Stability Assessment of a Critical Rock Slope Final Project Report Group Participants: Student Number: Rebeca Barja 71116107 Priyadarshi Hem 71071104 Pablo Urrutia 36638088 Shujing Zhang 71180103 Course: MINE 590W Instructor: Dirk Van Zyl Date: Dec. 14, 2010 2/21 Table of Contents 1 Summary ................................................................................................................................... 3 2 Objective .................................................................................................................................... 3 3 Introduction ................................................................................................................................ 3 4 Statistics .................................................................................................................................... 4 4.1 Normal Distribution .............................................................................................................. 4 4.2 Beta Distribution .................................................................................................................. 5 4.3 Exponential Distribution ....................................................................................................... 5 5 GoldSim Model ........................................................................................................................... 6 5.1 Base Case ........................................................................................................................... 8 5.2 Remedial Measures ........................................................................................................... 13 6 Costs ....................................................................................................................................... 14 7 Discussion of Results and Conslusions ..................................................................................... 15 8 References ............................................................................................................................... 21 List of Figures Figure 1: Sau Mau Ping Slope across from endangered apartment blocks ........................................ 4 Figure 2: GoldSim Model Layout ........................................................................................................ 7 Figure 3: Sau Mau Ping Slope Diagram ............................................................................................. 9 Figure 4: Probabilistic Distribution of the Tension Crack .................................................................... 9 Figure 5: Friction Angle - Beta and Truncated Normal Distributions ................................................. 10 Figure 6: Cohesion - Beta and Truncated Normal Distributions ........................................................ 10 Figure 7: Relationship between friction angles and cohesive strengths mobilised at failure of slopes in various materials .......................................................................................................................... 11 Figure 8: Probabilistic Distribution of the Earthquake ....................................................................... 12 Figure 9: Probabilistic Distribution of Water Level in the Tension Crack ........................................... 13 Figure 10: CDF of FS for Base Case Scenario (Beta Distribution).................................................... 16 Figure 11: CDF of FS for Base Case Scenario (Truncated Normal Distribution)............................... 16 Figure 12: Change in p(FS<1.5) and Cost for the Reducing Height Alternative ................................ 18 Figure 13: Change in p(FS<1.5) and Cost for the Reducing Slope Angle Alternative ....................... 18 Figure 14: Change in p(FS<1.5) and Cost for the Reinforcement Alternative ................................... 19 Figure 15: Combined Plot Showing p(FS<1.5) and Cost .................................................................. 20 List of Tables Table 1: Material Properties ............................................................................................................. 10 Table 2: Drainage Alternative Results .............................................................................................. 19 Appendixes: APPENDIX A: Calculations 3/21 1 SUMMARY A probabilistic stability analysis was carried out using GoldSim on the case study “A slope stability problem in Hong Kong” (Hoek, 2007), to assess the short-term and long-term stability of the Sau Mau Ping slope. Four remedial measures (reduction of slope height, reduction of slope angle, reinforcement and drainage) were considered for improving the long-term stability of the slope; short-term stability was not considered an issue under the modelled conditions. Costs were estimated for each of the remedial measures. Reducing the slope angle was selected as the preferred alternative due to a technical and economic point of view. 2 OBJECTIVE The objective of this project is to carry out a probabilistic analysis on a widespread problem that threatens the safety of people and their homes, slope stability. A probabilistic approach aided by GoldSim, a Monte Carlo simulation software, will be used to model the slope stability problem, generate a distribution for the factor of safety of the slope and finally, generate a distribution for the probability of failure and probability of acceptance of the slope. All of these distributions, along with estimated costs, will be used in making a decision on the most appropriate remedial measure that should be taken in order to ensure the long term stability of the slope. The problem chosen for this analysis will be introduced in the next section. 3 INTRODUCTION A literature review for slope stability problems lead to a very interesting case study written by Everet Hoek titled “A slope stability problem in Hong Kong”. In this article, the case of a rock slope located along the Sau Mau Ping Road in Kowloon is discussed. This rock slope posed potential danger to the apartment blocks located across a road which housed approximately 10,000 people. The critical concern for engineers was that a major rock slide could occur due to exceptionally heavy rains and cross the road causing damage to the apartment blocks. Before making a decision on whether to evacuate residents, some crucial questions needed to be answered. These questions were the following: • What was the factor of safety of the slope under earthquake and heavy rain conditions • What was the acceptable factor of safety for short term and long term conditions • What steps would be required to achieve these factors of safety Figure 1 shows the Sau Mau Ping Slope and the apartment blocks located across the road. The original solution to this case took a traditional deterministic approach. However, for the purposes of this project, a probabilistic approach will be taken to evaluate and compare the 4/21 original engineering design decision to the decision taken when the weight of influence from the problem variables are taken into account. A second case study by E. Hoek, “Factor of safety and probability of failure”, describes the recommended methodology for carrying out a sensitivity analysis of the probability of failure. The probability distributions used in this approach (i.e. truncated normal and exponential), as well as the probability functions that were chosen by our group for experimental purposes (i.e. beta function), will be described in the statistics section that follows. The Monte Carlo method of simulation will be used in order to generate a final distribution of results for the factor of safety calculation and the probability of failure and acceptance. Figure 1: Sau Mau Ping Slope across from endangered apartment blocks (Source: Practical Rock Engineering, E.Hoek) 4 STATISTICS 4.1 Normal Distribution The normal distribution is a continuous probability distribution with two parameters, mean µ and variance σ2, denoted by N(µ, σ2). The shape of the normal distribution is like a bell with a single peak at the mean of data. The spread of a normal distribution is controlled by the standard deviation σ. The smaller the standard deviation, the more concentrated the data. The normal density function is = eμ. 5/21 It is symmetric around the mean value, and inflection points occur at µ±σ. Normal distributions are one of the most common distributions used for probabilistic analysis in geotechnical engineering due to its simplicity and wide applicability. Given that its range from −¥ to ¥ can cause problems when used in conjunction with a Monte Carlo analysis, usually they are truncated by minimum and maximum values. In this particular case, a truncated normal distribution was used to define the following variables: • Friction angle • Cohesion • Tension crack depth 4.2 Beta Distribution The beta distribution is a continuous probability distributions having two positive shape parameters, denoted by α and β. The Beta distribution, in its standard form, ranges from zero to one, and takes a wide range of shapes. The beta density function is ; , = 1 − u1 − udu The beta distribution is often used to describe the uncertainty or random variation of a probability value. It can rescale and shift to create distributions with a wide range of shapes and over any finite range. Beta distributions are very flexible and can be used to replace many other common distributions. A second model with beta distributions assigned to friction angle and cohesion was also developed for comparison purposes for this analysis. 4.3 Exponential Distribution The exponential distribution is a continuous probability distribution. It describes the time between events in a Poisson process. For a given Poisson process, the time T between consecutive occurrences of events has an exponential distribution with the following density function: = ! λ exp−% &' ≥ 0 0 otherwise 1 Exponential distributions were used to model the effect of earthquake and heavy rain on the slope. Given the need of creating truncated exponential distributions for modelling the earthquake and storm (see Section 5.1.3 for more on this), and considering that GoldSim cannot truncate an exponential distribution, a truncated gamma distribution was used instead. This was possible due to the fact that an exponential distribution is a special case of a gamma distribution with the shape parameter k=1. Since a gamma distribution is defined by: • Mean = k*t 6/21 • Standard deviation = k ∗ t By substituting k=1 both the mean and standard deviation will become equal to t. Therefore, an exponential distribution with mean equal to X is equivalent to a gamma distribution with mean equal to the standard deviation (which is equal to X). 5 GOLDSIM MODEL As mentioned above, GoldSim (Version 10.11 (SP4), Academic License) was used to assess the probability of acceptance of the Sau Mau Ping slope. The Monte Carlo analysis was run considering 5,000 realizations in every case. Figure 2 illustrates the layout of the model created to accomplish this. 7/21 Figure 2: GoldSim Model Layout 8/21 5.1 Base Case The Factors of Safety for the Base Case were calculated based on the same procedures followed on the case study “A slope stability problem in Hong Kong” (Hoek, 2007), which are summarized by the following equation: Where, Symbol Parameter c Cohesive strength along sliding surface A Base area of wedge W Weight of sliding mass Ψp Angle of failure surface, measured from Horizontal α Horizontal earthquake acceleration U Uplift force due to water pressure on failure Surface V Horizontal force due to water in tension crack ø Friction angle of sliding surface For more information regarding the equations used in the Factor of Safety calculations, please refer to Appendix A. 5.1.1 Geometry of the Slope The Sau Mau Ping road was cut in a mass of unweathered granite, with sheet joints parallel to the exposed face of the cut slope. These sheet joints are the most probable surface of failure of the slope, and were estimated with a dip of 35°. According to the investigation carried out, the rock slope can be represented by the diagram shown in Figure 3. As illustrated there, the 60 m high granite slope is divided in three 20 m high benches with inclinations of 70° to the horizontal, and an overall slope angle of 50°. There were also tension cracks at some places behind the crest of the slope, with variable depths. The GoldSim model developed for this project assessed only the stability of the overall slope by including the tension crack, and not a case without it. Individual evaluation of the stability of each bench was not carried out. 9/21 Figure 3: Sau Mau Ping Slope Diagram (Source: Practical Rock Engineering, E.Hoek) The depth of the tension crack was modelled as a stochastic variable with a normal distribution, following the recommendation on “Factor of safety and probability of failure” (Hoek, 2007). Figure 4 illustrates the data used to create the probabilistic distribution. For more detailed information regarding the equations used in estimation of the standard deviation, minimum and maximum depth of the tension crack please refer to Appendix A. The mean depth of the tension crack was estimated as half the maximum depth. Figure 4: Probabilistic Distribution of the Tension Crack 5.1.2 Material Properties Due to the lack of shear strength information available on the Sau Mau Ping slope, unit weight, friction angle and cohesion of the rock slope had to be assumed to complete the 10/21 stability analysis. These assumptions were based on published information for similar rocks, as described in the case study “A slope stability problem in Hong Kong” (Hoek, 2007). Table 1 shows the parameters selected for the probabilistic analysis carried out as part of this project. Figures 5 and 6 illustrate the stochastic distributions used to model friction angle and cohesion. Table 1: Material Properties Mean CoV Std. dev Min Max Distribution Rock friction angle 40° 10% 3.75° 30° 45° Beta & Normal Rock cohesion 125 kN/m2 30% 37.5 kN/m2 50 kN/m2 200 kN/m2 Beta & Normal Unit weight 25.5 kN/m3 - - - - Deterministic Figure 5: Friction Angle - Beta and Truncated Normal Distributions (beta distribution) (truncated normal distribution) Figure 6: Cohesion - Beta and Truncated Normal Distributions (beta distribution) (truncated normal distribution) The unit weight was estimated as a deterministic value because no information was found with regard to probabilistic distributions that could be used to estimate this value. Friction angle and cohesion where estimated for the original analysis using the plot on Figure 7, from the case study “A slope stability problem in Hong Kong” (Hoek, 2007). The plot also shows 11/21 the range of estimated shear strength values for the sheet joints in unweathered granite, developed for the original analysis. Figure 7: Relationship between friction angles and cohesive strengths mobilised at failure of slopes in various materials (Source: Practical Rock Engineering, E.Hoek) The original analysis used the envelope of shear strength shown on Figure 7 to obtain mean values for the friction angle and cohesion. Considering that these strength parameters were estimated from the available literature, and that the reported parameters are likely reductions of a variety of field and lab tests as well as observations from the field, a more appropriate approach to assess the stability of the Sau Mau Ping slope would be to assume the shear strength as a random variable among a given range, rather than a fixed one. Following this line of thought, we used the same envelope on Figure 7 to generate our probabilistic distributions, setting up the minimums, maximums and means after it. The standard deviations used for our analysis were calculated with a typical coefficient of variation for each 12/21 parameter. These coefficients of variation were selected after revising some literature (Park, 1999; Park and West, 2001; Rétháthi, 1988). 5.1.3 Earthquake and Water in Tension Crack Besides geometry and inherent material parameters, other factors could also affect the stability of the slope, such as earthquakes (inducing external forces to the system) and storms (changing the saturation conditions of the system). In this specific case, earthquakes were not perceived as a mayor threat given that the region of study was not deemed as highly seismic. After discussing the subject with local experts, the developers of the original stability analysis (Hoek, 2007) considered appropriate to include a minor acceleration (in the form of a pseudo-static force) due to earthquake loading into the system. This acceleration was estimated to be 0.08 g. As a pseudo-static force, the earthquake was input into the model as a fraction of the gravity acceleration, applied to the weight of the sliding mass. The pseudo-static force was treated as a probabilistic variable, in this case with a truncated gamma distribution (see section 4.3) with a mean and standard deviation of 0.08 g and a maximum value of 0.16 g, as shown in Figure 8. Figure 8: Probabilistic Distribution of the Earthquake Typhoons, on the other hand, are very common in Hong Kong and one of these storms could easily fill the tension cracks with water and saturate the whole slope. Storm events are usually represented by exponential distributions, but given that we needed to model the amount of water in the tension crack, and the tension crack has a maximum depth, the exponential distribution representing the water level in the tension crack should also be truncated with a maximum value equal to the whole depth of the tension crack. The Gamma distribution shown on Figure 9 was used to model the water level in the tension crack, where “Z” is the depth of the tension crack (modelled as a normal distribution, as explained above). 13/21 Figure 9: Probabilistic Distribution of Water Level in the Tension Crack 5.2 Remedial Measures As the probability of acceptance of the Base Case exceeded our acceptance criteria (see Section 7), four different and independent remedial measures were assessed to improve the stability of the Sau Mau Ping slope. These measures were: • Reduction of the slope height • Reduction of the slope angle • Reinforcement of the slope (by cablebolts) • Drainage of the slope 5.2.1 Reduction of the Slope Height In order to assess the reduction of the slope height, changes in the height of the slope were evaluated using the GoldSim model. Reductions of 5 m were assessed up to a total height of 30 m, changing also the tension crack depth distribution with a new maximum (with consistent mean and standard deviation) while keeping the rest of the model inputs the same as with the Base Case. Results are discussed in Section 7. For further detailed information about the calculations of the new tension crack depth parameters please refer to Appendix A. 5.2.2 Reduction of the Slope Angle Similar as in the previous case, the overall angle of the slope was changed by intervals of first 5° and then 1° up to a dip of 40°. The tension crack depth distribution was also modified to reflect the change in the slope angle. The rest of the GoldSim model parameters were kept the same as in the Base Case. Please see Section 7 for discussion of the results. For more information regarding the calculations for the new tension crack depth parameters please refer to Appendix A. 14/21 5.2.3 Reinforcement of the Slope (by Cablebolts) Cable bolts of 25 m long were included in the model as a new force in the Factor of Safety equation shown on Section 5.1. The new equation is as follows: Where “T” is the force applied by the cable bolt and “Ѳ” is the angle of the cable bolt with respect to the horizontal. Four different “T” values, from 2,800 kN to 4,000 kN, were assessed, all of them for a “Ѳ” equal to 35°, and four factors of safety were calculated accordingly. Results are discussed in Section 7. 5.2.4 Drainage of the Slope To assess the effect of draining the slope, no water pressure was included in the Factor of Safety calculations for this remedial measure. Please see Section 7 for discussion of the results. 6 COSTS Costs were estimated based on literature review and professional engineering judgement of the authors of this report. The costs presented here are just an approximation based on several assumptions, given that the information available in the original case study does not focus on costing aspects. The costs used for this project are used simply for comparison of different alternatives. These should not be used for a different purpose. Excavation volumes, number of bolts and length of the drainage structures also had to be estimated in order to obtain comparable costs between different alternatives. The costs have been calculated for a section of the slope of 1.0 m width. Cost of material excavation was estimated as 50.00 $CAD/m3 of material. This amount includes transportation of the cut material to a waste dump 5 km away from the site and is based on previous experience and on the SME Mining Engineering Handbook (SME website, online source). The excavation volume estimated for this alternative was calculated by assuming excavations up to 50 m upstream of the crest of the slope. Cost of cable bolt installation was obtained from the book Cablebolting in Underground Mines (Hutchinson and Diederichs, 1996), which suggested a value of $CAD30.00 per meter of cablebolt. The length of the cable bolts was estimated as the length from the face of the slope up to the slip surface (i.e. sheet joints), which was 20 m, plus an additional 5 m for grouting and proper anchoring of the bolts. The number of cable bolts required was calculated assuming a yield stress of 400 kN per bolt (GIA industri ab webpage, online source). 15/21 Cost of draining the slope was estimated based on information from the book Slope Stability in Surface Mining (Hustrulid, McCarter and Van Zyl, 2000), which suggested US$45.00 per meter of drain hole (75 mm diameter). This reference was published in 2000 and refers to the construction cost of a drainage system in South America. With these in mind, we estimated the cost of the slope drainage as $CAD100.00 per meter of drain hole. The length of the drain was estimated as 25 m per section of slope (1 m wide), by assuming sets of 100 m long drains every 5 m. 7 DISCUSSION OF RESULTS AND CONSLUSIONS Factors of safety for the short term and long term stability of the Sau Mau Ping slope were assessed as part of this probabilistic analysis. Two different criteria were defined to accomplish this: • Probability of failure: Probability of the FS being less than unity, or p(FS<1.0) • Probability of acceptance: o Short term: Probability of the FS being less than 1.2, or p(FS<1.2) o Long term: Probability of the FS being less than 1.5, or p(FS<1.5) Considering that the consequence of slope failure would threaten the lives of approximately 10,000 people living downstream from the slope as well as material damage to the building blocks, the tolerance factor selected to assess the results was 5%. In other words, a probability of failure below 5% or a probability of acceptance below 5% would require remedial measures to improve the stability of the slope. Cumulative distribution functions (CDF) of the analysis for the Base Case scenario are shown in Figures 10 and 11. 16/21 Figure 10: CDF of FS for Base Case Scenario (Beta Distribution) Figure 11: CDF of FS for Base Case Scenario (Truncated Normal Distribution) First and foremost, Figures 10 and 11 show that the selection of a beta or truncated normal distribution does not significantly affect the outcome of the stability analysis. The resulting probabilities of acceptance for the beta distribution as well as the truncated normal distribution differ in value by 1 – 3 %. The beta distribution results are higher (i.e. more conservative). This is negligible for a stability evaluation like the one carried out in this project, along with many other forms of geotechnical analysis, where many assumptions and 17/21 simplifications are made. It is worth noting that given the flexibility of the beta distribution, it can take different shapes. Therefore, the similarity in the results of this specific analysis between the beta and truncated normal distributions is due to the shape of the current beta distribution (which is similar to the truncated normal), since it is related to the parameters (mean, standard deviation, min and max) used to define it. Figures 10 and 11 also show that the probability of failure for both distributions is about 1%, which does not imply a high risk; the probability of acceptance for the short term scenario is 8.71% for the beta distribution and 6.77% for the truncated normal. Even if the p(FS<1.2) is higher than 5%, the scenario modelled in GoldSim, where an earthquake and typhoon occur at the same time, is unlikely and its highly pessimistic given the site conditions. This leads to the developers of the original stability analysis, who obtained equivalent results with their deterministic analysis, to conclude that the stability of the Sau Mau Ping slope was good enough to comply with short-term stability criteria. Keeping in mind that we are considering a FS of 1.2 in the evaluation (where the original analysis assessed the short-term stability for a FS=1), we think the resulting p(FS<1.2) are acceptable and no short-term measures are required. On the other hand, p(FS<1.5) are significantly higher for both distributions and remedial measures should be considered to improve the stability of the rock slope. Figures 12, 13 and 14 illustrate the effect of the first three remedial measures in the stability of the slope, while showing the cost of each. Table 2 shows the results of the drainage alternative assessment. Figure 15 shows a combined plot of the results for each of the remedial measures (the plot includes only the results from the beta distribution analysis). 18/21 Figure 12: Change in p(FS<1.5) and Cost for the Reducing Height Alternative Figure 13: Change in p(FS<1.5) and Cost for the Reducing Slope Angle Alternative 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 $0 $10,000 $20,000 $30,000 $40,000 $50,000 $60,000 $70,000 $80,000 $90,000 $100,000 25 30 35 40 45 50 55 60 p (FS<1.5) Cost (CAD) Slope Height (m) Cost p(FS=1.5) -Beta p(FS=1.5) -Normal 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 $0 $10,000 $20,000 $30,000 $40,000 $50,000 $60,000 $70,000 $80,000 $90,000 $100,000 40 41 42 43 44 45 46 47 48 49 50 p (FS<1.5) Cost (CAD) Slope Angle (°) Cost p(FS=1.5) -Beta p(FS=1.5) -Normal19/21 Figure 14: Change in p(FS<1.5) and Cost for the Reinforcement Alternative Table 2: Drainage Alternative Results p(FS<1.5) - Beta p(FS<1.5) - Normal Cost (CAD) No drainage 40.77% 39.35% 0.00 Drainage 26.87% 24.01% 2,000.00 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 0 2 4 6 8 10 12 $0 $10,000 $20,000 $30,000 $40,000 $50,000 $60,000 $70,000 $80,000 $90,000 $100,000 Number of Anchors Cost (CAD) p (FS<1.5) Cost p(FS=1.5) -Beta p(FS=1.5) -Normal20/21 Figure 15: Combined Plot Showing p(FS<1.5) and Cost Of all the remedial measures proposed, only the drainage option will not reduce the p(FS<1.5) to less than 5%. The original stability analysis also obtained similar results for this. This is probably due to a combination of factors, like the geometry of the slope and the geotechnical parameters of the failure surface (Hoek, 2007). Simplifications and assumptions made during the stability analysis should not be forgotten, which have a significant impact on the results, and which were required in some cases due to GoldSim not being a program designed to compute stability evaluations. Drainage would usually be an effective measure to improve the stability of a slope and further investigation with other programs should be completed before further conclusions are made. The other three remedial measures proposed accomplish the objective of reducing p(FS<1.5) to a value lower than 5%, thus cost will be the next decision making factor. However, corrosion and creep failure of cablebolts can be an issue in the long term. Therfore, further research (considering corrosive site conditions and different bolts characteristics, which are beyond the scope of this project) should be carried out to completely understand the behaviour of this alternative in the long term. The life of a cable bolt varies from less than a year in case of highly corrosive environments to up to 25 years in case of complete corrosion free environment; the later is due to its creep failure. Figure 13 shows that even if the reinforcement and drainage remedial measures are the less costly, because of the former not being enough to lower p(FS<1.5) below 5% and the potential corrosion and creep issues of the latter, the preferred alternative to improve the stability of the Sau Mau Ping slope is reducing the slope angle. The difference in cost 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 $0 $10,000 $20,000 $30,000 $40,000 $50,000 $60,000 p(FS<1.5) Cost (CAD) Reducing Height Reducing Slope Reinforcement Drainage21/21 between reducing slope height and slope angle is due to the excavation volumes required for each of them, being the first one almost twice as much. Moreover, Figure 10 shows that to reach a p(FS<1.5) below 5% for the reducing height alternative, the slope would have to be cut to half of its height, which does not look like a viable option. Reduction of slope angles is an effective and widespread alternative when dealing with slope stability issues. Results of the probabilistic analysis carried out as part of this project are consistent with the ones obtained from the original deterministic analysis (Hoek, 2007). Differences are definitely present but they can be attributed to the material properties assumptions made for both evaluations. Results should be seen in the light of the limitations of the level of expertise of the authors with regard to the software. As well, limitations of the formulations used for this stability analysis should be taken into account. 8 REFERENCES 1) Hoek, E., 2007. Practical Rock Engineering, Chapter 7: A slope stability problem in Hong Kong. www.rocscience.com, (2010). 2) Hoek, E., 2007. Practical Rock Engineering, Chapter 8: Factor of safety and probability of Failure. www.rocscience.com, (2010). 3) Rétháti, 1988. L. Rétháti, Probabilistic Solutions in Geotechnics; Elsevier, Amsterdam-Oxford-New York-Tokyo (1988), pp. 45. 4) Park, 1999. Park, H.J., 1999. Risk analysis of rock slope stability and stochastic properties of discontinuity parameters in western North Carolina. PhD thesis. Purdue University. 5) Park and West, 2001. H.J. Park and T.R. West, Development of a probabilistic approach for rock wedge failure, Eng. Geol. 59 (2001), pp. 233–251. 6) GIA industri ab webpage: Products section. http://www.gia.se/eng/meth_cable.htm (accessed: Dec-09-2010). 7) SME website: Mining Engineering Handbook, Chapter 6.3 Costs and Cost Estimation, pp. 14, www.smenet.org (accessed: Dec-10-2010). 8) Hutchinson and Diederichs, 1996. D.J. Hutchinson, M.S. Diederichs, Cablebolting in Underground Mines; Hutchinson and Diederichs; BiTech Publishers Ltd., Ritchmond, British Columbia, Canada (1996), pp. 20. 9) Hustrulid, McCarter and Van Zyl, 2000. W.A. Hustrulid, M.K. McCarter, D.J.A. Van Zyl, Slope Stability in Surface Mining; Society for Mining, Metallurgy, and Exploration (2001), pp. 6. APPENDIX A: - CALCULATIONS - 1 FACTOR OF SAFETY CALCULATIONS Factor of Safety calculations were done following the method shown on the case study “A slope stability problem in Hong Kong” (Hoek, 2007). Figure A-1 shows the diagram of forces used to elaborate the FS equation, based on limit equilibrium concepts. It is worth noting that only forces were considered when calculating FS (not momentums), and that all the forces pass through the center of gravity of the sliding mass. This is a simplification, but it was required to use Goldsim to solve the problem. Furthermore, given the assumptions made to model the characteristics of the slope, it will not be very influential in the final results. Figure A-1 Where, Symbol Parameter c Cohesive strength along sliding surface A Base area of wedge W Weight of sliding mass Ψf Angle of the slope, measured from Horizontal Ψp Angle of failure surface, measured from Horizontal α Horizontal earthquake acceleration U Uplift force due to water pressure on failure Surface b Distance behind tension crack and crest z Tension crack depth zw Water level on tension crack and failure surface W Weight of sliding mass T Force representing the cablebolt reinforcement q Inclination of reinforcement V Horizontal force due to water in tension crack ø Friction angle of sliding surface Equations used to complete the FS calculations were taken from “A slope stability problem in Hong Kong” (Hoek, 2007), and are shown on Figure A2. Figure A-2 Equation for calculating “z” was obtained by minimization of the following equation (hoek and Bray, 1974): This minimization was done considering dry conditions on the slope, which is a simplification but is acceptable for the scope of this analysis (Hoek, 2007). 2 TENSION CRACK DEPTH DISTRIBUTION A truncated normal distribution was selected to model the uncertainties on the depth of the tension crack, following the recommendation on the case study “Factor of safety and probability of failure” (Hoek, 2007). The maximum depth of the tension crack occurs when the crack is located at the crest of the slope (Hoek, 2007), and was estimated by: z = H(1− tanψp / tanψf ) = 24.75 m The mean for the tension crack distribution was selected as half of the maximum crack depth, while an arbitrary standard deviation of 3 m was estimated, again following the recommendation on the case study “Factor of safety and probability of failure” (Hoek, 2007). A minimum depth of 0.01 m was selected for the tension crack to avoid numerical issues. The above described parameters were used for the Base Case scenario. When changing the model to assess the reduction in height and slope angle remedial measures, the maximum, mean, and standard deviation of the tension crack distribution were changed as well, to reflect the modifications made on the slope. The new values for max, mean and std. deviation were estimated as a fraction of the original ones, keeping the same ratio the change in height or slope angle, as shown in tables A-1 and A-2. Table A-1 H (m) Max crack depth (m) Std. deviation (m) 60 24.75 3.00 55 22.69 2.75 50 20.62 2.50 45 18.56 2.25 40 16.50 2.00 35 14.44 1.75 30 12.37 1.50 Table A-2 Slope angle (°) Max crack depth (m) Std. deviation (m) 50 24.75 3.00 45 17.99 2.18 44 16.49 2.00 43 14.95 1.81 42 13.34 1.62 41 11.67 1.41