SlopeStability_Theory_Manual

[Virtual Presenter] PLAXIS LE Slope Stability 2D/3D LIMIT EQUILIBRIUM SLOPE STABILITY ANALYSIS Theory Manual Written by: The Bentley Systems Team Last Updated: Wednesday, October 09, 2024 Bentley Systems Incorporated.

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[Audio] BENTLEY SYSTEMS Slope Stability Introduction 3 of 158 1 SLOPE STABILITY INTRODUCTION .................................................................................................................... 8 2 OVERVIEW OF SLOPE STABILITY ....................................................................................................................... 9 3 LIMIT EQUILIBRIUM METHODS ........................................................................................................................ 10 3.1 INTRODUCTION ........................................................................................................................................................................................................................................................10 3.2 DEFINITION OF FACTOR OF SAFETY ..................................................................................................................................................................................................................12 3.2.1 Unsaturated Soil Phi-b Method ............................................................................................................................ 12 3.3 GENERAL LIMIT EQUILIBRIUM METHOD .........................................................................................................................................................................................................12 3.3.1 Static Equilibrium Equations ............................................................................................................................... 13 3.3.2 Factor of Safety Equations for Moments and Horizontal Equilibrium ................................................................. 13 3.3.3 Normal Force at the Base of each Slice ............................................................................................................... 14 3.3.4 Interslice Normal Force and Interslice Shear Force ........................................................................................... 14 3.3.5 The Relationship between General Limit Equilibrium Method and other Methods of Slices ............................... 14 3.4 ORDINARY OR FELLENIUS METHOD .................................................................................................................................................................................................................15 3.4.1 Factor of Safety for a Circular Slip Surface ......................................................................................................... 16 3.4.2 Factor of Safety for a Composite Slip Surface ..................................................................................................... 17 3.5 BISHOP'S SIMPLIFIED METHOD ..........................................................................................................................................................................................................................18 3.5.1 Factor of Safety for a Circular Slip Surface ......................................................................................................... 19 3.5.2 Factor of Safety for Bishop's Simplified Method for Composite Slip Surface ...................................................... 20 3.5.3 Iterative Procedure Used in Solving Bishop's Simplified Method ....................................................................... 20 3.6 JANBU'S SIMPLIFIED METHOD ...........................................................................................................................................................................................................................21 3.7 SPENCER METHOD .................................................................................................................................................................................................................................................21 3.8 MORGENSTERN-PRICE METHOD ........................................................................................................................................................................................................................21 3.9 CORPS OF ENGINEERS METHOD .........................................................................................................................................................................................................................22 3.10 LOWE-KARAFIATH METHOD ..............................................................................................................................................................................................................................23 3.11 SARMA METHOD (1973) .......................................................................................................................................................................................................................................23 3.12 SARMA NON-VERTICAL SLICES METHOD (1979) .........................................................................................................................................................................................23 4 STRESS-BASED METHODS ................................................................................................................................... 27 4.1 ENHANCED LIMIT METHOD (KULHAWY METHOD) ......................................................................................................................................................................................27 4.1.1 Definition of Factor of Safety ............................................................................................................................... 27 4.1.2 Stress Transfer from the Finite Element Analysis to the Center of the Base of a Slice ........................................ 27 4.1.3 The Normal and Shear Stresses at the Center of the Base of a Slice .................................................................... 28 4.2 SAFE-DP METHOD ...............................................................................................................................................................................................................................................29 4.2.1 Definition of the Factor of Safety ......................................................................................................................... 29 4.2.2 Stress Transfer from the Finite Element Analysis to the Grid Point .................................................................... 30 4.2.3 The Normal and Shear Stresses on a Segment ..................................................................................................... 30 5 SEISMIC METHODS ............................................................................................................................................... 31 5.1 CALCULATE THE YIELD COEFFICIENT FOR ALL SLIP SURFACES SEISMIC ANALYSIS ........................................................................................................................31 5.2 NEWMARK PERMANENT DISPLACEMENT SEISMIC ANALYSIS ..................................................................................................................................................................31 5.3 DYNAMICS – TECHNICAL PREVIEW ..................................................................................................................................................................................................................32 6 TENSION CRACKS ................................................................................................................................................. 33 6.1 TENSION CRACK LINE ...........................................................................................................................................................................................................................................33 6.2 TENSION CRACK ANGLE ......................................................................................................................................................................................................................................33 6.3 HYDROSTATIC HORIZONTAL FORCE IN TENSION CRACK ...........................................................................................................................................................................34 7 SLIP SURFACES ...................................................................................................................................................... 35 7.1 CIRCULAR SLIP SURFACES ..................................................................................................................................................................................................................................35 7.1.1 Slope Search ......................................................................................................................................................... 35 Auto Refine Search ............................................................................................................................................................. 36 7.2 COMPOSITE CIRCULAR SLIP SURFACE .............................................................................................................................................................................................................38 7.3 NON-CIRCULAR SLIP SURFACES ........................................................................................................................................................................................................................38 7.3.1 Block Search ......................................................................................................................................................... 39 7.3.1.1 Block Search – Block .................................................................................................................................................... 39 7.3.1.2 Block Search – Point ..................................................................................................................................................... 39 7.3.1.3 Block Search – Line ...................................................................................................................................................... 39 7.3.1.4 Block Search - Polyline ................................................................................................................................................. 39 7.3.2 Fully Specified (Segments) ................................................................................................................................... 39 7.3.3 Path Search ...........................................................................................................................................................

[Audio] BENTLEY SYSTEMS Slope Stability Introduction 4 of 158 7.3.4.2 Objective Function of the Problem ................................................................................................................................ 43 7.3.4.3 Solution of the Greco (1996) Formulation .................................................................................................................... 43 7.3.4.4 Implementation of Greco (1996) Method ...................................................................................................................... 46 7.3.5 Dynamic Programming Method ........................................................................................................................... 47 7.3.5.1 General .......................................................................................................................................................................... 48 7.3.5.2 Definition of the Factor of Safety .................................................................................................................................. 48 7.3.5.3 Formulation of the Dynamic Programming Method...................................................................................................... 49 7.3.5.4 Procedure of the Dynamic Programming Search ........................................................................................................... 52 7.3.5.5 Research Applied to the Shape of the Critical Slip Surface ........................................................................................... 53 7.3.6 Cuckoo Search ...................................................................................................................................................... 54 7.4 OPTIMIZE SURFACES .............................................................................................................................................................................................................................................54 8 MATERIAL STRENGTH......................................................................................................................................... 56 8.1 MOHR-COULOMB STRENGTH..............................................................................................................................................................................................................................56 8.2 CURVED-SURFACE ENVELOPE MOHR-COULOMB .........................................................................................................................................................................................57 8.3 UNDRAINED SHEAR STRENGTH ..........................................................................................................................................................................................................................58 8.4 NO-STRENGTH .........................................................................................................................................................................................................................................................59 8.5 BEDROCK ..................................................................................................................................................................................................................................................................59 8.6 ANISOTROPIC STRENGTH .....................................................................................................................................................................................................................................59 8.7 ANISOTROPIC FUNCTION ......................................................................................................................................................................................................................................60 8.8 ANISTROPIC LINEAR MODEL (ALM1) .............................................................................................................................................................................................................60 8.8.1 Computation of Anisotropic Shear Strength ......................................................................................................... 61 8.9 MODIFIED ANISTROPIC LINEAR MODEL 2 (ALM2) .....................................................................................................................................................................................61 8.10 MODIFIED ANISOTROPIC LINEAR MODEL 3 (ALM3)...................................................................................................................................................................................62 8.11 MODIFIED ANISOTROPIC LINEAR MODEL 4 (ALM4)...................................................................................................................................................................................63 8.12 BILINEAR MODEL ...................................................................................................................................................................................................................................................64 8.13 POWER CURVE 1 .....................................................................................................................................................................................................................................................64 8.14 POWER CURVE 2 .....................................................................................................................................................................................................................................................64 8.15 UNDRAINED STRENGTH RATIO ..........................................................................................................................................................................................................................65 8.16 COMBINED FRICTIONAL-UNDRAINED MODEL ...............................................................................................................................................................................................65 8.17 SHEAR-NORMAL STRESS FUNCTION .................................................................................................................................................................................................................65 8.18 GENERALIZED HOEK-BROWN MODEL .............................................................................................................................................................................................................65 8.19 HOEK-BROWN MODEL ..........................................................................................................................................................................................................................................66 8.20 UNSATURATED SOIL MODELS ............................................................................................................................................................................................................................67 8.20.1 Linear Model (Phi-b) ....................................................................................................................................... 67 8.20.2 Nonlinear Models ............................................................................................................................................ 68 8.20.2.1 Fredlund Unsaturated Shear Strength Equation ............................................................................................................. 68 8.20.2.2 Vanapalli Unsaturated Shear Strength Equation ............................................................................................................ 69 8.20.2.3 Vilar Unsaturated Shear Strength Equation ................................................................................................................... 70 8.20.2.4 Khalili Unsaturated Shear Strength Equation ................................................................................................................ 70 8.20.2.5 Bao Unsaturated Shear Strength Equation ..................................................................................................................... 71 8.21 BARTON-BANDIS STRENGTH MODEL ...............................................................................................................................................................................................................71 8.22 SHANSEP STRENGTH MODEL ...........................................................................................................................................................................................................................71 9 INITIAL CONDITIONS ........................................................................................................................................... 73 9.1 PORE-WATER PRESSURE (PWP) ........................................................................................................................................................................................................................73 9.1.1 Water Surfaces ..................................................................................................................................................... 73 9.1.1.1 Water Table ................................................................................................................................................................... 73 9.1.1.2 Piezometric line ............................................................................................................................................................. 74 9.1.2 Ru Coefficient ....................................................................................................................................................... 75 9.1.3 Discrete Points ..................................................................................................................................................... 75 9.1.3.1 Discrete Point Types ..................................................................................................................................................... 76 9.1.3.2 Discrete Points Interpolation Method ............................................................................................................................ 76 9.1.4 Groundwater File ................................................................................................................................................. 76 9.1.5 Stress File ............................................................................................................................................................. 76 9.1.6 Allow Per Material Ru Coefficients to Override Selected PWP Method .............................................................. 76 9.1.7 B-Bar Coefficient .................................................................................................................................................. 76 9.2 SUCTION AND NEGATIVE PORE-WATER PRESSURE .....................................................................................................................................................................................77 9.2.1 Water Table Suction ............................................................................................................................................. 77 9.3 STRESSES FROM STRESS .......................................................................................................................................................................................................................................77 9.3.1 Gridlines for Dynamic Programming Method ..................................................................................................... 78 10 SPECTRAL PSEUDO-STATIC LOADING ............................................................................................................ 79 11 SUPPORTS ................................................................................................................................................................ 80 11.1 END ANCHORED......................................................................................................................................................................................................................................................80 11.2 GEOTEXTILE ............................................................................................................................................................................................................................................................81.

[Audio] BENTLEY SYSTEMS Slope Stability Introduction 5 of 158 11.2.1 Shear Strength Models ..................................................................................................................................... 81 11.2.2 Geotextile Support Theory ............................................................................................................................... 83 11.3 GROUTED TIEBACK ................................................................................................................................................................................................................................................85 11.4 GROUTED TIEBACK WITH FRICTION .................................................................................................................................................................................................................87 11.5 MICRO-PILE .............................................................................................................................................................................................................................................................88 11.6 SOIL NAIL .................................................................................................................................................................................................................................................................90 11.7 SOIL NAIL - HONG KONG PRACTICE .................................................................................................................................................................................................................91 11.8 USER DEFINED ........................................................................................................................................................................................................................................................92 12 RAPID DRAW-DOWN (3-STAGE METHOD) ....................................................................................................... 93 12.1 GENERAL ..................................................................................................................................................................................................................................................................93 12.2 EFFECTIVE STRESS METHODS AND TOTAL STRESS METHODS .................................................................................................................................................................93 12.3 THREE-STAGE TOTAL STRESS METHOD ..........................................................................................................................................................................................................93 12.3.1 Three-stage procedure ..................................................................................................................................... 93 12.3.1.1 First-stage ...................................................................................................................................................................... 93 12.3.1.2 Second stage .................................................................................................................................................................. 94 12.3.1.3 Third stage ..................................................................................................................................................................... 95 13 PROBABILISTIC ..................................................................................................................................................... 96 13.1 MONTE CARLO METHOD......................................................................................................................................................................................................................................96 13.2 LATIN HYPERCUBE METHOD ..............................................................................................................................................................................................................................97 13.2.1 Sampling .......................................................................................................................................................... 97 13.2.2 Grouping.......................................................................................................................................................... 97 13.2.3 Example ........................................................................................................................................................... 97 13.3 ALTERNATIVE POINT ESTIMATE METHOD (APEM) ....................................................................................................................................................................................99 13.3.1 Point Estimated Method .................................................................................................................................. 99 13.3.2 The Alternative Point Estimate Method ........................................................................................................... 99 13.3.3 Development of the Alternative Point Estimate Method ................................................................................ 100 13.3.4 Efficiency of the Alternative Point Estimate Point ........................................................................................ 103 13.3.5 Tornado Diagram .......................................................................................................................................... 104 13.3.6 Implementation of the Alternative Point Estimate Method ............................................................................ 106 13.4 NUMBER OF SAMPLES ........................................................................................................................................................................................................................................ 106 13.5 CRITICAL SLIP SURFACE LOCATION .............................................................................................................................................................................................................. 107 13.5.1 Fixed .............................................................................................................................................................. 107 13.5.2 Floating ......................................................................................................................................................... 107 14 SPATIAL VARIABILITY ...................................................................................................................................... 108 14.1 NONE ....................................................................................................................................................................................................................................................................... 108 14.2 1D SPATIAL VARIABILITY ................................................................................................................................................................................................................................ 108 14.2.1 Each Slice ...................................................................................................................................................... 109 14.2.2 Distance ......................................................................................................................................................... 109 14.3 2D SPATIAL VARIABILITY ................................................................................................................................................................................................................................ 110 14.3.1 General .......................................................................................................................................................... 110 14.3.2 Covariance Function ..................................................................................................................................... 110 14.3.3 Parameters .................................................................................................................................................... 113 14.3.4 Random Field ................................................................................................................................................ 114 15 SENSITIVITY ANALYSIS ..................................................................................................................................... 115 15.1 GENERAL ............................................................................................................................................................................................................................................................... 115 15.2 PARAMETERS ........................................................................................................................................................................................................................................................ 115 15.3 ONE-WAY SENSITIVITY ANALYSIS ................................................................................................................................................................................................................ 115 15.4 TWO-WAY SENSITIVITY ANALYSIS ............................................................................................................................................................................................................... 115 15.5 CRITICAL SLIP SURFACE LOCATION .............................................................................................................................................................................................................. 116 16 THEORY FOR 3D ANALYSIS .............................................................................................................................. 117 16.1 ONE-DIRECTIONAL 3D GENERAL LIMIT EQUILIBRIUM METHOD ......................................................................................................................................................... 117 16.2 TWO-DIRECTIONAL 3D GENERAL LIMIT EQUILIBRIUM METHOD ........................................................................................................................................................ 122 16.3 DISCUSSION OF ONE-DIRECTIONAL AND TWO-DIRECTIONAL 3D METHODS .................................................................................................................................... 126 16.4 DERIVATION OF 3D ENHANCED LIMIT METHOD (KULHAWY METHOD) ............................................................................................................................................ 126 16.5 SLIDING SURFACES FOR 3D MODELS ............................................................................................................................................................................................................ 127 16.5.1 Ellipsoidal Sliding Surface ............................................................................................................................ 127 16.5.2 Multi-planar Wedges ..................................................................................................................................... 128 16.5.3 Moving Wedges ............................................................................................................................................. 128 16.5.4 Fully-specified General Sliding Surface ........................................................................................................ 128.

[Audio] BENTLEY SYSTEMS Slope Stability Introduction 6 of 158 16.5.5 Composite Sliding Surface ............................................................................................................................. 128 16.6 BEDROCK ............................................................................................................................................................................................................................................................... 128 16.7 TENSION CRACKS ................................................................................................................................................................................................................................................ 128 16.8 DISCONTINUITIES ................................................................................................................................................................................................................................................ 128 16.9 INITIAL CONDITIONS .......................................................................................................................................................................................................................................... 129 16.10 POINT LOADS ........................................................................................................................................................................................................................................................ 129 16.11 SUPPORTS .............................................................................................................................................................................................................................................................. 129 16.12 MATERIAL STRENGTH MODELS ...................................................................................................................................................................................................................... 129 16.13 VERTICAL SIDE SHEAR RESISTANCE .............................................................................................................................................................................................................. 129 17 REFERENCES ........................................................................................................................................................ 132 18 APPENDIX A .......................................................................................................................................................... 136 18.1 DISCRETE POINTS INTERPOLATION METHODS ........................................................................................................................................................................................... 136 18.1.1 Spline Method ................................................................................................................................................ 136 18.2 CALCULATION BASED ON A DAT FILE ......................................................................................................................................................................................................... 137 18.3 LOCAL AVERAGE SUBDIVISION ...................................................................................................................................................................................................................... 137 18.3.1 General .......................................................................................................................................................... 137 18.3.2 One-Dimensional Local Average Subdivision ............................................................................................... 137 18.3.3 Multi-Dimensional Extensions ....................................................................................................................... 141 19 APPENDIX B – DYNAMICS – TECHNICAL PREVIEW .................................................................................... 144 19.1 DYNAMICS INTRODUCTION .......................................................................................................................................................................................................................... 144 19.1.1 Analysis Types ............................................................................................................................................... 144 19.1.2 Constitutive Models ....................................................................................................................................... 144 19.1.3 Boundary Conditions ..................................................................................................................................... 144 19.2 BASIC PRINCIPLES AND EQUATIONS .............................................................................................................................................................................................................. 145 19.2.1 Stress/Strain Conditions and Coordinate Systems ......................................................................................... 145 19.2.1.1 Spatial Coordinate System .......................................................................................................................................... 145 19.2.1.2 Time Dimension .......................................................................................................................................................... 146 19.2.2 Sign Conventions ........................................................................................................................................... 146 19.2.2.1 Load and Displacement ............................................................................................................................................... 146 19.2.2.2 Stresses ........................................................................................................................................................................ 146 19.2.2.3 Strains .......................................................................................................................................................................... 146 19.2.3 Units .............................................................................................................................................................. 146 19.2.4 Stress and strain Tensors ............................................................................................................................... 147 19.2.5 Displacement Derivatives .............................................................................................................................. 147 19.2.5.1 Spatial Derivatives ...................................................................................................................................................... 147 19.2.5.2 Temporal Derivatives .................................................................................................................................................. 148 19.2.6 Equations of motion ....................................................................................................................................... 148 19.2.6.1 General 3D .................................................................................................................................................................. 148 19.2.6.2 2D Plane Strain ............................................................................................................................................................ 149 19.2.7 Final Partial Differential Equations.............................................................................................................. 149 19.2.7.1 General 3D .................................................................................................................................................................. 150 19.2.7.2 2D Plane Strain ............................................................................................................................................................ 150 19.3 CONSTITUTIVE RELATIONSHIPS ...................................................................................................................................................................................................................... 150 19.3.1 General .......................................................................................................................................................... 150 19.3.2 Isotropic Linear Elastic Law (Total Stress) ................................................................................................... 150 19.4 DISCRETIZATION AND NUMERICAL SOLUTION ........................................................................................................................................................................................... 152 19.4.1 Discrete form of governing equations ........................................................................................................... 152 19.4.2 Mass matrix ................................................................................................................................................... 152 19.4.3 Damping matrix ............................................................................................................................................. 153 19.4.4 Time domain discretization and Integration schemes.................................................................................... 154 19.4.4.1 Newmark Method ........................................................................................................................................................ 154 19.4.4.2 HHT- method ............................................................................................................................................................ 155 19.5 INITIAL CONDITIONS .......................................................................................................................................................................................................................................... 155 19.5.1 Initial Stresses ............................................................................................................................................... 155 19.6 BOUNDARY CONDITIONS .................................................................................................................................................................................................................................. 156 19.6.1 Dynamic forces .............................................................................................................................................. 156 19.6.2 Constraints .................................................................................................................................................... 156 19.6.2.1 Free, fixed and motion boundary conditions ............................................................................................................... 156 19.6.2.2 Non-reflecting boundary conditions ............................................................................................................................ 157 19.7 DYNAMICS REFERENCES ............................................................................................................................................................................................................................... 158.

[Audio] BENTLEY SYSTEMS Slope Stability Introduction 7 of 158.

[Audio] BENTLEY SYSTEMS Slope Stability Introduction 8 of 158 1 SLOPE STABILITY INTRODUCTION Slope stability problems in geotechnical and geo-environmental engineering involve the solution of equilibrium equations of force and moment. This is traditionally accomplished through traditional method of slices techniques or more progressive stressbased methods. The PLAXIS LE - Slope Stability software was designed to be a technically comprehensive slope stability set of software tools which implements cutting-edge searching and solution techniques as well as classic limit equilibrium methods of slices. As such, the software combines the best of the old and new technologies and provides a comprehensive set of tools to the geotechnical consultant. The purpose of the theory manual is to provide the user with details regarding the theoretical formulation of each of the solution techniques implemented in the PLAXIS LE - Slope Stability software. The intent is not to provide an exhaustive summary of all theories associated with slope stability. Rather, the intent is to clearly describe specific details of the theory used in the PLAXIS LE - Slope Stability software. Throughout this theory manual the term "the software" will refer to the software product "PLAXIS LE - Slope Stability", unless otherwise noted. The software allows input from various finite element software modules in the PLAXIS product line. The user may import porewater pressures from PLAXIS LE - Groundwater or stress states from PLAXIS 2D. Some of the features of the software are as follows: Advanced stochastic analysis (Monte Carlo, Latin Hypercube, APEM) Spatial variability (1D, 2D) Unsaturated soil analysis Advanced critical slip surface searching algorithms Finite element analysis methods Comprehensive help systems Covering the historical development of slope stability calculation methods and the latest advances, the software provides the following list of solution methodologies: Ordinary Method (also referred as Conventional or Swedish Method) Bishop Simplified Method Janbu Simplified Method Corps of Engineers Methods 1 and 2 Lowe-Karafiath Method Spencer Method GLE (Fredlund) Method Morgenstern-Price Method Sarma Method Sarma Non-Vertical Slices Method Kulhawy Method SAFE-DP Method.

[Audio] BENTLEY SYSTEMS Overview of Slope Stability 9 of 158 2 OVERVIEW OF SLOPE STABILITY Slope Stability is designed as a comprehensive and theoretically advanced slope stability package. The package is designed to contain a mix of classic limit equilibrium methods of slices as well as other newer, more rigorous slope stability methods that use optimization and other computational techniques. Figure 1 provides an overall classification of slope stability methods of analysis. A distinction should be made between analysis methods and searching techniques. In 1977, Fredlund and Krahn classified the limit equilibrium methods of slices according to the elements of static equilibrium that were satisfied when solving for the factor of safety. This included categorizing the assumptions used to render the analysis determinate. In 1981, Fredlund, Krahn and Pufahl further extended the comparison of slope stability method of slices to include additional methods (Fredlund et al, 1981). Most of the limit equilibrium methods of slices made an assumption regarding the interslice forces (e.g., the interslice force function). Consequently, most of the methods of slices differed in the manner by which the normal force at the base of a slice was calculated. Common to all the methods of slices was the way the factor of safety was defined and the fact that the normal force was computed from static considerations of one slice through a potential sliding mass. Figure 1 indicates that it is now possible to also take into consideration the search technique associated with the determination of the shape and location of the critical slip surface. The finite element stress analysis method can also be used to determine the normal force at the base of a slice, giving rise to the Enhanced (Kulhawy) Limit method as well as other optimization techniques (e.g., Dynamic Programming). Figure 1 Overall classification of slope stability methods of analysis.

[Audio] BENTLEY SYSTEMS Limit Equilibrium Methods 10 of 158 3 LIMIT EQUILIBRIUM METHODS 3.1 INTRODUCTION Limit equilibrium methods of slope stability analysis have been widely used for computing the factor of safety since the advent of the digital computer in the 1960s. The methods can be classified and understood in terms of the principles involved and assumptions used in formulating each analysis. The general principles that have historically applied to all limit equilibrium methods of slices state that: 1. Plasticity conditions are developed only along the slip surface and the moving slope mass behaves as a solid, 2. The shape of the slip surface is to be selected by the end-user, and 3. The location of the critical slip surface is found by trial and error. The assumptions common to all limit equilibrium methods of slices are as follows: 1. The materials behave in a Mohr-Coulomb manner (i.e., each material has friction,  ', and cohesion, c', component of the shear strength) and the material fails in accordance with the Mohr-Coulomb failure criterion, 2. The factor of safety, Fs, is the same for all slices, and 3. The factor of safety, Fs, for the cohesive component is equal to the factor of safety for the frictional component as shown in the following shear force mobilized equation for a saturated soil. u ( )   w n F c S      − + =  tan [1] m F s s where: c' = effective cohesion, ' = effective angle of internal friction uw = pore-water pressure at the base of a slice,  = length along the base of a slice, n = normal stress acting on the base of a slice, Fs = overall factor of safety, Sm = shear force mobilized at the base of a slice, Ssoil = shear strength of the soil expressed as a force (i.e., Ssoil = ), Sc = cohesion strength expressed as a force (i.e., Sc = c'), and Sf = frictional strength expressed as a force (i.e., Sf = [( n - uw) tan']). All the methods of slices implemented in Slope Stability are based on the limit equilibrium principles and assumptions except for the SAFE method, which evaluates the factor of safety based on stress conditions determined from an independent finite element stress analysis (i.e., normal stresses and actuating shear stresses). The shape of a potential slip surface can be assumed to be circular, composite (i.e., combined circular and linear portions) or non-circular (i.e., fully specified or block specified slip surfaces). The slip surface passes through the material mass and the inscribed portion of material is subdivided into slices. Figure 2 and Figure 3 show the forces involved for circular and composite slip surfaces, respectively. Note that the fully specified slip surfaces and the block specified surfaces are somewhat similar in the linear portion of a composite slip surface. The variables are defined as follows:.

[Audio] BENTLEY SYSTEMS Limit Equilibrium Methods 11 of 158 Figure 2 Definition of forces and slope geometry variables for a circular slip surface Figure 3 Definition of forces and slope geometry variables for a composite slip surface where: W = weight of a slice, N = normal force on the base of the slice, Sm = shear force mobilized on the base of each slice, E = horizontal interslice normal forces. The 'L' and 'R' subscripts indicate the left and right sides of the slices, respectively, X = vertical interslice shear forces. The 'L' and 'R' subscripts indicate the left and right sides of the slices, respectively, D = external line load is applied only to the slice on which it acts. A distributed load can also be converted into an equivalent line load for each slice, k = horizontal seismic load coefficient applied through the centroid of each slice, r = radius (or the moment arm) associated with the mobilized shear force, Sm, f = perpendicular offset of the normal force from the center of rotation (or from the axis point). It is assumed that the 'f' distance on the right side of the center of rotation of a negative slope (i.e., a left-to-right slope) is negative and those on the left side of the center of rotation are positive, x = horizontal distance from the centroid of each slice to the center of rotation (or to the axis point), e = vertical distance from the centroid of each slice to the center of rotation (or to the axis point), d = perpendicular distance from a line load to the center of rotation (or to the axis point),.

[Audio] BENTLEY SYSTEMS Limit Equilibrium Methods 12 of 158 dr = perpendicular distance from a reinforcement load to the center of rotation (or the axis point), h = vertical distance from the center of the base of each slice to ground surface, aL,R = perpendicular distance from the resultant external water force to the center of rotation (or the axis point). The 'L' and 'R' subscripts indicate the left and right sides of the slope, respectively, AL,R = resultant external water forces. The 'L' and 'R' subscripts indicate the left and right sides of the slope, respectively,  = angle of the line load from the horizontal. This angle is measured counterclockwise from the positive x-axis,  = angle of the reinforcement load from the horizontal. This angle is measured counterclockwise from the positive x-axis, and  = angle between the tangent to the center of the base of each slice and the horizontal. The sign conversion for a slope is as follows: positive when the angle of the slice base is in the same direction as overall slope of the geometry, and negative when the angle of the slice base is opposite to the overall slope of the geometry. 3.2 DEFINITION OF FACTOR OF SAFETY The factor of safety is defined as: "that factor by which the shear strength must be reduced to bring a material mass into a state of limit equilibrium along a specified slip surface". The shear force mobilized for a saturated soil can be written in terms of the failure criterion for any given slice. ( )  ' tan '    S − + = [2] m u c w n F s where: Sm = shear force mobilized, c' = effective cohesion of the material, ' = effective angle of internal friction, uw = pore-water pressure at base of a slice,  = length along the base of a slice, and n = normal stress acting on the base of a slice. 3.2.1 Unsaturated Soil Phi-b Method For an unsaturated soil, the effect of matric suction needs to be included in the equation for the mobilized shear force (Fredlund et al., 1978). S     )tan ( tan ' ' − + − + = [3] ( )   b m u u u c w a a n F s where: ua = pore-air pressure at the base of a slice, ua - uw = matric suction, and b = the angle defining the rate of increase of strength due to an increase in suction. Equation [3] is a linear form for designating the shear strength of an unsaturated soil. The shear force mobilized equation is later written in terms of the soil-water characteristic curve, SWCC, for an unsaturated material. The component of shear strength related to matric suction then becomes nonlinear. This equation has the effect of increasing the shear strength in the unsaturated zone of a soil profile. 3.3 GENERAL {LIMIT EQUILIBRIUM METHOD.

[Audio] BENTLEY SYSTEMS Limit Equilibrium Methods 13 of 158 the incorporation of a variety of interslice force assumptions. The GLE method can then be specialized to other methods of slices available in the software. 3.3.1 Static Equilibrium Equations The following equations of Newtonian statics are used to formulate two factors of safety equations.  0 = 0 M : equation is applied for the overall slope and is the basis of the moment equilibrium factor of safety, (Fs)m equation,  h = 0 F : equation is applied for the overall slope and is the basis of the force equilibrium, (Fs) f equation,  v = 0 F : equation is applied to each slice for defining the normal force on the base of each slice, N, and  Fh = 0 : equation is applied for each slice when using an integration across the slope for defining the interslice normal forces, E. The GLE method can be viewed as independently solving two factors of safety equations. One equation gives the factor of safety with respect to moment equilibrium, (Fs) m, and the other equation provides the factor of safety with respect to overall horizontal force equilibrium, (Fs)f. 3.3.2 Factor of Safety Equations for Moments and Horizontal Equilibrium The factor of safety equation with respect to moment equilibrium is: ∑[𝑐′𝛽𝑟 𝑐𝑜𝑠 𝑎 + [𝑊 + (𝑋𝐿 − 𝑋𝑅) + [𝐷 𝑠𝑖𝑛 𝜔] + [𝐷𝑟 𝑠𝑖𝑛 𝜃] − 𝑢𝑤𝛽 𝑐𝑜𝑠 𝑎]𝑟 tan 𝜑′]/𝑚𝑎 (𝐹𝑠)𝑚 = ±𝐴𝑎 + ∑ 𝑊𝑥 − ∑ 𝑃𝑓 + ∑ 𝑘𝑊𝑒 ± [𝐷𝑑] ± [𝐷𝑟𝑑𝑟] [4] The factor of safety equation with respect to horizontal force equilibrium is: ∑ [𝑐′𝛽 + ( 𝑊 𝑐𝑜𝑠 𝛼 + (𝑋𝐿 − 𝑋𝑅) 𝑐𝑜𝑠 𝛼 + [𝐷 𝑠𝑖𝑛 𝜔] 𝑐𝑜𝑠 𝛼 + [𝐷𝑟 𝑠𝑖𝑛 𝜃] 𝑐𝑜𝑠 𝛼 − 𝑢𝑤𝛽) tan 𝜑′] /𝑚𝛼 (𝐹𝑠)𝑓 = ±𝐴 + ∑ 𝑡𝑎𝑛 𝛼 + ∑ 𝑘𝑊 − [𝐷 𝑐𝑜𝑠 𝜔] − [𝐷𝑟 𝑐𝑜𝑠 𝜃] [5] where: c' = effective cohesion of the material at the base of a slice, uw = pore-water pressure, N = normal force on the base of a slice, W = self-weight of each slice, D = load (line load and/or distributed load), Dr = reinforcement load, A = hydrostatic water force acting at the left, L, or right, R, extremities of the slip surface, r = radius (or moment arm) associated with the mobilized shear force, Sm, x = horizontal distance from the centroid of each slice to the center of rotation (or to the axis point), f = perpendicular offset of the normal force from the center of rotation (or from the axis point), d = perpendicular distance from a line load to the center of rotation (or the axis point), dr = perpendicular distance from a reinforcement load to the center of rotation (or the axis point), XR = vertical interslice shear forces on the right side of a slice, ER = horizontal interslice normal forces on the.

[Audio] BENTLEY SYSTEMS Limit Equilibrium Methods 14 of 158  = lambda value representing the percentage of the interslice force used in the analysis, ' = effective angle of internal friction of the material at the base of a slice,  = length along the base of a slice,  = angle of the reinforcement load from the horizontal. This angle is measured counterclockwise from the positive x-axis, and  = angle between the tangent to the center of the base of each slice and the horizontal. 3.3.3 Normal Force at the Base of each Slice The normal force at the base of each slice is obtained by summing forces in a vertical direction. u w        c X X W ( )     R L sin sin tan 'sin sin ' + + + − − + r F D D F s s = [6] N m  3.3.4 Interslice Normal Force and Interslice Shear Force The interslice normal force for each slice is obtained by summing forces horizontally on each slice in an integration manner progressing across the slope. m r L R L R D D kW S D D X X W E E − − + − + + − − + = [7] ( )                cos cos cos tan sin sin r The interslice shear forces, (XR or XL), are a function of the interslice normal forces. The interslice shear forces are computed from the interslice normal forces after each iteration, using the following equation. The calculation of the interslice shear forces is repeated after each iteration, until convergence is obtained for the overall factor of safety equations. ( ) f x E X R R  = [8] where: f(x) = mathematical equation describing the form of the interslice force function.  = percentage (in decimal form) of the function used. Figure 4 shows a typical half-sine function. The upper curve is the specified function while the lower curve shows the percentage of the function used in satisfying force and moment equilibrium  represents a ratio between the two curves. Figure 4 Half-sine interslice force function 3.3.5 The Relationship between General Limit Equilibrium Method and other Methods of Slices The normal force at the base of a slice is dependent on the interslice shear forces, XL and XR, on either side of the slice. The interslice shear forces are determined from the interslice normal force and the interslice force function. Different methods of slices have been proposed based on different assumptions related to the interslice force functions..

[Audio] BENTLEY SYSTEMS Limit Equilibrium Methods 15 of 158 In the Bishop Simplified method (1955) the interslice shear forces are set equal to zero. In other words, the interslice normal forces are assumed to be horizontal. Applying these conditions to the interslice force equations, gives the following conditions that must be met: E = E =  X 0 ( ) x = 0 f [9] where, f(x) cannot be equal to zero. The function f(x) should be a positive function, preferably extending between 0 and 1. The Bishop Simplified method becomes a special case of the GLE method by setting lambda,  equal to zero and setting the arbitrary function, f(x), to any value and satisfying only the moment equilibrium equation. In the Janbu's Simplified method (1954), the interslice shear force is also assumed to be equal to zero or the  value can be set equal to zero. Janbu's Simplified method satisfies the force equilibrium equation. In the Spencer method (1967), the direction of the interslice forces is assumed to have a constant direction throughout the sliding mass. This condition is simulated by setting  equal to the tangent of the selected angle for the interslice forces. That is, f(x) is equal to 1.0, and  is equal to [tan , where  is the slope of the interslice forces. Substituting these conditions into the general interslice force equation gives, ( ) tan f x E X  =  = [10] The interslice force function can be viewed as a constant for the Spencer method. When f(x) is set as a constant, the GLE method solves the Spencer method. The Spencer method satisfies both moment and force equilibrium conditions. The Corps of Engineers method internally computes the interslice force function based on the "average slope" of the geometry and sets  equal to 1.0. Only the force equilibrium equation is solved. There are two interpretations of "average slope" in the Corps of Engineers method. The Lowe-Karafiath method internally computes the interslice force function based on the average slope between the ground surface and the slip surface.  is set to 1.0 and only the force equilibrium equation is solved. Morgenstern-Price method is essentially the same as the GLE method in terms of the interslice force used in the analysis assumptions and the equations of static equilibrium solved. A slight difference between the two methods will be discussed later. The GLE method can be specialized to all other methods of slices regardless of slip surface shape. The GLE method in the software utilizes a wide range of interslice force functions such as: Constant function Half-sine function Clipped-sine function Trapezoid function Specified function Wilson-Fredlund (1983) function The GLE method satisfies both force and moment equilibrium by finding the intersection point of the moment and force equilibrium factors of safety (i.e., Fm and Fs), curves plotted versus lambda, . 3.4 ORDINARY OR FELLENIUS METHOD The.

[Audio] BENTLEY SYSTEMS Limit Equilibrium Methods 16 of 158  0 = 0 M : Moment equilibrium equation is applied to the overall slope about the center of rotation and is the basis for calculating the factor of safety, Fs, and  ⊥ = 0 F : Force equilibrium equation sums forces perpendicular to the base of each slice, and defines the normal force on the base of each slice, N, in terms of the weight of the slice, W. There is a fundamental theoretical limitation associated with the Swedish method or the Ordinary method and it can be explained as follows. First, it is assumed that the inter-slice forces cancel between slices. The normal force on the base of each slice is then calculated by summing forces perpendicular to the base of the slice. This means that the interslice forces must be parallel to the base of the slice to NOT appear in the statics equation. However, when moving to the next slice, the direction for summing forces at the base of the slice must change. Now, the interslice forces need to be in a slightly different direction to put two slices together. Consequently, the methodology has done something fundamentally wrong from a statics standpoint. Consequently, the Swedish (Ordinary) method should not be considered as an acceptable methodology because of the fundamental statics flaw. In other words, it is unacceptable Newtonian static equilibrium. The Ordinary method is only implemented in the software to provide a reasonable transition from the earliest methods to subsequent methods. The use of the Ordinary method in geotechnical engineering practice is not recommended. 3.4.1 Factor of Safety for a Circular Slip Surface Figure 5 defines the variables associated with a circular slip surface. The simplest form of the Ordinary method factor of safety when considering pore-water pressures is as follows. Figure 5 Forces associated with the Ordinary method using a circular slip surface Applying the moment equilibrium equation, Mo = 0, gives:   = − 0 sin S R Wr m  [11] where: ( )  ' tan '    w n [12] m u c F S − + = s   cos W n = [13]  where: c' = effective cohesion, ' = effective angle of internal friction, uw = pore-water pressure,  = length along the base of a slice, Fs = overall factor of safety, W = self-weight of each slice, Sm = shear force mobilized on the base of a slice,.

[Audio] BENTLEY SYSTEMS Limit Equilibrium Methods 17 of 158  = angle of slice base, r = radius (or moment arm) associated with the mobilized shear force, Sm, and n = normal stress on the base of a slice. Substituting the shear force mobilized, Sm, into Equation [11] gives:     u W c ) tan ' cos ( ' w  − +   = F s [14]  W sin  3.4.2 Factor of Safety for a Composite Slip Surface A composite slip surface is shown in Figure 6. In the analysis of a composite slip surface, the radius associated with the shear force mobilized, Sm, does not cancel. Therefore, the radius, r, remains as a variable in the factor of safety equation. In addition, the normal force at the base of the slice, N, produces a moment about the center of rotation (or center of moments). Figure 6 Forces acting on a composite slip surface The following equations need to be solved for the more general case of a non-circular slip surface and complex loading conditions involving a line load, distributed load, and reinforcement load. The moment equilibrium equation for a composite slip surfaces, Mo = 0, can be written,    = − − 0 sin S r Nf W r m  [15] Substituting the shear force mobilized Sm, into Equation [15] and solving for Fs gives:    r u N c F w ( )   tan [16] s −    − + =    Nf Wr sin The normal force, N, can be substituted into Equation [16] to give,     tan ' cos ' w  r u W c F ( )   s [17] − + =   − Wf Wr cos sin  .

[Audio] BENTLEY SYSTEMS Limit Equilibrium Methods 18 of 158 The effect of partial submergence of the slope by a fluid can also be added to the above equation. The normal force remains the same; however, additional moment terms are added to the denominator to account for the moments resulting from the left and right resultant water force about the center of rotation (or center of moments). The factor of safety equation then becomes,     tan ' cos ' w  − + = r u W r c F ( )   s [18] − + + −   Wf W r A a a A cos sin R R L L   The effect of external line loads, D, changes the calculation of the normal forces on the base of a slice, for the slice on which the load is acting. Summing forces perpendicular to the base of the slice,  F⊥= 0, gives, ( )    − + + = 90 cos cos a D W N   [19] A distributed load can also be converted to an equivalent line load on a slice. If the distributed load is due to gravity, it will act in a vertical direction and can be input as an equivalent material layer with a unit weight. The load due to gravity can either have associated shear strength parameters or no shear strength parameters. The use of a distributed load is discussed later. The effect of a reinforcement load, Dr, changes the calculation of the normal force in a manner like that of a line load. However, the effect of reinforcement can be assumed to act immediately or to develop with strain. Each assumption will lead to different expressions for the factor of safety. Further details are presented in Chapter 7. The effect of earthquake loading also affects the normal force calculated for the base of each slice. The horizontal earthquake force can be written as follows: W kW Fe =  = [20] g where: Fe = horizontal earthquake force, k = a/g, seismic coefficient, a = horizontal acceleration caused by the earthquake, and g = acceleration caused by gravity, 9.81 m/s2. The normal force, N, can be rewritten as: ( )    − + + − = 90 cos sin cos a D kW W N    [21] Substituting the expanded normal force equation into the factor of safety Equation [18] and including the effect of positive and negative moment terms due to external line loads and earthquake loads gives, o tan ' 90 cos( sin cos '          ( )    r u D kW W r c F s + − +   − − + + − + = kWe Nf Wx Aa Dd [22]      The user should note that the Ordinary method is generally only used to provide an approximate starting value for calculating other factors of safety. In the software, the Ordinary method is used to provide an initial or starting value for the factor of safety since the factor of safety equation is linear in form. The Ordinary method is not extensively used in engineering practice due to its lack of accuracy. The above equations for a composite.

[Audio] BENTLEY SYSTEMS Limit Equilibrium Methods 19 of 158  Mo = 0 : Equation representing moment equilibrium about the center of rotation for an overall slope. This equation forms the basis for deriving the factor of safety, Fs, and  Fv = 0 : Equation representing vertical force equilibrium for each slice. This equation is used to obtain the normal force, N, at the base of each slice. 3.5.1 Factor of Safety for a Circular Slip Surface The factor of safety equation with respect to the moment equilibrium about the center of rotation (or center of moments) is the same as that used for the Ordinary method and is shown in Figure 7. The force equilibrium equation is written in the vertical direction for each slice. Figure 7 Forces associated with Bishop's Simplified method of slices The moment equilibrium equation, Mn = 0, can be written as follows:   = − 0 sin S r Wr m  [23] Where the shear force mobilized equation can be written as: ( ) w u N c S     − =  + tan [24] m F s Substituting the shear force mobilized Equation [24] into Equation [23] gives the factor of safety equation:    u N c tan ' ' w  − + ( )   = F s [25]  W sin  The normal force at the base of a slice can be obtained by substituting the shear force mobilized, Sm, into the vertical force equilibrium equation, Fv = 0.   0 )tan ' sin ( ' cos = − + + − u N c W N      [26] w F s Rearranging, the normal force can be written as,      u w tan ' sin sin ' + − c W F F s = [27] N s m .

[Audio] BENTLEY SYSTEMS Limit Equilibrium Methods 20 of 158 where: Fs m ' tan sin cos     + = [28] Substituting the normal force, N, into the factor of safety equation and rearranging the equation gives the following equation for a circular slip surface. tan ' / cos cos ' ( )         w m u W c F s [29]  − + =  W sin  Since m involves the factor of safety, Fs, variable, Equation [29] is nonlinear. Therefore, an iterative technique is required when solving for the factor of safety, Fs. The proposed iterative technique is discussed in Section 2.5.3. 3.5.2 Factor of Safety for Bishop's Simplified Method for Composite Slip Surface The Bishop's Simplified Method has been extended to a composite slip surface in a manner like that shown for the Ordinary method. The derivation of the factor of safety equation is shown for the case including external loads, earthquake loads and water submergence. The moment equilibrium equation about an axis approximating the center for the slip surface, Mo = 0, can be written as follows:         =   + − − 0 Aa Dd kWe S r Nf Wx m [30] Vertical forces equilibrium on each slice, Fv = 0, can be written,   0 sin sin cos = + − −    D S N W m [31] Substituting the mobilized shear force at the base of a slice, Sm, into Equation [31] gives the normal force, N,  w u +  − c W D       sin sin tan sin +   s F s F = [32] N a m Substituting the normal force, N, (Equation [32] into the moment equilibrium Equation [30] gives the factor of safety, Fs, equation for a composite slip surface.   ( )   w  m r u D W r c F s        ' / tan cos sin cos ' [33] + − +   − + + = kWe Nf Wx Aa Dd      3.5.3 Iterative Procedure Used in Solving Bishop's Simplified Method The iterative procedure used in the software to solve for the factor of safety for Bishop Simplified method is as follows: 1. Assume an initial factor of safety Fs = 1.0, or use the factor of safety computed by the Ordinary method multiplied by 1.17, 2. Calculate m using the initial factor of safety and solve the factor of safety equation for a new factor of safety, Fs, corresponding to the first iteration, 3. The newly computed factor of safety, Fs, can now be used to compute new m values, 4. Compute a new factor of safety, Fs, using the new m values, 5. Repeat Steps 3) and 4) until the difference between consecutive factor of safety computations is less than the designated tolerance (i.e., generally about 0.001)..

[Audio] BENTLEY SYSTEMS Limit Equilibrium Methods 21 of 158 3.6 JANBU'S SIMPLIFIED METHOD The equilibrium equations used for Janbu's Simplified Method are like those used for Janbu's Generalized method. Janbu's Simplified method assumes that the resultant interslice forces are horizontal. Therefore, the interslice shear forces are removed from the factor of safety equations. The factor of safety equation can be written as follows:    m u D W c sin '        tan ' / cos        − + +   [34] Fs     − + + = A D kW D W cos tan sin         An iterative procedure is required when solving the factor of safety Equation [34]. The iterative technique is like that used when solving Bishop's Simplified method. In the original publication of the Janbu Simplified Method, the computed factor of safety was multiplied by a correction factor. The correction factor was always slightly greater than 1.0 and depended on the depth of the slip surface. The basis for the correction factor has been somewhat contested and therefore it has not been included in the software code. Consequently, Janbu's Simplified method satisfies horizontal force equilibrium when using the designated assumptions. 3.7 SPENCER METHOD The Spencer Method assumes that the ratio of the interslice shear force to the interslice normal force is constant throughout the sliding mass. This method independently satisfies horizontal force and moment equilibriums giving rise to force equilibrium factor of safety, Ff, and a moment equilibrium factor of safety, Fm. The interslice force assumption can be written as follows: X = E tan [35] where: X = interslice shear force, E = interslice normal force, and  = angle of the resultant interslice force from the horizontal. This relationship between the interslice shear force, X, and interslice normal force, E, can be considered as a specific case of the General Limit Equilibrium, GLE method. In the GLE method, the interslice forces are related using the empirical equation proposed by Morgenstern-Price (1965): f (x) E X  = [36] The Spencer method corresponds to the case where lambda, , is equal to 1.0 and the function, f(x) is a constant equal to (tan ) in the GLE method. 3.8 MORGENSTERN-PRICE METHOD The Morgenstern-Price Method (M-P method) is like the GLE method in terms of the assumptions made and the static equilibrium equations solved. The M-P method satisfies both moment equilibrium and horizontal force equilibrium equations when computing the factor of safety. There are a variety of assumptions that can be made with respect to the interslice force functions that can be used. Several functions were suggested by Morgenstern and Price (1965) and Wilson and Fredlund (1983) suggested the use of an empirical interslice force function based on finite element stress analysis are: Constant function Half-sine function Clipped-sine function Trapezoid function Specified function Wilson-Fredlund (1983) function.

[Audio] BENTLEY SYSTEMS Limit Equilibrium Methods 22 of 158 The original Morgenstern-Price (1965) method used an integration type solution with a modified Newton-Raphson solver. The solution method used in the software is a summation approach that is like that used for other methods of slices. A Rapid solver is used in the software for computing the factor of safety corresponding to the M-P method. The Rapid Solver procedure can be described as follows: 1. Set an initial value for  = 2/3 'cord slope' where the 'cord slope' is defined as shown in Figure 8, 2. Obtain the initial factor of safety using the Ordinary method and increase the computed value by 17%. The estimated value for Fs, is used as the initial Fs, for solving the M-P method,  Compute a set of (Fs) m and (Fs) f (i.e., moment and force equilibrium factors of safety)f using the initial,  4. Compare the two factors of safety and select a second  value, + =    then F F if 1.0 , ( ) ( ) f s s m [37] − =    then F F if ( ) ( ) 1.0 , f s m s 5. Compute a new set of factors of safety (i.e., (Fs)m and (Fs)f using the newly estimated lambda value, , and compare the difference in the two factors of safety to the tolerance. If the tolerance criterion is satisfied then the computations can be stopped, 6. If the tolerance criterion is not satisfied, repeat Step 3 using a new  value estimated from the two previous sets of factor of safety and  calculations. Figure 8 Definition of the chord slope 3.9 CORPS OF ENGINEERS METHOD The U.S. Army Corps of Engineers Method satisfies overall horizontal force equilibrium in calculating the factor of safety, Fs. Moment equilibrium is not satisfied. The assumption regarding the direction of the resultant interslice forces has led to two interpretations of the Corps of Engineers method: namely, Corps of Engineers 1 and Corps of Engineers 2. Assumption #1 for Corps of Engineers: the direction of resultant interslice forces is assumed to be equal to the average slope between the extreme entrance and exit of the slip surface and the ground surface. Assumption #2 for Corps of Engineers: the direction of resultant interslice forces is assumed to be equal to the slope at the ground surface at the top of each slice. Figure 9 Assumptions regarding the interslice force direction for the Corps of Engineers method.

[Audio] BENTLEY SYSTEMS Limit Equilibrium Methods 23 of 158 3.10 LOWE-KARAFIATH METHOD The Lowe-Karafiath Method is like the Corps of Engineers method and only differs in terms of the assumption regarding the direction of the resultant interslice forces. The Lowe-Karafiath assumption regarding the interslice force direction can be described as follows. The direction of the resultant interslice forces is assumed to be equal to the average of the ground surface slope and slip surface slope (Figure 10). Figure 10 Lowe-Karafiath interslice force direction assumption. 3.11 SARMA METHOD (1973) Sarma et al. (1973) proposed a limit equilibrium method of slices that calculated the "Critical Acceleration" required to force the factor of safety to become 1.0. Sarma et al. (1979) also made some additional comments related to calculating the internal stresses within the sliding mass by using non-vertical slices or general blocks. However, only vertical slices are used in the current implementation of the Sarma method in the software. Another difference between the Sarma method and the other general methods defined by the GLE formulation is related to the relationship between the interslice shear and normal forces. The following equation was used to evaluate the interslice shear forces, )tan ' ( '  uw E c h X − + = [38] where: c' = effective cohesion, ' = effective angle of internal friction, h = height of the slice, E = interslice normal force, and uw = pore-water pressure. Calculating the "Critical Acceleration" for a factor of safety of 1.0 can be accomplished by using the GLE Method and using several trial horizontal acceleration values. A plot of Factor of Safety versus the "Assumed Acceleration" reveals the "Critical Acceleration" for a factor of safety of 1.0. In the original Sarma method (1973), the average of the material properties was used to compute the interslice shear forces. In The software, the material properties and pore-water pressure at the 1/3 point upward from the base of each slice are used to compute the interslice shear forces. 3.12 SARMA NON-VERTICAL SLICES METHOD (1979) Sarma (1979) developed an extended wedge approach of the limited equilibrium method. In this method, the mass contained within the slip surface and the free ground surface is divided into n slices like in vertical slice methods, but the slices need not be vertical or even parallel as shown in Figure 11(a). The forces acting on the ith slice are shown in Figure 11(b). It assumes that under the influence of the force, KWi, the factor of safety on the slip surface is equal to one in which case K is the critical acceleration factor, 𝑘𝑐. The critical horizontal acceleration factor 𝑘𝑐 can be calculated in a closed form. The solution obtained in the form of 𝑘𝑐 can be used to determine the factor of safety, and the angle of each slice is found as part of the solution. The effect of the shear strength on the internal shear surfaces can be.

[Audio] BENTLEY SYSTEMS Limit Equilibrium Methods 24 of 158 Figure 11 Forces acting on individual non-vertical slice. From the vertical equilibrium of the slice, 𝑁𝑖 cos 𝛼𝑖 + 𝑇𝑖 sin 𝛼𝑖 = 𝑊𝑖 + 𝑋𝑖+1 cos 𝛿𝑖+1 − 𝑋𝑖 cos 𝛿𝑖 − 𝐸𝑖+1 sin 𝛿𝑖+1 [39] + 𝐸𝑖 sin 𝛿𝑖 From the horizontal equilibrium of the slice, 𝑇𝑖 cos 𝛼𝑖 − 𝑁𝑖 sin 𝛼𝑖 [40] = 𝐾𝑐𝑊𝑖 + 𝑋𝑖+1 sin 𝛿𝑖+1 − 𝑋𝑖 sin 𝛿𝑖 + 𝐸𝑖+1 cos 𝛿𝑖+1 − 𝐸𝑖 cos 𝛿𝑖 Mohr Coulomb failure criterion gives, ′ sec 𝛼𝑖 ′ + 𝐶𝑖 𝑇𝑖 = (𝑁𝑖 − 𝑈𝑖) tan ∅𝑖 [41] Mohr Coulomb failure criterion gives, ′ sec 𝛼𝑖 ′ + 𝐶𝑖 𝑇𝑖 = (𝑁𝑖 − 𝑈𝑖) tan ∅𝑖 [42] When the sliding mass is in a state of limiting equilibrium, the mass will not be able to move unless shear surfaces are formed within the body. It is assumed that the normal body force E and shear body force X on the slicing boundaries are such that they are also in a state of limiting equilibrium, so that, ′𝑑𝑖 ′ + 𝐶̅𝑖 𝑋𝑖 = (𝐸𝑖 − 𝑃𝑊𝑖) tan ∅̅𝑖 [43] ′ 𝑑𝑖+1 ′ + 𝐶̅𝑖+1 𝑋𝑖+1 = (𝐸𝑖+1 − 𝑃𝑊𝑖+1) tan ∅̅𝑖+1 [44] Where ∅̅′ is the average friction angle along the inclined plane; 𝐶̅′ is the average cohesion along the same plane; d is the length of the inclined plane; and PW is the water pressure force on that plane. From the above equations, we can get a closed form solution of 𝑘𝑐,.

[Audio] BENTLEY SYSTEMS Limit Equilibrium Methods 25 of 158 𝑘𝑐 = 𝑎𝑛 + 𝑎𝑛−1𝑒𝑛 + 𝑎𝑛−2𝑒𝑛𝑒𝑛−1 + ⋯ + 𝑎1𝑒𝑛𝑒𝑛−1 … 𝑒3𝑒2 [45] 𝑝𝑛 + 𝑝𝑛−1𝑒𝑛 + 𝑝𝑛−2𝑒𝑛𝑒𝑛−1 + ⋯ + 𝑝1𝑒𝑛𝑒𝑛−1 … 𝑒3𝑒2 In which, 𝑎𝑖 ′ − 𝑎𝑖) + 𝑅𝑖 cos ∅𝑖 ′ + 𝑆𝑖+1 sin(∅𝑖 ′ − 𝑎𝑖 − 𝛿𝑖+1) − 𝑆𝑖 sin(∅𝑖 ′ − 𝑎𝑖 − 𝛿𝑖) [46] = 𝑊𝑖 sin(∅𝑖 ′ cos(∅𝑖 ′ − 𝑎𝑖 + ∅̅𝑖+1 ′ − 𝛿𝑖+1) sec ∅̅𝑖+1 ′ − 𝑎𝑖) 𝑝𝑖 = 𝑊𝑖 cos(∅𝑖 ′ [47] cos(∅𝑖 ′ − 𝑎𝑖 + ∅̅𝑖+1 ′ − 𝛿𝑖+1) sec ∅̅𝑖+1 ′ ′ − 𝛿𝑖) sec ∅̅𝑖 ′ − 𝑎𝑖 + ∅̅𝑖 𝑒𝑖 = cos(∅𝑖 ′ [48] cos(∅𝑖 ′ − 𝑎𝑖 + ∅̅𝑖+1 ′ − 𝛿𝑖+1) sec ∅̅𝑖+1 ′ 𝑅𝑖 = 𝐶𝑖 ′𝑏𝑖 sec 𝛼𝑖 − 𝑈𝑖 tan ∅𝑖 [49] ′ 𝑅𝑖 = 𝐶𝑖 ′𝑏𝑖 sec 𝛼𝑖 − 𝑈𝑖 tan ∅𝑖 [50] 𝛼 - the angle of the base of the slice (relative to the horizontal axis); 𝛿 - the angle of the side of the slice (relative to the vertical axis); 𝑊 - the weight of the slice; 𝑈 - the water pressure force on the base of slice. The stability analysis solution obtained in the form of the acceleration factor 𝑘𝑐 is straightforward and easy. Factor 𝑘𝑐 itself can be used as a measure of the stability. After 𝑘𝑐 is obtained, the factor of safety (FOS) is calculated through an iterative process where the shear strength parameters are reduced until the 𝑘𝑐 is equal to zero. 𝐶 𝐶′ = 𝐹 [51] 𝑡𝑎𝑛∅ 𝑡𝑎𝑛∅′ = 𝐹 [52] The relationship between FOS and Acceleration Factor K is shown in Figure 2..

[Audio] BENTLEY SYSTEMS Limit Equilibrium Methods 26 of 158 Figure 12 Relationship between FOS and acceleration factor, K..

[Audio] BENTLEY SYSTEMS Stress-Based Methods 27 of 158 4 STRESS-BASED METHODS The following sections outline methods which use a finite element method of calculating the appropriate stresses along the slip surface. 4.1 ENHANCED LIMIT METHOD (KULHAWY METHOD) The Enhanced Limit Method utilizes the finite element method for the calculation of the stress states in the material mass. The fundamental conditions associated with limit equilibrium methods of analysis are retained in the Enhanced Limited Method. The method uses a finite element analysis to compute the stresses in the material mass and then computes the factor of safety, Fs, along a prescribed slip surface. The minimum factor of safety is obtained through a series of trial slip surfaces. 4.1.1 Definition of Factor of Safety There are several definitions that have been proposed for the factor of the safety based on the finite element stress analysis method. For example, Kulhawy (1969) proposed an equation for the factor of safety in term of the material strength. Zienkiewicz et al (1975) derived a factor of safety, based on stress level. Adikari and Commins (1985) published a factor of safety based on both material strength and stress level. The definition of safety factor proposed by Kulhawy (1969) appears to be the most reasonable and has been most widely adopted in geotechnical engineering practice. The Kulhawy (1969) Enhanced Limit method has been implemented in the software. The equation for the factor of safety for a saturated material can be rewritten as:    u c r w n  s S F ]' ) tan [ ' ( [53] S m m   − + = =    where: m = mobilized shear stress along the base of a slice, Sr = resisting shear force, Sm = mobilized shear forces, and  = length across the base of the slice. The factor of safety equation can be extended to accommodate unsaturated soils where matric suction is considered: b     u u u c ( )   r w a a n  S F tan ) tan ' ' ( [54] s S m m   − + − + = =    where: ua = pore-air pressure, uw = pore-water pressure, (negative) and b = angle defining the increase in strength due to matric suction, (ua - uw). Equation [54] has also been extended to accommodate the situation where shear strength with respect to matric suction is nonlinear (i.e., nonlinear  b ). Details related to the use of a nonlinear unsaturated shear strength envelope are presented in section 8.20.2. 4.1.2 Stress Transfer from the Finite Element Analysis to the Center of the Base of a Slice The stresses computed in a finite element analysis generally correspond to the Gauss points. These stresses need to be transferred to the nodes of each finite element. To compute the stresses at the center of the base of a slice, a search must.

[Audio] BENTLEY SYSTEMS Stress-Based Methods 28 of 158 S S S xp xp xp 3 3 2 2 1 1     + + = [55] xp S S S S yp yp yp 3 3 2 2 1 1     + + = [56] yp S S S S xyp xyp xyp 3 3 2 2 1 1     + + = [57] xyp S where: xp, xp, xp = stresses at point P within a triangular element, and xpi,xpi, xpi = stresses at each node point (i = 1, 2, 3). Figure 13 Designation of point, P, (i.e., center of the base of a slice) within a triangular finite element 4.1.3 The Normal and Shear Stresses at the Center of the Base of a Slice Once the stress, x, y, xy, are known at the center of the base of each slice, the normal stress, n, and mobilized shear stress, m, can be calculated using Equations [58] and [59].     + − y x y x = [58] +    n + xy  sin 2 cos 2 2 2 y x − − = [59] xy m       sin 2 2 2 cos where: x = normal stress in the x-direction at the center of the base of a slice, y = normal stress in the y-direction at the center of the base of a slice, xy = shear stress in the x- and y-direction at the center of the base of a slice,  = angle measured from the positive x-direction to the line of application of the normal stress, n = normal stress perpendicular to the base of a slice, and m = shear stress parallel to the base of a slice. The normal stress, n, and the shear stress, m, can be used in the calculation of the factor of safety when using Equations [53] or [54]..

[Audio] BENTLEY SYSTEMS Stress-Based Methods 29 of 158 Figure 14 The relationship between the finite element stresses and stresses at the center of the base of a slice 4.2 SAFE-DP METHOD The SAFE-DP Method utilizes the finite element method for the calculation of the stress states in the material mass. The method uses a finite element analysis to compute the stresses in the material mass and then search for the critical slip surfaces using Dynamic Programming Search method. 4.2.1 Definition of the Factor of Safety For an arbitrary slip surface AB, as shown in Figure 15, the equation for the factor of safety can be defined as: B  dL f  A F s = B [60]  dL  A where:  = mobilized shear stress along the slip surface, ƒ = shear strength of the material, and dL = an increment of length along the slip surface. Y "Stage" A "State point" B "1" "i" "n+1" X Figure 15 An arbitrary slip surface AB in a discretized form It is assumed that the critical slip surface can be approximated by an assemblage of linear segments. Each linear segment connects two state points located in two successive stages. The stage-state system forms a grid consisting of rectangular elements called the search grid. The rectangular elements formed by the search grid are called grid elements. In this discretized form, the overall factor of safety for the slip surface AB is defined as follows:.

[Audio] BENTLEY SYSTEMS Stress-Based Methods 30 of 158 n   L i f i  = i 1 [61] F s = n   L i i  = i 1 where: n = number of discrete segments, i = shear stress actuated, if  = shear strength, and iL  = length of the ith segment. 4.2.2 Stress Transfer from the Finite Element Analysis to the Grid Point The calculation of the stresses is the like the procedure used in the Enhanced Limit Method. The most significant difference is that the: Enhanced Limit Method calculates the stress at each base center point, and the SAFE-DP method calculates the stress at each Grid Point. 4.2.3 The Normal and Shear Stresses on a Segment The normal and shear stresses acting on a segment can be computed from the results of a stress analysis as follows: 2 2 [62]        sin 2 cos sin xy y x n − + = [63] − − − = ) ( ) cos (sin 2 2 x y xy n        sin 2 2 where: n = normal stress acting on a segment, n = shear stress acting on a segment,  = inclined angle of the segment with the horizontal direction, x = normal stresses acting in the x- coordinate direction, y = normal stresses acting in the y- coordinate direction, and = shear stresses acting in the x- and y-coordinate directions. xy .

[Audio] BENTLEY SYSTEMS Seismic Methods 31 of 158 5 SEISMIC METHODS Methods developed to date to calculate the stability or performance of slopes under earthquake loading fall into three general categories (Jibson R. W. 2011): 1. Pseudo-static analysis, i.e. Calculate the Yield Coefficient (Ky) for all slip surfaces, 2. Permanent-displacement analysis (Newmark method), 3. Stress-deformation analysis (finite-element method). The Calculate the Yield Coefficient for All Slip Surfaces method and the Newmark Permanent Displacement method are outlined here. If an advanced seismic analysis method is selected in the software, then the regular horizontal seismic load options will be unavailable. 5.1 CALCULATE THE YIELD COEFFICIENT FOR ALL SLIP SURFACES SEISMIC ANALYSIS Pseudo-static analysis models the seismic shaking as a permanent body force added to each slice/column in limit equilibrium analysis; normally, only the horizontal component of earthquake shaking is modeled because the effects of vertical forces tend to average out to near zero (Jibson R. W. 2011). To calculate the yield coefficient, Ky, for all slip surfaces is to compute the critical pseudo-static horizontal seismic coefficient for each slip surface. In other words, the method will calculate the horizontal pseudo-static seismic coefficient required to lower the trial slip surface Factor of Safety (FOS) to the Target FOS. By default, the Target FOS = 1, but the user can enter a different value. This analysis is performed for all trial slip surfaces and the slip surface with the lowest value of Ky found to reach the Target FOS is the critical slip surface, and the Ky is the critical seismic coefficient. Note: If all trial slip surfaces have an initial FOS less than the Target FOS, then all the Ky values will be zero. The output will be in terms of Ky values rather than FOS. 5.2 NEWMARK PERMANENT DISPLACEMENT SEISMIC ANALYSIS Newmark (1965) introduced a method to assess the performance of slopes during earthquakes that fills the gap between oversimplistic pseudo-static analysis and over-complex stress-deformation analysis. In recent years, the Newmark analysis is becoming more popular in standard engineering practice. In simple pseudo-static seismic slope stability methods, the seismic force is considered as a permanent (static) body force, and the assumption is made that the slope will fail if the peak ground acceleration exceeds the critical acceleration. In reality the analysis shows that slopes with significantly lower critical acceleration can survive higher earthquake acceleration without significant damage. The reason for this is that seismic ground accelerations are a transient phenomenon and that some permanent deformation of the slope may precede any damage of practical significance (Newmark 1965). Therefore, the Newmark analysis is less conservative than the pseudo-static analysis (Wilson & Keefer, 1983). Newmark's method models a slip surface as a rigid block under seismic loading, the steps are Step 1. Calculate the critical seismic coefficient ky for each trial slip surface Step 2. A seismic acceleration record of Acceleration vs. Time is selected. The record can be loaded from Dynamics {analysis.

[Audio] BENTLEY SYSTEMS Seismic Methods 32 of 158 Figure 16 Illustration of the Newmark integrations algorithm (Jibson, R.W., 2011) 5.3 DYNAMICS – TECHNICAL PREVIEW Refer to the Appendix of this document for theory related to the dynamic seismic functionality by the finite element method (FEM), currently in technical preview..

[Audio] BENTLEY SYSTEMS Tension Cracks 33 of 158 6 TENSION CRACKS Tension cracks can be defined in the software in one of two ways. One way is to specify a tension crack zone and the other way is to define a tension crack angle. 6.1 TENSION CRACK LINE A tension crack zone is defined by specifying a line defining the bottom of the tension cracks. The slip surface goes vertically upward to the ground surface as shown in Figure 17. In other words, slices above the slip surface, in the tension crack zone, will be ignored. Figure 17 Tension crack line at the bottom of the cracks The tension crack line cannot be set above the ground surface or within a material that is represented as having zero shear strength (i.e., c = 0 and  = 0). The material is assumed to be a fluid when the shear strength parameters are zero. 6.2 TENSION CRACK ANGLE Tension cracks can be considered when the angle of slip surface is steeper than a specified limiting tension crack angle. Therefore, a tension crack angle must be specified to designate the use of the tension crack zone. In other words, the tension crack zone is implemented when the angle of the slip surface is steeper than the specified limiting tension crack angle. The depth of the tension crack zone is difficult to predict and is usually established on a somewhat empirical basis. For example, it might be decided that the steepest slip surface angle should be 45 degrees to be horizontal and therefore an angle can be used to designate the tension crack zone. Figure 18 Definition of the limiting angle for tension cracks, c.

[Audio] BENTLEY SYSTEMS Tension Cracks 34 of 158 6.3 HYDROSTATIC HORIZONTAL FORCE IN TENSION CRACK If a tension crack is filled with water, there will be a hydrostatic horizontal force that needs to be taken into consideration in the slope stability analysis. The magnitude of the hydrostatic force is defined as follows:   =  [64] rH 2 F w w 2 where: w = unit weight of the fluid in the tension crack (e.g., water) H = depth of the tension crack from ground surface, and r = percentage of the crack filled with water from the bottom of the crack. Figure 19 Diagram showing the horizontal water force generated in the tension crack.

[Audio] BENTLEY SYSTEMS Slip Surfaces 35 of 158 7 SLIP SURFACES A factor of safety can be computed for every slip surface that is defined. The overall factor of safety of a slope is the minimum factor of safety of all the trial slip surfaces and is called the critical slip surface. In other words, each factor of safety is tied to a specific slip surface. It is necessary to specify the assumed shape of the slip surface before the factor of safety is calculated. There are numerous procedures that can be used to define the slip surfaces. The types of slip surfaces can be classified as: 1. circular slip surfaces, 2. composite slip surfaces and, 3. non-circular slip surfaces. Circular slip surfaces have been most commonly associated with earlier limit equilibrium formulations (e.g., Ordinary method or Bishop Simplified method). Composite slip surfaces are the combination of circular slip surfaces and linear line slip surfaces and were introduced by Fredlund and Krahn (1977). The non-circular slip surfaces consist of a series of linear line segments. Details pertaining to the three types of slip surface shapes are discussed in the following sections. 7.1 CIRCULAR SLIP SURFACES This section describes the theory behind some of the circular slip surfaces. All the Circular Slip Surface and Composite Slip Surface search methods involve finding trial circles with a circle center and circle radius. For example, for the Grid & Tangent search, Grid & Point search, and Grid & Line search, the circle center will be located on the Grid points, the circle radius will be the distance between the grid points and tangent points. 7.1.1 Slope Search Slope Search method is one of the search methods for locating the Global Minimum safety factor for CIRCULAR slip surfaces. 1) Parameters 1. Number of Surfaces This is the total number of valid surfaces generated by the slope Search. The invalid surfaces generated by the Slope Search are discarded and are not included in this number. Invalid slip surfaces include: The Entry or Exit point is outside the slope limits, Incorrect Initial Angle at Toe 2. Initial Angle at Toe The Initial Angle at Toe is the orientation of the line tangential to the circular surface at toe. The initial angle is calculated by the following equation: Initial Angle = Lower Angle+ (Upper Angle-Lower Angle) *Random where: Lower Angle: if the Lower angle is checked then the Lower angle input by user is used, Otherwise, the Lower Angle = -90 deg. Upper Angle: if the Upper angle is checked then the Upper angle input by user is used, Otherwise, Upper Angle = the angle of the slope segment at the initiation point of the slip.

[Audio] BENTLEY SYSTEMS Slip Surfaces 36 of 158 3. Generate the center point The center point is generated using the Initial Angle at Toe and the generated two points. Figure 20 Single Slope Limit for Slope Search Auto Refine Search Auto Refine Search is one of the search methods for locating the Global Minimum factor of safety for CIRCULAR slip surfaces. 1) Parameters 1. Division Along Slope. The Division Along Slope is the number of divisions into which the slope surface is divided for Each Iteration of the Search. i. In the initial iteration, the length of slope surface is defined by the Slope Limits. ii. In each subsequent iteration, the length of the slope surface to be analyzed is defined by the value of Division to Use in Next iteration. NOTE: The length of the slope surface is always measured along the slope polyline. They are not measured horizontally. This means the length of the slope surface is independent of the angle of the slope segments. 2. Radius per Division This number shows how many slip circles are generated for each pair of divisions along the slope. The procedure is: i. For each pair of division along the slope, a straight line joining the mid-points (e.g., A and B) of the two divisions is generated. ii. The angle of this line is considered as a Minimum angle. iii. Vertical angle (i.e., 90 degrees) is considered as a Maximum angle. iv. The angular range between the Minimum angle and the Maximum angle is then divided equally with the number of Circles Per Division. NOTE: To generate the valid slip surfaces, a small offset is applied to the Minimum and Maximum angles, respectively. v. A set of slip surfaces is then generated with the two points (e.g., A and B and a set of the initial angles generated in iv)..

[Audio] BENTLEY SYSTEMS Slip Surfaces 37 of 158 Figure 21 Radii Per Division Iterations. The Number of Iterations used for the Auto Refine Search is not for convergent purpose. The analysis is always carried out for the specified Number of Iterations. 3. Divisions to Use in Next Iteration. The number of divisions along the slope, with the lowest factor of safety is used as the basis for the next iteration in the Auto Refine Search. For example, Divisions Along Slope = 10 Divisions to Use in Next Iteration = 50% 5 divisions with the lowest factors of safety will be used to divide by 10 in the next iteration. 4. Total number of surfaces. Total number of surfaces is not an input parameter. It is calculated from the input parameters described above. Number of circles generated Per Iteration = yx(x-1)/2 Total number of surfaces = z[yx(x-1)/2] where: = Division Along Slope x = Radius per Division y z = Iterations. 2) Procedure for Generation of Slip Surfaces 1. The slope surface is divided into several divisions according to the value of the Divisions Along the Slope, NOTE: If the second set of limits is selected, the segments within the two limits will be used. 2. Slip Circles are generated between Each Pair of Divisions according to value of Radius per division, 3. The factors of safety are calculated for these slip circles. The average factor of safety associated with Each Division along the slope is recorded, 4. This constitutes one iteration of the Auto Refine Search, 5. In the subsequent iteration, the percentage of Divisions to Use In Next Iteration is used. For example: The Division Along Slope = 10 Divisions To Use In Next Iteration = 50% 5 divisions (10*50% = 5) with the lowest average factors of safety will be used and another 5 divisions with higher average factors of safety will be discarded. 6. The selected divisions in step 5) are used to form a new slope polyline and repeat Step 1) to Step 4) until the Number of Iterations is satisfied..

[Audio] BENTLEY SYSTEMS Slip Surfaces 38 of 158 7.2 COMPOSITE CIRCULAR SLIP SURFACE Circular slip surfaces are quite realistic for most homogeneous or slightly non-homogeneous material slopes. The circular slip surface cannot, however, satisfy cases where there are adjacent materials with highly contrasting shear strength parameters. Figure 22 shows a bedrock layer with high shear strength parameters. The slip surface will pass along the top of the bedrock through a softened zone. Accordingly, some portion of the slip will be circular, and another portion of the slip will tend to be linear. The combination of circular and linear lines is called a "composite" slip surface. Figure 22 "Composite" slip surface formed as a result of a bedrock layer The "composite" slip surfaces procedure is quite effective for slopes with a low shear strength material layer immediately overlying the top of the bedrock as shown in Figure 23. Figure 23 "Composite" slip surface within a weak layer "Composite" slip surfaces are created in a manner similar to the creation of circular slip surfaces. The differences are as follows: 1. When the slip surfaces pass into the bedrock layer, the slip surface will force the positioned immediately above the bedrock layer. 2. The shear strength properties at the bottom (or composite portion) of the slip surface are taken from the layer of material just above the bedrock layer. NOTE: That the defined bedrock layer does not always means a bedrock material. Rather, a material layer with high shear strength can simply be specified as "Bedrock" layer to force the slip surface into a composite mode. 7.3 NON-CIRCULAR SLIP SURFACES Non-circular slip surfaces are designated in quite a different manner from circular or composite slip surfaces. The differences are as follows: 1. Non-circular slip surfaces use a common rotation center referred to as the Axis point. This point is used as the center for moment equilibrium (Circular slip surfaces use the center of rotation for the center of moment equilibrium). 2. Non-circular slip surfaces are specified using a series of points (Circular slip surfaces are specified using a set of rotation points and a set of radii). In the software, non-circular slip surfaces can be specified using one of two procedures; namely, fully specified and block specified..

[Audio] BENTLEY SYSTEMS Slip Surfaces 39 of 158 7.3.1 Block Search A block search is one of the powerful non-circular slip surface search methods available in 2D. For example, it is especially useful to perform non-circular analysis along a thin weak layer. To define a non-circular slip surface with a block search, one or more block search objects must be defined. There are four block search objects available: 1. Block 2. Point 3. Line 4. Polyline 7.3.1.1 Block Search – Block A Block Search – Block is an arbitrary four-sided convex polygon as shown in Figure 24 where two Blocks are drawn. The software in 2D randomly generates ONE slip surface vertex inside each Block Search – Block for each slip surface. Figure 24 The grids generated in the Block specified method Figure 25 Slip surface designation using the Block specified method 7.3.1.2 Block Search – Point A Block Search – Point is a single point, each slip surface generated by a Block Search must pass through this point. This block search object is very useful to force the slip surface pass through the toe of a slope, or the bottom of a tension crack, etc. 7.3.1.3 Block Search – Line A Block Search – Line is a line segment defined by two end points. The software randomly generates ONE slip surface vertex along each Block Search – Line for each slip surface. 7.3.1.4 Block Search - Polyline A Block Search - Polyline consists of one or more line segments. The software randomly generates TWO slip surface vertexes along the polyline. The polyline points will be part of the slip surface points between these two generated slip surface vertexes. Block Search - Polyline is useful to define slip surfaces that pass through complex thin weak material layer. 7.3.2 Fully Specified (Segments) Fully specified slip surfaces can be designated using a set of points. This allows for considerable flexibly in specifying the location and the shape of the slip surfaces. Fully specified slip surfaces are useful for situations when the slip surface can be predetermined to some extent such as in the case of a retaining wall (Figure 26)..

[Audio] BENTLEY SYSTEMS Slip Surfaces 40 of 158 Figure 26 Fully specified slip surfaces It should be noted that the specified slip surface starts and ends outside the geometry. The entry point and exit point are then automatically computed. This procedure avoids difficulties associated with placing points directly on the ground surface. 7.3.3 Path Search A Path Search is one of the Search Methods that can be used for locating the Global Minimum factor of safety for NON-CIRCULAR slip surfaces. 1) Parameters 1. Number of Surfaces. This is the total number of valid surfaces generated by the Path Search. Invalid surfaces are NOT included in this number. The invalid surfaces may occur when either of the following factors is involved. Initial Angle at Toe is incorrect. The endpoint does not intersect the slope within the slope Limits. 2. Initial Angle at Toe The Initial Angle at Toe is the orientation of the FIRST line segment of a slip surface generated by the Path Search. Note: that the slip surfaces for the Path Search are generated from the toe of the slope to the crest of the slope regardless of the slip direction (Left to Right or Right to Left). Initial Angle = Lower Angle + (Upper Angle-Lower Angle) *Random where: Lower Angle: if the Lower angle is checked then the Lower angle input by user is used, Otherwise, Lower Angle = -90 degrees. Upper Angle: if the Upper angle is checked then the Upper angle input by user is used, Otherwise, Upper Angle = the angle of the slope segment at the initiation point of the slip surface Random: a random value between 0 and 1..

[Audio] BENTLEY SYSTEMS Slip Surfaces 41 of 158 Figure 27 Path Search The user can define angular limits through selecting the Upper Angle and/or Lower Angle. The Angle measurement is from –90 degrees to 90 degrees: Figure 28 Angular limit (Left Facing Slope) 3. Segment Length. Segment length is used for each segment of a non-circular surface generated by using the Path Search. Note that: By default, the segment length is automatically calculated based on the geometry boundaries. Segment length = 0.3H H = Maximum height of the slope. The user can specify a segment length by selecting the Use Segment Length Checkbox and entering a value into the editable box. Improper segment length (e.g., too small, or too large) will lead irrational results. 4. Random Number Generation For the Path Search, random numbers are used to generate: slip surface initiation points based on the Slope Limits. angle of the first slip surface segment based on Upper Angle and/or Lower Angle. angle of all subsequent slip surface segments. 2) Procedure for Generation of Slip Surfaces 1. A starting point is randomly generated on the slope surface based on the Slope Limits. If the second set of limits is selected, the right range will be used for Left-to-Right slip direction and the left range will be used for Right-to-Left slip direction. Possible Range of Path Search Initial Points.

[Audio] BENTLEY SYSTEMS Slip Surfaces 42 of 158 Figure 29 Possible Range of Path Search Initial points (Right Facing Slope) 2. Starting from the first point, the first segment is randomly generated according to the Initial Angle at Toe and the Segment Length specified. 3. The subsequent segments are randomly generated according to the segment length, and the previous segment angle. A restriction is to generate concave surfaces. 4. Calculations will continue until the segment intersects the ground surface. If the intersect point (i.e., endpoint) is within Slope Limits then this surface is valid, otherwise, invalid. 5. Repeat Steps 1 to 4, until the number of valid slip surfaces generated is equal to the Number of Surfaces specified. 7.3.4 Greco Method There are several Optimization methods that have been proposed in the research literature. The optimization methods remove the necessity for assuming the shape and location of the critical slip surface. The assumptions related to shape and location of the slip surface become part of the solution. 7.3.4.1 Definition of the Geometry The Monte Carlo technique proposed by Greco (1996) is one of the Optimization methods used in the software for locating the critical slip surface. The critical slip surface is defined using a series of straight lines segments. The concepts of the method are described as follows: Figure 30 Typical cross-section of a slope used in the Greco Optimization method (Greco 1996) The following variables used in Figure 30 are defined as follows: y = t(x) : a function describing the topographic profile of the material, y= s(x) : a function describing the slip surface, y= z(x) : a function describing the water table, and y = lj(x) : a function describing a discontinuity surface in a layered material system. The following practical constraints can be introduced. max min x x x   [65] ( ); ( ) ( ) t x s x h x   for all the x when max min x x x   [66].

[Audio] BENTLEY SYSTEMS Slip Surfaces 43 of 158 7.3.4.2 Objective Function of the Problem A potential slip surface is approximated by a series of line segments with n vertices: V1, V2, V3, Vn, whose coordinates (x1, y1), (x2, y2), (xn, yn) are unknowns in the problem, but can be used for locating the critical slip surface. These coordinates can be considered as components of a two-dimensional array:  Vn  V V S , , , 2 1  = [67] or  xn yn  y x y x S , , , , , , 2 2 1 1  = [68] The critical slip surface can be found by minimizing the objective function for the factor of safety, Fs, with respect to the array of coordinates, S:  ) ( min Fs S [69] 7.3.4.3 Solution of the Greco (1996) Formulation Searching for the minimum factor of safety, Fs, requires a nonlinear programming procedure. The factor of safety is calculated by starting with an initial assumed slip surface location and proceeding towards the minimized location through use of an iterative procedure. Consequently, a sequence of feasible slip surfaces, S1, S2, …., Sk, Sk+1, are generated along with the corresponding factors of safety in a decreasing sequence.         + ) ( ) ( ) ( ) ( 1 1 0 k k F S F S F S F S [70] where: k k k k  k  n k n k y x y x y x S , , , , , , 2 2 1 1   = [71] k k k k 1 1 1  1 1 1 k y x y x y x S [72] +   = k n k n 1 1 , , , , , , + + + + + + 1 2 2 where: ( k ) i k i y x , = coordinates of vertex i at stage k of the minimization procedure, and ik y x = coordinates of same vertex i at next stage k+1. k i     + + 1 1 , There are two stages involved in generating a new slip surface, Sk+11 from Sk in the current Greco (1996) method. The two steps are referred to as the "Exploration" stage and the "Extrapolation" stage. During the "Exploration" stage, each slip surface vertex is shifted to a new position using a random technique. If the factor of safety of the modified slip surface is smaller than the factor of safety for the previous location, then the vertex is fixed at the new position. If the opposite is true, its position is returned to the previous position. In the "Extrapolation" stage, the total displacement obtained in the "Exploration" stage is repeated, and the slip surface is updated if the corresponding factor of safety is smaller than that obtained at the end of the "Exploration" stage. "Exploration" Stage k+1). To attempt to reduce the factor of safety, vertex Vertex i is randomly moved from point (xi k, yi k) to.

[Audio] BENTLEY SYSTEMS Slip Surfaces 44 of 158 ) ( 1 1 + + = k i ik t x y for i = 1 and i = n [74] k i k i ik y y + +1 = for i = 2 to n-2 [75] where: ik = random displacement of vertex i in the x-direction, and ik = random displacement of vertex i in the y-direction. The displacements are given as follows: k i x x ik = N R Dx  [76] ik = N R Dy  [77] k i y y where: Rx = random number in the range [-0.5, 0.5], Ry = random number in the range [-0.5, 0.5], Dxik = width of the search steps in x-direction for vertex i at stage k, Dyik = width of the search steps in y-direction for vertex i at stage k, Nx = defined number, and Ny = defined number. The combination of Nx, Ny provides various movements of vertex i for the same pair of random numbers Rx and Ry. The following eight combinations are given for parameters Nx and Ny: 1 1 = = y x N N 1 1 = − = − y x N N 1 1 = − = y x N N 1 1 = = − y x N N 1 0 = = y x N N 1 0 = − = y x N N 0 1 = = y x N N 0 1 = = − y x N N Eight random displacements are tested for every vertex of the slip surface (see Figure 31). Effective directions for the displacements also depend on the sign of Rx and Ry..

[Audio] BENTLEY SYSTEMS Slip Surfaces 45 of 158 Figure 31 The moving directions of vertex i If one of the trials is successful, no further trials are made for a particular vertex, and the width of the search is increased as follows: ik x x Dx Dx − + = + + 1 1 [78] k i k i k i k i k i k i ik y y Dy Dy − + = + + 1 1 [79] If no trial is successful for vertex i, then the width of the search step for the successive step, k+1, is reduced as follows: ( −  ) + = 1 1 k i ik Dx Dx [80] ( −  ) + = 1 1 k i ik Dy Dy [81] where:  = a number [0,1], a value of 0.5 is recommended. o are fixed as: The initial width of the search steps, Dxi o and Dyi 1 0 x x Dx n i [82] − − = n ( )1 2 min max 0 − = [83] y y Dyi n where: ymax = maximum value of the vertex ordinates of the slip surface, and ymin = minimum value of the vertex ordinates of the slip surface. Extrapolation Stage k i k i ie x x x − = +1 2 for i = 1 and i = n [84] k i k i ie y y y − = +1 2 for i = 2 to i = n-1 [85] ) ( e i ie t x y = for i = 1 and i = n [86] The boundary conditions for Equation [84] are checked against the computed slip surface and then modified if required. If the factor of safety of the computed slip surface is less than the previous minimum factor of safety value, then the current slip surface is updated to the computed slip surface as follows: e i ik x x +1 = for i = 1 to i = n [87].

[Audio] BENTLEY SYSTEMS Slip Surfaces 46 of 158 e i ik y y +1 = for i = 1 to i = n [88] Otherwise, the new "Exploration" stage begins by starting with the slip surface obtained at the end of the previous trial. Criterion for searching when using the iterative process The iterative procedure involved the "Exploration" and "Extrapolation" stages is stopped if the following conditions are satisfied: Dxik+1   for i = 1 to i = n [89] Dyik+1   for i = 1 to i = n [90]   − + ) ( ) ( K 1 K F S F S [91] where:  = specified width for search range, and  = tolerance between subsequent values of factor of safety. The tolerance parameter  varies from 0.001 to 0.00001. The search range parameter, , is determined in terms of the size of sliding mass and the number of vertices, n, by using the following equation: 1 x xn [92] − −  = n ( )1 4 7.3.4.4 Implementation of Greco (1996) Method To avoid too many vertices along the slip surface, Greco (1996) suggested adopting a 3-step procedure when searching for the critical slip surface. The Greco recommendation has been implemented in the software. Step 1 Randomly generate a 4-vertices slip surface as follows: a. The abscissas of vertices 1 and 4 are randomly generated within the range [xmin, xmax] ( ) 4 / min max 1 min 1 x R x x x − + = [93] ( ) 4 / min max 4 max 4 x R x x x − − = b. The ordinates of vertices 1 and 4 are given by, ) ( ) ( 4 4 1 1 t x y t x y = = [94] c. The inclinations of the first and the last segments of the slip surface are randomly generated as, 6 3 5 1 15 45 15 45 R R     + = − =   [95] d. Point P with coordinates, (xp, yp), is determined as an intersection between the line passing through vertex 1 at an inclined angle of 1, and the line through vertex 4 and at an inclined angle of 3. e. Vertices 2 and 3 are then generated as, ( 1 ) 2 1 2 x R x x x p − + = [96] ( ) 1 1 2 1 2 x tan x y y − + = [97].

[Audio] BENTLEY SYSTEMS Slip Surfaces 47 of 158 ( xp ) R x x x − − = 4 3 4 3 [98] ( ) 3 4 3 4 3 x tan x y y − + = [99] Step 2 In this step, a 7-vertex slip surface is generated in which 4 vertices are those obtained at the end of Step 1 and the other 3 vertices are introduced at the midpoint of the straight line jointing adjacent vertices. Then the two-phase procedure is used to search for the critical slip surface with a minimum factor of safety. Step 3 In the same way, a 13-vertex slip surface is generated and the two-phase procedure is used to obtain the critical slip surface. Slip surfaces with seven vertices are relatively smooth and satisfactory for most engineering studies. 7.3.5 Dynamic Programming Method A conventional slope stability analysis involving limit equilibrium methods of slices consists of the calculation of the factor of safety for a specified slip surface of predetermined shape, and the determination of the location of the critical slip surface with the lowest factor of safety through a trial-and-error analysis. To render the inherently indeterminate analysis determinate, conventional limit equilibrium methods generally make use of assumptions regarding the relationship between the interslice forces. These assumptions become disadvantages to limit equilibrium methods since the actual stresses acting along the slip surface are quite approximate, and the location of the critical slip surface depends on the shape assumed by the analyst. The assumptions related to the interslice force function in limit equilibrium methods are unnecessary when a finite element stress analysis is used to obtain the normal and shear stresses acting at the base of slices (Fredlund and Scoular, 1999). A stress analysis can provide the normal and shear stresses through use of finite element numerical method with a "switch on" of the gravity body forces. Subsequently, the normal stress at the base of a slice is known and the equation for the factor of safety becomes linear. Assumptions regarding shape of the critical slip surface can be by-passed when an appropriate optimization technique is introduced into the analysis. Optimization techniques have been developed by several researchers for over two decades and have provided a variety of approaches to determine the shape and the location of the critical slip surface (Celestino and Duncan, 1981; Nguyen, 1985; Chen and Shao, 1988; Greco, 1996). Each approach has its own advantages and shortcomings. The main shortcoming associated with these approaches is that the actual stresses within a slope may be quite approximate. However, it would appear that these stresses should be more accurate than those computed by the methods of slices. In 1957, Bellman introduced a mathematical method called the "dynamic programming" method. One of the objectives of the dynamic programming method was to maximize or minimize a function. The dynamic programming method has been widely used in various fields other than geotechnical engineering. Baker (1980) appears to be the first to apply the optimization technique to the analysis of the stability of slopes. In 1980, Baker introduced an optimization procedure that utilized the dynamic programming algorithm to determine the critical slip surface. In this.

[Audio] BENTLEY SYSTEMS Slip Surfaces 48 of 158 The dynamic programming method can be combined with a finite element stress analysis to provide a more complete solution for the analysis of slope stability because the technique overcomes the primary difficulties associated with limit equilibrium methods. However, the disadvantage of the dynamic programming approach is that there are more variables to specify for the analysis; namely, the Poisson's ratio and the elastic moduli associated with the stress analysis. Fortunately, the computed stresses associated with the "switch-on" of gravity are quite insensitive to the elastic parameters of the soil. The minimum factor of safety and critical slip surface are obtained by combining the dynamic programming method and the stress fields obtained from a finite element analysis. The stress fields input into the software can be generated by the finiteelement method (FEM). The dynamic programming method is mainly based on the research of Yamagami and Ueta (1988) and Ha Pham (1999). Numerous example problems have been solved using the dynamic programming method. Examples include homogenous slopes, layered slopes as well as case histories. The results obtained from the analyses have been compared with results from several well-known limit equilibrium methods of slices (Fredlund and Krahn, 1977). While the factor of safety computations are quite similar by both methodologies for circular slip surfaces, the dynamic programming technique opens the way for solving more complex problems. 7.3.5.1 General In 1957, Bellman introduced a mathematical theory involving multi-stage decision process. The "decision process" was defined as a system whose state at any time t is specified by a vector P that undergoes transformations in the course of time. The transformation of the variable P is equivalent to a decision. If a single decision is made, the process is called a single-stage decision process. On the other hand, the system is defined as a multistage decision process if there are sequences of decisions to be made. The originality of the terminology "dynamic programming" is derived from the nature of the solution. The problem treated is a "programming" problem and the adjective "dynamic" indicates the significant involvement of time. However, the essential feature of the approach is the re-interpretation of many static processes as dynamic processes in which time can be artificially introduced. Features of the dynamic programming theory can be summarized as follows: 1. The purpose of the dynamic programming method is to maximize or minimize a function, 2. The function can be described as a system containing "stages". The system is characterized at any "stage" by a set of parameters called "state" variables, 3. At each "stage" of the process, there is a number of decisions to be made, 4. The effect of a decision is a transformation of the "state" variables, and 5. The past history of the system is not of importance in determining future actions. At this point, it is of interest to quote classic "principle of optimality", introduced by Bellman (1957). Principle of optimality: An optimal policy has the property that whatever the initial state and initial decision are the remaining decision must constitute an optimal policy regarding the state resulting from the first.

[Audio] BENTLEY SYSTEMS Slip Surfaces 49 of 158 where:  = mobilized shear stress along the slip surface, ƒ = shear strength of the material, and dL = an increment of length along the slip surface. Y "Stage" A "State point" B "1" "i" "n+1" X Figure 33 An arbitrary slip surface AB in a discretized form It is assumed that the critical slip surface can be approximated by an assemblage of linear segments. Each linear segment connects two state points located in two successive stages. The stage-state system forms a grid consisting of rectangular elements called the search grid. The rectangular elements formed by the search grid are called grid elements. In this discretized form, the overall factor of safety for the slip surface AB is defined as follows: n [101]   L i f i  = i 1 F s = n   L i i  = i 1 where: n = number of discrete segments, i = shear stress actuated, if  = shear strength, and  iL = length of the ith segment. 7.3.5.3 Formulation of the Dynamic Programming Method A minimization is necessary to determine the value of the factor of safety, Fs, in Equation [101]. Baker (1980) showed that the minimization of the factor of safety in Equation [101] may be found by using an auxiliary function, G. The auxiliary function is also known as the return function and can be defined as follows (Figure 34):.

[Audio] BENTLEY SYSTEMS Slip Surfaces 50 of 158 stage "i" stage "i+1" stage "i" stage "i+1" stage "i" stage "i+1" j j j f  ij ij Sij (ij) ij ij R lij lij  ij f ij Sij (ij) ij  Si Rij i R  ij ij f (ij) ij R S ij ij  Li S ij ij lij lij  ij f ij R ij  (ij)   k k  k lij lij lij lij stage "i" stage "i+1" stage "i" stage "i+1" stage "i" stage "i+1" Figure 34 Actuating and resisting forces acting on the ith segment. n [102] − = ) ( i s i F S R G  = i 1 where: Si = actuating forces acting on the ith segment of the slip surface, Ri = resisting forces acting on the ith segment of the slip surface, and n = total number of discrete segments comprising the slip surface. The minimum value of the auxiliary function, Gmin is defined as: n [103] − = i s i F S R G min ) ( min  = i 1 Along the line segment, the shear strength for a saturated-unsaturated soil can be calculated using the following equation (Fredlund and Rahardjo, 1993): b [104] w a a n f u u u c i     )tan ( )tan ' ( ' − + − + = where: c' = effective cohesion of the material at the base of a slice, ' = effective angle of internal friction of the material at the base of a slice, b = angle defining the increase in shear strength due to matric suction (or negative pore-water pressure), ) ( a n −u  = net normal stress acting on the ith segment, and ) ( w a u u − = matric suction on the ith segment. The normal and shear stresses acting on the ith segment can be computed from the results of a stress analysis as follows:        sin 2 cos sin 2 2 [105] xy y x n − + = [106] − − − = ) ( ) cos (sin 2 2 x y xy n        sin 2 2 where: n = normal stress acting on the ith segment,.