nd

Rock mass strength

scr

ies

f roc

e ap

f t

gen

y fu

y v

results suggest that it is promising to represent the random characters of rock mass mechanical

avoidaaling wre inhinform

of fome raby virt

probabilistic considerations for assessing the characteristics of

tion of joints in rock mass statistically, Priest and Hudson [5],

) andness-erentsamees of

It is widely recognized that the variability of some rock mass

ARTICLE IN PRESS

Contents lists availabl

.els

al&

International Journal of Rock Mechanics & Mining Sciences 46 (2009) 613626distributions of these parameters; this is primarily due to theEinstein and Baecher [7], and La Pointe and Hudson [8] revealedthat the distributive character of joint spacing follows theexponential distribution. Yegulalp and Mahtab [9], Yegulalp and

mechanical properties are difcult to measure experimentally.Rock engineers frequently utilize empirical methods to estimaterock properties with only limited site data [1719]. Theseempirical methods provide only mean values, not probability

difculty of handling the inherent uncertainties of the componentvariables. To overcome this constraint, discontinuity parameters,

Tel.: +903822150953; fax: +903822150592.

E-mail address: mehmetsari@aksaray.edu.tr1365-16

doi:10.1rock mass and the applications in rock engineering have gainedmore attention [47]. Following the analysis of spatial distribu-

rock mass, but assuming that all three input parameters can berepresented by normal distributions.must be described.The statistical approach was rst used to explain the inherent

variability of strength of brittle materials, in accordance with theweakest-link theory propounded by Weibull [3]. Subsequently,

distribution of the smallest values (extreme value statisticsMonte Carlo simulation; they used the chi-square (w2) goodof-t test to prove that the distribution reects inhvariability of the basalt properties. Hoek [16] applies themethod to estimate variation in the HoekBrown propertinature. For example, rock slope stability is highly dependenton discontinuity characteristics, and the random properties ofeach parameter have an important effect on the probabilisticanalysis [1,2]. When the variables in a process demonstrateuncertainty and variability, the nature of random characteristics

there are a limited number of studies that solely consider thestochastic evaluation of strength and deformability characteristicsof rock masses [11,15,16]. Two papers by Kim and Gao [11,15]present a probabilistic method of estimating the mechanicalcharacteristics of rock mass, using the third-type asymptoticMonte Carlo simulation

Stochastic assessment

1. Introduction

Uncertainty and variability are unand engineering geology studies deThis exists because rocks and soils athere is an insufcient amount ofconditions and the understandingincomplete. There will always be svarious characteristics of rock mass09/$ - see front matter & 2008 Elsevier Ltd. A

016/j.ijrmms.2008.07.007ble in rock engineeringith natural materials.

erently heterogeneous;ation available for siteailure mechanisms isndom variation in theue of its heterogeneous

Kim [10], and Kim and Gao [11] analyzed the mechanical propertydata obtained from the laboratory core tests using extreme valuestatistics and concluded that the mechanical properties follow thethird-type asymptotic distribution of the smallest values.

There have been some studies on the reliability analysis of rockslope stability and stochastic properties of discontinuity para-meters [1,2,1214]. These studies generally evaluate the factor ofsafety and probability of failure of rock slopes using randomvariables in probabilistic modeling. However, in the literatureChi-square testproperties when used with proper empirical or analytical relationships.

& 2008 Elsevier Ltd. All rights reserved.The stochastic assessment of strength apyroclastic rock mass

Mehmet Sari

Department of Mining Engineering, Aksaray University, 68100 Aksaray, Turkey

a r t i c l e i n f o

Article history:

Received 29 February 2008

Received in revised form

14 July 2008

Accepted 29 July 2008Available online 28 August 2008

Keywords:

Kizilkaya ignimbrite

UCS

a b s t r a c t

A practical procedure is de

and deformability propert

discontinuity properties o

the intact rock samples. Th

the chi-square goodness o

input parameters of the

dened probability densit

strength and deformabilit

journal homepage: www

InternationRock Mechanicsll rights reserved.deformability characteristics for a

ibed for implementing probabilistic determination of rock mass strength

of a pyroclastic rock medium. In order to carry out this procedure, the

k mass are collected in the eld and laboratory studies are conducted on

propriate statistical distributions are tted on the available data by using

technique. For the estimation of strength and deformability of rock mass,

eralized HoekBrown empirical equations are replaced by previously

nctions. After running Monte Carlo simulations, potential ranges of the

alues of the investigated rock mass are accomplished. The simulation

e at ScienceDirect

evier.com/locate/ijrmms

Journal ofMining Sciences

including orientation, length, aperture, and spacing are mea-sured in the eld and their random properties determined onthe basis of physical considerations and goodness-of-t testing.Then, introducing perceived uncertainty to the variables of thecalculation, the Monte Carlo method, which can be utilizedeffectively in probabilistic analysis, is carried out to assess therock mass strength and deformability distributions. This gives amuch more complete vision of rock mass, as it reects not onlythe expected value but also possible deviations arising due to alack of knowledge of the exact value of each variable.

The aim of this study is to demonstrate the use of stochasticanalysis theory, which incorporates the uncertainties of the intactrock strength and discontinuity parameters, in evaluation ofstrength and deformability of an ignimbritic rock mass. Walker[20] dened an ignimbrite as a pyroclastic rock consistingpredominantly of pumiceous material, which shows evidence ofhaving been formed from a hot and concentrated pyroclastic ow.The extent to which the ow is uidized inuences the amount ofsorting that occurs. Accordingly, ignimbrites possess a widevariety of geomechanical characteristics attributable to the modeof eruption, transportation, and deposition.

and travertine masses that are deposited with hot springsemerging from these ssures around the valley.

The rocks in the study area are basically classied aspyroclastic, called Selime tuffs, Kizilkaya ignimbrites, and Hasan-dag ashes [22]. The Pliocene Kizilkaya (red stone in Turkish)ignimbrite occurs in a complex tectonic environment within thetalc-alkaline Central Anatolian Volcanic Province [23]. Thegeomorphologic structure of the area was rst formed throughvolcanic activity, which can be divided into three main periods.The rst is represented by mostly andesitic activity, followed bythe second period that is characterized by the emplacement of athick ignimbritic sequence, and the third period during whichgreat andesiticbasaltic strato-volcanos developed [24].

The Kizilkaya ignimbrite well outcrops at Kizilkaya village andin the Ihlara Valley (Fig. 1). Le Pennec et al. [25] showed that theKizilkaya ignimbrite covers an area exceeding 10,600 km2, whichis the most extensive unit in Cappadocia, and has a volume of180km3. Its thickness changes drastically from 4 to 60m with afairly constant thickness on the plateau (1520m), and there is anash-fall layer with a maximum thickness of 20 cm at the base.According to Innocenti et al. [26], the age of the ignimbrite is4.45.5Ma. The Kizilkaya ignimbrite generally consists of two orthree distinct ow units, usually welded with a well-developed

which sometimes exceeds 30 cm. The pumice clasts are porphyri-

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M. Sari / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 6136266142. The geology of the study area

The Ihlara Special Environmental Protection Area is located inthe Cappadocia region in Central Anatolia, Turkey. The area isbordered by two large Quaternary central stratovolcanoes: MountErciyes (3917m) on the eastern margin and Mount Hasandag(3257m) on the southwestern margin of Cappadocia, nearAksaray. The average topography of the area is about 1300m.The region, possessing geological, archaeological, ethnographic,biological, agricultural, and tourism features and activities, has avast amount of natural and cultural values. The study area coversthe historical and touristic Ihlara Valley, which is about 14 km inlength and covers 52 km2 [21]. With the inuence of both waterand wind erosion, interesting rock shapes and morphologicalgures have been formed in the slopes of the valley. There are alsojoint systems that have developed on different lithological unitsFig. 1. A view of the Ihlara Valletic and contain between 30% and 35% amount of crystalsconsisting of plagioclase, biotite, quartz, and minor amounts ofamphibole and oxides. The whole-rock composition is rhyolitic(7173% SiO2) [28].

3. Laboratory and eld studies

A rock mass consists of two components: intact rock anddiscontinuities, each of which has a different effect on the rockcolumnar jointing, forming cliffs and precipitous canyon walls.The Kizilkaya ignimbrite contains pumice fragments, crystals,

glass shards, and lithic fragments [27]. The fragments aremainly of lapilli size and are embedded in ash-size material. Thedeposits display an inverse grading of pumice clasts, the size ofy and Kizilkaya ignimbrites.

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7 %8 %9 %10 %

joint sets in the Kizilkaya ignimbrites.

chanN

Fig. 2. Stereographic projection of

M. Sari / International Journal of Rock Mestrength and deformability. The properties of rock discontinuitiesgovern the overall behavior of the rock masses; these disconti-nuities include joints, fractures, faults, and other geologicalstructures. Rock joints are by far the most common discontinuityencountered in rock masses [29].

The engineering geological properties of the exposed Kizilkayaignimbrite in the study area were determined on the basis of eldobservations/measurements and laboratory tests. The mainorientation, spacing, persistence, aperture, lling, weathering,and roughness of the discontinuities were described using thescan-line survey method following the ISRM [30] descriptioncriteria. A total of 260 discontinuities were measured along astraight outcrop surface using a measuring tape and compass.A total of 18 rock blocks were collected from the eld, then 112cube specimens were prepared from the blocks for laboratorytesting. The uniaxial compressive strength (UCS) tests wereconducted according to Turkish Standards of methods of testingfor natural building stones [31]. This standard requires thepreparation of cube samples with dimensions of 70mm. Foruniaxial testing, a 1000kN servo-hydraulic testing machine wasused and all tests were carried out in force control with a loadingrate of 0.25 kN/s.

Measurements on joint sets are usually carried out byobservation and orientation measurements at outcrops. In thestudy area, a total of 260 joint measurements were taken from theKizilkaya ignimbrites. Discontinuity orientations were processedutilizing commercially available software based on an equal-anglelower hemisphere stereographic projection, and major joint setswere distinguished for the Kizilkaya ignimbrites (Fig. 2). Due to awell-developed columnar jointing systematically controlled byslower cooling of the ignimbrites, the joints in the rock mostly dipvertically. The major orientations of the joint sets for ignimbritesare as follows: joint set 1: 035/86, joint set 2: 085/86, joint set 3:143/85, joint set 4: 175/88.1 %2 %3 %4 %5 %6 %ics & Mining Sciences 46 (2009) 613626 615Table 1 summarizes the main characteristics of spacing,persistence, aperture, roughness, inlling, and weathering of thediscontinuities. A more detailed description of each will be givenin the following section. According to the ISRM [30], the joint setsin the Kizilkaya ignimbrites have very wide spacing, very highpersistence, and are smoothrough undulating. While these jointsets are very wide open and are slightly weathered near thesurface, they are tight and fresh at deeper levels.

4. Stochastic modeling of rock mass parameters

In this section, random properties of geological and geotechni-cal parameters are determined. The information obtained fromthe sampled data is used to make generalizations about thepopulations from which the samples were collected [32]. In thisstudy, the orientation, length, spacing, persistence, aperture, andstrength parameters of discontinuities are considered to be

Table 1Main properties of discontinuities observed in the Kizilkaya ignimbrites

Spacing of discontinuities (m) 0.512m (Average 3.41)

Persistence of discontinuities (m) Generally 420m, depending on thethickness of rock column

Aperture of discontinuities (cm) Generally 42 cm, however occasionallyo0.5 cm

Roughness of discontinuities Generally smooth, however occasionally

rough and undulating

Inlling Generally free from inll material,

however occasionally have material of

sand and gravel size at lower levels

Weathering degree Generally slightly weathered, occasionally

moderately weathered

Groundwater conditions Generally dry, occasionally damp

random variables and their random properties are found. Due tothe heterogeneous nature of the rock mass, it is important toquantify the variability of these parameters. A stochastic modelingapproach is introduced in this paper to quantify this variation.This technique takes the input variables as statistical distributionsrather than constants and through several thousand iterations,generates a statistical representation of the rock mass mechanicalproperties.

In general, probabilistic analysis is performed by two proce-dures [33]: the rst step consists of an analysis of the availablegeotechnical data to determine the basic statistical parameters(mean and variance) in order to represent and predict the randomcharacter of these geotechnical parameters. A probability densityfunction (PDF) describes the relative likelihood that a randomvariable will assume a particular value. In cases where it isbelieved that a given set of measured data represents a set ofrepresentative sample values of the variable, and no otherinfo

represents the best estimate of the random variable, and the

ii. Probability distribution functions are dened for eachparameter, which represent the range of values that wouldbe expected in the eld and laboratory.

iii. Monte Carlo simulations are executed to determine a statis-tical representation of the Rock Mass Rating (RMR) and rockquality designation (RQD).

iv. Geological Strength Index (GSI), mi, mb, s, and a are de-ned as probability distributions and are included instrength and deformability equations dened in a spreadsheetmodel.

v. The mean, standard deviation, and condence intervals(range) of the rock mass strength and deformability para-meters are then accomplished by running Monte Carlosimulation.

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M. Sari / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 613626616standard deviation or coefcient of variance (COV) of the PDFrepresents an assessment of the uncertainty.

In the second step, the stochastic assessment of rock massproperties is accomplished using the basic statistical parametersand the probability density distributions resulting from theprevious step. That is, once the probabilistic properties of inputparameters are assumed, the characteristics of the rock mass can beevaluated by many different risk analysis procedures. The MonteCarlo simulation method is commonly used to evaluate themechanical properties of rock mass when direct integration ofthe system function is not practical, but the PDF of each componentvariable is completely described. In this procedure, the values ofeach component are generated randomly by its respective PDFs;then these values are used to evaluate the rock mass properties.This estimation is reasonably accurate only if the number ofsimulations is very large. The advantage of this method is that thecomplete probability distribution for the mechanical properties canbe obtained if the PDF of input parameters is assessed precisely andthe correlation between the input parameters is estimated.

Fig. 3 is a simple illustration of the stochastic modelingmethodology in the context of rock mass strength and deform-ability assessment with the following comments about each step.

i. Data are collected for discontinuity parameters and intact rockstrength characterization.

INPUT1

SigcSigtSigcmEm

OUTPUT

UCS

RQD

Spacing

Aperture

Persistencerep

rmation is available, a probability density distribution isresentative of the random variable. The mean value of the PDFFig. 3. Diagram of the stochastic mRockmassStrength

andDeformability

Modela

s4.1. The RMR rock mass classication system

Rock Mass Rating (RMR) system [34] is one of the most widelyused rock mass classications. It was originally based on casehistories drawn from civil engineering applications. This methodincorporates geological, geometric, and design/engineering para-meters in arriving at a quantitative value of the rock mass quality.This engineering classication system, developed by Bieniawski[29], utilizes the following six rock mass parameters: (1) UCS ofintact rock material, (2) RQD, (3) spacing of discontinuities, (4)condition of discontinuities (persistence, roughness, inlling,weathering), (5) groundwater conditions, and (6) orientation ofdiscontinuities.

To apply the RMR classication, the ratings are assigned toeach of the six parameters listed above and they are summed togive a value of RMR. For each site the typical, rather than theworst, conditions are used. Furthermore, it should be noted thatthe ratings, which are given for discontinuity spacing, apply torock masses having three sets of discontinuities. Thus, when onlytwo sets of discontinuities are present, a conservative assessmentis obtained. Bieniawski modied the RMR ratings in 1974, 1976,1979, and 1989.

4.2. Input parameters

In the probabilistic analysis, input parameters can be sub-divided by their randomness into two groups, deterministic andprobabilistic parameters. Deterministic parameters are thoseconsidered to be known and having a single value for all rock

GSI

mi

mb

INPUT2odeling methodology.

mass. In the current study, the groundwater condition wasconsidered to be the only deterministic parameter.

For probabilistic parameters, the PDF and the values ofstatistical parameters for random variables are selected on thebasis of physical properties, test results, and evaluation of themeasured data. In this study, joint parameters, such as length,spacing, aperture, lling, and roughness, were considered to beprobabilistic in nature; however, they are assumed to beindependent. The covariance between random parameters playsan important role in probabilistic analysis although there arelimited conclusions from the research results involving theaccurate evaluation of covariance between random parametersin a rock mass [11,16].

4.2.1. Unconned compressive strength (UCS)

recovery since a better picture of the rock mass can be obtained

RQD 100e 0:1l 1 (1)where l is the average number of discontinuities per meter, i.e.,l 1/(joint spacing).

4.2.3. Discontinuity trace length

Discontinuity lengths determine the size of the rock blocksthat form within a rock mass. Furthermore, they may also affectjoint persistence, which is dened as the areal extent or size of adiscontinuity along a plane and can be crudely quantied byobserving the trace lengths of discontinuities on exposed surfaces[30]. Persistence is one of the most signicant joint parametersaffecting rock mass strength, but it is difcult to quantify. Einsteinet al. [41] suggested that persistence can be quantied onlyroughly by observing the discontinuity trace length on a rockexposure surface. This is because rock exposures are small andonly two dimensional. It is impossible in practice to measure the

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7075 1 1.27 0.057

M. Sari / International Journal of Rock MechanThe unconned (or uniaxial) compressive strength (UCS) of arock is frequently used as one of the key inputs in mostclassication schemes and design applications for the character-ization of rock material strength. It is normally measured oncylinders of rock core by compressing the core between twoplatens and measuring the maximum load at which failure occurs.Several researchers have suggested a normal distribution ortruncated normal distribution for the UCS [16,3538].

A comparison graph is superimposed on the UCS data for the112 cube samples of the Kizilkaya ignimbrites (Fig. 4) and thedistribution curve tted on the same graph, allowing a visualassessment of the quality of the t. As can be seen from the graph,the UCS of the ignimbrite rock varies considerably for the cubesamples from different blocks collected from the study area. Thefull nature of the UCS for this rock cannot be characterized in thecalculations by a single value, e.g. the average of all strengthvalues. Also, the frequency distribution of the sample may bestatistically compared to the hypothetical distribution to ascertainif this assumption is really warranted. This can be conrmed usinga w2 goodness-of-t test on the uniaxial strength testing data.

It is possible to test how well the distribution of sample valuesconforms to a theoretical distribution by the goodness-of-t or w2

procedure. The expected frequency of occurrence within eachinterval of a theoretical distribution can be compared to thefrequency of the actual observations that fall within the sameintervals. If the actual number of observations in each intervaldeviates signicantly from that expected it seems unlikely thatthe sample was drawn from a distributed population [39]. The teststatistics are calculated from the equation w2 P(oiei)2/ei,where oi is the number of actual observations within the ithclass, and ei the number of observations expected in that class.There are k classes or intervals.

0

5

10

15

20

25

30

20 25 30 35 40 45 50 55 60 65 70 75

Freq

uenc

yUCS (MPa)Fig. 4. Histogram of the observed and expected frequency in the UCS.from scan-line or area mapping. For scan-line data, an averagejoint spacing can be obtained (number of features divided bytraverse length). Bieniawski [34], relying on previous work byPriest and Hudson [5], has linked average joint spacing to RQD. Itshould be noted that the maximum possible RQD based on jointspacing given by Bieniawski actually corresponds to the best-trelationship proposed by Priest and Hudson [5]. The RQD can beestimated from average joint spacing based on the followingequation:

0:1lIn the rst column of Table 2, 112 UCS values are arbitrarilysubdivided into 12 classes, each having a width of 5MPa. Thesecond column lists the observed frequencies for each class, whilethe third column lists the expected frequencies for each class as aresult of the best-tting normal curve in Fig. 4. The fourth columnthen provides the w2 term for each class calculated fromw2 P(oiei)2/ei and the overall w2 statistic is calculated as11.76, with 11 degrees of freedom (number of classes1) at thebottom of Table 2.

4.2.2. Rock quality designation (RQD)

The RQD index was developed by Deere et al. [40] to provide aquantitative estimate of rock mass quality from drill core logs.RQD is dened as the percentage of intact core pieces longer than100mm (4 in) in the total length of core. An estimate of the RQD isoften needed in areas where line mapping or area mapping hasbeen conducted. In these areas it is not necessary to use core

7580 0 0.27 0.266

Total 112 111.9 11.76Table 2Chi-square test results for goodness-of-t in the UCS data

Interval Observed frequency oi Expected frequency ei (oiei)2/ei

1520 0 0.12 0.124

2025 2 0.68 2.545

2530 3 2.70 0.034

3035 3 7.69 2.864

3540 18 15.83 0.297

4045 29 23.50 1.287

4550 22 25.18 0.401

5055 15 19.46 1.024

5560 16 10.86 2.433

6570 3 4.37 0.430

ics & Mining Sciences 46 (2009) 613626 617discontinuity area accurately in a eld survey. Therefore, dis-continuity length was considered to be a form of persistence inthis study.

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10

han00 5 10 15 20 25 30 35 40 45 50 55 60

Joint persistence (m)Fig. 5. Histogram of the observed and expected frequency in joint persistence.

Table 3Chi-square test results for goodness-of-t in persistence data20

30

40

50

60

Freq

uenc

yM. Sari / International Journal of Rock Mec618Many studies of eld measurements have showed that thenegative exponential probability density distribution is suitable torepresent the discontinuity trace length distribution [13,4245].However, in this study, normal distribution (Fig. 5) best describesthe persistence of the joints, which is also supported by the w2

goodness-of-t test (Table 3). A truncated (060m) normaldistribution for this variable is proposed because, in some cases,the complete normal distribution provides unreasonably low orhigh values for the persistence of joints, which should naturally bebounded with the thickness of rock column.

4.2.4. Discontinuity spacing

Although mean discontinuity spacing provides a direct mea-sure of spacing data, several previous studies have attempted torepresent the distribution of measured spacing data by statisticalanalysis and description with the spacing data considered asrandom variables [5,6]. On the basis of data collected for theKizilkaya ignimbrites, w2 goodness-of-t tests were performed forlognormal and negative exponential distributions, which are thetwo distribution models commonly used for spacing evaluation.This is because those theoretical distributions are bounded at zeroand are skewed to the right and those characteristics are similar tothe properties of the spacing distribution.

Table 4 shows the results of the w2 tests for joint spacing in theKizilkaya ignimbrites. Priest and Hudson [5] stated that if thediscontinuities are fairly evenly spaced in cases of uniform

Interval Observed frequency oi Expected frequency ei (oiei)2/ei

05 3 2.39 0.158

510 11 7.38 1.777

1015 26 17.67 3.924

1520 39 32.78 1.181

2025 43 47.08 0.354

2530 48 52.39 0.367

3035 42 45.14 0.219

3540 30 30.13 0.001

4045 12 15.58 0.821

4550 4 6.24 0.801

5055 2 1.93 0.002

5560 0 0.46 0.464

Total 260 259.2 10.07columnar jointing basalt and evenly bedded sandstone, a normaldistribution of spacing values would be appropriate. Although it issimilar to the current case, where a well-developed columnarjointing due to cooling has developed in the Kizilkaya ignimbrites,the lognormal probability density distribution (Fig. 6) was chosenas appropriate distribution model to represent the randomproperty of the discontinuity spacing. The literature also proposesthe use of a lognormal probability distribution for discontinuityspacing [2,13,4547].

4.2.5. Joint aperture

Aperture is the perpendicular distance separating adjacentrock walls of an open discontinuity in which the intervening spaceis lled with air or water [30]. Aperture is thereby distinguishedfrom the width of a lled discontinuity. The apertures of realdiscontinuities are likely to vary widely over the extent of the

Table 4Chi-square test results for goodness-of-t in spacing data

Interval Observed frequency oi Expected frequency ei (oiei)2/ei

0.00.5 8 3.40 6.243

0.51.0 9 6.92 0.626

1.01.5 34 37.50 0.326

1.52.0 42 52.05 1.940

2.02.5 52 48.21 0.297

2.53.0 53 37.03 6.884

3.03.5 19 25.90 1.839

3.54.0 13 17.26 1.053

4.04.5 10 11.23 0.135

4.55.0 5 7.23 0.685

5.05.5 5 4.64 0.029

5.56.0 6 2.98 3.065

6.06.5 3 1.92 0.604

6.57.0 1 1.25 0.050

7.07.5 0 0.82 0.817

7.58.0 0 0.54 0.539

8.08.5 0 0.36 0.358

8.59.0 0 0.24 0.240

9.09.5 0 0.16 0.162

9.510.0 0 0.11 0.111

Total 260 259.7 26.00

ics & Mining Sciences 46 (2009) 613626discontinuity. Clearly, the variation of aperture will have aninuence on the shear strength of the discontinuity [19]. Moreimportant is the inuence of aperture on permeability orhydraulic conductivity of the discontinuity of the rock mass. Atoutcrop mapping, the joint aperture can be estimated onlyroughly, through the direct observation of joint exposed at theoutcrop, according to the ISRM [30]-suggested description. Thehistogram of joint aperture is best represented by an exponentialdistribution as shown in Fig. 7. The w2 goodness-of-t test(Table 5) also validates the assumption regarding the randomnature of joint aperture in the Kizilkaya ignimbrites.

4.2.6. Fitness of the distributions selected for input parameters

Table 6 illustrates the results of the w2 goodness-of-t test forthe input parameters. It should be noted that the lower the w2

calculated value with respect to the w2 tabulated value, the closerthe theoretical distribution appears to t the data. The nullhypothesis and the alternative one used in the test can be given asfollows. H0: the data have come from the proposed (normal,lognormal, exponential, etc.) distribution. H1: the data have notcome from the proposed distribution.

The w2 test is based on the w2 distribution. This distribution hasno negative value and is skewed to the right. Given a value of aand the degrees of freedom, a p-value can be calculated from thisdistribution by using the Excel command CHIINV(a, df). Forexample, if a 0.05 and df 11, then the critical value is

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0

10

20

30

40

50

60

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5Joint s

Freq

uenc

y

Fig. 6. Histogram of the observed and e

0

10

20

30

40

50

60

70

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Joint aperture (cm)

Freq

uenc

y

Fig. 7. Histogram of the observed and expected frequency in joint aperture.

Table 5Chi-square test results for goodness-of-t in aperture data

Interval Observed frequency oi Expected frequency ei (oiei)2/ei

0.01.0 70 65.15 0.362

1.02.0 50 43.09 1.108

2.03.0 38 31.85 1.185

3.04.0 21 23.55 0.276

4.05.0 24 17.41 2.496

5.06.0 23 12.87 7.975

6.07.0 13 9.51 1.278

7.08.0 10 7.03 1.252

8.09.0 2 5.20 1.969

9.010.0 4 3.84 0.006

10.011.0 1 2.84 1.193

11.012.0 0 2.10 2.100

12.013.0 0 1.55 1.553

13.014.0 0 1.15 1.148

14.015.0 1 0.85 0.027

Total 257 228.0 22.93

M. Sari / International Journal of Rock Mechan5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10pacing (m)xpected frequency in joint spacing.

Table 6Results of chi-square goodness-of-t tests for input variables

ics & Mining Sciences 46 (2009) 613626 619CHIINV(0.05,11) 19.7. If the calculated w2-value is less than thecorresponding tabulated w2-value for the test level of a andn N1 degrees of freedom then the null hypothesis cannot berejected and the rock mass parameters follow the proposeddistribution. All of the variables are adequately represented byanticipated theoretical PDF as veried by reasonably higherp-values (40.05), which indicate the probability that we willhave such a high value of w2 when H0 is true.

4.2.7. Joint roughness

A classication of discontinuity roughness has been suggestedby ISRM [30]. It describes the roughness, rst in meter scale (step,undulating, and planar) and then in centimeter scale (rough,smooth, and slickensided). The classication is useful to describethe joint surface but does not give any quantitative measure. Theroughness of discontinuities was considered as a probabilisticparameter in this research. Roughness is a potentially importantcomponent of strength and, therefore, it was observed in the eldfor each discontinuity, a qualitative description of joint roughnesswas accomplished, and a percentage score was attained fordifferent levels of surface roughness (5%, very rough; 20%, rough;40%, less rough; 20%, smooth; 5%, slickensided) based on thefrequencies observed in the eld.

4.2.8. Inlling

Filling is the term for material separating adjacent rock walls ofdiscontinuities. The joint can be clean or lled with weatheredproducts and deposits, ranging from sandy particles to swellingclays. Usually, most of the discontinuities in the study area arefree from any ll material, but sometimes sand and gravel size

Variable Distribution type d.f. w2-calculated w2-tabulated p-Value

UCS (MPa) Normal 11 11.76 19.7 0.382

Spacing (m) Lognormal 19 26.00 30.1 0.130

Persistence (m) Normal 11 10.07 19.7 0.524

Aperture (cm) Exponential 14 22.93 23.7 0.057

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hanparticles were observed, especially in the lower parts of the joints.Accordingly, a percentage score is obtained for different levels ofinlling (50%, unlled; 25%, hard llings; 25%, soft llings) basedon the frequencies observed in the eld.

4.2.9. Weathering

Natural discontinuities normally suffer weathering and altera-tion, which, in turn, also change the degree of matching of thediscontinuity surfaces. Weathering and alternation is usuallyvisible at the outcrops and/or from the cores. When the jointsurface is weathered, it often shows a change in color andappearance. Often, weathered products, such as grain particles,may also remain inside the joint. A detailed description of theweathering degree in the study area is carried out according to thedescriptive terms suggested by ISRM [30]. Sari and C- omlekc- iler[48] have found that the joint surfaces in the study area aremostly slightly weathered but sometimes fresh surfaces are alsoencountered due to the exposure of newly formed surfaces causedby the falling of old rock columns from the face of the cliffs. Sinceweathering can be extremely variable, different percentagesare also obtained for different levels of weathering (25%, fresh;50%, slightly; 25%, moderately) consistent with the observedfrequencies in the eld.

4.2.10. Material constant miThe empirical criterion formulated by Hoek and Brown [49]

allows using an approximate value of material constant mi for aparticular rock. The constant mi is changed with the type of rock,its mineral composition, interlocking of grains, grain size, etc. andHoek and Brown [49] recommend a value of 1375 for the materialconstant mi of tuffs. Accordingly, a truncated normal distributionhas been assumed with a mean value of 13.0 and standarddeviation of 2 bounded between a lower of 8.0 and upper of 18.0for this parameter. The physical meaning of the constantmi can beexpressed roughly as the ratio between the UCS and the tensilestrength of intact rock material [18]. Binal [50] carried out severalexperiments on the Kizilkaya ignimbrites and found a mean valueof 48.6MPa for UCS and 3.7MPa for Brazilian tensile strength. Thisgives an overall ratio of 48.6/3.7 13.15, which, in fact, is veryclose to mi that is assumed in this research.

4.3. The generalized HoekBrown criterion

The last version of generalized HoekBrown failure criterion[51] is applied to determine the rock mass properties of theKizilkaya ignimbrite. In order to use the HoekBrown criterion toestimate the strength and deformability of jointed rock masses,the following three properties of the rock mass have to beestimated: the UCS sci of the intact rock pieces in the rock mass,the value of the HoekBrown constant mi for these intact rockpieces, and the value of the GSI for the rock mass.

The generalized HoekBrown failure criterion for jointed rockmasses is dened by

s1 s3 sci mbs3sci

s a

(2)

where s1 and s3 are, respectively, the maximum and minimumeffective stresses at failure, mb is the value of the HoekBrownconstant m for the rock mass, s and a are constants which dependon the characteristics of the rock mass, and sci is the UCS of theintact rock samples.

Hoek et al. [51] suggested the following equations forcalculating the rock mass constants (i.e., mb, s, and a):

M. Sari / International Journal of Rock Mec620mb mi expGSI 10028 14D

(3)s exp GSI 1009 3D

(4)

a 12 16eGSI=15 e20=3 (5)

The uniaxial compressive and tensile strengths of the jointed rockmasses are calculated from the following equations suggested byHoek et al. [51]:

scMPa scisa (6)

stMPa sscimb

(7)

The UCS of the rock mass sc is given by Eq. (6). Whenconsidering the strength of a pillar, it is useful to have an estimateof its overall strength rather than a detailed knowledge of theextent of the fracture propagation in the pillar. This leads to theconcept of a global rock mass strength and Hoek et al. [51]proposed that this could be estimated from the MohrCoulombrelationship as

scm scimb 4s amb 8smb=4 sa1

21 a2 a (8)

The static modulus of deformation is among the parametersthat best represents the mechanical behavior of a rock and a rockmass, in particular for underground excavations. This is why mostnumerical nite element and boundary element analyses forstudies of the stress and displacement distribution aroundunderground excavations are based on this parameter. Seramand Pereira [52] proposed a relationship between the in situmodulus of deformation and Bieniawskis RMR classication. Thisrelationship is based on the back analysis of dam foundationdeformations and it has been found to work well for good-qualityrocks. However, for many of the poor-quality rocks it appears topredict deformation modulus values that are too high. Based onpractical observations and the back analysis of excavationbehavior in poor-quality rock masses, the following modicationto Seram and Pereiras equation was proposed by Hoek et al. [51]for scio100:

EmGPa 1 D2

sci100

r10 GSI10=40 (9)

where D is a factor that depends on the degree of disturbance towhich the rock mass has been subjected by blast damage andstress relaxation. Since the rock mass in this study is naturallyoutcropping, it is plausible to assume that it has been undisturbedand the value D 0 is used in the above equations. For a morecomprehensive denition of these parameters, and the equationsthat dene them, the reader is directed to the latest version of theHoekBrown failure criterion [51].

In the HoekBrown criterion, the GSI is the most importantinput parameter in terms of the relationship between the strengthand deformation properties determined in the laboratory andthose assigned to the eld scale rock mass [16]. In earlier versionsof this criterion, Bieniawskis RMR was used for this scalingprocess. Due to the usage of different rating scales for eachparameters, the RMR is more suitable for numerical computa-tions. It is possible to objectively estimate a frequency distributionof RMR for computing purposes in a probabilistic analysis. On theother hand, GSI can be determined only by eld observations ofblocks and discontinuity surface conditions. It does not permit us

ics & Mining Sciences 46 (2009) 613626to compute the variability of rock mass properties and therepresentative distribution of rock mass characteristics can bemade only subjectively [16].

5. Results and analysis

Following the denition of random properties of discontinuityparameters and intact rock strength, a stochastic analysis isgiven in this section. The term stochastic is used when thereis a random component to a model, e.g. values for the inputvariables are sampled from a statistical population. Themost common sampling technique is termed the Monte Carlo

method. The name was introduced during World War II as acode name for simulation of problems associated with thedevelopment of the atomic bomb [33]. The technique uses

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A B C D E F G H I J1 Input1 Rating Roughness Infilling Weathering2 UCS (MPa) 46.06 4 x p x p x p3 RQD (%) 95.0 20 6 0.05 6 0.50 6 0.254 Spacing (m) 2.575 20 5 0.20 2 0.25 5 0.505 Persistence (m) 27.09 0 3 0.40 0 0.25 3 0.256 Aperture (cm) 3.31 0 1 0.207 Roughness 3 0 0.158 Infilling 2 Input1 Formulae9 Weathering 5 B2 = Normal(46.06; 8.63;Truncate(0; 100))

10 Groundwater 15 B3 = 100*Exp(-0.1*1/B4)*(0.1*1/B4+1)11 RMR 69 B4 = Lognormal(2.575; 1.295)12 B5 = Normal(27.09; 9.79;Truncate(0; 60))13 Input2 B6 = Exponential(3.31)14 GSI 64 C7 = Discrete(E3:E7;F3:F7)15 Mi 13.0 C8 = Discrete(G3:G5;H3:H5)16 Mb 3.686 C9 = Discrete(I3:I5;J3:J5)17 s 0.022 C10 = 1518 a 0.502 C11 = Sum(C2:C10)19 Input2 Formulae20 Output B14 = C11-521 SigC 6.591 MPa B15 = Normal(13.0; 2.0;Truncate(8; 18))

6 = B15*Exp((B14-100)/28)7 = Exp((B14-100)/9)8 = 1/2+1/6*(Exp(-B14/15)-Exp(-20/3))

Output Formulae123

Table 7Input and output summary statistics

Cell Minimum Maximum Mean Std. Dev. 5% 95%

Input1 name

UCS (MPa) $B$2 10.87 87.28 46.06 8.633 31.86 60.25

Rqd (%) $B$3 72.27 99.36 95.01 2.58 90.12 97.99

Spacing (m) $B$4 0.39 15.72 2.58 1.295 1.05 5.02

Persistence (m) $B$5 0.01 59.74 27.16 9.648 11.23 43.17

Aperture (cm) $B$6 0.00 38.64 3.31 3.315 0.17 9.91

Roughness $C$7 0 6 2.7 1.819 0 5

Inlling $C$8 0 6 3.5 2.598 0 6

Weathering $C$9 3 6 4.75 1.090 3 6

RMR $C$11 51.0 84.0 69.34 4.62 62.0 77.0

Input2 name

GSI $B$14 46.0 79.0 64.34 4.62 57.0 72.0

Mi $B$15 8.01 18.00 13.00 1.91 9.81 16.19

Mb $B$16 1.619 7.777 3.688 0.816 2.463 5.146

s $B$17 0.002 0.097 0.022 0.011 0.008 0.045

a $B$18 0.501 0.508 0.502 0.001 0.501 0.504

Output name

SigC (MPa) $B$21 1.163 20.558 6.591 2.387 3.409 11.164

SigT (MPa) $B$22 1.048 0.040 0.266 0.126 0.510 0.110SigCm (MPa) $B$23 2.585 28.746 12.643 3.167 7.938 18.353

Em (GPa) $B$24 4.792 41.905 16.074 4.898 9.193 25.308

G(F(x)) => x

0.0

0.2

0.4

0.6

0.8

1.0

10 20 30 40 50 60G(F(x)) = x

F(x)

x => F(x)

Fig. 8. The relationship between x, F(x), and G(F(x)).

M. Sari / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 613626 62122 SigT -0.266 MPa B123 SigCm 12.643 MPa B124 Em 16.074 GPa B12526 B227 B228 B229

30 B24

Fig. 9. Spreadsheet model used fo= B2*B17^B18= -B17*B2/B16= B2*(B16+4*B17-B18*(B16-8*B17))*(B16/4+B17)^(B18-1)/(2*(1+B18)*(2+B18))

= (B2/100)^1/2*10^((B14-10)/40)

r @RISK simulation analysis.

random or pseudo-random numbers to sample from a probabilitydistribution. Any given sample may fall anywhere withinthe range of the input distribution but of course, samples aremore likely to be selected from regions of the distribution thathave higher probabilities of occurrence. With enough iterations,Monte Carlo sampling recreates input distributions and, withthe aid of a computer, it is very efcient to conduct hundreds orthousands of what-if scenarios, modeling most combinations

of input parameters and quantifying statistical distributions ofoutcomes [53].

5.1. How Monte Carlo simulation works

This section looks at the technical aspects of how MonteCarlo risk analysis software generates random samples for input

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10 15 20 25 30 35 40 45 50 55 60 65 70 75 80UCS (MPa)

Freq

uenc

y

0

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800

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80 82 84 86 88 90 92 94 96 98 100RQD (%)

Freq

uenc

y

0

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2100

0 1 2 3 4 5 6 7 8 9 10Spacing (m)

Freq

uenc

y

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0 5 10 15 20 25 30 35 40 45 50 55 60Persistence (m)

Freq

uenc

y

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0 2 4 6 8 10 12 14 16 18 20Aperture (cm)

Freq

uenc

y

0

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0 1 2 3 4 5 6Roughness

Freq

uenc

y

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4000

5000

M. Sari / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 6136266220

1000

2000

3000

0 1 2 3 4 5 6

Freq

uenc

yInfilling

Fig. 10. Histograms for Input1 variable00 1 2 3 4 5 62000

3000

Freq

uenc

yWeathering

s generated from simulation runs.

distributions of a model. The explanations are taken fromVose [53].

Consider the distribution of an uncertain variable x. Thecumulative distribution function F(x) gives the probability P thatthe variable X will be less than or equal to x, i.e.

Fx PXpx (10)The function F(x) obviously ranges from 0 to 1. Now, we canlook at this equation in the reverse direction: what is the valueof F(x) for a given value of x? This inverse function G(F(x)) iswritten as

GFx x (11)It is this concept of inverse function G(F(x)) that is used in thegeneration of random samples from each distribution in a riskanalysis model. Fig. 8 provides a graphical representation of therelationship between F(x) and G(F(x)).

To generate a random sample for a probability distribution, arandom number r is generated between 0 and 1. This value is thenfed into the equation to determine the value to be generated forthe distribution:

Gr x (12)

The random number r is generated from a Uniform(0,1) distribu-tion to provide equal opportunity of an x-value being generated inany percentile range. The inverse function concept is employed ina number of sampling methods. For a more comprehensivedescription of statistical distributions and how Monte Carlosimulation works, the reader is directed to [5460].

@RISK is a system that processes these calculations in standardspreadsheet packages; it explicitly includes the uncertaintypresent in inputs to generate outputs that show all possiblealternatives [61]. The program provides a simple and intuitiveimplementation of a Monte Carlo simulation together with thegeneralized HoekBrown failure criterion, allowing users to easilyobtain reliable estimates of rock mass properties, and to obtain avisualization of the changing effects of input variables on rockmass parameters.

A spreadsheet model for carrying out @RISK simulations, witha listing of all cell formulae for input and output parameters, isgiven in Fig. 9. For the model, 10,000 iterations are performedwith Latin Hypercube sampling to closely resemble the resultingprobability distribution. This means that every run of thesimulation yields 10,000 different possible combinations of inputvariables, which are sampled randomly from the deneddistributions. One problem in Monte Carlo simulation is the

ARTICLE IN PRESS

0

300

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54 58 62 66 70 74 78 82 86RMR

Freq

uenc

y

0

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48 52 56 60 64 68 72 76 80GSI

Freq

uenc

y

1200 1250

M. Sari / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 613626 6230

200

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8 9 10 11 12 13 14 15 16 17 18Mi

Freq

uenc

y

0

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1500

2000

2500

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Freq

uenc

ys

Fig. 11. Histograms for Input2 variable0

250

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1000

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7Mb

Freq

uenc

y

0

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1800

2400

3000

0.5 0.502 0.504 0.506 0.508

Freq

uenc

ya

s generated from simulation runs.

difculty of incorporating the covariance between input variables.Distributions in a model will often have to be correlated to ensurethat only meaningful scenarios are generated for each iteration ofthe model. For instance, a weathered discontinuity could havewider openings and completely lled with weathering products.A strong rock would be expected to have a shorter trace lengthand narrower opening. A closer spacing would result in a lowerRQD and shorter persistence. Although @RISK allows us toincorporate the rank order correlations observed between inputparameters as a correlation matrix in the spreadsheet model, it ishard to estimate the exact nature of relationships betweenrock material strength and discontinuity parameters and onlysubjective estimates can be made.

To solve this difculty, Ades and Lu [62] has proposed a one-stage alternative, in which the correlation structure is estimatedfrom the data directly by Bayesian Markov Chain Monte Carlomethods. Samples from the posterior distribution of the outputsthen correctly reect the correlation between parameters, giventhe data and the model. Besides its computational simplicity,this approach utilizes the available evidence from a wide varietyof structures, including incomplete data, and correlated anduncorrelated repeat observations. The major advantage of aBayesian approach is that, rather than assuming the correlationstructure is xed and known, it captures the joint uncertaintyinduced by the data in all parameters, including variances andcovariances, and correctly propagates this through the decision or

Since the 1989 version of Bieniawskis RMR classication isused in this research, in the strength equations, the GSI is settledRMR895 as suggested by Hoek and Brown [49], where RMR89has the groundwater rating set to 15 and the adjustment forjoint orientation set to zero. These ratings can then be assembledto acquire a GSI distribution with a mean value of 64 and standarddeviation of 4.6, which closely resembles a normal distribution.Typically, variables that occur as a sum of a number ofrandom effects, none of which dominate the total, are normallydistributed [53].

The model calculates the mean values for UCS, tensile strength,global strength, and deformation modulus of the Kizilkayaignimbrites as 6.591, 0.266, 12.643, and 16.074GPa, respectively.The summary statistics computed from the @RISK simulations forboth input and output variables are presented in Table 7. The RMRranges from 51 to 84 with a mean value of 69.34. While the UCS ofthe rock mass ranges from 1.163 to 20.558MPa, the unconnedtensile strength of rock mass ranges from 1.148 to 0.048MPa.The global rock mass strength changes from a minimum of2.585MPa to a maximum of 28.746MPa. The deformationmodulus of the rock mass ranges from 4.792 to 41.905GPa.

One of the most common graphical representations of aprobability distribution is a histogram in which the fraction ofall observations falling within a specied interval is plotted as abar above that interval. The histograms of input and outputvariables evaluated by stochastic modeling are given in Figs. 1012.

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M. Sari / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 613626624risk model.In the current view of the model spreadsheet, only average

values of the Input1 parameters are shown in the related cellsand they result in a total of 69 RMR ratings. In fact, theseratings are replaced every time with a new suitable valuecomputed randomly from any of the specied parent distribution.The cells, C1C10, containing the Input1 parameters are recom-puted automatically according to the appropriate range ofthe RMR ratings given in the Bieniawskis 1989 classicationscheme [34].

0

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0 2 4 6 8 10 12 14 16 18 20SigC (MPa)

Freq

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y

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4 6 8 10 12 14 16 18 20 22 24 26

Freq

uenc

ySigCm (MPa)Fig. 12. Histograms for Output variableIt is important to note that the graphs of input and outputvariables rarely have a single value; in fact, all exhibit consider-able variations between some specied intervals. Informationobtained from the simulation is still relevant, and gives furtherinsight into the strength parameters estimated from the HoekBrown criterion. In general, the strength and deformabilitycharacteristics of the Kizilkaya ignimbrite can be seen toapproximately resemble asymptotic distributions skewing tolarger values similar to intact rock materials [911]. Kim andGao [11] obtained that the laboratory and eld test data showed

0

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yFr

eque

ncy

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4 8 12 16 20 24 28 32 36 40 44

Em (GPa)

s generated from simulation runs.

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60 pri

chanthe same type of distributive character although the test methodsand the volumes of the test specimens were substantiallydifferent.

0

20

40

-5 0 5 10 15 20 25Minor principal stress 3 (MPa)

Majo

r

3

1

3

1

Fig. 13. Generalized HoekBrown s1s3 failure envelope for the Kizilkayaignimbrites.80

100

120

140

160nc

ipal s

tress 1

(MPa

)Rock masssigci=46.06 MPamb=3.688s=0.022a=0.502

Intact rocksigci=46.06 MPami =13.0s=1a=0.5

M. Sari / International Journal of Rock MeFig. 13 illustrates the graph of the generalized HoekBrownfailure envelope in principal stress space for the Kizilkayaignimbrites. It is accomplished by separately substituting pre-viously estimated mean values of the rock mass parameters intoHoekBrown empirical equation for intact rock and rock mass.A more rational approach in reecting the actual behavior of therock mass should be to include all the random variations in thecharacteristics of input parameters in the HoekBrown empiricalequation. This can be made to succeed only by dening the PDFsof the rock mass properties in the related equation instead ofusing only mean values. Therefore, it seems possible to representthe failure envelope of the Kizilkaya ignimbrite with a condenceinterval rather than a line. However, this requires more compre-hensive work and exceeds the scope of the current study.

6. 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 model the variation in the strength and deform-ability characteristics of a pyroclastic rock mass. Variations inmechanical properties of rock mass can be statistically modeledby considering the frequency distribution of the random inputparameters. The probabilistic analysis is more representative ofthe actual behavior of the random variables and provides results,that most likely will more closely represent the actual rock massconditions. The simulated results of output parameters are seen toapproximately resemble asymptotic distributions, skewing tolarger values that are the third-type asymptotic distribution ofthe smallest values (extreme value statistics) found by Kim andGao [11]. Overall, the rock mass properties affected by the

[7] Einstein HH, Baecher GB. Probabilistic and statistical methods in engineeringgeology: specic methods and examplespart I: exploration. Rock Mech Rock

Eng 1983;16:3972.

[8] La Pointe PR, Hudson JA. Characterization and interpretation of rock massjointing patterns. Geol Soc Am Spec Pap 1985;199:137.

[9] Yegulalp TM, Mahtab MA. A proposed model for statistical representation ofmechanical properties of rock. In: Proceedings of the 24th US rock mechanicssymposium, Texas A&M; 1983. p. 619.

[10] Yegulalp TM, Kim K. Statistical assessment of scale effect on rock propertiesusing the theory of extremes. Trans AIME 1992;294:18347.

[11] Kim K, Gao H. Probabilistic approaches to estimating variation in themechanical properties of rock masses. Int J Rock Mech Min Sci GeomechAbstr 1995;34:11120.

[12] Park HJ. Risk analysis of rock slope stability and stochastic properties ofdiscontinuity parameters in western North Carolina. PhD thesis, PurdueUniversity, 1999.

[13] Park HJ, West TR. Development of a probabilistic approach for rock wedgefailure. Eng Geol 2001;59:23351.

[14] Duzgun HSB, Yucemen MS, Karpuz C. A methodology for reliability baseddesign of rock slopes. Rock Mech Rock Eng 2003;36:95120.

[15] Kim K, Gao H. Probabilistic site characterization strategy for naturalvariability assessment of rock mass properties. In: Proceedings of the 10thASCE conference on engineering mechanics, University of Colarado, vol. 1;1995. p. 214.

[16] Hoek ET. Reliability of the HoekBrown estimates of rock mass properties andtheir impact on design. Int J Rock Mech Min Sci 1998;35:638.discontinuity characteristics and intact rock properties, whichare widely scattered and variable, cannot be sufciently repre-sented by single value input parameters and a single output value.Therefore, it is recommended that the probabilistic analysisshould be applied, particularly in cases where there is signicantscatter in the data of discontinuity and intact rock parameters.

The proposed approach provides a viable means of assessingthe variability of rock mass properties that cannot be achieved byexperimental means. Compared to traditional design parametersobtained from deterministic approaches these computed rockmass characteristics will help rock engineers to arrive at morerational design parameters for structures in/on rocks.

The task of determining rock mass properties is not usually anend in itself. It is carried out in order to provide input fornumerical analysis programs, which require material properties inorder to perform stability or stress analysis. The rock massproperties determined stochastically by @RISK can be used asinput for numerical analysis programs.

Acknowledgement

This study is partially based on the data collected from thenal projects of senior students in the Geological EngineeringDepartment of Aksaray University. The author would like to thankthose students and wish them success in their future careers. Theauthor would like to express his gratitude to the anonymousreviewers for their valuable comments.

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The stochastic assessment of strength and deformability characteristics for a pyroclastic rock massIntroductionThe geology of the study areaLaboratory and field studiesStochastic modeling of rock mass parametersThe RMR rock mass classification systemInput parametersUnconfined compressive strength (UCS)Rock quality designation (RQD)Discontinuity trace lengthDiscontinuity spacingJoint apertureFitness of the distributions selected for input parametersJoint roughnessInfillingWeatheringMaterial constant mi

The generalized Hoek-Brown criterion

Results and analysisHow Monte Carlo simulation works

ConclusionsAcknowledgementReferences