An analytical probability distribution for the factor of safety in underground rock mechanics

International Journal of Rock Mechanics & Mining Sciences 48 (2011) 597–605

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An analytical probability distribution for the factor of safety in underground rock mechanics P.P. Nomikos n, A.I. Sofianos School of Mining Engineering and Metallurgy, Division of Mining Engineering, National Technical University, 9 Iroon Polytechneiou Street, Zografou, Athens 15780, Greece

a r t i c l e i n f o

abstract

Article history: Received 26 April 2010 Received in revised form 5 November 2010 Accepted 12 February 2011 Available online 11 March 2011

In many underground rock mechanics cases, the factor of safety may be defined as the ratio of the capacity, of the rock or its support elements, to the pertinent demand. By representing the capacity and the demand as uniform random variables, an analytic solution is obtained for the probability distribution of the factor of safety and its probability density and cumulative distribution functions. Closed form solutions for the calculation of the mean value, the standard deviation and the minimum and maximum values of the factor of safety, are thus provided. Four relative positions are possible for the demand and capacity density functions, according to their limits. Closed form solutions are provided for the probability of failure for the identified relative positions. Application of the developed analytical solutions for the probabilistic analysis of two cases of underground roofs and four cases of room pillars follows straight forward. This methodology proves also useful for the utilization of field observations or for the extrapolation of any specific design. It allows, finally, for the parametric evaluation of the effect of specific design variables to the distribution of the safety factor. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Factor of safety Probability of failure Density function Underground rock mechanics Pillar stability Suspended roofs

1. Introduction Modeling in mining engineering is the idealization of a structure, which admits to simple but logical mathematical solution and still contains the essential elements of the prototype. Traditionally, induced loadings are modeled by completely defined, simple geometric or analytical representations. Material characterization is assumed to be complete, and inherent properties are taken to be stable and uniquely defined.

1.1. Reliability of the systems However, neither geometric configurations nor inherent material properties are completely known. Among these are uncertainties with respect to the extent of the strata, the rock strength, structure, alteration, seepage, natural stress, mineralogy, dynamic actions, toxic and hazardous materials, heat and cold, freezing and thawing, chemical and environmental factors, workmanship, etc. Further, engineering materials contain microcrystalline imperfections, which initiate cracks or permit their propagation. Almost all geometric configurations, induced loadings and material properties and imperfections are random, and all systems may often be subjected to overloads. n

Corresponding author. Tel.: +30 210 772 2194; fax: + 30 210 772 2199. E-mail address: [email protected] (P.P. Nomikos).

1365-1609/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2011.02.015

The failure of a system is assessed by its inability to perform its intended function adequately on demand for a period of time and under specified conditions. Its opposite, the measure of success, is called reliability. Failure is qualitative, whereas reliability can be defined and quantified, as the probability of an object performing its required function adequately for a specified period of time under stated conditions. The purpose of reliability based design is to produce an engineering system whose failure would be an event of very low probability. The acceptance of a level of reliability must be viewed within the context of possible costs, risks and associated social benefits. Traditionally in underground mining and tunnelling, assessments of the risk of failure are made on the basis of allowable factors of safety, learned from previous experiences for the considered system in its anticipated environment. The factor of safety fs is usually defined as the ratio of the capacity C of the object upon the demand D, and failure is assumed to occur when it is less than 1. In general, the demand function is the resultant of many uncertain components of the system under consideration, mainly geological [1], and similarly the capacity function will depend on the variability of engineering material parameters, testing errors, construction procedures, inspection supervision, ambient conditions, etc. 1.2. Early approaches in mining The concept of the factor of safety in underground exploitations has found particular application in the design of pillars of

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naturally supported mines, and in the bolting arrangement of suspended layered roofs. 1.2.1. Pillar dimensioning The safety factor of pillars, for pillar supported underground mining, is conventionally calculated by relating the pillar strength to the average axial stress [2] acting on the pillar. In flat-dipping ore bodies, a usual assumption made is that pillars are loaded according to the tributary area theory, where the average axial pillar stress is calculated from the vertical component sv of the natural stress field and the room and pillar dimensions wo and wp, respectively. The pillar strength at any particular mine operation, or at any array of mines with the same or similar geomechanical characteristics, is usually derived through back analysis of failed pillar cases resulting in empirical formulae, such as those of Salamon and Munro [3] developed for South African coal seams. They provided histograms illustrating the frequencies of intact pillar performance and pillar failure as a function of the factor of safety, and suggested a design value of the safety factor equal to 1.6. In the early 1970s, Coates [4] presented interpretation results from stress measurements at pillars of one of the Elliot Lake mines. There, a coefficient of variation more than 20% was found for the pillar stresses, while the variation of pillar strength was estimated to be between 15% and 25% [5]. With such a variation, difficulties arise in applying a deterministic design based on a safety factor of average strength against average stress. Probabilistic design was thus suggested by Coates [6], although the production of any data for probabilistic calculations was too laborious due to the poor computational tools of the time. Further, Hedley and Grant [7] applied similar analyses for the Elliot Lake uranium mines, and Sheorey et al. [8] for Indian coal pillars. 1.2.2. Roof bolting Suspension is one of the main reinforcement mechanisms of roof bolts in layered roofs, provided that strong rock lies above the immediate roof. Traditional dead-weight loading design is appropriate for such cases, when the immediate roof thickness is less than 1 m [9]. In Fig. 1, thin layers of the weak immediate roof of average unit weight g and total thickness ts are shown fully suspended from an overlying thick competent layer. Each roof bolt supports an area of sc  sl and carries weight W¼ gscslts. If the loading capacity of the bolt is Tbf, the safety factor is given by Fs ¼

Tbf Tbf ¼ W sc sl ts g

ð1Þ

Stillborg [10] suggests a factor of safety between 1.5 and 3, depending on the damage which would result from falling rock and also on whether permanent or temporary reinforcement is to

Competent rock layer

be considered. The dead-weight loads calculated for g ¼ 25 kN/m3 vary from 12.5 to 200 kN for bolt spacing sc ¼sl varying from 1.0 to 2.0 m and for weak rock layer thickness ts varying from 0.5 to 2.0 m. Mark [9] noticed that in some mines such thickness of the weak, immediate roof layer can vary by as much as 1.0 m over very short distances. Similar treatment may be applied for suspended rock wedges formed in the roof of underground openings. There, the weight of the wedge is the demand random variable taking values in a wide range, whereas the capacity is the ability of the bolts to carry load. Stillborg [10] suggests a factor of safety between 2 and 5. Further, he comments his experience of failure cases, where safety factors of 2 or more were apparently used in the design of the reinforcement system, which however should be attributed to non uniform bolt stressing. Hoek et al. [11] suggest a factor safety between 1.3 and 1.5, the lower factor being appropriate for temporary mining openings such as drilling drives, while the higher factor should be used in more permanent access openings such as ramps. 1.2.3. Ongoing research Such observations trigger ongoing pertinent discussions and a constantly increasing interest on the use of reliability based analysis for rock engineering design, such as by Harr [12], Pine [13], Skipp [14], Hoek et al. [11], James [15], and Hoek [16,17]. A complete reference to probabilistic methods for geotechnical analysis is given by Baecher and Christian in [18]. Modern numerical tools like finite elements and neural networks combined with probabilistic analysis rationalize the design as presented by Deng et al. [19]. With the application of elastoplastic finite elements, and the random field theory, Griffiths et al. [20] account for both the variability of the rock properties and the spatial correlation lengths. The evolving availability of computational tools and computational power, allowed slowly the employment of such methods for the endeavor of reliability analyses.

2. Formulation The capacity C and the demand D may be considered as random variables, whose probability density functions (PDF), fC and fD, respectively, are shown in Fig. 2. The safety factor fs is defined as the ratio of capacity to demand, i.e. fs ¼

C D

ð2Þ

fs is also a random variable, with cumulative distribution function (CDF) Ffs. For C 40 and D 40, Ffs is defined as   C ofs ¼ pðC oD fsÞ ð3Þ Ffs ðfsÞ ¼ p D It is calculated by the integral of the joint (bivariate) PDF f (C, D), of the variables C and D, in the shaded region of the C–D plane, shown in Fig. 3a, such that C/D rfs.

W

sl

sc ts sc Fig. 1. Suspension of thin rock layers from a thick competent rock layer (modified from [10]).

Probability density

Weak rock layers fD (D)

fC (C )

Capacity (C )or Demand (D) Fig. 2. Probability density functions of the random variables capacity C and demand D.

P.P. Nomikos, A.I. Sofianos / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 597–605

599

where FD is the CDF of the random variable D. If the PDFs of the random variables C and D are defined within the positive limits [LC, UC] and [LD, UD], respectively, the safety factor takes values within the positive interval [min fs¼LC/UD, max fs¼ UC/LD]. Outside these limits, by definition, the above three PDF take zero value. In that case, the integration region is shown in Fig. 3b and the CDF is calculated as Z minfUC ,UD fsg Z UD Ffs ðfsÞ ¼ fC ðCÞ dC ð5Þ   fd ðDÞ dD

D

C=fs·D C
max LD , fsC

LC

D=C/fs

3. Uniform independent random variables Let us suppose for simplicity that fC and fD are functions of the uniform distribution, such as shown in Fig. 4, with fC ¼ 1/(UC  LC), and fD ¼1/(UD LD).

C

C

3.1. Cumulative distribution and probability density function for the factor of safety

D The CDF of the factor of safety is calculated from Eq. (5) and is given by 8 0, fs o ULCD > > > > 2 2 > < U ðF L ÞL ðF L Þ F1 F2 D 1 D 2 C C 2fs ð6Þ FðfsÞ ¼ , ULCD r fs o ULDC > ðUC LC ÞðUD LD Þ > > > > : 1, fs Z UC

C=fs·D UD

LD

D=C/fs

LD

LC

C

UC

C

Fig. 3. (a) Integration area for positive random variables C and D and (b) integration area for random variables C and D defined within the limits [LC, UC] and [LD, UD], respectively.

In most of such analyses, the PDFs are assumed to have infinite extent and usually to be normally distributed. Negative or infinite extents are unreasonable for rock mechanics problems, and therefore truncations are necessary to retain the values of the variables within realistic limits. Other distributions may also be used, such as the LogNormal or Beta, whose support is limited to non-negative values. Such distributions generally complicate the subsequent analytical derivations, and oblige for the use of numerical tools. The uniform distribution overcomes such difficulties, as it has by definition limited extents and allows for the development of closed form solutions and further analytical mathematical treatment. Although other distributions may be more appropriate for some cases, if sufficient data are available, the uniform distribution may be considered at least as an approximation, and seems to be particularly suitable for any data limited problems encountered in rock mechanics analysis. If the random variables C and D are independent, then the joint PDF is simply the product of the PDF of the two random variables. The CDF Ffs is written as Ffs ðfsÞ ¼

Z

1

fC ðCÞ dC 0

Z

1

¼ 0

Z

1 C fs

D

According to the value of fs with respect to the ratios LC/LD and UC/UD four segments may be distinguished, which are shown in Table 1. The arrangement of the segments may be achieved in two permutations, according to the relative values of the ratios LC/LD and UC/UD. These are drawn in the abscissa of the two PDF diagrams of Fig. 5. When LC/LD 4UC/UD, the segments are arranged as in the first PDF diagram of Fig. 5. For fsrUC/UD (segment a), the PDF increases according to row 1 of Table 1, having a peak at fs¼UC/UD. Then, f(fs) decreases, according to row 4 of Table 1,

C

D

LD

fD ðDÞ dD

     Z 1 C C ¼ 1 dC fC ðCÞ dC 1FD fC ðCÞFD fs fs 0

where F1 ¼ minfUC ,UD fsg, F2 ¼ maxfLC ,LD fsg. The PDF f(fs) is the derivative of the CDF. Differentiating Eq. (6) with respect to fs yields 8 > 0, fs o ULCD > > > > 2 2 < F1 F2 , LC rfs o ULDC ð7Þ f ðfsÞ ¼ > 2fs2 ðUC LC ÞðUD LD Þ UD > > > U > C : 0, fs Z L

fC(C) or fD(D)

C
LC

UD

UC

Capacity (C) or Demand (D) ð4Þ

Fig. 4. Probability density functions for uniformly random capacity C and demand D variables.

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Solution of the above equation holds for both permutations, i.e. LC/LD 4UC/UD and LC/LD oUC/UD, shown in Fig. 5, and gives a common mean value for the factor of safety. The variance of the safety factor is given by

Table 1 CDF and PDF of the safety factor.

a

Segment of fs

F(fs)

f(fs)

fso min(LC/LD;UC/UD)



2 UD LC =fs fs 2 ðUC LC ÞðUD LD Þ

2 UC =fsLD fs 1 2 ðUC LC ÞðUD LD Þ fsðUD þ LD Þ2LC 2ðUC LC Þ UD ðUC þ LC Þ=ð2fsÞ UD LD

1 UD2 ðLC =fsÞ2 2 ðUC LC ÞðUD LD Þ

2 UC =fs L2D 1 2 ðUC LC ÞðUD LD Þ 1 UD þ LD 2 UC LC 1 UC þ LC 2fs2 UD LD

b

fs 4maxðLC =LD ; UC =UD Þ

c

LC/LD r fsr UC/UD

d

UC/UD r fsr LC/LD

s2fs ¼

Z

UC =LD

LC =UD

fs2 f ðfsÞ dfsm2fs ¼

ðUC þ LC Þ2 UC LC m2fs 3UD LD

ð9Þ

where sfs is the standard deviation. From (8) and (9) the coefficient of variation Vfs is calculated as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u uðUC þ LC Þ2 UC LC 1 ð10Þ Vfs ¼ t 3UD LD m2fs

Probability density, f ( fs)

If the coefficients of variation of C and D are VC and VD, respectively, the limits of the random variables can be expressed in terms of their mean value mC and mD as pffiffiffi pffiffiffi LC ¼ mC ð1 3VC Þ, UC ¼ mC ð1 þ 3VC Þ pffiffiffi pffiffiffi ð11Þ LD ¼ mD ð1 3VD Þ, UD ¼ mD ð1 þ 3VD Þ

8 7

UC UD

LC LD

d

b

UC LD

fs fs

LC a UD

(Permutation i: LC/LD>UC/UD)

VD = 0.1

6

max fs

5

µfs

4

VC=0.1 VC=0.2 VC=0.3

min fs

3 2

fs =1.0

Probability density, f ( fs)

1 0

8 7

1.0

1.5

2.0

2.5

3.0 µ C / µD

3.5

4.0

4.5

VD = 0.2 max fs

6 fs

5

LC UD

a

LC c LD

UC UD

b

UC LD

VC=0.1 VC=0.2 VC=0.3

min fs

2 fs=1.0

1

(Permutation ii: LC/LD
0

Fig. 5. Density function for the factor of safety.

8 7

1.0

1.5

2.0

2.5

3.0 µC / µD

3.5

4.0

4.5

VD = 0.3 max fs µfs

5 4 min fs

3

VC=0.1 VC=0.2 VC=0.3

2 3.2. The mean and the variance The mean value of the random variable fs may be calculated from Z UC =LD ðUC þLC ÞlnðUD =LD Þ ð8Þ mfs ¼ fsf ðfsÞ dfs ¼ 2ðUD LD Þ LC =UD

5.0

6 fs

for fsrLC/LD (segment d). For fs4LC/LD (segment b) f(fs) decreases according to row 2 of Table 1. When LC/LD oUC/UD, the segments are arranged as in the second PDF diagram of Fig. 5. For fs rLC/LD (segment a) the PDF increases according to row 1 of Table 1, reaching a maximum value at fs ¼LC/LD. Then, in segment c, f(fs) is constant according to row 3 of Table 1 and for fs ZUC/UD (segment b) f(fs) decreases according to row 2 of Table 1.

µfs

4 3

fs

5.0

1 0

fs=1.0 1.0

1.5

2.0

2.5

3.0 µC / µ D

3.5

4.0

4.5

5.0

Fig. 6. Mean, minimum, and maximum value of the safety factor for various values of the ratio mC/mD, VC ¼ 0.1, 0.2, 0.3 and VD ¼ 0.1, 0.2, 0.3.

P.P. Nomikos, A.I. Sofianos / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 597–605

Then, Eq. (8) is written as pffiffiffi pffiffiffi ! mC 3 1 þ 3VD pffiffiffiffiffiffiffiffiffi mfs ¼ ln mD 6VD 1 3VD

ð12Þ

It is observed that mfs is a linear function of the ratio mC/mD and a function of VD; however, it is independent of VC. The extreme values of the factor of safety are evaluated in terms of mC/mD, VC and VD from pffiffiffi pffiffiffi m 1 3VC m 1 þ 3VC pffiffiffi , maxfs ¼ C pffiffiffi minfs ¼ C ð13Þ mD 1 þ 3VD mD 1 3VD In Fig. 6 the mean value of the factor of safety and its extremes are drawn as a function of the ratio mC/mD, for VC ¼0.1, 0.2, 0.3 and VD ¼0.1, 0.2, 0.3.

In Fig. 8 the minimum required conventional safety factor for practically zero probability of failure, is plotted with respect to the coefficient of variation of the random variables C and D, for VC and VD in the range 0–0.5. 4. Applications Underground rock pillars for which failure is not related to rock structural features and bolt suspended rock roofs, are particularly prone for design based on the concept of the factor of safety. Below, example cases, some of which are taken from the literature, are investigated with the suggested reliability analysis. Uniform random variables for the capacity and the demand are employed with no correlation between them. Such an assumption of independence is

0.5

3.3. Probability of failure

0.4 2.5

0.3

4

3

5

6 7 8 9 10

2.0

VD

Failure is attained if the demand surpasses the capacity, i.e. if the factor of safety is less than 1. The probability of failure is calculated by the CDF of the factor of safety for fs¼1. Four relative positions may be distinguished, as shown in Fig. 7, with respect to the limits of the capacity and the demand. Closed form solutions for the probability of failure are provided in Fig. 7 for the identified relative positions. The conventional factor of safety may be defined as the ratio of the mean capacity, mC, to the mean demand, mD. Then, the minimum required conventional safety factor for the elimination of any probability of failure, may be calculated by setting LC ¼ UD (belongs to position 1) and solving for mC/mD: pffiffiffi mC 1 þ 3VD pffiffiffi ¼ ð14Þ mD 1 3VC

601

0.2

µC/µD=1.5

0.1 0

0

0.1

0.2

0.3

0.4

0.5

VC Fig. 8. Required conventional safety factor mC/mD to mitigate the probability of failure for VC and VD in the range 0–0.5.

Fig. 7. Relative positions of capacity and demand random variables, according to their limits, and respective probability of failure.

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commonly adopted in the literature (e.g. Coates [6], Hoek et al. [11], Hoek [16], Martin and Christiansson [21]). Nevertheless, this may sometimes not be the case. A higher overall bolting performance may be expected in competent roofs than in fractured roofs, although higher loadings are often caused by fractured rock masses

and lower loadings by more competent ones. Similarly, rock pillars at higher depth may be stronger and subjected to higher loadings. To obtain adequate such data may be costly and rather difficult. However, if available, they might be considered appropriately. 4.1. Stability of rock pillars

Table 2 Summary data for the application examples of paragraph 4.

4.1.1. Conventional factor of safety For one of Eliot Lake mines, Coates [4] estimated the coefficient of variation of pillar stresses equal to 26%. Coefficient of variation of the pillar strength is estimated in the range 15–25%. For these values, Eq. (14) gives mC/mD ¼1.96–2.56. These values compare generally well with the data, presented by Madden et al. [22], for the frequency of failure and the corresponding design safety factor of 90 pillar failures of South African collieries. There, the majority of failed pillars had conventional factors of safety less than 2.0, apart from some cases where failure was due to the presence of rock structural features such as slips or joints, soft floor, very shallow depth and weak roof.

Example of paragraph

mC sC VC

mD sD VD LC UC LD UD Position Min{fs} Max{fs}

mfs s2 s Permutation

4.1.2

4.1.4

4.2.1

4.2.2

138 MPa 21 Mpa 0.152 76 Mpa 21 Mpa 0.276 101.63 MPa 174.37 MPa 39.63 MPa 112.37 MPa 1 0.90 4.40 1.98 0.47 0.68 i

113 MPa 17.18 MPa 0.152 95 MPa 26.22 MPa 0.276 83.25 MPa 142.75 MPa 49.59 MPa 140.41 MPa 1 0.59 2.88 1.29 0.20 0.45 i

78.5 kN 3.7 kN 0.047 61.5 kN 28.2 kN 0.459 72.09 kN 84.91 kN 12.66 kN 110.34 kN 4 0.65 6.71 1.74 1.39 1.18 i

400 kN 115.47 kN 0.289 300 kN 57.74 kN 0.193 200 kN 600 kN 200 kN 400 kN 3 0.5 3.0 1.39 0.24 0.49 ii

0.8

1.0

Probability density

0.7

0.8

F ( fs)

0.6 0.5

0.6 f (fs)

0.4

LC/LD

0.3

0.4

0.2

0.2

0.1 LC/UD

Cumulative probability

UC/UD

UC/UD

0

0.0 0.5

1.0

1.5

2.0 2.5 3.0 Safety factor, fs

3.5

4.0

4.5

Fig. 9. PDF and CDF for the factor of safety of the rock pillars, and comparison with the histogram of numerical results.

4.1.2. Probabilistic factor of safety In a hard rock mine, Coates [6] reports the estimated mean pillar strength (capacity, C) as mC ¼138 MPa with a standard deviation sC ¼21 MPa. The mean stress acting on the pillar (demand, D) is equal to mD ¼76 MPa with a standard deviation sD ¼21 MPa. Using normal random variables Coates determined a probability of failure Pf ¼0.017, meaning that about two pillars in every 100 will fail. The same example is used herein assuming that the pillar strength and pillar stress are uniform random variables. The limits of the capacity and the demand random variables, the range of the safety factor and its mean value and variance are summarized in the pertinent column of Table 2. The relative position of capacity and demand random variables, according to their limits, is position 1 of Fig. 7. Permutation i, according to Fig. 5, is pertinent, as LC/LD ¼2.5641.55¼UC/UD. In order to plot in Fig. 9 the CDF and the PDF of fs, its values are evaluated at the appropriate a, d and b segments (Table 3). Segment a, holds for 0.90¼min fsrfso UC/UD ¼1.55. Segment d, holds for 1.55¼UC/UD rfsoLC/LD ¼2.56. Finally, Segment b, holds for fsZLC/LD ¼2.56. The probability of fsr1, calculated as for position 1, is Ffs(1)¼0.011. Therefore, for a large number of pillars, approximately one pillar in every 100 will fail. To compare with, Coates calculated a probability of failure 0.017 for normal random variables, which is quite close to that calculated herein using uniform random variables.

Table 3 PDF and CDF of the safety factor for the application examples of paragraph 4. Segment

F(fs)

a b c d

f(fs)

a b c d

Example of paragraph 4.1.2

4.1.4

4.2.1

4.2.2

  0:99 2 fs 1:09 fs  2 1:70 0:39 1fs fs – 1:90 1:55 fs

  0:80 2 fs 1:35 fs  2 1:37 0:48 1fs fs – 1:24 1:55 fs

  1:44 2 fs 2:20 fs  2 1:70 0:25 1fs fs – 0:80 1:13 fs

  0:5 2 fs 1:0 fs  2 1:5 0:5 1fs fs 0.75fs  0.5 –

1:19

0:98 fs2

2:87 0:15 fs2 – 1.90fs  2

0:64 1:82 2 fs 1:89 0:23 2 fs – 1.24fs  2

4:86

2:08 fs2

2:88 0:064 fs2 – 0:80 fs2

0:25 1:0 2 fs 2:25 0:25 2 fs 0.75 –

P.P. Nomikos, A.I. Sofianos / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 597–605

4.1.3. Utilization of field observations Let us suppose, that the field observations at the mine of the previous example suggest that about 10% of the pillars show signs of failure (e.g. spalling). Assume that the stress estimation is based on field measurements and is considered reliable both in terms of mean value and variation. Further, assume that pillar strength dispersion was validated through extensive laboratory and field testing. Then, the mean value of the pillar strength can be estimated from the field observations in the current mine. Using the equation for the probability of failure provided in Fig. 7 for position 1, substituting Ffs(1)¼0.1, VC ¼0.152, mD ¼76 MPa, VD ¼0.276 and solving for mC, yields one solution valid for position 1, mC ¼113 MPa. Then, the limits of the pillar strength, evaluated from Eq. (11), are LC ¼83.25 MPa and UC ¼142.75 MPa.

4.1.4. Extrapolation to a nearby stope The probabilistic factor of safety design is applied to a neighboring mine situated 25% deeper. There, it is reasonable to assume that the average pillar stress is increased by 25% so that mD ¼1.25  76¼ 95 MPa. Then the limits of the demand are evaluated from Eq. (11) and the relative position of the random variables, according to Fig. 7 is position 1. The range of the factor of safety and its mean value and variance are given in Table 2. Permutation i, according to Fig. 5, is pertinent, as LC/LD ¼1.6841.02¼UC/UD. In order to plot in Fig. 10, the CDF and the PDF of fs, its values are evaluated at the appropriate a, d, and b segments. Segment a, holds for 0.59¼min fsrfso UC/UD ¼1.02. Segment d, holds for 1.02¼UC/UD rfsoLC/LD ¼1.68. Segment b, holds for fsZLC/LD ¼1.68.

Probability density

1.2 1.0

f (fs)

0.6 0.4

UC/LD

0.2 0.0 0.5 L /U 1.0 C D

1.5 2.0 Safety factor, fs

2.5

1.4 1.2

0.80

1

0.60 LC/LD

4.2.1. Suspension of layered roof An application of the probabilistic analysis for the evaluation of the factor of safety for roof suspension is presented by Hoek et al. [11]. There, a roof layer of unit weight g ¼27 kN/m3 has a mean thickness mt ¼1.0 m with a standard deviation st ¼0.5 m. The layer is fully suspended using rock bolts installed in a square pattern of sc  sl ¼1.5 m  1.5 m. These values are used to calculate the demand D, that is represented by a truncated normal distribution with mean value mD ¼61.5 kN, and standard deviation sD ¼28.2 kN. Based on hypothetical pull out test results, the loading capacity C of the bolts is also represented by a truncated normal random variable with mC ¼78.5 kN, sC ¼ 3.7 kN. Assuming a uniform distribution for the capacity and the demand functions, their pertinent parameters are evaluated in Table 2. The relative position of the random variables is a position 4 according to Fig. 7. Permutation i, according to Fig. 5, is pertinent, since LC/LD ¼ 5.6940.77¼UC/UD. In order to plot in Fig. 11 the CDF and the PDF of fs, its values are evaluated at the appropriate a, d, and b segments. Segment a, holds for 0.65rfs r0.77. Segment d, holds for 0.77rfs r5.69. Segment b, holds for 5.69 rfsr6.71. Comparison of the PDF of fs with the numerical results of Latin Hypercube sampling is also shown in Fig. 11; practically identical results may be observed. The probability of failure for position 4 is Ffs(1) ¼0.33. Therefore, for a large number of roofs, approximately 33 bolts out of 100 will fail and such a design should be rejected. The numerical analysis provided by Hoek et al. [11], for truncated normal random variables for capacity and demand, evaluates for the factor of safety a mean value of 1.41, a standard deviation of 0.71, and a probability of failure approximately 30%, which is close to 33% calculated herein for uniform random variables. In order to mitigate the probability of failure, the demand density function might be shifted to the left. The required bolt

1.00

F (fs)

0.8

4.2. Stability of underground roofs

0.40 0.3 0.20

0.00 3.0

Fig. 10. PDF and CDF for the factor of safety of the rock pillars, if the current design is extended to the 25% deeper mine.

Probability density

UC/UD

Cumulative probability

1.4

The probability that fs r1, calculated for position 1, is Ffs(1)¼0.3. Approximately 30 pillars in every 100 will fail and a domino type failure is possible. This design should be rejected. To achieve a probability of failure equal to 1%, the capacity of the pillar should be increased, or the demand should be decreased, or both. The required ratio of mean capacity to the mean demand, assuming that the coefficients of variation do not change, may be calculated by setting Ffs(1)¼0.01, VC ¼0.152, and VD ¼ 0.276 for position 1: mC/mD ¼1.82.

1.0 UC/UD 0.8 F (fs) 0.6

0.8 f (fs)

0.6

0.4 0.33

0.4 0.2

LC/LD

LC/UD

0.2

Cumulative probability

The minimum required conventional safety factor mC/mD, to practically exclude any probability of failure, may be calculated from Eq. (14) (or from Fig. 8) as mC/mD ¼ 2.0, with VC ¼ sC/mC ¼0.152 and VD ¼ sD/mD ¼0.276. Comparison of the results of the analytical formulae for the distribution and density functions of the factor of safety can be performed by simulation with the Latin Hypercube sampling technique. A MatlabTM script implementing a simulation procedure has been prepared. A built-in Matlab function is used for Latin Hypercube generation of n¼20,000 samples for the Capacity and Demand variables. The n values for each variable are randomly distributed within each interval (0,1/n), (1/n,2/n), y, (1 1/n,1), and they are randomly permuted. Then, n ¼20,000 calculations are performed for the random variable fs, that is the ratio of capacity to demand. In Fig. 9, the simulation results, represented by the histogram, are compared to the results of the analytical solution; practically identical results may be observed.

603

UC/UD

0

0.0 0.0

1.0

2.0

3.0 4.0 Safety factor, fs

5.0

6.0

7.0

Fig. 11. PDF and CDF for the factor of safety of the rock bolts, and comparison with the histogram of numerical results.

604

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spacing to practically eliminate any probability of failure (i.e. min(fs)¼1) may then be calculated by setting mD ¼ gs2mt; VD ¼ sD/mD ¼ st/mt; VC ¼ sC/mC in Eq. (14) and solving for s: pffiffiffi mC 1 3sC =mC 2 pffiffiffi s ¼ ¼ 1:43 ) s ¼ 1:20 m gmt 1 þ 3st =mt Such a value may be evaluated by Fig. 12, where the probability of fs o1.0, fs o1.5, and fso2.0 is plotted with respect to the bolt spacing sc ¼sl ¼s. The numerical analysis provided by Hoek et al. [11] with truncated normal random variables for capacity and demand, suggested similarly a bolt grid spacing of 1.25 m which gives a minimum factor of safety 1.04 and a zero corresponding probability of failure. 4.2.2. Wedge stability Potentially unstable roof rock wedges could fall into the excavation under the influence of gravity and need to be supported. Provided that the only external forces acting on such a wedge are its own weight and the support force, the factor of safety against falling is given by the ratio of support capacity to the wedge weight demand. The weight of the wedge (demand D) is considered as a uniform random variable with limits LD ¼200 kN and UD ¼400 kN. The wedge is to be supported with rock bolts and the support capacity C is considered also as a uniform random variable with

0.9 0.8

Probability

0.7 0.6 0.5

p (fs<2.0)

p (fs<1.5)

p (fs<1.0)

1.4 1.5 1.6 Bolt spacing, s (m)

1.7

0.4 0.3 0.2 0.1 0 1

1.1

1.2

1.3

1.8

1.9

2

Fig. 12. Probability of the safety factor being less than 1.0, 1.5, and 2.0 for varying the bolt spacing.

1.0

UC/UD

LC/LD

F (fs)

0.6

0.8 0.6

0.4

f (fs) 0.4 0.25 0.2

0.2 LC/UD

Cumulative probability

Probability density

0.8

UC/LD

0

0.0 1.0

2.0 Safety factor, fs

3.0

Fig. 13. PDF and CDF for the safety factor of the roof rock wedge, and comparison with the histogram of numerical results.

limits LC ¼200 kN and UC ¼600 kN. The relative position of the random variables is a position 3 according to Fig. 7. Permutation ii, according to Fig. 5, is pertinent, as LC/LD ¼1.0 o1.5¼ UC/UD. In order to plot in Fig. 13 the PDF and the CDF of the safety factor, its values are evaluated at the appropriate a, c, and b segments (Table 3). Segment a, holds for 0.5 ofso1.0. Segment c, holds for 1.0 ofso1.5. Segment b, holds for 1.5ofs o3.0. The probability of failure for position 3 is Ffs(1)¼0.25. There is a 25% probability of failure, and such a design should be rejected, although the conventional safety factor mC/mD ¼1.33 is marginally in the range of suggested conventional safety factors in the relevant literature. Comparison of the results of the analytical formulae for the distribution and density functions of the factor of safety is performed by simulation with the Latin Hypercube sampling method. In Fig. 13, the simulation results represented by the histogram are compared to the results of the analytical solution that is represented by the solid line; practically identical results may be observed.

5. Conclusions The inability of a system to perform adequately, its failure, is not easily defined. Failure may be spontaneous or gradual, reversible or remnant, and crisp or fuzzy. Further, judgment of the decision makers shall be added, who must balance potential benefits against cost. Reliability based design allows for accounting the pertinent risks. Design based on reliability is concerned primarily with computing the probability of occurrence of events about which there is only partial information. Prototype tests, prior to mining, for underground rock mechanics systems, are rare, and seldom may full scale tests be conducted. Therefore, generalizations must be made from very limited observations of the performance of samples of similar systems or scaled experiments. The knowledge represents a small only portion of the population about which some truth has to be discovered. Several probabilistic methods have been developed that yield measures of the distribution of functions of random variables, from the so-called exact methods that require computer oriented numerical treatment, to approximate procedures that can be treated by relatively simple algebraic calculations. The exact methods require that the probability distribution functions of all component variables be known initially. Because of the complexity of the solution process, the unknown component distributions are assumed usually to be between the few common distributions, and evaluation is performed numerically. Therefore, the treatment of such exact methods analytically seems intriguing. The traditional concept of the allowable factor of safety of a rock structure is a frequent problem treated with exact methods. In many underground rock mechanics cases, the factor of safety random variable may be defined directly as the ratio of the capacity to the demand random variables. These are usually represented as normal random variables that may be truncated to avoid unrealistic negative or extremely large values. However, the mathematical formulation of these probability distributions does not allow for analytic solutions and therefore numerical tools are necessary. Significant simplification to the problem is attained if the capacity and the demand are considered as uniform random variables. Although such an assumption is uncommon in numerical evaluations, in which practitioners resort to powerful and easy Monte Carlo methods, it is not unrealistic, especially when the input data are limited and a rough estimate of the possible extent of the capacity and the demand may only be made. Therefore, analytical formulae for the cumulative probability

P.P. Nomikos, A.I. Sofianos / International Journal of Rock Mechanics & Mining Sciences 48 (2011) 597–605

and density functions of the factor of safety are derived. This mathematical formulation, and the pertinent drawn diagrams, allows for parametric analysis and for the identification of the trends of the safety factor with changes in the capacity and the demand functions. Thus, it is found, that by increasing the conventional safety factor (i.e. the ratio of the mean capacity to the mean demand), the minimum, maximum and mean values of the factor of safety increase linearly. Four relative positions are possible for the demand and capacity density functions, according to their limits. The probability of failure depends on the pertinent relative position. A closed form solution is provided, for each of the four positions identified. From these, the minimum required conventional safety factor for the elimination of any probability of failure is calculated. In the diagrams drawn, it is shown that the required conventional safety factor increases asymptotically with the coefficient of variation of the capacity and the demand. Application of the developed analytical solutions for the reliability analysis of two cases of underground roofs and four cases of room pillars follows straight forward. The variability of the capacity and the demand is approximated with uniform random variables, and the distribution of the factor of safety and the probability of failure are defined analytically. This methodology proves also useful for the utilization of field observations or for the extrapolation of any specific design to other stopes. It allows also the evaluation of the effect of specific design variables to the distribution of the safety factor. The application examples presented in this article, apart from illustrating the applicability of the reliability based design, are used also for the numerical validation of the formulae, for identified random variable positions, using simulation with the Latin Hypercube sampling technique. The numerical evaluations are practically identical to the analytical ones. The analytical formulae developed for the distribution of the factor of safety in underground rock mechanics problems, allow for easy hand calculation, by an exact method, of the reliability of the supporting structures in underground mining. Decisions may then be taken accordingly, based on the acceptable risk.

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