I.J.M.G.E., University of Tehran, Vol. 46, No. 1, June 2012, PP. 33-42 33 Probabilistic - The Future of Rock

Nielen van der Merwe* Professor in the Centennial Chair for Rock Engineering University of the Witwatersrand Johannesburg, South Africa (Received 29 March 2011, Received in Revised form 27 June 2011, Accepted 28 November2011)

Abstract A brief background to the development of the rock engineering design process is given, showing that since the development of the science of mathematics, deterministic methods have been used to perform various calculations. The variability of rock properties and support characteristics have always been known. However, they were not explicitly used in design but compensated for by the use of a safety factor, i.e. making a design more stringent than required by the calculations. The problem with this procedure is that the effect of increasing a safety factor on the overall stability of the design cannot be known because the of variability is not incorporated in the design. The only way to overcome this problem is to make use of the science of probability. In doing that, the ranges of rock qualities are explicitly included in the design and the probability of failure is exposed. Examples of common rock engineering calculations in mining are provided, showing that the probabilistic are not difficult or time consuming to perform and yield much more useful outcomes than merely using a safety factor.

Keywords: Rock Engineering, Deterministic Design, Reliability, Point Estimate method (PEM)

Introduction Rock control has been practised ever It became possible to calculate the weight since the first hole in the ground was made of blocks of rock and then to calculate the by man. Even in the very ancient times, it strength required for supports to withstand is for instance conceivable that loose that weight. fragments of rock would have been Once computers became available and removed before they could cause injury. accessible, it became possible to develop The subterranean city of Derinkuyu in and use highly sophisticated mathematical Turkey, excavated more than 3000 years routines and complex mining or other ago, had arched roofs and shaped pillars. excavation lay-outs. Yet, all the In the 2000 year old Roman limestone mathematical routines that are used are mines underneath Paris, roof control pillars based on explicitly defined singular input built of stone still offer support today. conditions. There is no evidence today that those It is universally appreciated that rock is designs were based on mathematics. That variable in nature, given to imperfections. only came much later. The most common way to accommodate With the development of the science of that, is to incorporate a safety factor in mathematics and the wider teaching of designs, in other words to multiply the mathematics, it became possible to outcome of a resistance calculation by develop quantified laws of nature and some factor to compensate for unknown based on that, the extensions to predict the deviations from the average input numbers behaviour of rock. that were used.

*Corresponding author: Tel: +27 11 680 5213, Fax: +27 11 433 4561, Email: [email protected] (Nielen van der Merwe)

34 I.J.M.G.E., University of Tehran, Vol. 46, No. 1, June 2012

This is a reasonable and under most Background of Probabilistic Design conditions a safe approach, but not an The notion of using probabilistic optimum one. In this way, we do design procedures in rock engineering is somewhat account for variability, but in a not new. There are several examples of manner that we cannot quantify: a safety scientific procedures to use probabilistic factor of 2, for instance, that the concepts in rock engineering design in average support is twice as strong as the literature. average load, but it certainly does not In South Africa, Stacey [1] presented a follow that the overall design is twice as simple probabilistic approach to handle likely to be stable. In those areas where the rockfalls in deep gold mine workings load is greater than the average, and the while Esterhuizen and Streuders [2] support resistance less than the average, presented a probabilistic procedure to we simply don’t know how stable the evaluate the probability of instability design is. arising due to keyblock failure, based on With the deterministic approach, we can the distribution of discontinuities in the only say for certain what the situation is at roof. Hoek [3] gives an example of using the point where the average load coincides the probability of failure concept to with the average of the support resistance, evaluate the stability of a rock slope. Hantz but without even knowing where that point et al [4] advocate the combined use of is! fundamental rock engineering concepts, For too long, as geotechnical engineers, we history and probabilistic procedures to have been reluctant to incorporate the assess the rockfall hazard, using a potential science of probability in our designs. This rock slope failure in the Grenoble area as may in part be due to a suspicious view of an illustrative example. Tono et al [5] caused by the irresponsible use discuss the application of statistics to thereof in advertising and other areas of handle imprecise to evaluate the society. It may also simply be due to the reliability of tunnel linings. Esterhuizen fact that it is not included in all rock (2003)’s JBlock program [6] has been engineering training programs and available for some time and is included in consequently, rock engineers are not rock engineering curricula at certain always equipped with the necessary skills. universities. Harr [7] presents several Yet, if properly used, it is a vital tool in statistical procedures used for civil our arsenal. It is the only science which we engineering design – this work is unique in can use to quantify uncertainty and to shed the sense that it explains statistics from a light on the probability of failure of our practical engineering perspective as designs. opposed to a classical statistical This paper will show examples of how the perspective. science of probability can be used in There are several more examples; the common designs on a more rational basis above are included to indicate that than with the deterministic design examples of several applications of procedures based on average values with a probabilistic procedures in several compensating safety factor. branches of the broader rock engineering discipline already exist. Yet, after several decades, even though the application has

Probabilistic Design - The..... 35 been demonstrated to be superior to simple and even adherence to design dimensions. deterministic design in several areas, The design is then based not on average reliability based design is not in general values as in deterministic design processes, use in the field of rock engineering. There but on the distributions of the input could be several reasons for this. variables. The output is then also in the The most important obstacle may well be form of a distribution instead of a single that the application thereof will force the number. The output distribution is then general public as well as engineers to very simply used to determine the come to terms with the fact that explicit probability that failure will occur. probabilities of failure will be stated in In order to perform a reliability based designs. The probability of failure per se is design, the variabilities of the various not new. It has always been there, it has input parameters also have to be known. always been an integral part of any design, This does not necessarily require more but numbers have not been calculated and work to be done for . In brought into the open. most cases, the data have already been Stacey [8] points out that the wider gathered in order to calculate average adoption of the concept of the probability numbers for the deterministic input. It of failure will require the participation of merely requires one or two additional management at executive level to state calculations to be performed, which is no design objectives in those terms, but that trouble at all with commonly used this has not happened. Hoek [9] offers software like Microsoft Excel. some hope by stating “…..there does At the core of the process, the fundamental appear to be a slow but steady trend in deterministic are still there. society to accept the concepts of risk Essentially, the input ranges are merely analysis more readily than has been the used with the same as would be case in the past”. used for the deterministic calculation, to The definition of acceptable risk levels calculate a number of outputs and then to will require serious debate. In the mining base decisions on the range of outputs. The industry, the trend internationally is to aim single number resulting from a for zero harm. It is doubtful whether deterministic calculation is merely “zero” in the public perception coincides extended into a range, based on the range with the true meaning of the word in of input variables. scientific terms. It is perhaps more The nature of the output distribution is a appropriate to equate zero harm to function of the nature of the input acceptable harm levels in the home variables. For instance, the wider the environment – which is not true zero and spread of the input variables, the wider the in itself difficult to determine as those spread of the output. The wider the range risks are not the same the world over. of the outputs, the less certain the design and the higher the probability of failure. Basis of Reliability based Design This is the fundamental improvement Reliability based design in rock brought about by using a probabilistic engineering is based on the fundamental approach: with the deterministic approach, acceptance that rock qualities are variable the range of possible outcomes cannot be and so are the qualities of artificial support quantified and hence the uncertainty is not

36 I.J.M.G.E., University of Tehran, Vol. 46, No. 1, June 2012 known while with the probabilistic PEM Example approach, it is highlighted. The probability of planar failure for the These concepts will become clearer with following situation is to be investigated, the examples which follow. see Figure 1. The area of the plane per running metre is Handy Techniques 6.9 m2. The plane has an inclination of 600 There are a number of handy, and the weight of the material that may fail relatively simple techniques that can be is 275.26 kN per running metre. used to handle the probabilistic routines. The equation for the safety factor, Most of these can be performed at least at assuming that the slope is dry and entry level with commonly available unsupported, is: software like Microsoft Excel. There are Ac W cos() tan() also a number of tailors made statistical SF  (1) packages available, one of the more W sin() popular ones being @Risk, an add-on where, module for Excel. A = area of plane There are several good handbooks that W = weight of plane explain the procedures in detail, Harr [7] c = cohesion  = inclination of plane being particularly useful for engineering = angle of friction of plane. applications. In this paper, only some In this example, the cohesion and angle of examples are presented as of friction are variables. The and the ease with which the procedures can be of the cohesion is 29 performed and the usefulness of the kPa and 5 kPa respectively and that of the approach. angle of friction 320 and 40 respectively.

As there are two independent variables, the The Point Estimate Method (PEM) number of calculations required is 22 = 4. The PEM is useful in situations where For each calculation, the constants (in this the deterministic algorithms to be used are relatively simple and where the case A, W and ) will not be changed, distributions of the input parameters are while the following will be used for c and known. It can then be used to determine : the distribution of the safety factor and the c + standard deviation: 34 kPa probability of failure. c – standard deviation: 24 kPa 0 In essence, a small number of safety factor  + standard deviation: 36 0 calculations are performed by varying the  – standard deviation: 32 input according to the distributions. The The calculations are then tabulated as number of calculations to be performed are follows: n 2 , where n is the number of variable Table 1: Calculations for PEM example inputs. The calculations are performed 2 2 SF(c,f) SFi,j SFi,j /4 (SFi,j ) (SFi,j ) /4 using the mean plus or minus one standard FS++ 1.404 0.351 1.970 0.493 deviation for each variable. The following FS+- 1.291 0.323 1.667 0.417 example will illustrate the process. FS-+ 1.114 0.279 1.242 0.310 FS-- 1.002 0.250 1.003 0.251  1.203 1.471

Probabilistic Design - The..... 37

The mean safety factor is the mean of the 4 area, is then the probability that failure will calculated safety factors, or 1.2. The occur. This can be quantified. standard deviation of the safety factors is The area of overlap, or the probability of found by: failure, is

2   SF 2  SF   1.4712 1.2032  0.024  0.155 A  p  .5 () SF f (3) (2) C  D With the deterministic calculation, the   S 2  S 2 safety factor can be known but the c d (4) distribution of safety factors brought about by the variability of the input, is not For > 2.2, known.  2  1 0,5     ()  .5  (2 ) exp  The probability of failure is the probability   2  (5) that the safety factor is less than 1. This is easily found as a standard Excel function, and for  ≤ 2.2 statistical tables are used. NORMDIST (x, mean, standard deviation) The mathematical description of the tables,  NORMDIST (1.0,1.203,0.155)  0.095 albeit less academically pure, can also be With the deterministic procedure, the used instead of the tables with better failure probability of 9.5% is not known, accuracy: although it is real. It is merely hidden.  ()  .0006 6  .0096 5  .0563 4  .1379 3

 .0423 2  .3893  .0004 Overlapping Distributions of Capacity (6) and Demand Once it is recognised that both the Overlapping Distributions Example load on any system (the demand) and the Consider the case where mine pillars are strength (capacity) are variable, and the used as the primary support. If the mean means and standard deviations are known, strength of the pillars is 12 MPa and the it is possible to construct standard deviation, caused by variability of distributions for them, such as in Figure 3 pillar height and width, is 3 MPa while the in the example to follow. In the area where mean pillar load is 8 MPa and the standard the distributions overlap, the demand deviation 2 MPa, then the frequency exceeds the capacity and failure will occur. distributions will be as shown in Figure 2. The area of overlap relative to the total

A = 6.9 m2

W = 275.26 kN

 = 600

Figure 1: Cross section through a plane that has the potential to fail.

38 I.J.M.G.E., University of Tehran, Vol. 46, No. 1, June 2012

Frequency Distributions of Load and Strength

0.25

0.2

0.15 Load Strength

0.1 Frequency

0.05

0 0 5 10 15 20 25 Load / Strength Mpa

Figure 2: Frequency distributions for pillar load and strength.

According to Equation 4,  = 1.11 and distribution and for each set of input then Equation 7 yields a value for () of numbers, an output using a known 0.37. The area of overlap is 0.13 according algorithm is calculated, see Figure 3. This to Equation 3 and the probability of failure process is repeated several hundreds of is then 13%. Note that the deterministic thousand times. The outputs are then safety factor, the ratio of strength to load, collated in a distribution and the final using just the average values, is 1.5. outcome is the distribution of outputs with a known mean and standard deviation. For Monte Carlo Technique the purpose of determining a failure The Monte Carlo technique had its probability, this simply means that the origins with the development of the cumulative frequency of safety factors less nuclear bomb in the United States during than 1.0 need to be found. the second world war, where scientists had This technique would be close to to consider the effect of a large number of impossible to perform without the use of variables on the probability of a chain modern computers, but with computers it reaction occurring. is relatively simple. Custom made It is fundamentally simple. Assuming that software, like the popular Excel add-on @ the distribution characteristics of each of Risk, are available. Even without those, it the input parameters is known, random is possible to perform the procedure in input numbers are chosen from each Excel with some limitations.

Probabilistic Design - The..... 39

The most important limitations when using a commonly used computer without Excel are that the algorithm has to be special additions. written such that each variable only occurs once in the equation and that the standard Monte Carlo Example deviation of the output cannot be Where a mine roof is supported using calculated due to the limitation of that the suspension principle, i.e. weak material procedure in Excel. However, the mean, in the immediate roof is simply suspended most frequent value and importantly, the from a stronger layer higher up by means probability of failure, can be calculated. of roof bolts, see Figure 4, the safety factor In Excel, the process entails defining the of the support system can be expressed by: characteristics of the input distributions  l  (these need not be normal distributions)  b  nd h  1 and then the basic algorithm to calculate  t w  (7) SF  for instance the safety factor has to be gBs provided. Then, the safety factor is where, calculated any number of times using the n = number of bolts per row random function in conjunction with the dh = diameter of hole input distribution characteristics and stored lb = length of hole in a matrix. The of tw = thickness of weak layer the output can then be plotted, basic  = shear strength of resin/rock contact parameters like the mean and most plane frequent value determined and the B = road width probability of a safety factor being less s = spacing between rows of roofbolts than 1.0 (in other words the probability of Note that Equation 7 is based on the failure) can be determined. pessimistic assumption that the sidewalls The author uses this process for a number of the roadways offer no support to the of applications and even with more than 2 roof, to cater for the situation where joints million output calculations, the execution occur at the edges of the roadway. time is minutes and certainly not hours on

Input 1 Input 4

Input 2

Input 3 Output

Figure 3: Schematic explanation of the Monte Carlo technique

40 I.J.M.G.E., University of Tehran, Vol. 46, No. 1, June 2012

dh

ta

Anchor zone ta l t b Layer to be w supported

Roadway

L

Cross Section

Figure 4: Simplified cross section through a roadway showing the installed support and explaining the symbols used in Equation 8.

In Equation 7, all the parameters except deterministic safety factor was 1.5, which the number of bolts per row, n, can be is regarded as acceptable for roof support regarded as variable. Table 2 contains design. reasonable numbers for the distributions of The results are only valid for the situation the input. where joints occur at the edges of the roadways, which is not the general case, Table 2: Distribution characteristics of input rather a worst case scenario. If the parameters for roof support design sidewalls are indeed regarded as support,

Parameter Mean Standard in other words in the situation where joints value Deviation do not occur at the edges, a simple way of Hole length 1.55 m 0.1 m Thickness of weak 1.0 m 0.2 m performing the calculation is to artificially layer increase the number of bolts per row by 2, Hole diameter 0.028 m 0.001 m in other words add one roofbolt in the Resin/rock shear 2 100 450 kPa strength kPa place of each sidewall. Row spacing 1.7 m 0.3 m This more realistic general view results in Road width 6.6 m 1.1 m an increased safety factor for the situation, to a value of 2.2. The distribution of safety The easiest way to handle a number of factors is also more favourable, see Figure variables in a single equation is to perform 6. However, the probability of failure is a Monte Carlo simulation. This was done still significant at 14%. for the distributions in Table 2 as input into Equation 7 with 4 bolts per row. The distribution of safety factors resulting from this is shown in Figure 5. The probability of failure was found to be 29%, notwithstanding the fact that the

Probabilistic Design - The..... 41

Distribution of Safety Factors 0.04

0.035

0.03

0.025

0.02

Frequency 0.015

0.01

0.005

0

0

2

0.4

0.8

1.2

1.6

2.4

2.8

3.2

3.6

0.08

0.16

0.24

0.32

0.48

0.56

0.64

0.72

0.88

0.96

1.04

1.12

1.28

1.36

1.44

1.52

1.68

1.76

1.84

1.92

2.08

2.16

2.24

2.32

2.48

2.56

2.64

2.72

2.88

2.96

3.04

3.12

3.28

3.36

3.44

3.52

3.68

3.76

3.84 3.92 Safety Factor

Figure 5: Distribution of roof support safety factors. The safety factors less than 1.0, where failre can be expected, are shown in red.

Distribution of Safety Factors 0.03

0.025

0.02

0.015 Frequency 0.01

0.005

0

0

2

0.4

0.8

1.2

1.6

2.4

2.8

3.2

3.6

0.08

0.16

0.24

0.32

0.48

0.56

0.64

0.72

0.88

0.96

1.04

1.12

1.28

1.36

1.44

1.52

1.68

1.76

1.84

1.92

2.08

2.16

2.24

2.32

2.48

2.56

2.64

2.72

2.88

2.96

3.04

3.12

3.28

3.36

3.44

3.52

3.68

3.76

3.84 3.92 Safety Factor

Figure 6: Distribution of roof support safety factors for the situation where the sidewalls are also regarded as supports. The safety factors less than 1.0, where failure can be expected, are shown in red.

These examples indicate that even with an used, although the failure probability will acceptable deterministic safety factor of exist in the background. It will be there, 1.5, significant failure can still be expected but it will not be known. The only way to using reasonable values for the variability expose it, is to perform the probabilistic of the input parameters. This will not be procedure described here. known if the probabilistic procedure is not

42 I.J.M.G.E., University of Tehran, Vol. 46, No. 1, June 2012

Conclusions include most of the impact of the input The most popular method to cater for variability. the variability in rock engineering design The methods to incorporate variability as has been to include a . With an integral part of calculations are not the factor of safety, a design is simply overly complex or time consuming. The made a number of times as strong as a statistical methods are proven, well calculation based on average values would documented and easy to use. require. In the examples provided in this paper, it However, this does not really compensate was shown that day-to-day calculations for variability, mainly because the range of can easily be performed using the input is not used to determine a suitable probabilistic methods. The output is factor of safety. Therefore the effect of substantially improved and much more variability is not quantified, and real information becomes available to importantly, it is not known whether or not make rational decision making much more the output is increased sufficiently to defensible.

References 1- Stacey, R.T. (1989). “Potential rockfalls in conventionally supported stopes – a simple probabilistic approach.” Journal of South African Institute of Mining and Metallurgy, PP. 111 – 115. 2 - Esterhuizen, G.S. and Streuders, S.B. (1998). “Rockfall hazard evaluation using probabilistic keyblock analysis.” Journal of South African Institute of Mining and Metallurgy, Vol. 98, PP. 59 – 63. 3 - Hoek, (2011a). “Factor of safety and probability of failure.” Rock Engineering Course Notes. http://www.rocscience.com/education/hoeks_corner. Accessed May 2011. 4- Hantz, D, Vengeon, J. M. and Dussauge-Peisser, C. (2003). “An historical, geomechanical and probabilistic approach to rock-fall hazard assessment.” Natural Hazards and Earth Sciences (2003), Vol. 3, No. 6, PP. 693 – 701. 5- Tonon, F, Mammino, A. and Bernardini, A. (1996). “A random set approach to the uncertainties in rock engineering and tunnel lining design.” In Proceedings of Eurock ’96, Barla (Ed). Balkema, Rotterdam, ISBN 90 5410 843 6. 6 - Esterhuizen, G. S. (2003). JBlock User’s Manual. 7- Harr, M. E. (1987). “Reliability-based design in .” McGraw-Hill, New York. 8 - Stacey, R.T. (2007). “Is rock engineering addressing risk appropriately?.” In Proceedings of SANIRE Symposium 2007: Redefining the Boundaries Part II.” South African National Institute of Rock Engineering. 9 - Hoek (2011b). “When is a rock engineering design acceptable.” Rock Engineering Course Notes. http://www.rocscience.com/education/hoeks_corner. Accessed May 2011.