Landslides and Engineered Slopes – Chen et al. (eds)
© 2008 Taylor & Francis Group, London, ISBN 978-0-415-41196-7
The evaluation of failure probability for rock slope based on fuzzy set theory
and Monte Carlo simulation
Hyuck-Jin Park
Department of Geoinformation Engineering, Sejong University, Republic of Korea
Jeong-gi Um
Department of Environmental Exploration Engineering, Pukyung National University, Republic of Korea
Ik Woo
Department of Ocean System Engineering, Kunsan National University, Republic of Korea
ABSTRACT: Uncertainty is pervasive in rock slope stability analysis due to various reasons and sometimes it
causes serious rock slope failures. Therefore, since 1980’s the importance of uncertainty has been recognized and
subsequently the probability theory has been used to quantify the uncertainty. However, not all uncertainties are
objectively quantifiable. Some uncertainties, due to incomplete information, cannot be handled satisfactorily in
the probability theory and the fuzzy set theory is more appropriate. In this study the random variable in rock slope
stability analysis is considered as fuzzy number and the fuzzy set theory is employed. In addition, the Monte
Carlo simulation technique is utilized to evaluate the probability of failure for rock slope. This overcomes the
shortcomings of the previous studies, which are employed vertex method, first order second moment method and
point estimate method. Since the previous studies used only the representative values from membership function
to evaluate the stability of rock slope, the approximated analysis results were obtained in the previous studies.
With Monte Carlo simulation technique, more complete analysis results can be secured in the proposed method.
The proposed method was applied to the practical example. According to the analysis results, the probabilities
of failure obtained from the fuzzy Monte Carlo simulation coincide with the probabilities of failure from the
probabilistic analysis.
1
INTRODUCTION
One of the difficulties in slope stability analysis is
uncertainty inevitably involved in the variability of
the material properties and the geotechnical model.
The natural materials comprising most slopes have an
innate variability difficult to establish and to predict
and therefore, the variability of geologic material is
one of the major sources of uncertainties. In addition,
insufficient amount of information for site conditions
and incomplete understanding of failure mechanism
are also another sources of uncertainties. Therefore,
the presence and the significance of uncertainties
in slope stability analysis has been appreciated for
long time. Consequently several approaches such as
observation method (Peck, 1969) have been suggested
to deal properly with uncertainty. The probabilistic
approach has been proposed as an objective tool for
representing uncertainty in failure model and material characteristics. Many probabilistic analyses have
been published in literature (Einstein and Baecher,
1982; Mostyn and Small, 1987; Mostyn and Li, 1993;
Nilsen, 2000; Park and West, 2001; El-Ramly et al.,
2002; Pathak and Nilsen, 2004; Park et al., 2005).
However, the probabilistic analysis requires the statistical parameters and distribution type for random
variables in order to quantify the uncertainty. That is,
the mean, standard deviation and probability density
function for uncertain parameters are prerequisite in
order to carry out the appropriate probabilistic analysis. However, a large amount of information and data
are required to obtain statistical parameters and distribution type for random variable but in many practical
conditions, the amount of data is frequently limited.
Consequently it is difficult to secure statistical parameters and distribution type of the uncertain variable, and
this situation makes the application of probabilistic
analysis difficult.
The uncertainties caused by limited or incomplete information cannot be handled satisfactorily
1943
2
FUZZY MONTE CARLO SIMULATION
METHOD
2.1
Fuzzy set theory
In classical set theory, an element either belongs or
does not belong to the set. That is, the membership of
classical set theory is defined in strict sense. When a
certain element x belongs to set A, x is a member or
element of a set A and can be written
x|A
(1)
Whenever x is not an element of a set A, we write
x|A
1.0
Membership function
in the probability theory and the fuzzy set is more
appropriate (Dodagoudar and Venkatachalan, 2000).
Therefore, the present study proposed the utilization
of fuzzy set theory in order to overcome the limitations
of the probabilistic approach. Fuzzy set theory has
been proposed by Zadeh (1965) and it has been known
as appropriate approach for dealing with uncertainty
mainly caused by incomplete information. Consequently, fuzzy set theory has been employed in many
slope stability analyses (Juang & Lee, 1992; Lee and
Juang, 1992; Davis and Keller, 1997; Juang et al.,
1998; Dodagoudar and Venkatachalan, 2000; Giasi
et al., 2003; Li and Mei, 2004). However, the previous studies combined the fuzzy set theory with the
approximate method such as point estimate method or
first order second moment method. Since the approximate methods use only few representative values
from uncertain parameters, the analysis cannot provide accurate analysis results. Therefore, this study
proposed the new approach incorporating the Monte
Carlo simulation which provides complete analysis
results with fuzzy set theory.
0.5
HEIGHT
CORE
0.0
SUPPORT
x
Figure 1.
Concept of membership function.
membership function in a fuzzy set may admit some
uncertainty, its membership is a matter of degree. The
membership function can be manifested by many different types of function and different shapes of their
graphs. Triangular and trapezoidal shapes are most
common types in the membership function. Fig. 1
shows the concept for support, core and height in a
trapezoidal shaped fuzzy set. The support is the set of
all elements of set x that have nonzero membership
in A. In addition, core is the set of all elements of x
for which the degree of membership in A is 1. The
height of a fuzzy set A may be defined as the largest
membership grade obtained by an element in that set.
If the height of a fuzzy set A is 1, set A is called
normal and otherwise, it is called subnormal.
There are two commonly used ways of denoting
fuzzy sets.
A = x, μA
x
1
or
(2)
A set can be defined by membership function that
declares which elements of x are members of the set
and which are not.
1, x | A
μA (x) =
(3)
0, x | A
For each x | A, when μA (x) = 1, x is declared to be
a member of A. When μA (x) = 0, x is declared to be
a nonmember of A.
However, in fuzzy sets, which is introduced by
Zadeh (1965), more flexible sense of membership is
possible. That is, the membership function can be generalized such that the values assigned to the elements
fall within a specified range. In fuzzy set, the degree
of membership to a set is indicated by a number of
between 0 and 1.
In fuzzy set theory, each fuzzy set is uniquely
defined by a membership function. Since an element’s
A=
μA (x)
x
2.2
Fuzzy Monte Carlo simulation
(4)
The probabilistic analysis has been known as an
effective tool to quantify and model uncertainty.
However, limited information for uncertain parameters makes the application of the probabilistic analysis difficult. This is because the probabilistic analysis
is carried out on the premise that the precise mean and
standard deviation and the appropriate probability density function for uncertain parameter can be obtained.
However, in order to obtain the adequate statistical parameters and distribution function for uncertain
parameter, a large amount of data is required but it
is often not practically possible. Frequently only the
maximum and minimum values for uncertain parameter can be obtained and therefore, uncertain parameter
can be expressed only with interval between minimum
1944
and maximum. Under this condition, uncertain parameter may be expressed as a fuzzy set, if there is some
reason to believe that not all values in the interval
have the same degree of support (Juang et al., 1998).
Since uncertainties due to incomplete information are
pervasive in the procedure of slope stability analysis,
several researches utilized fuzzy set theory in slope
stability analysis (Juang et al., 1998; Dodagoudar and
Venkatachalan, 2000; Giasi et al., 2003). However, the
previous researches utilized the vertex method (Dong
and Wong, 1987) to evaluate fuzzy input parameters
in slope stability analysis. The vertex method is based
on the α-cut concept of fuzzy numbers and involves an
interval analysis. The basic idea of the vertex method is
to discretize a fuzzy number into a group of α-cut intervals. By replacing fuzzy numbers in the slope model
with intervals, the fuzzy computation obtains factor
of safety in the deterministic slope model. However,
when the factor of safety is evaluated from the deterministic slope model using two interval values, the
first order second moment method (Giasi et al., 2003)
or point estimate (Dodagoudar and Venkatachalam,
2000) has been applied. According to Harr (1987), the
first order second moment method and point estimate
method are considered as approximate method since
the methods do not utilize complete information for
random variables to evaluate performance function.
The approximate method has been proposed to evaluate the probability using simple calculation with only
few representative values of random variable without
distribution information. However, since the previous
researches used incomplete information in the analysis, there is a possibility that approximate results would
be obtained instead of the precise analysis results.
Therefore, this study proposed the new approach
evaluating the reliability of rock slope with fuzzy number and Monte Carlo simulation. The Monte Carlo
simulation is the most complete method of the probabilistic analysis since all the random variables are represented by their statistical parameters and probability
density function. In addition, the complete information is employed to evaluate performance function in
Monte Carlo simulation. In order to combine Monte
Carlo simulation with fuzzy set theory, uncertain
parameter is considered as fuzzy number and its membership function is decided by means of available
information and engineering judgment. Then Monte
Carlo simulation is utilized to evaluate the probability
of slope failure from fuzzy numbers of uncertain parameters.
In most rock slope stability analyses, the friction angle of discontinuity is considered as uncertain
parameters. This is because the number of the direct
shear tests which are carried out to acquire shear
strength of discontinuity is always limited and therefore, the true value of friction angle cannot be evaluated. Consequently, in the present study the friction
angle is considered as fuzzy number and its membership function is decided on the basis of analysis
for laboratory test results. However, in the Monte
Carlo simulation, the cumulative density function for
uncertain parameter is required. In the present study,
the membership function is adapted to cumulative
density function in the calculation of performance
function. Then in Monte Carlo simulation, the process
takes a single value selected randomly from its cumulative distribution. The randomly selected parameter
is used to generate a single random value for factor of
safety. By repeating this process many times to generate a large number of different factors of safety, a
cumulative density function for factor of safety can be
obtained and then probability of failure is evaluated.
3
CASE STUDY
The proposed method in the present study has been
applied to practical example in order to check the feasibility and validity of the proposed approach and compare with the probabilistic analysis results. A slope has
been selected and the detailed field investigation has
been carried out. The dip direction and dip angle of
the slope are 325 degree and 65 degree, respectively
and its height is 40.8 m. The slope is composed of
Precambrian metasedimentary rock. Approximately
350 discontinuity data has been obtained on scanline survey and 6 discontinuity sets were identified
by means of clustering process (Table 1). Among
6 discontinuity sets, 2 sets (set 2 and set 4) are analyzed as kinematically unstable for planar failure. In
this study, the analysis for only planar failure is performed since the analysis results of planar failure are
easy to compare to other analysis results. In addition,
the direct shear test is carried out in order to acquire
the shear strength parameter for discontinuity. Based
on the 19 direct shear test results, the friction angle
ranges from 20.9 to 46.3 and their mean and standard
deviation are 34.6 and 8.2, respectively (Fig. 2). However, even if 19 tests were performed, the probability
density function cannot be determined due to severe
scattering as can be seen in Fig. 2. Even the previous
Table 1.
gation.
Discontinuity sets observed from field investi-
Discontinuity sets
Representative orientation
Set 1
Set 2
Set 3
Set 4
Set 5
Set 6
217/77
320/30
061/66
311/40
196/56
183/05
1945
0.12
5
0.10
4
0.08
Frequency
Frequency
6
3
2
0.04
0.02
1
0
0.06
0.00
20 - 25
25 - 30
30 - 35
35 - 40
40 - 45
0
45 - 50
Figure 2.
1
2
3
Factor of safety
Friction Angle
Figure 3.
Results of direct shear tests.
Results of probabilistic analysis for joint set 2.
0.16
3.1
Results of probabilistic analysis
In order to compare to other analysis results, the
deterministic analysis based on the limit equilibrium
approach has been carried out for joint set 2 and 4,
which are analyzed as kinematically unstable on the
stereonet analysis. This analysis has been performed
with same input values for all the deterministic parameters used in the probabilistic analysis and mean value
of the distribution for random parameter. The factor
of safety for set 2 is evaluated as 1.20 and the factor of safety for set 4 is 0.82. That is, joint set 2 has
been analyzed as stable in the deterministic analysis
but unstable for joint set 4.
The probabilistic analysis is also carried out for set 2
and set 4 using the procedure proposed by Park et al.
(2005). In the present study, the orientation of discontinuity is taken into account for the deterministic
parameter and therefore, the single fixed dip direction
and dip angle for discontinuity orientation is employed
in the probabilistic analysis. On the other hand, the
friction angle for discontinuity is considered as the random variable. The mean value and standard deviation
have been used and subsequently normal distribution
0.12
Frequency
researches proposed normal distribution for probability density function of friction angle (Mostyn and Li,
1993; Nilsen, 2000; Pathak and Nilsen, 2004; Park
et al., 2005), it is not easy to decide normal distribution as probability density function for friction angle
in this study due to uncertainty. Therefore, the friction
angle is considered as fuzzy number whose support
is between 20.9 and 46.3 in this study. The triangular shape is chosen for membership function of the
friction angle and the core value of membership function is decided to 34.6 which is mean value of the test
results. In addition, on the basis of Hoek’s suggestion
(1997) in rock slope stability analysis, cohesion is not
considered in slope stability analysis.
0.08
0.04
0.00
0
1
2
3
Factor of safety
Figure 4.
Results of probabilistic analysis for joint set 4.
has been chosen for probability density function of
friction angle. In order to evaluate the probability
of failure, the Monte Carlo simulation approach is
employed in the probabilistic analysis. Total 16,000
repeated calculations are carried out. Figs. 3 and 4
show the results of analysis, which show the distributions of the factor of safety. For joint set 2, the
probability of failure is evaluated as 29.3% and the
probability of failure for set 4 is 73.5%. In case of
set 2, the result of the deterministic analysis indicate
stable but the result of the probabilistic analysis shows
quite high probability of failure. This is because the
deterministic analysis does not reflect the variability
and uncertainty in input parameter.
However, the coefficient of variation for friction
angle used in this study is calculated as 23.3% and
this is quite high value compared to the other previous
researches, which show 10% (Park and West, 2001).
It means that the dispersion of direct shear test
results used in the present study is too large. That
is, the randomly generated friction angle from Monte
Carlo simulation ranges from 10 to 59.2 in the confidence interval of 99.8%. Consequently the uncertainty
of friction angle is too large and subsequently in
1946
1.0
0.21
0.8
Membership function
Frequency
0.28
0.14
0.07
0.6
0.4
0.2
0.00
0
1
2
3
0.0
Factor of safety
10
20
30
40
50
Internal friction angle
Figure 5. Results of probabilistic analysis for joint set 2
when COV = 10%.
Figure 7.
0.36
0.16
0.27
0.12
Frequency
Frequency
Triangular membership function.
0.18
0.09
0.08
0.04
0.00
0
1
2
0.00
3
0
Factor of safety
1
2
3
Factor of safety
Figure 6. Results of probabilistic analysis for joint set 4 on
when COV = 10%.
Monte Carlo simulation, too small value of friction
angle could be generated and used in factor of safety
calculation. This could cause serious error in the evaluation of the probability of failure. Therefore, in order
to check out the influence of uncertainty in input
parameters, the dispersion of friction angle is reduced
to 10% of C.O.V (coefficient of variation) and the
probability of failure is recalculated (Fig. 5 and 6).
Figures 5 and 6 show that the dispersion of factor
of safety is reduced comparing to Figures 3 and 4.
The probability of failure for joint set 2 is reduced to
8.8% but the probability of failure for joint set 4 is
increased to 94.0%. This is because the lower half
of the dispersion for factor of safety in joint set 2
is reduced but the upper half of the dispersion in
joint set 4 is reduced. This shows that uncertainty
and dispersion of input parameter affect the analysis
result. Consequently if the number of data is limited and subsequently the random properties cannot
be recognized precisely, the results of the probability
analysis can be affected by the dispersion of the input
parameters.
Figure 8.
Results of FMC analysis for joint set 2.
3.2 Fuzzy Monte Carlo simulation
As can be seen previously, the friction angle obtained
from direct shear test includes a large amount of uncertainty. This uncertainty is usually caused by a lack of
test results. A lack of test results prevents the precise understanding of random properties for uncertain
parameters and also makes the application of the probabilistic analysis difficult. Therefore, in the present
study the friction angle is considered as fuzzy number. The friction angle is considered as triangular fuzzy
number and the minimum and maximum values of the
membership function are decided as 20.9 and 46.3,
respectively on the basis of test results. In addition, the
mean value, 34.6 is decided as core in the membership
function (Fig. 7). This means that the randomly generated value from Monte Carlo simulation ranged from
20.9 to 46.3. This shows the dispersion of fuzzy number is much smaller than the dispersion used in the
probabilistic analysis. The C.O.V of the fuzzy number is calculated as 13.3% on the confidence level
of 99.8% and this value is smaller than the C.O.V
used in the probabilistic analysis. The distribution of
1947
0.12
Frequency
0.10
0.08
0.06
0.04
0.02
0.00
0
1
2
3
Factor of safety
Figure 9.
Results of FMC analysis for joint set 4.
factor of safety evaluated from fuzzy Monte Carlo
simulation is given in Figures 8 and 9. Comparing
the factors of safety distribution obtained from fuzzy
Monte Carlo simulation with the factors of safety distribution obtained from the probabilistic analysis, the
dispersion of factor of safety distribution in fuzzy
Monte Carlo simulation is reduced. In accordance
with the analysis results, the probability of failure for
joint set 2 is 33.5% and the probability of joint set 4
is 72.9%. In case of joint set 2, the probabilities of
failure evaluated from the probabilistic analysis and
fuzzy Monte Carlo simulation are 29.3% and 33.5%
respectively and the analysis results are somewhat different. But in case of joint set 4, the probabilities of
failure evaluated from the probabilistic analysis and
fuzzy Monte Carlo simulation are 73.5% and 72.9%
respectively and the analysis results are quite similar.
Consequently, even if the application of fuzzy set
theory reduced the dispersion in input value, the
probabilities of failure obtained from two different
approaches are similar. As a result, the application
of fuzzy set theory manages the uncertainty of input
parameter effectively.
4
CONCLUSIONS
Uncertainty is pervasive in rock slope stability analysis due to various reasons and sometimes it causes
serious rock slope failures. Therefore, the probability
theory has been used to quantify the uncertainty. However, not all uncertain-ties are objectively quantifiable.
Some uncertainties, due to incomplete information,
cannot be handled satisfactorily in the probability theory and the fuzzy set theory is more appropriate. In
this study the random variable in rock slope stability
analysis is considered as fuzzy number and the fuzzy
set theory and Monte Carlo simulation are employed.
In order to verify the feasibility and validity of the
proposed approach, the proposed method was applied
to the practical example. In the deterministic analysis results, joint set 2 is analyzed as stable but joint
set 4 is analyzed as unstable. On the contrary in the
probabilistic analysis results, the probability of failure for joint set 2 is 29.3% and the probability for
joint set 4 is 73.5%. The data used in the probabilistic
analysis are widely scattered since the COV of friction
angle is evaluated as 23.3%. The widely scattered data
may cause serious miscalculation in the evaluation of
the failure probability since impractical data could be
used in the calculation. Therefore, the probability of
failure is recalculated with the modified data whose
COV is reduced to 10.0% and the probabilities has
been changed. The analysis results of the proposed
method using fuzzy Monte Carlo simulation are 33.5%
for joint set 2 and 73.5% for join set 4 and the COV of
the data that considered as fuzzy number is 13.3%. The
probabilities from the probabilistic analysis and the
proposed method are somewhat similar but the COV
of data in the proposed method is smaller than the COV
in the probabilistic analysis. Therefore, the fuzzy set
theory managed uncertainty in data more effectively
than the probabilistic analysis.
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