Strength of Intact Rock and Rock Masses

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 short commentary on various methods of fitting failure criteria to experimental data follows, it is demonstrated that the method of fitting the criterion to the test data has a major effect on the estimates obtained of the material properties. The results of a recent analysis of a large data base of test results is then presented. presented. This demonstrates that there are inadequacies in the Hoek-Brown empirical failure criterion as currently proposed for intact rock rock and, by inference, inference, as extended to rock mass strength. strength. The parameters m i and s c  are not material properties if the exponent is fixed at 0.5. Published values of m i can be misleading as m i does not appear to be related to rock type. The Hoek-Brown criterion can be generalised by allowing the exponent to vary. This change results in a better model of the experimental data. Analysis of individual data sets indicates that the exponent, a , is a function of m i which is, in turn, closely related to the ratio of s c/s t t . A regression analyis of the entire data base provides a model to allow the triaxial strength of an intact rock to  be estimated estimat ed from reliable reliabl e measurement measurem ent of its uniaxial uniaxi al tensile tensil e and compressive compres sive strengths. strengt hs. The method  proposed  propos ed is the most accurate of those methods method s that do not require triaxial triaxia l testing testin g and is adequate adequa te for   preliminary analysis. Analysis is presented that shows applying the Hoek-Brown criterion to most rocks results in systematic errors. Simple relationships for triaxial strength that are adequate for slope design are  presented. The second part of this paper presents a discussion of the application of the Hoek-Brown criterion to estimating the shear strength of rock masses for slopes. The current methodology methodolo gy is presented. Problems with the estimation of GSI   and scale dependency for slopes are discussed. A critical review of the  parameters  s, mb and a   is presented with a view to improving the application of the criterion to slopes. A new approach to parameter parameter estimation is introduced. 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 slope engineering, therefore it is important that it is validated in both applications. This paper presents a detailed analysis of the application of the criterion to intact rock and suggests modifications that provide improved  predicti  pred ictions ons of triaxial tria xial strength stre ngth based base d on easily easi ly measured meas ured material mate rial properti prop erties. es. Modifica Modi fication tionss to the application 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 these are well presented in most rock mechanics textbooks, with good examples being Jaeger and Cook (1976) and Brady and Brown (1993). (1993). Most practical engineering engineering relies on a linear linear MohrÕs envelope being being fitted to experimental data or to the relevant portion of a theoretical or empirical empirical criterion. Notwithstanding this it is  becoming increasingly common for computer software to be able to deal directly with one or more non linear  criteria. Thus while the limits and pitfalls of linearisation are well understood, it is now important to assess the accuracy of non linear criteria. 1

Garry Mostyn, Pells Sullivan & Meynink, 11/10 East Pde, Eastwood, NSW, 2122, Australia, [email protected] 2 Kurt Douglas, School of Civil & Environmental Engineering, The University of NSW, Sydney, 2052, Australia, [email protected]

Equation 1 presents a generalised criterion where a1, a2, etc are material properties. Figure 1 presents a generalised failure criterion in the s 1-s 3 plane and shows the locations of the s c, s  Bt , s ut  ut  and s   pt . (1)

0 = fn(s 1, s 2 , s 3 , a1 , a2 ,....)

It is well known that the theoretical criteria do not accurately predict the failure strength of  rock and often rely on parameters that are difficult to measure. For this reason many criteria have been developed that seek to capture the important elements of measured rock strengths or seek to modify theoretical approaches to accommodate experimental  S  i    g m evidence, several of these empirical criteria are  a 1   /   listed in Hudson and Harrison (1997) and  S  i    g m  a Sheorey (1997). Most of these share a  C  reasonably similar structure and all have Sigma 3 / Sigma C elements that are likely to fail at the extremes. Given the variability typical of rock test results it is likely that any one criteria is as suitable Figure 1 : Generalised failure criterion overall as any of the alternatives. The HoekBrown empirical failure criterion (Hoek & Brown, 1980) was developed in the early 1980s for intact rock and rock masses, it has been subject to continual refinement refinement for rock masses. For intact rock its form has has not changed and is given in Equation 2. 3

2

1

s 

c

s 

Brazillian t

s 

0

uniaxial t

s 

pure t

-1

-0.25

æ m s  ö s 1 = s 3 + s c çç i 3 + 1÷÷ è s c ø

0. 00

0.25

0.50

0. 5

(2)

In common with most of the empirical failure criteria, the Hoek-Brown criterion is formulated in terms of  s 1 and s 3  and is independent of s 2. The authors do not consider this a major impediment impediment for practical practical  purposes. It is the authorsÕ experience that the Hoek-Brown criterion is virtually the only non linear criterion now used by practicing engineers and and is now in almost universal use. Further it forms the basis for extension into rock mass strength. It thus has been adopted as the basis of examination for the rest of this paper. Further  for these reasons it is important to establish that the criterion does accurately represent actual intact rock   behaviour. LABORATORY TEST DATABASE FOR INTACT ROCK 

The authors have assembled a large data base of test results on a wide variety of rocks; tests include uniaxial tensile strength, Brazilian tensile strength, unconfined compressive strength and triaxial compression 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 test information 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 results forming 485 sets and is being continuously extended. 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?

Equation 1 presents a generalised criterion where a1, a2, etc are material properties. Figure 1 presents a generalised failure criterion in the s 1-s 3 plane and shows the locations of the s c, s  Bt , s ut  ut  and s   pt . (1)

0 = fn(s 1, s 2 , s 3 , a1 , a2 ,....)

It is well known that the theoretical criteria do not accurately predict the failure strength of  rock and often rely on parameters that are difficult to measure. For this reason many criteria have been developed that seek to capture the important elements of measured rock strengths or seek to modify theoretical approaches to accommodate experimental  S  i    g m evidence, several of these empirical criteria are  a 1   /   listed in Hudson and Harrison (1997) and  S  i    g m  a Sheorey (1997). Most of these share a  C  reasonably similar structure and all have Sigma 3 / Sigma C elements that are likely to fail at the extremes. Given the variability typical of rock test results it is likely that any one criteria is as suitable Figure 1 : Generalised failure criterion overall as any of the alternatives. The HoekBrown empirical failure criterion (Hoek & Brown, 1980) was developed in the early 1980s for intact rock and rock masses, it has been subject to continual refinement refinement for rock masses. For intact rock its form has has not changed and is given in Equation 2. 3

2

1

s 

c

s 

Brazillian t

s 

0

uniaxial t

s 

pure t

-1

-0.25

æ m s  ö s 1 = s 3 + s c çç i 3 + 1÷÷ è s c ø

0. 00

0.25

0.50

0. 5

(2)

In common with most of the empirical failure criteria, the Hoek-Brown criterion is formulated in terms of  s 1 and s 3  and is independent of s 2. The authors do not consider this a major impediment impediment for practical practical  purposes. It is the authorsÕ experience that the Hoek-Brown criterion is virtually the only non linear criterion now used by practicing engineers and and is now in almost universal use. Further it forms the basis for extension into rock mass strength. It thus has been adopted as the basis of examination for the rest of this paper. Further  for these reasons it is important to establish that the criterion does accurately represent actual intact rock   behaviour. LABORATORY TEST DATABASE FOR INTACT ROCK 

The authors have assembled a large data base of test results on a wide variety of rocks; tests include uniaxial tensile strength, Brazilian tensile strength, unconfined compressive strength and triaxial compression 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 test information 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 results forming 485 sets and is being continuously extended. 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?

The authors have devoted a moderate portion of this paper to these questions as it is of little value undertaking a comprehensive test program program or detailed analysis of the results if the methodology is flawed. flawed. In fact, parameters determined can vary from very conservative to quite the opposite, both situations have consequences in analysis and design. Turning to the first question, the data included is that described described in the previous section. section. It has been common practice for researchers fitting empirical failure criterion to intact rock to exclude results thought to exhibit ductile behaviour; this approach has been adopted by Hoek and Brown (1981), Shah (1992), Johnston (1985) and Sheorey (1997). (1997). In general these researchers have adopted the brittle-ductile transition suggested  by Mogi (1966) of s 1=3.4 s 3. The exclusion of ductile data would be appropriate if (i) only brittle behaviour  was of interest, (ii) the boundary was clear and (iii) the failure criterion was disjoint across across the transition. In the case of a criterion based solely on GriffithÕs theory, exclusion of ductile tests results would be appropriate. 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 wide range of stresses. Further this same research shows that the failure envelope is not necessarily necessarily disjoint at the  brittle-ductile  brittle-ducti le transition. transition . Thus an appropriate criterion can model strength on both sides of the transition. transition . The authors have included as many test results as possible and only excluded results for which there is significant doubt as to their accuracy. There are several forms of the Hoek-Brown criterion that can be adopted for data analysis, these include equation 2 and:

æ m s  ö s 1 = s 3 + s c çç i 3 + 1÷÷ è s c ø s 1 = s 3

0.5

ü for s 3 > - s c mi ï ý for s 3 £ - s c mi ïþ

(s 1 - s 3 )2 = mis 3s c + s c2

(

log(s 1 - s 3 ) = 0.5 log mis 3s c + s c2

(3)

(4)

)

(5)

Equation 2 is strictly the Hoek-Brown criterion, but is undefined for s 3  less than -s c/m i. Equation 3 ensures that the criterion is defined over the full range of s 3. Equations 4 and 5 are linearisations linearisation s of the criterion. The impact of adopting these these different forms 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 sound statistical reasons to so do. Shah (1992) suggests that the simplex simplex method with the function (observed(observed2  predicted)  is a better method than least squares. In fact, the method presented by Shah is least squares, the simplex is purely a numerical method to optimise some function, in this case minimising the sum of squared differences (ie errors). The authors have verified that the resulting parameter estimates estimates are the same as those from other robust least squares procedures. If the departure of the measured s 1 from the predicted s 1  (ie the error) is normally distributed with a variance that is independent of the predictor variables (here s 3), then the predictions obtained with least squares, either with a simplex or otherwise, will be uniform minimum variance unbiased estimators; this is highly desirable. But consideration of data with multiple measurements measurements of s ut  ut  or s  B  B t   will indicate that straight least squares is not appropriate for fitting the Hoek-Brown criterion. criterion. Consider an experimental program with multiple measurements of s ut  ut , it is clear that if a failure criterion is to be fitted to the test data it is desirable that the estimated tensile strength should be the average of these measurements (ie the fitted curved should pass through the middle of the measured values). Equation 2 is not defined for measured values of s ut  ut  less than the fitted value (ie larger tensile strengths) and this forces many fitting methods to fit the maximum measured (ie most negative) tensile strength as the estimated tensile strength. Equation 3 overcomes this problem, but reference to Figure 1 shows that the slope of the equation to the left of the estimated s ut  ut  is much less than that to the right; the figure is drawn for an m i of 8 and the slope to the right is much steeper for higher mi. Given that a general least squares approach assesses the error as the observed s 1 (ie zero) minus the predicted s 1, then data a given distance to the right of the estimated s ut  ut  will have a much larger ÒerrorÓ 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 s ut   and high m i, ie the opposite effect to adopting equation 2 A resolution of the above problem comes about by recognising that in a uniaxial tensile strength test, the controlled variable is s 1  and the measured variable is s 3 , thus the real error is observed s 3  minus the  predicted s 3. But this error is scaled in s 3  and needs to be adjusted if it is to have equal status with measurements in s 1. The authors suggest that scaling by m i  is a convenient and accurate approach. Given this they recommend a least squares procedure where the error is defined as:

(measured s 1 - predicted s 1 ) (measured s 3 - predicted s 3 )´ mi

for s 1 > -3s 3 ü ý for s 1 £ -3s 3 þ

(6)

This has been found to provide very good fits for a wide variety of data. It is the authorsÕ 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 of  test data. Table 1 and Figure 2 show the results of analysis of a simulated test program with results generated for a material with a Hoek-Brown failure criterion, s c and m i are both normally distributed with mean/standard deviation of 10/2 MPa and 12/2 respectively. Results generated were 10 uniaxial tensile strength tests, 20 unconfined compressive strength tests, and 4 each triaxial strength tests at confining  pressures of 1, 2, 5, 10, 20, 40 and 80 MPa. Thus there were 58 data points in all, simulating a very comprehensive test program from which it should be possible to determine accurate estimates of material  properties. Table 1 : Results of different regression methods on artificial data Case

Equation

Fitting method

1 2 3

Actual data Normal equation Extended equation

2 3

4

Extended equation

3

5 6 7 8 9

Adopting known s c and normal equation Excluding s t  results and normal equation Excluding s c & s t  results and normal equation Stress difference squared Stress difference squared and known s c

Least squares Least squares Modified least squares, eqn 6

s c (MPa)

mi 

r   (%)

58 58 58

10.0 14.9 8.46

12.0 7.75 15.6

na 97.88 99.12

58

10.7

12.0

99.00

2

Least squares

58

na

5.21

91.69

2

Least squares

48

9.53

13.7

99.06

2

Least squares

28

6.20

21.4

98.80

4

Least squares

58

3.97

35.2

95.53

4

Least squares

58

na

13.8

95.47

10

Stress difference squared

4

11

Logarithms Logarithms and excluding s t  results

5

Least sum of  absolute differences Least squares

5

Least squares

12

2

Number

58

9.18

15.4

95.45

58

8.09

4.12

55.97

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 two exceptions, the methods provide a reasonable fit for the majority of the data. But reference to Figure 2b shows that most methods provide a very poor fit to the data at low stresses, that is over the stress range of  interest in slope analysis. The following comments are offered on the various analyses undertaken, listed in the same order as in Table 1.

1. The generated data, the authors consider that this is a 250 reasonable representation of a  Artificial data 10.012.0 Normal eqn & LS14.97.75 comprehensive test program in a E xt en de d e qn & LS 8 .4 61 5. 5 Ext eqn & m od LS 1 0. 712. 0 Fix UCS & LS 10.05.21 200 moderately variable unit. Excl Sc or St & LS 6 .1921. 4 DS^2 & LS 3.9735.2 Log & LS 8.094.12 2. The strict application of  least squares to Equation 2, ie 150 the usual form of the HoekBrown criterion, results in the 100 uppermost curve in Figure 2b.  S  i    g m The criterion cannot be  a Excl St & LS 9.5213.7 50 1  DS^2 with UCS fixed 10.013.8 evaluated for s 3  less than the  (   M DS^2 & Least abs sum 9.1815.4 Log with excl St 9.6712.2 P   a estimated tensile strength. This  )   0 results in large estimates of s ut  -10 10 30 50 70 and s c and thus a low mi. From Sigma 3 (MPa) 5 to 80 MPa the curve passes through the middle of the data. Below 1 MPa the estimated 40  Artificial data 10.012.0 strength is nearly 50% higher  Normal eqn & LS14.97.75 E xt en de d e qn & LS 8 .4 61 5. 5 than the true strength even Ext eqn & m od LS 1 0. 712. 0 Fix UCS & LS 10.05.21 2 Excl Sc or St & LS 6 .1921. 4 30 though the regression r  is DS^2 & LS 3.9735.2 Log & LS 8.094.12 nearly 98%. This problem could be partially fixed by 20 including only the average measured tensile strength in the  S  analysis but this ignores i    g m 10  a considerable readily obtained Excl St & LS 9.5213.7 1  DS^2 with UCS fixed 10.013.8  (   M DS^2 & Least abs sum 9.1815.4 and economic data and Log with excl St 9.6712.2 P   a  )   disguises the true variability. 0 3. Least squares applied to -2 0 2 4 6 Equation 3 results in vastly Sigma 3 (MPa) improved parameter estimation Figure 2 : Fits to artificial data (a) full range (b) low stress range  but the lower slope to the left of  the estimated s ut  produces a low estimate of s ut   and thus somewhat low estimated s c and high estimated m i. A good fit overall with the highest r 2, but approximately 15% underestimate of true strength for low s 3. 4. Modified least squares, Equation 6, applied to Equation 3 results in accurate estimation of the  parameters and does so in almost all circumstances. The fact that r 2 is slightly less than for method 3 is a necessary consequence of the treatment of variability of the measured tensile strengths. 5. Least squares applied to Equation 2 with s c  fixed at the average of the test results. It might be thought that knowing one property should help with estimating a second unknown property, this is not the case here. The problem in 2 above is now magnified to produce almost the worst fit imaginable. It shows that an r 2 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, results in a good fit. Again the problem is that good economic data is ignored and the fit at low stress will be more variable. 7. Least squares applied to either Equation 2 or 3 with both the tensile and unconfined compression test results excluded. In this case more than half the data is ignored and, in the present case, the fit at low s 3 is more than 30% out. This is a random error and the fit could be low or high. The problem with this approach is 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 to data and estimating s c and m i. This method virtually minimises ÒerrorsÓ to the fourth power, hence the lowish r 2, and dramatically overweights the larger values of s 1. Errors in parameter estimates are not UCS

mi  

Not shown

UCS

mi  

Not shown

 predictable, but in this example, estimated s c and mi are 40% and 300% of the true values respectively, even though the corrupted r 2 is over 95%. Over most of the range of the test results it is a very good fit but not over that portion of interest in slope design. It is not recommended. 9. As for 8 above but with s c fixed at the mean value, in contrast to 5 above this results in a good fit across the range but relies on a good estimate of s c and increased faith that this accurately represents triaxial  behaviour. 10. Least sum of absolute differences applied to Equation 4. This in large measure compensates for the overweighting of large s 1 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. As for Equation 2 this equation is not defined for s 3 less than s  pt . This method has major problems fitting any data which includes a moderate spread of tensile testing. 12. Least squares applied to Equation 5 with the tensile strength test results excluded. A robust method weighted to low stress results and good for slope analysis but unable to take advantage of economic and readily available data. From Table 1 it can be seen that r 2 is not a useful indicator of accuracy of estimates of the parameters and that these estimates can vary widely depending on the method of analysis. Methods with r 2 in excess of 95% and that model the data very well over most of the range have estimates of s c varying from 3.97 to 14.9 MPa and m i  from 7.75 to 35.2 and this is for artificial data that follows exactly the criterion with only test variability. Thus many of these methods are very poor estimators of strength in the low stress region that is of interest in slope analysis. HOEK-BROWN CRITERION FOR INTACT ROCK 

Modified least squares, Equation 6, was combined with the extended formulation of the Hoek-Brown criterion, Equation 3, to estimate s c and m i  for all test data in the data base. Discussion in the previous section indicates that the fit is poorly controlled at low stresses for sets with little data, particularly s c and s t . Small changes in the data can lead to wildly varying estimates of both s c and mi, in general with s c becoming very small and m i very high but with the fit being almost identical over the range of the test results. In fact for many data sets s c and m i  are not independent but s c® 0 as m i®¥ . The best solution to this issue is to  place plausibility limits on the parameters. A number of limits were considered and the following ones adopted: · As all the test results were taken from materials described as rock, s c was limited to be not less than 1 MPa. · Published values of mi fall in the range of 4 to 33 (Hoek & Brown, 1998). As will be discussed later  m i is very closely related to the ratio - s c/s ut , reference to the figures in Lade (1993) indicates that this ratio varies from less than 2 to over 50. This 40 limits mi to the range 1 to 50. Further m i is related to the i   30  t   angle of friction at  e  s   t   s 3=0 (ie f 0 ), which is of great interest in slope analysis. It 20 was considered that f 0  should be limited to the range of 15 to f   10 r   o 65°, which for the m m f   i    t   Hoek-Brown  t   i   n  g criterion further  H B  0 limits mi to the range  e 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  q  u 1.4 to 40.  a m  from literature, m  t   ipub i    o n The process was  , m completed for 475 Figure 3 : mi from literature against mi from test results and Hoek-Brown Equation i   

i 

data sets involving 3779 test results. The results of the analysis are provided in Figures 3 to 6. Figure 3 shows a Òbox and whiskerÓ plot of the values of m i estimated from the data, m itest , against the values of mi  provided in Hoek & Brown (1998) and Hoek et al (1995), m ipub. Several such figures are presented in this  paper, the whiskers show the range of test results, the box shows the upper and lower quartiles and the bar  the median value. Also shown on this figure is a linear regression between published m ipub  and m itest  weighted for the number of data points supporting each estimate. The regression equation is: mitest  = 7.58 + 0.441 mipub

(7)

This is a very poor relation, r 2=16.4%, between m i determined on the basis of actual testing and that obtained from the literature. Figure 4 presents m itest   against rock type 40 ordered in increasing mipub, it can be seen that it is difficult to ascribe a single or even small 30 range to m i on the basis of rock type. It should  be remembered th at 20 50% of the test data falls outside the range m indicated by the box for  each rock type, thus for  10 example 50% of the values for sandstone fall  below 11 or above 19, and for granite below 0 19 or above 31.  S  C T   S   a  o  u Figure 5 presents the  C  A B  N D  S  S  l   l   a f   M f    t    G  S  a P  D h  h  C   p  a T A  F   c  a P  R  t   D D  S  G o  Q u E  G i   Q n  G  S  l    a  a i    e A  o A   C  D h  h  B  l    e i   A   s  r  L  i   l   r  s c  determined from  G r   a  o  G e  a r  n M  t    t    g l    Q  y  i   h   e l    c  r  r  i   l    u i    a r  n  a  e  e  e L   G  t    s  i   W l   i    o k  r  n  a  o  s   b   b  i   l   i    u m  t    o l   h  r  y  l   a  c  g i    a  a  c  e r   u  p l   b  d  m t   r   t    e  u  a  a  e  a  t   r   s  i   n  o  o  b   d  r   e l    t   r   e  a  o  c  n  y  a  o  o  e  o h  r  n i   h  i   r  l    y  m  e  a  d  e  g r  t   p r  n  s  t    c  l    e  a i   r   d  i    t    t    s   e fitting the Hoek-Brown m  u  y  i   i   e i   t   z   t    s  n  a l   t   i   i   o i   h   t    t   n  s  s  t   n  s   e  t    t   n  o  e l   i   d   a  p  t    s  z  f    e  e  c   o  y  s  r  w i    t    t    e  e f    e z  i   i    s   t    d  d   o  s  i   mi from r  results  t    b equation 1   a  t   n  t   Figure 4 t   : Rock  o type against test i    oHoek-Brown  e  t    d  t    e and  o  a  t   i   n n  o i   o equation against the s c  o  e  o n  p  c  o l   h  k  n determined from s t  testing or, at least, as reported in the literature from 1000 700 which the data was obtained. Again several figures in this paper  400 are presented in this style. The upper and lower dashed lines 100 represent 1.5 and 0.67 times the 70 reported s c values. Further, the 40 symbols represent the number of  test results used to determine the fit, a small cross is 4 or less data 10  points, a small circle is 7 or less, 7  U large circle is 12 or less and a  C  4  S  f   square is more than 12 data points. r   o m It can be seen that virtually all the H B   e data lies in a very narrow band, 1  q  u 1 4 7 10 40 70 100 400 700 1000  a  t   such that the fitted s c is quite close i    o n Unconfined compressive strength (MPa)  (   M to the reported s c. P  Figure 6 is a similar   a )   Figure 5 : Unconfined compressive strength against that  presentation to Figure 5 except that  predicted by the Hoek-Brown equation i     t     e  s   t   

it presents fitted tensile strength 100.0 70.0 versus reported tensile strength. It 40.0 can be seen that the fitting method adopted provides a very good estimate of s t   for those data sets 10.0 which do have reported tensile 7.0 strengths. Most of the other  4.0 methods fail for such data, so much so that often practitioners are 1.0 forced to ignore the valuable 0.7 information available from 0.4 inexpensive tensile testing. This is T   e  particularly a problem as such data n  s  i   l    e forms a good control on the failure 0.1  s   t   r   0.1  0.7  4.0 10.0 70.0  e envelope over the low stress range n  0.4  1.0  7.0 40.0 100.0  g  t   h  (Lade, 1993). f   Tensile strength (MPa) r   o m In summary the proposed H B  method results in good fits of the  e Figure 6 : Uniaxial tensile strength against that predicted by the  q  u Hoek-Brown criterion to the data  a Hoek-Brown equation  t   i    o n and, in particular, results in good  (   M P  fits in the low stress region. It appears  athat published values of the parameter m i might be quite misleading  )   as 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 behaviour  observed in soils and soft rocks. Fixing the exponent at a half means that at best the criterion is a poor model of soft rocks. This is not surprising as it was developed for brittle rocks but it is a limitation which is often overlooked by practitioners who apply it to all rocks. Further it is a severe limitation on the extension of the criterion to rock mass strength. · Lade (1993) in comparing the theories and the evidence regarding rock strength criteria finds that an appropriate criterion should have three independent characteristics Ð the opening angle, the curvature and the tensile strength. The fixed exponent on the Hoek-Brown criterion limits it to modelling only two of these characteristics. In fact as often used, m i  is varied to model the curvature over the stress range of the test results and neither the opening angle nor the tensile strength are modelled. Lade also states that it may be an advantage to include the tensile strength in determination of material parameters as it stabilises the fit at low stresses. This is particularly important for slope analysis. If the exponent, a , is allowed to vary the Hoek-Brown criterion can model widely varying curvatures and opening angles. It is also able to include an accurate representation of the tensile strength. The authors have applied this ÒgeneralisedÓ 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: a 

æ m s  ö s 1 = s 3 + s c çç i 3 + 1÷÷ è s c ø s 1 = s 3

ü for s 3 > - s c mi ï ý for s 3 £ - s c mi ïþ

(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, s c>1, m i in the range 1 to 50, and a  mi in the range 0.7 to 20 (this is the equivalent limit on f 0). In addition, a  is limited to the range 0.2 to 1.

Allowing a    to vary 50  provides the ability to obtain a much better fit over the low stress range 40 which is of greatest interest in slope analysis. The results of the 30 analysis are presented in a series of figures. Figure 7 20  presents a box and whisker  f    plot of mi determined from  or m m f   i   the data against the  t    t   10 i   n  g  published values of mi.  e  u Again it can be seen that  q  a  t   there is little relationship  oi   0 n 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  b e t w e e n the two. m  from literature Likewise, there was found to be no relationship Figure 7 : mi from literature against mi from test results and generalised equation  between mi and rock type. The slope of the generalised criterion at s 3=0 is 1+a  mi, and is related to f 0 by: i   

i 

( (

f 0 = 2 atan (1 + a  mi )

0.5

)- 45)

(9)

25 If a classification of  samples, say by m i or  rock type, is predictive 20 of the triaxial envelope at low stresses then it will be apparent on a 15  p l o t of that classification against a  m i . Figures 8 and 9 10 f   r   o  present plots of a  mi m f   i    t   against published m i  t   i   n 5 and rock t y p e   p al   h  g  e  q  a  u respectively. From *    m a  t   i    o n Figure 8 it can be seen 0 that published mi is not 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 a good predictor of the m  from literature triaxial envelope at a  low s t r e s s . Figure 8 : mi from literature against mi from test results and generalised equation Examination of Figure 9 shows that there is a weak correlation of rock type with a  m i in that fine grained rocks tend to have the lowest values, medium to coarse grained higher and rocks with tightly interlocked crystals the highest. The authors do not believe that the relationship is strong enough to be used predictively. Figure 10 presents the s c obtained from fitting the generalised equation against the reported s c. It can be seen that virtually all the data lies in a very narrow band, such that the fitted s c is quite close to the reported s c. 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 the uniaxial tensile strength s ut  is bound as: i   

i 

-

s c

mi

< s ut  £

- s c (mi + 1)

(10)

20

15

i   

A  l      p h   a *    m

10

5

0  S  C T   S   a  o  u  C  A B  N D  S  S  l   l   a f   M f    a  t    G  C   S  P  D h  h  F   c   a T A  P R p  t   D D  S  G o  Q u E  G i   Q n  G  S   a l    a  a i   A   C i   D h  h  B  l    o  e A   s  r  L  i   l   r   G r   a  e  o  G r  M e i   e n A   t    t    g l    Q  y  i   h  l    c  r  r  i   l    u  a r   e  a n  a  e  e L   G  t    s  i   W l   i    o k  o m b  t    o h  r  e  a  t   n  a  e i   m t   l   u  a  a i   o r   y   t   i   u  a  a  c  e r   u  p l   b  r  s   d  r  s   c  g r   d  i   n l    a l    o  t    b  r   e  b  l    e  a  o  c  n  y  a  o  o r  a  d  n  t   i   h  e  a i   i   e  e n  a m y  r  m h   d  e  c  l    t   r  o r  n  s  t    s  l   e  u  e  e i   s   t   p i   t   i   z   g r   s  t    y  i   l   t   i   i   o h   t   n  s  s  t    e  t    t   y  n  s  n l   i   d   a  p  t   z  o  e  t    e f    c   o r  w s  i    t    t    t    e  e i   b  f    e z  i   a  i    s   t    d   o  s   t    d   o  t   i   r   t   1   a n  t    o generalised  e  t   Figure 9n i  : Rock type against test a results and mi from o  d   t   e i   i   o n  o equation  o  e  o n  p  c  o l   h  k  n

1000

100.0

700

70.0

400

40.0

10.0

100

7.0

70

4.0

40

1.0

10

0.7

7

 U  C   S  f   r   o m f   i    t    t   i   n  g  e  q  u  a  t   i    o n  (   M P   a  )  

0.4 4

1 1

4

7

10

40

70

100

Unconfined compressive strength (MPa)

400

T   e n  s  i   l    e  s   t   r   e n 700 1000  g  t   h  f   r   o m H B   e  q  u  a  t   i    o n  (   M P   a  )  

Figure 10 : Unconfined compressive strength against that predicted by generalised equation

0.1  0.1

 0.7  0.4

 4.0  1.0

10.0  7.0

70.0 40.0

100.0

Tensile strength (MPa)

Figure 11 : Uniaxial tensile strength against that  predicted by generalised equation

Table 2 presents the errors involved in Table 2 : Error in approximating s ut  as -s c/(mi+1) adopting Рsc  /( m i+1) as s t . For simplicity the lower bound has been adopted in plotting a mi Error (%) Figures 6 and 11, the error in doing this is quite 1 All 0 small. It should be noted that it is likely that 0.8 All <7.6 many of the reported s t  are likely to be Brazilian 0.5 1 19 tensile strengths. The authors are attempting to 0.5 >8 <10 resolve as many of these as possible in 0.4 1 23 continuing work. 0.4 >9 <10 The fit in Figure 11 is extremely good. Figures 10 and 11 provide considerable confidence that the fitted curves provide a very good model of triaxial strength at low stresses. In both cases the unexplained variance of the generalised fits is about half that of the Hoek-Brown fits.

Figure 12 presents a 1.2  plot of a   against m i as f 0 determined for each data 35 45 55 65 set from the generalised 1.0 criterion. Also shown on the figure are hyperbolae 0.8 showing lines of constant a  m i, ie f 0, for 15 ° to 65°. Inspection of the figure 0.6 shows that the constraints on m i, a   and a  mi did not   pA  l    h   a often limit the regression 0.4  procedure. An interesting and useful observation from 0.2 Figure 12 is that there 25 f 15 0 appears to be a 0.0 relationship between a  0 10 20 30 40 50 and m i . Such a mi  relationship is derived in the next section and is Figure 12 : a  against mi shown on Figure 12. It is often thought that the curvature of the strength envelope, ie a   against m i  in the current context, should be greater for strong rocks than weak rocks. The data set and analysis do not support this contention. Figure 13 shows the relationship between a   and m i plotted for the data divided into four categories depending on the s c. It can be seen that the relationship is independent of strength. High strength rocks can have linear flat failure envelopes and low strength rocks can have steep curved envelopes. Thus s c  is a truly independent  parameter in a rock failure criterion.

1.0

0.5

0.0 0

20

40

0

40

40
UCS<= 40  a l      p h   a

20

1.0

0.5

0.0 0

20

40

0

100
20

40 200
mi 

Figure 13 : a  against mi categorised by s c

A consequence of allowing a   to vary at all, is that a failure envelope with a high f 0 can have a low f   at high stresses and thus failure envelopes for different rocks normalised on s c, can cross at high stresses. This cannot happen with a fixed at 0.5 in which case all envelopes cross only once at s c. Figure 14 shows a family of curves, normalised by s c, for various m i and the a   typical of the relationship shown on Figure 12. It can be seen that the m i equals 40 curve crosses the m i equal 10 and 3 curves at 1.4 and 2.5 times s c respectively. This implies that high frictional strength at low stresses is often associated with low frictional strength at higher stresses. Figure 15 shows some examples of test data that illustrate this  point. A relationship between a   and m i  implies that the triaxial failure envelope can be estimated from s c  and either m i or s t , with a   being determined by the relationship. Thus there are no more parameters to be determined than for the usual Hoek-Brown criterion. The  parameters can be based on simple testing and  provide a more accurate prediction of strength than published m i  values, particularly in the low stress region typical of slopes.

10

8

6

4

mi 

 S  i    g m  a 1   /    S  i    g m  a  C 

40

3 1

10 2

0

-2 -0.5

0.0

0.5

1.0

1.5

2.0

2.5

Sigma 3 / Sigma C

Figure 14 : Family of failure envelopes

20 Set 382 Sandstone 18

Set 305 Granite Set 425 Gabbro

16

14

12

10

8

 S  i    g m  a 1   /    S  i    g m  a  C 

6

4

2

0 -1

0

1

2

3

Sigma 3 / Sigma C

Figure 15 : Results showing failure envelopes crossing

4

5

3.0

GLOBAL PREDICTION

A single equation could be fitted to the entire data base on the following assumptions: · The value of s c obtained from fitting the generalised Hoek-Brown criterion is the best estimate of s c for each data set. · A reasonable estimate of | s t | is obtained for each data set by dividing s c by the value of m i 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 Рsc  /s t  and Equations 3 and 8 can be rewritten as: a 

æ s  ö = + çç1 - 3 ÷÷ s c s c è s t  ø s 3

s 1

s 1 s c

=

æ ç a+b s 3 öçè ÷

æ + çç1 - ÷ s c è s t  ø s 3

(11)

æ æ ööö ç1+ expç æç m0 -s c ö÷ c ÷ ÷ ÷ çç ç s t  ÷ø ÷ø ÷ ÷ èè è øø

(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 a general function that varies from a to b with a midpoint at Рm0 and a variable length of step. These equations can 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 Equation 11, ie a constant exponent, resulted in a  being estimated as 0.439 and an r 2 of 83.5%. This is a reasonable fit when the range of rocks to which it applies is considered. Examination of the residuals reveals that a better  fit will be possible as the residual is a function of mi. This is illustrated on Figure 16, for mi<10 the residuals are positive and increase with s 1. The residuals gradually reduce until for m i  greater than 40 they are  predominantly negative. This is strong evidence that a  is not constant. 10

5

0

-5

-10 0

5

10

15

20

0

5

10

15

20

0

5

(5,10]

<= 5

10

15

20

15

20

(10,20]

10

5

0

-5

R  e  s  i    d   u  a l   f    o r   g l    o  b   a l   r   e  g r   e  s   s  i    o n w i    t   h   c   o n  s   t    a

Number under graph is estimated ratio of  -Sigma c / Sigma t

-10 0

5

10 (20,30]

15

20

0

5

10

15

(30,40]

20

0

5

10 > 40

Sigma 1 / Sigma c

Figure 16 : Residuals for global fit with a  constant against s 3/s c categorised by -s c/s t 

Fitting Equation 12 resulted in the following estimate for the exponent: a  = 0.4032 + 1.08585 (1 + exp(mi 7.455))

(13)

This equation is shown on Figure 12 and models the results of the analysis of the individual data sets very well. This analysis resulted in an r 2 of 94.8%, which is extremely good for such a global fit. The residuals are plotted on Figure 17, there is no or little trend with mi or s 1 and it can be seen that this is a much better fit than Equation 11 and Figure 16. 10

5

0

-5

-10 0

5

10

15

20

0

<= 5

5

10

15

20

0

5

(5,10]

10

15

20

15

20

(10,20]

10

5

0

-5

Number under graph is estimated ratio of 

-Sigma c / Sigma t R  e -10  s  i    d  0 5 10 15 20 0 5 10 15  u  a (20,30] (30,40] l   f    o r  Sigma 1 / Sigma c  g l    o  b   a Figure 17 : Residuals for global fit with variable a  against l   r   e  g r   e  s  Figure 18 shows a three  s  i    o dimensional plot of the failure n w criterion i   described by Equations 12  t   h  and 13, av ie s 1 as a function of s 3 and r  i    /s  ). It can be seen that for  m i (ie Р a s c t   b  l    e m i<8 the failure envelope is close to  a l   linear  ph and then becomes more  a

curved. The ridge at m i  equals 8 is well supported in the data and may reflect ÒmoreÓ or ÒlessÓ than Griffith  behaviour. Figures 19 and 20 show slices through the model for high and low stress ranges respectively with the equation for the midrange of each slice also shown. Thus the upper left subgraph on Figure 19 presents all the s 1 versus s 3 data, normalised by s c, for sets with s c /|s t |<=5 together  with the equation for s c /|s t |=3.

20

0

5

10 > 40

s  /s c categorised by -s c/s t  3

Figure 18 : 3D plot of global fit

Figure 19 provides the data for s 3 up to three times s c and Figure 20 for s 3 to half of s c. It can be seen that the fits are very good. The ridge at high stress and mi=8 are apparent on Figure 19 with uniform behaviour at low stress seen on Figure 20. 15

Number under graph is estimated ratio of  -Sigma c / Sigma t 10

5

0 0

1

2

3

0

1

2

3

0

1

(5,10]

<= 5

2

3

2

3

(10,20]

15

 S  i    g m  a 1   /    S  i    g m  a  c 

10

5

0 0

1

2

3

0

1

(20,30]

2

3

0

1

(30,40]

> 40

Sigma 3 / Sigma c

Figure 19 : s 1/s c with fits for variable a  against s 3/s c categorised by -s c/s t  for high stress

Figure 21 presents a  against m i   including Equation 13 and showing those data for which there

1.2 Data with compressive and tensile strengths Other data 1.0

5

0.8

Number under graph is estimated ratio of  -Sigma c / Sigma t

4

0.6

3

A  l      p h   a

2

0.4

1

0 0.0

0.1

0.2

0.3

0.2 0.4

0.5

0.0

0.1

0.2

0.3

0.4

0.5

<= 5 5

0.1

0.2

0.3

0.4

0.5

(10,20]

0.0 0

10

20

30

4

 S  i    g m  a 1   /    S  i    g m  a  c 

0.0

(5,10]

40

50

mi 

Figure 21 : a  against mi showing cases with measured or reported s 3 and s t 

3

2

1

0 0.0

0.1

0.2

0.3

(20,30]

0.4

0.5

0.0

0.1

0.2

0.3

0.4

(30,40]

0.5

0.0

0.1

0.2

0.3

0.4

0.5

> 40

Sigma 3 / Sigma c

Figure 20 : s 1/s c with fits for variable a  against s 3/s c categorised by -s c/s t  for low stress

is actual, not estimated, values for both s c and s t . It can be seen that these sets are distributed similarly to those for which at least one of these parameters has been estimated by fitting the generalised Hoek-Brown criterion.

COMPARISON OF CRITERIA

A comparison of the various criterion as fitted to the data base is provided in Table 3. The variance explained approximates r 2. As would be expected, the generalised Hoek-Brown criterion provides by far the  best fit, r 2 of 99.5%, as it has three parameters and is fitted to the individual data sets. None the less, the fit obtained is considerably better than the fit obtained by the Hoek-Brown criterion (ie with a   fixed at 0.5) , r 2 of 98.9%. The unexplained variance for the generalised criterion is less than half that of the Hoek-Brown criterion with a  of 0.5. Table 3 : Comparison of predictions Variable/prediction

s1/sc Global regression with a constant Hoek-Brown with published mi Global regression with a variable Hoek-Brown fitted to individual sets Generalised Hoek-Brown fitted to individual sets

Variance

Variance explained

16.93 2.78 2.00 0.846 0.186 0.077

0 83.6 88.2 95.0 98.9 99.5

The above methods compare different ways of fitting triaxial data, ie different criteria applied to actual triaxial data. Table 3 also allows a comparison of three methods of prediction of triaxial strength that are not  based on having actual data but are based on parameters estimated in some other manner. The methods are discussed in the following points: ·  Prediction based on global equation with variable a . This method is based on Equations 12 and 13 and estimates of s c and s t . It has an r 2 of 95.0% when used to predict the test results in the data base. The 15 predictions are illustrated on Figures 19 and 20. 5 accuracy of the Categorised by published based values on global equation with constant a . This method, Categorised based onbyofEquation 11, is the least ·  Prediction published values mi 4 of miis not discussed further. accurate of the10three and 3 based on published values of mi. This method is based on Equation 3 with values of m i ·  Prediction estimated from those widely published in the literature. The method has an r 2 of 88.2% when used to predict 2 the test results5 in the data base. On average this method predicts the strengths well but with considerably more scatter than that from the global equation. Figures 22 and 23 present the data categorised by published 1 mi, these figures are in a similar form and can be compared to Figures 19 and 20. It is clear from the figures 00 m i the m i it 0.4 that at low published triaxial strength is under predicted and 0.5 at3 high published is over predicted. 0.00 0.1 0.2 1 0.3 2 0.4 0.5 3 0.0 0 0.1 10.2 0.3 2 0.4 0.0 0 0.1 1 0.2 0.3 2 0.5 3 In effect what this means is<=<= that triaxial strength is poorly i values and the method is (7,9] (7,9]predicted by published m(9,18] (9,18] 77 15 5  predicting the average strength for all tests. 4

 S  i    g m  a 1   /    S  i    g m  a  c 

10 3

2 5 1

00 0.00

0.1

0.2 1

0.3

(18,19] (18,19]

2 0.4

0.5 3 0.0 0

0.1

10.2

0.3

2 0.4

(19,24] (19,24]

0.53 0.0 0

0.1

1 0.2

0.3 2

0.4

0.5 3

> 24

Sigma 3 / Sigma c

s //s  s   with fits s //s  s   categorised by Figure Figure 22 23 :: s  fits for for published published m  against s  by m  for high low stress stress mii against mii for 11 cc with 33 cc categorised

SYSTEMATIC ERROR IN HOEK-BROWN CRITERION

If a   for a particular rock is not equal to 0.5 then there is a systematic error in fitting the Hoek-Brown criterion to any triaxial test results obtained on that rock. The error is illustrated on Figure 24, two data sets are shown, the upper  one is for s c, m i and a   of 30 MPa, 24 and 0.4 respectively and the lower one for 8 MPa, 5 and 0.8. The different s c were chosen to separate the curves, and the m i  and a  are typical combinations determined in the analysis of the entire data base. The solid lines represent the Hoek-Brown fits to these data. The residuals are shown on the bottom graph, if a   is less than 0.5 then there are negative residuals Figure 24 : Pattern of residuals for  at both the low and high end of the range of s 3 tested with Hoek-Brown fits  positive residuals in the middle range. The sign of the residuals is reversed if a   is greater than 0.5. While the fits in the upper graph might look satisfactory for engineering  purposes, the errors in estimates of s c  and m i   can be significant. Table 4 shows the parameters estimated from fitting the Hoek-Brown criterion to these data, it can be seen that errors in the estimates vary from one half to five times the correct values. Thus the parameters of  this model cannot be considered material properties. These errors are discussed in more detail below. 160

140

120

100

80

60

 S  i     g  m  a  1   (   M P   a   )  

40

20

0

0

5

10

15

20

25

30

35

Sigma 3 (MPa)

20 10

0

-10

R   e   s  i     d   u   a  l     (   M P   a   )  

-20

0

5

10

15

20

25

30

35

Table 4 : Errors in fitting Hoek-Brown criterion to materials with a  ¹ 0.5 Actual parameters 2

Material

s c

mi 

a 

r   (%)

Upper Lower

30.0 8.0

24.0 5.0

0.4 0.8

100.0 100.0

Parameters determined by fitting Hoek-Brown criterion 2

Material

Estimated s c

Estimated mi 

r   (%)

Upper Lower

33.0 3.94

11.7 48.5

99.65 97.47

Consider programs of triaxial testing on rocks with s c 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 rock  slope. Further consider that different test programs are undertaken in which the maximum s 3 is determined  by the capacity of the triaxial apparatus. The following test programs could result: · Program A - 12 stages to a maximum s 3 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 if  the Hoek-Brown criterion is fitted to these test programs. The estimated s c and m i  are given in Table 5. Estimated s c varies from 1.66 to 4.20 MPa and mi from 8.0 to 29.6 with very high r 2.

Table 5 : Variation of s c and mi with s 3max for exact simulated results mi  =

Material (a) s c = 3 MPa, Program

s 3max  (MPa)

A B C D E

10 5 3 1.5 0.7

40 and a  = 0.4

Estimated s c

Stages

r   (%)

12 4.20 11.5 10 3.77 14.2 8 3.47 16.9 6 3.22 20.3 4 3.07 23.4 Material (b) sc = 3 MPa, mi = 4 and a = 0.8

s3max

Program

(MPa)

A B C D E

10 5 3 1.5 0.7

99.43 99.41 99.38 99.52 99.74 2

Stages

Estimated sc

Estimated mi

r  (%)

12 10 8 6 4

1.66 2.23 2.59 2.85 2.96

29.6 17.5 12.6 9.60 8.01

97.96 98.65 98.96 99.46 99.85

40

40

 Artificial data for 

 Artificial data for 

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

 S  i    g m  a 1 

2

Estimated mi 

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

35

35

30

30

25

25

20

20

 S  i    g m  a 1 

15

15

10

10

5

5

0

0 -2

0

2

4

6

8

10

12

-2

0

2

Sigma 3

4

6

8

10

12

Sigma 3

Figure 25 : Hoek-Brown fits to artificial data The dashed lines on Figure 25 show the envelopes fitted to cases A, C and E. It is emphasised that while the upper line for material (a) and the lower line for material (b) (ie Case E) do not look like good fits, they are in fact very good fits for the 4 test results, below s 3 equal 0.7 MPa, that form their basis with r 2 of  99.74% and 99.85% respectively. Such results might erroneously be taken to support the contention that the material was well modelled by the Hoek-Brown criterion. It can be concluded that the estimated parameters are as much a function of the test program as of the material tested. These errors would generally be obscured 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 analysis has assumed different maximum possible s 3. This further illustrates the errors that can occur if a HoekBrown envelope is fitted to material for which a   does not equal 0.5. Depending on the test program,

estimates of s c obtained by fitting the generalised criterion vary from 85.4 to 57.7 MPa and of mi from 6.35 to 13.1, variations of 150% and 205%. If the Hoek-Brown criterion is fitted, the estimates vary from 23.8 to 55.5 MPa (230%) and 31 to 138 (445%). Again the r 2 determined for the fits are very good. Table 6 : Variation of s c and mi with s 3max for data set 434 For generalised Hoek-Brown criterion s 3max  (MPa)

N

Estimated s c

All data 400 200 100

20 16 9 5

85.4 59.2 57.7 64.1

Estimated

mi 

2

Estimated a 

r   (%)

0.75 0.65 0.66 0.87

99.70 99.63 99.50 98.18

6.71 13.1 13.0 6.35

For Hoek-Brown criterion

 S  i    g m  a 1 

2

s 3max  (MPa)

N

Estimated s c

Estimated mi 

r   (%)

All data 400 200 100

20 16 9 5

23.8 35.1 44.9 55.5

138 75.2 49.7 31.0

97.00 98.49 98.20 93.99

2500

2500

2000

2000

1500

1500

 S  i    g m  a 1 

1000

500

1000

500

0

0 0

100

200

300

400

500

600

700

0

100

200

Sigma 3

300

400

500

600

700

Sigma 3

Figure 26 : Hoek-Brown fits to actual data Figure 27 shows the residuals from fitting the Hoek-Brown criterion to data sets with s c less than 20 MPa  plotted against s 3 divided by the maximum test s 3. There are four graphs showing cases where the estimated a   from fitting the generalised criterion is (a) less than 0.4, (b) between 0.4 and 0.6, (c) between 0.6 and 0.8 and (d) greater than 0.8. On each graph the residuals have been fitted with a quadratic relationship. It can be seen that these residuals conform almost perfectly to those predicted on Figure 24. This is very strong

evidence that the Hoek-Brown model is not appropriate. Figure 28 shows the residuals obtained from fitting the generalised Hoek-Brown criterion. It can be seen that these show little or no trend. Data with estimated UCS less than 20 MPa 30

20

10

0

-10

-20

-30 -0.2

0.0

0.2

0.4

0.6

0.8

1.0

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.6

0.8

1.0

 Alpha (.4,.6]

 Alpha <= .4 30

20

10

R  e  s  i    d   u  a l   f   r   o m f   i    t    t   i   n  g H B   e  q  u  a  t   i    o n

0

-10

-20

-30 -0.2

0.0

0.2

0.4

0.6

0.8

1.0

-0.2

0.0

0.2

 Alpha (.6,.8]

0.4

 Alpha > .8

Sigma 3 / Maximum test sigma 3

Figure 27 : Residuals for Hoek-Brown fits for weak rock against s 3/s 3max categorised by a 

Data with estimated UCS less than 20 MPa 30

20

10

0

-10

-20

-30 -0.2

0.0

0.2

0.4

0.6

0.8

1.0

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

0.6

0.8

1.0

 Alpha (.4,.6]

 Alpha <= .4 30

20

10

Figure

R  e  s  i    d   u  a l   f   r   o m f   i    t    t   i   n  g  g  e n  e r   a l    e 28 q  u:  a  t   i    o n

0

-10

-20

-30 -0.2

0.0

0.2

0.4

0.6

0.8

1.0

-0.2

0.0

0.2

 Alpha (.6,.8]

0.4

 Alpha > .8

Sigma 3 / Maximum test sigma 3

Residuals for generalised fits for weak rock against s 3/s 3max categorised by a 

APPLICATION TO SLOPE ENGINEERING

In general the triaxial strength of intact rock is not particularly important in the analysis or design of rock  slopes, even large rock slopes. The maximum depth of the failure surface for even a 500 m high slope is generally only 100 to 150 m deep, and thus the maximum s 3  of interest is around 4 MPa. The relative contribution of the triaxial component of strength for various rocks (ie s c, mi and a ) and overburden stresses is given in Table 7. It can be seen that for high f 0 rocks triaxial strength is generally significant, even for  quite low slopes and high strength, but for most of these situations intact rock strength will not be critical except in forming the basis of rock mass strength. For low f 0  rocks triaxial strength is important for low strength or high stress conditions and it is these situations for which intact strength may be critical in design. Table 7 : Triaxial component of strength

s c

s 3

High f 0 case mi  = 50, a  = 0.4 %

300 100 30 10 3 300 100 30 10 3

4.0 4.0 4.0 4.0 4.0 0.5 0.5 0.5 0.5 0.5

24 59 140 280 570 3.4 9.8 29 70 160

Low f 0 case mi  = 0.8, a  = 0.9 %

2.3 6.9 23 68 230 0.3 0.9 2.9 8.6 29

Table 8 and Figure 29 provide a comparison of different methods of predicting the triaxial strength of low strength rocks at low stress. The predicted strengths are compared with the measured strengths for all cases in the data base for which s c  is less than 20 MPa and s 3  is between 0 and 5 MPa (excluding UCS test results). The variances of the residuals scaled on s c are given in Table 8 and the scaled residuals plotted on Figure 29. In general the order of accuracy of the various prediction methods is the same as discussed for the overall predictions. It is of interest to note that the global generalised equation (r 2 of 88.6%) is almost as accurate a prediction of the triaxial strength of these rocks as that obtained by fitting the Hoek-Brown criterion directly to triaxial test data ( r 2 of 90.3%). This reflects the fact that there is abundant evidence that a   does not equal 0.5 for a large proportion of the rocks tested Ð see Figure 12 and thus there will be systematic errors at low stress. Table 8 : Comparison of predictions for weak rocks at low stress Variable/prediction s 1/s c

Global regression with a   constant Hoek-Brown with published mi Global regression with a  variable Hoek-Brown fitted to individual sets Generalised Hoek-Brown fitted to individual sets

Variance 7.69 1.91 1.99 0.88 0.74 0.12

Variance explained % 0 75.1 74.1 88.6 90.3 98.5

The above situation arises because the range of s 3  over which the tests were performed hardly ever  corresponded to 5 MPa and thus there were almost always systematic errors at the lower stresses tested. It is sometimes argued that the solution to this is to test over a range of s 3 that represents the field conditions, but this 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 to determine the parameters as a function of stress, but this virtually defeats the purpose of adopting a non linear failure envelope and confirms they are not a material property. A useful approximation of the effective stress  parameters, c0  and f 0, at low stress can be obtained in the following manner. Equation 10 can be rearranged to  provide an estimate of mi based on s c and s t as:   10

5

s c s t 

- 1 £ mi <

s c s t 

(14)

In addition, a   can be estimated from Equation 13, f o is given by Equation 9 and: c0 = s c

(2(1 + a m ) ) 0.5

i

0

 G  e n  e r   a l   i    s  e  d H B

-5 0

2

4

Sigma 3 10

(15)

The writers do not argue that this approximation is a substitute for triaxial testing but, in the absence of such testing, the approximation should be more accurate than other methods of estimation such as using s c and  published values of mi. Further it results in a linear failure envelope which is exact at the origin and as such is convenient to use in many slope stability programs.

5

H B

0

-5 0

2

4

Sigma 3 10

5

 G l    o  b  a l  

0

-5 0

2

4

Sigma 3 10

5

0

 G l    o  b  a l   w i    t  h f  i   x  e  d  a l    p h  a

-5 0

2

4

Sigma 3 10

5

0

H B  a n  d  p  u  b l   i    s h  e  d m i  

-5 0

2

4

Sigma 3

Figure 29 : Residuals against s 3 for various fits

FAILURE CRITERIA FOR ROCK MASS

A rock mass criterion should only be used where Òthere are a sufficient number of closely spaced discontinuities that isotropic behaviour involving failure on discontinuities can be assumedÓ (Hoek and Brown, 1997). Such a situation for slopes is illustrated on Figure 30, it should be noted that the concept of closely spaced should be defined in terms of the scale of the failure surface. Slope failures in which the failure surface is entirely through the rock mass are not common. This is due to the low stresses typically acting in a slope. Failure usually requires large scale (relative to the slope in question) features concentrating stresses into regions of weak rock mass. For  example a long vertical joint may lead to stressing of weak  material at the toe of the slope (Figure 31).

Figure 30 : Heavily jointed rock mass

ÔlongÕ subvertical  joint

Rock mass failure at toe of  slope

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 in design increases. Methods used for assessing this shear strength are based on empirical criteria. As a general rule such criteria are based on laboratory scale specimens with very little, and often no, field validation. 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. The modification process has typically been based on model tests, small sample testing and limited experience. The criteria all assume a non-zero unconfined compressive strength, s cm, and hence tensile strength, s tm, for  the rock mass. These criteria would therefore be expected to overpredict the strength for poor quality rock  masses at the low stresses common to failure surfaces in slopes.

a 

æ s  ö = am + bçç 3 ÷÷ s c è s c ø s 1

æ s  ö s 1 = s 3 + ams 3 ç cm ÷ ç s  ÷ è 3 ø æ s  ö s 1 = s cm çç1 + 3 ÷÷ è s tm ø

(16)

bm

(17)

bm

(18)

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

æ s ¢ ö s 1¢ = s 3¢ + s c çç mb 3 + s ÷÷ è s c ø

a

(19)

Estimating the parameters in the Hoek-Brown criterion was very difficult, thus correlations with rock  mass 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 m i and mb are intact and mass material constants; a and  s are constants that depend on the rock mass characteristics; and s c is the uniaxial compressive strength of the intact rock.

æ GSI  - 100 ö = expç ÷ mi 28 è ø

mb

(20)

for GSI >25: æ GSI  - 100 ö ÷ 9 è ø

 s = expç a

(21)

= 0. 5

(22)

=0

(23)

for GSI <25:  s

a = 0.65 -

GSI 

200

(24)

Hoek (1997) provides Figure 32 to determine the GSI   directly. Hoek et al (1995) say that the GSI  may also be calculated using BieniawskiÕs (1976 and 1989) rock mass rating ( RMR ), GSI  RMR, or BartonÕs (1974) Q-system, GSI Q.

Figure 32 : Estimation of GSI (Hoek, 1997) DISCUSSION OF THE HOEK-BROWN CRITERION FOR USE WITH SLOPES Calculation of GSI

GSI  RMR and GS I Q are derived from the rating parameters for several rock mass properties (Equations 25

and 26). GSI  RMR = å Ratings(intact strength + RQD + defect spacing + defect condition )

(25)

æ RQD  J r  ö ÷÷ + 44 GSI Q = 9 loge çç  J   J  è n aø where

(26)

 J r  = joint roughness number   J n = joint set number   J a = joint alteration number 

 RM R  and the Q -system were developed for underground applications of limited size and thus could be

expected 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 scale slopes.  R Q D   has a heavy weighting in both rating systems (in  particular the Q -system). Since the  RQ D  is based on a fixed cutoff length of 100mm the ability of the  R Q D   to give meaningful information reduces as slopes get larger. For  slopes of several hundred metres the  RQD   (particularly if  estimated from borehole data) has questionable value. On the scale of a large pit slope it is unlikely that all the defects encountered in boreholes would be of significance to the rock  mass stability. The defect spacing parameter suffers from a similar   problem to that of  RQD . The maximum rating is applied for a spacing interval of Ògreater than 3mÓ and Ògreater than 2mÓ for  Bieniawski (1976) and Bieniawski (1989) respectively. The spacing increments given by Bieniawski (1976, 1989) were derived for and on the basis of underground tunnels that were of the order of 10 Ð 20m in span. Where a slope is in the order  of several hundred meters these spacing increments are unlikely to be valid. Figure 33 shows blocks from a 400m high slope failure. A tunnel of 15m span is unlikely to have rock  mass strength problems with a block size as big as those in the Figure 33 : Slope failure block size figure. When assessing the  RQ D   and discontinuity spacing from  boreholes, all discontinuities are included however, for large rock slopes those discontinuities that are large in area will play the major role in the rock mass strength. Without careful orientation techniques, it is difficult to get either the ÔtrueÕ spacing or the number of discontinuity sets. It is well known that intact rock  Smooth exhibits a strength scale effect. This Very rough scale effect exists up to block sizes of at least one metre. Therefore the Defect A  parameter for intact rock strength should be adjusted to account for  scale for large block sizes as in Very rough Figure 33. When assessing the rating  parameters for defect condition,  J r  and  J a, the analyst should take into Defect B account the Ôlarge scaleÕ (i.e. scale of  rock mass) joint characteristics as Smooth & well as those on the small scale. The infilled thickness of joint infilling should be considered proportionately to the length and shape of the Figure 34 : Effect of scale on defect properties discontinuities. Figure 34 shows

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 lower  strength. Figure 32 shows estimates of GSI   provided by Hoek (1997). The main components affecting the strength of the rock mass are covered (ie structure and surface conditions). It is not clear how scale is to be interpreted on this figure. The authors believe that GSI   should be interpreted as being on the scale of the rock mass under assessment. Using judgement the user can estimate the condition of their rock mass at the scale of their slope. For example, a ÔblockyÕ rock mass at a scale of 10m is vastly different to a ÔblockyÕ rock mass at the scale of 500m. Smaller relative block size leads to more freedom for block rotation and a greater chance for mass failure. Liao & Hencher (1997) showed that relative block size was critical in deciding the mode of failure. The smaller the block size (when compared to slope height) the more likely rock 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 GS I  RMR and GSI Q from boreholes should only be used for preliminary strength estimates. It should be remembered that the key to the structure column is Ôdegree of interlockingÕ. The degree of interlocking should be assessed on the scale of the slope under consideration. For example, the rock mass controlling the slopes for an ultimate  pit may be considered interlocked whilst the rock mass may be considered as very well interlocked on the scale of individual benches of the same slope. It should also be remembered that Òwhere block size is of the same order as that of the structure being analysed, the Hoek-Brown criterion should not be used. The stability of the structure should be analysed by considering the behaviour of blocks and wedges defined by intersecting structural featuresÓ (Hoek, 1997). Estimation of parameters from GSI

Figure 35 shows the variation of the Hoek1.0 GSI table max GSI min Brown parameters mb/mi, a and s with GSI  based on Equations 20-24. The figure also indicates the GSI =25 0.8 lower bound for GSI  and the upper bound for GSI  if      a the Hoek (1997) table, Figure 32, is used. mb/mi,     s 0.6 which mainly accounts for friction, varies gradually  , a      i     m from unity as could be expected for a rock mass. 0.4       b The value of  s  (which mainly accounts for      m  s m b /m i cohesion) diminishes rapidly with a reduction in 0.2 GSI   thus, indicating a rapid reduction in compressive strength and an even more rapid 0.0 reduction in tensile strength as the quality of the rock mass decreases. This is as expected. As 0 20 40 60 80 100 rockmass defects become more cohesive it would GSI  be expected that  s would be non-zero so as to avoid Figure 35 : Variation of a, s and mb/mi with GSI zero compressive strength. But GSI   reduces for  increasing cohesion and if  s  is predicted from GSI  then  s approaches zero not a finite value. This may  be why the Hoek-Brown criterion will underpredict the shear strength of clayey bench slopes. It should be remembered at this point that the initial Hoek-Brown criterion was developed for hard rocks and has only recently been accepted for use with very poor quality rock masses by Hoek and Brown (1997). Thus, it could be expected that the experience with using the criterion for poor quality rock masses (particularly for  slopes) would be very limited. The value of a remains relatively constant and has a maximum value of 0.6 (using Figure 32). This is not consistent with what is known about compacted rockfill strength (a material that could represent a lower   bound 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 slight curvature to the failure envelope ( a = 0.90). Where the intact rock approaches that of a soft rock or hard soil the 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  ,

   /

reasonable estimate of 0.4 to 0.9 depending on m i. It could be concluded from this that the Hoek-Brown criterion 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 the other parameters ( mb  and  s ) and therefore a  cannot simply be changed without addressing the other   parameters as well. VALIDATION OF CRITERION

Douglas & Mostyn (1999) show that the  prediction of strength using G S I   derived from Figure 32 (ie scale sensitive) gives  better results compared to using G S I   predicted from  R M R   and the Q-system. Figure 36 shows back analysed shear  strengths for two slope failures and several large scale in-situ shear tests divided by shear strengths estimated from the HoekBrown strength criterion using Figure 32. The Nattai and Katoomba escarpment failures were natural slope failures of  approximately 300m and 200m height respectively. These failures were caused by the opening of joints in strong sandstones and the shearing of weaker rock mass in the underlying claystones (Mostyn et al., 1997).

1.4 1.2      e        l        b      a       t

      t

       /

     u       t        i      s         n        i

      t

1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

s

n (MPa)

Katoomba escarpment failure Aviemore shear tests  Nattai escarpment Failure

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

HOEK-BROWN FOR SLOPES Ð PROPOSED CHANGES

These are proposals for the use of the Hoek-Brown criterion for slopes that the authors are in the process of 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 of  the parameters (Equations 28 to 32). The basic assumption is that the rock mass parameters will be factored versions of the intact parameters developed in the previous section. a b

æ m s  ö s 1 = s 3 + s c çç b * 3 + sb ÷÷ è s c ø *

(28)

mb = Ami

(

 A =  f  GSI f 

(27)

)

(29)

 sb = Bsi

(30)

 B =  f (GSI c )

(31)

(

a  =  f  mb , GSI f 

)

(32)

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

GSI f  is calculated using the original GSI   (Figure 32) by Hoek (1997) and is used to estimate mb/m i. It is  proposed that GSI c  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. GSI c would be used to estimate  sb. The value of  sb  is expected to be zero for non-cohesive, non-interlocked rock mass, unity for 

intact rock and finite for a cohesive rock  mass. Analysis of rockfill materials indicates an  sb  of about 0.002 for Ôblocky mass/good surface qualityÕ rock mass. Figure 38 shows a diagrammatic plot of a  versus m   curves. It is expected that m b/mI will decrease as GSI f  decreases from 100 and hence a   will increase from the intact value and eventually approach 0.9 for very low GSI  or very large slopes. Douglas and Duran (2000) determine large scale slope angles for values of G S I   based on an analysis of failed and stable slopes. These can be used to put bounds on mb  for large scale slopes. Using these, an approximate range of m b   for large rock  masses is one to six.

Decreasing defect cohesion   s   e   c  s ˆ  1   e    i   p    k   c D      o   r e  c      f r    e  a     o s  i      g n     n  g      i    k   c G      S       o  I        l c      r   e    t   n    i  s = 0.002   n    i   e   s   a   e   r   c   e    D

 s = 0

Figure 37 : Proposed method for estimating GSI c

Large scale

@ 0.90

mb = 2

mb = 4

a 

Intact, mi

@ 0.2

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

CURRENT RESULTS

The authors are attempting to develop bounds on rock mass strength for application to slope design. Preliminary bounds scaled by s c  are shown on Figure 39. Intact rock provides an upper bound and curves are shown for high and low f 0 rock. Good quality rock fill can be adopted as a lower bound for the strength of rock masses in which the block strength is not important. The bounds to large slopes derived from Duran and Douglas (2000) also form limits to rock mass strength. The authorsÕ analysis of the Nattai escarpment is shown with respect to these bounds and indicates that this poor quality rock mass is correctly located with respect to the bounds. 3

2

 S  i    g m  a 1   /    S  i    g m  a  c 

1

Intact mi =40 Intact mi =1 Rockfill Large slope mi =4 0 0.0

S l   o   p e  f   a i   l   u r  0.1 e 

Large slope mi =2 0.2

0.3

0.4

0.5

Sigma 3 / Sigma c

Figure 39 : Preliminary bounds for rock mass strength CONCLUSION

The first part of this paper presented an overview of the strength of intact rock. It was demonstrated that the method of fitting the criterion to the test data has a major effect on the estimates obtained of the material  properties. The results of a recent analysis of a large data base of test results demonstrated that there are inadequacies in the Hoek-Brown empirical failure criterion as currently proposed for intact rock and, by inference, as extended to rock mass strength. The parameters m i  and s c  are not material properties if the exponent is fixed at 0.5. Published values of m i can be misleading as m i did not appear to be related to rock  type. The Hoek-Brown criterion can be generalised by allowing the exponent to vary. This change resulted in a better model of the experimental data. Analysis of individual data sets indicated that the exponent, a , is a function of m i which is, in turn, closely related to the ratio of s c/s t . A regression analyis of the entire data  base provided a model to allow the triaxial strength of an intact rock to be estimated from reliable measurement of its uniaxial tensile and compressive strengths. The method proposed is the most accurate of  those methods that do not require triaxial testing and is adequate for preliminary analysis. An analysis was  presented that showed applying the Hoek-Brown criterion to most rocks results in systematic errors. Simple relationships 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 for  slopes. It is concluded that the estimation of the parameters  s, mb and a can be improved for application of  the criterion to slopes. A new approach to parameter estimation was introduced. Work is on going to validate the method. REFERENCES

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