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CHAPTER 6: SLOPE STABILITY ANALYSIS BY NUMERICAL MODELLING

6.0 Introduction

Numerical models are mathematical models that use some sort of numerical time-stepping procedure

to obtain the models behavior over time. These are computer programs that represent the mechanical

response of a rock mass subjected to a set of initial conditions such as in situ stresses and water

levels, boundary conditions and induced changes such as slope excavation. Numerical models divide

the rock mass into zones. Each zone is assigned a material model and properties. The result of a

numerical model can be extrapolated confidently outside its database in comparison to empirical

methods in which the failure mode is explicitly defined. It can also incorporate geologic features

such as faults and ground water, providing more realistic approximations of behavior of real slopes

than analytical models.

Numerical modelling techniques have been widely used to solve complex slope problems, which

otherwise, could not have been possible using conventional techniques. These models are used to

simulate rock slope as well soil slope with complex conditions. All rock slopes involve many

discontinuities such as joint, fault, bedding plane, etc. Precise representation of discontinuities in

numerical models depends on the type of model.

Numerical methods of analysis used for rock slope stability investigations may be divided into three

approaches:

• Continuum modeling

• Discontinuum modeling

• Hybrid modeling

6.1 Continuum modeling

Continuum modeling is best suited for the analysis of slopes that are comprised of massive, intact

rock, weak rocks, and soil-like or heavily fractured rock masses. Figure1 shows the discritiesed view

of slope by continuum code using phase2 software. Continuum codes assume that material is

continuous throughout the body. Discontinuities are treated as special cases by introducing interfaces

between continuum bodies. This model cannot handle many intersecting joints. It can typically

simulate less then 10 non-intersecting discontinuities. The fundamental importance to continuum

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models and deformable blocks in discrete element models is representation of the rock mass

behaviour. Discrete fractures such as faults and bedding planes can be incorporated in most

continuum models. However, these models cannot be used to simulate highly fracture rock mass.

Finite difference, finite element and boundary element methods are based on this modeling theory. In

this method, the problem domain is discretized into a set of sub-domains or elements. The solution

procedure may be based on numerical approximations of the governing equations. Two-dimensional

continuum codes assume plane strain conditions, which are frequently not valid in inhomogeneous

rock slopes with varying structure, lithology and topography.

Complex behaviour of slope can be modeled using continuum codes. Groundwater, pore pressures

and dynamic interaction can also be simulated. It requires input properties such as constitutive model

(e.g. elastic, elasto-plastic, creep etc.), groundwater characteristics, shear strength of surfaces and in

situ stress state (figure 2 & 3). During modeling, effects of boundary, mesh aspect ratios, symmetry,

hardware memory restrictions are important factors. Figure1 shows the discritiesed view of mne

dump slope in 3D by continuum method. Some softwares based on continuum modeling like Phase2

(rocscience), FLAC2D, FLAC3D (Itasca 1997) and VISAGE (VIPS, 2001), PLAXIS () are well

suited for slope stability problems.

Figure 1: Finite-element model showing descritised slope.

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Figure 2: Finite-element mesh of a rock slope having a coal seam.

Figure 3: Simulation of rain water infiltration by Finite-element method.

Figure 4: simulation of mine waste dump by FLAC based on finite difference method

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6.2 Discontinuum Modeling

Discontinuum modeling methods treat the rock slope as a discontinuous rock mass by considering it

as an assemblage of rigid or deformable blocks. The analysis includes sliding and opening of rock

discontinuities controlled by the normal and shear stiffness of joints. It allows the deformation and

movement of blocks relative to each other so it can model complex behaviour and mechanisms. It

requires representative slope and discontinuity geometry, intact constitutive criteria, discontinuity

stiffness and shear strength, groundwater characteristics and the in situ stress state. The major

limitation of discontinuum modeling is that it requires representative discontinuity geometry

(spacing, persistence, etc.) along with joint data and properties of each block. Discontinuities divide

the problem domain into blocks that may be either rigid or deformable while continuum behavior is

assumed within deformable blocks. The most widely used discrete element codes for slope stability

studies are UDEC (Universal Distinct Element Code; Itasca Consulting Group, 2000) and 3DEC (3-

Dimensional Distinct Element Code; Itasca Consulting Group, 2003). Several variations of the

discrete element methodology are:

• Distinct element method;

• Discontinuous deformation analysis

• Particle flow codes.

Discontinuum methods treat the problem domain as an assemblage of distinct, interacting bodies or

blocks subjected to external loads and expected to undergo significant motion with time. This

methodology is collectively referred as the discrete element method (DEM) (Figure 5 to 7). The

development of discrete-element procedures represents an important step in the modelling and

understanding of the mechanical behaviour of jointed rock masses. This method of analysis permits

sliding along and opening or closure between blocks or particles. The equation of dynamic

equilibrium for each block in the system is formulated and repeatedly solved until the boundary

conditions and laws of contact and motion are satisfied. The method thus accounts for complex non-

linear interaction phenomena between blocks.

UDEC (11) uses a force-displacement law specifying interaction between the deformable joint

bounded blocks and Newton’s second law of motion, providing displacements induced within the

rock slope. It is particularly well suited to problems involving jointed media and has been used

extensively in the investigation of both landslides and surface mine slopes. The influence of external

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factors such as underground mining, earthquakes and groundwater pressure on block sliding and

deformation can also be simulated.

The discontinuous deformation analysis, DDA, developed by Shi (18) has also been used with

considerable success in the modelling of discontinuous rock masses, both in terms of rockslides (19)

and rockfalls (20).

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DDA) is a type of discrete element method originally proposed by Shi in 1988. DDA is somewhat

similar to the finite element method for solving stress-displacement problems, but accounts for the

interaction of independent particles (blocks) along discontinuities in fractured and jointed rock

masses. DDA is typically formulated as a work-energy method, and can be derived using the

principle of Minimum Potential Energy (e.g., Shi) or by using Hamilton's principle. Once the

equations of motion are discretized, a step-wise linear time marching scheme in the Newmark family

is used for the solution of the equations of motion. The relation between adjacent blocks is governed

by equations of contact interpenetration and accounts for friction. DDA adopts a stepwise approach

to solve for the large displacements that accompany discontinuous movements between blocks. The

blocks are said to be "simply deformable". Since the method accounts for the inertial forces of the

blocks' mass, it can be used to solve the full dynamic problem of block motion.

DDA models a discontinuous material as a system of individually deformable blocks that move

independently without interpenetration (Shi, 1988 and 1993). Its formulation is based on a dynamic

equilibrium that considers the kinematics of individual blocks as well as friction along the block

interfaces. The displacements and deformations of the blocks are the result of the accumulation of a

number of small increments, corresponding to small time steps. The transient formulation of the

problem, which is based on minimization of potential energy, makes it possible to investigate the

progression of block movements with time.

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Particle flow code allows the rock mass to be represented as a series of spherical particles that

interact through frictional sliding contacts. Clusters of particles may be bonded together through

specified bond strengths in order to simulate joint bounded blocks. High stresses induced in the rock

slope breaks the bonds between the particles simulating the intact fracture of the rock in an

approximate manner.

Figure 5: Simulation of highly intersecting joint by discontinuum code

Figure 6: Rock slope distinct-element model showing discretization of geometry blocks into finite-

difference elements.

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Figure 7: 3D Simulation of rock slope by 3Dec software based on finite difference method

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6.3 Hybrid approach

Hybrid approaches are increasingly being adopted in rock slope analysis. This may include combined

analyses using limit equilibrium stability analysis and finite-element groundwater flow and stress

analysis as adopted in GEO-SLOPE (Geo-Slope 2000). These models have been used for a

considerable time in underground rock engineering including coupled boundary finite element and

coupled boundary-distinct element solutions. Recent advances include coupled particle flow and

finite-difference analyses using PFC3D and FLAC3D (Itasca 1999).

The coupling of finite-distinct element codes, for example in ELFEN (Rockfield 2001), allows

modeling of both the intact rock behaviour and the development and behaviour of fractures (figure

8). These methods use a finite element mesh to represent either the rock slope or joint bounded

blocks coupled together with discrete elements to model deformation involving joints. If the stresses

within the rock slope exceed the failure strength within the finite-element continuum, a discrete

fracture is initiated. Adaptive remeshing allows the propagation of the cracks through the finite-

element mesh to be simulated. Coupled finite-/distinct element models are able to simulate intact

fracture propagation and fragmentation of jointed and bedded rock (figure 9). However, complex

problems require high memory capacity, comparatively little practical experience in use and

requirement of ongoing calibration are major constraints.

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Figure 8: Hybrid finite-/discrete-element rockslide analysis showing several progressive stages of

brittle failure (from Eberhardt et al. 2002).

Figure 9: 3D Simulation of Rock fall using hybrid approach

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The important points and basic issues that should be considered during the numerical modeling of

slope are as follows:

6.4 Important considerations

6.4.1 Two-dimensional analysis versus three-dimensional analysis

The slope can be simulated in 2D or 3D by numerical modeling (figure 10 & 11). It depends on

many factors such as time require for simulation, critical parameter, requirement of simulation, field

condition and computer configuration. Most design analyses for slopes assume a two- dimensional

geometry comprising a unit slice through an infinitely long slope under plane strain conditions, i.e.

the radius of both the toe and the crest are assumed to be infinite. However, three-dimensional

analyses are required when the direction of major geological discontinuities does not strike within

20–30◦ of the strike of the slope or the distribution of geomechanical units varies along the strike of

the slope. This also becomes necessary when the slope geometry in plan cannot be represented by

two-dimensional analysis, which assumes axisymmetric or plain strain condition.

Figure 10: 3D Simulation by Ansys software based on Finite element method

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Figure 11: 2D Simulation by Geoslope software based on Finite element method

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6.4.2 Continuum versus discontinuum models

In general, slope stability problems involve discontinuities at one scale or another. If rock mass of

slope can be represented as an equivalent continuum, continuum models should be used to solve

these types of problems (figure 12). Therefore, many analyses begin with continuum models. If the

slope under consideration is unstable without structure, there is no point in going to discontinuum

models (figure 13).

In contrast to the above, if the discontinuities can be represented in terms of equivalent rock mass

properties and the number of intersecting joints are large, than discontinuum code may provide better

solution. However, these models require in-depth knowledge of the properties of discontinuities of

rock and their mutual interaction and are very difficult to determine. Therefore, one can go for

simulation of rock slope mass by using continuum code in many cases.

Figure 12: 2D simulation of bench slope by FLAC based on finite difference method

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Figure 13: 3D simulation of slope 3DEC software based on discontinuum modeling

6.4.3 Selecting appropriate zone size

The next step in the process is to select an appropriate zone size. The finite difference zones assume

that the stresses and strains within each zone do not differ with position within the zone. It is

necessary to use relatively fine discretizations in order to capture stress and strain gradients within

the slope adequately. Finite element programs using higher-order elements require less zones than the

constant strain/constant stress elements common in finite difference codes. As shown in figure 14 the

coarse model(a) yields a very crude approximation of the factor of safety. A properly descritized

model as shown in figure 14(b) provides a fair estimate of the safety factor without compromise is

solution accuracy and time as compared to an intensively descritized model (c).

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Figure 14: Shows the different view discritized view of internal dump slope

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6.4.4 Initial conditions

Initial conditions are those conditions that existed prior to mining. The important initial conditions at

mine sites are the in situ stress field and the ground water conditions. Numerical models can be used

to evaluate the effect of in situ on stability of rock and soil slopes. Three-dimensional models should

be used for solving slope stability problem under anisotropic in situ conditions.

6.4.5 Boundary conditions

Boundaries are either real or artificial. Real boundaries in slope stability problems correspond to the

natural or excavated ground surface that is usually stress free. All problems in geomechanics

including slope stability problems require that the infinite extent of a real problem domain be

artificially truncated to include only the immediate area of interest. Figure 15 shows typical

recommendations for locations of the artificial far-field boundaries in slope stability problems.

Artificial boundaries can be of two types: prescribed displacement or prescribed stress. Prescribed

displacement boundaries inhibit displacement in either the vertical or the horizontal direction, or

both. Prescribed displacement boundaries are used to represent the condition at the base and the toe

of the slope model. Displacement at the base of the model is always fixed in both the vertical and

horizontal directions to inhibit rotation of the model.

The far-field boundary location and its condition must be specified in any numerical model for slope

stability analyses. The general notion is to select the far-field location so that it does not significantly

influence the results. In most slope stability studies, a prescribed-displacement boundary is used.

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Figure 15: Typical recommendations for locations of artificial far-field boundaries in slope stability

analyses.

6.4.6 Water pressure

The effect of water pressure in reducing effective stresses is very important in slope stability analysis.

As the effect of various assumptions regarding specification of pore pressure distributions in slopes is

not well understood, two methods are commonly used to specify pore pressure distributions within

slopes. The most rigorous method is to perform a complete flow analysis, and use the resultant pore

pressure in the stability analyses (figure 16). A less rigorous but more common method is to specify a

water table, and the resulting pore pressures are given by the product of the vertical depth below the

water table, the water density and gravity.

The water table method under-predicts the actual pore pressure concentrations near toe of a slope,

and slightly over-predicts the pore pressure behind toe by ignoring the inclination of equipotential

lines. Seepage forces must also be considered in the analysis. Flow analysis automatically accounts

for seepage forces resulting due to difference in hydraulic gradient between any two points at the

elevation.

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Figure 16: Simulation of rain water infiltration and generation of water table

6.4.7 Excavation sequence

Simulating excavations in numerical models poses no conceptual difficulties. However, the amount

of effort required to construct a model depends on the number of excavation stages to be simulated.

The most accurate solution is obtained using the largest number of excavation steps, because the real

load path for any zone in the slope is closely followed (Figure 17). A reasonable approach regarding

the number of excavation stages has evolved over the years. Using this approach, only one, two or

three excavation stages are modeled. For each stage, two calculation steps are taken. In the first step,

the model is run elastically to remove any inertial effect caused by sudden removal of a large amount

of material. Second, the model is run allowing plastic behavior to develop. Following this approach,

reasonable solutions to a large number of slope stability problems have been obtained.

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Figure 17: Show the sequential excavation

Important factors that control the stability of rock slope are jointed discontinuities and water whereas,

in soil the cohesion and water. Numerical modeling depends on both site conditions and the potential

mode of failure. Numerical techniques can be applied to complex translational rock slope

deformations where step-path failure necessitates degradation and failure of intact rock bridges along

basal, rear and lateral release surfaces.

Initiation or trigger mechanisms may involve sliding movements which can be analysed using limit

equilibrium theory. However, creep in rock, progressive deformation, and extensive internal

disruption of the slope mass cannot determined by using limit equilibrium approach. The factors

initiating eventual sliding may be complex and not easily allowed for in simple static analysis.

Recent advances in the characterization of complex rock slope deformation and failure using

numerical techniques have demonstrated significant potential for broadening our understanding of the

mechanisms and the associated risk. Numerical modelling of slopes is now used routinely in the civil

and mining engineering sectors as well as in academic research. The major benefits of numerical

modeling are that

• Both the stress and the displacements can be calculated,

• Different constitutive relations can be employed.

The advantages of a Numerical approach to slope stability analysis over traditional limit equilibrium

methods can be summarized as follows:

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(a) 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. Numerical method preserve global equilibrium until failure is reached.

(c) If realistic soil compressibility data are available, numerical solutions can give information about

deformations at working stress levels.

(d) Numerical methods are able to monitor progressive failure including overall shear failure.

Basic outline in simulating the slope problem using numerical modeling

• Define the problem

• Understand the failure mechanism

• Select appropriate method

• Model the problem

• Define the boundary conditions

• Initial in situ conditions

• Input the material properties

• Solve the problem

• Analysis the results

6.5 Important failure indicators in slope analysis

6.5.1 Factor of Safety

In conventional limit equilibrium methods, the factor of safety is defined as the ratio of resisting

movements to the overturning movement. It can also be defined as the ratio of actual shear strength to

the assumed value of reduced shear strength leading to failure. When the uncertainty and the

consequences of failure are both small, it is acceptable to use small factor of safety, of the order of 1.3

or even smaller in some circumstances. However, when the uncertainties or the consequences of failure

increase, larger factor of safety is necessary. Large uncertainties coupled with large consequences of

failure represent an unacceptable condition irrespective of the calculated value of the factor of safety.

Typical minimum acceptable value of factor of safety is about 1.3 for end of construction and

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multistage loading, 1.5 for normal long-term loading conditions, and 1.1 to 1.3 for rapid drawdown in

cases where it represents an infrequent loading condition (ref). In cases where rapid drawdown

represents a frequent loading condition, as in pumped storage projects, the factor of safety should be

higher. The use of a factor of safety greater than 1.5 for static analyses is recommended. Fractured or

jointed cemented slope could be analyzed using peak strength parameters which are derived from high

quality samples of unfractured material (Hoek and Bray 1981).

The shear strength reduction technique has two advantages over the conventional approach. The critical

failure surface is found automatically and it is not necessary to specify the shape of the failure surface.

In general, the failure mode of slopes is more complex than simple circles or segmented surfaces.

Further, numerical methods automatically satisfy the translational and the rotational equilibrium

conditions, whereas, not all the limit equilibrium methods satisfy these conditions. To 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 (φ ) are

reduced for each trial using equations (1) and (2). If multiple materials are present, reductions are 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.

CF

C trialtrial 1

= ….(1)

( )φφ tan1arctan trialtrial

F=

…..(2)

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6.5.2 Unbalanced Force

A grid point in a numerical model is surrounded by the adjoining 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

(figure 18). A very low value of unbalanced forces indicates balance of forces at all grid points,

however, steady plastic flow may occur without acceleration. In order to distinguish between these two

conditions, other indicators of slope failure should also be examined.

Figure 18: unbalance force solving slope stability by FLAC based on finite difference method

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6.5. 3 Gridpoint Velocities

The gridpoint velocities may be assessed either by plotting the complete field of velocities or by

selecting certain key points in the grid and tracking their velocities with histories/time solution. 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 (figure 18). 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.

Figure 18: Result of slope in terms of Gridpoint Velocities and velocity vector

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6.5.4 Plastic Indicators

For plasticity models, the those zones/elements satisfying the yield criterion can be displayed. 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. In such cases it is important to examine

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 (figure 19).

Figure 19: Simulation Result of slope in terms of plastic and yield zones or elements

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6.5.5 Displacement

An unstable model is usually characterized by a non-zero, often fluctuating, maximum unbalanced

force, as well as increasing velocities and displacements (figure 19). The model can also collapse due to

displacements becoming very large, thus distorting the individual elements badly and prohibiting

further timestepping.

Figure 19: Result of slope in terms of X and Y displacement

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6.5.6 Failure Surface

Once unstable or steady plastic flow has been identified, the location of 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 where failure surface can develop. By looking at the

displacement pattern in the model, a more precise estimate can be made (figure 20).

Figure 20: shear strain rate in slope after simulation by finite element method