Stability analysis of open pit slope by finite difference method

IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
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Volume: 03 Issue: 05 | May-2014, Available @ http://www.ijret.org 326
STABILITY ANALYSIS OF OPEN PIT SLOPE BY FINITE DIFFERENCE
METHOD
K. Soren 1
, G. Budi 2
, P. Sen3
1
Research Scholar, Mining Engineering, Indian School of Mines, Jharkhand, India
2
Assistant Professor, Mining Engineering, Indian School of Mines, Jharkhand, India
3
Professor, Mining Engineering, Indian School of Mines, Jharkhand, India
Abstract
Open pit slope stability analysis has been performed to assess the slope deformation, state of stresses at critically instable failure
zones, different failure modes and safe & functional design of excavated slopes using the numerical modeling technique, finite
difference method. The purpose of slope stability analyses by numerical modeling in rock mechanics is not only to provide specific
values of stress and strain at specific points but is also to enhance our understanding of the processes involved to finding instable
zones, investigation of potential failure mechanisms, designing of optimal slopes with regard to safety, reliability, economics, and
designing possible remedial measures. As part of the rock mass, the discontinuities inherently affect the strength and the
deformational behavior of slopes in rock mass containing sets of ubiquitous joints. In this paper the finite difference method, FLAC
(Fast Lagrangian Analysis of Continua) of numerical modeling technique is used to predict the stress-strain behavior of pit slope and
to evaluate the stability analysis of open pit slope. Ubiquitous joint model, which is an anisotropic plasticity model that includes weak
planes of specific orientation embedded in Mohr-Coulomb solids, has been adopted to account for the presence of weak planes, such
as weathering joints, bedding planes, faults, joints in FLAC Mohr-Coulomb model. Thus by studying the distribution of stress,
displacement and factor of safety, the stability analysis of pit slope has been evaluated and suggestions based on the numerical
analysis has been presented.
Keywords: open pit slope stability; Numerical model; Ubiquitous joint model; Jointed rock mass; Finite Difference
Method.
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1. INTRODUCTION
The primary objectives of the open pit slope stability analyses
are performed to investigate the pit slope stability conditions,
potential failure mechanism, slope sensitivity or susceptibility
and to design optimum pit slope angles in terms of safety,
reliability and economically profitable. In the process of
stability analysis of open pit slopes in jointed rock mass, a
number of steps and levels of analyses are required, from local
bench design to overall stability of the high wall open pit rock
slopes. This process requires the use of a variety of methods of
analyses and software ranging from limit equilibrium methods
to more involved numerical analyses methods such as distinct
elements, finite elements and finite difference methods which
can capture detailed geology input parameters and different
types of failure modes. Stability of open pit slope depends on
geometry of slope, rock mass characteristics and shear
strength behaviour of the joints [1]. Since a rock mass is not a
continuum, its behaviour is dominated by discontinuities such
as joints, bedding planes and faults. In general, the presence or
absence of discontinuities has a profound influence on the
stability of rock slopes.
The discontinuities such as various geological structures
joints, weak bedding planes, faults, weak schistocity planes
and weak zones has been long recognized in rock mechanics
which significantly influence the response of rock masses to
loadings and surface excavations [2]. Generally, at smaller
scales discontinuities exert greater influence on behaviour than
do intact rock properties. In small slopes, failure mechanisms
such as planar and wedges, which are controlled by joints, are
common. Of the various numerical methods used for the stress
analysis today, the family of Discrete Element Methods
(DEMs) and Discontinuous Deformation Analysis (DDA)
have been considered to be the well-suited to the problems of
blocky and joint network rock masses. Recently however, it
has been demonstrated that the Finite Element Method (FEM)
with explicit representation of discontinuities with joint
elements is a credible alternative [3, 4, 5].
The objective of this paper is to present the numerical analysis
method of FDM, FLAC software which has been used for the
optimization of the open pit slope design and the analysis of
different modes of slope failures occurred in an open pit mines
have also been discussed using the FLAC slope models of the
various slope failure mechanism. This paper also deals with
the evaluation of factor of safety of slope by FLAC software,

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FLAC/Slope using shear strength reduction method in order to
understand the instability failure mechanism of open pit slope
and stress deformation response of slope in a typical open pit
mine. Through the analysis of various modes of slope failure
examples in open pit mine, it shows how these mechanisms
depend on slope geometry and change with increasing slope
angles.
2. NUMERICAL MODELLING
Numerical modelling technique is an efficient method that is
widely used in various fields of science and rock engineering.
Numerical modeling for rock mechanics has been developed
for the design of rock engineering structures in or on it at
different circumstances and for different purposes. Numerical
modelling techniques provide an approximate solution to
problems which otherwise cannot be solved by conventional
methods, e.g. complex geometry, material anisotropy, non-
linear behaviour, in situ stresses etc. Numerical
analysis allows for material deformation and failure;
modelling of pore pressures, creep deformation, dynamic
loading, assessing effects of parametric variations
etc. Numerical modelling is used to investigate a variety of
problems in rock slope behaviour, underground mining and
tunnelling engineering which can assist the geo-technical
engineer in designing underground excavations, surface
excavations and support systems. If extensive geological and
geo-technical data are available, then comprehensive
predictions of deformations, stability and support loads can be
made by numerical stress analyses and the model can be used
to perform parametric studies, providing insight into the
possible range of responses of a system, given the likely
ranges for the various parameters. This understanding of the
key parameters can then help set priorities for site
investigation and material testing, which in turn will produce
data that are used in design of slopes.
Due to the widespread availability of powerful computers, the
FDM with explicit modelling of the behaviour of individual
joints can be used for practical engineering in jointed rock
masses. This has also been facilitated by the development of
techniques for generating networks of discrete fractures, and
the development of the Shear Strength Reduction (SSR)
method. SSR analysis [6, 7, 8] allows factors of safety of
slopes to be calculated with numerical methods. Although
FDM-based SSR analysis is an alternative to conventional
limit equilibrium methods (LEM) in many cases, its ability to
readily combine slip along joints with failure through intact
material offers several advantages in the modelling of jointed
rock mass problems. Perhaps, the greatest benefit of FDM-
based SSR analysis is that it can automatically determine the
broad variety of failure mechanisms with no prior assumptions
regarding the type, shape or location.
Strength reduction FLAC method has been used to analyze
slope stability which can be judged by reducing residual shear
strength, changing yield strength criterion of rocks, calculating
the stress and strain of slope rocks, checking the displacement
development of landslide [9, 10]. Application of limit
equilibrium method to solve 3D problems is rather limited due
to several simplifying assumptions [11, 12, 13]. It must be
noted, however, that an increasing number of investigators use
3D numerical calculations for estimating slope stability [14,
15]. The factor of safety (FoS) for slope may be computed by
reducing shear strength of rock or soil in stages until the slope
fails. This method is called shear strength reduction technique
(SSR). FLAC code is often applied for estimating factor of
safety for rock slopes [16, 17a,b] or even foliated rock slopes
[18]. It is also applied in evaluating stability of soft rock slope
perforated by underground openings [19]. FLAC is also
widely used for analyzing stability of soil slopes [14, 15].
Sometimes FLAC is even used for slope stability engineering
in combination with other methods. SSR technique is often
used with FEM to solve quite sophisticated problems such as
estimating stability of slope reinforces by piles [20, 21] or
slope with horizontal drains [22]. A good overview of FEM
application for slope stability engineering may be found in
[23]. Advantages and disadvantages of SSR and LEM are
presented in [24, 25]. The majority of investigators prefer
using FEM or FDM for estimation of factor of safety for
slopes.
With regard to the analysis of slopes in a homogeneous rock
mass, the use of rock mass classification methods has become
common for the determination of rock mass strength and
deformation parameters. The rock mass classification numbers
Q [26] and RMR [27] and, more recently, the Geological
Strength Index GSI [28] have been correlated with the rock
mass modulus, rock mass strength parameters both (Hoek-
Brown and Mohr-Coulomb) and the unconfined compressive
strength of the rock mass. Correlations of significance are with
the rock mass deformation modulus [29] and with the rock
mass strength parameters. On the basis of [30] and [31] the
rock mass input parameters on rock mass classification data,
and made use of a finite difference method to evaluate the
stability of rock slopes in open pit mines. Numerical modeling
is finding increasing application, particularly for defining
potential failure modes. However, some degree of calibration
is required before the numerical models can be considered
predictive in the design sense. Prediction of behaviour, which
is very significantly involved in the analysis and design of
open pit mine slopes, requires a very thorough understanding
of the mechanisms of deformation and behaviour of the
slopes.
3. OPEN PIT SLOPE FAILURE MECHANISMS
A realistic design of open pit rock slopes requires (1) a good
understanding of the dominant failure mechanisms in open pit
rock slopes, and (2) a design tool that can be used to assess the
stability of a slope subjected to various failure mechanisms.
Methods for jointed rock slope stability analysis span a very
wide range in the simplest form; they can be purely empirical
guideline. More complex analysis methods can explicitly

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account for slope geometry, geological structure, rock stress
conditions, geo-hydrological conditions and rock mass
characteristics. In this group, one finds both limit equilibrium
methods and numerical modeling. The basis for design must
be an understanding of the failure mechanisms in jointed rock
slopes. Unfortunately, failure mechanisms of high rock slopes,
especially in hard jointed rock, are generally poorly
understood and/or known [32].
3.1 Pit Slope Stability Analysis
In this section, typical pit slope stability analyses for a variety
of failure modes have been discussed. The objective of this
section is to show how numerical models can be used to
simulate the pit slope behavior and compute safety factors for
typical field problems. Modeling issues important in each of
the following failure modes of jointed rock masses such as
plane failure, toppling failure (block and flexural) and circular
failure. Plane failure modes that involve rigid blocks sliding
on planar joints that daylight in the slope face and toppling
failure modes involve rotation and thus usually are difficult to
solve using limit equilibrium methods. Block toppling
involves free rotation of individual blocks, whereas flexural
toppling involves bending of rock columns or plates. Block
toppling occurs where narrow slabs are formed by joints
dipping steeply into the face, combined with flatter cross-
joints. The cross-joints provide release surfaces for rotation of
the blocks. In the most common form of block toppling, the
blocks, driven by self-weight, rotate forward out of the slope.
Flexural toppling occurs when there is one dominant, closely
spaced, set of joints dipping steeply into the face, with
insufficient cross-jointing to permit free rotation of blocks
(Figure 6). The columns bend out of the slope like cantilever
beams as shown in (Figure 4) in the case of toppling failure.
Following are the different modes of slope failure in open pit
mines.
Fig.1 Schematic diagram depicting Circular failure in
weak/weathered rock of an open pit mines.
Fig. 2 Schematic diagram depicting circular failure in heavily
jointed rock mass
Fig. 3 Schematic diagram depicting the Plane failure
mechanism in open pit
Fig. 4 Schematic diagram depicting the Toppling failure
mechanism in open pit
(Block type) mechanism in open pit.

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(Flexural type) mechanism in open pit mines.
The three main components which are to be considered for an
open pit slope design in the view of pit slope stability
conditions are as follows (Figure 7). First, the overall pit slope
angle from crest to toe that incorporates all ramps and
benches. Second, the inter-ramp angle is the slope (i.e the
angle of crest of the ram to the toe of the slope), or slopes,
lying between each ramp that will depend on the number of
ramps and their widths. Third, the face angle of individual
benches depends on vertical spacing between benches, or
combined multiple benches, and the width of the benches
required to contain minor rock falls [33].The factors that may
influence stability of slope designing conditions are the slope
height, geology, rock strength, ground water pressures and
damage to the face by blasting etc.
Fig.7 Typical open pit slope geometry showing relationship
between overall slope angle, inter-ramp angle and bench
geometry.
4. NUMERICAL ANALYSIS
4.1 Finite Difference Method FLAC/Slope
FLAC/Slope is a numerical modelling code which is based on
finite difference method for advanced geotechnical analyses of
soil, rock, and structural support in two and three dimensions.
It is used in analyzing, testing, and designing of pit slopes by
geotechnical, civil, and mining engineers. It is designed to
accommodate any kind of geotechnical engineering projects
where continuum analysis is necessary [34]. FLAC/Slope
utilizes an explicit finite difference formulation that model
complex behaviors are not readily suited to FEM codes, such
as problems that consist of several stages, large displacements
and strains, non-linear material behavior and unstable systems.
FLAC/Slope is an abbreviation of calculated software “Fast
Lagrangian Analysis of Continua”. The open pit slope has
been simulated using FLAC/Slope (version 5.0) developed by
Itasca consulting group [35].
4.2 Arithmetic Introduction for FLAC/Slope
FLAC applies explicit finite difference method substituting for
concealed finite difference method which was originally
extensively used. Programming divides the medium of
computing domain as several 2-dimension units which are
connected by nodes with each other [36, 37]. The difference
between the gridding divisions and the finite method is that
the grids are divided as physical grid and mathematical grid,
and the two kinds are mutual projection.
4.3 Generation of Finite Difference Mesh
FLAC has been used for analyzing the movements of rock
mass and deformable analysis of slope failure in a typical open
pit mines. For analysis of slope stability in open pit mine, the
slope model is first divided into rock blocks that are then
internally discretized into finite difference square elements
(Figure 8). In fact, the entire model geometry, including
excavations, is created using discontinuities, which makes it
fairly easy to create complex geometries in FLAC.
Consequently, FLAC may overestimate the collapse load.
Fig. 8 Cross-section of open pit slope model geometry with
domain descritization (finite difference mesh)

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4.4 Model Set Up
An entire cross-section through an open pit mines was
modeled. A typical open pit slope model is shown in Figure 9
with 120 meters depth having 60˚ slope angles of footwall and
70˚ slope angle of hanging wall. Bp is the excavation
boundary of the open pit mine. The majority of the models
were mined in only one mining step.
Fig. 9 Cross-section through a typical orebody mined as an
open pit.
Fig.10 Model geometry for a 120 meter deep open pit mine.
Roller boundaries were applied on the left and right sides of
the model, as well as on the bottom boundary was fixed to
restrict movement perpendicular to each boundary. The
ground surface was left as a free boundary (see Figure 11).
The virgin stress state was initialized in the model, which was
then allowed to come to initial (pre-mining) equilibrium. The
vertical virgin stress was assumed to be solely due to the
weight of the overlying rock mass. An out-of-plane horizontal
stress was also specified, to avoid yield failure in the out-of-
plane direction due to loss of confinement. The out-of-plane
stress was set equal to the in-plane horizontal virgin stress.
The influence zone due to the creation of surface excavation is
generally taken as the 5-6 times of the boundary of the open
pit geometry.
Fig.11 Required model sizes to minimize the influence from
artificial boundaries.
4.5 FDM Shear Strength Reduction Method
In this paper the shear strength reduction method is
implemented into the FLAC/Slope numerical code and its
principal is that the reduction factor is divided simultaneously
by values of intension parameter of slope, cohesion and
friction angle, then obtaining a set of new values, and
computing again with these new values as new input
parameters. When the computing is not convergent, the
corresponding values are called as the minimal safety factor of
slope which approaches the extreme state. The SSR technique
for slope stability analysis involves systematic use of finite
difference analysis to determine a stress reduction factor
(SRF) or factor of safety value that brings a slope to the verge
of failure. The shear strengths of all the materials in a FDM
model of a slope are reduced by the SRF. Conventional FDM
analysis of this model is then performed until a critical SRF
value that induces instability is attained. A slope is considered
unstable in the SSR technique when its FDM model does not
converge to a solution (within a specified tolerance).
Based on theory of elasticity and plasticity, FLAC/Slope finite
difference method is applied to iterate for calculation of the
stress and strain in term of certain load and boundary
conditions of open pit slope. Here the slope is represented by
an equivalent continuum in which the effect of discontinuities
has been considered by reducing the properties and strength of
intact rock to those of the rock mass. The slope has been
analyzed for plane strain condition in small strain mode. At
the base of the model boundary, both horizontal (x) and
vertical (y) displacements are arrested by fixing the nodes.
Along left and right of the boundary horizontal displacements
are arrested. Initial stresses of magnitude xx = yy = zz =
10MPa are applied to all the zones. Stability analysis is carried
out using Mohr - Coulomb failure criterion in FLAC/Slope
model. FLAC/Slope calculates the factor of safety
automatically using the shear strength reduction technique
[38].

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4.6 Calculation of factor of safety (FoS) and strength
reduction factor (SRF)
The factor of safety, which is defined as the ratio of the total
force available to resist sliding to the total force tending to
induce sliding along any discontinuity surfaces [39]. Slope of
open pit mines fails because of its material shear strength on
the sliding surface is insufficient to resist the actual shear
stresses. Factor of safety (FoS) is a value that is used to
examine the stability state of slopes. For FoS values greater
than 1 means the slope is stable, while values lower that 1
means slope is unstable. In accordance to the shear failure, the
factor of safety against slope failure is simply calculated as:
f
FOS



(1)
Where  is the shear strength of the slope material of a
typical open pit mine, which is calculated through Mohr-
Coulomb criterion as given below.
 tannC  (2)
And, f is the shear stress on the sliding surface along the two
discontinuity plane and which can be calculated by the
equation as given below:
fnff C  tan (3)
Where, C is cohesion and σn is normal stress. This method
allows finding the safety factor of an open pit slope by
initiating a systematic reduction sequence for the available
shear strength parameters c and  to just cause the slope to
fail. The reduction values of shear strength parameters trialc
and trial are defined as:
SRF
C
Ctrial  (4)






 
SRF
trial


tan
tan 1 (5)
The values of „ trialc ‟ and „ trial ‟ at which slope will have
instability i.e. failure is calculated by FLAC/Slope using the
strength reduction technique in which SRF is the strength
reduction factor. The safety factor of the slope, FOS, is the
value of SRF to bring the slope to failure.
An example of the FLAC/Slope model showing three different
slope layers of a typical open pit mine has been considered for
the analyses of mechanical behaviour of pit slope stability
using the finite difference method. Thus factor of safety
obtained (FoS= 0.33) indicates the critically instable failure
zones as shown in Figure 14, Figure 15 and Figure 17.
FLAC/Slope, FDM Software has been illustrated, which
presents domain discretization, critically instable failure zones
and contours of maximum shear - strain rate in the slopes
respectively as given below.
Fig. 12 FLAC/slope model showing the three different layers
of an open pit slope.
Fig. 13 FLAC/slope model with domain discretization.
Fig. 14 FLAC/slope model showing critically instable failure
zones.

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Fig. 15 FLAC/slope model showing critically instable failure
zones as well as maximum shear-strain rate.
If multiple materials and/or joints are present, the reduction is
made simultaneously for all materials. The trial factor of
safety is increased gradually until the slope fails. At failure,
the factor of safety equals the trial factor of safety (i.e. f =
FoS). [6] shows that the shear strength reduction factors of
safety are generally within a few percent of limit analysis
solutions when an associated flow rule, in which the friction
angle and dilation angle are equal, is used.
The shear strength reduction technique has two main
advantages over limit equilibrium slope stability analyses.
First, the critical slide surface is found automatically, and it is
not necessary to specify the shape of the slide surface (e.g.
circular, log spiral, piecewise linear) in advance. In general,
the failure surface geometry for slopes is more complex than
simple circles or segmented surfaces. Second, numerical
methods automatically satisfy translational and rotational
equilibrium, whereas not all limit equilibrium methods do
satisfy equilibrium. Consequently, the shear strength reduction
technique usually will determine a safety factor equal to or
slightly less than limit equilibrium methods.
Table - 1: Input parameters for the study of open pit slope
stability analysis.
Parameters Values Parameters Values
Slope height 120m Slope angle 45̊
Water pressure 0 Density (kg/m3
) 2660
Rock mass
friction angle
43̊ Cohesion (kPa) 675
Rock mass
tension
0
Bulk modulus
(GPa)
6.3
Shear modulus
(GPa)
3.6 Joint friction angle 40˚
4.7 Numerical Analysis using FLAC/Slope
Here, a Mohr-Coulomb FLAC/Slope model to simulate
ubiquitous joint model has been discussed. The ubiquitous
joint model in FLAC is a Mohr-Coulomb plastic solid with an
embedded penetrative weak plane (ubiquitous joint) of
specific orientation. In a ubiquitous joint model, plastic
yielding may occur either in the Mohr-Coulomb solid or along
the weak plane, or both, depending on the stress state, the
orientation of the weak plane and the material properties of the
solid and the weak plane. The yield criteria of the weak plane
(ubiquitous joint) are:
jnjs c  tan (3)
Where, s is shear strength, jc is ubiquitous joint cohesion,
n is normal stress and j is ubiquitous joint friction angle
In the recent years, the 2-D finite difference code, FLAC has
become very popular for design and analyses of slopes in
continuum rock. This code allows a wide choice of
constitutive models to characterize the rock mass and
incorporates time dependent behaviour, coupled hydro-
mechanical and dynamic modelling. An example of
FLAC/Slope with single interface model that illustrates the
circular type failure mechanism of weathered/jointed rock
slopes has been given below in Figure 16 and Figure 17,
which show the domain discretization and maximum shear
strain rates occurred along the critically failure surfaces.
Fig. 16 FLAC/Slope single interface model with domain
discretization.

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Fig. 17 FLAC/Slope model showing circular type failure.
5. CONCLUSIONS AND SUMMARY
Stability analyses of an open pit slope represents the critically
failure mechanism of pit slopes in which finite element
method, distinct element and finite difference methods provide
a lot of benefits over limit equilibrium because these
techniques are suitable for indication of the stress and strain
distribution within critically instable failure zones, element
displacement vectors and the plastic state of slopes. In this
paper, the authors applied finite difference code, FLAC/Slope
numerical analysis method for the slope stability analysis of a
typical open pit mine and has found appropriate for the
analysis of slope failure having similar geo-mining conditions.
The numerical analysis technique of shear strength reduction
method has been implemented into the finite difference
method and found to be appropriate for the pit slope stability
analysis. Intensity of mesh for simulation domain of all the
applied techniques has been selected as a fine level for an
acceptable accuracy and economical computation time. The
various numerical models provided in this paper demonstrate
that such modelling can be used to understand the behaviour
of open pit slopes at the similar geo-mining conditions much
better, on the other hand, captures the complicated variations
of slope failure mechanisms. So that a geo-technical engineer
can have a predictive capability for design of open pit slopes,
and that predictive capability can only be achieved if the key
features of the rock reality have indeed been captured in the
slope models. This paper also examines the ability of the
numerical technique (finite difference method) to capture the
variation of factor of safety and failure mechanism of open pit
slopes.
ACKNOWLEDGEMENTS
Authors express their acknowledgement to the Department of
Mining Engineering, Indian School of Mines, Dhanbad for
allowing the paper to be published, and also express their
sincere gratitude to all those who help directly or indirectly in
preparing the paper.
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