Rock Slope Engineering: Civil and Mining, 4th Edition Analysis.pdf · rock bounded by speciﬁed slide planes. In con-trast, this chapter discusses numerical analysis methods to calculate

Chapter 10

Numerical analysis

Dr Loren Lorig and Pedro Varona1

10.1 Introduction

The previous four chapters discussed limit equi-librium methods of slope stability analysis forrock bounded by specified slide planes. In con-trast, this chapter discusses numerical analysismethods to calculate the factor of safety withoutpre-defining slide planes. These methods are morerecent developments than limit equilibrium meth-ods and, at present (2003), are used predomin-ately in open pit mining and landslides studies,where interest often focuses on slope displace-ments rather than on the relative magnitude ofresisting and displacing forces.

Numerical models are computer programs thatattempt to represent the mechanical response of arock mass subjected to a set of initial conditionssuch as in situ stresses and water levels, bound-ary conditions and induced changes such as slopeexcavation. The result of a numerical model sim-ulation typically is either equilibrium or collapse.If an equilibrium result is obtained, the resultantstresses and displacements at any point in the rockmass can be compared with measured values. If acollapse result is obtained, the predicted mode offailure is demonstrated.

Numerical models divide the rock mass intozones. Each zone is assigned a material modeland properties. The material models are idealizedstress/strain relations that describe how thematerial behaves. The simplest model is a linearelastic model, which uses the elastic properties

1 Itasca Consulting Group, Inc., Minneapolis, Minnesota55401 USA.

(Young’s modulus and Poisson’s ratio) of thematerial. Elastic–plastic models use strengthparameters to limit the shear stress that a zonemay sustain.2

The zones may be connected together, termed acontinuum model, or separated by discontinuities,termed a discontinuum model. Discontinuummodels allow slip and separation at explicitlylocated surfaces within the model.

Numerical models tend to be general purpose innature—that is, they are capable of solving a widevariety of problems. While it is often desirable tohave a general-purpose tool available, it requiresthat each problem be constructed individually.The zones must be arranged by the user to fit thelimits of the geomechanical units and/or the slopegeometry. Hence, numerical models often requiremore time to set up and run than special-purposetools such as limit equilibrium methods.

There are several reasons why numericalmodels are used for slope stability studies.

• Numerical models can be extrapolated confid-ently outside their databases in comparison toempirical methods in which the failure modeis explicitly defined.

• Numerical analysis can incorporate key geo-logic features such as faults and ground waterproviding more realistic approximations ofbehavior of real slopes than analytic models.

2 In numerical analysis the terms “elements” and “zones” areused interchangeably. However, the term element is usedmore commonly in finite element analysis, and the termzone in finite difference analysis.

Numerical analysis 219

In comparison, non-numerical analysis meth-ods such as analytic, physical or limit equi-librium may be unsuitable for some sites ortend to oversimplify the conditions, possiblyleading to overly conservative solutions.

• Numerical analysis can help to explainobserved physical behavior.

• Numerical analysis can evaluate multiple pos-sibilities of geological models, failure modesand design options.

Many limit equilibrium programs exist todetermine factors of safety for slopes. Theseexecute very rapidly, and in the case of the methodof slices for circular failure, use an approximatescheme in which a number of assumptions aremade, including the location and angle of inter-slice forces (see Section 8.2). Several assumed slidesurfaces are tested, and the one giving the lowestfactor of safety is chosen. Equilibrium is satis-fied only on an idealized set of surfaces. Withnumerical models, a “full” solution of the coupledstress/displacement, equilibrium and constitutiveequations is made. Given a set of properties, thesystem is found either to be stable or unstable.By performing a series of simulations with vari-ous properties, the factor of safety can be foundcorresponding to the point of stability.

The numerical analysis is much slower, butmuch more general. Only since the late 1990s,with the advent of faster computers, has it becomea practical alternative to the limit equilibriummethod. Even so, while limit equilibrium solu-tions may require just a few seconds, numericalsolutions to large complex problems can take anhour or more, particularly when discontinuumbehavior is involved. The third section of thischapter presents typical safety factor analyses forthe most common discontinuum failure modes inrock slopes.

For slopes, the factor of safety often is definedas the ratio of the actual shear strength to the min-imum shear strength required to prevent failure.A logical way to compute the factor of safety witha finite element or finite difference program is toreduce the shear strength until collapse occurs.The factor of safety is the ratio of the rock’s actual

strength to the reduced shear strength at failure.This shear-strength reduction technique was usedfirst with finite elements by Zienkiewicz et al.(1975) to compute the safety factor of a slopecomposed of multiple materials.

To perform slope stability analysis with theshear strength reduction technique, simulationsare run for a series of increasing trial factorsof safety (f ). Actual shear strength properties,cohesion (c) and friction angle (φ), are reducedfor each trial according to the equations

ctrial =(

1f

)c (10.1)

φtrial = arctan(

1f

)tan φ (10.2)

If multiple materials and/or joints are present, thereduction is made simultaneously for all materi-als. The trial factor of safety is increased graduallyuntil the slope fails. At failure, the factor of safetyequals the trial factor of safety (i.e. f = FS).Dawson et al. (1999) show that the shear strengthreduction factors of safety are generally within afew percent of limit analysis solutions when anassociated flow rule, in which the friction angleand dilation angle are equal, is used.

The shear strength reduction technique has twomain advantages over limit equilibrium slope sta-bility analyses. First, the critical slide surface isfound automatically, and it is not necessary tospecify the shape of the slide surface (e.g. circular,log spiral, piecewise linear) in advance. In general,the failure surface geometry for slopes is morecomplex than simple circles or segmented sur-faces. Second, numerical methods automaticallysatisfy translational and rotational equilibrium,whereas not all limit equilibrium methods do sat-isfy equilibrium. Consequently, the shear strengthreduction technique usually will determine asafety factor equal to or slightly less than limitequilibrium methods. Itasca Consulting Group(2002) gives a detailed comparison of four limitequilibrium methods and one numerical methodfor six different slope stability cases.

220 Numerical analysis

Table 10.1 Comparison of numerical and limit equilibrium analysis methods

Analysis result Numerical solution Limit equilibrium

Equilibrium Satisfied everywhere Satisfied only for specific objects,such as slices

Stresses Computed everywhere using fieldequations

Computed approximately on certainsurfaces

Deformation Part of the solution Not consideredFailure Yield condition satisfied everywhere;

slide surfaces develop “automatically”as conditions dictate

Failure allowed only on certainpre-defined surfaces; no check onyield condition elsewhere

Kinematics The “mechanisms” that develop satisfykinematic constraints

A single kinematic condition isspecified according to the particulargeologic conditions

A summary of the differences between anumerical solution and the limit equilibriummethod is shown in the Table 10.1.

10.2 Numerical models

All rock slopes involve discontinuities. Represent-ation of these discontinuities in numerical modelsdiffers depending on the type of model. There aretwo basic types of models: discontinuum modelsand continuum models. Discontinuities in discon-tinuum models are represented explicitly—thatis, the discontinuities have a specific orientationand location. Discontinuities in continuum mod-els are represented implicitly, with the intentionthat the behavior of the continuum model is sub-stantially equivalent to the real jointed rock massbeing represented.

Discontinuum codes start with a methoddesigned specifically to model discontinua andtreat continuum behavior as a special case. Dis-continuum codes generally are referred to asDiscrete Element codes. A Discrete Element codewill typically embody an efficient algorithm fordetecting and classifying contacts, and maintaina data structure and memory allocation schemethat can handle many hundreds or thousandsof discontinuities. The discontinuities divide theproblem domain into blocks that may be eitherrigid or deformable; continuum behavior is

assumed within deformable blocks. The mostwidely used discrete element codes for slopestability studies are UDEC (Universal DistinctElement Code; Itasca Consulting Group, 2000)and 3DEC (3-Dimensional Distinct ElementCode; Itasca Consulting Group, 2003). Of fun-damental importance in discontinuum codes isthe representation of joint or discontinuity beha-vior. Commonly used relations for represent-ing joint behavior are discussed later in thissection.

Continuum codes assume material is con-tinuous throughout the body. Discontinuitiesare treated as special cases by introducinginterfaces between continuum bodies. Finiteelement codes such as PHASE2 (Rocscience,2002c) and its predecessor PHASES (P lasticHybrid Analysis of Stress for Estimation ofSupport) and finite difference codes such as FLAC(Fast Lagrangian Analysis of Continua; ItascaConsulting Group, 2001) cannot handle gen-eral interaction geometry (e.g. many intersectingjoints). Their efficiency may degenerate drastic-ally when connections are broken repeatedly.Typical continuum-based models may have lessthan ten non-intersecting discontinuities. Of fun-damental importance to continuum codes anddeformable blocks in discrete element codesis representation of the rock mass behavior.Continuum relations used to represent rock massbehavior are discussed later in this section.


Finite element programs are probably morefamiliar, but the finite difference method isperhaps the oldest numerical technique used tosolve sets of differential equations. Both finiteelement and finite difference methods producea set of algebraic equations to solve. While themethods used to derive the equations are differ-ent, the resulting equations are the same. Finitedifference programs generally use an “explicit”time-marching scheme to solve the equations,whereas finite element methods usually solvesystems of equations in matrix form.

Although a static solution to a problem is usu-ally of interest, the dynamic equations of motionare typically included in the formulation of finitedifference programs. One reason for doing this isto ensure that the numerical scheme is stable whenthe physical system being modeled is unstable.With non-linear materials, there is always thepossibility of physical instability—for example,the failure of a slope. In real life, some of thestrain energy is converted to kinetic energy. Expli-cit finite-difference programs model this processdirectly, because inertial terms are included. Incontrast, programs that do not include inertialterms must use some numerical procedure totreat physical instabilities. Even if the procedureis successful at preventing numerical instability,the path taken may not be realistic. The con-sequence of including the full law of motion infinite difference programs is that the user musthave some physical feel for what is happening.Explicit finite-difference programs are not blackboxes that will “give the solution.” The behaviorof the numerical system must be interpreted.

FLAC and UDEC are two-dimensional finite-difference programs developed specifically forgeomechanical analysis. These codes can simu-late varying loading and water conditions, andhave several pre-defined material models for rep-resenting rock mass continuum behavior. Bothcodes are unique in their ability to handlehighly non-linear and unstable problems. Thethree-dimensional equivalents of these codes areFLAC3D (Fast Lagrangian Analysis of Continuain 3 Dimensions; Itasca Consulting Group, 2002)and 3DEC (Itasca Consulting Group, 2003).

10.2.1 Joint material models

The material model used most commonly torepresent joints is a linear-elastic–perfectly-plasticmodel. The limiting shear strength is defined bythe usual Mohr–Coulomb parameters of frictionangle and cohesion (see Section 4.2). A peak andresidual shear strength relation can also be spe-cified for the joints. The residual strength is usedafter the joint has failed in shear at the peakstrength. The elastic behavior of the joints is spe-cified by joint normal and shear stiffnesses, whichmay be linear or piece-wise linear.

10.2.2 Rock mass material models

It is impossible to model all discontinuities ina large slope, although it may be possible tomodel the discontinuities for a limited number ofbenches. Therefore, in large slopes much of therock mass must be represented by an equivalentcontinuum in which the effect of the discontinu-ities is to reduce the intact rock elastic propertiesand strength to those of the rock mass. Thisis true whether or not a discontinuum modelis used. As mentioned in the introduction tothis chapter, numerical models divide the rockmass into zones. Each zone is assigned a mater-ial model and material properties. The materialmodels are stress/strain relations that describehow the material behaves. The simplest modelis a linear elastic model that uses only the elasticproperties (Young’s modulus and Poisson’s ratio)of the material. Linear elastic–perfectly plasticstress–strain relations are the most commonlyused rock mass material models. These modelstypically use Mohr–Coulomb strength paramet-ers to limit the shear stress that a zone maysustain. The tensile strength is limited by the spe-cified tensile strength, which in many analysesis taken to be 10% of the rock mass cohe-sion. Using this model, the rock mass behavesin an isotropic manner. Strength anisotropy canbe introduced through a ubiquitous joint model,which limits the shear strength according to aMohr–Coulomb criterion in a specified direction.The direction often corresponds to a predominantjointing orientation.


A more complete equivalent-continuum modelthat includes the effects of joint orientationand spacing is a micropolar (Cosserat) plasticitymodel. The Cosserat theory incorporates a localrotation of material points as an independentparameter, in addition to the translation assumedin the classical continuum, and couple stresses(moments per unit area) in addition to the clas-sical stresses (forces per unit area). This model, asimplemented in FLAC, is described in the contextof slope stability by Dawson and Cundall (1996).The approach has the advantage of using a con-tinuum model while still preserving the abilityto consider realistic joint spacing explicitly. Themodel has not yet (as of 2003) been incorporatedinto any publicly available code.

The most common failure criterion for rockmasses is the Hoek–Brown failure criterion (seeSection 4.5). The Hoek–Brown failure criterionis an empirical relation that characterizes thestress conditions that lead to failure in intact rockand rock masses. It has been used successfullyin design approaches that use limit equilibriumsolutions. It also has been used indirectly innumerical models by finding equivalent Mohr–Coulomb shear strength parameters that providea failure surface tangent to the Hoek–Brown fail-ure criterion for specific confining stresses, orranges of confining stresses. The tangent Mohr–Coulomb parameters are then used in traditionalMohr–Coulomb type constitutive relations andthe parameters may or may not be updated duringanalyses. The procedure is awkward and time-consuming, and consequently there has been littledirect use of the Hoek–Brown failure criterionin numerical solution schemes that require fullconstitutive models. Such models solve for dis-placements, as well as stresses, and can continuethe solution after failure has occurred in some loc-ations. In particular, it is necessary to develop a“flow rule,” which supplies a relation betweenthe components of strain rate at failure. Therehave been several attempts to develop a full con-stitutive model from the Hoek–Brown criterion:for example, Pan and Hudson (1988), Carteret al. (1993) and Shah (1992). These formulationsassume that the flow rule has some fixed relation

to the failure criterion and that the flow rule isisotropic, whereas the Hoek–Brown criterion isnot. Recently, Cundall et al. (2003) has proposeda scheme that does not use a fixed form of the flowrule, but rather one that depends on the stresslevel, and possibly some measure of damage.

Real rock masses often appear to exhibit pro-gressive failure—that is, the failure appears toprogress over time. Progressive failure is a com-plex process that is understood poorly and diffi-cult to model. It may involve one or more of thefollowing component mechanisms:

• Gradual accumulation of strain on principalstructures and/or within the rock mass;

• Increases in pore pressure with time; and• Creep, which is time-dependent deformation

of material under constant load.

Each of these components is discussed briefly laterin the context of slope behavior.

Gradual accumulation of strain on principalstructures within the rock mass usually resultsfrom excavation, and “time” is related to theexcavation sequence. In order to study the pro-gressive failure effects due to excavation, onemust either introduce characteristics of the post-peak or post-failure behavior of the rock massinto a strain-softening model or introduce similarcharacteristics into the explicit discontinuities. Inpractice, there are at least two difficulties asso-ciated with strain-softening rock mass models.The first is estimating the post-peak strength andthe strain over which the strength reduces. Thereappear to be no empirical guidelines for estimat-ing the required parameters. This means that theproperties must be estimated through calibration.The second difficulty is that, for a simulation inwhich the response depends on shear localizationand in which material softening is used, the res-ults will depend on the zone sizes. However, itis quite straightforward to compensate for thisform of mesh-dependence. In order to do this,consider a displacement applied to the boundaryof a body. If the strain localizes inside the body,the applied displacement appears as a jump acrossthe localized band. The thickness of the band


contracts until it is equal to the minimum allowedby the grid, that is a fixed number of zone widths.Thus, the strain in the band is

ε = u/n�z (10.3)

where n is a fixed number, u is the displacementjump, and �z is the zone width.

If the softening slope is linear, the change in aproperty value �p is proportional to strain, thechange in property value with displacement is:

�p

�u= s

n · �z (10.4)

where s is the softening slope.In order to obtain mesh-independent results, a

scaled softening slope s can be input, such that

s = s′�z (10.5)

where s′ is constant.In this case, (�p/�u) is independent of �z. If

the softening slope is defined by the critical strain,εscrit, then

εscrit ∝1

�z(10.6)

For example, if the zone size is doubled, thecritical strain must be halved for comparableresults.

Strain-softening models for discontinuities aremuch more common than similar relations forrock masses. Strain-softening relations for dis-continuities are built into UDEC and 3DEC, andcan be incorporated into interfaces in FLAC andFLAC3D via a built-in programming languagesuch as FISH functions. Strain-softening modelsrequire special attention when computing safetyfactors. If a strain-softening constitutive modelis used, the softening logic should be turned offduring the shear strength reduction process or thefactor of safety will be underestimated. Whenthe slope is excavated, some zones will haveexceeded their peak strength, and some amountof softening will have taken place. During the

strength reduction process, these zones should beconsidered as a new material with lower strength,but no further softening should be allowed dueto the plastic strains associated to the gradualreduction of strength.

Increases in pore pressure with time are notcommon in rock slopes for mines. More com-monly, the pore pressures reduce due to deepen-ing of the pit and/or drainage. However, there arecases in which the pore pressures do increase withtime. In such cases, the slope may appear to failprogressively.

Creep, which is time-dependent deformation ofmaterial under constant load, is not commonlyconsidered in the context of slope stability. Itis much more common in underground excav-ations. Several material models are available tostudy creep behavior in rock slopes. These includeclassical viscoelastic models, power law models,and the Burger-creep viscoplastic model. Applic-ation of a creep model to the study of slope beha-vior at Chuquicamata mine in Chile is discussedlater in this chapter (see Section 10.5.2).

10.3 Modeling issues

Modeling requires that the real problem be ideal-ized, or simplified, in order to fit the constraintsimposed by factors such as available materialmodels and computer capacity. Analysis of rockmass response involves different scales. It isimpossible—and undesirable—to include all fea-tures, and details of rock mass response mechan-isms, into one model. In addition, many of thedetails of rock mass behavior are unknown andunknowable; therefore, the approach to model-ing is not as straightforward as it is, say, in otherbranches of mechanics. This section discusses thebasic issues that must be resolved when setting upa numerical model.

10.3.1 Two-dimensional analysis versusthree-dimensional analysis

The first step in creating a model is todecide whether to perform two-dimensionalor three-dimensional analyses. Prior to 2003,


three-dimensional analyses were uncommon, butadvances in personal computers have permittedthree-dimensional analyses to be performedroutinely. Strictly speaking, three-dimensionalanalyses are recommended/required in thefollowing:

1 The direction of principal geologic structuresdoes not strike within 20–30◦ of the strike ofthe slope.

2 The axis of material anisotropy does not strikewithin 20–30◦ of the slope.

3 The directions of principal stresses are notneither parallel nor perpendicular to theslope.

4 The distribution of geomechanical units variesalong the strike of the slope.

5 The slope geometry in plan cannot be rep-resented by two-dimensional analysis, whichassumes axisymmetric or plain strain.

Despite the forgoing, most design analysis forslopes assumes a two-dimensional geometry com-prising a unit slice through an infinitely longslope, under plane strain conditions. In otherwords, the radii of both the toe and crest areassumed to be infinite. This is not the conditionencountered in practice—particularly in open pitmining where the radii of curvature can have animportant effect on safe slope angles. Concaveslopes are believed to be more stable than plainstrain slopes due to the lateral restraint providedby material on either side of a potential failure ina concave slope.

Despite its potential importance in slope sta-bility, very little has been done to quantify thiseffect. Jenike and Yen (1961) presented the resultsof limit theory analysis of axisymmetric slopes ina rigid, perfectly plastic material. However, Hoekand Brown (1981) concluded that the analysisassumptions were not applicable to rock slopedesign.

Piteau and Jennings (1970) studied the influ-ence of curvature in plan on the stability of slopesin four diamond mines in South Africa. As aresult of caving from below the surface, slopeswere all at incipient failure with a safety factor

of 1. The average slope height was 100 m. Piteauand Jennings (1970) found that the average slopeangle for slopes with radius of curvature of 60 mwas 39.5◦ as compared to 27.3◦ for slopes witha radius of curvature of 300 m.

Hoek and Bray (1981) summarize their experi-ence with the stabilizing effects of slope curvatureas follows. When the radius of curvature ofa concave slope is less than the height of theslope, the slope angle can be 10◦ steeper thanthe angle suggested by conventional stability ana-lysis. As the radius of curvature increases to avalue greater than the slope height, the correc-tion should be decreased. For radii of curvaturein excess of twice the slope height, the slope anglegiven by a conventional stability analysis shouldbe used.

To better quantify the effects of slope curvatureon stability, a series of generic analyses were per-formed. All analyses assumed a 500 m high dryslope with a 45◦ face angle excavated in an iso-tropic homogeneous material with a density of2600 kg/m3. Initial in situ stresses are assumedto be lithostatic, and the excavation was made in40 m decrements beginning from the ground sur-face. For these conditions, pairs of friction angleand cohesion values were selected to produce afactor of safety of 1.3 using circular failure chartnumber 1 (see Section 8.3). A factor of safety of1.3 is a value that is frequently used in the designof slopes for open pit mines. The actual valuesused are shown in Table 10.2.

A series of analyses was performed using FLACfor different radius of curvature for both concaveand convex slopes. For concave slopes, the radiusof curvature is defined as the distance between

Table 10.2 Janbu’s Lambda coefficient for variouscombinations of friction angle and cohesion

Friction angle Cohesion(MPa)

λ = γH tan φ/c

45 0.22 5935 0.66 1425 1.18 515 1.8 2


–10 –2–8 0 2 4 6 8 10–6

Slope height/radius of curvature

�= 15°�= 25°

�= 35°

�= 45°

–4

Convex slopes Concave slopes

FS

curv

ed s

lope

s

FS

plan

e st

rain

Figure 10.1 Results of FLAC axisymmetric analyses showing effect on factor of safety of slope curvature.

the axis of revolution and the toe of the slope.For the convex slopes, the radius of curvature isdefined as the distance between the axis of revolu-tion and the crest of the slope—not the toe. Underboth definitions, cones have a radius of curvatureof zero.

Figure 10.1 shows the results withFS/FSplane strain versus height/radius of curvature(H/Rc), which is positive for concave and negat-ive for convex slopes. The figure shows that thefactor of safety always increases as the radius ofcurvature decreases, but the increase is faster forconcave slopes. One unexpected result is that asthe friction angle increases, the effect of curvaturedecreases. One possible explanation is that asJanbu’s lambda coefficient (λ = γH tan φ/c)increases, the slide surface is shallower with onlya skin for purely frictional material. This, makesthe slope less sensitive to the confining effect inconcave slopes, and to the ratio of active/passivewedges for the convex ones.

One reason that designers are reluctant totake advantage of the beneficial effects of con-cave slope curvature is that the presence ofdiscontinuities can often negate the effects.However, for massive rock slopes, or slopeswith relatively short joint trace lengths, the

beneficial effects of slope curvature should not beignored—particularly in open pit mines, wherethe economic benefits of steepening slopes can besignificant. The same is true for convex slopes,which are also more stable than plane strainslopes. This goes against observed experiencein rock slopes. If the slide surface is definedin terms of active (top) and passive (bottom)wedges, the ratio of the surface (and weight)of the passive wedge to the active wedge in aconvex slope is greater than the plane straincondition. However, this only applies to a homo-geneous Mohr–Coulomb material that mightbe found, for example, in waste dumps. Thereason why “noses” in rock slopes are usu-ally less stable may be related to the fact thatthey are more exposed to structurally controlledfailures.

10.3.2 Continuum versus discontinuummodels

The next step is to decide whether to use acontinuum code or a discontinuum code. Thisdecision is seldom straightforward. There appearto be no ready-made rules for determining whichtype of analysis to perform. All slope stability


problems involve discontinuities at one scale oranother. However, useful analyses, particularlyof global stability, have been made by assum-ing that the rock mass can be represented as anequivalent continuum. Therefore, many analysesbegin with continuum models. If the slope underconsideration is unstable without structure, thereis no point in going to discontinuum models. If,on the other hand, a continuum model appearsto be reasonably stable, explicit incorporation ofprincipal structures should give a more accurateestimation of slope behavior.

Selection of joint geometry for input to a modelis a crucial step in discontinuum analyses. Typic-ally, only a very small percentage of joints canactually be included in a model in order to createmodels of reasonable size for practical analysis.Thus, the joint geometry data must be filteredto select only those joints that are most crit-ical to the mechanical response. This is done byidentifying those that are most susceptible to slipand/or separation for the prescribed loading con-dition. This may involve determining whether suf-ficient kinematic freedom is provided, especiallyin the case of toppling, and calibrating the ana-lysis by comparing observed behavior to modelresponse.

10.3.3 Selecting appropriate zone size

The next step in the process is to select anappropriate zone size. The finite difference zonesassume that the stresses and strains within eachzone do not differ with position within the zone—in other words, the zones are the lowest-orderelements possible. In order to capture stress andstrain gradients within the slope adequately, it isnecessary to use relatively fine discretizations. Byexperience, the authors have found that at least20 (and preferably 30) zones are required overthe slope height of interest. As discussed later, ifflexural toppling is involved, a minimum of fourzones across the rock column are required. Finiteelement programs using higher-order elementslikely would require less zones than the constantstrain/constant stress elements common in finitedifference codes.

10.3.4 Initial conditions

Initial conditions are those conditions that existedprior to mining. The initial conditions ofimportance at mine sites are the in situ stressfield and the ground water conditions. The roleof stresses has been traditionally ignored in slopeanalyses. There are several possible reasons forthis:

• Limit equilibrium analyses, which are widelyused for stability analyses, cannot include theeffect of stresses in their analyses. Neverthe-less, limit equilibrium analyses are thoughtto provide reasonable estimates of stabilityin many cases, particularly where structure isabsent, such as soil slopes.

• Most stability analyses have traditionally beenperformed for soils, where the range of pos-sible in situ stresses is more limited than forrocks. Furthermore, many soil analyses havebeen performed for constructed embankmentssuch as dams, where in situ stresses do notexist.

• Most slope failures are gravity driven, andthe effects of in situ stress are thought to beminimal.

• In situ stresses in rock masses are not routinelymeasured for slopes, and their effects arelargely unknown.

One particular advantage of stress analysis pro-grams such as numerical models is their ability toinclude pre-mining initial stress states in stabilityanalyses and to evaluate their importance.

In order to evaluate the effects of in situ stressstate on stability, five cases were run using asimple model similar to the model shown inFigure 10.2. For each of the five cases, the slopeangle was 60◦, the slope height was 400 m, thematerial density was 2450 kg/m3, the frictionangle was 32◦, and the cohesion was 0.92 MPa.The results of FLAC analyses are shown inTable 10.3.

In general, it is impossible to say what effectthe initial stress state will have on any partic-ular problem, as behavior depends on factors


Density = 2600 kg/m3

Friction angle = 35°Cohesion = 660 kPa

45°Step –138.9 < x < 2639 –1264 < y < 1514

0 500

1500500

Horizontal axis (m)

1000 2000

Ver

tical

axi

s (m

)

–1000

0

1000

–500

500

Water table

Grid plot

FLAC (Version 3.40)

Legend

Figure 10.2 Problem geometry used to determine the effect of in situ stresses on slope stability.

Table 10.3 Effect of in situ stress on slope stability[x—horizontal in-plane direction; y—vertical in-planedirection; z—out-of-plane direction]

In-planehorizontal stress

Out-of-planehorizontal stress

Factor ofsafety

σxx = σyy σzz = σyy 1.30σxx = 2.0 σyy σzz = 2.0 σyy 1.30σxx = 0.5 σyy σzz = 0.5 σyy 1.28σxx = 2.0 σyy σzz = 0.5 σyy 1.30σxx = 0.5 σyy σzz = 2.0 σyy 1.28

such as orientation of major structures, rock massstrength and water conditions. However, someobservations on the effects of in situ stress onstability can be made:

• The larger the initial horizontal stresses, thelarger the horizontal elastic displacements.This is not much help, as elastic displacements

are not particularly important in slopestudies.

• Initial horizontal stresses in the plane of ana-lysis that are less than the vertical stressestend to slightly decrease stability and reducethe depth of significant shearing with respectto a hydrostatic stress state. This observa-tion may seem counter-intuitive; smaller hori-zontal stresses would be expected to increasestability. The explanation lies in the fact thatthe lower horizontal stresses actually provideslightly decreased normal stress on potentialshearing surfaces and/or joints within theslope. This observation was confirmed in aUDEC analysis of a slope in Peru wherein situ horizontal stresses lower than the ver-tical stress led to deeper levels of joint shear-ing in toppling structures compared to casesinvolving horizontal stresses that were equalto or greater than the vertical stress.


• It is important to note that the regionaltopography may limit the possible stressstates, particularly at elevations aboveregional valley floors. Three-dimensionalmodels have been very useful in the past inaddressing some regional stress issues.

10.3.5 Boundary conditions

Boundaries are either real or artificial. Realboundaries in slope stability problems corres-pond to the natural or excavated ground surfacethat is usually stress free. Artificial boundariesdo not exist in reality. All problems in geo-mechanics, including slope stability problems,require that the infinite extent of a real prob-lem domain be artificially truncated to includeonly the immediate area of interest. Figure 10.3shows typical recommendations for locations ofthe artificial far-field boundaries in slope sta-bility problems. Artificial boundaries can be oftwo types: prescribed displacement, or prescribedstress. Prescribed displacement boundaries inhibitdisplacement in either the vertical direction orhorizontal direction, or both. Prescribed displace-ment boundaries are used to represent the condi-tion at the base of the model and toe of the slope.

Displacement at the base of the model is alwaysfixed in both the vertical and horizontal directionsto inhibit rotation of the model. Two assumptions

W

H

�

>H/2

>W

Figure 10.3 Typical recommendations for locationsof artificial far-field boundaries in slope stabilityanalyses.

can be made regarding the displacement bound-aries near the toe of any slope. One assumption isthat the displacements near the toe are inhibitedonly in the horizontal direction. This is the mech-anically correct condition for a problem that isperfectly symmetric with respect to the plane oraxis representing the toe boundary. Strictly speak-ing, this condition only occurs in slopes of infinitelength, which are modeled in two-dimensionsassuming plane strain, or in slopes that are axi-ally symmetric in which the pit is a perfect cone.In reality, these conditions are rarely satisfied.Therefore, some models are extended laterally toavoid the need to specify any boundary condi-tion at the toe of the slope. It is important tonote that difficulties with the boundary conditionnear the slope toe are usually a result of the two-dimensional assumption. In three-dimensionalmodels, this difficulty generally does notexist.

The far-field boundary location and condi-tion must also be specified in any numericalmodel for slope stability analyses. The generalnotion is to select the far-field location so thatit does not significantly influence the results.If this criterion is met, whether the boundaryis prescribed-displacement or prescribed-stress isnot important. In most slope stability studies, aprescribed-displacement boundary is used. Theauthors have used a prescribed-stress bound-ary in a few cases and found no significantdifferences with respect to the results from aprescribed-displacement boundary. The mag-nitude of the horizontal stress for the prescribed-stress boundary must match the assumptionsregarding the initial stress state in order for themodel to be in equilibrium. However, followingany change in the model, such as an excavationincrement, the prescribed-stress boundary causesthe far-field boundary to displace toward theexcavation while maintaining its original stressvalue. For this reason, a prescribed-stress bound-ary is also referred to as a “following” stress, orconstant stress boundary, because the stress doesnot change and follows the displacement of theboundary. However, following stresses are mostlikely where slopes are cut into areas where the


topography rises behind the slope. Even whereslopes are excavated into an inclined topography,the stresses would flow around the excavationto some extent, depending on the effective widthof the excavation perpendicular to the downhilltopographic direction.

A summary of the effects of boundary condi-tions on analysis results is as follows:

• A fixed boundary causes both stresses anddisplacements to be underestimated, whereasa stress boundary does the opposite.

• The two types of boundary condition“bracket” the true solution, so that it is pos-sible to conduct tests with smaller modelsto obtain a reasonable estimate of the truesolution by averaging the two results.

A final point to be kept in mind is that all open pitslope stability problems are three-dimensional inreality. This means that the stresses acting in andaround the pit are free to flow both beneath andaround the sides of the pit. Therefore, it is likelythat, unless there are very low strength faultsparallel to the analysis plane, a constant stressor following stress boundary will over-predict thestresses acting horizontally.

10.3.6 Incorporating water pressure

The effect of water pressure in reducing effectivestresses and, hence, slope stability is well under-stood. However, the effect of various assump-tions regarding specification of pore pressuredistributions in slopes is not as well understood.Two methods are commonly used to specifypore pressure distributions within slopes. Themost rigorous method is to perform a completeflow analysis, and use the resultant pore pres-sures in the stability analyses. A less rigorous,but more common method is to specify a watertable, and the resulting pore pressures are givenby the product of the vertical depth below thewater table, the water density and gravity. In thissense, the water table approach is equivalent tospecifying a piezometric surface. Both methodsuse similar phreatic surfaces. However, the watertable method under-predicts actual pore pressure

concentrations near the toe of a slope, and slightlyover-predicts the pore pressure behind the toe byignoring the inclination of equipotential lines.

Seepage forces must also be considered in theanalysis. The hydraulic gradient is the differencein water pressure that exists between two pointsat the same elevation, and results from seepageforces (or drag) as water moves through a porousmedium. Flow analysis automatically accountsfor seepage forces.

To evaluate the error resulting from specifyinga water table without doing a flow analysis, twoidentical problems were run. In one case, a flowanalysis was performed to determine the porepressures. In the second case, the pressures weredetermined using only a piezometric surface thatwas assumed to be the phreatic surface taken fromthe flow analysis. The material properties andgeometry for both cases are shown in Figure 10.2.The right-hand boundary was extended to allowthe far-field phreatic surface to coincide with theground surface at a horizontal distance of 2 kmbehind the toe. Hydraulic conductivity within themodel was assumed to be homogeneous and iso-tropic. The error caused by specifying the watertable can be seen in Figure 10.4. The largesterrors, under prediction of up to 45%, are foundjust below the toe, while over prediction errorsin pore pressure values behind the slope are gen-erally less than 5%. The errors near the phreaticsurface are insignificant, as they result from therelatively small pore pressures just below thephreatic surface where small errors in small valuesresult in large relative errors.

For a phreatic surface at the ground surfaceat a distance of 2 km, a factor of safety of 1.1is predicted using circular failure chart number3 (refer to Section 8.3). The factor of safetydetermined by FLAC was approximately 1.15 forboth cases. The FLAC analyses give similar safetyfactors because the distribution of pore pressuresin the area behind the slope where failure occursis very similar for the two cases. The conclusiondrawn here is that there is no significant differ-ence in predicted stability between a completeflow analysis and simply specifying a piezometricsurface. However, it is not clear if this conclusion


–4.5

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Contour interval = 5.00E-02(zero contour omitted)

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Horizontal axis (m)

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s (m

)

200 600 1400 1800

–800

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Pore pressuresunderpredicted bywater table in thisarea

Pore pressuresoverpredicted bywater table in thisarea

FLAC (Version 4.00)

Legend

Step 20000

Pore pressure error

Cons. time 1.3380E + 11

Figure 10.4 Error in pore pressure distribution caused by specifying water table compared to performing a flowanalysis.

can be extrapolated to other cases involving, forexample, anisotropic flow.

10.3.7 Excavation sequence

Simulating excavations in numerical modelsposes no conceptual difficulties. However, theamount of effort required to construct a modeldepends directly on the number of excavationstages simulated. Therefore, most practical ana-lyses seek to reduce the number of excavationstages. The most accurate solution is obtainedusing the largest number of excavation steps,because the real load path for any zone in theslope will be followed closely. In theory, it isimpossible to prove that the final solution is inde-pendent of the load path followed. However, formany slopes, stability seems to depend mostlyon slope conditions, such as geometry and porepressure distribution at the time of analysis, andvery little on the load path taken to get there.

A reasonable approach regarding the numberof excavation stages has evolved over the years.Using this approach, only one, two or threeexcavation stages are modeled. For each stage,two calculation steps are taken. In the first step,the model is run elastically to remove any iner-tial effects caused by sudden removal of a largeamount of material. Second, the model is runallowing plastic behavior to develop. Followingthis approach, reasonable solutions to a largenumber of slope stability problems have beenobtained.

10.3.8 Interpretation of results

As noted in the introduction, finite differenceprograms are not black boxes that “give thesolution.” The behavior of the numerical systemand results from finite difference models mustbe interpreted in much the same way as slopemovement data are interpreted. Finite difference


programs record displacements and velocities atnominated points within the rock mass. Duringthe analysis, the recorded values can be examinedto see if they are increasing, remaining steady, ordecreasing. Increasing displacements and velocit-ies indicate an unstable situation; steady displace-ments and decreasing velocities indicate a stablesituation. In addition, velocity and displacementvectors for every point in the model can be plot-ted. Fields of constant velocity and displacementindicate failure.

The authors have found that velocities below1e−6 indicate stability in FLAC and FLAC3D;conversely, velocities above 1e−5 indicateinstability. Note that no units are given for velocit-ies. This is because the velocities are not real, dueto the damping and mass scaling used to achievestatic solutions. While the displacements are real,the velocities are not, and there is no informationon the “time” the displacement occurs.

It is also possible to examine the failure (plas-ticity) state of points within the model, wherefailure is defined as failure in tension or shear.Care must be used in examining the failure stateindicators. For example, local overstressing atthe base of toppling columns can appear to forma deep-seated slip surface when, in reality, it isjust compressive failure of the columns. There-fore, the failure (plasticity) indicators must bereviewed in the context of overall behavior beforeany definitive conclusions can be drawn.

10.4 Typical stability analysis

In this section, typical stability analyses for a vari-ety of failure modes are discussed. The objectiveof this section is to show how numerical modelscan be used to simulate slope behavior, and com-pute safety factors for typical problems. Modelingissues important in each of the following failuremodes are discussed:

• Rock mass failure;• Plane failure—daylighting and

non-daylighting;• Wedge failure—daylighting and

non-daylighting;

• Toppling failure—block and flexural; and• Flexural buckling failure.

The two-dimensional distinct element code,UDEC, was used for most of the analyses, and thecharacteristics of the slopes that were modeled areas follows:

Slope height 260 mSlope angle 55◦Water pressure none/dryDensity 2660 kg/m3

(26.1 kN/m3)Rock mass friction angle 43◦Rock mass cohesion 675 kPaRock mass tension 0Rock mass bulk modulus 6.3 GPaRock mass shear modulus 3.6 GPaJoint friction angle 40◦Dilation angle 0◦

A maximum zone size of 15 m was used exceptwhere noted. In all cases, the factor of safetywas estimated by simultaneously reducing therock mass properties and the joint friction untilfailure occurs using the procedure shown in equa-tions (10.1) and (10.2). The safety factor wasassumed to be the reciprocal of the reductionrequired to produce failure. For example, if thestrengths must be reduced by 25% (i.e. 75% oftheir “real” strength) in order to achieve failure,the safety factor is 1.33.

10.4.1 Rock mass failure

Numerical analysis of slopes involving purelyrock mass failure is studied most efficiently usingcontinuum codes such as PHASE2, FLAC orFLAC3D. As mentioned in the previous section,discontinuities are not considered explicitly incontinuum models; rather, they are assumed tobe smeared throughout the rock mass. Assumingthat rock mass shear strength properties can beestimated reasonably, the analysis is straightfor-ward. The process for initially estimating the rockmass properties is often based on empirical rela-tions as described, for example, by Hoek and


–100

200

500

0

300

100

400

200 4001000

Horizontal axis (m)

300 500 600

UDEC (Version 3.20)

Legend

Boundary plotUser defined grid valueDisplacement contour interval = 0.5

Ver

tical

axi

s (m

)

0.51.01.52.02.5

0.0

Figure 10.5 Rock mass failure mode for slope determined with UDEC.

Brown (1997). These initial properties are thenmodified, as necessary, through the calibrationprocess.

Failure modes involve mainly shearing throughthe rock mass. For homogeneous slopes wherethe slide surface is often approximately circular,intersecting the toe of the slope and becom-ing nearly vertical near the ground surface. Thefailure mode for the parameters listed earlier isshown in Figure 10.5. The calculated safety factoris 1.64.

The following is a comparison of a slopestability analysis carried out using limit equilib-rium circular failure analysis (Bishop method)and numerical stability analysis. In Chapter 8, thestability of a benched slope in strong, but closelyfractured, sandstone including a water table andtension crack is described (see Figure 8.19). Therock mass is classified as a Hoek–Brown materialwith strength parameters:

mi = 0.13GSI = 20

σc = 150 MPaDisturbance factor, D = 0.7

The tensile strength is estimated to be 0.012 MPa.For the Bishop’s analysis method, the Mohr–Coulomb strength is estimated by fitting a straightline to the curved Hoek–Brown failure envelopeat the normal stress level estimated from the slopegeometry. Using this procedure, friction angleand cohesion were

φ = 43◦c = 0.145 MPa

The mass density of the rock mass and waterwere 2550 kg/m3 (25.0 kN/m3) and 1000 kg/m3

(9.81 kN/m3) respectively. The phreatic surfaceis located as shown in Figure 8.19. Based uponthese parameters, the Bishop method produces alocation for the circular slide surface and tensioncrack, as shown in Figure 8.19, and a factor ofsafety of 1.39.


In using FLAC to analyze the stability of theslope in Figure 8.19, the slide surface can evolveduring the calculation in a way that is represent-ative of the natural evolution of the physical slidesurface in the slope. It is not necessary to makean estimate for the location of the circular slidesurface when beginning an analysis, as it is withlimit equilibrium methods. FLAC will find theslide surface and failure mechanism by simulat-ing the material behavior directly. A reasonablyfine grid should be selected to ensure that the slidesurface will be well defined as it develops. It is bestto use the finest grid possible when studying prob-lems involving localized failure. Here, a zone sizeof 2 m was used.

The FLAC analysis showed a factor of safety of1.26, with the slide surface closely resembling thatproduced from the Bishop solution (Figure 10.6).However, the tensile failure extends farther upthe slope in the FLAC solution. It is importantto recognize that the limit equilibrium solutiononly identifies the onset of failure, whereas the

FLAC solution includes the effect of stress redis-tribution and progressive failure after movementhas been initiated. In this problem, tensile failurecontinues up the slope as a result of the tensilesoftening. The resulting factor of safety allowsfor this weakening effect.

10.4.2 Plane failure—daylighting andnon-daylighting

Failure modes that involve rigid blocks sliding onplanar joints that daylight in the slope face aremost efficiently solved using analytical methods.For comparison purposes, a UDEC analysis is per-formed for blocks dipping at 35◦ out of the slope.The joints are assumed to have a cohesion of100 kPa and a friction angle of 40◦. The resultingsafety factor is 1.32, which agrees with the ana-lytic value given by equation (6.4) in Chapter 6,assuming that no tension crack forms. The planefailure mode in the UDEC analysis is shown inFigure 10.7.

– 40

20

80

–20

40

0

60

– 40 –20 20 600 40 80 100

Contour interval = 0.002

Max. shear strain increment

Boundary plot

200

Horizontal axis (m)

FLAC (Version 4.00)

Ver

tical

axi

s (m

)

Legend

0 0.004 0.008 0.012 0.016 0.020 0.024

Figure 10.6 Failure mode and tension crack location determined with FLAC for slope in closely fracturedsandstone slope (refer to Figure 8.19).


Cycle 487840

100 200 400 600300

Horizontal axis (m)

500 700

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tical

axi

s (m

)

–100

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500

0

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100

400

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Block plotVelocity vectors

Maximum = 0.043

0.20

UDEC (Version 3.10)

Legend

Figure 10.7 Plane failure mode with rigid blocks determined with UDEC.

If a tension crack does form, then the factorof safety is slightly reduced. Deformable blockswith elastic–plastic behavior are required to formtension cracks within the UDEC analysis. Whendeformable zones are used, the resultant safetyfactor is 1.27, similar to the value of 1.3 given bythe analytic solution. The difference may be thatthe analytic solution assumes a vertical tensioncrack, whereas the UDEC analysis indicates thatthe tension crack curves where it meets the slidingplane (see Figure 10.8).

Similar analyses can be performed for non-daylighting failure planes. In this case, failureinvolves sliding on discontinuities and shearingthrough the rock mass at the toe of the slope, asshown in Figure 10.9. Here, the cohesionless slid-ing planes dip at 70◦ and are spaced 20 m apart.The resultant safety factor is about 1.5.

10.4.3 Wedge failure—daylighting andnon-daylighting

Analyses involving wedge failures are similar tothose involving plane failures, except that the

analyses must be performed in three dimensions.As with plane failure, sliding analysis of day-lighting rigid blocks is best solved using analyticmethods, as described in Chapter 7. Analysesinvolving formation of tension cracks and/or non-daylighting wedges require numerical analysis.Candidate codes include FLAC3D and 3DEC.The plasticity formulation in FLAC3D uses amixed discretization technique and presentlyprovides a better solution than 3DEC in caseswhere rock mass failure dominates. On the otherhand, setting up problems involving more thanone sliding plane in FLAC3D is more difficult andtime consuming than similar problems in 3DEC.

10.4.4 Toppling failure—block and flexural

Toppling failure modes involve rotation and thususually are difficult to solve using limit equilib-rium methods. As the name implies, block top-pling involves free rotation of individual blocks(Figure 9.3(a)), whereas flexural toppling involvesbending of rock columns or plates (Figure 9.3(b)).


X displacement contoursContour interval = 0.2

Cycle 541651

(zero contour line omitted)

Block plot

100 200 400 600300

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500 700

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tical

axi

s (m

)

–100

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500

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300

100

400

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UDEC (Version 3.20)

Legend

0.20.40.60.81.01.21.41.61.8

Figure 10.8 Plane failure mode with deformable blocks determined with UDEC.

Y displacement contoursContour interval = 0.2

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500 700

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tical

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s (m

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UDEC (Version 3.20)

Cycle 1246860 Time 1.286E + 03 sec

Legend

–0.6–0.4–0.2 0.2


Block plot

Figure 10.9 Non-daylighting plane failure mode determined with UDEC.



100 200 400 600300

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Cycle 1153501Time 1.451E + 03 sec

Legend

–10–12

–8–6–4–20


Block plot

Figure 10.10 Forward block toppling failure mode determined with UDEC.

Block toppling occurs where narrow slabsare formed by joints dipping steeply into theface, combined with flatter cross-joints (seeSection 9.4). The cross-joints provide releasesurfaces for rotation of the blocks. In the mostcommon form of block toppling, the blocks,driven by self-weight, rotate forward out ofthe slope. However, backward or reverse top-pling can also occur when joints parallel to theslope face and flatter cross-joints are particularlyweak. In cases of both forward and backwardtoppling, stability depends on the location ofthe center of gravity of the blocks relative totheir base.

Figure 10.10 shows the results of an analysisinvolving forward block toppling. The steep jointset dips at 70◦ with a spacing of 20 m. The cross-joints are perpendicular and are spaced at 30 m.The resultant safety factor is 1.13. Figure 10.11shows the result of an analysis involving back-ward block toppling. In this case, the face-paralleljoints are spaced at 10 m, and the horizontal jointsare spaced at 40 m. The factor of safety for thisfailure mode is 1.7.

Flexural toppling occurs when there is onedominant, closely spaced, set of joints dip-ping steeply into the face, with insufficientcross-jointing to permit free rotation of blocks.The columns bend out of the slope like cantileverbeams. Figure 10.12 shows the results of analysiswith joints spaced at 20 m. The factor of safetyis 1.3, with the safety factor being reduced asthe joint spacing decreases. Problems involvingflexural toppling require finer zoning than prob-lems involving block toppling. Because flexuraltoppling involves high stress gradients across anyrock column, it is necessary to provide sufficientzones to represent accurately the stress gradientsdue to bending. In the modeling of centrifugetests reported by Adhikary and Guo (2000),UDEC modeling required four zones across eachcolumn, resulting in a model with nearly 20,000three-noded triangular zones. In contrast, a finiteelement model with Cosserat plasticity elementsrequired only about 1200, eight-noded isopara-metric quadrilateral elements. Both models pro-duced good agreement with the laboratory results(see also Section 9.5).


Figure 10.11 Reverse(backward) block topplingfailure mode determined withUDEC; arrows showmovement vectors.

10.4.5 Flexural buckling failure

Buckling failures (see Figure 10.13) are alsodifficult to reproduce in numerical modelsbecause of the large number of zones requiredto represent the high stress gradients involved inbuckling. One of the most complete studies onthe topic of numerical analysis of buckling inrock slopes is given by Adhikary et al. (2001),who provide design charts based on numericalanalysis using the proprietary finite element pro-gram AFENA (Carter and Balaam, 1995) and aCosserat material model to simulate behavior offoliated rock.

10.5 Special topics

10.5.1 Reinforcement

Reinforcement is often used to stabilize civilslopes, and occasionally critical mine slopes.

Three different types of reinforcement can berepresented in numerical models:

• fully grouted rock bolts (local reinforcement);• cable bolts; and• end-anchored rock bolts.

The basic formulation for each type of reinforce-ment is discussed briefly.

The local reinforcement formulation considersonly the local effect of reinforcement where itpasses through existing discontinuities. This con-dition immediately implies that some form ofdiscontinuous behavior is being modeled in therock mass. The formulation results from obser-vations of laboratory tests of fully grouted unten-sioned reinforcement in good quality rock withone discontinuity, which indicate that strains inthe reinforcement are concentrated across the dis-continuity (Bjurstrom, 1974; Pells, 1974; Spangand Egger, 1990). This behavior can be achieved



100 200 400 600300

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Cycle 750880Time 1.083E + 03 sec

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–10–12

–8–6–4–20


Block plot

Figure 10.12 Flexural toppling failure mode determined with UDEC.

(a) (b)

Figure 10.13 A schematicrepresentation of slopes in a foliatedrock mass: (a) flexural toppling; and(b) flexural buckling (Adhikary et al.,2001).

in the computational model by calculating, foreach zone, the forces generated by deformation ofan “active length” of the element where it crossesa discontinuity (see Figure 6.9). This formula-tion exploits simple force–displacement relationsto describe both the shear and axial behaviors ofreinforcement across discontinuities. Large sheardisplacements are accommodated by consider-ing the simple geometric changes that developlocally in the reinforcement near a discontinu-ity. Although the local reinforcement model canbe used with either rigid blocks or deformable

blocks, the representation is most applicable tocases in which deformation of individual rockblocks may be neglected in comparison withdeformation of the reinforcing system. In suchcases, attention may be focused reasonably onthe effect of reinforcement near discontinuities.The original description of a local reinforcementmodel is given by Lorig (1985).

In assessing the support provided by rockreinforcement, two components of restraintshould be considered. First, the reinforce-ment provides local restraint where it crosses


m

m

m

Excavation

Reinforcing element (steel)

Grout annulus

Shear stiffness of grout

Axial stiffnessof steel

Slider (cohesivestrength of grout)

Reinforcement(nodal point)

Figure 10.14 Conceptual mechanicalrepresentation of fully bondedreinforcement, accounting for shearbehavior of the grout annulus.

discontinuities. Second, there is restraint to intactrock due to inelastic deformation in the failedregion surrounding an excavation. Such situ-ations arise in modeling inelastic deformationsassociated with failed rock and/or reinforcementsystems such as cable bolts, in which the cementor resin grout bonding agent may fail in shearover some length of the reinforcement. Cable ele-ments allow the modeling of a shearing resistancealong their length, as provided by the shear res-istance generated by the bond between the groutand either the cable or the rock. The cable isassumed to be divided into a number of segmentsof length L, with nodal points located at each seg-ment end. The mass of each segment is lumpedat the nodal points, as shown in Figure 10.14.Shearing resistance is represented by spring/sliderconnections between the structural nodes and therock in which the nodes are located.

End-anchored rock bolts are the simplest tomodel. They simply supply axial restraint to theportions of the model in which they are anchored.The axial stiffness K, is given by

K = AEL

(10.7)

where A is the cross-sectional area of the bolt,E, the modulus of the steel and L the distancebetween the anchoring points

10.5.2 Time-dependent behavior

The issue of time-dependent behavior is dis-cussed in reference to behavior of the west wallof Chuquicamata mine, which experiences largeon-going displacements of the order of 2–4 mper year. The slope behavior is affected by thepresence of a pervasive fault and an adjacentzone of sheared rock near the toe of the slope(Figure 10.15). Deformation of these mater-ials is expressed in toppling further upslope.Previous analyses attempted to estimate safetyfactors for the slope using UDEC. However,difficulties in identifying both a clear point offailure and a failure mode suggested that othercriteria should be considered in assessing theacceptability of west-wall slope designs. Slopedisplacement and displacement rates were con-sidered as other criteria. However, these criteriarequired use of material models that could repres-ent time-dependent behavior. Such models werenot available in programs used to study discon-tinuum slope behavior. The modeling described


P3

P5

N-5500

N-5000

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N-3500

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E-2

000

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000

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N-5500E

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0

E-3

500

E-2

500

E-3

000

Moderateshear zone

West fault

Intense shear zone

Figure 10.15 Plan of Chuquicamataopen pit showing shear zone (modeled asa time-dependent material), and ProfilesP3 and P5.

here demonstrates that the time-dependent beha-vior of the west wall can be reasonably sim-ulated when the sheared zone is representedby a two-component power-law creep modelcombined with a Mohr–Coulomb elasto-plasticmodel. Measured slope movements and changesin displacement rates over a period of six years arecompared to model predictions for two profiles(see Figure 10.16).

Profile P3 is an east–west section located atmine coordinate N3600; it is a good section forcalibration of behavior of the west wall for tworeasons. First, the location of P3 is near the

middle of the west wall and therefore is rep-resentative of conditions in this wall. Second,its orientation is perpendicular to the west walland the geology, allowing representative two-dimensional analysis. Profile P5 is a good sectionfor calibration because mining with steep slopeangles within the shear zone led to a slope fail-ure in February 2002. Both profiles have goodhistorical information about slope movementfrom prism monitoring, as described in the nextsection.

Records of slope movement were availablefrom monitoring records. The prism locations for


Pit advance 1995–2000

S-221-2000

S-223-1996

Prism Start FinishLevel(m)

S-221

S-222

S-223

S-225S-224

S-221

S-222

S-223

S-224

S-225

S-221-2000

S-223-1996

2844.16

2820.18

2672.23

2593.32

2414.34

2852.52

2691.55

28 Mar 98

26 Aug 00

9 Jun 96

14 Feb 01

14 Feb 01

14 Feb 01

14 Feb 01

21 Mar 95

21 Mar 95

21 Mar 95

1 Aug 99

21 Mar 95

12 Mar 00

10 Jul 96

Figure 10.16 Location of movement monitoring prisms in Profile P3 (see Figure 10.15 for location ofProfile P3).

Profile P3 are shown in Figure 10.16, which alsoshows the position of the pit at the end of years1995–2000. The prism records for both profilesshow the following important characteristics:

• horizontal displacements greater than verticaldisplacements, which is characteristic oftoppling behavior;

• decreasing displacement rates from mid-1995to the end of 1999; and

• increasing displacement rates starting near theend of 1999 in Profile P3 and the end of 2000in Profile P5; these periods correspond to thetimes when the shear zone was being mined.

UDEC models for Profiles P3 and P5 are shownin Figures 10.17 and 10.18, respectively, and havethe following features.

• Rock mass behavior for all units except theshear zone is represented by an elasto-plasticmodel, with a bilinear Mohr–Coulomb failuresurface. The bilinear model approximates aHoek–Brown failure surface, and is easier to

use than the non-linear Hoek–Brown failureenvelope.

• Faults and other major discontinuities wereincluded explicitly in the models. Predominatejointing structure was included implicitlythrough a ubiquitous joint model inGranodiorita Fortuna, which is locatedupslope from the sheared zone.

• Water pressures were taken from MINEDW(a three-dimensional finite element developedby Hydrologic Consultants Inc.), and trans-ferred to the UDEC model at yearlyintervals.

• Lithologic units were obtained from Chuquica-mata’s block model and imported into theUDEC model using a recently developedtransfer algorithm.

• Hydrostatic initial in situ stresses wereassumed. Initial in situ stresses with deviatoriccomponents would induce creep under initialpre-mining conditions, a condition that is notbelieved to be correct.

• Small deformation logic was used to avoidproblems with poor zone geometry resultingfrom large deformations within the shear zone.


Intense shear zone

Figure 10.17 UDEC model for Profile P3 with lithology, discontinuities and annual pit geometries.

Moderate shear zone

Figure 10.18 UDEC model of Profile P5 with lithology, discontinuities and annual pit geometries(see Figure 10.15 for location of Profile P5).

Creep behavior was believed to be concentratedwithin the sheared zone located just to thewest of the West Fault. In the model discussedhere, the behavior of the sheared rock wasrepresented using a viscoplastic model thatcombined the behavior of the viscoelastic two-component Norton Power Law model andthe Mohr–Coulomb elasto-plastic model. Thestandard form of the Norton Power Law (Norton,1929) is

εcr = Aσn (10.8)

σ =(

32

)1/2 (σdijσ

dij

)1/2 (10.9)

where εcr is the creep rate, σdij is the deviatoricpart of σij, and A and n are material propertiesthat were found by calibration.

Modeling was performed for conditions repres-entative of the years 1996 through 2002. Initially,the model was brought to equilibrium under con-ditions representative of January 1996. At thispoint, the creep model was turned on and run forone year of simulated time. For each subsequentyear, new water pressures and slope geometrywere introduced, and the model was run foranother year of simulated time.

Calibration was performed by adjusting thepower law parameters until reasonable agree-ment was reached between the prism records and


Prism east

UDEC east

UDEC vertical

Prism vertical

Time–10

–5

0

5

Dis

plac

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t (m

)

10

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20

May-00Aug-99Dec-98Apr-98Aug-97Dec-96 Jan-01 Sep-01Mar-96

Figure 10.19 Comparison of movement record for prism S-223 records with UDEC results for Profile P3;A = 1e–23 and n = 2.5 (equations (10.8) and (10.9)).

Prism

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Figure 10.20 Comparison of actual prism movement record with UDEC results for Profile P5; A = 1e–24 andn = 2.5 (equations (10.8) and (10.9)).

the numerical results. Representative results areshown in Figures 10.19 and 10.20 for P3 and P5,respectively. The sharp increase in displacementP5 resulted in slope failure in the upper portionof the slope.

In the analyses reported here, it has beenassumed that creep behavior initiates as soon asdeviatoric stresses are present, and the state ofstress is not hydrostatic. However, it is more

likely that creep behavior starts after a thresholddeviatoric stress is reached. Evidence for this canbe seen by examining the pre-mining in situ stressstate, which has been shown through measure-ments to include deviatoric stresses. There was noevidence of creep behavior in the pre-mining con-dition. Thus, it can be concluded that a thresholddeviatoric stress exists below which no creepbehavior occurs.


10.5.3 Dynamic analysis

Traditional approaches to dynamic analysis arebased on a pseudo-static approach in whichthe effects of an earthquake are representedby constant horizontal and/or vertical accelera-tions. The first explicit application of the pseudo-static approach to the seismic slope stability hasbeen attributed to Terzaghi (1950). The applic-ation of horizontal and/or vertical accelerationscan be made in limit equilibrium methods andnumerical methods alike. The results of pseudo-static analyses depend on the value of the seis-mic coefficient as discussed in Section 6.5.4.Difficulty in assigning appropriate pseudo-staticcoefficients and in interpretation of pseudo-static safety factors, coupled with the advanceof numerical models has provided an alternat-ive to the use of the pseudo-static approachfor seismic slope stability analyses. Numericalmethods, in addition to the Newmark methoddiscussed in Section 6.5.5, allow permanent slope

deformations resulting from seismic excitation tobe computed.

Both finite-element and finite-differenceapproaches can be used to compute permanentdeformations. Typical analyses involve applica-tion of a seismic record to the base of a modeland propagating the wave through the model.Small amounts of damping are sometimes appliedto account for real energy losses that are notrepresented by either the joint behavior or therock mass behavior.

Although there are no documented cases oflarge-scale failures of open pits under seismicloads, there are many instances of failure of nat-ural slopes during earthquakes (see Section 6.5.1).In open pits, smaller-scale failures comprisingrock fall and bench-scale structurally controlledfailures may occur under severe shaking. Wheresuch failures are an operational hazard, mitig-ation can usually be provided by suitable catchbench configurations.

Book CoverTitleCopyrightContents1 Principles of rock slope design2 Structural geology and data interpretation3 Site investigation and geological data collection4 Rock strength properties and their measurement5 Ground water6 Plane failure7 Wedge failure8 Circular failure9 Toppling failure10 Numerical analysis11 Blasting12 Stabilization of rock slopes13 Movement monitoring14 Civil engineering applications15 Mining applicationsAppendix I Stereonets for hand plotting of structural geology dataAppendix II Quantitative description of discontinuities in rock massesAppendix III Comprehensive solution wedge stabilityAppendix IV Conversion factorsReferencesIndex

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Rock Slope Engineering: Civil and Mining, 4th Edition Analysis.pdf · rock bounded by speciﬁed slide planes. In con-trast, this chapter discusses numerical analysis methods to calculate