2008 probabilistic approach of rock slope stability analysis using mcm radhi et al

ICCBT 2008 - E- (37) – pp449-468

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: [email protected]

http://www.uniten.edu.my/newhome/content_list.asp?contentid=4017

Probabilistic Approach of Rock Slope Stability Analysis Using Monte Carlo Simulation

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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.

M. S. Mat Radhi et. al.

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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.


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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,


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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).


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S

S

S

S

S S

Figure 2. Pos Selim Area with granite and schist formation

Table 1: Study Locations and Lithology

Slope Location Lithology S1 Ch 2 + 960 Granite S2 Ch 9 + 100 Granite S3 Ch 17 + 600 Granite S4 Ch 18 + 280 Schist S5 Ch 18 + 800 Schist S6 Ch 20 + 750 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


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0

20

40

60

80

100

120

S1 S2 S3 S4 S5 S6

Joint Fault Foliation

(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.

Figure 3. Discontinuity data showing percentage of joint, fault and foliation at S1, S2, S3, S4, S5 and S6.


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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.


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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

S2

S3

S4

S5

S6

89

98

91

85

81

94

10

7

4

6

9

3

1

1

1

1

3

0

100

106

96

92

93

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.


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Figure 7. Rosette Plot

Table 3. No of plotted plane in Rosette Plot for each study slope

Slope Plotted

Planes

Maximum Plot

(Bin No)

Minimum

Plot (Bin No)

Total

Discontinuity

S1 92 6a & 24a 4c, 12c, 16c, 22c, 29c, & 34c 112

S2 98 7b & 25b 1c, 5c, 13c, 19c, 23c,& 31c 115

S3 92 13a & 31a 6c, 10c, 24c, & 28c 102

S4 79 7a & 25a 4c, 5c, 9c, 21c, 22c, & 27c

100

S5 84 6a & 24a 4d, 11d, 14d, 22d, 29d, & 32d 108

S6 80 18a & 36a 8c, 16c, 26c, & 34c 100

Total 525 118 38 637

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].


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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

Location

Fisher

Concentration

Discontinuity set

(dip/dip direction) Marked

Types of

discontinuity Lithology

4-7% 77°/170° J1 Joint Granite

S1 4-7% 80°/318° J2 Joint Granite

CH 2960 4-7% 33°/13° J3 Joint Granite

4-7% 72°/244° J4 Joint Granite

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


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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.


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Table 5. Results of Factor of Safety for Planar Failure

Slope Joint Set I.D.

Slope Face (deg)

Planar Failure (deg)

Mean Friction Angle (deg)

Factor of Safety (Wet)

Factor of Safety (Dry)

J1 77 30 Stable Stable J2 80 30 Stable Stable J3 33 30 Stable Stable

S1

J4

63

72 30 Stable Stable J1 59 30 0.879 0.953 J2 82 30 Stable Stable J3 80 30 Stable Stable

S2

J4

63

48 30 Stable Stable J1 86 30 Stable Stable J2 68 30 Stable Stable J3 52 30 Stable Stable

S3

J4

73

71 30 Stable Stable S4 J1 63 45 30 0.899 0.991

J2 65 30 Stable Stable J3 74 30 Stable Stable J4 66 30 Stable Stable

S5 J1 102 35 30 Stable Stable J2 71 30 Stable Stable J3 87 30 Stable Stable J4 62 30 Stable Stable

S6 J1 63 62 30 0.767 0.891 J2 19 30 Stable Stable J3 68 30 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 Intersection Joint

Slope Face (deg)

Dip of intersection

(deg)

Mean Friction

Angle (deg)

Factor of Safety

Factor of Safety (Dry)

J1J2 58 30 1.296 Stable J1J3 36 30 1.411 Stable

S2

J1J4

63

33 30 Stable Stable J3J1 43 30 0.931 Stable S3 J3J4

73 48 30 0.999 Stable

S4 J1J3 63 56 30 1.395 Stable J2J3 17 30 1.434 Stable S5 J3J4

102 38 30 0.831 1.572

S6 J1J3 63 30 30 Stable Stable


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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

Iteration Mean Min Max Med

Standard

Deviation Pf

WS 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

DS 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

Legend: WS = Wet slope, DS = Dry slope


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0

0.5

1

1.5

2

2.5

10 100 500 1000 5000 10000

No of Iteration

Mea

n of

FoS

Wet Slope Dry slope

Figure 9. Mean of FOS at J1J2, S2 for each iteration.

0

0.2

0.4

0.6

0.8

1

10 100 500 1000 5000 10000

No of Iteration

Pf

Wet Slope Dry Slope

Figure 10. Probability of wedge failure at J1J2, S2 for each iteration.


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0

0.5

1

1.5

2

2.5

1.441.56

1.68 1.81.92

2.042.16

2.28 2.42.52

2.64

Factor of Safety

Freq

uenc

y

0

0.5

1

1.5

2

2.5

3

3.5

0.270.39

0.510.63

0.750.87

0.991.11

1.231.35

1.47

Factor of Safety

Freq

uenc

y

0

2

4

6

8

10

12

14

1.381.54 1.7

1.862.02

2.182.34 2.5

2.662.82

2.98

Factor of Safety

Freq

uenc

y

0

2

4

6

8

10

12

0.170.33

0.490.65

0.810.97

1.131.29

1.451.61

1.77

Factor of Safety

Freq

uenc

y

0

10

20

30

40

50

60

70

1.291.47

1.651.83

2.012.19

2.372.55

2.732.91

3.09

Factor of Safety

Freq

uenc

y

0

10

20

30

40

50

60

70

0.070.25

0.430.61

0.790.97

1.151.33

1.511.69

1.87

Factor of Safety

Freq

uenc

y

(a) Wedge Failure Wet Slope (left) and Dry Slope (right) for 10 iteration

(b) Wedge Failure Wet Slope (left) and Dry Slope (right) for 100 iteration

(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.


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0

200

400

600

800

1000

1200

1400

1.24 1.44 1.64 1.84 2.04 2.24 2.44 2.64 2.84 3.04 3.24Factor of Safety

Freq

uenc

y

0

200

400

600

800

1000

1200

1400

0.04 0.24 0.44 0.64 0.84 1.04 1.24 1.44 1.64 1.84 2.04Factor of Safety

Freq

uenc

y

0

20

40

60

80

100

120

1.311.49

1.671.85

2.032.21

2.392.57

2.752.93

3.11

Factor of Safety

Freq

uenc

y

0

20

40

60

80

100

120

140

0.060.26

0.460.66

0.861.06

1.261.46

1.661.86

2.06

Factor of Safety

Freq

uenc

y

0

100

200

300

400

500

600

700

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2

Factor of Safety

Freq

uenc

y

(d) Wedge Failure Wet Slope (left) and Dry Slope (right) for 1000 iteration

0

100

200

300

400

500

600

700

-0.010.19

0.390.59

0.790.99

1.191.39

1.591.79

1.99

Factor of Safety

Freq

uenc

y

(e) Wedge Failure Wet Slope (left) and Dry Slope (right) for 5000 iteration

(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.


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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

J1 No Stable Stable 0 0

J2 No Stable Stable 0 0 S1

J3 Planar Stable Stable 0 0 J4 No Stable Stable 0 0

J1 Planar 0.879 0.953 0.763 0.516 S2 J2 No Stable Stable 0 0





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Table 8 continue S3 J2 No Stable Stable 0 0

J3 Planar Stable Stable 0 0


J1 Planar 0.899 0.991 0.38 0.178 S4 J2 No Stable Stable 0 0



J1 Planar Stable Stable 0 0 S5 J2 No Stable Stable 0 0



J1 Planar 0.767 0.891 0.629 0.49

S6 J2 No Stable Stable 0 0 J3 No Stable Stable 0 0

Table 9. Results of wedge failure for the deterministic analysis and the probabilistic analysis (iteration of 10000)

Deterministic Probabilistic Set Set Potential

Slope No. I No. 2 Instability Analysis (FOS) Analysis (Pf)

Wet Dry Wet Dry

S1 J2 J3 No Stable Stable 0 0

J1 J2 Wedge 1.296 Stable 0.752 0

S2 J1 J3 Wedge 1.411 Stable 0.579 0 J1 J4 Wedge Stable Stable 0.203 0 J3 J4 Wedge 0.999 Stable 0.482 0 S3 J3 J1 Wedge 0.931 Stable 0.203 0 S4 J1 J3 Wedge 1.395 Stable 0.43 0

J3 J2 Wedge 1.434 Stable 0.77 0 S5 J3 J4 Wedge 0.831 1.572 0.386 0

S6 J1 J3 Wedge Stable Stable 0.32 0


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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

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2008 probabilistic approach of rock slope stability analysis using mcm radhi et al

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rock slope analysis

rock slope engineering

rock slope design

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slope s2

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instability of rock

slope s1