STRENGTH OF INTACT ROCK AND ROCK MASSES

Garry Mostyn1 and Kurt Douglas2

ABSTRACT

This paper is in two parts. The first part presents an overview of the strength of intact rock. A shortcommentary on various methods of fitting failure criteria to experimental data follows, it is demonstratedthat the method of fitting the criterion to the test data has a major effect on the estimates obtained of thematerial properties. The results of a recent analysis of a large data base of test results is then presented. Thisdemonstrates that there are inadequacies in the Hoek-Brown empirical failure criterion as currently proposedfor intact rock and, by inference, as extended to rock mass strength. The parameters mi and sc are notmaterial properties if the exponent is fixed at 0.5. Published values of mi can be misleading as mi does notappear to be related to rock type. The Hoek-Brown criterion can be generalised by allowing the exponent tovary. This change results in a better model of the experimental data. Analysis of individual data setsindicates that the exponent, a, is a function of mi which is, in turn, closely related to the ratio of sc/s t. Aregression analyis of the entire data base provides a model to allow the triaxial strength of an intact rock tobe estimated from reliable measurement of its uniaxial tensile and compressive strengths. The methodproposed is the most accurate of those methods that do not require triaxial testing and is adequate forpreliminary analysis. Analysis is presented that shows applying the Hoek-Brown criterion to most rocksresults in systematic errors. Simple relationships for triaxial strength that are adequate for slope design arepresented.

The second part of this paper presents a discussion of the application of the Hoek-Brown criterion toestimating the shear strength of rock masses for slopes. The current methodology is presented. Problemswith the estimation of GSI and scale dependency for slopes are discussed. A critical review of theparameters s, mb and a is presented with a view to improving the application of the criterion to slopes. Anew approach to parameter estimation is introduced. Work is on going to validate the method.

INTRODUCTION

The application of the Hoek-Brown criterion to intact rock and rock masses is common in slopeengineering, therefore it is important that it is validated in both applications. This paper presents a detailedanalysis of the application of the criterion to intact rock and suggests modifications that provide improvedpredictions of triaxial strength based on easily measured material properties. Modifications to theapplication of the criterion to rock mass are introduced and the way ahead discussed.

FAILURE CRITERIA FOR INTACT ROCK

There are two approaches to the selection of a failure criterion for intact rock, theoretical and empirical.The base of the most commonly adopted theoretical approaches are those of Coulomb or Griffiths and theseare well presented in most rock mechanics textbooks, with good examples being Jaeger and Cook (1976) andBrady and Brown (1993). Most practical engineering relies on a linear Mohrs envelope being fitted toexperimental data or to the relevant portion of a theoretical or empirical criterion. Notwithstanding this it isbecoming increasingly common for computer software to be able to deal directly with one or more non linearcriteria. Thus while the limits and pitfalls of linearisation are well understood, it is now important to assessthe accuracy of non linear criteria.

1Garry Mostyn, Pells Sullivan & Meynink, 11/10 East Pde, Eastwood, NSW, 2122, Australia,g.mostyn@psmsyd.com.au2Kurt Douglas, School of Civil & Environmental Engineering, The University of NSW, Sydney, 2052, Australia,k.douglas@unsw.edu.au

Equation 1 presents a generalised criterion where a1, a2, etc are material properties. Figure 1 presents ageneralised failure criterion in the s1-s3 plane and shows the locations of the sc, sBt, sut and spt.

,....),,,,(fn0 21321 aasss= (1)

It is well known that the theoretical criteriado not accurately predict the failure strength ofrock and often rely on parameters that aredifficult to measure. For this reason manycriteria have been developed that seek tocapture the important elements of measuredrock strengths or seek to modify theoreticalapproaches to accommodate experimentalevidence, several of these empirical criteria arelisted in Hudson and Harrison (1997) andSheorey (1997). Most of these share areasonably similar structure and all haveelements that are likely to fail at the extremes.Given the variability typical of rock test resultsit is likely that any one criteria is as suitableoverall as any of the alternatives. The Hoek-Brown empirical failure criterion (Hoek &Brown, 1980) was developed in the early 1980s for intact rock and rock masses, it has been subject tocontinual refinement for rock masses. For intact rock its form has not changed and is given in Equation 2.

5.0

331 1

++=

c

ic

m

ss

sss (2)

In common with most of the empirical failure criteria, the Hoek-Brown criterion is formulated in terms ofs1 and s3 and is independent of s2. The authors do not consider this a major impediment for practicalpurposes.

It is the authors experience that the Hoek-Brown criterion is virtually the only non linear criterion nowused by practicing engineers and is now in almost universal use. Further it forms the basis for extension intorock mass strength. It thus has been adopted as the basis of examination for the rest of this paper. Furtherfor these reasons it is important to establish that the criterion does accurately represent actual intact rockbehaviour.

LABORATORY TEST DATABASE FOR INTACT ROCK

The authors have assembled a large data base of test results on a wide variety of rocks; tests includeuniaxial tensile strength, Brazilian tensile strength, unconfined compressive strength and triaxialcompression and tension. Many of the results were sourced from Sheorey (1997), Hoek and Brown (1980),Shah (1992) and Johnston (1985) and checked against original sources where possible. Further testinformation from other sources was obtained. Full details of the data are contained in Douglas and Mostyn(2000). At present, the data consists of 3817 test results forming 485 sets and is being continuouslyextended.

ANALYSIS OF ANALYSIS OF DATA

When confronted with a set of data, there are a number of questions that have to be addressed: What data should be included in the analysis? What equation should be fitted? What method of fitting should be adopted?

Sigma 3 / Sigma CS

igma 1 / S

igma C

-1

0

1

2

3

-0.25 0.00 0.25 0.50

sc

spure t

suniaxial t

sBrazillian t

Figure 1 : Generalised failure criterion

The authors have devoted a moderate portion of this paper to these questions as it is of little valueundertaking a comprehensive test program or detailed analysis of the results if the methodology is flawed. Infact, parameters determined can vary from very conservative to quite the opposite, both situations haveconsequences in analysis and design.

Turning to the first question, the data included is that described in the previous section. It has beencommon practice for researchers fitting empirical failure criterion to intact rock to exclude results thought toexhibit ductile behaviour; this approach has been adopted by Hoek and Brown (1981), Shah (1992), Johnston(1985) and Sheorey (1997). In general these researchers have adopted the brittle-ductile transition suggestedby Mogi (1966) of s1=3.4s3. The exclusion of ductile data would be appropriate if (i) only brittle behaviourwas of interest, (ii) the boundary was clear and (iii) the failure criterion was disjoint across the transition. Inthe case of a criterion based solely on Griffiths theory, exclusion of ductile tests results would beappropriate. It is not necessary, is counter productive and is arbitrary, for an empirical criterion. Research(Evans et al., 1990) shows that the transition is not well defined for all rocks and certainly occurs over a widerange of stresses. Further this same research shows that the failure envelope is not necessarily disjoint at thebrittle-ductile transition. Thus an appropriate criterion can model strength on both sides of the transition.The authors have included as many test results as possible and only excluded results for which there issignificant doubt as to their accuracy.

There are several forms of the Hoek-Brown criterion that can be adopted for data analysis, these includeequation 2 and:

-=

->

++=

ic

icc

ic

m

mm

ssss

ssss

sss

331

3

5.0

331

for

for 1 (3)

( ) 232

31 ccim sssss +=- (4)

( ) ( )2331 log5.0log ccim sssss +=- (5)

Equation 2 is strictly the Hoek-Brown criterion, but is undefined for s3 less than -sc/mi. Equation 3ensures that the criterion is defined over the full range of s3. Equations 4 and 5 are linearisations of thecriterion. The impact of adopting these different forms is discussed at the end of this section.

The method of least squares is very widely adopted in fitting models to data; there are often very soundstatistical reasons to so do. Shah (1992) suggests that the simplex method with the function (observed-predicted)2 is a better method than least squares. In fact, the method presented by Shah is least squares, thesimplex is purely a numerical method to optimise some function, in this case minimising the sum of squareddifferences (ie errors). The authors have verified that the resulting parameter estimates are the same as thosefrom other robust least squares procedures.

If the departure of the measured s1 from the predicted s1 (ie the error) is normally distributed with avariance that is independent of the predictor variables (here s3), then the predictions obtained with leastsquares, either with a simplex or otherwise, will be uniform minimum variance unbiased estimators; this ishighly desirable. But consideration of data with multiple measurements of sut or sB t will indicate thatstraight least squares is not appropriate for fitting the Hoek-Brown criterion.

Consider an experimental program with multiple measurements of sut, it is clear that if a failure criterionis to be fitted to the test data it is desirable that the estimated tensile strength should be the average of thesemeasurements (ie the fitted curved should pass through the middle of the measured values). Equation 2 isnot defined for measured values of sut less than the fitted value (ie larger tensile strengths) and this forcesmany fitting methods to fit the maximum measured (ie most negative) tensile strength as the estimatedtensile strength. Equation 3 overcomes this problem, but reference to Figure 1 shows that the slope of theequation to the left of the estimated sut is much less than that to the right; the figure is drawn for an mi of 8and the slope to the right is much steeper for higher mi. Given that a general least squares approach assessesthe error as the observed s1 (ie zero) minus the predicted s1, then data a given distance to the right of theestimated sut will have a much larger error than data the same distance to the left. Thus a standard least

squares procedure will result in a very poor fit at low stresses and force a small sut and high mi, ie theopposite effect to adopting equation 2

A resolution of the above problem comes about by recognising that in a uniaxial tensile strength test, thecontrolled variable is s1 and the measured variable is s3, thus the real error is observed s3 minus thepredicted s3. But this error is scaled in s3 and needs to be adjusted if it is to have equal status withmeasurements in s1. The authors suggest that scaling by mi is a convenient and accurate approach. Giventhis they recommend a least squares procedure where the error is defined as:

( )( )

--

->-

3133

3111

3for predicted measured

3for predicted measured

ssss

ssss

im(6)

This has been found to provide very good fits for a wide variety of data.It is the authors experience that the method of parameter estimation can, and often does, have a large

impact on parameters derived from experimental data but the effect is often camoflaged by the variability oftest data. Table 1 and Figure 2 show the results of analysis of a simulated test program with resultsgenerated for a material with a Hoek-Brown failure criterion, sc and mi are both normally distributed withmean/standard deviation of 10/2 MPa and 12/2 respectively. Results generated were 10 uniaxial tensilestrength tests, 20 unconfined compressive strength tests, and 4 each triaxial strength tests at confiningpressures of 1, 2, 5, 10, 20, 40 and 80 MPa. Thus there were 58 data points in all, simulating a verycomprehensive test program from which it should be possible to determine accurate estimates of materialproperties.

Table 1 : Results of different regression methods on artificial data

Case Equation Fitting method Number sc (MPa) mi r2 (%)

1 Actual data 58 10.0 12.0 na2 Normal equation 2 Least squares 58 14.9 7.75 97.883 Extended equation 3 Least squares 58 8.46 15.6 99.12

4 Extended equation 3Modified leastsquares, eqn 6

58 10.7 12.0 99.00

5Adopting known sc andnormal equation

2 Least squares 58 na 5.21 91.69

6Excluding st results andnormal equation

2 Least squares 48 9.53 13.7 99.06

7Excluding sc & st resultsand normal equation

2 Least squares 28 6.20 21.4 98.80

8 Stress difference squared 4 Least squares 58 3.97 35.2 95.53

9Stress difference squaredand known sc

4 Least squares 58 na 13.8 95.47

10 Stress difference squared 4Least sum of

absolutedifferences

58 9.18 15.4 95.45

11 Logarithms 5 Least squares 58 8.09 4.12 55.97

12Logarithms and excludingst results

5 Least squares 48 9.67 12.2 95.00

The entire generated data and selected fits are shown on Figure 2a. It can be seen that, with twoexceptions, the methods provide a reasonable fit for the majority of the data. But reference to Figure 2bshows that most methods provide a very poor fit to the data at low stresses, that is over the stress range ofinterest in slope analysis.

The following comments are offered on the various analyses undertaken, listed in the same order as inTable 1.

1. The generated data, theauthors consider that this is areasonable representation of acomprehensive test program in amoderately variable unit.

2. The strict application ofleast squares to Equation 2, iethe usual form of the Hoek-Brown criterion, results in theuppermost curve in Figure 2b.The criterion cannot beevaluated for s3 less than theestimated tensile strength. Thisresults in large estimates of sutand sc and thus a low mi. From5 to 80 MPa the curve passesthrough the middle of the data.Below 1 MPa the estimatedstrength is nearly 50% higherthan the true strength eventhough the regression r2 isnearly 98%. This problemcould be partially fixed byincluding only the averagemeasured tensile strength in theanalysis but this ignoresconsiderable readily obtainedand economic data anddisguises the true variability.

3. Least squares applied toEquation 3 results in vastlyimproved parameter estimationbut the lower slope to the left ofthe estimated sut produces a lowestimate of sut and thussomewhat low estimated sc and high estimated mi. A good fit overall with the highest r

2, but approximately15% underestimate of true strength for low s3.

4. Modified least squares, Equation 6, applied to Equation 3 results in accurate estimation of theparameters and does so in almost all circumstances. The fact that r2 is slightly less than for method 3 is anecessary consequence of the treatment of variability of the measured tensile strengths.

5. Least squares applied to Equation 2 with sc fixed at the average of the test results. It might bethought that knowing one property should help with estimating a second unknown property, this is not thecase here. The problem in 2 above is now magnified to produce almost the worst fit imaginable. It showsthat an r2 of over 90% can be obtained with a fit that bears virtually no relationship to the data.

6. Least squares applied to either Equation 2 or 3, with the tensile strength test results excluded, resultsin a good fit. Again the problem is that good economic data is ignored and the fit at low stress will be morevariable.

7. Least squares applied to either Equation 2 or 3 with both the tensile and unconfined compression testresults excluded. In this case more than half the data is ignored and, in the present case, the fit at low s3 ismore than 30% out. This is a random error and the fit could be low or high. The problem with this approachis that it is poorly controlled at the stresses of interest in slope analysis.

8. Least squares applied to Equation 4. This is a common form of fitting the Hoek-Brown criterion todata and estimating sc and mi. This method virtually minimises errors to the fourth power, hence thelowish r2, and dramatically overweights the larger values of s1. Errors in parameter estimates are not

Sigma 3 (MPa)

Sigm

a 1 (MP

a)

0

50

100

150

200

250

-10 10 30 50 70

UCS miArtificial data 10.012.0Normal eqn & LS14.97.75Extended eqn & LS 8.4615.5Ext eqn & mod LS 10.712.0Fix UCS & LS 10.05.21Excl Sc or St & LS 6.1921.4DS^2 & LS 3.9735.2Log & LS 8.094.12

Not shownExcl St & LS 9.5213.7DS^2 with UCS fixed 10.013.8DS^2 & Least abs sum 9.1815.4Log with excl St 9.6712.2

Sigma 3 (MPa)

Sigm

a 1 (MP

a)

0

10

20

30

40

-2 0 2 4 6

UCS miArtificial data 10.012.0Normal eqn & LS14.97.75Extended eqn & LS 8.4615.5Ext eqn & mod LS 10.712.0Fix UCS & LS 10.05.21Excl Sc or St & LS 6.1921.4DS^2 & LS 3.9735.2Log & LS 8.094.12

Not shownExcl St & LS 9.5213.7DS^2 with UCS fixed 10.013.8DS^2 & Least abs sum 9.1815.4Log with excl St 9.6712.2

Figure 2 : Fits to artificial data (a) full range (b) low stress range

predictable, but in this example, estimated sc and mi are 40% and 300% of the true values respectively, eventhough the corrupted r2 is over 95%. Over most of the range of the test results it is a very good fit but notover that portion of interest in slope design. It is not recommended.

9. As for 8 above but with sc fixed at the mean value, in contrast to 5 above this results in a good fitacross the range but relies on a good estimate of sc and increased faith that this accurately represents triaxialbehaviour.

10. Least sum of absolute differences applied to Equation 4. This in large measure compensates for theoverweighting of large s1 values of method 8. The resulting estimates are good.

11. Least squares applied to Equation 5, again a common form of fitting the Hoek-Brown criterion. Asfor Equation 2 this equation is not defined for s3 less than spt. This method has major problems fitting anydata which includes a moderate spread of tensile testing.

12. Least squares applied to Equation 5 with the tensile strength test results excluded. A robust methodweighted to low stress results and good for slope analysis but unable to take advantage of economic andreadily available data.

From Table 1 it can be seen that r2 is not a useful indicator of accuracy of estimates of the parameters andthat these estimates can vary widely depending on the method of analysis. Methods with r2 in excess of 95%and that model the data very well over most of the range have estimates of sc varying from 3.97 to 14.9 MPaand mi from 7.75 to 35.2 and this is for artificial data that follows exactly the criterion with only testvariability. Thus many of these methods are very poor estimators of strength in the low stress region that isof interest in slope analysis.

HOEK-BROWN CRITERION FOR INTACT ROCK

Modified least squares, Equation 6, was combined with the extended formulation of the Hoek-Browncriterion, Equation 3, to estimate sc and mi for all test data in the data base. Discussion in the previoussection indicates that the fit is poorly controlled at low stresses for sets with little data, particularly sc and st.Small changes in the data can lead to wildly varying estimates of both sc and mi, in general with sc becomingvery small and mi very high but with the fit being almost identical over the range of the test results. In factfor many data sets sc and mi are not independent but sc0 as mi. The best solution to this issue is toplace plausibility limits on the parameters. A number of limits were considered and the following onesadopted:

As all the test results were taken from materials described as rock, sc was limited to be not less than 1MPa.

Published values of mi fall in the range of 4 to 33 (Hoek & Brown, 1998). As will be discussed latermi is very closely related to the ratio -sc/sut, reference to the figures in Lade (1993) indicates that this ratiovaries from less than2 to over 50. Thislimits mi to the range1 to 50. Further miis related to theangle of friction ats3=0 (ie f0), whichis of great interest inslope analysis. Itwas considered thatf0 should be limitedto the range of 15 to65, which for theHoek-Browncriterion furtherlimits mi to the range1.4 to 40.

The process wascompleted for 475

mi from literature, mipub

mi from

fitting HB

equation, mitest

0

10

20

30

40

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

Figure 3 : mi from literature against mi from test results and Hoek-Brown Equation

data sets involving 3779 test results. The results of the analysis are provided in Figures 3 to 6. Figure 3shows a box and whisker plot of the values of mi estimated from the data, mitest, against the values of miprovided in Hoek & Brown (1998) and Hoek et al (1995), mipub. Several such figures are presented in thispaper, the whiskers show the range of test results, the box shows the upper and lower quartiles and the barthe median value. Also shown on this figure is a linear regression between published mipub and m itestweighted for the number of data points supporting each estimate. The regression equation is:

ipubitest mm 441.058.7 += (7)

This is a very poor relation, r2=16.4%, between mi determined on the basis of actual testing and thatobtained from the literature.

Figure 4 presentsmitest against rock typeordered in increasingmipub, it can be seen thatit is difficult to ascribe asingle or even smallrange to mi on the basisof rock type. It shouldbe remembered that50% of the test datafalls outside the rangeindicated by the box foreach rock type, thus forexample 50% of thevalues for sandstone fallbelow 11 or above 19,and for granite below19 or above 31.

Figure 5 presents thesc determined fromfitting the Hoek-Brownequation against the scdetermined from s ttesting or, at least, asreported in the literature fromwhich the data was obtained.Again several figures in this paperare presented in this style. Theupper and lower dashed linesrepresent 1.5 and 0.67 times thereported sc values. Further, thesymbols represent the number oftest results used to determine thefit, a small cross is 4 or less datapoints, a small circle is 7 or less,large circle is 12 or less and asquare is more than 12 data points.It can be seen that virtually all thedata lies in a very narrow band,such that the fitted sc is quite closeto the reported sc.

Figure 6 is a similarpresentation to Figure 5 except that

mitest

0

10

20

30

40

Clayston

FireclayG

reenstoM

udstoneS

erpentiS

chistS

haleC

halkC

hloritiLim

estonM

arbleS

iltstonS

late Biocalca

Dolom

iteA

nhydritS

altC

oalT

uff Pyroclas

Rhyolite

Aplite

Basalt

Lamproph

Trachite

Agg tuffG

reywack

Whinston

Andesite

Diabase

DoleriteQ

uartzdoS

andstonG

ranite1N

oriteQ

uartzitD

uniteE

clogiteG

abbroP

eridotiA

mphibol

DioriteQ

uartzdiG

ranodioG

neissG

ranite

Figure 4 : Rock type against mi from test results and Hoek-Brown equation

Unconfined compressive strength (MPa)

UC

S from

HB

equation (MP

a)

1

4

7

10

40

70

100

400

700

1000

1 4 7 10 40 70 100 400 700 1000

Figure 5 : Unconfined compressive strength against thatpredicted by the Hoek-Brown equation

it presents fitted tensile strengthversus reported tensile strength. Itcan be seen that the fitting methodadopted provides a very goodestimate of s t for those data setswhich do have reported tensilestrengths. Most of the othermethods fail for such data, so muchso that often practitioners areforced to ignore the valuableinformation available frominexpensive tensile testing. This isparticularly a problem as such dataforms a good control on the failureenvelope over the low stress range(Lade, 1993).

In summary the proposedmethod results in good fits of theHoek-Brown criterion to the dataand, in particular, results in goodfits in the low stress region. It appears that published values of the parameter mi might be quite misleadingas mi does not appear to be related to rock type.

GENERALISED CRITERION FOR INTACT ROCK

There are a number of concerns regarding the formulation of the Hoek-Brown criterion: Several authors, including Johnston (1985), note that soil, soft rock, and brittle rock form a

continuum and thus a failure criterion should be able to accommodate the linear or near linear behaviourobserved in soils and soft rocks. Fixing the exponent at a half means that at best the criterion is a poor modelof soft rocks. This is not surprising as it was developed for brittle rocks but it is a limitation which is oftenoverlooked by practitioners who apply it to all rocks. Further it is a severe limitation on the extension of thecriterion to rock mass strength.

Lade (1993) in comparing the theories and the evidence regarding rock strength criteria finds that anappropriate criterion should have three independent characteristics the opening angle, the curvature and thetensile strength. The fixed exponent on the Hoek-Brown criterion limits it to modelling only two of thesecharacteristics. In fact as often used, mi is varied to model the curvature over the stress range of the testresults and neither the opening angle nor the tensile strength are modelled. Lade also states that it may be anadvantage to include the tensile strength in determination of material parameters as it stabilises the fit at lowstresses. This is particularly important for slope analysis.

If the exponent, a, is allowed to vary the Hoek-Brown criterion can model widely varying curvatures andopening angles. It is also able to include an accurate representation of the tensile strength. The authors haveapplied this generalised Hoek-Brown criterion for intact rock to the full data set. As would be expected,adding an extra parameter or property always improves the fit but has many other benefits as well.

The equation becomes:

-=

->

++=

ic

icc

ic

m

mm

ssss

ssss

sssa

331

33

31

for

for 1(8)

The modified least squares, Equation 6, is adopted.The limits, given above for fitting the Hoek-Brown criterion, are also placed on the parameters. For the

generalised criterion these become, sc>1, mi in the range 1 to 50, and a mi in the range 0.7 to 20 (this is theequivalent limit on f0). In addition, a is limited to the range 0.2 to 1.

Tensile strength (MPa)

Tensile strength from

HB

equation (MP

a)

0.1

0.4

0.7

1.0

4.0

7.0

10.0

40.0

70.0

100.0

0.1

0.4

0.7

1.0

4.0

7.0

10.0

40.0

70.0

100.0

Figure 6 : Uniaxial tensile strength against that predicted by theHoek-Brown equation

Allowing a to varyprovides the ability toobtain a much better fitover the low stress rangewhich is of greatest interestin slope analysis.

The results of theanalysis are presented in aseries of figures. Figure 7presents a box and whiskerplot of mi determined fromthe data against thepublished values of mi.Again it can be seen thatthere is little relationshipb e t w e e n t h e t w o .Likewise, there was foundto be no relationshipbetween mi and rock type.

The slope of the generalised criterion at s3=0 is 1+a mi, and is related to f0 by:

( )( )( )451atan2 5.00 -+= imaf (9)If a classification of

samples, say by mi orrock type, is predictiveof the triaxial envelopeat low stresses then itwill be apparent on ap l o t o f t h a tclassification against am i. Figures 8 and 9present plots of a miagainst published m ia n d r o c k t y p erespectively. FromFigure 8 it can be seenthat published mi is nota good predictor of thetriaxial envelope atl o w s t r e s s .Examination of Figure9 shows that there is a weak correlation of rock type with a mi in that fine grained rocks tend to have thelowest values, medium to coarse grained higher and rocks with tightly interlocked crystals the highest. Theauthors do not believe that the relationship is strong enough to be used predictively.

Figure 10 presents the sc obtained from fitting the generalised equation against the reported sc. It can beseen that virtually all the data lies in a very narrow band, such that the fitted sc is quite close to the reportedsc. As would be expected the overall correlation is better than that shown on Figure 5.

Figure 11 presents the fitted tensile strength versus reported tensile strength. It can be shown that theuniaxial tensile strength sut is bound as:

( )1+-

Table 2 presents the errors involved inadopting sc/(m i+1) as s t. For simplicity thelower bound has been adopted in plottingFigures 6 and 11, the error in doing this is quitesmall. It should be noted that it is likely thatmany of the reported st are likely to be Braziliantensile strengths. The authors are attempting toresolve as many of these as possible incontinuing work.

The fit in Figure 11 is extremely good.Figures 10 and 11 provide considerableconfidence that the fitted curves provide a very good model of triaxial strength at low stresses. In both casesthe unexplained variance of the generalised fits is about half that of the Hoek-Brown fits.

Table 2 : Error in approximating sut as -sc/(mi+1)

a mi Error (%)1 All 0

0.8 All 8 9

Figure 12 presents aplot of a against m i asdetermined for each dataset from the generalisedcriterion. Also shown onthe figure are hyperbolaeshowing lines of constanta mi, ie f0, for 15 to 65.Inspection of the figureshows that the constraintson mi, a and a mi did notoften limit the regressionprocedure.

An interesting anduseful observation fromFigure 12 is that thereappears to be arelationship between aand m i . Such arelationship is derived inthe next section and isshown on Figure 12. It isoften thought that the curvature of the strength envelope, ie a against mi in the current context, should begreater for strong rocks than weak rocks. The data set and analysis do not support this contention. Figure 13shows the relationship between a and mi plotted for the data divided into four categories depending on thesc. It can be seen that the relationship is independent of strength. High strength rocks can have linear flatfailure envelopes and low strength rocks can have steep curved envelopes. Thus sc is a truly independentparameter in a rock failure criterion.

mi

Alpha

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 10 20 30 40 50

f0 15

25

35 45 55 65

f0

Figure 12 : a against mi

mi

alpha

UCS

A consequence of allowing a to vary at all,is that a failure envelope with a high f0 canhave a low f at high stresses and thus failureenvelopes for different rocks normalised on sc,can cross at high stresses. This cannot happenwith a fixed at 0.5 in which case all envelopescross only once at sc. Figure 14 shows afamily of curves, normalised by sc, for variousmi and the a typical of the relationship shownon Figure 12. It can be seen that the mi equals40 curve crosses the mi equal 10 and 3 curvesat 1.4 and 2.5 times sc respectively. Thisimplies that high frictional strength at lowstresses is often associated with low frictionalstrength at higher stresses. Figure 15 showssome examples of test data that illustrate thispoint.

A relationship between a and mi impliesthat the triaxial failure envelope can beestimated from sc and either mi or st, with abeing determined by the relationship. Thusthere are no more parameters to be determinedthan for the usual Hoek-Brown criterion. Theparameters can be based on simple testing andprovide a more accurate prediction of strengththan published mi values, particularly in thelow stress region typical of slopes.

Sigma 3 / Sigma C

Sigm

a 1 / Sigm

a C

-2

0

2

4

6

8

10

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

mi 40

10

3

1

Figure 14 : Family of failure envelopes

Sigma 3 / Sigma C

Sigm

a 1 / Sigm

a C

0

2

4

6

8

10

12

14

16

18

20

-1 0 1 2 3 4 5

Set 382 Sandstone

Set 305 Granite

Set 425 Gabbro

Figure 15 : Results showing failure envelopes crossing

GLOBAL PREDICTION

A single equation could be fitted to the entire data base on the following assumptions: The value of sc obtained from fitting the generalised Hoek-Brown criterion is the best estimate of sc

for each data set. A reasonable estimate of |st| is obtained for each data set by dividing sc by the value of mi obtained

by fitting the generalised Hoek-Brown criterion.Figures 10 and 11 show that the above assumptions are quite reasonable for those cases where there is

data to confirm them. On this basis mi can be set to sc/st and Equations 3 and 8 can be rewritten as:

a

ss

ss

ss

-+=

tcc

331 1 (11)

-++

-+=

cmba

tcc

t

c

ss

ss

ss

ss

0exp1

331 1 (12)

Equation 11 is equivalent to the Hoek-Brown criterion but with an exponent not necessarily equal to 0.5.Equation 12 is equivalent to the generalised Hoek-Brown criterion. The exponent of Equation 12 is ageneral function that varies from a to b with a midpoint at m0 and a variable length of step. These equationscan be fitted to the entire data set using Equation 6.

Two of the data sets produced extremely large residuals and were ignored in reanalysis. Fitting Equation11, ie a constant exponent, resulted in a being estimated as 0.439 and an r2 of 83.5%. This is a reasonable fitwhen the range of rocks to which it applies is considered. Examination of the residuals reveals that a betterfit will be possible as the residual is a function of mi. This is illustrated on Figure 16, for mi

Fitting Equation 12 resulted in the following estimate for the exponent:

( )( )455.7exp108585.14032.0 im++=a (13)

This equation is shown on Figure 12 and models the results of the analysis of the individual data sets verywell. This analysis resulted in an r2 of 94.8%, which is extremely good for such a global fit. The residualsare plotted on Figure 17, there is no or little trend with mi or s1 and it can be seen that this is a much better fitthan Equation 11 and Figure 16.

Figure 18 shows a threedimensional plot of the failurecriterion described by Equations 12and 13, ie s1 as a function of s3 andmi (ie sc/st). It can be seen that formi

Figure 19 provides the data for s3 up to three times sc and Figure 20 for s3 to half of sc. It can be seen thatthe fits are very good. The ridge at high stress and mi=8 are apparent on Figure 19 with uniform behaviour atlow stress seen on Figure 20.

Figure 21 presents aagainst m i includingEquation 13 and showingthose data for which there

Sigma 3 / Sigma c

Sigm

a 1 / Sigm

a c

40

0 1 2 3

Number under graph isestimated ratio of

-Sigma c / Sigma t

Figure 19 : s1/sc with fits for variable a against s3/sc categorised by -sc/st for high stress

Figure 20 : s1/sc with fits for variable a against s3/sc categorised by -sc/st for low stress

Sigma 3 / Sigma c

Sigm

a 1 / Sigm

a c

40

0.0 0.1 0.2 0.3 0.4 0.5

Number under graph isestimated ratio of

-Sigma c / Sigma t

mi

Alpha

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 10 20 30 40 50

Data with compressive and tensile strengths

Other data

Figure 21 : a against mi showing cases with measured or reported s3 and st

is actual, not estimated, values for both sc and st. It can be seen that these sets are distributed similarly tothose for which at least one of these parameters has been estimated by fitting the generalised Hoek-Browncriterion.

COMPARISON OF CRITERIA

A comparison of the various criterion as fitted to the data base is provided in Table 3. The varianceexplained approximates r2. As would be expected, the generalised Hoek-Brown criterion provides by far thebest fit, r2 of 99.5%, as it has three parameters and is fitted to the individual data sets. None the less, the fitobtained is considerably better than the fit obtained by the Hoek-Brown criterion (ie with a fixed at 0.5) , r2

of 98.9%. The unexplained variance for the generalised criterion is less than half that of the Hoek-Browncriterion with a of 0.5.

Table 3 : Comparison of predictions

Variable/prediction VarianceVarianceexplained

s1/sc 16.93 0Global regression with a constant 2.78 83.6Hoek-Brown with published mi 2.00 88.2Global regression with a variable 0.846 95.0Hoek-Brown fitted to individual sets 0.186 98.9Generalised Hoek-Brown fitted to individual sets 0.077 99.5

The above methods compare different ways of fitting triaxial data, ie different criteria applied to actualtriaxial data. Table 3 also allows a comparison of three methods of prediction of triaxial strength that are notbased on having actual data but are based on parameters estimated in some other manner. The methods arediscussed in the following points:

Prediction based on global equation with variable a. This method is based on Equations 12 and 13and estimates of sc and st. It has an r

2 of 95.0% when used to predict the test results in the data base. Theaccuracy of the predictions are illustrated on Figures 19 and 20.

Prediction based on global equation with constant a. This method, based on Equation 11, is the leastaccurate of the three and is not discussed further.

Prediction based on published values of mi. This method is based on Equation 3 with values of miestimated from those widely published in the literature. The method has an r2 of 88.2% when used to predictthe test results in the data base. On average this method predicts the strengths well but with considerablymore scatter than that from the global equation. Figures 22 and 23 present the data categorised by publishedmi, these figures are in a similar form and can be compared to Figures 19 and 20. It is clear from the figuresthat at low published mi the triaxial strength is under predicted and at high published mi it is over predicted.In effect what this means is that triaxial strength is poorly predicted by published mi values and the method ispredicting the average strength for all tests.

Figure 22 : s1/sc with fits for published mi against s3/sc categorised by mi for high stress

Sigma 3 / Sigma c

Sigm

a 1 / Sigm

a c

24

0 1 2 3

Categorised by publishedvalues of mi

Figure 23 : s1/sc with fits for published mi against s3/sc categorised by mi for low stress

Sigma 3 / Sigma c

Sigm

a 1 / Sigm

a c

24

0.0 0.1 0.2 0.3 0.4 0.5

Categorised bypublished values

of mi

SYSTEMATIC ERROR IN HOEK-BROWN CRITERION

If a for a particular rock is not equal to 0.5 then there isa systematic error in fitting the Hoek-Brown criterion toany triaxial test results obtained on that rock. The error isillustrated on Figure 24, two data sets are shown, the upperone is for sc, mi and a of 30 MPa, 24 and 0.4 respectivelyand the lower one for 8 MPa, 5 and 0.8. The different scwere chosen to separate the curves, and the mi and a aretypical combinations determined in the analysis of theentire data base. The solid lines represent the Hoek-Brownfits to these data. The residuals are shown on the bottomgraph, if a is less than 0.5 then there are negative residualsat both the low and high end of the range of s3 tested withpositive residuals in the middle range. The sign of theresiduals is reversed if a is greater than 0.5. While the fitsin the upper graph might look satisfactory for engineeringpurposes, the errors in estimates of sc and m i can besignificant. Table 4 shows the parameters estimated fromfitting the Hoek-Brown criterion to these data, it can beseen that errors in the estimates vary from one half to five times the correct values. Thus the parameters ofthis model cannot be considered material properties. These errors are discussed in more detail below.

Table 4 : Errors in fitting Hoek-Brown criterion to materials with a 0.5

Actual parameters

Material sc mi a r2 (%)

Upper 30.0 24.0 0.4 100.0Lower 8.0 5.0 0.8 100.0

Parameters determined by fitting Hoek-Brown criterion

Material Estimated sc Estimated mi r2 (%)

Upper 33.0 11.7 99.65Lower 3.94 48.5 97.47

Consider programs of triaxial testing on rocks with sc equal to 3 MPa, and mi and a of (a) 40 and 0.4 and(b) 4 and 0.8. Both are weak rocks and their triaxial strength could be of interest in design of a large rockslope. Further consider that different test programs are undertaken in which the maximum s3 is determinedby the capacity of the triaxial apparatus. The following test programs could result:

Program A - 12 stages to a maximum s3 of 10 MPa, Program B - 10 stages to 5 MPa (ie omitting the last two stages), Program C - 8 stages to 3 MPa, Program D - 6 stages to 1.5 MPa, and Program E - 4 stages, including UCS, to 0.7 MPa.If there was no variability in the test apparatus or material, and measurement was perfect, the test results

would be as shown on Figure 25, that is there is no sample or test error. Quite different envelopes result ifthe Hoek-Brown criterion is fitted to these test programs. The estimated sc and mi are given in Table 5.Estimated sc varies from 1.66 to 4.20 MPa and mi from 8.0 to 29.6 with very high r

2.

Sigma 3 (MPa)

Sigma 1 (M

Pa)

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30 35

Residual (MPa)

-20

-10

0

10

20

0 5 10 15 20 25 30 35

Figure 24 : Pattern of residuals forHoek-Brown fits

Table 5 : Variation of sc and mi with s3max for exact simulated results

Material (a) sc = 3 MPa, mi = 40 and a = 0.4

Programs3max

(MPa)Stages Estimated sc Estimated mi r

2 (%)

A 10 12 4.20 11.5 99.43B 5 10 3.77 14.2 99.41C 3 8 3.47 16.9 99.38D 1.5 6 3.22 20.3 99.52E 0.7 4 3.07 23.4 99.74

Material (b) sc = 3 MPa, mi = 4 and a = 0.8

Programs3max

(MPa)Stages Estimated sc Estimated mi r

2 (%)

A 10 12 1.66 29.6 97.96B 5 10 2.23 17.5 98.65C 3 8 2.59 12.6 98.96D 1.5 6 2.85 9.60 99.46E 0.7 4 2.96 8.01 99.85

The dashed lines on Figure 25 show the envelopes fitted to cases A, C and E. It is emphasised that whilethe upper line for material (a) and the lower line for material (b) (ie Case E) do not look like good fits, theyare in fact very good fits for the 4 test results, below s3 equal 0.7 MPa, that form their basis with r

2 of99.74% and 99.85% respectively. Such results might erroneously be taken to support the contention that thematerial was well modelled by the Hoek-Brown criterion. It can be concluded that the estimated parametersare as much a function of the test program as of the material tested. These errors would generally beobscured by the material variability but they are still present.

Figure 26 and Table 6 present the results of analysis of data set 434, a sandstone, in which the analysishas assumed different maximum possible s3. This further illustrates the errors that can occur if a Hoek-Brown envelope is fitted to material for which a does not equal 0.5. Depending on the test program,

Sigma 3

Sigm

a 1

0

5

10

15

20

25

30

35

40

-2 0 2 4 6 8 10 12

Artificial data for

sc=3 MPa, mi=40 and alpha=0.4

Sigma 3

Sigm

a 1

0

5

10

15

20

25

30

35

40

-2 0 2 4 6 8 10 12

Artificial data for

sc=3 MPa, mi=4 and alpha=0.8

Figure 25 : Hoek-Brown fits to artificial data

estimates of sc obtained by fitting the generalised criterion vary from 85.4 to 57.7 MPa and of mi from 6.35to 13.1, variations of 150% and 205%. If the Hoek-Brown criterion is fitted, the estimates vary from 23.8 to55.5 MPa (230%) and 31 to 138 (445%). Again the r2 determined for the fits are very good.

Table 6 : Variation of sc and mi with s3max for data set 434

For generalised Hoek-Brown criterion

s3max (MPa) N Estimated sc Estimated mi Estimated a r2 (%)

All data 20 85.4 6.71 0.75 99.70400 16 59.2 13.1 0.65 99.63200 9 57.7 13.0 0.66 99.50100 5 64.1 6.35 0.87 98.18

For Hoek-Brown criterion

s3max (MPa) N Estimated sc Estimated mi r2 (%)

All data 20 23.8 138 97.00400 16 35.1 75.2 98.49200 9 44.9 49.7 98.20100 5 55.5 31.0 93.99

Figure 27 shows the residuals from fitting the Hoek-Brown criterion to data sets with sc less than 20 MPaplotted against s3 divided by the maximum test s3. There are four graphs showing cases where the estimateda from fitting the generalised criterion is (a) less than 0.4, (b) between 0.4 and 0.6, (c) between 0.6 and 0.8and (d) greater than 0.8. On each graph the residuals have been fitted with a quadratic relationship. It can beseen that these residuals conform almost perfectly to those predicted on Figure 24. This is very strong

Sigma 3

Sigm

a 1

0

500

1000

1500

2000

2500

0 100 200 300 400 500 600 700

Sigma 3

Sigm

a 1

0

500

1000

1500

2000

2500

0 100 200 300 400 500 600 700

Figure 26 : Hoek-Brown fits to actual data

evidence that the Hoek-Brown model is not appropriate. Figure 28 shows the residuals obtained from fittingthe generalised Hoek-Brown criterion. It can be seen that these show little or no trend.

Data with estimated UCS less than 20 MPa

Sigma 3 / Maximum test sigma 3

Residual from

fitting HB

equation

Alpha .8

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

Figure 27 : Residuals for Hoek-Brown fits for weak rock against s3/s3max categorised by a

Figure 28 : Residuals for generalised fits for weak rock against s3/s3max categorised by a

Data with estimated UCS less than 20 MPa

Sigma 3 / Maximum test sigma 3

Residual from

fitting general equation

Alpha .8

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

APPLICATION TO SLOPE ENGINEERING

In general the triaxial strength of intact rock is not particularly important in the analysis or design of rockslopes, even large rock slopes. The maximum depth of the failure surface for even a 500 m high slope isgenerally only 100 to 150 m deep, and thus the maximum s3 of interest is around 4 MPa. The relativecontribution of the triaxial component of strength for various rocks (ie sc, mi and a) and overburden stressesis given in Table 7. It can be seen that for high f0 rocks triaxial strength is generally significant, even forquite low slopes and high strength, but for most of these situations intact rock strength will not be criticalexcept in forming the basis of rock mass strength. For low f0 rocks triaxial strength is important for lowstrength or high stress conditions and it is these situations for which intact strength may be critical in design.

Table 7 : Triaxial component of strength

sc s3High f0 case

mi = 50, a = 0.4%

Low f0 casemi = 0.8, a = 0.9

%300 4.0 24 2.3100 4.0 59 6.930 4.0 140 2310 4.0 280 683 4.0 570 230

300 0.5 3.4 0.3100 0.5 9.8 0.930 0.5 29 2.910 0.5 70 8.63 0.5 160 29

Table 8 and Figure 29 provide a comparison of different methods of predicting the triaxial strength of lowstrength rocks at low stress. The predicted strengths are compared with the measured strengths for all casesin the data base for which sc is less than 20 MPa and s3 is between 0 and 5 MPa (excluding UCS testresults). The variances of the residuals scaled on sc are given in Table 8 and the scaled residuals plotted onFigure 29. In general the order of accuracy of the various prediction methods is the same as discussed for theoverall predictions. It is of interest to note that the global generalised equation (r2 of 88.6%) is almost asaccurate a prediction of the triaxial strength of these rocks as that obtained by fitting the Hoek-Browncriterion directly to triaxial test data (r2 of 90.3%). This reflects the fact that there is abundant evidence thata does not equal 0.5 for a large proportion of the rocks tested see Figure 12 and thus there will besystematic errors at low stress.

Table 8 : Comparison of predictions for weak rocks at low stress

Variable/prediction VarianceVariance

explained %s1/sc 7.69 0Global regression with a constant 1.91 75.1Hoek-Brown with published mi 1.99 74.1Global regression with a variable 0.88 88.6Hoek-Brown fitted to individual sets 0.74 90.3Generalised Hoek-Brown fitted to individual sets 0.12 98.5

The above situation arises because the range of s3 over which the tests were performed hardly evercorresponded to 5 MPa and thus there were almost always systematic errors at the lower stresses tested. It issometimes argued that the solution to this is to test over a range of s3 that represents the field conditions, butthis is hardly ever possible as generally the one set of testing is used to design low and high slopes. Further

for a given slope different portions of the failure surface are at different stresses. Another solution is todetermine the parameters as a function of stress, but this virtually defeats the purpose of adopting a nonlinear failure envelope and confirms they are not a material property.

A useful approximation of the effective stressparameters, c0 and f0, at low stress can be obtained in thefollowing manner. Equation 10 can be rearranged toprovide an estimate of mi based on sc and st as:

t

ci

t

c mss

ss

FAILURE CRITERIA FOR ROCK MASS

A rock mass criterion should only be used where there area sufficient number of closely spaced discontinuities thatisotropic behaviour involving failure on discontinuities can beassumed (Hoek and Brown, 1997). Such a situation for slopesis illustrated on Figure 30, it should be noted that the conceptof closely spaced should be defined in terms of the scale of thefailure surface.

Slope failures in which the failure surface is entirelythrough the rock mass are not common. This is due to the lowstresses typically acting in a slope. Failure usually requireslarge scale (relative to the slope in question) featuresconcentrating stresses into regions of weak rock mass. Forexample a long vertical joint may lead to stressing of weakmaterial at the toe of the slope (Figure 31).

Figure 31 : Example of shear failure through rock mass at the toe of a slope - Nattai Escarpment Failure

As mine slopes become higher and longer, the necessity to account for the strength of rock masses indesign increases. Methods used for assessing this shear strength are based on empirical criteria. As ageneral rule such criteria are based on laboratory scale specimens with very little, and often no, fieldvalidation.

Yudhbir et al (1983), Ramamurthy et al (1994) and Sheorey (1997) (Equations 16 to 18 respectively)present rock mass criteria that have been developed as extensions of the strength criteria for intact rock. Themodification process has typically been based on model tests, small sample testing and limited experience.The criteria all assume a non-zero unconfined compressive strength, scm, and hence tensile strength, stm, forthe rock mass. These criteria would therefore be expected to overpredict the strength for poor quality rockmasses at the low stresses common to failure surfaces in slopes.

Figure 30 : Heavily jointed rock mass

longsubvertical

joint

Rock massfailure at toe of

slope

ass

ss

+=

cm

c

ba 31 (16)

mb

cmma

+=

3331 ss

sss (17)

mb

tmcm

+=ss

ss 31 1 (18)

The most commonly used strength criterion, having received widespread interest and use over the last twodecades, is the Hoek-Brown empirical rock mass failure criterion, the most general form of which is given inEquation 19. Hoek and Brown (1980) developed this criterion as there was no suitable alternate empiricalstrength criterion. The equation, which has subsequently been updated by Hoek and Brown (1988), Hoek etal. (1992) and Hoek et al. (1995), was based on their criterion for intact rock discussed earlier in this paper.The only rock mass tested and used in the original development of the Hoek-Brown criterion was 152mmcore samples of Panguna Andesite from Bougainville in Papua New Guinea (Hoek and Brown, 1980). Hoekand Brown (1988) later noted that it was likely this material was in fact disturbed. The validation of theupdates of the Hoek-Brown criterion have been based on experience gained whilst using this criterion. Tothe authors knowledge the data supporting this experience has not been published.

a

cbc sm

+

+=

ss

sss 331 (19)

Estimating the parameters in the Hoek-Brown criterion was very difficult, thus correlations with rockmass rating parameters were developed. The most current of these is the Geological Strength Index (GSI)(Hoek et al. 1995). These correlations are given in Equations 20 to 24. The parameters mi and mb are intactand mass material constants; a and s are constants that depend on the rock mass characteristics; and sc is theuniaxial compressive strength of the intact rock.

-=

28

100exp

GSI

m

m

i

b (20)

for GSI>25:

-=

9

100exp

GSIs (21)

5.0=a (22)

for GSI

Hoek (1997) provides Figure 32 to determine the GSI directly. Hoek et al (1995) say that the GSI mayalso be calculated using Bieniawskis (1976 and 1989) rock mass rating (RMR), GSIRMR, or Bartons (1974)Q-system, GSIQ.

Figure 32 : Estimation of GSI (Hoek, 1997)

DISCUSSION OF THE HOEK-BROWN CRITERION FOR USE WITH SLOPES

Calculation of GSI

GSIRMR and GSIQ are derived from the rating parameters for several rock mass properties (Equations 25and 26).

( ) +++= conditiondefect spacingdefect strengthintact Ratings RQDGSIRMR (25)

44log9 +

=

a

r

neQ J

J

J

RQDGSI (26)

where Jr = joint roughness numberJn = joint set numberJa = joint alteration number

RMR and the Q-system were developed for underground applications of limited size and thus could beexpected to be reasonable indicators of rock mass properties for underground tunnels and caverns. Douglas& Mostyn (1999) discuss the following problems with the estimation of the GSI with regards to large scaleslopes.

RQD has a heavy weighting in both rating systems (inparticular the Q-system). Since the RQD is based on a fixedcutoff length of 100mm the ability of the R Q D to givemeaningful information reduces as slopes get larger. Forslopes of several hundred metres the RQD (particularly ifestimated from borehole data) has questionable value. On thescale of a large pit slope it is unlikely that all the defectsencountered in boreholes would be of significance to the rockmass stability.

The defect spacing parameter suffers from a similarproblem to that of RQD. The maximum rating is applied for aspacing interval of greater than 3m and greater than 2m forBieniawski (1976) and Bieniawski (1989) respectively. Thespacing increments given by Bieniawski (1976, 1989) werederived for and on the basis of underground tunnels that wereof the order of 10 20m in span. Where a slope is in the orderof several hundred meters these spacing increments areunlikely to be valid. Figure 33 shows blocks from a 400m highslope failure. A tunnel of 15m span is unlikely to have rockmass strength problems with a block size as big as those in thefigure.

When assessing the RQD and discontinuity spacing fromboreholes, all discontinuities are included however, for large rock slopes those discontinuities that are largein area will play the major role in the rock mass strength. Without careful orientation techniques, it isdifficult to get either the true spacing or the number of discontinuity sets.

It is well known that intact rockexhibits a strength scale effect. Thisscale effect exists up to block sizesof at least one metre. Therefore theparameter for intact rock strengthshould be adjusted to account forscale for large block sizes as inFigure 33.

When assessing the ratingparameters for defect condition, Jrand Ja, the analyst should take intoaccount the large scale (i.e. scale ofrock mass) joint characteristics aswell as those on the small scale. Thethickness of joint infilling should beconsidered proportionately to thelength and shape of thediscontinuities. Figure 34 shows

Figure 33 : Slope failure block size

Very roughSmooth

Very rough

Smooth&

infilled

Defect A

Defect B

Figure 34 : Effect of scale on defect properties

two defects, on the small scale (borehole) defect A would have a high rating and defect B a low rating.However, when one looks at the large scale (large slope), defect A would be expected to have a lowerstrength.

Figure 32 shows estimates of GSI provided by Hoek (1997). The main components affecting the strengthof the rock mass are covered (ie structure and surface conditions). It is not clear how scale is to beinterpreted on this figure. The authors believe that GSI should be interpreted as being on the scale of therock mass under assessment. Using judgement the user can estimate the condition of their rock mass at thescale of their slope. For example, a blocky rock mass at a scale of 10m is vastly different to a blockyrock mass at the scale of 500m. Smaller relative block size leads to more freedom for block rotation and agreater chance for mass failure. Liao & Hencher (1997) showed that relative block size was critical indeciding the mode of failure. The smaller the block size (when compared to slope height) the more likelyrock mass failure would be the dominant failure mechanism.

The authors recommend the use of Figure 32 for calculations of GSI for slopes. The use of GSIRMR andGSIQ from boreholes should only be used for preliminary strength estimates. It should be remembered thatthe key to the structure column is degree of interlocking. The degree of interlocking should be assessed onthe scale of the slope under consideration. For example, the rock mass controlling the slopes for an ultimatepit may be considered interlocked whilst the rock mass may be considered as very well interlocked on thescale of individual benches of the same slope. It should also be remembered that where block size is of thesame order as that of the structure being analysed, the Hoek-Brown criterion should not be used. Thestability of the structure should be analysed by considering the behaviour of blocks and wedges defined byintersecting structural features (Hoek, 1997).

Estimation of parameters from GSI

Figure 35 shows the variation of the Hoek-Brown parameters mb/mi, a and s with GSI based onEquations 20-24. The figure also indicates thelower bound for GSI and the upper bound for GSI ifthe Hoek (1997) table, Figure 32, is used. mb/mi,which mainly accounts for friction, varies graduallyfrom unity as could be expected for a rock mass.The value of s (which mainly accounts forcohesion) diminishes rapidly with a reduction inGSI thus, indicating a rapid reduction incompressive strength and an even more rapidreduction in tensile strength as the quality of therock mass decreases. This is as expected. Asrockmass defects become more cohesive it wouldbe expected that s would be non-zero so as to avoidzero compressive strength. But GSI reduces forincreasing cohesion and if s is predicted from GSIthen s approaches zero not a finite value. This maybe why the Hoek-Brown criterion will underpredict the shear strength of clayey bench slopes. It should beremembered at this point that the initial Hoek-Brown criterion was developed for hard rocks and has onlyrecently been accepted for use with very poor quality rock masses by Hoek and Brown (1997). Thus, itcould be expected that the experience with using the criterion for poor quality rock masses (particularly forslopes) would be very limited.

The value of a remains relatively constant and has a maximum value of 0.6 (using Figure 32). This is notconsistent with what is known about compacted rockfill strength (a material that could represent a lowerbound to poor quality rock masses) and the strength of intact rock. Thus it is not correct at two known limits.

A statistical analysis of a large number of rockfill tests conducted by the authors indicates only a slightcurvature to the failure envelope (a = 0.90). Where the intact rock approaches that of a soft rock or hard soilthe curvature is also likely to be much less pronounced than an exponent of 0.65 would suggest (Johnston &Chiu, 1984). The previous section on intact rock indicates that a actually varies from 0.2 to 1.0, with a

m b /m is

a

GSI min GSI table max

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100

GSI

m b/

m i,

s,

a

GSI =25

Figure 35 : Variation of a, s and mb/mi with GSI

reasonable estimate of 0.4 to 0.9 depending on mi. It could be concluded from this that the Hoek-Browncriterion may over predict the curvature of the strength envelope of poor quality rock masses.

As has been shown for intact rock, fixing or limiting a has a very large impact on the estimation of theother parameters (mb and s ) and therefore a cannot simply be changed without addressing the otherparameters as well.

VALIDATION OF CRITERION

Douglas & Mostyn (1999) show that theprediction of strength using GSI derivedfrom Figure 32 (ie scale sensitive) givesbetter results compared to using G S Ipredicted from R M R and the Q-system.Figure 36 shows back analysed shearstrengths for two slope failures and severallarge scale in-situ shear tests divided byshear strengths estimated from the Hoek-Brown strength criterion using Figure 32.The Nattai and Katoomba escarpmentfailures were natural slope failures ofapproximately 300m and 200m heightrespectively. These failures were caused bythe opening of joints in strong sandstonesand the shearing of weaker rock mass in theunderlying claystones (Mostyn et al., 1997).

HOEK-BROWN FOR SLOPES PROPOSED CHANGES

These are proposals for the use of the Hoek-Brown criterion for slopes that the authors are in the processof validating and intend to publish with fuller details in due course.

The authors propose to use the form of the Hoek-Brown criterion (Equation 27) and to modify some ofthe parameters (Equations 28 to 32). The basic assumption is that the rock mass parameters will be factoredversions of the intact parameters developed in the previous section.

b

bc

bc s

ma

ss

sss

++= *

3*31 (27)

ib Amm = (28)

( )fGSIfA = (29)

ib Bss = (30)

( )cGSIfB = (31)

( )fa GSImf b ,= (32)

In general, *cs is sc of the intact rock; unless the scale of discontinuities affects strength (Medhurst, 1996)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

sn (MPa)

t in-s

itu/t ta

ble

Katoomba escarpment failure

Aviemore shear tests

Nattai escarpment Failure

Figure 36 : Back-analysis results using Figure 32 for GSI

GSIf is calculated using the original GSI (Figure 32) by Hoek (1997) and is used to estimate mb/mi. It isproposed that GSIc be calculated using a similar approach but using degree of interlocking (or blockiness)and cohesional, rather than frictional, properties of the defects, as shown in Figure 37. GSIc would be used toestimate sb. The value of sb is expected to be zero for non-cohesive, non-interlocked rock mass, unity forintact rock and finite for a cohesive rockmass. Analysis of rockfill materials indicatesan sb of about 0.002 for blocky mass/goodsurface quality rock mass.

Figure 38 shows a diagrammatic plot of aversus m curves. It is expected that mb/mIwill decrease as GSIf decreases from 100 andhence a will increase from the intact valueand eventually approach 0.9 for very low GSIor very large slopes.

Douglas and Duran (2000) determinelarge scale slope angles for values of GSIbased on an analysis of failed and stableslopes. These can be used to put bounds onmb for large scale slopes. Using these, anapproximate range of m b for large rockmasses is one to six.

mb = 2

Intact, mi

a

m

@ 0.90

@ 0.2

mb = 4Large scale

Figure 38 : Variation of m and a from values for intact rock to those for large scalemass

Figure 37 : Proposed method for estimating GSIc

Decreasing GSIc

Dec

reas

e in

inte

rloc

king

of

rock

pie

ces

Decreasing defect cohesion

s 1

s = 0

s = 0.002

CURRENT RESULTS

The authors are attempting to develop bounds on rock mass strength for application to slope design.Preliminary bounds scaled by sc are shown on Figure 39. Intact rock provides an upper bound and curvesare shown for high and low f0 rock. Good quality rock fill can be adopted as a lower bound for the strengthof rock masses in which the block strength is not important. The bounds to large slopes derived from Duranand Douglas (2000) also form limits to rock mass strength. The authors analysis of the Nattai escarpment isshown with respect to these bounds and indicates that this poor quality rock mass is correctly located withrespect to the bounds.

Figure 39 : Preliminary bounds for rock mass strength

Sigma 3 / Sigma c

Sigm

a 1 / Sigm

a c

0

1

2

3

0.0 0.1 0.2 0.3 0.4 0.5

Intact mi=40Intact mi=1

RockfillLarge slope mi=4

Large slope mi=2Slope failure

CONCLUSION

The first part of this paper presented an overview of the strength of intact rock. It was demonstrated thatthe method of fitting the criterion to the test data has a major effect on the estimates obtained of the materialproperties. The results of a recent analysis of a large data base of test results demonstrated that there areinadequacies in the Hoek-Brown empirical failure criterion as currently proposed for intact rock and, byinference, as extended to rock mass strength. The parameters mi and sc are not material properties if theexponent is fixed at 0.5. Published values of mi can be misleading as mi did not appear to be related to rocktype. The Hoek-Brown criterion can be generalised by allowing the exponent to vary. This change resultedin a better model of the experimental data. Analysis of individual data sets indicated that the exponent, a, isa function of mi which is, in turn, closely related to the ratio of sc/st. A regression analyis of the entire database provided a model to allow the triaxial strength of an intact rock to be estimated from reliablemeasurement of its uniaxial tensile and compressive strengths. The method proposed is the most accurate ofthose methods that do not require triaxial testing and is adequate for preliminary analysis. An analysis waspresented that showed applying the Hoek-Brown criterion to most rocks results in systematic errors. Simplerelationships for triaxial strength that are adequate for slope design were presented.

The second part of this paper discussed problems with the estimation of GSI and scale dependency forslopes. It is concluded that the estimation of the parameters s, mb and a can be improved for application ofthe criterion to slopes. A new approach to parameter estimation was introduced. Work is on going tovalidate the method.

REFERENCES

Barton, N., Lien, R. and Lunde, J. (1974) Engineering classification of rock masses for the design of tunnelsupport. Rock Mechanics, Vol. 6 pp. 189-236.

Bieniawski, Z.T. (1976) Rock mass classifications in rock engineering. Proceedings of the Symposium onExploration for Rock Engineering, Johannesburg pp. 97-107.

Bieniawski, Z.T. (1989) Engineering Rock Mass Classifications. Wiley, New York.Brady, B.H.G. and Brown, E.T. (1993) Rock Mechanics for Underground Mining. Chapman & Hall.Brown, E.T. and Hoek, E. (1988) Discussion on Paper No. 20431 by R. Ucar, entitled: 'Determination of

shear failure envelope in rock masses'. A.S.C.E., Journal of the Geotechnical Engineering Division, Vol.114 (3), pp. 371-373.

Douglas, K.J. and Mostyn, G. (2000) Strength of Intact Rock, UNICIV Report in prep, School of Civil andEnvironmental Engineering, The University of New South Wales, Sydney.

Douglas, K.J. and Mostyn, G. (1999) Strength of large rock masses field verification. Rock Mechanicsfor Industry, Proceedings of the American Rock Mechanics Association, Vail, Colorado pp. 271-276.Balkema, Rotterdam.

Duran, A. and Douglas, K. (2000) Experience with empirical rock slope design. GEOENG2000,Melbourne, Australia.

Evans, B., Fredrich, J.T. and Wong, T. (1990) The brittle-ductile transition in rocks: recent experimentaland theoretical progress. The Brittle-Ductile Transition in Rocks, Geophysical Monograph 56, Duba etal. Eds, pp. 1-20, American Geophysical Union, Washington, D.C.

Hoek, E. (1983) Strength of jointed rock masses. Geotechnique, Vol. 33 (3), pp. 187-223.Hoek, E. (1997) Reliability of Hoek-Brown estimates of rock mass properties and their impact on design.

Technical Note. International Journal of Rock Mechanics and Mining Sciences.Hoek, E. and Brown, E.T. (1980a) Empirical strength criterion for rock masses. A.S.C.E., Journal of the

Geotechnical Engineering Division, Vol. 106 (GT9), pp. 1013-1035.Hoek, E. and Brown, E.T. (1980b) Underground Excavations in Rock. The Institution of Mining and

Metallurgy, London.Hoek, E. and Brown, E.T. (1988) The Hoek-Brown failure criterion - a 1988 update. Proceedings of the

15th Canadian Rock Mechanics Symposium, Toronto.Hoek, E. and Brown, E.T. (1997) Practical estimates of rock mass strength. International Journal of Rock

Mechanics and Mining Sciences. Vol. 34(8), pp. 1165-1186.Hoek, E., Kaiser, P.K. and Bawden, W.F. (1995) Support of Underground Excavations in Hard Rock.

A.A.Balkema,Hoek, E., Wood, D. and Shah, S. (1992) A modified Hoek-Brown failure criterion for jointed rock masses.

Eurock '92 pp. 209-213.Hudson, J.A. and Harrison, J.P. (1997) Engineering Rock Mechanics: An Introduction to the Principles.

Pergamon.Jaeger, J.C. and Cook, N.G.W. (1979) Fundamentals of Rock Mechanics. Chapman & Hall.Johnston, I.W. (1985a) Comparison of two strength criteria for intact rock. A.S.C.E., Journal of the

Geotechnical Engineering Division, Vol. 111 (12), pp. 1449-1454.Johnston, I.W. (1985b) Strength of intact geomechanical materials. A.S.C.E., Journal of the Geotechnical

Engineering Division, Vol. 111 (6), pp. 730-749.Lade, P.V. (1993) Rock strength criteria: the theories and the evidence. In Comprehensive Rock

Engineering (E. T. Brown ed.), pp. 255-284, Pergamon, Oxford, New York.Liao, Q.H. and Hencher, S.R. (1997) The effect of discontinuity orientation and spacing on failure

mechanisms in rock slopes results from systematic numerical modelling. CIM Vancouver 97,Vancouver, Canadian Institute of Mining.

Medhurst, T.P. (1996) Estimation of the in situ strength and deformability of coal for engineering design.PhD thesis, University of Queensland.

Mogi, K. (1966) Pressure dependance of rock strength and transition from brittle fracture to ductile flow.Bulletin of the Earthquake Research Institute, University of Tokyo, Vol. 44 (1), pp. 215-232.

Mostyn, G., M.D. Helgstedt & K.J. Douglas (1997) Towards field bounds on rock mass failure criteria.International Journal of Rock Mechanics and Mining Sciences Vol. 34 (3-4): Paper No. 208.

Ramamurthy, T. and Arora, V.K. (1994) Strength predictions for jointed rocks in confined and unconfinedstates. International Journal for Rock Mechanics, Mining Sciences and Geomechanical Abstracts, Vol.31 (1), pp. 9-22.

Shah, S. (1992) A Study of the Behaviour of Jointed Rock Masses. PhD, University of Toronto.

Shah, S. and Hoek, E. (1992) Simplex reflection analysis of laboratory strength data. CanadianGeotechnical Journal, Vol. 29, pp. 278-287.

Sheorey, P.R. (1997) Empirical Rock Failure Criteria. A.A.Balkema.Yudhbir, Lemanza, W. and Prinzl, F. (1983) An empirical failure criterion for rock masses. Proceedings 5th

Congress for I.S.R.M., Melbourne.