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- 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

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