probabilistic approach of rock slope stability

1. ICCBT2008
Probabilistic Approach of Rock Slope Stability Analysis Using
Monte Carlo Simulation
M. S. Mat Radhi, Universiti Putra Malaysia, MALAYSIA
N. I. Mohd Pauzi*, Universiti Tenaga Nasional, MALAYSIA
H. Omar, Universiti Putra Malaysia, MALAYSIA
ABSTRACT
___________________________________________________________________________
Probabilistic analysis has been used as a tool to analyze and model variability and
uncertainty for rock slope analysis. Uncertainty in rock slope may appear as scattered values
of discontinuity length and persistence. This study is to develop the probabilistic approach of
rock slope stability based on discontinuity parameters using Monte Carlo simulation. The
probabilistic analysis was done using kinematic and kinetic analysis. Kinematic analysis is
based on stereographic projection analysis and kinetic analysis is based on the deterministic
analysis. Factor of Safety (FOS) is determined for each type of failure i.e. planar and wedge
failure. The slope that has FOS less than 1.00 is considered as not stable and FOS more than
1.00 is considered as stable. Data of six slopes which is denoted as Slope S1, S2, S3, S4, S5
and S6 show that, Slope S2, Slope S4, and Slope S6 have FOS of 0.953, 0.991, and 0.891
respectively which show the slope as not stable. Whilst for wedge failure analysis, all the
slopes show FOS greater than 1.00 which is stable, although the kinematic analysis
(stereographic projection) shows otherwise. Probabilistic analysis is developed for rock slope
stability using Monte Carlo Simulation. Monte Carlo simulation calculate the probability of
failure for planar and wedge type of failure. The probability of failure (Pf) for planar failure
at slope S2, S4, and S6 are 51.6%, 17.8%, and 49% respectively. Wedge failure analysis show
0% probability of failure for dry slope cases while for wet slope cases, all slopes excluded the
S1 has the probability of failure (Pf) varies from 7.7% to 75.2%. This shows that the
probabilistic analysis will give relevant and enhance results which can help to determine
instability of rock slope. The development of probabilistic analysis using Monte Carlo
simulation is useful tool to get an accurate data in stability analysis of rock slope which have
great values of uncertainty.
Keywords: Probabilistic Approach, Monte Carlo Simulation, Rock slope stability
*Correspondence Authr: Nur Irfah Mohd Pauzi, Universiti Tenaga Nasional, Malaysia. Tel: +60389212020 ext
6254, Fax: +60389212116. E-mail: irfah@uniten.edu.my
ICCBT 2008 - E- (37) – pp449-468

2. Probabilistic Approach of Rock Slope Stability Analysis Using Monte Carlo Simulation
1.
INTRODUCTION
Uncertainty and variability are common in engineering geology studies dealing with natural
materials. This is because of rocks and soils are inherently heterogeneous, insufficient amount
of information for site conditions are available and the understanding of failure mechanism is
incomplete. There are many researcher have made efforts to limit or quantify uncertainty of
input data and analysis results. Perhaps, slope engineering is the geotechnical subject most
dominated by uncertainty since slopes are composed of natural materials [2]. Uncertainty in
rock slope engineering may occur as scattered values for discontinuity orientations and
geometries such as discontinuity length and persistence. Therefore, one of the greatest
challenges for rock slope stability analysis is the selection of representative values from
widely scattered discontinuity data. Since geotechnical engineering problems are
characterized by uncertain variables, design is always subjected to uncertainties
Application of probabilistic analysis has provided an objective tool to quantify and model
variability and uncertainty. It makes the rock slope stability possible to consider uncertainty
and variability in geotechnical and geological parameters. There is several commercial
available limit equilibrium codes (such as SWEDGE, ROCKPLANE, SLIDE, SLOPE/W)
often incorporate probabilistic tools, in which variations in discontinuity properties can be
assessed.
Various probabilistic studies of rock slopes and mining areas have been carried out by these
researchers [10, 11, 1, 9, 5, 6, 7, 14]. Though in Malaysia, such research are very few and
limited.
In summary, this study is to determine probabilistic analysis of rock slope stability based on
discontinuity parameters which is analyze and simulate probabilistic analysis method. Hence,
it would become helpful for the engineers to design and monitor the rock slope. The main aim
of this research is to determine probabilistic analysis of rock slope stability based on
discontinuity parameters which will help the slope engineers in rock slope design and stability
analysis.
2.
BASIC THEORY
The development of road and highway constructions involved deep cutting into the slope, in
order to minimize the traveling time and distance between the two places. Besides, the need of
development on hilly areas for building and residential purpose has also increased and these
lead to the concern of safety and stability of the slope for the public. The slope failure occur
due to human and natural causes which consist of improper planning, design and
implementation of the projects for human error while natural causes may be result of
weathering process, weak material and geological setting of the area.
In this research, cutting of rock slope is the major concern to be studied since a high degree of
reliability is required because slope failure or even rock falls can rarely be tolerated. Rock
slope stability is concern about analyzing the structural fabric of the site to determine if the
orientation of the discontinuities could result in instability of the slope under consideration.
Basically, there are four types of rock slope failures that always occurred at the rock slope
which are planar failure, wedge failure, toppling failure, and circular failure.
450
ICCBT 2008 - E- (37) – pp449-468

3. M. S. Mat Radhi et. al.
Planar failure is movement occurs by sliding on a single discrete surface that approximates a
plane and it is analyzed as two-dimensional problems which additional discontinuities may
define the lateral extent of planar failures, but these surfaces are considered to be release
surfaces, which do not contribute to the stability of the failure mass. Wedge failure happened
when rock masses slide along two intersection discontinuities both of which dip out of the cut
slope at an oblique angle to the cut face, forming a wedge-shaped block. Toppling failure
happened most commonly in rock masses that are subdivided into a series of slabs or columns
formed by a set of fractures that strike approximately parallel to the slope face and dip steeply
into the face. Circular failure is defined as a failure in rock for which the failure surface is not
predominantly controlled by structural discontinuities and that often approximately the arc of
a circle. Rock types that are susceptible to circular failures include those that are partially to
highly weathered and those that are closely and randomly fractured.
Applications of probabilistic analysis in geotechnical engineering have increased remarkably
in recent years. This is ranging from practical design and construction problems to advanced
research publications. A lot of study and research have been conducted regarding probabilistic
analysis since geotechnical and geological engineering deal with material whose properties
and spatial distribution are poorly known. Consequently, a somewhat different philosophical
approach is necessary to overcome the uncertainty occurs in geotechnical and geological
engineering.
This paper explains on determining probability of failure of the rock slope which deals with
the uncertainty in geotechnical and geological engineering parameter using Monte Carlo
simulation. The Monte Carlo simulation used the extensive computational effort involved in
the simulations required researchers to develop their own software to solve slope stability
problems. The limitations and sometimes the complexities of probabilistic methods combined
with the poor training of most engineers in statistic and probabilistic theory have substantially
inhibited the adoption of probabilistic slope stability analysis in practice.
3.
METHODOLOGY
Methodology of the research can be described by four stages; the first one is literature search
and formulation of objective, the second one is data collection at the field, the third one is
analysis and overview on the data collection and finally, the fourth stage is the development
of the probabilistic approach using Monte Carlo simulation and finally suggestion for further
work to be done for this research. The flow chart of the methodology is shown in Figure 1.
ICCBT 2008 - E- (37) – pp449-468
451

4. Probabilistic Approach of Rock Slope Stability Analysis Using Monte Carlo Simulation
Figure 1. Methodology of the research
The data collections at site are such as discontinuity data on the cut slope, classification and
identification of grade weathering and lithology of the selected cut slope. Then, the analysis
on the fieldwork data is done using kinematic analysis and kinetic analysis. The kinematic
analysis is done using the data collection of geological structural at site. The DIPS 3.0
software is used to give the analysis in Rosette Plot, Scatter Plot, Pole Plot, and Potential
Instability.
Limit equilibrium method which is also known as kinetic analysis is carried out to determine
factor of safety for each cases of potential instability. The probabilistic approach is developed
using Monte Carlo Simulation Method for rock slope stability analysis. The development of
this probabilistic approach is done using spreadsheet software such as EXCEL and
probabilistic simulation is done using RISKAMP. These two software are link together and
the analysis is simulated which is called Monte Carlo Simulation.
Probabilistic approach is carried out by simulating the results according to number of
iteration. For each number of iteration, it would give the Factor of Safety (FOS) from where
the probability of failure can be obtained. The simulation carried out here are for 10, 100, 500,
452
ICCBT 2008 - E- (37) – pp449-468

5. M. S. Mat Radhi et. al.
1000, 5000, and 10000 no of iteration. The input data needed for the development of
probabilistic approach is the kinematic and kinetic analysis data. The kinematic analysis
inputs are the dip and dip direction data, friction angle, cohesion, and slope angle. The outputs
data are the type of failure of the slope whether it is planar, wedge, toppling or combination of
the two type of failure.
The kinetic analysis requires input data such as slope properties, cohesion, friction angle, and
groundwater table. The outputs from this analysis are the FOS of the slope. If FOS is less than
1, the slope is considered fail and if the FOS is more than 1, the slope is stable. The output
from kinetic analysis only gives one value of FOS. Then, when the simulation is carried out
for 10 times, 100 times, 500 times, 1000 times, 5000 times and 10000 times, the RiskAMP
software would gives many results of FOS and probability of failure, Pf for that particular
slope.
4.
RESULTS AND ANALYSIS
4.1 Pos Selim Area
The probabilistic analysis which is developed for this study has been tested using data from
Pos Selim Highway. This probabilistic analysis is used to get accurate result and to determine
the uncertainties in geotechnical data. The probabilities of failure of the slope are the outcome
of this research.
Pos Selim Highway is located in Perak and can be accessed from Simpang Pulai or Cameron
Highland. The highway is part of the Malaysian Plan for East West second Link and divided
into eight packages. The highway starts from Simpang Pulai in Perak and ends at Kuala
Berang in Terengganu. Package two has been awarded to MTD Construction Sdn. Bhd. under
a Fixed Turnkey Lump Sum contract of total RM 282 million. The construction of the
highway in package two has started in May 1997 and was scheduled for completion in April
2000 [8]. Due to continuous cut slope failure along the highway, the construction of this
project was delayed [12]. Now in the year of 2005, the project has been opened to be used for
the public.
Along the terrain of Pos Selim Highway, there are two types of main lithological units which
are igneous and metasediment rock (Figure 4.1). The igneous rocks consist of granite and
metasediment rocks consist of quartz mica schist, quartz schist and closely foliated phylite.
Granite rock covers over 65% while metasediment is about 35% [8].
Locations of slope study are distributed over three granite slopes and three schist slopes.
These slopes have been labeled as a S1, S2, and S3, which cover the granitic areas and S4, S5,
and S6 cover the schist areas (Table 1).
ICCBT 2008 - E- (37) – pp449-468
453

6. Probabilistic Approach of Rock Slope Stability Analysis Using Monte Carlo Simulation
S
S
S
S
S
S
Figure 2. Pos Selim Area with granite and schist formation
Table 1: Study Locations and Lithology
Slope
S1
S2
S3
S4
S5
S6
Location
Ch 2 + 960
Ch 9 + 100
Ch 17 + 600
Ch 18 + 280
Ch 18 + 800
Ch 20 + 750
Lithology
Granite
Granite
Granite
Schist
Schist
Schist
4.2 Kinematic Analysis
Kinematic analysis is done to plot the discontinuity data such as dip and dip direction into
graphical method. The graphical method which are discussed in this analysis are pole plot,
rosette plot, scatter plot. From these plots, the potential instability for each slope can be
determined.
4.2.1
Pole plot
From the six slopes that have been chosen for study, about 637 of discontinuity data have
been collected for analysis. From Figure 3, it can be seen that joint is the most dominant and
common at field, followed by fault and lastly foliation. The percentage occurrence of joint
from field measurement is 79.7%, while fault is 10.9% and another 9.4% is foliation. S1, S2
454
ICCBT 2008 - E- (37) – pp449-468

7. M. S. Mat Radhi et. al.
(Figure 4), and S3 show that joint is dominant and higher which are 97%, 81%, and 97%
respectively from field measurement. S4, S5 (Figure 5), and S6 show that percentage of joint
decrease which is 63%, 70%, and 67% respectively.
120
100
80
60
40
20
0
S1
S2
Joint
S3
Fault
S4
S5
S6
Foliation
Figure 3. Discontinuity data showing percentage of joint, fault and foliation at S1, S2, S3, S4,
S5 and S6.
ICCBT 2008 - E- (37) – pp449-468
455

8. Probabilistic Approach of Rock Slope Stability Analysis Using Monte Carlo Simulation
Figure 4. Pole Plot at S2
Figure 5. Pole Plot at S5
4.2.2 Scatter Plot Analysis
Scatter plot data (Figures 6) collected for six numbers of slopes, the total number of plot is
584. Table 2 has summarized scatter plot for each slope which shows three category of plot;
one, two, and three plots. The importance of the scatter plot is to distinguish any discontinuity
data that have the same value of dips and dips direction, where in pole plot it does not show
the poles that share the same value.
456
ICCBT 2008 - E- (37) – pp449-468

9. M. S. Mat Radhi et. al.
Figure 6: Scatter plot data
Table 2. No. of Scatter plot for each study slope
Slope
One Plot(■)
Two Plot(▲)
Three Plot(►)
Total Plot
S1
89
10
1
100
S2
98
7
1
106
S3
91
4
1
96
S4
85
6
1
92
S5
81
9
3
93
S6
94
3
0
97
Total
538
39
7
584
4.2.3 Rosette Plot Analysis
There are 525 of planes out of 637 discontinuities that have been plotted into rosette plot
(Figure 7) covering S1, S2, S3, S4, S5, and S6. Dips direction of each slope is plotted into
respected bin at interval of 10 degrees. Table 3 shows the maximum and minimum
frequencies of plotted plane for each slope, which are 10 and 1 respectively. Each slope has
only two bin of maximum frequency, whilst the number of bin for minimum frequency varies
between two and four.
ICCBT 2008 - E- (37) – pp449-468
457

10. Probabilistic Approach of Rock Slope Stability Analysis Using Monte Carlo Simulation
Figure 7. Rosette Plot
Table 3. No of plotted plane in Rosette Plot for each study slope
Plotted
Maximum Plot
Minimum
Total
Planes
(Bin No)
Plot (Bin No)
Discontinuity
S1
92
6a & 24a
112
S2
98
7b & 25b
S3
92
13a & 31a
S4
79
7a & 25a
S5
84
6a & 24a
S6
80
18a & 36a
4c, 12c, 16c, 22c, 29c, &
34c
c
c
c
1 , 5 , 13 , 19c, 23c,&
31c
6c, 10c, 24c, & 28c
4c, 5c, 9c, 21c, 22c, &
27c
d
d
4 , 11 , 14d, 22d, 29d, &
32d
8c, 16c, 26c, & 34c
Total
525
118
Slope
115
102
100
108
100
637
38
a = frequency of 10, b = frequency of 9, c = frequency of 1, d = frequency of 2.
4.2.4 Potential Instability
Potential instability analysis is determined using stereoplot computer software, DIPS. This
analysis facilitates the determination of possible kinematic sliding of weathered rock slope in
types of planar, wedge, and toppling failure. Planar and wedge failure analysis is referred to
the work by Hoek and Bray [4], but toppling failure analysis is referred to Goodman and Bray
[3].
458
ICCBT 2008 - E- (37) – pp449-468

11. M. S. Mat Radhi et. al.
The discontinuity sets obtained from the geological mapping are then plotted for potential
instability analysis together with geometry of the slope and its friction angle. Planar failure
analysis describe that any pole (discontinuity) falling outside of pole friction cone represents a
plane which could slide if kinematically possible. The crescent shape zone formed by the
Daylight Envelope and the pole friction circle therefore encloses the region of planar sliding.
Any poles in this region represent planes that can and will slide (Figure 8).
Figure 8. Planar sliding zone represented by crescent shaped region.
The discontinuity set will be plotted together with the slope face and friction angle to
determine the type of potential failure. Major discontinuities sets and stereographic intensities
for each location of slope are shown in Table 4.
Table 4. Major discontinuities set and Fisher Concentration
Slope and
Fisher
Discontinuity set
Location
Concentration
(dip/dip direction)
4-7%
77°/170°
J1
Joint
Granite
4-7%
80°/318°
J2
Joint
Granite
J3
Joint
Granite
J4
Joint
Granite
S1
CH 2960
4-7%
°
33 /13
°
°
°
Marked
Types of
discontinuity
Lithology
4-7%
72 /244
4.5-6%
59/231
J1
Joint
Granite
S2
>12%
82/336
J2
Joint
Granite
CH 9100
3-4.5%
80/301
J3
Joint
Granite
3-4.5%
48/351
J4
Joint
Granite
ICCBT 2008 - E- (37) – pp449-468
459

12. Probabilistic Approach of Rock Slope Stability Analysis Using Monte Carlo Simulation
Table 4 Continue
5-6%
86/122
J1
Joint
Granite
S3
>10%
68/220
J2
Joint
Granite
CH 17600
3-4%
52/79
J3
Joint
Granite
7-8%
71/313
J4
Joint
Granite
8-9%
45/87
J1
Joint
Schist
S4
6-7%
65/322
J2
Joint
Schist
CH 18280
6-7%
74/208
J3
Joint
Schist
6-7%
66/237
J4
Joint
Schist
>12%
35/98
J1
Foliation
Schist
S5
10.5-12%
71/252
J2
Joint
Schist
CH 18800
4.5-6%
87/176
J3
Joint
Schist
4.5-6%
62/315
J4
Joint
Schist
>10%
62/261
J1
Joint
Schist
S6
7-8%
19/191
J2
Joint
Schist
CH 20750
6-7%
68/292
J3
Joint
Schist
4.3 Kinetic Analysis
Kinetic analysis is carried out by applying direct formula using single fixed values (typically,
mean values). Therefore, the stability analysis is carried out using only one set of geotechnical
parameter. Factor of safety, based on limit equilibrium is widely used to evaluate slope
stability because of its simple calculation and results.
From this study, three slopes out of six slopes have been identified potential planar failure
which is S2, S4, and S6. Through these three slopes, only three joint sets have fall and satisfy
with conditions for planar failure. Table 5 below shows factor of safety for each joint set for
wet and dry case for planar failure.
460
ICCBT 2008 - E- (37) – pp449-468

13. M. S. Mat Radhi et. al.
Table 5. Results of Factor of Safety for Planar Failure
Slope
S1
S2
S3
S4
S5
S6
Joint Set
I.D.
J1
J2
J3
J4
J1
J2
J3
J4
J1
J2
J3
J4
J1
J2
J3
J4
J1
J2
J3
J4
J1
J2
J3
Slope
Face
(deg)
63
63
73
63
102
63
Planar
Failure
(deg)
77
80
33
72
59
82
80
48
86
68
52
71
45
65
74
66
35
71
87
62
62
19
68
Mean Friction
Angle (deg)
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
Factor of
Safety
(Wet)
Stable
Stable
Stable
Stable
0.879
Stable
Stable
Stable
Stable
Stable
Stable
Stable
0.899
Stable
Stable
Stable
Stable
Stable
Stable
Stable
0.767
Stable
Stable
Factor of
Safety
(Dry)
Stable
Stable
Stable
Stable
0.953
Stable
Stable
Stable
Stable
Stable
Stable
Stable
0.991
Stable
Stable
Stable
Stable
Stable
Stable
Stable
0.891
Stable
Stable
For wedge planar failure case, five slopes have been identified potential to fail which are S2,
S3, S4, S5, and S6. Nine intersections of joint sets have been identified and satisfied for this
type of failure. The results for factor of safety for each intersection joints are shown in Table
6 below for wet and dry case.
Table 6. Results of Factor of Safety for Wedge Failure
Slope
S2
S3
S4
S5
S6
Intersection
Joint
Slope Face
(deg)
J1J2
J1J3
J1J4
J3J1
J3J4
J1J3
J2J3
J3J4
J1J3
63
73
63
102
63
ICCBT 2008 - E- (37) – pp449-468
Dip of
intersection
(deg)
58
36
33
43
48
56
17
38
30
Mean
Friction
Angle (deg)
30
30
30
30
30
30
30
30
30
Factor of
Safety
1.296
1.411
Stable
0.931
0.999
1.395
1.434
0.831
Stable
Factor of
Safety
(Dry)
Stable
Stable
Stable
Stable
Stable
Stable
Stable
1.572
Stable
461

14. Probabilistic Approach of Rock Slope Stability Analysis Using Monte Carlo Simulation
4.4 Probabilistic Analysis
Probabilistic analysis is carried out using Monte Carlo Simulation method which simulates
results according to number of iteration. Each number of iteration would give factor of safety
(FOS) which from this probability of failure can be calculated for that particular plane of
failure. The simulation carried out here are for 10, 100, 500, 1000, 5000, and 10000 no of
iteration. Same like deterministic analysis, probabilistic analysis has been carried out for
planar failure and wedge failure analysis to determine the factor of safety for each case.
For the purpose of this paper, only the results for slope S2 are shown. The type of wedge
failure has been identified for slope S2. Intersection of J1 and J2 at S2 for wedge failure
analysis shows that, probability of failure is 75% and 0% for wet and dry slope respectively
for the case of iteration of 10000 (Table 7). The mean of FOS for this analysis in Figure 9
shows that the values are constant and maintain at the rate of 0.68 to 0.716 for wet slope
cases. However, Figure 10 shows that the Pf of each number of iteration varies and this is
support by the Figure 11 which shows more details about histogram of factor of safety for
each number of iteration.
Table 7. Probabilistic Analysis on Wedge Failure, S2, J1J2 showing the values of Factor
of Safety
No of
Standard
Mean
Min
Max
Med
10
0.68
0.22
1.36
0.7
0.36
0.9
100
0.75
0.09
1.64
0.75
0.33
0.76
500
0.71
0
1.83
0.67
0.32
0.798
1000
0.71
0
1.9
0.69
0.32
0.769
5000
0.72
0
1.89
0.68
0.32
0.802
10000
0.716
0
1.983
0.683
0.328
0.752
10
1.896
1.41
2.572
1.921
0.364
0
100
1.967
1.304
2.85
1.979
0.33
0
500
1.921
1.21
3.033
1.888
0.318
0
1000
1.925
1.22
3.105
1.894
0.323
0
5000
1.927
1.207
3.093
1.897
0.323
0
10000
1.927
1.145
3.186
1.893
0.327
0
Iteration
WS
DS
Deviation
Pf
Legend: WS = Wet slope, DS = Dry slope
462
ICCBT 2008 - E- (37) – pp449-468

15. M. S. Mat Radhi et. al.
2.5
Mean o f F o S
2
1.5
1
0.5
0
10
100
500
1000
5000
10000
No of Iteration
Wet Slope
Dry slope
Figure 9. Mean of FOS at J1J2, S2 for each iteration.
1
0.8
Pf
0.6
0.4
0.2
0
10
100
500
1000
5000
10000
No of Iteration
Wet Slope
Dry Slope
Figure 10. Probability of wedge failure at J1J2, S2 for each iteration.
ICCBT 2008 - E- (37) – pp449-468
463

16. Probabilistic Approach of Rock Slope Stability Analysis Using Monte Carlo Simulation
2.5
3.5
3
2
2.5
1.5
Frequency
Frequency
2
1.5
1
1
0.5
0.5
2.
64
2.
52
2.
4
2.
28
2.
16
2.
04
1.
92
1.
8
1.
68
1.
44
1.
47
1.
35
1.
23
1.
11
0.
99
0.
87
0.
75
0.
63
0.
51
0.
39
0.
27
1.
56
0
0
Factor of Safety
Factor of Safety
(a) Wedge Failure Wet Slope (left) and Dry Slope (right) for 10 iteration
12
14
10
12
10
Frequency
6
4
8
6
4
Factor of Safety
2.
98
2.
82
2.
66
2.
5
2.
34
2.
18
2.
02
1.
86
1.
7
1.
38
1.
77
1.
61
1.
45
1.
29
1.
13
0.
97
0.
81
0.
65
0.
49
0
0.
33
2
0
0.
17
2
1.
54
Frequency
8
Factor of Safety
(b) Wedge Failure Wet Slope (left) and Dry Slope (right) for 100 iteration
70
60
60
50
50
40
40
Frequency
Frequency
70
30
30
20
20
Factor of Safety
3.
09
2.
91
2.
73
2.
55
2.
37
2.
19
2.
01
1.
83
1.
65
1.
47
1.
29
1.
87
1.
69
1.
51
1.
33
1.
15
0.
97
0.
79
0.
61
0.
43
0
0.
25
10
0
0.
07
10
Factor of Safety
(c) Wedge Failure Wet Slope (left) and Dry Slope (right) for 500 iteration
Figure 11. Histogram of FOS calculated in probabilistic analysis for combination of joint set 1
and 2 at S2.
464
ICCBT 2008 - E- (37) – pp449-468

17. M. S. Mat Radhi et. al.
140
120
120
100
80
80
Frequency
60
40
60
40
3.
11
2.
93
2.
75
2.
57
2.
39
2.
21
2.
03
1.
85
1.
67
1.
31
2.
06
1.
86
1.
66
1.
46
1.
26
1.
06
0.
86
0.
66
0
0.
46
0
0.
26
20
0.
06
20
1.
49
Frequency
100
Factor of Safety
Factor of Safety
(d) Wedge Failure Wet Slope (left) and Dry Slope (right) for 1000 iteration
700
600
600
500
500
400
400
Frequency
Frequency
700
300
300
200
200
100
100
0
Factor of Safety
3.
2
3
2.
8
2.
6
2.
4
2.
2
2
1.
8
1.
6
1.
4
1.
2
1.
99
1.
79
1.
59
1.
39
1.
19
0.
99
0.
79
0.
59
0.
39
0.
19
-0
.0
1
0
Factor of Safety
(e) Wedge Failure Wet Slope (left) and Dry Slope (right) for 5000 iteration
1200
1200
1000
1000
800
800
Frequency
1400
Frequency
1400
600
400
200
600
400
200
0
0
0.04 0.24 0.44 0.64 0.84 1.04 1.24 1.44 1.64 1.84 2.04
1.24 1.44 1.64 1.84 2.04 2.24 2.44 2.64 2.84 3.04 3.24
Factor of Safety
Factor of Safety
(f) Wedge Failure Wet Slope (left) and Dry Slope (right) for 10000 iteration
Figure 11: Histogram of FOS calculated in probabilistic analysis for combination of joint set 1
and 2 at S2 (continued).
The result could be interpreted that the increase in the number of iteration in Monte Carlo
simulation, the result becomes even details and thus increase the accuracy of calculation of
factor of safety of the rock slope. Probability of failure for dry slope at slope S2 is zero which
means the slope is stable and when the slope in wet condition, the probability of failure is in
the range of 0.752 to 0.802.
ICCBT 2008 - E- (37) – pp449-468
465

18. Probabilistic Approach of Rock Slope Stability Analysis Using Monte Carlo Simulation
5.
CONCLUSIONS
For the planar failure shown in Table 8 for 10000 number of iteration, deterministic
analysis of J1 at S2 gives 0.879 and 0.953 for wet and dry slope cases, while in probabilistic
analysis it gives 76.3% and 51.6% respectively. For J1 at S4, deterministic analysis
shows the lowest values of FOS; 0.899 and 0.991. But probabilistic analysis gives
the result of 38% and 17.8% for wet and dry slope cases. For planar failure of J1 at S6 gives
the high values of probability failure which is 62.9% and 49% and deterministic analysis
results are 0.767 and 0.891 respectively. These indicate that the slope has high possibility
of planar failure at J1 for S2 and S6, compare to J1 of S4. The Slope of S2, S4, and S6 also
show that probability of failure is high even the slopes are in dry condition.
For wedge analysis shown in Table 9 for 10000 number of iteration, high probability of
failure are determined at J1J2 and JIJ3 of S2, with 75.2 % and 57.9% respectively,
J3J4 of S3 with 48.2%, and J1J3 of S4 with 43%. Even deterministic values show
the FOS is more and equal to 1.00, it still has a higher probability to fail in these
circumstances. For example in slope S2 for wedge failure in Table 9, the FOS values is 1.434
but the probabilistic analysis result show otherwise where its probability of failure is 0.77 for
wet slope. This means that although factor of safety calculation said the slope is stable but the
probabilistic analysis run using Monte Carlo has detailed out the calculation and indicates the
slope is not stable.
Others intersection shows the lower results of probabilistic analysis with less than 40% for
each cases. The probabilistic analysis for wedge failure show that in dry condition, the value
of Pf is equal to 0, which mean that the slope is stable.
Table 8. Comparison of results for the deterministic and probabilistic analysis
(iteration of 10000) for planar failure
Deterministic
Probabilistic
Joint Set
Potential
Slope
Analysis (FOS)
Analysis (Pf)
Instability
I.D
Wet
Dry
Wet
Dry
Stable
0
0
No
Stable
Stable
Stable
0
0
J3
Planar
Stable
Stable
0
0
J4
No
Stable
Stable
0
0
J1
Planar
0.953
0.763
J2
J3
No
No
J4
No
0.879
Stable
Stable
Stable
Stable
Stable
Stable
0
0
0
0
0
0
J1
No
Stable
Stable
0
0
J1
J2
S1
S2
466
No
0.516
ICCBT 2008 - E- (37) – pp449-468

19. M. S. Mat Radhi et. al.
Table 8 continue
S3
J2
J3
Stable
Stable
Stable
No
Planar
Stable
Stable
Stable
0
0
0
0
0
0
0.991
0.38
Stable
Stable
Stable
0
0
0
0
0
0
J4
J1
No
0.899
Stable
Stable
Stable
J1
Planar
Stable
Stable
0
0
J2
J3
No
No
Stable
Stable
J4
No
Stable
Stable
Stable
Stable
0
0
0
0
0
0
J1
Planar
0.767
J2
No
0.629
0
No
Stable
Stable
0.891
Stable
J3
S6
No
No
J4
S5
Planar
J2
J3
S4
No
Stable
0
0.178
0.49
0
0
Table 9. Results of wedge failure for the deterministic analysis and the
probabilistic analysis (iteration of 10000)
Deterministic
Probabilistic
Set
Set
Potential
Analysis (Pf)
Slope
No. I
No. 2 Instability Analysis (FOS)
Wet
Dry
Wet
Dry
S1
S4
S5
S6
No
Stable
Stable
J2
Wedge
1.296
Stable
0.752
0
J1
J3
J4
Wedge
Wedge
1.411
Stable
Stable
Stable
0.579
0.203
0
J1
S3
J3
J1
S2
J2
J3
J4
Wedge
0.999
Stable
0.482
0
J3
J1
Wedge
0.931
Stable
0.203
0
J1
J3
Wedge
1.395
Stable
0.43
0
J3
J2
Wedge
1.434
Stable
0.77
0
J3
J4
Wedge
0.831
1.572
0.386
0
J1
J3
Wedge
Stable
Stable
0.32
0
ICCBT 2008 - E- (37) – pp449-468
0
0
0
467

20. Probabilistic Approach of Rock Slope Stability Analysis Using Monte Carlo Simulation
Acknowledgements
The authors would like to thanks Universiti Putra Malaysia (UPM) for the funding of this
project and Universiti Tenaga Nasional (UNITEN) for their constant support and
encouragement.
REFERENCES
[1]. Baecher, G.B. and Einstein H.H., 1978, Slope Stability models in pit optimisation, Proc. 16
Apcom Symp., Tucson, AZ, USA, Golden Press, 16: 501-512
[2]. El-Ramly, H.H., Morgenstern N.R. and Cruden, D.M., 2002, Probabilistic slope stability analysis
for practice, Can. Geotech. J., 39: 665-685
[3]. Goodman R.E. and Bray J.W., 1976, Toppling of Rock Slopes, In Proc. Specialty Conference on
Rock Engineering for Foundation and Slopes, Boulder, Colo., American Society of Civil
Engineers, New York, Vol. 2, 201-234
[4]. Hoek, E. and Bray, J.W., 1981, Rock slope engineering. The Institution of Mining and
Metallurgy, London
[5]. Hoerger S. F., and Young, D. S., 1987, Predicting local rock mass behavior using geostatistics,
in: Proc. of 28th Symp. Rock Mech., Rotterdam, Balkema, 99-106
[6]. Kulatilke P.H.S.W., 1988, State-of-the-art in stochastic joint geometry modeling, Proc. 29th US
Symp. on Rock Mech., University of Minnesota, Minneapolis, A. A. Balkema, Rotterdam,
Netherlands, 29: 155-169
[7]. Leventhal A.R., Barker, C. S., and Ambrosis, L. P., 1992, Malanjk-hand copper project-overview
of the geotechnical investigation for optimum mining exploration, in: Regional Symp. On Rock
Slope, India, 69-78
[8]. Madun. A., 2002, Stability Analysis of Weathered Rock Cut Slope using Geological Mapping
and Laboratory Tests., Master thesis, Universiti Putra Malaysia
[9]. Marek J. and Savely J.P., 1978, Probabilistic analysis of plane shear failure mode in: Proc. 19th
US Symp. On Rock Mech., Nevada, USA. A.A. Balkema, Rotterdam, Netherlands, 40-44
[10]. McMahon B.K., 1971, A statistical method for the design of rock slopes, in: Proc. Of 1st
Australia -New Zealand Conf. on Geomech., Melbourne, Australia, 314 – 321.
[11]. McMahon B.K., 1975, Probability of Failure and expect volume of failure in high rock slopes,
Proc. 2nd Australia – New Zealand Conf. on Geomech., Brisbane, Australia, 308 – 314.
[12]. Omar H., 2002, Development of Risk Assessment And Expert Systems For Cut Slope, PhD
Thesis, Universiti Putra Malaysia
[13]. Piteu D.R. and Martin D.C., 1977, Slope stability analysis and design based on probability
techniques at Cassiar Mine., Can. Min. Metall. J. (March), 1 – 12
[14]. Young D.S., 1993, Probabilistic slope analysis for structural failure. Int. J. Rock. Mech. Min. Sci.
Geomech. Abstr. 30 (7), 1623-1629
468
ICCBT 2008 - E- (37) – pp449-468