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Computers & Geosciences 36 (2010) 959–969

Contents lists available at ScienceDirect

Computers & Geosciences

0098-30

doi:10.1

n Corr

E-m

(C. Karp

journal homepage: www.elsevier.com/locate/cageo

Estimating rock mass properties using Monte Carlo simulation:Ankara andesites

Mehmet Sari a,n, Celal Karpuz b, Can Ayday c

a Department of Mining Engineering, Aksaray University, Adana Road, 68100 Aksaray, Turkeyb Department of Mining Engineering, Middle East Technical University, 06531 Ankara, Turkeyc Aerospace Sciences Research Institute, Anadolu University, 26470 Eskisehir, Turkey

a r t i c l e i n f o

Article history:

Received 19 November 2009

Received in revised form

24 February 2010

Accepted 26 February 2010

Keywords:

Ankara andesites

Stochastic estimation

Rock mass strength

Hoek–Brown failure criterion

Monte Carlo method

04/$ - see front matter & 2010 Elsevier Ltd. A

016/j.cageo.2010.02.001

esponding author: Tel.: +90 382 2801345; fa

ail addresses: mehmetsari@aksaray.edu.tr (M

uz), cayday@anadolu.edu.tr (C. Ayday).

a b s t r a c t

In the paper, a previously introduced method (Sari, 2009) is applied to the problem of estimating the

rock mass properties of Ankara andesites. For this purpose, appropriate closed form (parametric)

distributions are described for intact rock and discontinuity parameters of the Ankara andesites at three

distinct weathering grades. Then, these distributions are included as inputs in the Rock Mass Rating

(RMR) classification system prepared in a spreadsheet model. A stochastic analysis is carried out to

evaluate the influence of correlations between relevant distributions on the simulated RMR values. The

model is also used in Monte Carlo simulations to estimate the possible ranges of the Hoek–Brown

strength parameters of the rock under investigation. The proposed approach provides a straightforward

and effective assessment of the variability of the rock mass properties. Hence, a wide array of

mechanical characteristics can be adequately represented in any preliminary design consideration for a

given rock mass.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

A rock mass consists of two components: intact rock anddiscontinuities, each of which has a significant effect on the rockmass strength and deformability. It is also known that weatheringsignificantly influences the engineering properties of rocks in-situ(Karpuz and Pasamehmetoglu, 1997; Ehlen, 2002; Gurocak andKilic, 2005; Park et al., 2005). Therefore, reliable estimates of rockmass strength and deformability are very important in rock masscharacterization. While near surface workings in quarries aremost strongly influenced by the structural conditions and thedegree of weathering of the rock mass, the properties of rockformations vary greatly, both spatially and randomly, in thequarries, they can rarely be predicted with confidence.

Since natural materials like soil and rock are inherentlyheterogeneous and variable, their nature of random character-istics needs to be appropriately described in preliminary designinvestigations. Therefore, Muspratt (1972) emphasized the ex-istence of vast areas of potential application of probabilisticmethods in geosciences because natural phenomena occur withsuch variation that a stochastic rather than a deterministic systemdefinition is more realistic. In the deterministic estimation of rock

ll rights reserved.

x: +90 382 2801365.

. Sari), karpuz@metu.edu.tr

mass properties, only a unique value is used, usually the averageof the investigated property. However, in a stochastic estimation,it is possible to consider the full range of data concerning thespecific random characteristic. This can be easily achieved withprobability distributions, which give both the range of values thatthe variable could take and relative frequency of each valuewithin the range (Evans et al., 1993).

It is difficult to measure experimentally the variability of rockmass mechanical properties. Rock engineers frequently utilizeempirical methods to estimate rock mass properties with onlylimited site data (Bieniawski, 1978; Hoek and Brown, 1980;Barton, 1983). These empirical methods require just a single valuefor each input characteristic and give a single output value. Theydo not provide the probability distributions of the investigatedrock mass parameters; this is primarily due to the difficulty ofhandling the inherent uncertainties of the component variables inthe empirical model.

Earlier research used statistical and probability methods toestimate rock strength and deformability from laboratory tests ordetermine the minimum number of specimens for laboratorytesting in rock mechanics (Yegulalp and Mahtab, 1983; Grassoet al., 1992; Gill et al., 2005; Sari and Karpuz, 2006; Ruffolo andShakoor, 2009). Stochastic models are also commonly employedto deal with uncertainties, due to the stochastic nature of thegeometry of rock masses and the variability of their mechanicalproperties (Dershowitz and Einstein, 1988; Kim and Gao, 1995a;Kulatilake et al., 1997; Meyer and Einstein, 2002; Sari, 2009).

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In the literature, there are a limited number of studieswhich considers only the stochastic estimation of strength anddeformability characteristics and variations of rock masses (Kimand Gao, 1995a; 1995b; Hoek, 1998; Sari, 2009). Two papers byKim and Gao (1995a, 1995b) presented a probabilistic method ofestimating the mechanical characteristics of a rock mass, usingthe third type asymptotic distribution of the smallest values(extreme value statistics) and Monte Carlo simulation. They usedthe chi-square goodness-of-fit test to prove that the distributionreflects the inherent variability of the basalt properties. Hoek(1998) applied the same method to estimate variation in theHoek–Brown properties of a hypothetical rock mass, and assum-ing that all three input parameters of the criterion can berepresented by normal distributions. The paper by Sari (2009) wasbased upon traditional (not extreme) statistics, as Hoek (1998)used. However, instead of arbitrarily assuming normal distribu-tions for input variables used in the Hoek–Brown criterion, theactual probability distributions of discontinuity properties andintact rock strength parameters were obtained directly from thehistograms of the observed frequencies of an ignimbrite rockmass. All of the above mentioned studies about stochasticmodeling have regarded input parameters as independent vari-ables, but this is not true.

The aim of this study is to demonstrate the use of the Monte

Carlo (MC) method. This is the most common sampling techniqueused in stochastic modeling. In the MC method, the uncertaintiesof the intact rock strength and discontinuity parameters areincorporated into a spreadsheet model using statistical distribu-tions. Furthermore, the relationships between these parametersare also accounted for in the stochastic analysis, using acorrelation matrix. This significantly improves the current studyover the previous study applied to a pyroclastic rock mass by thefirst author (Sari, 2009). The new MC model is basically designedfor the probabilistic evaluation of the variations observed in therock mass strength and deformability properties of Ankaraandesites at three different weathering grades. In each case, thestrength and deformation characteristics of the rock mass areestimated by means of the Hoek–Brown procedure. Extensivefield investigations and laboratory experiments previously con-ducted on the Ankara andesites by two of the co-authors (Karpuz,1982; Ayday, 1989) are the starting point for the present study.

2. Ankara andesites

Ankara andesite is widely utilized for building and pavementstone production due to its appearance, color, and durability.Andesites, described by Kasapoglu (1980), Karpuz (1982), Ayday(1989), Doyuran et al. (1993), Karpuz and Pasamehmetoglu(1997), and Ercanoglu and Aksoy (2004), occur extensively inthe vicinity of Ankara (Fig. 1). This material has been used sinceRoman times and before in building works. There are manyandesite quarries (Fig. 2), and many important historicalstructures were built from the rocks gathered from these quarries.

Significant changes in discontinuity spacing or characteristicswithin the same rock type may necessitate the division of the rockmass into a number of structural regions. For that reason, Ayday(1989) subdivided the studied quarry into three regions contain-ing different types of rocks: A-type, pinkish andesites; B-type,dark pinkish-gray andesites; and C-type, black andesites. Thedifference of color in each region is attributed to the degree ofweathering of the rock. Black andesites are un-weathered and thepinkish are the most weathered. The different degrees of andesiteweathering have also been observed by Kasapoglu (1980). Basedon the thin-section studies of samples of each type, the rock maybe classified as hornblende–andesite (Ayday, 1989).

For the Golbasi district, Karpuz and Pasamehmetoglu (1997)have also distinguished three types of andesites similar to thosedetermined by Ayday (1989): (i) brownish gray-to-grayish purple,fine-grained glassy andesite (A-type); (ii) pale red, fine-grainedandesite (B-type); and (iii) gray, fine-grained andesite (C-type). Todetermine average physical and mechanical properties, Karpuz(1982) conducted extensive rock mechanics experiments on thedistinct weathering grades of Ankara andesites collected fromdifferent locations of the studied rock.

Studies by Karpuz (1982), Ayday (1989), Doyuran et al. (1993),and Karpuz and Pasamehmetoglu (1997) addressed changes inthe frequency, length, and appearance of joints in the rock mass asweathering progresses. This research on vertical or steeplydipping joints in Ankara andesite shows that with increasedweathering, not only there are statistically significant differencesin mean joint spacing and mean trace length with increasedweathering, but also the physical appearance of joints changes asweathering progresses.

Doyuran et al. (1993) investigated the most appropriatefrequency distributions of aperture, persistence, and spacing ofdiscontinuities measured in different weathering grades ofGolbasi andesite. The results of the statistical analyses showedthat the degree of weathering affects the type of frequencydistributions of the discontinuity parameters even in the samerock.

3. Collection of the necessary data for stochastic estimation

The engineering geological properties of the exposed faces ofquarries within the study area were evaluated extensively in fieldobservations/measurements and laboratory tests by two of theco-authors (Karpuz, 1982; Ayday, 1989). In this study, Golbasiandesites are treated broadly under three distinct categoriesaccording to their degree of weathering and field observations asA-type andesite, B-type andesite, and C-type andesite.

To quantify the variability of the investigated discontinuityproperties of Ankara andesites at different weathering grades,closed form (parametric) distributions fitted to actual observa-tions by Ayday (1989) are directly included in the stochasticmodeling. In the case of any specific distributions of themechanical properties are not described in the original work ofKarpuz (1982) and Ayday (1989), a truncated normal distributionis subjectively assumed for these properties of the rock. Manyauthors have shown that the frequency distributions of someresults of tests on the strength of rock can be well represented bya normal distribution (Yegulalp and Mahtab, 1983; Grasso et al.,1992; Hoek, 1998; Hsu and Nelson, 2002; Gill et al., 2005; Sari andKarpuz, 2006; Sari, 2009).

The mean value of the normal probability distributionrepresents the best estimate of the random variable, and thestandard deviation or coefficient of variance (COV) of this meanvalue represents an assessment of the uncertainty. Generally, astandard deviation equivalent to 15–20% of the mean value,which represents the typical range of COV applicable for mostnatural geotechnical materials (Rethati, 1988) was assumed as ameasure of variability in this study.

3.1. Rock mass rating (RMR) classification system

The RMR system (Bieniawski, 1989) is one of the most widelyused rock mass classifications. This method incorporates geolo-gical, geometric, and design or engineering parameters in arrivingat a quantitative measure of the rock mass quality. Thisengineering classification system, developed by Bieniawski(1973), utilizes the following six rock mass parameters:

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Fig. 2. A general view of an Ankara andesite rock quarry.

Fig. 1. Location and geologic map of study area (modified from Karpuz and Pasamehmetoglu, 1997).

M. Sari et al. / Computers & Geosciences 36 (2010) 959–969 961

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(1) uniaxial compressive strength (UCS) of intact rock material, (2)rock quality designation (RQD), (3) spacing of discontinuities, (4)condition of discontinuities (aperture, persistence, roughness,infilling, weathering), (5) groundwater conditions, (6) orientationof discontinuities.

3.1.1. Unconfined compressive strength (UCS)

The unconfined (or uniaxial) compressive strength (UCS) of arock is frequently used as one of the key inputs in mostclassification schemes and design applications for the character-ization of rock material strength. Sari and Karpuz (2006) suggest anormal distribution for 103 UCS tests on the Ankara andesitesamples. Ayday (1989) states that the strength values of theGolbasi andesite increases, starting from the A-type through tothe C-type. Accordingly, the normally distributed UCS valuesgiven in Table 1 are assumed for the three different weatheringgrades of this rock.

3.1.2. Rock quality designation (RQD)

The RQD was developed by Deere et al. (1967) to provide aquantitative estimate of rock mass quality from drill core logs. Inthe areas, where scan-line or area mapping can be conducted, it isnot necessary to use core recovery since a better picture of therock mass can be obtained from these measurements. For scan-line data, an average joint spacing can be obtained (number offeatures divided by traverse length). Bieniawski (1989) has linkedan average joint spacing to RQD relying on previous work by Priestand Hudson (1976). The RQD in this study was estimated from theaverage joint spacing based on the following equation:

RQD¼ 100:e�0:1lð0:1lþ1Þ ð1Þ

where l is the average number of discontinuities per meter, l¼1/(mean joint spacing). The distribution of spacing must benegatively exponential, if the theoretical RQD is to be applied toa particular rock.

3.1.3. Spacing of discontinuities

On the data collected for the Ankara andesites by Ayday(1989), chi-square goodness-of-fit tests were performed fornormal, gamma, lognormal, and negative exponential distribu-tions; the last two being the distribution models commonly usedfor spacing evaluation (Priest and Hudson, 1976; Hudson andPriest, 1983). This is because those theoretical distributions arebounded at zero and are skewed to the right, those characteristicsare similar to the properties of the spacing distribution. Thenegative exponential probability distribution (Table 1) waschosen as an appropriate parametric distribution model torepresent the discontinuity spacing in Ankara andesites.

Table 1Intact rock and discontinuity properties of Ankara andesites.

Parameter A_type B_type

UCS (MPa) Normal 53.0710.6n Normal 68.0710.2Joint spacing (cm) N. Exponential 54.4743.9 N. Exponential 125.97Joint persistence (m) N. Exponential 2.9272.1 Gamma 3.8872.68Joint aperture (mm) Normal 5.0870.98 Normal 4.0571.48

Roughness Slightly rough Slightly rough

Weathering Moderately weathered Slightly weathered

Infilling Soft fillings Hard fillings

Groundwater Dry DryEi (GPa) Normal 22.874.6 Normal 28.174.2mi Normal 4.171.0 Normal 7.071.4D Normal 0.770.1 Normal 0.570.1

n Mean7Std. Dev.

3.1.4. Joint persistence

Many studies of field measurements have shown thatthe negative exponential probability density distribution isappropriate to represent the discontinuity trace length distribu-tion (Wallis and King, 1980; Baecher, 1983; Park and West, 2001;Kulatilake et al., 2003). Ayday (1989) found that, in Ankaraandesites, the type of distribution changes from gamma tonegative exponential as the intensity of weathering increases.The frequency distributions given in Table 1 are assumed for jointpersistence characterization of Ankara andesites.

3.1.5. Joint aperture

The apertures of real discontinuities are likely to vary widelyover the extent of the discontinuity. Clearly, the variation ofaperture will have an influence on the shear strength of thediscontinuity (Barton, 1983). From outcrop mapping, the jointaperture can only be roughly estimated, through the directobservation of joint exposed at the outcrop, according to Brown(1981). Ayday (1989) recommended normal distributions for thejoint aperture at distinct weathering degrees of Ankara andesites(Table 1).

3.1.6. Joint roughness

A classification of discontinuity roughness has been suggestedby Brown (1981). Roughness is a potentially important compo-nent of strength and therefore, a qualitative description of jointroughness was first accomplished for each different weatheringgrades of Ankara andesites based on field observations (Table 1).Then these qualitative descriptions are converted to quantitativevalues using the relevant RMR score in the stochastic model.

3.1.7. Infilling

Filling is the term for material that separates the adjacent rockwalls of discontinuities. The joint can be clean or filled withweathered products and deposits, ranging from sandy particles toswelling clays. Again, a qualitative description of joint infillingwas accomplished for each distinct weathering grades of Ankaraandesites based on field observations (Table 1). Then, thesequalitative descriptions are converted to quantitative values usingthe relevant RMR score in the stochastic model.

3.1.8. Weathering

When the joint surface is weathered, it often shows a change incolor and appearance. Frequently, weathered products, such asgrain particles may remain inside the joint. According to the thin-section results of the andesites (Ayday, 1989), A-type ismoderately stained, B-type is slightly stained and C-type isvirtually no staining. It was also observed that the discontinuities

C_type Source

Normal 128.0712.8

Karpuz (1982) and Ayday (1989)

97.2 N. Exponential 73.0752.9

Gamma 1.8671.0

Normal 4.7271.10

Very rough

Unweathered

Unfilled

Dry

Normal 45.174.5

Normal 10.571.6

Normal 0.370.1

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were generally in dry condition. Table 1 describes the weatheringgrades for different types of Ankara andesites.

3.2. The Hoek–Brown criterion

The most recent version of the generalized Hoek–Brown failurecriterion (Hoek et al., 2002) is employed to estimate the rock massproperties of the weathered Ankara andesites. In order to use theHoek–Brown (HB) criterion to estimate the strength and deform-ability of jointed rock masses, the following three properties ofthe rock mass need to be described: the uniaxial compressivestrength sci of the intact rock pieces in the rock mass, the value ofthe Hoek–Brown constant mi for these intact rock pieces, and thevalue of the geological strength index (GSI) of the rock mass.

At failure, the generalized HB criterion (Hoek et al., 2002)relates the maximum effective stress, s1, to the minimumeffective stress, s3, through the functional relationship

s1 ¼ s3þsci mbs3

sciþs

� �a

ð2Þ

where mb extrapolates the intact rock constant mi to the rockmass

mb ¼mi expGSI�100

28�14D

� �ð3Þ

sci is the uniaxial compressive strength of the intact rock and s

and a are constants that depend upon the rock mass’s character-istics

s¼ expGSI�100

9�3D

� �ð4Þ

a¼1

2þ

1

6ðe�GSI=15�e�20=3Þ ð5Þ

The GSI introduced by Hoek (1994) and Hoek et al. (1995)provides a number which, when combined with the intact rockproperties, can be used to estimate the reduction in rock massstrength for different geological conditions. GSI takes into accountthe geometrical shape of the intact rock fragments as well as thecondition of the joint surfaces. Angular rock pieces with clean,rough discontinuity surfaces will result in a much stronger rockmass than one which contain rounded particles surrounded byweathered and altered material.

The parameter D is a factor that quantifies the disturbance of rockmasses. It varies from 0 (undisturbed) to 1 (disturbed) depending onthe amount of stress relief, weathering, and blast damage resultingfrom nearby excavations. In order to allow some variations in thevalues of this parameter in the stochastic analysis and to treat it as arandom variable of the HB model, a narrow normal distribution isassigned for distinct grades of this rock (Table 1).

Table 2Correlation matrix incorporated into MC simulation model.

UCS RQD S P A R I W

UCS 1.0

RQD 0.9 1.0

S 0.9 0.9 1.0

P �0.6 �0.6 �0.6 1.0

A �0.6 �0.3 �0.6 0.6 1.0

R 0.6 0.6 0.3 �0.3 �0.6 1.0

I �0.6 �0.3 �0.3 0.6 0.9 �0.3 1.0

W �0.9 �0.6 �0.9 0.9 0.9 �0.6 0.9

GW �0.6 �0.6 �0.6 0.6 0.9 �0.3 �0.3

mi 0.6 0.6 0.6 �0.3 �0.3 �0.3 �0.3 �

Ei 0.9 0.6 0.6 �0.6 �0.9 0.6 �0.6 �

D �0.6 �0.6 �0.6 0.6 0.9 �0.6 0.6

The uniaxial compressive and tensile strengths of the jointedrock masses are calculated from the following equations assuggested by Hoek et al. (2002)

scðMPaÞ ¼ sci:sa ð6Þ

stðMPaÞ ¼ �s:sci

mbð7Þ

Hoek et al. (2005) re-examined existing empirical methods forestimating rock mass deformation modulus and concluded thatnone of these methods provided reliable estimates over the wholerange of rock mass conditions that are frequently encountered. Arelated modification for estimating the deformation modulus ofrock masses was created by Hoek and Diederichs (2006). Using acommercial curve fitting software, they derived the followingbest-fit equation for a new set of reliable rock mass deformationmodulus data from China and Taiwan.

Erm ¼ Ei 0:02þ1�D=2

1þeðð60þ15D�GSIÞ=11Þ

� �ð8Þ

in which Ei is the elastic modulus of intact rock. The Ei (GPa)values found by Karpuz (1982) are given in Table 1 for Ankaraandesites at different weathering grades.

3.2.1. Geological strength index (GSI)

In the HB criterion, the GSI is the most important inputparameter in terms of the relationship between the strength anddeformation properties determined in the laboratory and thoseassigned to the field scale rock mass (Hoek, 1998). In earlierversions of this criterion, Bieniawski’s RMR was used for thisscaling process and the use of different rating scores for eachparameter made the RMR more suitable for numerical computa-tions. Therefore, it is possible to directly obtain a frequencydistribution from the rock mass rating scores that is crucial for thepurpose of simulation in a probabilistic analysis. On the other hand,GSI can only be determined by field observations of the blocks anddiscontinuity surface conditions. The descriptive and largelyqualitative nature of the GSI tables does not allow a quantitativedefinition for the variability of the rock mass properties. As a result,the type of parametric distribution that represents the rock masscharacteristics can only be made in a subjective manner, such asassuming a normal distribution as Hoek (1998) did.

3.2.2. Material constant mi

The empirical criterion formulated by Hoek and Brown (1997)allows the use of an approximate value of material constant mi fora particular rock. The constant mi is changed with the type of rock,its mineral composition, interlocking of grains, grain size, etc.Karpuz (1982) recommended different material constants mi forthe distinct weathering grades of Ankara andesite samples based

GW mi Ei D

UCS—uniaxial compr. strength

RQD-rock quality designation

S-spacing

P-persistenc

A-aperture

R-roughness

I-infilling

1.0 W-weathering

0.6 1.0 GW-groundwater

0.6 �0.3 1.0 mi-material constant

0.9 �0.3 0.6 1.0 Ei-elasticity modulus

0.9 0.6 �0.6 �0.9 1.0 D-disturbance factor

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on triaxial strength data. Accordingly, a normal distribution wasassumed with different mean values and different COVs (15%, 20%,and 25% of the mean) for distinct types of this rock (Table 1).

4. Stochastic estimation of rock mass strength parameters

4.1. Monte Carlo (MC) simulation

The most common sampling technique used in stochasticanalysis is the Monte Carlo (MC) method. The term stochastic isused when there is a random component to a model, e.g. valuesfor the input variables are sampled from a statistical distribution.The technique uses random or pseudo-random numbers tosample from a probability distribution. Any given sample mayfall anywhere within the range of the input distribution but,samples are more likely to be selected from the regions of thedistribution that have higher probabilities of occurrence. Withsufficient iterations, MC sampling ‘‘recreates’’ input distributionsand with the aid of a computer, hundreds or thousands of ‘‘what-if’’ scenarios can be conducted, modeling most combinations ofinput parameters and quantifying the statistical distributions ofoutcomes (Vose, 2000). This estimation is reasonably accurateonly if the number of simulations is very large. The advantage ofthis method is that the complete probability distribution for themechanical properties can be successfully obtained if theprobability density function of input parameters are accuratelyassessed and the interdependence between these parameters arequantified correctly.

When using the MC simulation, it is not necessary to firstdescribe the input variables with a closed form distribution, thesecan be directly sampled from the frequency distributions(histograms). In fact, in this way inaccuracies can be avoided,including the normal approximation. Clearly, this is not as easily

Fig.3. Schematic diagram of stochastic modeling

achieved as in the use of parametric distributions. Also, since anormal distribution goes from plus to minus infinity, it should beused with truncation to describe variables belonging to naturalmaterials.

The core of the MC method is a random-number generator. Arandom number is a uniformly distributed random variable overthe interval [0,1]. Random numbers can be generated using theExcel RAND() function, which produces a new random numberevery time the spreadsheet is recalculated. The next step is totransform uniformly distributed random numbers into a non-uniform distribution. For a more comprehensive description ofstatistical distributions and how the MC simulation works, thereader is directed to Ripley (1987), Evans et al. (1993), Johnsonet al. (1993, 1994, 1995), Gentle (1998), Law and Kelton (2000),and Vose (2000).

Dependencies do exist between variables in a system beingmodeled and they will often need to be correlated to ensure thatonly meaningful scenarios are generated for each iteration of themodel. One problem in the MC simulation is the difficulty ofincorporating the covariance between input variables. Thecovariance between random parameters plays an important rolein probabilistic analysis, although there are limited conclusionsfrom the research involving the accurate evaluation of covariancebetween random parameters in a rock mass (Kim and Gao, 1995a;Hoek, 1998; Sari, 2009).

For instance, a weathered discontinuity could be expected tohave wider openings and be completely filled with weatheringproducts. A stronger rock would be expected to have a shortertrace length and narrower opening. A closer spacing would resultin a lower RQD and shorter persistence. Pairs of input variablessuch as UCS and mi, RQD and mean joint spacing, and GSI andmean joint spacing, certainly have interdependence in nature too.Simulation software allows the incorporation of the rank ordercorrelations observed between input parameters as a correlation

methodology applied to an Ankara andesite.

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matrix in the spreadsheet models. However, since it is challengingto establish the exact nature of relationships observed betweenrock material and discontinuity parameters only subjectiveestimates can be made.

In a previous study (Sari, 2009), intact rock and jointparameters were considered to be probabilistic in nature;however, they were also assumed to be independent. For thisstudy, to overcome the problem of selecting a representativerating of various parameters, a correlation matrix (Table 2) isconstructed intuitively for use in MC simulations. The correlationmatrix is a lower triangular matrix with ones along the diagonal.The construction of this matrix is solely based on the direction ofthe relationships and relative weights of the rock massparameters considered in the RMR classification scheme andengineering judgment of the authors. To the authors’ knowledge,the construction of such a matrix is probably one of the firstattempts in the field of rock engineering.

During the construction of the matrix, the direction of therelationship observed between input parameters is first defined asnegative or positive. Then, the strength of association anticipatedbetween these parameters is quantified using a descriptive scale.There are five classes into which the relationships can be assignedranging from 0 to 1, corresponding to ‘‘none’’ (0.0), ‘‘weak’’ (0.3),‘‘moderate’’ (0.6), ‘‘strong’’ (0.9), and ‘‘perfect’’ (1.0).

4.2. Results and discussion

Fig. 3 is a simple illustration of the stochastic modelingmethodology applied in this study. In the stochastic estimation ofrock mass strength and deformability properties of weatheredAnkara andesites, the following steps were taken intoconsideration:

Fig. 4. Spreadsheet model used in MC

i.

sim

The data for discontinuity parameters and intact rock strengthcharacterization was compiled from the studies of Karpuz(1982) and Ayday (1989).

ii.

Closed form distribution functions, which represent both theprobability and range of values that would be expected inthe field and laboratory were defined for each parameter ofthe RMR classification scheme.

iii.

In the third step, the stochastic assessment of rock massproperties was accomplished using the probability densitydistributions from the previous step. MC simulations wereexecuted to obtain a statistical representation of the RMR andRQD in the spreadsheet model.

iv.

UCS, GSI, Ei, mi, and D were defined as parametric distributionsand included in strength and deformability equations toestimate the mb, s, and a parameters of the Hoek–Brownfailure criterion for the rock mass.

v.

By running the MC simulation for the rock mass propertiesdescribed as statistical distributions in previous steps, themean, standard deviation, and confidence intervals (range) ofthe uniaxial compressive strength (Sigc), uniaxial tensilestrength (Sigt), and deformation modulus (Erm) of the rockmass were then achieved for each distinct type of the Ankaraandesites.

Using Excel add-in @RISK, that processes these calculations instandard spreadsheet packages; it is possible to explicitly includethe uncertainty present in inputs to generate outputs that showall probable alternatives (Palisade, 2000). The program provides asimple and intuitive implementation of an MC simulationtogether with prepared spreadsheet models. For this purpose,different spreadsheet models are prepared separately for threedistinct types of Ankara andesite. These models include thegeneralized Hoek–Brown failure criterion formulae and allowsusers to easily obtain reliable estimates of rock mass strength

ulation for A-type andesite.

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properties. One of the models prepared for the A-type andesite ispresented in Fig. 4 with related cell formulas.

In the spreadsheet model, cells B2:B6 include formulae for theprobability distributions defined previously for rock mass dis-continuity and strength parameters. Cells C2:C10 contain theappropriate RMR scores given in Bieniawski’s (1989) classificationscheme. Many different combinations of these ratings aresimulated to produce a distribution of RMR in cell C11. In Fig. 4,only average values of the discontinuity and intact rock strengthparameters are rated according to the RMR classification in the

GSI (%)

Freq

uenc

y (%

) AB C

a

Freq

uenc

y (%

)

C

B

A

Sigc (MPa)

Freq

uenc

y (%

)

A

B

C

0

5

10

15

20

25

30

35

40

05

10152025303540455055

05

10152025303540455055

0

5

10

15

20

25

30

35

40

Erm (G

Freq

uenc

y (%

)

A

B

0 5 10 15 20

0 10 20 30 40 50

0.5 0.502 0.504 0.506 0.508 0.51

20 30 40 50 60 70 80 90 100

Fig. 5. Frequency distributions of rock mass pa

related cells and they give a total of 57 RMR value. When a newsuitable value in the relevant cells is generated randomly from thespecified parametric distributions the spreadsheet recalculatesmany different combinations of these ratings according to thecorrelation matrix given in Table 2. Cells, C2:C10, which containthe input parameters for the RMR classification system willautomatically assign the scores according to the appropriaterange of the RMR values given in the 1989 version of Bieniawski’sclassification scheme. These ratings can then be combined toacquire a final RMR frequency distribution with an average value

mbFr

eque

ncy

(%)

A

B

C

s

Freq

uenc

y (%

)

A

BC

05

1015202530354045

010

2030

4050

6070

80

05

1015202530354045

Sigt (MPa)

Freq

uenc

y (%

)

A

B

C

Pa)

C

25 30 35 40 45

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

rameters simulated by spreadsheet model.

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M. Sari et al. / Computers & Geosciences 36 (2010) 959–969 967

of 60.9 and a standard deviation of 6.34, which typicallyresembles a normal distribution.

Since the 1989 version of the Bieniawski’s RMR classificationsystem is applied for the calculation of ratings in the cells, cell B13calculates the GSI as equal to RMR89-5 as suggested by Hoek andBrown (1997), where RMR89 has the groundwater rating set to 15(dry) and the adjustment for joint orientation is set to 0 (veryfavorable). It should be noted that when a rock mass classificationsystem is used for estimating rock mass strength (and deforma-tion properties), only the inherent parameters of the intact rockand discontinuities need to be considered for evaluation of theclassification index. Other parameters such as groundwater and

Table 3@Risk inputs summary statistics.

Input name Minimum Maximum MeanStandard

deviation5% 95%

UCS_A 16.53 104.58 53.02 10.70 35.50 70.42

UCS_B 31.34 109.99 68.01 10.26 51.17 84.74

UCS_C 83.70 198.93 128.03 12.96 106.91 149.04

Spacing_A 0.03 6.95 0.52 0.52 0.06 1.49

Spacing_B 0.09 16.35 1.27 1.25 0.15 3.59

Spacing_C 0.06 8.87 0.73 0.70 0.10 2.05

Persistence_A 0.50 17.80 3.21 2.51 0.68 8.21

Persistence_B 0.44 28.10 3.85 2.66 0.92 8.84

Persistence_C 0.81 9.77 1.98 1.23 0.21 4.15

Aperture_A 2.02 9.86 5.08 0.99 3.47 6.69

Aperture_B 0.10 11.77 4.07 1.48 1.65 6.49

Aperture_C 1.01 9.91 4.72 1.11 2.90 6.53

Ei_A 8.59 44.42 22.81 4.63 15.22 30.33

Ei_B 14.73 47.36 28.11 4.23 21.17 35.01

Ei_C 30.08 64.79 45.11 4.53 37.66 52.50

D_A 0.41 1.00 0.70 0.10 0.54 0.86

D_B 0.22 0.80 0.50 0.10 0.34 0.66

D_C 0.01 0.60 0.30 0.10 0.14 0.46

mi_A 0.84 8.87 4.10 1.01 2.45 5.74

mi_B 2.02 13.59 7.00 1.41 4.70 9.29

mi_C 5.28 19.24 10.50 1.62 7.86 13.13

Table 4@Risk outputs summary statistics.

Output name Minimum Maximum Mean

RQD_A 20.68 99.99 89.62

RQD_B 69.79 100.00 97.33

RQD_C 51.93 99.99 94.32

RMR_A 32 77 60.9

RMR_B 51 86 70.9

RMR_C 62 91 79.9

GSI_A 27 72 55.9

GSI_B 46 81 65.9

GSI_C 57 86 74.9

mb_A 0.03 1.62 0.42

mb_B 0.24 4.17 1.44

mb_C 1.06 7.51 3.76

s_A 0.000 0.027 0.00

s_B 0.000 0.094 0.01

s_C 0.003 0.210 0.05

a_A 0.501 0.527 0.50

a_B 0.501 0.508 0.50

a_C 0.500 0.504 0.50

Sigc_A 0.04 15.20 2.85

Sigc_B 0.53 33.66 7.99

Sigc_C 4.82 91.14 29.75

Sigt_A �3.04 �0.01 �0.38

Sigt_B �4.51 �0.03 �0.67

Sigt_C �7.23 �0.17 �1.88

Erm_A 0.28 18.72 4.47

Erm_B 1.12 33.24 11.02

Erm_C 6.32 55.02 28.65

in-situ stress should not be thought to modify the classificationindex, because they are considered in the design stage of rockstructures. For example, when RMR is used for rock mass strengthestimation, the rock mass should be assumed to be completelydry and have a very favorable discontinuity orientation to avoiddouble counting (Hoek et al., 1995, 2002).

For each model simulation, 1000 iterations are performedusing the Latin Hypercube sampling. This method uses a stratifiedsampling to better resemble the resulting probability distributionwith fewer iterations compared to MC sampling (Palisade, 2000).Every run of the simulation yields 1000 different possiblecombinations of input variables that are sampled randomly fromthe previously defined parametric distributions taking intoaccount the correlations provided in Table 2. Once the values ofthe GSI in cell B13 have been estimated from the previous steps,the truncated normal distributions located in cells B2 for UCS, B14for mi and B16 for D can be used to estimate the Hoek–Brown rockmass parameters mb, s, and a values in cells B18, B19, and B20,respectively. Then, possible combinations of the Hoek–Brownfailure parameters simulated in the previous step are entered intothe equations contained in cells B22, B23, and B24 for uniaxialcompressive strength (Sigc), uniaxial tensile strength (Sigt), anddeformation modulus (Erm) to obtain the final rock mass strengthand deformability distributions of the Ankara andesites.

All the frequency distributions of the output variablessimulated by stochastic modeling are given in Fig. 5. It isimportant to note that stochastically estimated rock massmechanical properties of the Ankara andesites never have asingle value, in fact, all exhibit significant variations betweensome specified intervals. Information obtained from the MC

simulation runs is still relevant, and gives further insight intovariability of the rock mass strength and deformability propertiesestimated from the HB criterion. The rock mass strength anddeformability characteristics of the Ankara andesites are generallyseen to resemble asymptotic distributions skewing to largervalues, a similar trend that has been observed in intact rockmaterials.

Standard deviation 5% 95%

16.38 48.64 99.78

5.11 85.40 99.96

9.25 71.79 99.88

8.56 42 72

6.08 59 79

5.73 69 89

8.56 37 67

6.08 54 74

5.73 64 84

0.25 0.09 0.91

0.57 0.59 2.48

1.14 1.98 5.76

3 0.004 0.000 0.010

5 0.013 0.002 0.039

8 0.041 0.010 0.140

5 0.004 0.502 0.514

2 0.001 0.501 0.504

1 0.001 0.500 0.502

2.13 0.31 6.63

4.43 2.12 16.69

13.21 11.21 51.53

0.33 �1.01 �0.03

0.49 �1.59 �0.14

1.09 �3.91 �0.50

2.99 0.76 10.06

5.00 3.79 19.90

8.21 14.95 41.60

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0

20

40

60

80

100

Normal stress,σn(MPa)

Shea

r stre

ss,τ

(MPa

)

A

B

C

Intact

-20 0 20 40 60 80

Fig. 7. Generalized Hoek–Brown failure envelopes for Ankara andesites in sn�t

M. Sari et al. / Computers & Geosciences 36 (2010) 959–969968

The summary statistics computed from the @RISK simulationfor the input and output variables are listed in Tables 3 and 4,respectively. For the A-type andesite, the RMR ranges 32–77 witha mean value of 60.9. For the B-type andesite, it ranges 51–86with a mean value of 70.9. For the C-type andesite, it ranges62–91 with a mean value of 79.9. The model simulates the meanvalues for Sigc of A-, B-, and C-type andesites as 2.85, 7.99, and29.75 MPa, respectively. It estimates the average values for Sigt ofA-, B-, and C-type andesites as �0.38, �0.67, and �1.88 MPa,respectively. The Erm averages of A-, B-, and C-type andesites arefound as 4.47, 11.02, and 28.65 GPa, respectively.

For different weathering stages of the Ankara andesites, thefailure envelopes in the principal stress space (Fig. 6) and shear-normal stress space (Fig. 7) were obtained as the final step in thestudy. The strength envelopes are drawn by entering the meanvalues for the parameters of each rock mass previously estimatedfrom the stochastic analysis. Necessary calculations for this arecarried out by solving the Hoek–Brown empirical equationsproposed for intact rock and rock masses. It is obvious in bothgraphs that as the weathering intensity of Ankara andesitesincreases, the relative change in the strength envelopes becomessmaller.

space.

5. Conclusions

Stochastic modeling is a technique where an existing model ofa system is used to quantify the random variation that is expectedin the system under investigation. In this study, this technique hasbeen used to estimate the variation in the rock mass strength anddeformability characteristics of weathered Ankara andesite. Theresults showed that:

�

Figspa

variations in the mechanical properties of the rock mass couldbe easily estimated, if appropriate techniques were appliedthat took into account the frequency distribution of the inputparameters.

0

50

100

150

200

250

300

350

Minor principal stress, σ3 (MPa)

Maj

or p

rinci

pal s

tress

,σ1 (

MPa

)

Intact

C

B

A

-15 -5 5 15 25 35 45

. 6. Generalized Hoek–Brown failure envelopes for Ankara andesites in s1�s3

ce.

�

MC simulated results of strength and deformability parameterswere found to show distributions skewing to larger values. � interdependence between rock mass parameters has to be

incorporated into any stochastic model to obtain meaningfulcombinations.

� values of the rock mass strength and deformability that are

found will most likely more closely characterize the complexgeological environments encountered in the real world.

Overall, the rock mass properties affected by the random natureof discontinuity characteristics and intact rock properties, whichare widely scattered and variable, cannot be sufficiently repre-sented by a single value for each input characteristic and a singleoutput value. Therefore, it is highly recommended that theprobabilistic analysis should be applied, particularly in caseswhere there is significant scatter in the data of discontinuity andintact rock parameters.

Note to readers: the interested readers who want to obtain theoriginal data associated with this paper should contact theauthors directly: karpuz@metu.edu.tr (C. Karpuz) and cayday@a-nadolu.edu.tr (C. Ayday). However, some valuable raw data canalso be obtained from open literature, such as Doyuran et al.(1993) and Karpuz and Pasamehmetoglu (1997).

Acknowledgements

The authors are grateful to Dr. Ali Shafiei from University ofWaterloo and an anonymous referee for critically reviewing themanuscript, and improving the quality of the paper.

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