probabilistic analysis of rock slope stability
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Probabilistic Analysis of Rock Slope StabilityTRANSCRIPT
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Engineering Geology 7
Probabilistic analysis of rock slope stability and random
properties of discontinuity parameters, Interstate Highway 40,
Western North Carolina, USA
Hyuck-Jin Parka,T, Terry R. Westb, Ik Wooa
aDepartment of Geoinformation Engineering, Sejong University, Gunja-dong, Gwangjin-gu, Seoul, KoreabDepartment of Earth and Atmospheric Sciences, Purdue University, West Lafayette, IN 47907, USA
Received 26 January 2004; received in revised form 30 December 2004; accepted 4 February 2005
Available online 7 April 2005
Abstract
Probabilistic analysis has been used as an effective tool to evaluate uncertainty so prevalent in variables governing rock
slope stability. In this study a probabilistic analysis procedure and related algorithms were developed by extending the Monte
Carlo simulation. The approach was used to analyze rock slope stability for Interstate Highway 40 (I-40), North Carolina, USA.
This probabilistic approach consists of two parts: analysis of available geotechnical data to obtain random properties of
discontinuity parameters; and probabilistic analysis of slope stability based on parameters with random properties. Random
geometric and strength parameters for discontinuities were derived from field measurements and analysis using the statistical
inference method or obtained from experience and engineering judgment of parameters. Specifically, this study shows that a
certain amount of experience and engineering judgment can be utilized to determine random properties of discontinuity
parameters. Probabilistic stability analysis is accomplished using statistical parameters and probability density functions for
each discontinuity parameter. Then, the two requisite conditions, kinematic and kinetic instability for evaluating rock slope
stability, are determined and evaluated separately, and subsequently the two probabilities are combined to provide an overall
stability measure. Following the probabilistic analysis to account for variation in parameters, results of the probabilistic analyses
were compared to those of a deterministic analysis, illustrating deficiencies in the latter procedure. Two geometries for the cut
slopes on I-40 were evaluated, the original 758 slope and the 508 slope which has developed over the past 40 years of
weathering.
D 2005 Elsevier B.V. All rights reserved.
Keywords: Probabilistic analysis; Kinematic analysis; Kinetic instability; Rock slope; Monte Carlo simulation
1. Introduction
0013-7952/$ - see front matter D 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.enggeo.2005.02.001
T Corresponding author. Fax: +82 2 462 7537.
E-mail address: [email protected] (H.-J. Park).
Uncertainty and variability are inevitable in engi-
neering geology studies dealing with natural materi-
als. This prevails because rocks and soils are
9 (2005) 230–250
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250 231
inherently heterogeneous, insufficient amount of
information for site conditions are available and the
understanding of failure mechanisms is incomplete.
Therefore, many early efforts have been made to limit
or quantify uncertainty of input data and analysis
results (Casagrande, 1965; Peck, 1969; Einstein and
Baecher, 1983; Whitman, 1984). Slope engineering is
perhaps the geotechnical subject most dominated by
uncertainty since slopes are composed of natural
materials (El-Ramly et al., 2002). Uncertainty in rock
slope engineering may occur as scattered values for
discontinuity orientations and geometries such as
discontinuity trace length and spacing, and in labo-
ratory or in situ test results. Therefore, one of the
greatest challenges for rock slope stability analysis is
the selection of representative values from widely
scattered discontinuity data.
Application of probabilistic analysis has provided
an objective tool to quantify and model variability and
uncertainty. In particular, the probabilistic approach to
rock slope stability makes it possible to consider
uncertainty and variability in geotechnical parameters
of rock masses. Although probabilistic analyses have
been applied to rock slope stability, a limited number
of examples applied to practical cases have been fully
described. Lately commercially available limit equili-
brium codes (such as SWEDGE, ROCKPLANE,
SLIDE, SLOPE/W) often incorporate probabilistic
tools, in which variations in discontinuity properties
can be assessed. In this study, an application of the
probabilistic method to practical problems in rock
slope stability analysis is provided, and improved
procedures for the evaluation of random properties of
discontinuity parameters are explained. For this
purpose, a rock cut in western North Carolina provides
the example where the probabilistic approach is
applied to analyze slope stability. In addition, random
properties of discontinuity parameters, which were
measured in the field, obtained through laboratory
testing and applied in the probabilistic analysis, are
discussed.
2. Probabilistic analysis
Slope stability analysis requires the kinematic and
kinetic evaluation. In the kinematic analysis the
question is whether slope failure of a rock mass is
possible based on the geometry of discontinuities and
slope orientation. Combinations of discontinuity
orientations and the slope face are examined to
determine if specific failure modes are possible.
Analysis is commonly conducted with the aid of
stereographic projections of the planar features. This
indicates whether kinematic instability is likely and
then kinetic stability is evaluated using forces acting
on the rock mass. This procedure should be carried
out for the probabilistic analysis as well as the
deterministic analysis.
For deterministic analysis, single fixed values
(typically, mean values) of representative orientation
and strength parameters are determined and then the
kinematic and kinetic analyses are conducted using
single representative values. Therefore, the stability
analysis is normally carried out with one set of
geotechnical parameters. Factor of safety, based on
limit equilibrium analysis, is widely used to evaluate
slope stability because of its simple calculation and
results. However, most input values measured in the
field or obtained by laboratory tests and used
subsequently to calculate a safety factor show a wide
scatter across a significant range rather than being a
fixed single value. Thus, each parameter should be
considered as a random variable and the analysis
involving different values for each parameter will
result in different factors of safety. Therefore, the
factor of safety itself is a random variable, depending
on many input variables. However, the deterministic
analysis is unable to account for variation in rock
mass properties and conditions.
The probabilistic analysis was developed to con-
sider the uncertainty in parameters and results. In the
probabilistic approach, the analysis carries out the
analysis of random properties of the discontinuities
and rock mass. Random properties of input parame-
ters are required for probabilistic analysis and are
obtained by statistical evaluation of available geo-
logical and geotechnical data. Basic statistical param-
eters are the mean and coefficient of variation, and the
probability density function (PDF) which are obtained
during this step. Subsequently, using random proper-
ties of input parameters determined previously,
probability of failure is evaluated. The Monte Carlo
simulation, First Order Second Moment method
(FOSM) and Point Estimate Method (PEM) are
commonly used, but for the current research, the
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250232
Monte Carlo simulation was used to calculate
probability of failure.
3. Geology of the study area
The study area consists of an extensive rock cut
along Interstate Highway 40 (I-40) in western North
Carolina, near the Tennessee border. This area along
Interstate 40 shows excellent exposures of a series of
metasedimentary rocks of Late Pre-Cambrian age
(Fig. 1). The area is located in the western Blue
Ridge province, one of several physiographic prov-
inces which comprise the Appalachian Highlands.
The Blue Ridge structural province includes on its
western boundary the Great Smoky Mountains and
associated thrust faults, and on the east, the Brevard
fault zone (Wiener and Merschat, 1975). This
province consists of high metamorphic grade, Middle
Proterozoic basement to early Paleozoic, off-shelf
cover sedimentaries and Paleozoic igneous intrusives.
Major rock types in this area are a gray, thin bedded to
laminated feldspathic meta-sandsone and a green slate
with thin interbeds of fine meta-sandstone. Bedding is
Fig. 1. Geological map
distinct and the rock is highly jointed. The I-40 site
has experienced several large landslides during and
after construction. An investigation for relocation of
the highway concluded that wedge failures were the
most common phenomena. On July 1, 1997, a large
rockslide occurred in this area after heavy rainfall,
when two discontinuities forming an unstable wedge,
failed. More than 100,000 m3 of rock were removed
during mitigation of this rockslide. In the current
study a large number of discontinuity orientations and
geometries were measured in the field and their
random properties evaluated by the authors.
4. Random properties of discontinuity parameters
In the following section random properties of
geological and geotechnical parameters are deter-
mined. Information obtained from sampled data is
used to make generalizations about the populations
from which the samples were collected (Ang and
Tang, 1975). This is an important procedure needed to
obtain accurate and proper stability analysis results.
However, from one study to another, the selection of
of the study area.
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250 233
random variables can be quite different. Some authors
have considered only the geometric parameters of
discontinuity and groundwater conditions to be
random variables, whereas others also include
strength parameters as random variables. In this study,
the orientation, length, spacing, persistence and
strength parameters of discontinuities are considered
to be random variables and their random properties
are found.
4.1. Discontinuity strength parameters
No detailed shear strength testing was provided for
the study area. The North Carolina Department of
Transportation (NCDOT, 1980) used 308 as the
internal friction angle for all discontinuities in the
area to calculate the factor of safety in their slope
stability investigation and analysis. According to
Glass (1998), this apparently was obtained using a
back analysis calculation based on observations made
in the field. Even though determined from a simple
calculation without shear strength testing, this value
seems to be reasonable. This holds true because
according to Barton (1973), approximate friction
angle values for siltstone, the major lithology in this
study area, lies in a range between 278 and 318.However, this value includes a high level of uncer-
tainty since it was obtained from a simple back
calculation without any shear strength tests and only
one value suggested for shear strength parameters. A
range of friction values is preferred in the stability
analysis when significant uncertainty is involved.
The probability density function of the shear
strength parameters, especially friction angle, can be
inferred from previous research, and from an analysis
performed by the authors in a nearby study area in
North Carolina (Park, 1999; Park and West, 2001).
Several researchers have suggested a normal distribu-
tion or truncated normal distribution for the friction
angle (Mostyn and Li, 1993; Hoek, 1998; Nilsen,
2000; Pathak and Nilsen, 2004). Based on experience
by the authors, this can be confirmed using a chi-
square goodness-of-fit test on direct shear strength
testing data. Therefore, a truncated normal distribution
is considered for the density distribution of the
internal friction angle for the study area. Also, the
authors assumed 308 as the mean value of the
distribution.
For a probability analysis, another factor is
required to represent the random parameter property
to delineate the dispersion of a parameter; the standard
deviation or coefficient of variation. As it develops,
the coefficient of variation is a fairly stable measure of
variability. The coefficient of variation varies for each
physical property (or geotechnical parameter) for a
geological material, even within the same layer, but
the coefficient of variation for the same physical
properties of geological materials in many parts of the
world has a value within a relatively narrow range
(Rethai, 1998). Thus, we can assume reliable data for
the expected standard deviation of a given physical
characteristic of a geological material even before
performing the laboratory tests (Harr, 1987).
In research by the authors (Park, 1999; Park and
West, 2001), internal friction angle data indicated a
coefficient of variation of approximately a 10%. This
value corresponds well to a representative coefficient
of variation for parameters commonly used in civil
engineering design by Harr (1987) and also agrees
with the coefficients of variations of the friction angle
for soil, according to Schultze (1975). Therefore the
authors used a value of 10%, to calculate the standard
deviation for the friction angle in this area.
4.2. Discontinuity orientation
Discontinuity orientation is an important parameter
affecting rock slope stability because failure type and
kinematic instability are influenced mainly by this
feature. The principal need is to identify the sets of
preferred orientations. Orientation of these sets, and
the degree of clustering within each set, has a major
influence on the engineering characteristics of the
rock mass. In this research, the clustering procedure,
proposed by Mahtab and Yegulalp (1982), was
adopted. The algorithm is based on the assumption
that a discontinuity set has a significantly greater
degree of clustering than would a totally random
distribution of orientations.
Orientation data for the study area were collected
using the scanline method. Subsequently, in order to
reduce sampling bias from the scanline sampling,
weighting factors were applied to the orientation
data. Fig. 2 shows the results of clustering in this
area after applying a weighting factor. A total of 6
discontinuity sets were identified and their represen-
Fig. 2. Results of clustering process of discontinuity normals on equal angle lower hemisphere projection in the study area.
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250234
tative orientations were 217/77 for J1, 183/05 for J2,
163/63 for J3, 196/56 for J4, 227/37 for J5 and 061/
66 for J6. After performing the clustering procedure,
the appropriate probability density function was
determined for a discontinuity orientation distribu-
tion. Owing to its simplicity and flexibility, the
Fisher distribution was selected. This distribution is
based on the assumption that a population of
orientation values is distributed about a btrueQ value(Fisher, 1953). This assumption is similar to the
concept of discontinuity normals being distributed
about some true value within a set.
In view of its simplicity and flexibility, the
Fisher distribution provides a valuable model to
evaluate discontinuity orientation data (Priest, 1993).
However, the distribution provides only an approx-
imation for asymmetric data because it is a sym-
metrical distribution. Therefore, some different
models have been proposed to provide better fits
for asymmetric orientation data. However, these
models are too complex in their parameter estima-
tion. Furthermore, because of their complexity,
generation of random values from those asymmetric
orientation distributions is difficult to accomplish
and subsequently the analyses based on that
probabilistic approach are difficult to perform.
Hence, the Fisher distribution is commonly adopted
for many probabilistic calculations and that was the
case in the current study.
4.3. Discontinuity trace length
Knowledge of discontinuity lengths for a rock
mass is important for predicting rock behavior and
analysis of rock slopes because discontinuity lengths
influence the size of blocks that may be formed.
Mean discontinuity length and length distribution
provide important data for each joint set which are
required for a probabilistic model of rock slope
analysis in a jointed rock mass. However, estimation
of the mean trace length is difficult because of bias
errors involved in trace length measurements. Bias
problems due to the scanline sampling procedure
have been discussed by several authors (Baecher and
(a) Joint set 1
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00
10
20
30
40
50
Occ
uren
ce fr
eque
ncy
(%)
Discontinuity trace length (m)
(b) Joint set 2
0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.00
5
10
15
20
25
30
35
40
Occ
uren
ce fr
eque
ncy
(%)
Discontinuity trace length (m)
Fig. 3. Histogram of the occurrence frequency in discontinuity trace
length.
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250 235
Lanney, 1978; Cruden, 1977; Priest and Hudson,
1981) and several researchers have attempted to
provide a procedure to provide an unbiased estima-
tion for the mean length of joint sets. According to
Mauldon et al. (2001), the approaches to circumvent
or correct for sampling biases and estimate correct
mean trace length of discontinuity are: (1) approach
that assumes a particular form for the trace length
distribution of the sampled population and (2)
methods that are distribution free (Martiz, 1981).
The latter approach is based on the commonly used
circular scanline or window mapping and the mean
trace length is obtained without consideration of the
underlying trace length distribution. Therefore, it is
difficult to apply the results of the second approach
to the simulation procedure in this study regardless
of many advantages for that approach. Regarding the
first approach, Priest and Hudson (1981) showed
that the corrected probability density distribution of
trace lengths would have a negative exponential
distribution. Moreover, they noted that if the actual
trace lengths have a negative exponential distribu-
tion, the distribution and the mean of semi-trace
lengths are the same as the actual trace lengths.
Therefore, by measuring the semi-trace lengths of
discontinuities in the field, the distribution and mean
value of the actual trace lengths can be obtained.
Consequently, by selecting a negative exponential
distribution as the actual trace length distribution,
adoption of semi-trace length measurements as a
basis for estimating mean trace lengths can be
supported. In addition, many studies of field
measurements have showed that the negative expo-
nential probability density distribution is suitable to
represent the discontinuity trace length distribution
(Wallis and King, 1980; Baecher, 1983; Kulatilake et
al., 1993, 2003).
Discontinuity trace length data were obtained by
the authors from a field survey on road cuts along
Interstate Highway 40. Approximately 300 data points
were sampled with the authors collecting the semi-
trace length of discontinuities using the detailed
scanline method. After obtaining this discontinuity
length data, lengths were reclassified on the basis of
discontinuity sets and the mean length for each set
was evaluated. Fig. 3 shows the histograms of
discontinuity trace length obtained for joint sets 1
and 2.
4.4. Discontinuity persistence
4.4.1. Traditional definition of joint persistence
Discontinuity lengths determine the size of the
rock blocks that form within a rock mass. Further-
more, they may also affect joint persistence, which is
defined as the areal extent or size of a discontinuity
along a plane (ISRM, 1978). Persistence is one of the
most significant joint parameters affecting rock mass
strength, but it is difficult to quantify. With reference
to a joint plane (a plane through the rock mass
containing a combination of discontinuities and intact
rock regions), joint persistence is defined as the
fraction of area that is actually discontinuous (Einstein
et al., 1983). Therefore, the persistence value (K) can
be expressed in the limit form:
K ¼ limAoYl
P
iaDi
AD
ð1Þ
(b)(a) (c)
Fig. 4. Traces of (a) intermittent, (b) impersistent and (c) persisten
joints (after Hudson and Priest, 1983).
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250236
in which D is a region of the plane with area AD and
aDi is the area of the ith joint in D. Einstein et al.
(1983) suggested that persistence can only be roughly
quantified by observing the discontinuity trace length
on a rock exposure surface. This is because rock
exposures are small and only two-dimensional. It is
impossible in practice to measure the discontinuity
area accurately in a field survey. Consequently, joint
persistence can be expressed as a limit length ratio
along a given line on a joint plane in terms of trace
length:
K ¼ limLSYl
P
ilSi
LSð2Þ
where LS is the length of the straight-line segment S,
and lSi is the length of the ith joint segment in S.
4.4.2. Importance of discontinuity persistence
The reason that discontinuity persistence is
important in slope stability analysis is because of
its major effect on rock mass strength. The shear
strength available for a rock bridge is one to two
orders of magnitude greater than the shear strength
available on the discontinuity. As Einstein et al.
(1983) and West (1996) suggested, joint persistence
can be used to estimate the strength of a rock mass
against sliding along a given plane. That is, if the
joint is not persistent, failure occurs through the
rock bridge. This causes a significant increase in
shear strength.
Singh and Sun (1989) and Scavia (1990) applied
a fracture mechanics concept to evaluate the
stability of rock slopes which do not have a
100% persistence failure plane. Kemeny (2003)
proposed a fracture mechanics model in slope
stability, which is considered the time dependent
degradation of rock bridge cohesion. Fracture
mechanics considers rock slope failure to be a
result of joint initiation and propagation. Therefore,
the joint tip stress intensity factor is the governing
parameter with respect to rock slope stability and
the factor of safety is defined in terms of stress
intensity factor (Whittaker et al., 1992). However,
this approach has the limitation that the factor of
safety defined as the stress intensity factor indicates
crack stability, but not the overall stability of the
slope. In addition, those approaches consider the
persistence as a fixed value, so it is not possible to
consider the persistence as a random variable in the
probabilistic analysis.
The serious problem concerning persistence is
that its extent is difficult to measure because direct
mapping of discontinuities within a rock mass is not
possible. In practice, 100% persistence is assumed.
However, the possibility of a 100% persistent
discontinuity on the shear planes is quite low under
field conditions. In addition, as Einstein et al.
(1983) suggested, because every joint in a set does
not have the same value and these values are
uncertain, persistence should be considered as a
random variable. Therefore, a new approach is
requisite for the probabilistic analysis, and random
properties must be evaluated to characterize joint
persistence.
4.4.3. A new concept of persistence proposed in this
study
Hudson and Priest (1983) recognized that two
kinds of persistence could be identified: intermittent
joints as in Fig. 4(a) and impersistent discontinuities
as in Fig. 4(b). Intermittent joints in Fig. 4(a)
require that the planes contain a patchwork of
discontinuities and intact rock regions through the
rock mass. As discussed previously, the previous
concept of persistence implies that two or more
joints occur on the same plane, so the previous
persistence concept is based on the concept of
intermittent joints.
However, from a practical point of view, it
appears that intermittent joints in Fig. 4(a) are
t
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250 237
geologically unlikely (Mauldon, 1994). That is,
Mauldon (1994) concluded that intermittent joints
would seem to imply existence of weakness planes
through the rock mass, but locally separated to form
visible joints. Consequently, the intermittent discon-
tinuity should be treated as persistent, i.e. continu-
ous across the region of interest, at least for the
purpose of mechanical analysis.
To this end, a new approach is proposed in this
research which accounts for persistence in spite of
the limitations of measurement, based on the
impersistent joint concept in Fig. 4(b). Joint
persistence is described in this study as a function
of the length of individual joints and the maximum
sliding dimension, determined by slope geometry,
joint orientation and joint dimension (Fig. 5). This
approach assumes that only one joint forms the
sliding surface (multiple joints do not line up end
to end) and that this joint is not offset from the
sliding surface, which is also proposed by Mauldon
(1994). In our field survey for this study, a
discontinuity is considered to occur within the
same plane.
To utilize this new approach, the probability that
the joint length is long enough to form a block
capable of sliding is evaluated. That is, using
information on statistical parameters and the prob-
ability distribution of discontinuity length consid-
ered previously, a large number of individual joint
length values are generated. Then each value of the
generated joint length is compared to the sliding
dimension and the probability that the joint length
is equal to or greater than the maximum sliding
length is calculated (Fig. 5). This is the probability
that a fully persistent discontinuity exists. Then this
value is multiplied by the probability of slope
Length of jointon sliding plane Maximum sliding dimension
Fig. 5. Geometrical feature of sliding dimension and joint on sliding
plane.
failure with the premise that joints are fully
persistent.
P rock slope failure½ �
¼ P rock slope failure j fully persistent joint½ �
� P fully persistent joint exists½ � ð3Þ
The assumption of the fully persistent joint in rock
slope stability is common in deterministic analysis as
well as the probabilistic analysis. This is quite
conservative approach in stability analysis. However,
this approach overcomes the limitation of a conserva-
tive analysis.
4.5. Discontinuity spacing
In rock slope stability analysis, spacing of dis-
continuity sets is part of the representation of geo-
metric characteristics for each discontinuity set.
Measurement of joint set spacing causes sampling
bias since scanlines are not positioned perpendicular
to discontinuities because of the limited rock face
exposure (Terzaghi, 1965). Therefore, the correction
of the sampling bias caused by a inclined scanline
orientation is accomplished using the acute angle
between the scanline orientation and the orientation of
the line normal to the mean orientation for the joint set
in question (Giani, 1992). Statistical parameters, i.e.
mean and standard deviation of spacing for each set,
are evaluated using the corrected data, and then the
probability density function for discontinuity spacing
was derived from these data.
Although mean discontinuity spacing provides a
direct measure of spacing data, several previous
studies have tried to represent the distribution of
measured spacing data by statistical analysis and
description with the spacing data considered as a
random variable.
In order to determine the appropriate spacing
distribution, the authors collected approximately 300
values of discontinuity data using a borehole sampling
method (approximately 60% of 300 data) and the
detailed scanline method (40% of 300 data) at the
Interstate 26 site in northwestern North Carolina near
the Tennessee border. This is because considerably
more data are provided for this adjacent area than are
available for the I-40 site (Park, 1999). On the basis of
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250238
data collected for I-26, chi-Square goodness-of-fit
tests were performed for lognormal and negative
exponential distributions, which are the two distribu-
tion models commonly used for spacing evaluation.
This is because those theoretical distributions are
bounded at zero and are skewed to the right and those
characteristics are similar to the properties of the
spacing distribution. Table 1 shows the results of chi-
square tests for joints in the Interstate 26, Area A.
Results show that both the lognormal distribution and
the exponential distribution appear to be valid models
for spacing at the 5% significance level. However,
because the calculatedP
(ni�ei)2 / ei value for
lognormal distribution is smaller than that for the
exponential distribution, the lognormal distribution is
the better of the two. Table 2 for data from Interstate
26, Area B shows similar results. Therefore, the
lognormal probability density distribution was used as
the distribution model to represent the random
property of discontinuity spacing. The literature also
proposes the use of a lognormal probability distribu-
Table 1
Chi-square test results for relative goodness-of-fit in spacing data in Inter
Interval Observed frequency (ni) Theoretical fre
Exponential
0–0.15 0 2.5855
0.15–1.00 18 12.4024
1.00–1.85 5 9.2896
1.85–2.70 7 6.9580
2.70–3.55 11 4.7233
3.55–4.40 4 3.9036
4.40–5.25 1 2.9238
5.25–6.10 1 2.1900
6.10–6.95 1 1.6403
6.95–7.80 0 1.2286
7.80–8.65 0 0.9203
8.65–9.50 0 0.6893
9.50–10.35 2 0.5163
10.35–11.20 1 0.3867
11.20–12.05 0 0.2896
12.05–12.90 1 0.2169
12.90–13.75 0 0.1625
13.75–14.60 1 0.1217
14.60–15.45 0 0.0912
15.45–16.30 0 0.0683
16.30–17.15 0 0.0511
17.15–18.00 0 0.0383
N18.00 0 0.0287
51.4260
tion for discontinuity spacing. Rouleau and Gale
(1985), Sen and Kazi (1984) and Kulatilake et al.
(2003) suggest that the lognormal probability density
distribution was appropriate, based on their goodness-
of-fit tests.
5. Probabilistic analysis of rock slope stability
5.1. Analysis procedure
After the random properties of discontinuity
parameters are defined, the probabilistic analysis is
accomplished. The Monte Carlo simulation was used
for the probabilistic analysis in this study. The Monte
Carlo technique is frequently applied to evaluate the
probability of failure of a mechanical system, in
particular, when the direct integration is not practical
or when the integration equation is difficult to solve
(Mostyn and Li, 1993). The simulation procedure
proceeds in two steps, the first being kinematic
state 26, Area A
quency (ei) (ni�ei)2 /ei
Lognormal Exponential Lognormal
0.5451 2.5855 0.5451
15.1551 2.5263 0.5341
11.5748 1.9808 3.7346
7.1535 0.0003 0.0033
4.6025 8.3411 8.8924
3.1028 0.0024 0.2594
2.1756 1.2658 0.6353
1.5754 0.6466 0.2101
1.1713 0.2500 0.0251
0.8904 1.2286 0.8904
0.6897 0.9203 0.6897
0.5429 0.6893 0.5429
0.4334 4.2640 5.6633
0.3502 0.9727 1.2056
0.2861 0.2896 0.2861
0.2360 2.8264 2.4732
0.1964 0.1625 0.1964
0.1647 0.1217 0.1647
0.1391 0.0912 0.1391
0.1183 0.0683 0.1183
0.1012 0.0511 0.1012
0.0870 0.0383 0.0870
0.0752 0.0287 0.0752
51.3667 29.3515 27.4725
Table 2
Chi-square test results for relative goodness-of-fit in spacing data in Interstate 26, Area B
Interval Observed frequency (ni) Theoretical frequency (ei) (ni�ei)2 /ei
Exponential Lognormal Exponential Lognormal
0.0–1.0 16 17.5246 17.9265 0.1326 0.2070
1.0–2.0 11 10.2124 10.9442 0.0671 0.0003
2.0–3.0 7 5.9513 5.1976 0.1848 0.6250
3.0–4.0 4 3.4681 2.7770 0.0816 0.5386
4.0–5.0 3 2.0210 1.6230 0.4742 1.1683
5.0–6.0 3 1.1777 1.0133 2.8195 3.9850
6.0–7.0 0 0.6863 0.6653 0.6863 0.6653
7.0–8.0 0 0.4000 0.4545 0.4000 0.4545
8.0–9.0 0 0.2331 0.3207 0.2331 0.3207
9.0–10.0 0 0.1358 0.2323 0.1358 0.2323
10.0–11.0 1 0.0792 0.1721 10.7132 3.9823
11.0–12.0 0 0.0461 0.1300 0.0461 0.1300
12.0–13.0 0 0.0269 0.0998 0.0269 0.0998
13.0–14.0 0 0.0157 0.0778 0.0157 0.0778
14.0–15.0 0 0.0091 0.0614 0.0091 0.0614
15.0–16.0 0 0.0053 0.0490 0.0053 0.0490
16.0–17.0 0 0.0031 0.0395 0.0031 0.0395
17.0–18.0 0 0.0018 0.0322 0.0018 0.0322
N18.0 0 0.0011 0.0264 0.0011 0.0264
41.9985 41.8426 16.0309 12.6056
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250 239
analysis, examining kinematic instability of a rock
body defined by discontinuities. Based on disconti-
nuity orientation, it is determined whether the rock
body is able to move or not. If the kinematic analysis
indicates that the geometric condition is potentially
unstable, then the kinetic stability is assessed by the
limit equilibrium method. This comprises the second
step.
5.2. Evaluation for probability of slope failure
To check the stability of rock slope systems,
both kinematic and kinetic analyses are required to
analyze the geometry and strength of discontinuities.
In a complete study this should be accomplished for
both probabilistic analysis as well as for determin-
istic analysis. However, only kinetic instability was
evaluated and it was assume to be the probability of
failure for the rock slope in some previous studies.
Difficulty in performing kinematic analysis is
considered as one reason why it is omitted. For
planar failure, the kinematic analysis is relatively
easy since clear criteria exist such that the dip
direction of discontinuity must be within 208 of dip
direction of the slope face. However, the kinematic
analysis is commonly accomplished using stereo-
graphic projections, and subsequently a calculation
of probability of the kinematic instability is not easy
accomplished, especially for wedge failure analysis.
The stereographic projection method is not suited to
conducting computational and repeated calculations
used in the Monte Carlo simulation which is the
typical procedure for probability analysis of wedge
failure. This is because the closed form of kinematic
analysis is not provided. That is, if the Monte Carlo
simulation is utilized, in order to obtain the input
values for each simulation, the great number of
stereographic projections is required. The large
number of stereographic projection is needed for
each set of parameter combinations and the input
values for each simulation must be measured from
the stereoplot if the Monte Carlo simulation is
utilized. Some years ago McMahon (1971) and
Herget (1978) proposed a probabilistic kinematic
analysis approach which can evaluate the proba-
bility of kinematic instability using stereographic
projection. However, the procedure did not provide
a closed form equation and was limited only to the
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250240
planar condition. Therefore, in this study a simple
equation for checking a kinematic instability for
rock wedges was used (Park and West, 2001;
Gunther, 2003).
X b e b aapparent ð4Þ
aapparent ¼ tan�1 tana cos bi � bsð Þ�½ ð5Þ
where e is the dip angle of the line of intersection
between two discontinuities, X is the dip angles of
the upper ground surface and aapparent is the
apparent dip of the slope face in the dip direction
of the intersection line, not in the dip direction of
slope face (Park and West, 2001; Gunther, 2003). bi
and bs are the dip directions of the lines of
intersection and slope face, respectively. In the
current study, the probability of kinematic instability
was evaluated using the apparent dip of slope face.
Also, in the current study, a step-by-step proce-
dure for evaluating the slope failure probability was
used. Procedural steps were accomplished separately
for kinematic stability and kinetic stability in the
probabilistic analysis. That is, this procedure
assesses the probability of kinematic instability, in
which a number of iterations form a block or wedge
that can kinematically move. Once the kinematically
unstable blocks or wedges have been identified and
evaluated as kinematically unstable, the kinetic
probability is evaluated as a conditional probability
which has a premise that the block is kinematically
unstable. Therefore, the overall probability of slope
failure will be
Pf ¼ P kinematically unstable½ �
� P kinetic unstablejkinematic unstable½ � ð6Þ
This concept was also proposed by Einstein (1996).
Therefore, the probability of slope failure is
Pf ¼ Pkm � Pkn=km ð7Þ
The probability of kinematic instability is defined as:
Pkm ¼ Nm
NT
ð8Þ
where Nm is the number of iterations which is
kinematically unstable and NT is total number of
iterations. Because the kinetic analysis is performed
only when the block is kinematically unstable, the
probability of kinetic instability is defined as
Pkn=km ¼ Nf
Nm
ð9Þ
where Nf is the number of iterations that a wedge
has factor of safety less than one. Therefore, the
probability of failure is
Pf ¼ Pkm � Pkn=km ¼ Nm
NT
� Nf
Nm
ð10Þ
6. Results of analysis
6.1. Input parameters
In the probabilistic analysis, input parameters can
be subdivided into two groups by their randomness:
deterministic and probabilistic parameters.
Deterministic parameters are those considered to
be known and having a single value for all sliding
blocks. In the current study, the orientation and height
of the cut slope and rock density were considered to
be deterministic parameters. In addition, roughness of
a discontinuity was considered as a deterministic
parameter. Roughness is a potentially important
component of shear strength and therefore, it was
measured in the field for each discontinuity using a
disk clinometer. This value was added to the friction
angle of the discontinuity.
For the probabilistic parameters, the probability
density function and the values of statistical param-
eters for random variables are chosen on the basis of
physical properties, test results and evaluation of the
measured data. In this study, joint parameters were
considered to be probabilistic in nature. In all, joint
orientation, geometric parameters, such as length and
spacing, and shear strength parameters were consid-
ered to be probabilistic parameters. In addition, pore
water pressure in the discontinuity is considered to be
a random variable since the groundwater table level
varies. However, those probabilistic parameters are
assumed to be independent. The covariance between
random parameters plays an important role in prob-
abilistic analysis. However, research results involving
the accurate evaluation of covariance between random
parameters in a rock mass are limited and some
Table 4
Comparison of results for the deterministic analysis and the
probabilistic analysis at cut slope angle of 758
Joint
set I.D.
Factor
of safety
Probability of failure Average volume
of possible
block (m3)Kinematic Kinetic Total
J1 Stable 0.345 0.018 0.006 5.3
J2 Stable 0 0 0
J3 Stable 0 0 0
J4 0.29 0.621 0.014 0.0087 18.9
J5 0.69 0.759 0.671 0.509 42.6
J6 Stable 0 0 0
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250 241
researchers propose that the assumption of independ-
ence is conservative (Cherubini et al., 1983; Li and
White, 1987).
Parameters used in this study are listed in Table 3.
Input data for slope geometry are included in Table
3(a) and the input for discontinuity properties is given
in Table 3(b). A point to note here is the length of
joint set 5. All other discontinuity lengths for
discontinuity sets were determined in the field, except
for J5 which was identified as a bedding plane. In
many cases, bedding plane lengths are assumed to be
infinite, but in this study, in order to show that the
length of bedding was much greater than that of other
joint sets, it was assigned a value of 60 m.
6.2. Results for planar failure
Table 4 shows the results of the deterministic and
probabilistic analyses of planar failure for each joint
set. In the deterministic analysis, mean values of each
random variable are used and the factor of safety is
calculated for each set. In contrast, random properties
of random variable are considered and the probability
of failure is evaluated using the simulation procedure.
In order to compare the results between the determin-
istic and probabilistic analysis, the same performance
function suggested by Hoek and Bray (1981) was
utilized. In Fig. 6, the histogram of safety factor is the
result of calculation using the repeated simulation
procedure. Based on results of the deterministic
analysis, joint sets 4 and 5 were found to be unstable
having a factor of safety of 0.29 and 0.69, respec-
tively. That is, kinematic analyses for J4 and J5
Table 3
Input values for I-40 area
(a) Input for slope geometry
Orientation of slope (dip direction/dip) Height
210/75 34
(b) Input for discontinuity properties
Set I.D J1 J2 J3
Mean orientation (dip direction/dip) 217/77 183/5 16
Fisher constant 42 53 29
Mean friction angle (deg) 30 30 30
S.D of friction angle 3 3 3
Mean length (m) 1 1 1
Roughness (deg) 1 1 1
Mean spacing (m) 1.6 3.5 0
indicate they are kinematically unstable and subse-
quently, factors of safety for those joint sets are
computed and found to be less than 1. In Table 4, the
word dstableT indicates that the kinematic analysis
found the set to be kinematically stable and sub-
sequently, the kinetic analysis was not conducted.
Therefore, all joint sets except J4 and J5 are kine-
matically stable.
However, the probabilistic analysis for the planar
failure shows different results. According to the
probability of planar failure in Table 4, the analysis
indicates that joint sets J1, J4 and J5 have a possibility
of failure. Especially for joint set 1, the probability of
kinematic instability calculated is 34.5% and the
probability of kinetic instability is 1.8% despite the
fact that J1 was found to be stable in the deterministic
analysis. For the deterministic analysis, the mean
orientation of J1 does not show a possibility of
kinematic instability because the mean dip angle for
J1 is greater than the slope cut angle and the join will
not daylight. However, when the variation in orienta-
of slope (m) Unit weight of rocks (t/m3)
2.56
J4 J5 J6 PDF
3/63 196/56 227/37 061/66 Fisher
119 36 106
30 30 30 Normal
3 3 3
0.5 60 1 Exponential
1 0 1
.43 4.5 1.0 2.7 Lognormal
0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0
0.5 1.0 1.5 2.0 2.5 3.0
0.5 1.0 1.5 2.0 2.5 3.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Fre
quen
cy
Factor of safety
Fre
quen
cy
Factor of safety
(a) Set 1 at 75 degree slope angle (b) Set 4 at 75 degree slope angle
0.5 1.0 2.0 3.01.5 2.50.00
0.05
0.10
0.15
0.20
0.25
0.00
0.05
0.10
0.15
0.20
0.25
0.00
0.05
0.10
0.15
0.20
0.25
Fre
quen
cy
Factor of safety
Fre
quen
cy
Factor of safety
(c) Set 5 at 75 degree slope angle (d) Set 4 at 50 degree slope angle
Fre
quen
cy
Factor of safety
(e) Set 5 at 50 degree slope angle
Fig. 6. Histogram of factor of safety calculated in probabilistic analysis for planar failure.
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250242
tion is considered, many of scattered orientations have
the possibility of kinematic instability even though the
mean orientation does not yield a kinematically
unstable condition. Consequently, the deterministic
analysis based on a fixed representative orientation of
discontinuities fails to show the possibility of kine-
matic instability.
The probability of kinetic instability for J4 of 1.4%
is much lower than the probability of kinetic
instability for J5, 67.1%. In addition, the probability
of slope failure for J4, 0.9% is lower than that for J5,
50.9%. Therefore, based on probabilistic analysis, J5
represents a greater risk and more a dangerous
condition. Discontinuity trace length is a possible
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250 243
reason. In this study, discontinuity length was
considered to be a form of persistence. This is used
when the probability of the kinetic instability is
evaluated. Therefore, J5, a bedding plane with a mean
length of 60 m, has a higher kinetic instability
probability than J1 whose mean length is only 0.5 m.
Regarding a specific aspect of probabilistic anal-
ysis, the mean volume of blocks for each case of
possible failure, is evaluated using discontinuity
orientation data and cut slope geometry. That is, if
each block whose dimension is calculated by ran-
domly selected discontinuity parameter is kinemati-
cally and kinetically unstable, the volumes of each
block are calculated using the Hoek and Bray (1981)
equation. Then the mean volume of possible blocks is
evaluated. For this, the daylight point (where possible
failure surface meets the slope face) is randomly
selected from slope face. Mean volumes of possible
blocks are 5.3 m3 for J1, 18.9 m3 for J4 and 42.6 m3
for J5. Therefore, joint set 5, which is a bedding plane,
exhibits a high possibility of plane failure and the size
of the failure block will also be great. In addition, J4, a
plane indicated as a failure possibility by NCDOT
(1980), shows in the current study, a maximum 19.6%
failure probability and 18.9 m3 volumes for block
size.
However, since the cut slopes were excavated more
than 40 years ago and several large and many small
slides occurred in this area, the slope cut has become
much flatter. Based on observation of the authors the
cut slope angle is now approximately 508 rather thanthe original 758. Using this slope angle, the factor of
safety and probability of failure were evaluated again
and the results compared to the previous calculations.
Table 5 includes results of both deterministic and
probabilistic analyses for the 508 slope angle. A
Table 5
Comparison of results for the deterministic analysis and the
probabilistic analysis at cut slope angle of 508
Joint
set I.D.
Factor
of safety
Probability of failure Average volume
of possible
block (m3)Kinematic Kinetic Total
J1 Stable 0 0 0
J2 Stable 0 0 0
J3 Stable 0 0 0
J4 Stable 0.053 0.014 0.0007 3.2
J5 0.61 0.663 0.669 0.444 19.1
J6 Stable 0 0 0
deterministic analysis of the current slope shows joint
set 4 is now stable following slope flattening.
Representative orientation data for joint set 4 are no
longer kinematically unstable because the dip of the
discontinuity is greater than the 508 slope dip angle.
Therefore, without further investigation of the shear
strength, joint 4 is designated to be stable because the
joint will not daylight kinematically. For joint 5,
however, the safety factor is not much different from
the previous value, 0.69. Hence, reducing the slope
angle does not significantly change the factor of safety
for joint set 5.
Considering the results of probabilistic analysis,
joint set 1 is stable for a 508 cut slope. That is, bothprobabilities of kinematic and kinetic instability are
zero. In addition, the probabilities of kinematic
instability for J4 and J5 are reduced from 62.1% and
75.9% to 5.3% and 66.3%, respectively, but the
probabilities of kinetic instability in both joint sets
are unchanged. However, the total probability of slope
failure is reduced somewhat because of the multi-
plication effect.
As the slope angle is reduced, volumes of possible
rock blocks are also reduced. For joint set 4, the mean
volume is reduced from 18.9 m3 to 3.2 m3 and the
mean volume for joint set 5 is reduced from 42.6 m3
to 19.1 m3. Therefore, the risk of a large slope failure
is significantly reduced. However, in practice, the
planar failure on joint set 5 will occur only when the
lateral extent of potential failure mass is isolated by a
lateral release surface, which is a prerequisite for
planar failure to occur (Hoek and Bray, 1981).
6.3. Results for wedge failure
Relative to wedge failure, for only two joint set
combinations (J2 and J3 and J2 and J6), do the results
of the deterministic analysis for kinematic stability
agree with those of the probabilistic analysis (Table 6;
Fig. 7). They are kinematically stable based on the
deterministic analysis and have zero probability of
kinematically instability in the probabilistic analysis.
However, some joint combinations indicated as
kinematically stable by the deterministic analysis are
shown to be kinematic unstable using probabilistic
analysis. Those combinations (J1 and J2, J1 and J5, J2
and J4, J2 and J5, J3 and J6, J5 and J6) have small
probabilities of kinematic instability, ranging from
1 2 3 4 5 1 2 3 4 5
1 2 3 4 5 1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
Fre
quen
cy
Factor of safety
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Fre
quen
cy
Factor of safety
(a) Combination of joint sets 1 and 2 (b) Combination of joint sets 1 and 3
0.00
0.01
0.02
0.03
0.04
0.05
0.00
0.01
0.02
0.03
0.04
0.05
Fre
quen
cy
Factor of safety
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Fre
quen
cy
Factor of safety
(c) Combination of joint sets 1 and 4 (d) Combination of joint sets 1 and 5
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Fre
quen
cy
Factor of safety
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Fre
quen
cy
Factor of safety
(e) Combination of joint sets 1 and 6 (f) Combination of joint sets 2 and 4
0.00
0.01
0.02
0.03
0.04
0.05
Fre
quen
cy
Factor of safety
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
Fre
quen
cy
Factor of safety
(g) Combination of joint sets 2 and 5 (h) Combination of joint sets 3 and 4
Fig. 7. Histogram of factor of safety calculated in probabilistic analysis for wedge failure at 758 slope angle.
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250244
1 2 3 4 5 1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Fre
quen
cy
Factor of safety
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Fre
quen
cy
Factor of safety
(i) Combination of joint sets 3 and 5 (j) Combination of joint sets 3 and 6
0.00
0.02
0.04
0.06
0.08
0.10
Fre
quen
cy
Factor of safety
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
Fre
quen
cy
Factor of safety
(k) Combination of joint sets 4 and 5 (l) Combination of joint sets 4 and 6
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Fre
quen
cy
Factor of safety
(m) Combination of joint sets 5 and 6
Fig. 7 (continued).
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250 245
0.1% to 3.7%. By contrast, the joint set combinations
shown to be kinematically unstable in the determin-
istic analysis show high probabilities of kinematic
instability in the probabilistic analysis. That is,
unstable joint combinations (J1 and J3, J1 and J4,
J1 and J6, J3 and J4, J3 and J5, J4 and J5, J4 and J6)
show high failure probabilities ranging from 18.9% to
93.5%.
For the kinetic analysis, a sliding mode for wedge
failure is determined and a factor of safety value is
calculated using a deterministic analysis. However, in
the probabilistic approach, four different kinetic
probabilities can be evaluated for four different sliding
modes because the scattered orientations of disconti-
nuities can produce these different sliding modes for
combinations of the two discontinuities. For example,
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250246
for the J3 and J5 combination in Table 6, the
deterministic analysis shows a factor of safety of
0.36 for the wedge sliding with contact on plane 2.
However, based on the probabilistic analysis, there is
56.1% possibility of sliding with contact on plane 2.
Also, there is 19.1% possibility of sliding without
contact (that is, the contact is lost on both planes since
water pressures on both planes is greater than the
normal force) on both planes. In fact, the combina-
tions of J2 and J5, J4 and J5 and J5 and J6 which are
stable in the deterministic analysis have three or four
different sliding modes with high probabilities of
failure. In Table 6, refer to the sliding mode where the
factor of safety for each joint set combination is
evaluated using a deterministic analysis. The sliding
mode with the highest probability of kinetic instability
among those four different sliding modes shows the
lowest factor of safety. For example, in the J3 and J5
combination, the deterministic analysis has a factor of
safety of 0.36 which is a sliding mode in contact on
plane 2, and this sliding mode has the highest
probability of kinetic instability (Pf =56.1%) of all
the other sliding modes.
Comparing these probabilities with the average
volume of the failure block, the combinations of J1
and J3, J1 and J5, J3 and J4, J4 and J5 and J5 and J6
indicate the possibility of small volume blocks, that is,
0.011 m3, 0.095 m3, 6.86 m3, 1.72 m3 and 0.041 m3,
respectively. However, for the J3 and J5 combination,
Table 6
Results of wedge failure for the deterministic analysis and the probabilist
Set no. 1 Set no. 2 Factor of
safety
Probability of failure
Kinematic No contact Plane 1
J1 J2 Stable 0.001 0.025 0
J1 J3 FS1=0 0.4 0.014 0
J1 J4 FS3=0.32 0.374 0.004 0
J1 J5 Stable 0.037 0.038 0
J1 J6 FS3=0.43 0.189 0.005 0
J2 J3 Stable 0 0 0
J2 J4 Stable 0.002 0 0
J2 J5 Stable 0.002 0.338 0.005
J2 J6 Stable 0 0 0
J3 J4 FS3=0.09 0.935 0.012 0
J3 J5 FS3=0.36 0.738 0.191 0
J3 J6 Stable 0.015 0.023 0
J4 J5 FS3=0.33 0.471 0.046 0.001
J4 J6 FS1=0 0.309 0.011 0
J5 J6 Stable 0.009 0.554 0.012
with a high probability of failure, 55.8%, the average
block volume is approximately 82.38 m3. These
results agree with the failure history for slopes in this
area. According to Glass (1998), several small slides
were reported each year and the large rockslide that
occurred on July 1, 1997 was a wedge failure formed
by J3 and J5. This finding for the site is verified by the
results of the current research, showing high proba-
bilities of failure for several small slides and for a
large wedge slide.
When the slope angle is assumed to be 508, theresults of a deterministic analysis are much different
from the previous deterministic results (Table 7; Fig.
8). Only the J4 and J5 combination is found to be
kinematically unstable and the factor of safety for the
sliding mode, with contact on plane 2, equals 0.33.
The other combinations show kinematically stable
conditions. However, based on probabilistic analysis,
a total of eight combinations show kinematic insta-
bility, even though the probabilities are reduced
significantly from the previous probabilities on the
758 cut slope.As can be observed in Table 7, the probabilities of
total kinetic instability (that is, the sum of four
probabilities of kinetic instability based on a sliding
mode) for J2 and J5, J3 and J5 and J4 and J5
combinations are still high, 68.3%, 75.5% and 69.8%,
respectively. As the results in Table 7 show, we
observe that J5 plays an important role in kinetic
ic analysis in I-40 area at 758 cut slope
Total probability
of failure
Average volume of
possible wedge (m3)Plane 2 Both planes
0.036 0 0 0.006
0.0125 0 0.011 0.011
0.0074 0 0.004 0.004
0.244 0 0.010 0.095
0.0067 0 0.002 0.002
0 0 0 0
0.0045 0 0 5.39
0.331 0.0029 0.001 15.35
0 0 0 0
0.013 0 0.023 6.86
0.561 0.0037 0.558 82.38
0 0 0.0004 0.13
0.514 0.0014 0.265 1.72
0 0 0.0034 0.019
0.193 0 0.0068 0.041
1 3 4 52
1 3 4 52
1 3 4 52
1 3 4 52
1 3 4 52
1 3 4 52
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Fre
quen
cy
Factor of safety
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Fre
quen
cy
Factor of safety
(a) Combination of joint sets 1 and 3 (b) Combination of joint sets 1 and 5
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Fre
quen
cy
Factor of safety
0.00
0.02
0.04
0.06
0.08
0.10
Fre
quen
cy
Factor of safety
(c) Combination of joint sets 2 and 5 (d) Combination of joint sets 3 and 4
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Fre
quen
cy
Factor of safety
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
Fre
quen
cy
Factor of safety
(e) Combination of joint sets 3 and 5 (f) Combination of joint sets 4 and 5
Fig. 8. Histogram of factor of safety calculated in probabilistic analysis for wedge failure at 508 slope angle.
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250 247
instability. However, there is an interesting conclu-
sion: as the slope angle is reduced, the probabilities of
kinetic failure when contact on both planes is lost
actually increase. This indicates that the effective
normal forces acting on both planes are reduced and
these values become smaller than the pore water
pressures on the sliding planes as the slope angle is
reduced (Duzgun et al., 2003). However, probabilities
of total kinetic instability are the same as the
probabilities for the 758 cut slope. That is, the
decrease in slope angle does not affect the distribution
and values of factor of safety, as shown previously for
planar failure. This can be confirmed by noting that
the factor for safety for the combination of J4 and J5
at the 758 slope angle in Table 6, 0.33, is the same as
the value for the 508 slope angle in Table 7.
Table 7
Results of wedge failure for the deterministic analysis and the probabilistic analysis in I-40 area at 508 cut slope
Set no. 1 Set no. 2 Factor of
safety
Probability of failure Total probability
of failure
Average volume of
possible wedge (m3)Kinematic No contact Plane 1 Plane 2 Both planes
J1 J2 Stable 0 0 0 0 0 0 0
J1 J3 Stable 0.003 0.015 0 0.012 0 0 0.005
J1 J4 Stable 0.003 0.005 0 0.011 0 0 0.001
J1 J5 Stable 0.004 0.17 0 0.22 0 0.0014 0.002
J1 J6 Stable 0 0 0 0 0 0 0
J2 J3 Stable 0 0 0 0 0 0 0
J2 J4 Stable 0.001 0.003 0 0.004 0 0 0.118
J2 J5 Stable 0.001 0.460 0 0.220 0.002 0.0006 0.441
J2 J6 Stable 0 0 0 0 0 0 0
J3 J4 Stable 0.190 0.017 0 0.008 0 0.0047 0.314
J3 J5 Stable 0.683 0.434 0 0.320 0.001 0.481 3.78
J3 J6 Stable 0 0 0 0 0 0 0
J4 J5 FS3=0.33 0.332 0.231 0 0.467 0 0.232 0.0381
J4 J6 Stable 0 0 0 0 0 0 0
J5 J6 Stable 0 0 0 0 0 0 0
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250248
Regarding the average volume of a possible block
for the 508 slope angle, volumes are reduced
significantly from those of the 758 slope, as expected.Among seven combinations which are analyzed as
stable by the deterministic but unstable in the
probabilistic analysis in Table 7, six joint combina-
tions have mean volume less than 0.6 m3. Many
combinations become stable and the mean volumes of
possible wedge are reduce as slope angle decreases. In
particular, the volume for the J3 and J5 combination is
reduced from 82.39 m3 to 3.78 m3, showing that the
probability of a large volume failure was significantly
reduced after the cut slope angle was flattened by
slope processes.
7. Summary and conclusions
Rock slope stability is highly dependent on
discontinuity characteristics, and the random proper-
ties of each parameter have an important effect in the
probabilistic analysis. Therefore, random properties
for geometric and strength parameters of discontinu-
ities play a critical role in the probabilistic analysis.
In this study, discontinuity parameters including
orientation, length and spacing were measured in the
field and their random properties determined on the
basis of physical considerations and goodness-of-fit
testing. In addition, the new concept of persistence,
which can be utilized effectively in the probabilistic
approach, was proposed. The proposed approach
simply uses joint length data rather than the persis-
tence value. This is significant because field deter-
mination of persistence is not possible on a practical
basis. Therefore, the proposed approach expresses the
probability that the joint length is equal to or greater
than the maximum sliding length, which is multiplied
by the probability of failure of the rock slope, the
latter being evaluated assuming a fully persistent
joint.
To evaluate rock slope stability, both kinematic and
kinetic conditions were examined. Both conditions
were evaluated simultaneously. This is because
examination of kinetic condition is conducted only
after the kinematic failure is indicated. Then the
probabilities of the two conditions were calculated
separately. They were combined to evaluate the
overall probability of slope failure. The Monte Carlo
simulation technique was utilized to analyze the
possibility of failure for planar and wedge features
in the study area. This probabilistic analysis was
applied to a study area on I-40 in western North
Carolina.
Comparisons between the deterministic analysis
and probabilistic analysis showed that results of the
probabilistic method yields significantly different
results from those of the deterministic analysis. In
some cases the deterministic analysis, based on a
fixed value for discontinuity and slope parameters, did
not indicate a slope failure condition whereas the
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250 249
probabilistic method did. Variations in discontinuity
orientation are one cause for this difference, as
discontinuity orientations have a wide scatter within
the same set. This illustrated by J1 in Table 4 and the
J3 and J5 combination of Table 7.
In this study, the study area has experienced several
small slope failures and a few large slope failures.
With the details of the slope failure history, the
probability of slope failure can be updated. According
to Bayesian approach, the probability of failure can be
updated by new information and subjective judgment.
In the approach, the probability of failure is treated as
a random variable and the probability of failure is
updated using all available information, both theoret-
ical and experimental (Powell and Pine, 1996).
Therefore, the probability of slope failure in this area
can be updated based on detailed failure history and
the authors will focus on this subject in a further
study.
Acknowledgements
This research is partly supported by Korea Minis-
try of Science and Technology (Project No. M1-0302-
00-0063). The authors would like to express their
gratitude to the anonymous reviewers for their
valuable comments.
References
Ang, A.H.S., Tang, W.H., 1975. Probability Concepts in Engineer-
ing Planning and Design, vol. 1. Wiley, New York.
Baecher, G.B., 1983. Statistical analysis of rock mass fracturing.
J. Math. Geol. 15 (2), 329–347.
Baecher, G.B., Lanney, N.A., 1978. Trace length biases in joint
surveys. Proceedings of 19th US Symposium on Rock
Mechanics, pp. 56–65.
Barton, N.R., 1973. Review of a new shear strength criteria for rock
joints. Eng. Geol. 7, 287–332.
Casagrande, A., 1965. Role of the bCalculated RiskQ in earthwork
and foundation engineering. J. Soil Mech. Found. Div. 91 (4),
1–40.
Cherubini, C., Cotecchia, V., Renna, G., Schiraldi, B., 1983. The
use of bivariate probability density functions in Monte Carlo
simulation of slope stability in soils. Proc. 4th Int. Conf. on the
Application of Statistics and Probability to Soil and Structural
Engineering, pp. 1401–1411.
Cruden, D.M., 1977. Describing the size of discontinuities. Int. J.
Rock Mech. Min. Sci. Geomech. Abstr. 14, 133–137.
Duzgun, H.S.B., Yucemen, M.S., Karpuz, C., 2003. A methodology
for reliability based design of rock slopes. Rock Mech. Rock
Eng. 36 (2), 95–120.
Einstein, H.H., 1996. Risk and risk analysis in rock engineering.
Tunn. Undergr. Space Technol. 11 (2), 141–155.
Einstein, H.H., Baecher, G.B., 1983. Probabilistic and statistical
methods in engineering geology; specific methods and
examples—Part 1: exploration. Rock Mech. Rock Eng. 16,
39–72.
Einstein, H.H., Veneziano, D., Baecher, G.B., O’Reilly, K.J., 1983.
The effect of discontinuity persistence on rock sloe stability. Int.
J. Rock Mech. Sci. Geomech. Abstr. 20, 227–236.
El-Ramly, H., Morgenstern, N.R., Cruden, D.M., 2002. Probabil-
istic slope stability analysis for practice. Can. Geotech. J. 39,
665–683.
Fisher, R.A., 1953. Dispersion on a sphere. Proc. R. Soc. Lond., A
217, 295–305.
Giani, G.P., 1992. Rock Slope Stability Analysis. A.A. Balkema.
Glass, F.R., 1998. A large wedge failure along Interstate 40 at North
Carolina–Tennessee State line. Proceedings of 48th Highway
Geology Symposium. Arizona Department of Transportation,
Prescott, Arizona, pp. 65–75.
Gunther, A., 2003. SLOPEMAP: programs for automated mapping
of geomaterical and kinematical properties of hard rock hill
slopes. Comp. Geosci. 29, 865–875.
Harr, M.E., 1987. Reliability Based on Design in Civil Engineering.
McGraw-Hill, New York.
Herget, G., 1978. Analysis of discontinuity orientation for a
probabilistic slope stability design. Proceedings of 19th U.S.
Symposium on Rock Mechanics, Reno, Nevada. University of
Nevada, pp. 42–50.
Hoek, E.T., 1998. Factor of safety and ptrobability of failure
(Chpater 8). Course notes, Internet edition, http://www.rockeng.
utoronto.ca/hoekcorner.htm.
Hoek, E.T., Bray, J.W., 1981. Rock Slope Engineering. Institute of
Mining and Metallurgy.
Hudson, J.A., Priest, S.D., 1983. Discontinuity frequency in rock
masses. Int. J. Rock Mech. Sci. Geomech. Abstr. 20, 73–89.
ISRM, 1978. Suggested methods for the quantitative description of
discontinuities in rock masses. Int. J. Rock Mech. Sci.
Geomech. Abstr. 15, 319–368.
Kemeny, J., 2003. The time reduction of sliding cohesion due to
rock bridge along discontinuities: a fracture mechanics
approach. Rock Mech. Rock Eng. 36 (1), 27–38.
Kulatilake, P.H.S.W., Wathugala, D.N., Stephansson, O., 1993.
Joint network modeling with a validation exercise in Strip mine,
Sweden. Int. J. Rock Mech. Sci. Geomech. Abstr. 30, 503–526.
Kulatilake, P.H.S.W., Um, J., Wang, M., Escandon, R.F., Varvaiz, J.,
2003. Stochastic fracture geometry modeling in 3-D including
validations for a part of Arrowhead East Tunnel, California,
USA. Eng. Geol. 70, 131–155.
Li, K.S., White, W., 1987. Probabilistic approaches to slope design.
Research Report, vol. 20. Dept. of Civil Engineering, Australian
Defense Force Academy, Canberra, Australia, p. 4.
Mahtab, M.A., Yegulalp, T.M., 1982. A rejection criterion for
definition of clusters in orientation data. Proceedings of 22nd
Symposium on Rock Mechanics. American Institute of
H.-J. Park et al. / Engineering Geology 79 (2005) 230–250250
Mining Metallurgical and Petroleum Engineers, New York,
pp. 116–123.
Martiz, J.S., 1981. Distribution-free Statistical Methods. Chapman
and Hall, London.
Mauldon, M., 1994. Intersection probabilities of impersistent joints.
Int. J. Rock Mech. Sci. Geomech. Abstr. 31 (2), 107–115.
Mauldon, M., Dunne, W.M., Rohrbaugh, M.B., 2001. Circular
scalines and circular windows: new tools for characterizing the
geometry of fracture traces. J. Struct. Geol. 23, 247–258.
McMahon, B.K., 1971. Statistical methods for the design of rock
slopes. 1st Australian–New Zealand Conference on Geome-
chanics, pp. 314–321.
Mostyn, G.R., Li, K.S., 1993. Probabilistic slope analysis—state of
play. Proceedings of Conference on Probabilistic Methods in
Geotechnical Engineering. A.A. Balkema, Canberra, Australia,
pp. 89–109.
Nilsen, B., 2000. New trend in rock slope stability analysis. Bull.
Eng. Geol. Environ. 58, 173–178.
NCDOT, 1980. I-40:Slope Stability Study Final Report, vols. 1,2,3.
North Carolina Department of Transportation.
Park, H.J., 1999. Risk analysis of rock slope stability and stochastic
properties of discontinuity parameters in western North Caro-
lina. PhD thesis. Purdue University.
Park, H.J., West, T.R., 2001. Development of a probabilistic
approach for rock wedge failure. Eng. Geol. 59, 233–251.
Pathak, S., Nilsen, B., 2004. Probabilistic rock slope stability
analysis for Himalayan condition. Bull. Eng. Geol. Environ. 63,
25–32.
Peck, R.B., 1969. Advantages of limitations of the observational
method in applied soil mechanics: 9th Rankie Lecture. Geo-
techniques 19 (2), 171–187.
Powell, N., Pine, R.J., 1996. Bayesian approach to slope stability
assessment by updating probability of failure treated as a
random variable. Trans. Inst. Min. Metall. 105, A31–A36.
Priest, S.D., 1993. Discontinuity Analysis For Rock Engineering.
Chapman and Hall, New York.
Priest, S.D., Hudson, J.A., 1981. Estimation of discontinuity
spacing and trace length using scanline surveys. Int. J. Rock
Mech. Min Sci. Geomech. Abstr. 18, 183–197.
Rethai, L., 1998. Probabilistic Solutions in Geotechnics. Elsevier,
Amsterdam.
Rouleau, A., Gale, J.E., 1985. Statistical characterization of the
fracture system in the Strip Granite, Sweden. Int. J. Rock Mech.
Min. Sci. Geomech. Abstr. 22 (6), 353–367.
Scavia, C., 1990. Fracture mechanics approach to stability analysis
of rock slopes. Eng. Fract. Mech. 35, 899–910.
Schultze, E., 1975. The general significance of statistics for the civil
engineer. Proceedings of the 2nd International Conference on
Application of Statistics and Probability in Soil and Structural
Engineering, Aachen.
Sen, Z., Kazi, A., 1984. Discontinuity spacing and RQD estimates
from finite length scanline. Int. J. Rock Mech. Min. Sci.
Geomech. Abstr. 21 (4), 203–212.
Singh, R.N., Sun, G.X., 1989. Fracture mechanics approach to slope
stability analysis. Int. Symp. on Surface Mining. University of
Nottingtton, England, pp. 93–97.
Terzaghi, R.D., 1965. Source of error in joint surveys. Geotechnique
15, 287–304.
Wallis, P.F., King, M.S., 1980. Discontinuity spacings in a
crystalline rock. Int. J. Rock Mech. Min. Sci. Geomech. Abstr.
17, 63–67.
West, T.R., 1996. The effects of positive pore pressure on sliding
and toppling of rock blocks with some considerations of intact
rock effects. Environ. Eng. Geosci. 2, 339–353.
Whitman, R.V., 1984. Evaluating calculated risk in geotechnical
engineering. J. Geotech. Eng., ASCE 110 (2), 145–186.
Whittaker, B.N., Singh, R.N., Sun, G., 1992. Rock Fracture
Mechanics; Principles, Design and Application. Elsevier.
Wiener, L.S., Merschat, C.E., 1975. Field Guidebook to the
Geology of the Central Blue Ridge of North Carolina and
Spruce Pine Mining District. Association of American State
Geologists.