + s_height  end  def x4  x4 = (y4 - y3)/(TAN(s_angle*PI/180)) + x3  end  def x5  x5 = x4 + s_height ;Distance behind the crest is equal to the slope height  end    set y5 = y4   set x6 = x5   set y6 = y1   ;CREATE OUTTER AREA OF THE SLOPE  ro 0.1 bl x1,y1 x2,y2 x3,y3 x4,y4 x5,y5 x6,y6  ;SET UP TO DIVIDE SLOPE INTO JOINT SPACING ZONATION  set y7 = y1  def x7 225  x7 = x3 + ((y7-y3)*(tan((90-s_angle)*PI/180))) ;Makes the joint region equal to bedding thickness x number layers end  def x9 x9 = x4 + ((bs*numlyr)/(sin(s_angle*PI/180))) end  def x8plusone x8plusone = x8 +1 end  def x8minusone x8minusone = x8 - 1 end  def y9 y9 = y4 end  def x9plusone x9plusone = x9 + 1 end  def x9minusone x9minusone = x9 - 1 end  def x8 x8 = x9 - ((y9-y1)*(tan((90-s_angle)*PI/180))) ;Joint region equal to bedding thickness x number layers end  set y8 = y1   ; ZONE THE MODEL INTO THREE REGIONS crack x7,y7 x3,y3 crack x8minusone,y8 x9minusone,y9  id = 10   ;ASSIGN JOINT REGIONS (3 TOTAL) jregion id 1 x7,y7 x4,y4 x9,y9 x8,y8 jregion id 2 x1,y1 x2,y2 x3,y3 200, 131 jregion id 3 x8,y8 x9,y9 x5,y5 x6,y6  ;ASSIGN Material REGIONS only to jointed area ;Material 1 will be elastic and is hard to define change mat=2 range region x7,y7 x4,y4 x9,y9 x8,y8  ;ORTHOGONAL JOINT SET GENERATION   set bx = x3  ;Input the x reference point on the slope (toe)   set by = y3  ;Input the y reference point on the slope (toe)  226  def j1d  j1d = -1*(90 - bd) ;Sets the joint normal to bedding  end    set j1t = bs  ;Joint trace is equal to the bedding spacing   set j1g = bs  ;Joint gap is equal to the bedding spacing   set j1x = x3  ;x reference point to x3   set j1y = y3  ;y reference point to y3    def j2x   ;The location of the x coordinate is defined based on the joint set geometry  j2x = bx-(1.5 * j1s * cos(bd*(PI/180)) + bs*(sin(bd*PI/180)))  end  def j2y   ;The location of the x coordinate is defined based on the joint set geometry  j2y = by-(1.5 * j1s * sin(bd*(PI/180)) + bs*(sin(j1d*PI/180)))  end  def bsa   bsa = 4 * bs  ;Input the spacing normal to bedding outside region of interest end    set j1at = bsa ;Joint trace is equal to the bedding spacing   set j1ag = bsa ;Joint gap is equal to the bedding spacing  def j1as   j1as = 4 * j1s ;This is the spacing of joints outside the region of interest end  def j2ax  ;The location of the x coordinate is defined based on the joint set geometry  j2ax = x8-(1.5 * j1as * cos(bd*(PI/180)) + bsa*(sin(bd*PI/180)))  end  def j2ay  ;The location of the x coordinate is defined based on the joint set geometry  j2ay = y8-(1.5 * j1as * sin(bd*(PI/180)) + bsa*(sin(j1d*PI/180)))  end  227  ;ASSIGN JOINT PROPERTIES PER REGION  ;joint sets joint region 1 jset bd,0 btr,0 bg,0 bs,0 x3,y3 range jregion 1 jset j1d,0 j1t,0 j1g,0 j1s,0 j1x,j1y range jregion 1 jset j1d,0 j1t,0 j1g,0 j1s,0 j2x,j2y range jregion 1  ;joint sets joint region 2 jset bd,0 btr,0 bg,0 bsa,0 bx,by range jregion 2 jset j1d,0 j1at,0 j1ag,0 j1as,0 x8,y8 range jregion 2 jset j1d,0 j1at,0 j1ag,0 j1as,0 j2ax,j2ay range jregion 2  ;joint sets joint region 3 jset bd,0 btr,0 bg,0 bsa,0 bx,by range jregion 3 jset j1d,0 j1at,0 j1ag,0 j1as,0 x8,y8 range jregion 3 jset j1d,0 j1at,0 j1ag,0 j1as,0 j2ax,j2ay range jregion 3  ; meshing gen edge 5 range mat=2 gen edge 30 range mat=1   change cons=3 range mat=2 change cons=3 range mat=1 change jmat=1 range ang bdminus1 bdplus1 change jmat=2 range ang j1minus1 j1plus1      ;CREATES THE LOW STRENGTH SLOPE change jmat=10 range id=10 ;COINCIDENT JOINT!  ; m-c properties prop mat 2 bulk=672e6 shear=350e6 prop mat 2 coh=12e6 fric=45 dil=0 prop mat 2 den=2273  prop mat 1 coh=12e6 fric=50 dil=0 prop mat 1 bulk=672e6 shear=350e6 den=2273   ;m-c discontinuity properties for jmat1 (Bedding) and jmat2 (joints) prop jmat=1 jfric 45 jcoh 0 ;KEEP THESE HIGH TO INITIALIZE MODEL prop jmat=1 jkn 5570e6 jks 557e6;KEEP THESE HIGH TO INITIALIZE MODEL prop jmat=2 jfric 35 jcoh 0 ;KEEP THESE HIGH TO INITIALIZE MODEL prop jmat=2 jkn 1e10 jks 1e9 ;KEEP THESE HIGH TO INITIALIZE MODEL  prop jmat=10 jfric 50 jcoh 0 ;KEEP THESE HIGH TO INITIALIZE MODEL prop jmat=10 jkn 1e10 jks 1e9 ;KEEP THESE HIGH TO INITIALIZE MODEL  def setup boundary_x1 = bx1 boundary_x2 = bx2 boundary_x3 = bx3 boundary_x4 = bx4 boundary_y1 = by1 228  boundary_y2 = by2 boundary_y3 = by3 boundary_y4 = by4 boundary_y5 = by5 end  def bx1 bx1 = x1-1 end  def bx2 bx2 = x1+1 end  def bx3 bx3 = x5+1 end  def bx4 bx4 = x5-1 end  def by1 by1 = y1-1 end  def by2 by2 = y1+1 end  def by3 by3 = y5-1 end  def by4 by4 = y5+1 end  def by5 by5 = y2+1 end  gravity 0 -9.81  ; boundary conditions boun bx1 bx3 by1 by2 yvel 0 xvel 0 boun bx4 bx3 by1 by4 xvel 0 boun bx1 bx2 by1 by5 xvel 0   hist unbal  solve  229  fluid density 1000  prop jmat=1 jperm=1e2 azero=0.002 ares=0.0005  prop jmat=2 jperm=1e2 azero=0.001 ares=0.0005  prop jmat=1 jperm=1e2 azero=0.002 ares=0.0005  bound imperm range bx1,bx3 by1, by2 bound pp 2e6 pygrad -1e4 range bx1,bx2 by1, 200 bound pp 3.2e6 pygrad -1e4 range bx4,bx3 by1,320  table 1 0,200 200,200 303.92,260 746,320  insitu yw table 1  set flow steady  reset disp jdisp rot  set delc off  fluid den 1000  prop mat 2 coh=0.200e6 fric=30 dil=0  prop jmat=1 jfric 30 jcoh 0 ;bedding  prop jmat=2 jfric 35 jcoh 0 ;joints  prop jmat=10 jfric 15 jcoh 0 ;slope coincident joint  solve   230  E.4 References Itasca. (2004) "Universal Distinct Element Code (UDEC) User's Manual."  Wyllie, D. C., and Mah, C. W. (2004) Rock slope engineering: civil and mining, Spon Press, New York, NY. 231  APPENDIX F. CHATSWORTH PROJECT LABORATORY TESTING F.1     Uniaxial Compressive Strength Testing of Intact Rock To estimate the uniaxial compressive strength (UCS) of intact sandstone of the Chatsworth Formation, 38 UCS tests were conducted on samples collected during the borehole drilling and coring activities.  Uniaxial compressive strength testing provides a “check” on the field logging of Rock Grade (Brown, 1981) and also is an input parameter used to estimate the rock mass strength of the formation (Hoek and Brown 1997). Fig. F. 1 is a histogram presenting the results of the UCS tests on intact rock samples. As part of the UCS testing, the unit weight of intact rock was recorded.  The results of the unit weight testing are presented in Fig. F. 2.   Fig. F. 1.  Summary of uniaxial compressive strength of the Chatsworth Formation Sandstone. 232   Fig. F. 2.  Summary of Chatsworth Formation Sandstone unit weight. F.2     Direct Shear Tests on Saw Cut Samples A series of seven direct shear tests were performed on saw cut rock samples to estimate the “basic” friction angle of the sandstone of the Chatsworth Formation.  The testing procedures generally followed those outlined in ASTM D5607.  Hoek et al. (1981) describe the basic friction angle as the strength envelope generated by conducting direct shear tests on saw cut rock. The strength envelope developed should theoretically show zero strength at zero normal stress (i.e. zero cohesion on a Mohr-Coulomb diagram) and should increase in a linear manner with increasing normal stress.  Because the tested saw cut surfaces are not “polished”, the peak strength generated (at low shear strains) may not be considered the residual shear strength. Although there is generally not a dramatic drop in shear strength at high shear strains, the residual friction angle should be slightly lower than the basic friction angle generated during this type of testing.  Fig. F. 3 shows the results of direct shear testing on saw cut samples. 233   Fig. F. 3.  Direct shear testing on saw cut surfaces of Chatsworth Sandstone. F.3     Direct Shear Tests on Natural Discontinuity Surfaces To estimate the peak and residual shear strength of natural discontinuities within sandstone of the Chatsworth Formation, thirteen direct shear tests were performed.  The natural discontinuities chosen for this testing were generally oxide coated and representative of discontinuities that do not contain clay infilling.  The testing procedures generally followed those outlined in ASTM D5607.  The direct shear testing along natural discontinuities provides an indication of the dilative nature of the discontinuity as it shears.  The testing is completed by mating opposing faces of a discontinuity surface together.  The sample is sheared to a large strain; in the cases presented, about 6 mm.  The stress strain curve from each test is plotted and peak and residual shear strengths are chosen based on the greatest and least post-peak shear stresses recorded during the testing procedures.  234  The peak friction angle measured on clean, rough, natural discontinuities is generally greater than those measured by using the saw cut samples.  The residual friction angles measured are usually identical to the residual friction angles measured via the same tests on saw-cut samples. At the strain and pressure at which the residual strength is achieved, any asperities on the discontinuity surfaces will be worn down and the sample will not dilate during the testing.  In general, where the discontinuity walls are weathered, not smooth and planar, and where the weathering zone is thin enough that the discontinuity walls can come in contact while shearing, the shear strength values measured are greater than those when direct shear tests are completed on saw-cut samples.  Results of direct shear tests on natural discontinuities are presented in Fig. F. 4.   Fig. F. 4.  Direct shear testing on natural discontinuity (clean) surfaces. 0 200 400 600 800 0 200 400 600 800 1,000 1,200 Sh ea r S tr es s,  (k Pa ) Normal Stress, (kPa) 235  F.4 Torsional Ring Shear Tests on Shale Bedding To estimate the fully softened and residual shear strength of shale layers interbedded within the sandstone of the Chatsworth Formation, two torsional ring shear tests were conducted.  Testing of the shale layers is considered to provide an estimate of the shear strength along bedding where bedding consists of shale interbeds.  Stark and Eid (1994, 1997) pioneered torsional ring shear testing and ASTM adopted this type of testing in ASTM D6467.  DMG Special Publication 117 states that the fully softened shear strength estimated via torsional ring shear testing is equivalent to the “ultimate” strength of the material.  The residual strength is the strength at high strains where clay particles are aligned.  Moreover, the residual strength from torsional ring shear testing is recommended to estimate the shear strength along bedding where flexural slip has occurred.  Direct shear testing of clay lined bedding planes were also conducted and compared to the results of the torsional ring shear tests.  Fig. F. 5 is a chart showing the shear strength of clay- lined and shale bedding planes.  Fig. F. 5.  Direct shear testing results on clay-filled discontinuities. F.5 Direct Shear Tests on Intact Rock Samples To estimate the peak, ultimate, and residual shear strength of intact sandstone of the Chatsworth Formation we conducted a series of 11 direct shear tests.  Samples that range in Rock Grade from R0 to R1 (according to Brown, 1981) were tested using this method. Grade 236  R0 has a uniaxial compressive strength ranging from 0.25 to 1 MPa.  Grade R1 rock has a uniaxial compressive strength ranging from one to 5 MPa.  In a very general sense, the R0 rock is considered decomposed and found at the surface of the soil/rock contact on site.  R1 rock is encountered beneath the decomposed rock and usually characterized as highly weathered. Localized zones of R1 and R0 rock in the slightly weathered or fresh rock were encountered well below the surficial oxidized zone.  The testing procedures generally followed those outlined in ASTM D3080.  SCEC (2002) recommends this type of testing to estimate “cross-bedding” strength of intact rock using samples collected from the rock core borings.  The direct shear testing provides an estimate of the peak, ultimate and residual shear strength of intact rock.  Although this type of testing (on intact rock) is seldom completed elsewhere, it has become a standard in Southern California (SCEC, 2002).  The sample is sheared to a large strain; in this case to about 6 mm of displacement.  The stress strain curve from each test is plotted and peak, ultimate, and residual shear strengths are chosen based on the greatest, median, and least shear stress recorded during the testing procedures.  As discussed in SCEC, (2002), ultimate shear stress was chosen as the inflection point between the peak and residual values.  In some cases, the ultimate shear strength is difficult to ascertain because at higher normal stresses, the rock becomes less ductile under shear loading.  Results of direct shear testing on Grade R1 intact rock samples are presented in Fig. F. 6. 237   Fig. F. 6.  Direct shear testing completed on Grade R1 intact rock samples. F.6 Triaxial Testing of Intact Rock Samples Triaxial tests were conducted on intact rock samples to estimate the peak shear strength envelope of the Chatsworth Formation, determined through the UCS tests to have an estimated UCS of five to 25 MPa.  This range of UCS strength represents Rock Grade R2 according to (Brown 1981).  A total of 15 tests were performed.  One of the tests was a “staged” test where one sample was subjected to three different confining pressures during the testing procedure. The goal of the triaxial testing was to establish a relationship between the confining pressure applied to the samples and the axial stress required causing failure.  This information was then used to estimate empirical curve fitting parameters to establish a failure envelope for the rock mass according to the Hoek Brown Failure Criterion (Hoek and Brown 1980).  The triaxial 238  testing is also useful for estimating a strength envelope for the intact rock.  The principal stresses recorded during the triaxial testing on intact rock core are presented in Fig. F. 7.   Fig. F. 7.  Principle stresses from triaxial tests on intact samples of the Chatsworth Sandstone – Grade R2 rock. 0 5 10 15 20 25 30 35 40 45 50 55 60 65 0 5 10  1,  M Pa 3, MPa Hoek-Brown Parameters: mi ~ 17 mb ~ 17 s = 1 D = 0  Mohr-Coulomb Parameter  ~ 42 deg. c ~ 3.6 MPa  Uniaxial Compressive Strength ~ 14 MPa  Tensile Strength ~ 0.85 MPa 239  F.7 References ASTM (2004) "Standard Test Method for Direct Shear Test of Soils Under Consolidated Drained Conditions": American Society for Testing Materials. Designation D-3080.  ASTM (2006) "Standard Test Method for Torsional Ring Shear Test to Determine Drained Residual Shear Strength of Cohesive Soils": American Society for Testing Material. Designation D-6467.  ASTM (2008) "Standard Test Method for Performing Laboratory Direct Shear Strength Tests of Rock Specimens Under Constant Normal Force": American Society for Testing and Materials. Designation Number D-5607.  Brown, E. T. (1981) Rock Characterization, Testing and Monitoring - ISRM Suggested Methods, Oxford: Pergamon.  Hoek, E., and E. T. Brown (1997) "Practical estimates of rock mass strength," International Journal of Rock Mechanics and Mining Sciences, Vol. 34, No. 8, pp 1165-1186.  Hoek, Evert, John Bray, and Institution of Mining and Metallurgy (Great Britain) (1981) Rock slope engineering, London: Institution of Mining and Metallurgy.  Hoek, Evert, and E. T. Brown (1980) Underground excavations in rock, London: The Institution of Mining and Metallurgy.  Southern California Earth Quake Center (2002) "Recommended Proceedings for Implementation of DMG Special Publication 117 Guidelines for Analyzing and Mitigating Landslide Hazards in California," In: T. F. Blake, R. A. Hollingsworth, and J. P. Stewart, Eds.: Southern California Earthquake Center, University of Southern California, Los Anglees, p 110 plus appendices.  Stark, Timothy D., and Hisham T. Eid (1994) "Drained residual strength of cohesive soils," Journal of Geotechnical Engineering, Vol. 120, No. 5, pp 856-871. 240   Stark, Timothy D., and Hisham T. Eid (1997) "Slope stability analyses in stiff fissured clays," Journal of Geotechnical and Geoenvironmental Engineering, Vol. 123, No. 4, pp 335-343. 

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