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Limit equilibrium Method (LEM) Advantage of LEM Limitation of LEM Numerical modeling Advantage Limitation STABILITY ANALYSIS OF SLOPE 1limit equilibrium methods still remain the most commonly adopted solution method in rock slope engineering, even though most failures involve complex internal deformation and fracturing which bears little resemblance to the 2-D rigid block assumptions required by most limit equilibrium back-analyses. Both the stress and the displacements can be calculated, Different constitutive relations can be employed. No assumption needs to be made in advance about the shape or location of the failure surface. Failure occurs `naturally' through the zones within the soil mass in which the soil shear strength is unable to sustain the applied shear stresses.(b) Since there is no concept of slices in the Numerical approach, there is no need for assumptions about slice side forces. The Numerical method preserves global equilibrium until `failure' is reached. (c) If realistic soil compressibility data are available, the Numerical solutions will give information about deformations at working stress levels. (d) The Numerical method is able to monitor progressive failure up to and including overall shear failure.Software based on Limit equilibrium Method

SLIDE (rocscience group)GALENA GEO-SLOPEGEO5GGUSOILVISION 2Software based on Numerical modeling

PHASES2PLAXIS FLAC-SLOPE / UDEC / PPFANSYSFEFLOWGEOSLOPE/SIGMASOIL-VISION3Required input properties

Young modulusPoisson ratioDensityFailure criterion: M-C H-BCohesion UCSFriction angle m & s4Numerical modeling

Type of failure mechanismPhysico-mechanical behaviour of slope materialTypes of analysis5Types of analysis: long or Short term analysis, static or dynamic analysisJoint analysis, water pressure analysis, fault or bedding plane, analysis jointed rock mass,

Numerical modeling

Continuum modelling FEM, BEM and FDM Discontinuum modelling DEM, UDEC Hybrid modelling PPF, 6What are the conditions of slope in the field

Simple slope with single, two or three jointsLarge number of joint sets present in the slopeHeavily jointed rock slope Waste dump / very weak rock / soil

7When simulating the mathematical model of slope that represent the nearly actual/real behaviour of the slope in the field. That requires the precise behavior of slope material i.e. constitute equations (stress- strain relationship) Simple slope with single, two or three joints

large number of joint sets present in the slope

Heavily jointed rock

Waste dump / very weak rock / soilProperties of each Joints strength

Properties of each joint set or combined properties

Properties of jointed rock mass

Properties of waste rock8When simulating the mathematical model of slope that represent the nearly actual/real behaviour of the slope in the field. That requires the precise behavior of slope material i.e. constitute equations (stress- strain relationship) Continuum modelling

Continuum modeling is best suited for the analysis of slopes that are comprised of massive, intact rock, weak rocks, and soil-like or heavily jointed rock masses. Discontinuum modeling is appropriate for slopes controlled by discontinuity behaviour.

Critical Parameters: shear strength of material, constitutive criteria, water condition, insitu stress state

Advantages: Allows for material deformation and failure, model complex behaviour, pore pressures, creep deformation and/or dynamic loading can be simulated

Limitations: inability to model effects of highly jointed rock9Continuum methods are best suited for the analysis of rock slopes that are comprised of massiveintact rock, weak rocks, or heavily fractured rock masses. For the most part, earlier studies wereoften limited to elastic analyses and as such were limited in their application. Most continuumcodes, however, now incorporate a facility for including discrete fractures such as faults andbedding planes. Numerous commercial codes are available, which often offer a variety ofconstitutive models including elasticity, elasto-plasticity, strain-softening and elastoviscoplasticity(allowing for the modelling of time-dependent behaviour).Continuum modelling

Typical Input requiredModuls of Elasticity Poision ratio DensityShear strength (cohesion and friction angle)Model Behavior 10 Continuum modelling

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Typical Input requiredModuls of Elasticity for rock and jointsPoision ratio for rock and joints DensityShear strength for rock and jointsJoint behaviourWater pressure13 Continuum modelling (water simulation)

Pore water pressure Ground water tableInfiltration of rain water 14

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20Mohr-Coulomb model; The Mohr-Coulomb model is the conventional model used to represent shearfailure in soils and rocks. Vermeer and deBorst (1984), for example, reportlaboratory test results for sand and concrete that match well with the Mohr-Coulomb criterion.

ubiquitous-joint model; The ubiquitous-joint model is an anisotropic plasticity model that includesweak planes of specific orientation embedded in a Mohr-Coulomb solid.

strain-hardening / softening model;The strain-hardening/softening model allows representation of non-linear materialsoftening and hardening behavior based on prescribed variations of theMohr-Coulomb model properties (cohesion, friction, dilation, tensile strength)as functions of the deviatoric plastic strain.double-yield model; The doube-yield model is intended to represent materials in which there maybe significant irreversible compaction in addition to shear yielding, such ashydraulically-placed backfill or lightly-cemented granular material.

21Mohr-Coulomb model; The Mohr-Coulomb model is the conventional model used to represent shearfailure in soils and rocks. Vermeer and deBorst (1984), for example, reportlaboratory test results for sand and concrete that match well with the Mohr-Coulomb criterion.

ubiquitous-joint model; The ubiquitous-joint model is an anisotropic plasticity model that includesweak planes of specific orientation embedded in a Mohr-Coulomb solid.

strain-hardening / softening model;The strain-hardening/softening model allows representation of non-linear materialsoftening and hardening behavior based on prescribed variations of theMohr-Coulomb model properties (cohesion, friction, dilation, tensile strength)as functions of the deviatoric plastic strain.double-yield model; The double-yield model is intended to represent materials in which there maybe significant irreversible compaction in addition to shear yielding, such ashydraulically-placed backfill or lightly-cemented granular material.

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23Discontinuum modelling

Discontinuum modeling is appropriate for slopes controlled by discontinuity behaviour

Critical Parameters: discontinuity stiffness and shear strength; groundwater characteristics; in situ stress state.

Advantages: Allows for block deformation and movement of blocks relative to each other, can modeled with combined material and discontinuity behaviour coupled with hydro - mechanical and dynamic analysis

Limitations: need to simulate representative discontinuity geometry (spacing, persistence, etc.); limited data on joint properties available

24Although 2-D and 3-D continuum codes are extremely useful in characterizin g rock slope failuremechanisms it is important to recognize their limitations, especially with regards to whether theyare representative of the rock mass under consideration. Where a rock slope comprises multiplejoint sets, which control the mechanism of failure, then a discontinuum modelling approach maybe considered more appropriate. Discontinuum methods treat the problem domain as anassemblage of distinct, interacting bodies or blocks that are subjected to external loads and areexpected to undergo significant motion with time. This methodology is collectively referred to asthe discrete-element method (DEM).

The development of discrete-element procedures represents an important step in the modellingand understanding of the mechanical behaviour of jointed rock masses. Although continuumcodes can be modified to accommodate discontinuities, this procedure is often difficult and timeconsuming. In addition, any modelled inelastic displacements are further limited to elastic ordersof magnitude by the analytical principles exploited in developing the solution procedures. Incontrast, discontinuum analysis permits sliding along and opening/closure between blocks orparticles. The underlying basis of the discrete-element method is that the dynamic equation ofequilibrium for each block in the system is formulated and repeatedly solved until the boundaryconditions and laws of contact and motion are satisfied (Fig. 17). The method thus accounts forcomplex non-linear interaction phenomena between blocks.

Discontinuum modelling

25The dip of the slope must exceed the dip of the potential slide planeThe potential slip plane must daylight on the slope planeThe dip of the potential slip plane must be such that the strength of the plane is reachedThe dip direction of the sliding plane should lie approximately 20 of the dip direction of the slopeDiscontinuum modelling

26The dip of the slope must exceed the dip of the potential slide planeThe potential slip plane must daylight on the slope planeThe dip of the potential slip plane must be such that the strength of the plane is reachedThe dip direction of the sliding plane should lie approximately 20 of the dip direction of the slope




cohesion joint dilation jointfriction jointjoint normal stiffnessjoint shear stiffness 30The dip of the slope must exceed the dip of the potential slide planeThe potential slip plane must daylight on the slope planeThe dip of the potential slip plane must be such that the strength of the plane is reachedThe dip direction of the sliding plane should lie approximately 20 of the dip direction of the slopeHybrid modelling

Hybrid codes involve the coupling of these two techniques (i.e. continuum and discontinuum) to maximize their key advantages.

Critical Parameters: Combination of input parameters Advantages: Coupled finite-/distinctelement models able to simulate intact fracture propagation and fragmentation of jointed and bedded rock. Limitations: high memory capacity;




34The dip of the slope must exceed the dip of the potential slide planeThe potential slip plane must daylight on the slope planeThe dip of the potential slip plane must be such that the strength of the plane is reachedThe dip direction of the sliding plane should lie approximately 20 of the dip direction of the slopeImportant considerations35Two-dimensional analysis versus three-dimensional analysis

3D Simulation by Ansys software based on Finite element method

2D Simulation by Geoslope software based on Finite element method 36Continuum versus discontinum models

2D simulation of bench slope by FLAC based on finite difference method

3D simulation of slope 3DEC software based on discontinum modeling37Selecting appropriate zone size

Different view discritized view of internal dump slope38Boundary conditions

Typical recommendations for locations of artificial far-field boundaries in slope stability analyses.39Water pressure

Simulation of rain water infiltration and generation of water table 40Excavation sequence

Show the sequential excavation 41Stability / failure indicators

Factor of safetyDisplacement ( x and Y)Shear StrainYield PointsPlastic Pointsunbalance force/ convergence of solution Velocity 422.7.2 Unbalanced ForceA grid point in a model is surrounded by up to eight zones that contribute forces to the grid point. At equilibrium, the algebraic sum of these forces is almost zero (i.e., the forces acting on one side of the grid point nearly balance those acting on the other). Unbalanced force approaching a constant non-zero value indicates elastic equilibrium and /or plastic flow occurring within the model. Only very low value of unbalanced forces indicates that force balance at all grid points; however, steady plastic flow may occur, without acceleration. In order to distinguish between these two conditions and true equilibrium, other indicators such as those described below should be examined. 2.7.3 Grid point VelocitiesThe grid velocities may be assessed either by plotting out the whole field of velocities or by selecting certain key points in the grid and tracking their velocities with histories. Steady-state conditions are indicated, if the velocity histories show horizontal traces in their final stages. If they have all converged to near zero (in comparison to their starting values), then absolute equilibrium has occurred. If a history has converged to a non-zero value, then steady plastic flow occurs at the grid point corresponding to the recorded history. If one or more velocity history plots show fluctuating velocities, then the system is likely to be in a transient condition. To confirm that continuing plastic flow is occurring, a plot of plasticity should be examined. When the model is stable, the gridpoint velocities decrease to zero and the velocity vectors often appear random in direction. However, for unstable model, the gridpoint velocities have converged to a non-zero value; it is likely that steady plastic flow is occurring in the model. In this case, the velocity vectors show some systematic orientation.2.7.4 Plastic IndicatorsFor the plasticity models, the FLAC code can display those zones in which the stresses satisfy the yield criterion. Such an indication usually denotes that plastic flow is occurring, but it is possible for an element to simply sit on the yield surface without any significant flow taking place. It is important to look at the whole pattern of plasticity indicators to see if a failure mechanism has developed. Two types of failure mechanisms are indicated by the plasticity state; shear failure and tensile failure. 2.7.5 Displacement The system can also be unstable, meaning that it is heading for ultimate failure or collapse. In addition to the above criterion for steady plastic flow, an unstable model is usually characterized by a non-zero, often fluctuating, maximum unbalanced force, as well as increasing velocities and displacements. The model can also collapse due to displacements becoming very large, thus distorting the individual elements badly and prohibiting further timestepping2.7.6 Failure SurfaceOnce unstable or steady plastic flow has been identified, the question of failure surface formation needs to be answered. The location of a failure surface can be judged by plasticity indicators, displacement field and localization of shear strain. The extent of the zone of actively yielding elements forms the outer limit as to where the failure surface can develop. By looking at the displacement pattern in the model, a more precise estimate can be made.

Stability / failure indicators Factor of safety

43To perform slope stability analysis with the shear strength technique, simulations are run for a series of increasing trial factor of safety, F, actual shear strength properties cohesion (c) and internal friction angle ( ) are reduced for each trial according to the equations 2.1 and 2.2. If the multiple materials are present, the reduction is made simultaneously for all materials. The trial factor of safety is gradually increased until the slope fails. At failure, the safety factor equals the trial safety factor. The factor of safety is defined according to the equation

Stability / failure indicators Displacement ( x and Y)

44Stability / failure indicators Shear Strain

45Stability / failure indicators Yield Points

46Stability / failure indicators Velocity Vector

47Stability / failure indicators unbalance force/ convergence of solution

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