An empirical failure criterion for rocks and jointed rock masses

Engineering Geology, 26 (1989) 141-159 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

141

AN EMPIRICAL FAILURE CRITERION FOR ROCKS AND JOINTED ROCK M A S S E S

P.R. SHEOREY 1, A.K. BISWAS 1 and V.D. CHOUBEY 2 ~Central Mining Research Station, Barwa Road, Dhanbad-826001 (India) 2Indian School of Mines, Dhanbad-826004 (India) (Received March 23, 1987; accepted after revision September 3, 1987)

ABSTRACT Sheorey, P.R., Biswas, A.K. and Choubey, V.D., 1989. An empirical failure criterion for rocks and jointed rock masses. Eng. Geol., 26:141 159. A new triaxial strength equation is proposed as fitting the available laboratory triaxial test data. It is shown how the five triaxial parameters - - unconfined compressive, tensile and shear strengths, coefficient of friction and exponent in the triaxial equation - - can be inter-related by three equations. A relation between intact rock strength/rock mass strength ratio and Barton's Rock Quality (1974) is next developed based on geomechanical data for four longwall panels and in situ coal strength data. This relation and the coefficient of internal friction from Barton's classification are used in the three inter-relations to evaluate the rest of the triaxial parameters for rock masses. Performance of the failure criterion is studied using practical cases of rock structures in coal and coal measure strata, viz. five longwall caving panels and four coal pillars.

INTRODUCTION M a n y e m p i r i c a l f a i l u r e c r i t e r i a for r o c k s h a v e b e e n f o r m u l a t e d i n t h e p a s t i n a n a t t e m p t to o v e r c o m e s o m e of t h e s h o r t c o m i n g s of t h e o r e t i c a l c r i t e r i a . O n e s u c h i m p o r t a n t c r i t e r i o n ( H o e k a n d B r o w n , 1980), b y n o w m u c h r e f e r r e d , w a s f o r m u l a t e d for j o i n t e d r o c k m a s s e s , s i n c e o b v i o u s l y it is n o t e n o u g h to k n o w o n l y t h e b e h a v i o u r of t h e i n t a c t l a b o r a t o r y s p e c i m e n for d i r e c t a p p l i c a t i o n to rock structures. F r o m a s t u d y of s e v e r a l t r i a x i a l d a t a , as o b t a i n e d f r o m t e s t s o n c o a l m e a s u r e s a n d as c o l l e c t e d for o t h e r r o c k s f r o m a v a i l a b l e l i t e r a t u r e , a n e w f a i l u r e e q u a t i o n is p r o p o s e d i n t h i s p a p e r as t h e b e s t fit. I t is t h e n s h o w n h o w t h i s r e l a t i o n m a y be m o d i f i e d for j o i n t e d r o c k m a s s e s , w i t h t h e h e l p of r o c k s t r u c t u r e s in coal measures. I t m u s t be m e n t i o n e d a t t h e o u t s e t t h a t t h e f a i l u r e c r i t e r i o n i n t h i s p a p e r r e l a t e s o n l y to b r i t t l e f a i l u r e , t h e l i m i t o f b r i t t l e n e s s b e i n g a p p r o x i m a t e l y d e f i n e d b y M o g i (1966) as a 1 = 3.4(r 3, a 1 a n d a3 b e i n g t h e p r i n c i p a l s t r e s s e s .

0013-7952/89/$03.50

© 1989 Elsevier Science Publishers B.V.

142

EARLIER EMPIRICAL CRITERIA

Several empirical equations are available for describing the triaxial behaviour of intact specimens:

gl =6~+63+a63 b 61 = 6~ + a63 h a l = 6o + aa3

(Hobbs, 1964)

(Murrel, 1965; Mogi, 1966) (Bodonyi, 1970)

(61 - a 3 ) = a (a 1+63) h 61 : 6 3 + j m 6 c 6

3 + s6 2

61/6c=a+b (63/6¢)~

(Franklin, 1971) (Hoek and Brown, 1980) (Yudhbir et al., 1983)

(1) (2) (3) (4) (5) (6)

in which 61, 63 are the major and minor principal stresses, a c is the unconfined compressive strength and a, b, m, s and ~ are empirical constants depending on the rock type. The first four were proposed for intact rock but can perhaps be modified to predict rock mass failure, provided the various empirical constants are suitably adjusted corresponding to the degree of jointedness. The last two criteria were given for jointed rock masses and include modifying the empirical constants (m, s and a, b) corresponding to RMR, Bieniawski's Rock Mass Rating (1976) or the Rock Quality Q of Bart on et al. (1974). The first two equations give, since the exponent b < 1,

da3/~3 =0 which means that the slope of the 61 - a3 curve becomes 90 ° and t hat it does not exist in the negative (tensile) quadrant. This also means t h a t the angle of internal friction (¢p) will be 90 °. These discrepancies preclude the adoption of these two equations as general failure criteria. The extensive curve-fitting done by Franklin and also by Hoek and Brown for triaxial test data of different rock types indicates t hat triaxial behaviour is best generalised as non-linear. The third equation, which is the Coulomb - Navier criterion, is thus not sufficiently accurate. The second drawback with a linear equation is t h a t its intercept on the negative a3-axis does not give the tensile strength (Jaegar and Cook, 1976). The fourth relation, proposed by Franklin, is difficult to handle in practice because a 1 cannot be obtained algebraically for a given value of 63 and numerical solution has to be resorted to. Since the last two criteria have been offered for jointed rock masses it was decided to study them in some detail. Triaxial test data on 8 Indian coals (Das and Sheorey, 1986) were used for obtaining best fit regressions of eqn. (5) to give the results of Table 1. The tensile strength 6, was obtained using 6c

6, = ~ ( m - ~

+ 4s)

143 TABLE 1 Triaxial test results from 8 coals for eqn. (5) Seam

Tested

Estimated values

ac (MPa) 12 17 top 15 top Kargali Uchitdih Jambad top Singhpur mid. Kenda

14.3 15.6 17.8 27.8 35.3 40.3 42.8 43.7

a, (MPa)

at (MPa)

ac/a ,

37.2 42.7 41.5 21.5 60.5 67.7 56.0 57.8

7.0 10.4 8.2 1.9 18.8 14.9 16.2 10.2

5.3 4.1 5.1 11.3 3.2 4.5 3.5 5.7

where s = 1 for intact rock specimens. It is observed t hat the unconfined compressive strength estimated from the regressions is significantly higher t h a n the tested value in all but one case. Also, the a¢/at ratio is much on the lower side than is seen in coals, again excepting one coal. Hoek and Brown have also given graphical relations for adjusting the constants m and s with respect to RMR or Q. These relations are such t hat the angle of internal friction increases with a fall in RMR or Q, a fact which is just the opposite of both of these engineering classifications. Equation (6) was proposed by Yudhbir et al. after tests on G y p s u m - c e l i t e water mixtures. The constant a has been related to the degree of jointedness and varies between 0 and 1. The other constant depends on rock type, ranges between 2 and 5 and is independent of jointedness. The exponent a is proposed to be constant irrespective of rock type or jointing. Thus the position of the a l - a 3 curve is controlled along the ordinate (al) by the value of a, but its slope for a given rock type, remains the same for any Rock Mass Rating since a is constant. In addition, this equation does not exist in the tensile quadrant. AN EQUATION FOR INTACT ROCK

As the first step for developing a rock mass failure criterion, it was necessary to collect and analyse as much triaxial test data as possible. For this purpose three coal measure rocks (sandstone, shale and shaly sandstone) were tested in a Hoek and Fr a nkl i n triaxial cell. In addition to this, data for eight coals were already available as mentioned. L i t e r a t u r e on this topic was then searched for gathering further information. In all 23 sets of testing data could be collected. The following general equation was then tried for least-squares curve fitting. a l=a c (l+aa3)

b

The condition (a3)¢1=0 = - a t gives a = 1/a t. Hence o', = o ' c ( 1 + a 3 ~ b

at,/

(7)

144

This equation shows that -

\do-3/,.3 = 0

-

o

(8)

Ot

Since eqn. (7) is not convertible to linear for regression analysis, curve fitting was done by an interative procedure. The results for the 23 sets are given in Table 2. The index of determination (r 2) was calculated from r 2 = 1 - ~ [y - f(x)12 ~ y 2 (~y)2 n

where

y n

f(x)

=observed value of a 1 = n u m b e r of data pairs = r i g h t hand side of eqn.7.

With reference to Table 2, special mention must be made of the triaxial results of Donath (1964) obtained for studying the influence of orientation of bedding planes on triaxial strength. These data are included merely to improve the generality of the new failure equation by taking as many data as possible, without assigning any special significance to joint orientation as such. The values of shear strength (~) and angle of internal friction (~0) were determined from Mohr circles drawn directly from the best fit equations, instead of from the test data, to give the mean envelope accurately. These are also given in Table 2. The following points may be observed from this table: (a) The values of r 2 indicate significant correlation in all cases. (b) The tested and estimated values of a~ are quite close. (c) The a¢/at ratio ranges between 3.7 and 27.8 which agrees fairly well with experimental observations (Vutukuri et al., 1978). RELATIONS BETWEEN TRIAXIAL PARAMETERS

It has been shown elsewhere (Sheorey et al., 1986) that the five triaxial parameters ac, a,, b, ~c and p (= tan ~0) may be inter-related for a given triaxial strength equation. A change in any of these parameters (with RMR, say) must not be made independently of each other and must be consistent with the relations between the parameters. A tentative failure criterion for coal seams was also given based on this principle. Using the same line of reasoning and trying several different relations, those given below were ultimately developed (Figs.1 and 2) a c = 1.8 ~¢(x/~ tt 2 + p)

(9)

a, = 1.67 %(x/f~ p2 _ ~)

(10)

Eqn. (9) is close to the well known Coulomb-Navier relation

145

TABLE 2 P a r a m e t e r s obtained from a n a l y s i s of triaxial data for eqn. (7) Material

Sandstone

Shale Shaley sandstone Marble Granite Slate c

Coal d

Name/ Location

Tested a¢ (MPa)

Estimated a c (MPa)

a t (MPa)

b

r2

r, (MPa)

o

ac/o t

Kusunda

62.0

65.4

5.2

12.6

0.551

0.984

11.3

59

Buchberg a Buchberg a Darley Dale b Kusunda Kusunda

72.5

55.5 56.9 74.3

2.2 2.4 3.1

25.3 24.2 23.8

0.511 0.508 0.488

0.995 0.969 0,993

6.5 7.0 9.6

67 66 65

64.0 70.0

57.5 75.6

9.9 8.3

5.8 9.1

0.708 0.544

0.952 0.946

13.3 15.0

44 51

Carrara a Gotherd a Martensb u r g (0 °) -do-(15 °) -do-(30 ° ) -do-(45 ° ) -do-(60 °) -do-(75 °) -do-(90 ° ) 12 Seam 17 Top 15 Top Kargali Uchitdih J a m b a d top Singhpur mid. Kenda

-

104.5 161.7 102.6

15.6 11.2 11.6

6.7 14.4 8.8

0.628 0.584 0.728

0.995 0.988 0.996

24.0 26.0 19.0

44.5 59 51

14.3 15.6 17.8 27.8 35.3 40.3 42.8

36.1 6.8 26.9 62.3 110.1 183.2 15.1 16.9 17.2 28.8 36.9 41.8 49.4

6.3 1.7 7.0 16.7 25.6 31.3 0.6 0.6 0.6 5.0 1.4 1.6 7.8

5.7 3.9 3.8 3.7 4.3 5.9 24.0 27.0 27.8 5.7 26.8 26.9 6.4

0.728 0.863 0.828 0.828 0.878 0.828 0.534 0.501 0.506 0.721 0.411 0.436 0.588

0.993 0.991 0.990 0.991 0.989 0.996 0.997 0.993 0.997 0.990 0.993 0.997 0.984

8.4 1.8 7.3 17.2 27.2 40.0 1.9 2.2 2.0 6.6 4.2 5.3 11.5

40 35 35 34 39 44.5 66 67 67 45 68 67 45

43.7

42.4

2.9

14.8

0.516

0.990

6.9

60

aKovari and Tisa (1975). bRamez (1967). CDonath (1964); value in p a r e n t h e s e s is angle b e t w e e n load axis and bedding plane. dDas & S h e o r e y (1986),

Equations (9) and (10) by themselves are not enough as there is no relation as yet in terms of the exponent b. The initial slope of the a l - a 3 curve given by eqn. (8) and the coefficient of friction ~ were found to be well correlated. It was, however, realised that any relation between ba¢/a t and ~ must satisfy certain theoretical conditions. Instead of the above two variables, it was found that the conditions could be more conveniently framed if the variables chosen are ac/a t and b/i~. To arrive at these conditions it was necessary to estimate the values of b/l~ for the two theoretically extreme values of a J a t, viz. unity (frictionless granular materials) and infinity (zero tensile strength). Ifa¢=l,~=0. 0-t

146 200

32 150

o/ /

z -

bo

100

50

i

A

i

1

0

O//

2/.

o

i

i

16

t

t

50 ~c( l*'f~-H2

t.

i

100

*p}.

i

i

0

i

~

i

i

8

J

J

i

16

MPo

Fig.1. Relation between unconfined compressive strength 0¢ shear strength zc and internal friction p. Fig.2. Relation between tensile strength a,, shear strength z¢ and internal friction p.

S o t h e r a d i u s of M o h r ' s c i r c l e s b e c o m e s c o n s t a n t at 0-1

--

0"3

2

_

0"e

_

0"t

2

2

which obviously means that

0-¢~ O"t and 0"1=0"¢+0"3 T h e n t h e f a i l u r e e q u a t i o n for t h i s e x t r e m e c a s e b e c o m e s

0"1=0"e(1+ 0-3~b 0"c} w h i c h c a n be t r u e o n l y if b = 1. H e n c e t h e first c o n d i t i o n w h e n at~o"t---*1 is w r i t t e n as w h e n - -0"c -~1; ~ 0 , 0"t

b~l,

b - ~ . /~

147

The second condition is then directly put down as when -a¢ -

---~ o0; /~ ---~ oO,

at

b -~ 0. P

The plot of Fig.3 shows that the aforementioned two conditions are satisfied by the general equation (with A and B as arbitrary constants)

1)

\at

which, after regression, becomes

20( 1)-°8

b

(11)

We thus get the third relation including the exponent b. C R I T E R I O N FOR ROCK M A S S E S

While considering jointed rock masses, eqn. (7) can be suitably rewritten as

al=a¢,ac(1 + a3 ~b

(12)

GtnO'c/

where

a¢ = unconfined intact rock strength Gcn = Gcj/G c O'tn = O'tj/G c

1.5

O

10

o

.zJ:a.

0-5

0

o

*

i

,

!

*

,

i

i

l

10

*

J

*

l

20

b/# r a t i o

vs. s t r e n g t h ratio

,

30

O"t

Fig.3. V a r i a t i o n of

*

a=/a~.

148

The rock mass strengths acj and atj are thus normalised with respect to the intact rock compressive strength.

Choice of a classification index Since the laboratory compressive strength is included in eqn. (12) it will be necessary to use a "strength-free" classification index to avoid double weightage to the intact rock strength. In Bieniawski's classification this presents some difficulty. In the Q - - system of Barton et al. this is achieved quite simply by putting the stress reduction factor SRF = 1 in the equation for Q

Q= R Q D . g~. Jw J~ J~ 8 R F in which

(13)

RQD = rock quality designation J,1 Jr J~ Jw

= j o i n t set number = joint roughness number = joint alteration number = water reduction number.

The relation between RMR and Q is given by Bieniawski (1976) as

R M R = 9 In Q + 44 For the two extreme possible values of Q, 0.001 and 1000, the corresponding values of R M R are - 18 and 106, which shows t hat the Q-system covers a wider range of rock classes. This was the second, though less important, reason for preferring the index Q.

Estimation of rock mass strength The real difficulty in framing a rock mass failure criterion is to estimate, with a reasonable degree of confidence, the normalised rock mass strength ac, or at,. Very little experimental evidence is available for this purpose. Kidybinski (1984) has proposed in situ-intact strength ratios (at.) for different core lengths (Table 3). This table also gives the corresponding RQD values estimated from the following relation between joint frequency and RQD (Farmer, 1985): Joint frequency (m 1)

RQD (%)

1

90-100

1 5 5 10 10 20 > 20

75-90 50-75 25 50 0-25

Classification and strength data for several longwall extraction panels are available from Indian Coalfields (Sarkar and Singh, 1985 and Sheorey, 1984).

149

TABLE 3 R e l a t i o n b e t w e e n a~,, a v e r a g e core l e n g t h a n d RQD Average core l e n g t h

Joint frequency

(m)

(m ')

1.5 1 1.5 0.5 1 0.1 0.5 <0.1

1 1 1 2 2-10 >10

RQD (°/o)

90 90 80 50 <50

100 100 90 80

0.9 0.8 0.6 0.4 <0.2

Four of these are chosen for their widely ranging RQD and Q values. Table 4 gives the Q values for SRF= 1 and the corresponding estimates of a~ obtained from Table 3. We thus get an approximate relation between a¢. and Q (SRF= 1) which is shown in Fig.4. The curve representing this relation is drawn by hand to pass approximately through the middle of the data. As a check on this relationship, help may be taken of the large amount of in situ strength data on Indian Coals (Sheorey et al., 1982). The in situ-laboratory strength ratio for Indian coals has been found to range from 0.11 to 0.45. This ratio is expected to be somewhat higher than the true a¢., because it is calculated from the in situ strength of 30-cm cubes. This ratio is therefore assigned the symbol a'¢,. The upper bound value of Q for coal seams may be approximately obtained by choosing the following values of classification indices from Barton's tables:

RQD Jn

Jr J, J~

(3 joint sets viz. bedding planes ÷ two cleat sets) (rough planar joints) (no alteration) (dry condition)

100% 9 1.5 1.0 1.0

Hence Q= 16.7 from eqn. (13) which may be taken to stand for the maximum of acn'=0.45. The lowest value of acn'=0.11 is from Jitpur colliery (XIV seam, TABLE 4 L o n g w a l l case s t u d i e s for e s t i m a t i n g ac, (see Table 6)

1. 2. 3. 4.

Colliery

RQD (%)

Q = (SRF= 1)

a,,

Dhemo Main Khottadih Moonidih Banki

65 92 46 17

32.5 138 23 8.5

0.4 0.8-0.9 0.2-0.4 < 0.2

150 10

•

Longwol(

o

In s i t u

cose

studies

OB tests

o n 30 cm

coQJ c u b e s

06

s

~S

0'-.

]

0.2

0

J

A

i

I

-2

-3

J

J

L

i

I

J

L

i

L

i

0

--I

i

,

i

i

~

J

i

L

i

i

,

i

h

J

2

[og100 ($RF = I 1

Fig.4. Rock mass compressive strength - - intact strength ratio ac, in relation to Rock Quality Q. J h a r i a coalfield) w h e r e the seam, a t t h e c o m p r e s s i o n t e s t sites, w a s p a r t l y f r a c t u r e d . T h e v a l u e of Q w a s e s t i m a t e d ' a s 10

1

1

. . . . .

Q=121

1

0.83

in w h i c h Jn s t a n d s for t h r e e j o i n t sets + r a n d o m j o i n t s a n d Jr for s m o o t h joints, since t h e r a n d o m j o i n t s w e r e f o u n d to be of this n a t u r e . T w o d a t a points w e r e t h u s p l o t t e d in Fig.4 for t h e s e coal cases. T h e s e t w o p o i n t s are f o u n d to be a b o v e t h e m e a n curve, w h i c h w a s t h e r e f o r e c o n s i d e r e d a c c e p t a b l e . T h e p o i n t for J i t p u r colliery in Fig.4 h a s a b r o k e n line g o i n g u p w a r d s , s h o w i n g t h a t a'c. is u n d e r - e s t i m a t e d as t h e s e a m w a s p a r t l y c r u s h e d a t t h e t e s t site. A b r o k e n c u r v e h a s b e e n d r a w n , w i t h t h e h e l p of t h e s e t w o d a t a p o i n t s for coal, w h i c h follows the s a m e g e n e r a l t r e n d as the c u r v e for ac,.

Estimation of other triaxial parameters B a r t o n et al. p r o p o s e t h e following r e l a t i o n for the a n g l e of i n t e r n a l f r i c t i o n ~b = t a n - ~ J~

J,

J,

or p - j ~

151

The highest value of Jr/Ja in this classification is 4/0.75 (=5.33) and the corresponding highest value of Q is 1000. It is clear that when JdJa is less, Q will acquire a highest value less than 1000. These upper limits of Q were found from = Qm~

Jr× ~

1000 5.33

for any given Jr/J~ ratio. Equations (9)-(11) can be written in normalised form as

(14)

~o.

~'" = 1.8(x/1 + It2 + it ) 1.67~cn at~ - x/1 + it2 + it

(

(15) -0.8

b = 2.6it i f ° ° - 1 \O'tn

(16)

Equation 15 uses the identity 1

Jl+p

2 --p=

x/i- + It2 +it Then for a chosen J,/J~ (= it) ratio, acn was measured from Fig.1 for different Q values such that Q < Qm~,. The other parameters at,, ~cn and b were then calculated from eqns. (14)-(16). Thus we get a separate curve for each value of JdJa in the case of at, and vc, (Fig.5 and 6). Since the ratio atn/a¢, depends only on p (eqn.9 and 10) the exponent b has been plotted against J~/Ja in Fig.7. More curves for at, and ~ . may be constructed for any JdJ~ values, if required, from Fig.1 and eqns. (14) and (15). Figure 7 shows that

Jr

b-~0 as ~-~(--- #1--*0 in spite of that fact that eqn. (11) fulfills the condition b ~ l as it--*0 This inconsistent behaviour (shown by the broken line) occurs because the curvilinear form of the failure criterion is retained for all values of b. This is clearly impossible and hence b must be assigned the value of 1.0 for tt = 0. This will make the failure criterion identical with the Coulomb-Navier criterion for this extreme case. This has been done by extending the curve to the maximum value of b-- 1, disregarding the broken curve. Figures 4-7 thus give the failure criterion for jointed rock masses. Besides geomechanical characteristics, the only parameter required to be determined is the unconfined compressive strength. No triaxial tests are necessary for using this criterion.

D

E

~

"fen

©

0

g

b7

-n

-

In o

rl~

o

153 1.0

0.8 i/

0.5

04.

0,2

0

I

2

3

I.

Jr IJa

Fig.7. Variation of exponent b with internal friction p ( = Jr/Ja). M o h r envelope f o r the c r i t e r i o n

It is s o m e t i m e s more c o n v e n i e n t to h a v e a failure criterion in the form of a Mohr envelope, e.g. in elasto-plastic numerical stress analyses. If we assume the general form of the e n v e l o p e as T=~¢ ( I + B a) c it must satisfy the conditions (~)~=

0 =

-

~,

o-=0

where v, a are the shear and normal stresses. These c o n d i t i o n s determine the two c o n s t a n t s B and C as B-

1 O-t

C - #°t ~c

154

The failure envelope then becomes

Z=Zc t l + - - O"~flat/tc or in normalised form

ff ~ttatn/rcn Z=ZcnO" c

1+

--

O'tnO'-c]

where # =

dr/da.

CASE S T U D I E S

Application of the failure criterion, eqn. (12) has been done to five cases of longwall panels and four of coal mine pillars. A simple elastic finite element analysis has been used for this purpose, mainly on the grounds of cost and speed and as such the results from these analyses are by no means rigorous. The intention was merely to see, as a first approximation, how far the criterion succeeded in practical cases.

Longwall caving panels In four of these panels, which have already been mentioned earlier, the distance of face travel required to c a u s e caving of the immediate roof is known. In the fifth case, the roof could not cave and the panel was subsequently abandoned. The longwall openings were thus r e c t a n g u l a r in plan. The dimensions of these panels and other relevant data are given in Table 5. Points to be noted about these cases are as below: (a) The thickness of the immediate roof is roughly t aken as five times the seam height, as is conventional in coal mine strata control practice. (b) Since the panels are r e c t a n g u l a r in plan, normally it would be necessary to use the three-dimensional finite element method to obtain the principal

TABLE 5 P a r t i c u l a r s of longwall case histories

1. 2. 3. 4. 5.

Colliery

Seam height (m)

Depth (m)

Caved area (m x m)

Equivalent span (m)

Dhemo M a i n Khottadih Moonidih Banki Bijuri

3.0 2.4 1.8 2.2 1.8

130 280 220 130 65

120 x 45 120 x 79 110 x 38 120 x 14 85 x 60 (no fall)

43 64 37 14 47

155

stresses. This was not necessary, however, as equivalent spans for the corresponding plane strain situation were available, as published elsewhere (Sheorey, 1984). These spans, given in Table 5, were used for constructing twodimensional finite element models. The weighted averages of RQD and compressive strength for the immediate roof together with other geomechanical parameters are given in Table 6. Rather than treating each rock bed individually, the average values were assigned to the entire immediate roof to obtain the average estimate of Q, which was then used in the failure criterion. The values of the various triaxial parameters estimated from Figs.4-7 are shown in Table 7. The failure criterion was directly applied to the principal stresses obtained from the finite element runs (although this is not strictly correct) for plotting the safety factor contours in the roof. The line of safety f a c t o r = 1.0 indicates the limit of failure initiation in the roof, as seen in Fig.8. It may be noted that reasonable roof caving is predicted in the finite element models. The anomalous behaviour in case 5 (Bijuri colliery, where no caving had taken place) can perhaps be attributed to an incorrect estimate of the intact rock strength a¢. The foregoing case studies have the following common features: (a) The rock mass may be classed as "hard, horizontally layered". (b) The openings are very oblong rectangular. (c) The roof failed mostly in tension. TABLE 6 Geomechanical data for immediate roofs of longwall panels Colliery

RQD

J.

Jr

Ja

Jw

Q

(%) 1. 2. 3. 4. 5.

Dhemo M a i n Khottadih Moonidih Banki Bijuri (no caving)

65 92 46 17 94

3 1 3 3 1

1.5 1.5 1.5 1.5 1.5

1 1 1 1 1

1 1 1 1 1

TABLE 7 Triaxial parameters of eqn. (12) for five longwall roofs

1. 2. 3. 4. 5.

ac (MPa)

Colliery

a¢ (MPa)

a¢,

a,,

b

Dhemo M a i n Khottadih Moonidih Banki Bijuri

86 27 67 15 14

0.32 0.57 0.28 0.20 0.57

0.027 0.048 0.024 0.017 0.048

0.605 0.605 0.605 0.605 0.605

32.5 138 23 8.5 141

86 27 67 15 14

156

.~ - :... ::...." t;',"i:., , " i : :.'~ ":i" .:' i:-

Dhemo

Main

Moonidih

d/'-

I ,:.

--4/---

I

':i:..:-"..-:',";~'.,:

~.-. '...:.: .:: : : . : . L.::::'." ".:;. : " . : ' " - . : : : "

•

...

..'.'

Banki

Khottadih

0 - 4 p

~

lore

_

[:.?:] Failed zone m

Coal seam Sandstone

[ t~--~ Shale l ~

5haly sandstone

rJi]3 Sandy shale Bijuri

Fig.8. Failure initiation zones in the roofs of five longwall panels.

Coal mine pillars The failure criterion has thus been tested for basically o n l y one type of rock formation and mostly in tension. It was, therefore, decided to apply it to coal mine pillars w h i c h typify failure in compression. Two collapsed and two stable cases of square pillars are given in Table 8 and the relevant g e o m e c h a n i c a l parameters in Table 9. Since the R Q D of these seams was not available, Q was determined from a'¢, (the in situ-lab strength ratio) using the broken curve of Fig.4. at. and b were estimated for Jr/Ja = 1.5 (for rough planar joints w i t h o u t alteration).

157 TABLE 8 Cases of collapsed and stable mine pillars Colliery (seam)

Depth to seam floor (m)

Seam height (m)

Pillar width (m)

Roadway width (m)

51

1.8

3.0

3.6

129

3.6

7.2

5.7

37.5

5.1

7.2

3.9

9

2.4

2.7

5.4

(a) Collapsed cases

Ramnagar (Begunia) Birsinghpur (Johilla top) (b) Stable cases

Lachipur (Lower Kajora) Rana 6 pit (Rana)

TABLE 9 Geomechanical parameters for the coal seams of Table 8

1. 2. 3. 4.

Colliery (seam)

Strength (MPa) In situ* Lab (at)**

a',n

Q

ac.

at,

b

Ramnagar (Begunia) Birsinghpur (Johilla top) Lachipur (Lower Kajora) Rana 6 Pit (Rana)

7.6

25.5

0.30

4.6

0.16

0.014

0.605

8.8

38.6

0.23

2.3

0.13

0.011

0.605

7.2

33.4

0.21

1.9

0.12

0.011

0,605

6.9

50.3

0.14

0.7

0.09

0.008

0.605

*Strength of 30cm cubes. **Strength of 2.5cm cubes.

Three-dimensional finite element runs for these cases were made with the h e l p o f t h e B M I N E S c o m p u t e r p r o g r a m o f t h e U.S. B u r e a u o f M i n e s . S i n c e t h e pillars could be treated as a more or less uniform array, only an eighth of the pillar, formed due to three orthogonal planes of symmetry passing through the pillar centre, was simulated for the run in each case. Using the data of Table 9 the failure criterion was next applied to the principal stresses. The minimum e l e m e n t s a f e t y f a c t o r i n t h e s t a b l e c a s e s 3 a n d 4 w a s f o u n d t o b e 2.0 a n d 3.2 respectively. F o r t h e c o l l a p s e d c a s e s 1 a n d 2, t h e f a i l u r e i n i t i a t i o n z o n e s b o u n d e d b y t h e contours of unit safety factor are shown in middle vertical section and sectional plan in Fig.9. Since coal pillars undergo "stiff" failure these are not

158 L....................................................

I

.tt C

°

F~ . . . . . . . . . . . . .

b

F~

. . . . . . . . . .

,,. .."~:.-:-:.-qa

I... r--~::f: i.

L

/W

-lb -_qc

E u~

co

i

"(:''!":

~....--b-b-:"__..:~".. • :, :..

Birsing_hl)ur

Romnaga!

Fig.9. Failure initiation zones in two cases of collapsed coal mine pillars.

the final failed zones, which in fact would be greater. Failure of these pillars is thus adequately indicated. DISCUSSION AND CONCLUSION The new triaxial failure criterion appears to work reasonably for both l a b o r ato r y triaxial data and rock masses. The failure initiation zones plotted for the different case studies should be treated only as a first approximation. For realistic application, the criterion must be employed iteratively as is commonly done in elasto-plastic finite element analyses. The criterion, when applied to jointed rock masses, implicitly assumes (as does the Hoek and Brown criterion) t h a t the joints are arbitrarily oriented. The internal friction values of Barton's classification are given for individual joints, but they have been t aken for the rock mass as a whole. This is not always correct but was done to keep the criterion essentially simple. All the case studies are related to coal seams and coal measures only. It will be interesting to see the results of applying the criterion to different rock formations. For problems in which joint orientation is of special importance, e.g. slope stability, it is tentatively suggested t h a t the stress-free Q value be subject to some adjustments. Since Barton's classification does not consider joint orien-

159

tation, this can perhaps be achieved by converting Q into the equivalent RMR, adjusting it according to Bieniawski's tables and converting it back to a new Q value. These adjustments, of course, must be tested against case studies. ACKNOWLEDGEMENTS

The authors are grateful to the Director, CMRS for permission to publish this paper. They also thank the Department of Coal (Ministry of Energy, New Delhi) and the Central Mine Planning and Design Institute, Ranchi for sponsoring this research. The opinions expressed are those of the authors and not necessarily of the institutions to which they belong. REFERENCES Barton, N., Lien, R. and Lunde, J., 1974. Engineering classification of rock masses for the design of tunnel support. Rock Mech., 6: 189-236. Bieniawski, Z.T., 1976. Rock mass classifications in rock engineering. In: Z.T. Bieniawski (Editor), Symposium on Exploration for Rock Engineering. Balkema, Johannesburg, 1: 97-106. Bodonyi, J., 1970. Laboratory tests of certain rocks under axially symmetrical loading conditions. In: 2nd ISRM Int. Congress on Rock Mechanics. Belgrade, Paper 2-17. Das, M.N. and Sheorey, P.R., 1986. Triaxial strength behaviour of some Indian coals. J. Mines Metals Fuels, 34: 118-122. Donath, F.A., 1964. Strength variations and deformational behaviour in anisotropic rock. In: W.R. Judd (Editor), State of Stress in the Earth's Crust. Elsevier, New York, 281-297. Franklin, J.A., 1971. Triaxial strength of rock materials. Rock Mech., 3:86 98. Hobbs, D.W., 1964. The strength and s t r e s s - s t r a i n characteristics of coal in triaxial compression. J. Geol., 72: 214-231. Hoek, E. and Brown, E.T., 1980. Underground Excavations in Rock. The Institution of Mining and Metallurgy, London. Jaegar, J.C. and Cook, N.G.W., 1976. Fundamentals of Rock Mechanics. Chapman and Hall, London. Kidybinski, A., 1984. State-of-art on rock mechanics research from the aspect of optimal exploitation of solid minerals. Review of recent proceedings of the Int. Bureau of Strata Mechanics. In: 12th World Mining Congress, New Delhi, 1: Paper 2,01. Kovari, K. and Tisa, A., 1975. Multiple failure state and strain controlled triaxial tests. Rock Mech., 7:17 33. Mogi, K., 1966. Pressure dependence of rock strength and transition from brittle fracture to ductile flow. Bull. Earthquake Res. Inst. Japan, 44: 215-232. Murrel, S.A.F., 1965. The effect of triaxial stress systems on the strength of rock at atmospheric temperature. Int. J. Rock Mech. Min. Sci., 3: 11-43. Ramez, M.R.H., 1967. Fractures and the strength of a sandstone under triaxial compression. Int. J. Rock Mech. Min. Sci., 4: 257-268. Sarkar, S.K. and Singh, B., 1985. Longwall mining in India. Mrs. S. Sarkar, Calcutta. Sheorey, P.R., Raju, N.M., Singh, B., Ghose, A.K. and Singh, R.D., 1982. Analysis of strata control practices in bord and pillar workings in India. In: 7th Int. Strata Control Confi, Liege. Sheorey, P.R., 1984. Use of rock classification to estimate root caving span in oblong workings. Int. J. Min. Eng., 2: 133-140. Sheorey, P.R., Das, M.N., Bordia, S.K. and Singh, B., 1986. Pillar strength approaches based on a new failure criterion for coal seams. Int. J. Min. Geol. Eng., 4: 273-290. Vutukuri, V., Lama, R.D. and Saluja, S.S., 1974. Handbook on Mechanical Properties of Rocks. Vol.1, Trans Tech, Clausthal. Yudhbir, Lemanza, W. and Prinzl, F., 1983. An empirical failure criterion for rock masses. In: 5th ISRM Int. Congress on Rock Mechanics, Balkema, Melbourne, 1: B1-B9.