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Support pressure estimation in failed rock conditions
Engineering Geology, 22 (1985) 127--140 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands
127
SUPPORT PRESSURE ESTIMATION IN FAILED ROCK CONDITIONS
P.R. SHEOREY Central Mining Research Station, Barwa Road, Dhanbad--8 26001, Bihar (India) (Received May 29, 1984; accepted after revision January 21, 1985)
ABSTRACT Sheorey, P.R., 1985. Support pressure estimation in failed rock conditions. Eng. Geol., 22: 127--140. Ultimate roof support pressure estimates using a modern rock mass classification (Barton et al., 1974) are compared with the observed values in seven cases of squeezing ground. Since the agreement is not good, a modification of the classification method is suggested. Also given is a tentative approximate guideline for modifying the classification for obtaining wall support pressures. Assuming the failed rock surrounding an underground opening to behave as a Coulomb material, a semi-analytical elastoplastic method is proposed for calculating the ultimate support pressure. An earlier closed-form solution to the problem of stresses in the failed rock annulus assigns constant material properties through the failed rock zone, giving a discontinuity in the tangential stress at the failed--intact rock boundary. The semianalytical method uses a step-wise gradual change of properties, eliminating the discontinuity. This method is shown to give reasonable estimates of support pressure if the peak and residual angles of internal friction are chosen from the classifications of Bieniawski (1976) and of Barton's (1974), respectively. INTRODUCTION A t u n n e l o p e n i n g o r m i n e r o a d w a y at d e p t h is generally s u r r o u n d e d b y a failed r o c k a n n u l u s w h i c h c a n be a s s u m e d t o b e h a v e as a C o u l o m b m a t e r i a l in t h e active R a n k i n e state. S u p p o r t s e r e c t e d in such o p e n i n g s h a v e t o w i t h s t a n d h i g h e r pressures d u e t o r o c k s q u e e z i n g t h a n in i n t a c t o r l o o s e n i n g r o c k c o n d i t i o n s . A n elastoplastic f i n i t e - e l e m e n t s i m u l a t i o n o f failed r o c k condit i o n s m a y p e r h a p s lead t o reliable s u p p o r t p r e s s u r e values, p r o v i d e d t h e i n p u t p r o p e r t i e s are s u f f i c i e n t l y realistic. A l t e r n a t i v e l y , e l a s t o p l a s t i c closedf o r m s o l u t i o n s are available, as r e v i e w e d b y B r o w n et al. ( 1 9 8 3 ) , giving a r e l a t i o n b e t w e e n t h e s u p p o r t p r e s s u r e a n d t h e radius o f t h e failed r o c k zone. Such s o l u t i o n s h a v e a s s u m e d d i f f e r e n t c o n s t i t u t i v e laws f o r b r o k e n r o c k b e h a v i o u r , t h e c h i e f o n e being C o u l o m b ' s law. As o p p o s e d t o t h e s e t w o m a t h e m a t i c a l m e t h o d s , t h e e m p i r i c a l f o r m u l a e p r o p o s e d b y B a r t o n et al. ( 1 9 7 4 ) o f f e r a m u c h s i m p l e r a n d q u i c k e r a p p r o a c h t o t h e p r o b l e m o f support pressure prediction. I f u n i f o r m p r o p e r t i e s are t a k e n t h r o u g h t h e failed r o c k , t h e c l o s e d - f o r m 0013-7952/85/$03.30
© 1985 Elsevier Science Publishers B.V.
128 solution suffers from some fundamental shortcoming, to be mentioned later, and the rock classification m e t h o d of Barton et al., in its present form, gives erroneous estimates of support pressure as seen from the case studies given. An alternative semi-analytical m e t h o d is proposed in this paper, after introducing the earlier solution employing Coulomb's law (Jaegar and Cook, 1976), and a comparison is made with observed support pressures as well as those calculated by Barton's m e t h o d , suggesting a possible modification of the latter method. Brown et al. (1983) have also employed a semi-analytical approach with a step-wise variation of failed rock properties based on the Hoek and Brown (1980) failure criterion, but n o t for estimating support pressure. EARLIER ELASTOPLASTIC APPROACH If a and b are the radii of the opening and the failed rock annulus, respectively, the radial and tangential stresses or and oe in the annulus, a < r < b, are related by the equilibrium equation: dot ~ or--o0 _ 0 dr r
(1)
in polar coordinates (r, 0 ). If the failed material obeys Coulomb's law (2)
ao = C + qor
where C = uniaxial compressive strength of the failed rock, q = [ ( 1 + / / 2 ) 1/2 + p ] 2, = tan ¢, coefficient of internal friction. Substituting eq.2 in eq.1 we have the solution: ar
=
C 1 -- q
~
+
Ar
~-1
(3)
where A is an arbitrary constant. Substituting eq.3 in eq.2, a0
-
C ~ 1 -- q
+ Aqr
q-1
(4)
In the intact zone, r ~ b, the rock is assumed to behave elastically, so the stresses are: f
B
Or = P0 - - ~
(5) B oo = Po + r:
in which B is a n o t h e r integration constant and Po is the hydrostatic virgin stress away from the opening.
129 For determining the constants A and B in eqs.3 and 5 and the radius of the broken rock zone b, three conditions are available: atr=a,
orfps
atr=b,
or=o'r
art=b,
o~
(6)
C'+q'a'r
where ps is the support pressure and C', q' have similar definitions as C and q, but for the intact zone. The last condition specifies that the intact rock must fail at r = b according t o Coulomb's law. These conditions give us the stresses and the radius of the broken zone b in terms of the support pressure Ps and opening radius a. Alternatively, if b is known, Ps can be obtained. When the b o u n d a r y conditions of eq.6 are applied to eqs.3--5, we get the expressions for A, B and the broken zone radius b as: A
aq-- 1
-
s
c)
1 -- q
b2
B = ~ q+l
F b =a
[Po(q--1)+C]
2[p0(q -- 1) + C]
] 1/(q-i)
L[ps --UT ] 67 1)J
A schematic plot of the stresses using eqs.3--5 is given as Fig.1.
-----
q(C2
~
q~>2
0
Radial
distance
r
Fig.l. Stre~es in the broken and intact rock zones around an opening for constant failed rock properties.
130 ALTERNATIVE SEMI-ANALYTICAL METHOD
The condition of continuity o f the radial stress or in eq.6 is essential for otherwise there would be differential radial displacements at the broken zone limit r = b. Continuity of the tangential stress oo is, however, not considered in eq.6 and this, in fact, leads to a discontinuity in a0 at r = b, as shown in Fig.1. If the condition o f continuity of ao is added to eq.6, this will lead to an over-determined system of equations -- more conditions than the constants to be determined -- ~vhich cannot give a unique solution. The o0discontinuity occurs because of a sudden change in the properties C, q from the broken to the intact rock, while in reality they should change gradually, the failure at r = b being incipient. There are thus two ways to eliminate this discrepancy, viz. adopt the same properties for the broken and intact rock, which is unrealistic, or to introduce a gradual variation of C, q through the broken zone, the limiting value being the same as for intact rock. The second alternative is incorporated in the stress analysis given below.
llLllli / fSt*~ ~ s ~ I ~ ~ P.
/
/
b
\
I
\
I
I I I I
i
n /
\ \
/
I
I
/i /
\
~'%
I
s ~'j
J
c'. ¥
ci.,t,i
n
C=O,sS o
Fig.2. S c h e m e for a semi-analytical m e t h o d and a m o d e l for variation o f properties in the failed rock annulus.
131
If the failed rock zone is discretised using n annular slices (Fig.2), the values of C, q can be considered to remain constant within each slice if n is large. The stresses can then be defined for each slice using the previous procedure and the usual condition of continuity of a~ between slices. In the absence of any information regarding variation of C, q within the failed rock zone, a m o d e l is proposed for this purpose as shown in Fig.2. In this model the variation in C, ~ (instead of q) follows the post-failure curve of the rock involved, the ultimate minimum values at the opening wall being C = 0 and q = q0, where q0 corresponds to the residual angle of friction ¢0. Then using eqs.2 and 3, the stresses in a slice i can be written down as: ors = ~ + A ~r~
(7)
005 = C i + q i o r i
where ai = C~/(1 {3i = q i -
-- qi)
(8)
1
F o r slice 1, or, = Ps at r = a and for subsequent slices continuity of ~ between slices must exist. These conditions completely determine the stresses in each slice: or, = ~z + (p, -- ~,) ( r ) ~
or: = ~ 2 +
(P~--~')
+~'-~
Ori = t~i +
\a+t
\a+
/
t/
""
+i--2t]
+ i - - l t ~ ",-~
+(~z-a2)\a+t
/
\a+2t/
"'"
+ i -- 2tJ
(aa_+i--lt~"i-' + ... + (~i-: --ai-1)
+ i--2t/
(9)
+(~i-,-~ ) 1 ( " a +ri - l i )~' where t = (b - - a ) / n or slice thickness. N o w at r = b, incipient failure will relate ar with a0 for the intact zone according to the Coulomb criterion, viz. the third condition in eq.6. Also. continuity o f or will exist between the nth slice and the intact zone. These t w o conditions give us from eqs.5 and 9: t
°
t
132
giving b2
B - - -
q'+l
[C'+
(q'--
1)po]
and
B
Po --~"~ = (or.)r= b
10
8
a.
=
6
4
2
0
/.
8
12
16
Distance from roadwoy centre,m
Fig.3. Stresses in the failed and intact zones w h e n properties are varied according to the m o d e l in Fig.2 ( e x a m p l e : case 5, Table I).
133 Substituting for B and using the general expression 9 for orn, transposing and simplifying, we get
Ps =~l + ( ~ 2 - ~ l ) \ a + t ]
+(~3-a2)
( )
+ ... + (~. --~.-1)
(2p_.~oz C '
~
~
\a + 2t} a
O, ( a + t ~ \a + 2t/
(: "'" "'"
\ a + 2t]
,,)"o'
+ n-- lt n - - it)~n b
This equation has been used for estimating the support pressure ps wherever the size of the broken zone b was available. The stress distribution in one of the cases considered later was plotted from eqs.5, 7 and 9 as shown in Fig.3. BARTON'S METHOD According to the engineering classification of rock masses b y Barton et al. (1974), the ultimate r o o f support pressure is given by: 2 ps = ~ Q-1/3
(11)
where
Q _ RQ___D J, Jw Jn J. SRF
(12)
and RQD = rock quality designation, Jn = joint set number, Jr = joint roughness number, Ja = joint alteration number, Jw = water reduction number, SRF = stress reduction factor. The values o f the parameters defining the rock mass quality Q are given b y Barton et al. for different rock descriptions. E q . l l fits their case records, b u t when the joint sets are less than three a modification of e q . l l is proposed by them: 2J~ n Q-1/3 P " - 3Jr In this paper only e q . l l is used throughout for simplicity. CASE STUDIES Application of the t w o methods discussed has been done for seven case studies, one of a coal mine and six.of hydroelectric tunnels in the Himalayan regions (J.L. Jethwa and A.K. Dube, pers. comm., 1983). Regarding these cases, whose data are given in Table I, the following points should be noted. (a) The depth of cover ranges between 240 m and 680 m.
134
TABLE
I
Data pertaining to seven case studies
Rock type Cover load (kPa/m)
Case 1 Jitpur colliery *~
2 Chibro-Khodri
3 4 5 Chibro-- Chibro-- Loktak Khodri Khodri
coal 25
shales 27
shales 27
clays 26
shales slates 27 25
phyllites 23
280 1.5
680 4.5
280 4.5
300 2.4
380 2.1
240 2.1
--
14.5
13.4
--
--
10 4 0.5 8 1 0.156 27
10--20 15 1 6 1 0.055 18
Depth (m) 450 O p e n i n g radius, 3.0 a (m) B r o k e n z o n e tad. 9.5--11.0
5.5--6.5
6 Giri *~
7 Giri *~
b (m)
RQD (%),2
10 12 1 1 1 0.833 37
Jn Jr Ja Jw
Q(SRF = 1) RMR
10--20 15 1.5 4 1 0.250 31
10--20 15 1.5 4 1 0.250 31
10--25 4 1 1 1 2.500 52
10--25 4 0.5 2 1 0.620 40
* ~B u c k l e d s u p p o r t s . *2Lower value o f RQD t a k e n f o r c a l c u l a t i n g Q.
(b) The roadways considered at the Jitpur colliery axe located in a shaft pillar in the 8.4 m thick seam XIV with longwall workings on three sides. The hydro-tunnels were driven through disturbed rock, sometimes intmthrust zones. (c) The radius of the failed rock annulus b was determined using multi-
10
Days
8
E .~" O
o
190~ 6
,
100 4
[
2
501
c3
I
I
I
Broken zone radius
11m
i
0
I
I
5
Distancefrom
I
I
i
~
10
I
" ~
l
15
roadway centre, m
Fig.4. E s t i m a t i o n o f failed r o c k z o n e b y b o r e h o l e e x t e n s o m e t e r o b s e r v a t i o n s ( e x a m p l e : case 1, T a b l e I).
135 point borehole extensometers, as shown by an example plot in Fig.4. The average ratio b/a ranged between 3.2 and 5.6. (d) The period o f support load observations ranged between 3 and 26 months. The steadied-off value o f load was taken as ultimate. (e) In cases 1, 6 and 7 the supports had buckled. (f) The rock classification parameters were determined from in situ exposures in consultation with geologists.
Estimation o f properties F o r a realistic analysis, it was necessary to estimate the uniaxial compressive strength C' and the peak and residual values o f C for the rock mass in situ. C' could be determined this w a y by in-situ crushing tests only in one place, viz. the Jitpur colliery (Sheorey et al., 1982), the value being 3.0 MPa. F o r the other cases C' was estimated approximately, using the following procedure. (a) Determine Q from eq.12 with the classification parameters of Table I and with SRF = 1. (b) Calculate the R o c k Mass Rating (RMR) of Bieniawski's classification (1976) from the relation:
RMR = 9 1 n Q + 4 4 (c) Multiply the value of 3.0 MPa b y the ratio RMR/37, where 37 is the RMR of Jitpur coal, to obtain C'. The underlying assumption that the compressive strength C' is proportional to RMR is considered reasonable since it is seen that the shear strength varies directly as RMR in Bieniawski's classification tables and in the Coulomb-Navier failure criterion: C' = 2 S[(1 + p2),n + p/ where S is the strength in shear. This relation was tried initially using S and C values from Bieniawski's tables, b u t the values of C' obtained were unusually low (for Jitpur the value was 0.53 MPa). Determination of C, on the other hand, was much more critical since eq.10 is more sensitive to b o t h peak and residual values q' and q0 (and hence C' and C0) than to C'. In the absence of experimental evidence, the t w o classifications mentioned were used. The peak values of C r e c o m m e n d e d b y Barton were f o u n d to be considerably on the higher side as far as this analysis was concerned. Bieniawski's values of the peak friction angle C' were therefore chosen corresponding to the RMR calculated as above. The residual friction angle ¢o was, however, taken from Barton's tables which give these values depending on the joint alteration number Ja, while Bieniawski's tables do not include them. In choosing the upper limit o f C0 from Barton's range it was decided that the minimum difference between ~b' and C0 should be 5 °, while the lower limit was considered acceptable. The relevant properties so estimated are given in Table II.
136
TABLE
II
Rock mass properties, support pressures and modified S R F values for the seven cases of Table I Case 1 Jitpur colliery *1
2
3
4
5
6
Chibro-Khodri
Chibro-Khodri
Chibro-Khodri
Loktak
Giri*t
7 Girl .1
7.28
8.10
9.50
5.52
27 12--16
18 16--24
31
29
Cover Pressure (MPa)
11.25
7.56
RMR ~0 (deg.) ~b' (deg.) q'
37 25--29 34 3.54 3.0 5--10
31 25--27
C' (MPa) SRF
32 3 . 2 5
2.5 3--5
Support pressure Ps ( k g / c m 2 ) Observed (r) *~ 1.5 3.1 (s)
Computed,
18.36 31 ----
2.5 15--25
3.12
2.88
2.2 2--4
1.5 1--2
52 ----
40 ----
4.2 10--15
3.2 3--5
0.5 3.5--5.0
1.7 1.9--2.4
6.5 7.1 --
11.5 12.2 8.0--10.4
5.4 5.4 2.8--7.3
2.0 2.4 --
2.0 5.0 --
4.5--5.8 3.6--4.5
3.6--4.6 3.0--3.6
4.6--5.7 5.2--6.2
12.7--15.9 9.4--11.8
9.0--11.0 5.2--6.6
2.5--3.2 3.2--3.6
8.1--10.1 6.8--8.0
eq.10
Barton Modified Barton
*1Buckled supports. * 2 ( r ) = r o o f , ( s ) = w a l l o r side.
Comparison of support pressures The support pressure was calculated from eq.10 for four cases (where b was measured) after calculating the properties of each slice using the postfailure model o f Fig.2 and eq.8. The post-failure characteristic could be determined for the coal but not for other rocks. The same post-failure model was therefore adopted in all four cases. It was found that the solution was sufficiently accurate if 7 or more slices were employed for discretisation. Slices greater than 11 or 12 were not necessary. These calculated estimates are given in Table II together with the observed pressures. Barton's e q . l l was also used for all the cases but the stress reduction factor SRF in the definition of Q, eq.12, was initially taken as recommended by Barton et al. (1974): Mild squeezing pressure, SRF = 5 to 10 (cases 2, 4, 5, 6, 7) Heavy squeezing pressure, SRF = 10 to 20 (cases 1, 3) The ultimate pressures with these SRF values are also given in Table II. The measured support pressures, in the case of buckled supports, were not expected to give the ultimate values but an underestimation. Therefore, agreement with only the stable cases was sought for in this analysis. The predicted pressures using the semi-analytical method c o m e reasonably close to the observed stable pressures, especially if we give some latitude to the insufficiency o f experimental information. The range predicted corresponds to the range of ¢0 values in Table II.
137 Barton's formula, however, does n o t give good corroboration if the aforesaid S R F values are taken. It is seen from eq.10 that the ultimate support pressure m u s t depend on the depth of cover or P0 as well as the broken zone size and the S R F values recommended do not consider this fact effectively. Cases 2 and 3 in Table II indicate t h a t in the same rock formation the support pressure increases with depth. At the same time, since eq.11 is simple to handle, it was decided to see if Barton's approach could be somewhat modified for squeezing rock. The extent o f the broken zone will depend on the cover pressure P0 vis-avis rock strength, but the support pressure will vary inversely with the broken zone radius. At the same time, the support pressure must increase with the depth of cover. Thus it is apparent t h a t the choice of S R F should consider both rock strength and cover pressure. Since the values of the rock mass strength C' were obtained by guess work in all except one case, t h e y can at best be approximate. An alternative reliable parameter, which could perhaps be accepted as a direct index of rock mass strength, is RMR. After some trial calculations it was seen that the stress reduction factor S R F could have reasonable values if it was varied against the product (RMR × P0) in the manner shown in Fig.5. It should be noted t h a t the RMR values are those from Table I. These S R F values, given in Table II, lead to ultimate (roof) support pressure estimates which are in good agreement with the observed pressures. The criterion in Fig.5 was constructed with emphasis on the stable support cases, the unstable cases giving higher estimates t h a n the 30
SRF chosen f o r case studies 20 LL n, t/1 "ID C
E o
el
10
0
200
400
600
RMR = c o v e r pressure. HPo
Fig.5. R e c o m m e n d e d values o f the stress reduction factor (SRF) for openings in squeezing rock.
138 TABLE IH R e c o m m e n d e d values of stress r e d u c t i o n factor (SRF) for squeezing rock conditions
RMR x cover pressure P0
R e c o m m e n d e d SRF
(in MPa) < 150 150--200 200--250 250--300 300--350 350--400 400--450 450--500 500--550 550--600
1--2 2--4 2.5--5 3.5--6.5 4--8 5--10 7--12 9--17 13--22 18--28
observed values. Two upper and lower curves enveloping the modified S R F values were constructed in the figure to give the S R F values of Table III recommended for squeezing conditions. Fig.6 shows a comparison between the estimates of ultimate support pressure obtained by using the modified SRF values in e q . l l , the observed pressures and those computed.from the semi-analytical approach, eq.10. 16
12
~ "~o~ $
s
\
.s
,~~
•
Measured (stable supports)
x T i r
Measured (buckled supports) Barton (modified SRF) Semi-anatytica[ method Roof pressure
s Sd i epressure_
' \T ~LT ~\
N
~eS
T,r
t
/*s-jr ~ ~-_0
r's I 1
4
,
I
I Sx~
rxe, s
rx xr xS
0
0.2
0./.
0.6
0-8
1
Jr 0~ Fig.6. C o m p a r i s o n o f u l t i m a t e s u p p o r t pressures as measured and as calculated f r o m B a r t o n ' s e q . l l ( w i t h m o d i f i e d S R F ) and f r o m the semi-analytical m e t h o d , eq.10.
139
Wall support pressure Barton et al. have suggested using t h e following substitution for Q in eq.11 for obtaining the side or wall support pressure: Q <0.1 0.1-10 >10
Replace Q b y 1.0 Q 2.5 Q 5.0 Q
In only one stable case (case 2, Table II) is the wall-support pressure distinctly different from the r o o f pressure. In the rest of the cases the t w o pressures are approximately equal. Taking S R F as 3--5 for case 2 as per Table II, Q becomes 0.083--0.05 which means that the side pressure should equal the r o o f pressure, which is contrary t o the observation. Table IV indicates that the criterion above is difficult to fit as far as Q values are concerned. The product JrQ '/3, on the other hand, offers a possible solution. It is suggested tentatively, for wall pressure in squeezing conditions, to adopt the following guideline:
JrQ '/3 <0.5 >O.5
Replace Q b y 1.0 Q 2.5 Q
More case studies, however, will be needed to corroborate this criterion. 'I'ADb 5 iV E f f e c t o f classification p a r a m e t e r s o n o b s e r v e d wall s u p p o r t pressure vis-A-vis r o o f pressure Case
Jr
SRF
Q
JrQ 1/3
Ps(wall) = P s ( r o o f ) ?
1.5 1.5 0.5 1.0
3--5 15--25 2--4 1--2
0.083--0.050 0.017-0.010 0.078-0.039 0.055-0.028
0.65-0.55 0.39---0.32 0.21-0.17 0.38-0.30
No Yes Yes Yes
no.
2 3 4 5
CONCLUSION
E q . l l proposed b y Barton et al. (1979) for ultimate r o o f support pressure does not agree with practical observations if their r e c o m m e n d e d values of the stress reduction factor S R F are used in calculating the rock quality Q. The modified values of S R F suggested in Table III and Fig.5 give support pressures in good agreement with the observed values. This modification in S R F considers the rock mass rating R M R and the cover pressure to account for the effects of depth and broken zone size on support pressure. Alterna-
140
tively, Table III can be constructed by incorporating the in-situ large-scale strength instead of R M R . The semi-analytical method proposed for stress analysis in the broken rock annulus also gives reasonable support pressure values. The method shows that Bieniawski's and Barton's rock classificationsgive good estimates of the peak and residual angles of internal friction respectively. The pre-requisite of the semi-analytical method is that the radius of the broken zone must be known from in-situ observations. If such a measurement is carried out with buckling supports, the radius obtained will be larger and the ultimate support pressure calculated will be in error. N o such measurement is required while using Barton's eq.11 with the recommended S R F values and therefore this becomes a simple formula giving reasonable estimates of the ultimate support pressure. The tentative criterion suggested for wall pressure in squeezing conditions needs further experimental verification. ACKNOWLEDGEMENTS T h a n k s are d u e t o t h e D i r e c t o r , CMRS f o r permission t o publish this p a p e r a n d t o t h e a u t h o r ' s colleagues Drs. J.L. J e t h w a and A.K. D u b e f o r their valuable d a t a o n t u n n e l s and f o r free discussions. The o p i n i o n s expressed are t h o s e o f t h e a u t h o r and n o t necessarily o f t h e CMRS.
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. Brown, E.T. et al., 1983. Ground response curves for rock tunnels. J. Geotech. Eng., 109: 15--39. Hock, 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. Sheorey, P.R., Das, M.N. and Singh, B., 1982. A numerical procedure for rock pressure problems in level seams. In: I.W. Farmer (Editor), Symposium on Strata Mechanics. Newcastle upon Tyne, pp.254--259.
127
SUPPORT PRESSURE ESTIMATION IN FAILED ROCK CONDITIONS
P.R. SHEOREY Central Mining Research Station, Barwa Road, Dhanbad--8 26001, Bihar (India) (Received May 29, 1984; accepted after revision January 21, 1985)
ABSTRACT Sheorey, P.R., 1985. Support pressure estimation in failed rock conditions. Eng. Geol., 22: 127--140. Ultimate roof support pressure estimates using a modern rock mass classification (Barton et al., 1974) are compared with the observed values in seven cases of squeezing ground. Since the agreement is not good, a modification of the classification method is suggested. Also given is a tentative approximate guideline for modifying the classification for obtaining wall support pressures. Assuming the failed rock surrounding an underground opening to behave as a Coulomb material, a semi-analytical elastoplastic method is proposed for calculating the ultimate support pressure. An earlier closed-form solution to the problem of stresses in the failed rock annulus assigns constant material properties through the failed rock zone, giving a discontinuity in the tangential stress at the failed--intact rock boundary. The semianalytical method uses a step-wise gradual change of properties, eliminating the discontinuity. This method is shown to give reasonable estimates of support pressure if the peak and residual angles of internal friction are chosen from the classifications of Bieniawski (1976) and of Barton's (1974), respectively. INTRODUCTION A t u n n e l o p e n i n g o r m i n e r o a d w a y at d e p t h is generally s u r r o u n d e d b y a failed r o c k a n n u l u s w h i c h c a n be a s s u m e d t o b e h a v e as a C o u l o m b m a t e r i a l in t h e active R a n k i n e state. S u p p o r t s e r e c t e d in such o p e n i n g s h a v e t o w i t h s t a n d h i g h e r pressures d u e t o r o c k s q u e e z i n g t h a n in i n t a c t o r l o o s e n i n g r o c k c o n d i t i o n s . A n elastoplastic f i n i t e - e l e m e n t s i m u l a t i o n o f failed r o c k condit i o n s m a y p e r h a p s lead t o reliable s u p p o r t p r e s s u r e values, p r o v i d e d t h e i n p u t p r o p e r t i e s are s u f f i c i e n t l y realistic. A l t e r n a t i v e l y , e l a s t o p l a s t i c closedf o r m s o l u t i o n s are available, as r e v i e w e d b y B r o w n et al. ( 1 9 8 3 ) , giving a r e l a t i o n b e t w e e n t h e s u p p o r t p r e s s u r e a n d t h e radius o f t h e failed r o c k zone. Such s o l u t i o n s h a v e a s s u m e d d i f f e r e n t c o n s t i t u t i v e laws f o r b r o k e n r o c k b e h a v i o u r , t h e c h i e f o n e being C o u l o m b ' s law. As o p p o s e d t o t h e s e t w o m a t h e m a t i c a l m e t h o d s , t h e e m p i r i c a l f o r m u l a e p r o p o s e d b y B a r t o n et al. ( 1 9 7 4 ) o f f e r a m u c h s i m p l e r a n d q u i c k e r a p p r o a c h t o t h e p r o b l e m o f support pressure prediction. I f u n i f o r m p r o p e r t i e s are t a k e n t h r o u g h t h e failed r o c k , t h e c l o s e d - f o r m 0013-7952/85/$03.30
© 1985 Elsevier Science Publishers B.V.
128 solution suffers from some fundamental shortcoming, to be mentioned later, and the rock classification m e t h o d of Barton et al., in its present form, gives erroneous estimates of support pressure as seen from the case studies given. An alternative semi-analytical m e t h o d is proposed in this paper, after introducing the earlier solution employing Coulomb's law (Jaegar and Cook, 1976), and a comparison is made with observed support pressures as well as those calculated by Barton's m e t h o d , suggesting a possible modification of the latter method. Brown et al. (1983) have also employed a semi-analytical approach with a step-wise variation of failed rock properties based on the Hoek and Brown (1980) failure criterion, but n o t for estimating support pressure. EARLIER ELASTOPLASTIC APPROACH If a and b are the radii of the opening and the failed rock annulus, respectively, the radial and tangential stresses or and oe in the annulus, a < r < b, are related by the equilibrium equation: dot ~ or--o0 _ 0 dr r
(1)
in polar coordinates (r, 0 ). If the failed material obeys Coulomb's law (2)
ao = C + qor
where C = uniaxial compressive strength of the failed rock, q = [ ( 1 + / / 2 ) 1/2 + p ] 2, = tan ¢, coefficient of internal friction. Substituting eq.2 in eq.1 we have the solution: ar
=
C 1 -- q
~
+
Ar
~-1
(3)
where A is an arbitrary constant. Substituting eq.3 in eq.2, a0
-
C ~ 1 -- q
+ Aqr
q-1
(4)
In the intact zone, r ~ b, the rock is assumed to behave elastically, so the stresses are: f
B
Or = P0 - - ~
(5) B oo = Po + r:
in which B is a n o t h e r integration constant and Po is the hydrostatic virgin stress away from the opening.
129 For determining the constants A and B in eqs.3 and 5 and the radius of the broken rock zone b, three conditions are available: atr=a,
orfps
atr=b,
or=o'r
art=b,
o~
(6)
C'+q'a'r
where ps is the support pressure and C', q' have similar definitions as C and q, but for the intact zone. The last condition specifies that the intact rock must fail at r = b according t o Coulomb's law. These conditions give us the stresses and the radius of the broken zone b in terms of the support pressure Ps and opening radius a. Alternatively, if b is known, Ps can be obtained. When the b o u n d a r y conditions of eq.6 are applied to eqs.3--5, we get the expressions for A, B and the broken zone radius b as: A
aq-- 1
-
s
c)
1 -- q
b2
B = ~ q+l
F b =a
[Po(q--1)+C]
2[p0(q -- 1) + C]
] 1/(q-i)
L[ps --UT ] 67 1)J
A schematic plot of the stresses using eqs.3--5 is given as Fig.1.
-----
q(C2
~
q~>2
0
Radial
distance
r
Fig.l. Stre~es in the broken and intact rock zones around an opening for constant failed rock properties.
130 ALTERNATIVE SEMI-ANALYTICAL METHOD
The condition of continuity o f the radial stress or in eq.6 is essential for otherwise there would be differential radial displacements at the broken zone limit r = b. Continuity of the tangential stress oo is, however, not considered in eq.6 and this, in fact, leads to a discontinuity in a0 at r = b, as shown in Fig.1. If the condition o f continuity of ao is added to eq.6, this will lead to an over-determined system of equations -- more conditions than the constants to be determined -- ~vhich cannot give a unique solution. The o0discontinuity occurs because of a sudden change in the properties C, q from the broken to the intact rock, while in reality they should change gradually, the failure at r = b being incipient. There are thus two ways to eliminate this discrepancy, viz. adopt the same properties for the broken and intact rock, which is unrealistic, or to introduce a gradual variation of C, q through the broken zone, the limiting value being the same as for intact rock. The second alternative is incorporated in the stress analysis given below.
llLllli / fSt*~ ~ s ~ I ~ ~ P.
/
/
b
\
I
\
I
I I I I
i
n /
\ \
/
I
I
/i /
\
~'%
I
s ~'j
J
c'. ¥
ci.,t,i
n
C=O,sS o
Fig.2. S c h e m e for a semi-analytical m e t h o d and a m o d e l for variation o f properties in the failed rock annulus.
131
If the failed rock zone is discretised using n annular slices (Fig.2), the values of C, q can be considered to remain constant within each slice if n is large. The stresses can then be defined for each slice using the previous procedure and the usual condition of continuity of a~ between slices. In the absence of any information regarding variation of C, q within the failed rock zone, a m o d e l is proposed for this purpose as shown in Fig.2. In this model the variation in C, ~ (instead of q) follows the post-failure curve of the rock involved, the ultimate minimum values at the opening wall being C = 0 and q = q0, where q0 corresponds to the residual angle of friction ¢0. Then using eqs.2 and 3, the stresses in a slice i can be written down as: ors = ~ + A ~r~
(7)
005 = C i + q i o r i
where ai = C~/(1 {3i = q i -
-- qi)
(8)
1
F o r slice 1, or, = Ps at r = a and for subsequent slices continuity of ~ between slices must exist. These conditions completely determine the stresses in each slice: or, = ~z + (p, -- ~,) ( r ) ~
or: = ~ 2 +
(P~--~')
+~'-~
Ori = t~i +
\a+t
\a+
/
t/
""
+i--2t]
+ i - - l t ~ ",-~
+(~z-a2)\a+t
/
\a+2t/
"'"
+ i -- 2tJ
(aa_+i--lt~"i-' + ... + (~i-: --ai-1)
+ i--2t/
(9)
+(~i-,-~ ) 1 ( " a +ri - l i )~' where t = (b - - a ) / n or slice thickness. N o w at r = b, incipient failure will relate ar with a0 for the intact zone according to the Coulomb criterion, viz. the third condition in eq.6. Also. continuity o f or will exist between the nth slice and the intact zone. These t w o conditions give us from eqs.5 and 9: t
°
t
132
giving b2
B - - -
q'+l
[C'+
(q'--
1)po]
and
B
Po --~"~ = (or.)r= b
10
8
a.
=
6
4
2
0
/.
8
12
16
Distance from roadwoy centre,m
Fig.3. Stresses in the failed and intact zones w h e n properties are varied according to the m o d e l in Fig.2 ( e x a m p l e : case 5, Table I).
133 Substituting for B and using the general expression 9 for orn, transposing and simplifying, we get
Ps =~l + ( ~ 2 - ~ l ) \ a + t ]
+(~3-a2)
( )
+ ... + (~. --~.-1)
(2p_.~oz C '
~
~
\a + 2t} a
O, ( a + t ~ \a + 2t/
(: "'" "'"
\ a + 2t]
,,)"o'
+ n-- lt n - - it)~n b
This equation has been used for estimating the support pressure ps wherever the size of the broken zone b was available. The stress distribution in one of the cases considered later was plotted from eqs.5, 7 and 9 as shown in Fig.3. BARTON'S METHOD According to the engineering classification of rock masses b y Barton et al. (1974), the ultimate r o o f support pressure is given by: 2 ps = ~ Q-1/3
(11)
where
Q _ RQ___D J, Jw Jn J. SRF
(12)
and RQD = rock quality designation, Jn = joint set number, Jr = joint roughness number, Ja = joint alteration number, Jw = water reduction number, SRF = stress reduction factor. The values o f the parameters defining the rock mass quality Q are given b y Barton et al. for different rock descriptions. E q . l l fits their case records, b u t when the joint sets are less than three a modification of e q . l l is proposed by them: 2J~ n Q-1/3 P " - 3Jr In this paper only e q . l l is used throughout for simplicity. CASE STUDIES Application of the t w o methods discussed has been done for seven case studies, one of a coal mine and six.of hydroelectric tunnels in the Himalayan regions (J.L. Jethwa and A.K. Dube, pers. comm., 1983). Regarding these cases, whose data are given in Table I, the following points should be noted. (a) The depth of cover ranges between 240 m and 680 m.
134
TABLE
I
Data pertaining to seven case studies
Rock type Cover load (kPa/m)
Case 1 Jitpur colliery *~
2 Chibro-Khodri
3 4 5 Chibro-- Chibro-- Loktak Khodri Khodri
coal 25
shales 27
shales 27
clays 26
shales slates 27 25
phyllites 23
280 1.5
680 4.5
280 4.5
300 2.4
380 2.1
240 2.1
--
14.5
13.4
--
--
10 4 0.5 8 1 0.156 27
10--20 15 1 6 1 0.055 18
Depth (m) 450 O p e n i n g radius, 3.0 a (m) B r o k e n z o n e tad. 9.5--11.0
5.5--6.5
6 Giri *~
7 Giri *~
b (m)
RQD (%),2
10 12 1 1 1 0.833 37
Jn Jr Ja Jw
Q(SRF = 1) RMR
10--20 15 1.5 4 1 0.250 31
10--20 15 1.5 4 1 0.250 31
10--25 4 1 1 1 2.500 52
10--25 4 0.5 2 1 0.620 40
* ~B u c k l e d s u p p o r t s . *2Lower value o f RQD t a k e n f o r c a l c u l a t i n g Q.
(b) The roadways considered at the Jitpur colliery axe located in a shaft pillar in the 8.4 m thick seam XIV with longwall workings on three sides. The hydro-tunnels were driven through disturbed rock, sometimes intmthrust zones. (c) The radius of the failed rock annulus b was determined using multi-
10
Days
8
E .~" O
o
190~ 6
,
100 4
[
2
501
c3
I
I
I
Broken zone radius
11m
i
0
I
I
5
Distancefrom
I
I
i
~
10
I
" ~
l
15
roadway centre, m
Fig.4. E s t i m a t i o n o f failed r o c k z o n e b y b o r e h o l e e x t e n s o m e t e r o b s e r v a t i o n s ( e x a m p l e : case 1, T a b l e I).
135 point borehole extensometers, as shown by an example plot in Fig.4. The average ratio b/a ranged between 3.2 and 5.6. (d) The period o f support load observations ranged between 3 and 26 months. The steadied-off value o f load was taken as ultimate. (e) In cases 1, 6 and 7 the supports had buckled. (f) The rock classification parameters were determined from in situ exposures in consultation with geologists.
Estimation o f properties F o r a realistic analysis, it was necessary to estimate the uniaxial compressive strength C' and the peak and residual values o f C for the rock mass in situ. C' could be determined this w a y by in-situ crushing tests only in one place, viz. the Jitpur colliery (Sheorey et al., 1982), the value being 3.0 MPa. F o r the other cases C' was estimated approximately, using the following procedure. (a) Determine Q from eq.12 with the classification parameters of Table I and with SRF = 1. (b) Calculate the R o c k Mass Rating (RMR) of Bieniawski's classification (1976) from the relation:
RMR = 9 1 n Q + 4 4 (c) Multiply the value of 3.0 MPa b y the ratio RMR/37, where 37 is the RMR of Jitpur coal, to obtain C'. The underlying assumption that the compressive strength C' is proportional to RMR is considered reasonable since it is seen that the shear strength varies directly as RMR in Bieniawski's classification tables and in the Coulomb-Navier failure criterion: C' = 2 S[(1 + p2),n + p/ where S is the strength in shear. This relation was tried initially using S and C values from Bieniawski's tables, b u t the values of C' obtained were unusually low (for Jitpur the value was 0.53 MPa). Determination of C, on the other hand, was much more critical since eq.10 is more sensitive to b o t h peak and residual values q' and q0 (and hence C' and C0) than to C'. In the absence of experimental evidence, the t w o classifications mentioned were used. The peak values of C r e c o m m e n d e d b y Barton were f o u n d to be considerably on the higher side as far as this analysis was concerned. Bieniawski's values of the peak friction angle C' were therefore chosen corresponding to the RMR calculated as above. The residual friction angle ¢o was, however, taken from Barton's tables which give these values depending on the joint alteration number Ja, while Bieniawski's tables do not include them. In choosing the upper limit o f C0 from Barton's range it was decided that the minimum difference between ~b' and C0 should be 5 °, while the lower limit was considered acceptable. The relevant properties so estimated are given in Table II.
136
TABLE
II
Rock mass properties, support pressures and modified S R F values for the seven cases of Table I Case 1 Jitpur colliery *1
2
3
4
5
6
Chibro-Khodri
Chibro-Khodri
Chibro-Khodri
Loktak
Giri*t
7 Girl .1
7.28
8.10
9.50
5.52
27 12--16
18 16--24
31
29
Cover Pressure (MPa)
11.25
7.56
RMR ~0 (deg.) ~b' (deg.) q'
37 25--29 34 3.54 3.0 5--10
31 25--27
C' (MPa) SRF
32 3 . 2 5
2.5 3--5
Support pressure Ps ( k g / c m 2 ) Observed (r) *~ 1.5 3.1 (s)
Computed,
18.36 31 ----
2.5 15--25
3.12
2.88
2.2 2--4
1.5 1--2
52 ----
40 ----
4.2 10--15
3.2 3--5
0.5 3.5--5.0
1.7 1.9--2.4
6.5 7.1 --
11.5 12.2 8.0--10.4
5.4 5.4 2.8--7.3
2.0 2.4 --
2.0 5.0 --
4.5--5.8 3.6--4.5
3.6--4.6 3.0--3.6
4.6--5.7 5.2--6.2
12.7--15.9 9.4--11.8
9.0--11.0 5.2--6.6
2.5--3.2 3.2--3.6
8.1--10.1 6.8--8.0
eq.10
Barton Modified Barton
*1Buckled supports. * 2 ( r ) = r o o f , ( s ) = w a l l o r side.
Comparison of support pressures The support pressure was calculated from eq.10 for four cases (where b was measured) after calculating the properties of each slice using the postfailure model o f Fig.2 and eq.8. The post-failure characteristic could be determined for the coal but not for other rocks. The same post-failure model was therefore adopted in all four cases. It was found that the solution was sufficiently accurate if 7 or more slices were employed for discretisation. Slices greater than 11 or 12 were not necessary. These calculated estimates are given in Table II together with the observed pressures. Barton's e q . l l was also used for all the cases but the stress reduction factor SRF in the definition of Q, eq.12, was initially taken as recommended by Barton et al. (1974): Mild squeezing pressure, SRF = 5 to 10 (cases 2, 4, 5, 6, 7) Heavy squeezing pressure, SRF = 10 to 20 (cases 1, 3) The ultimate pressures with these SRF values are also given in Table II. The measured support pressures, in the case of buckled supports, were not expected to give the ultimate values but an underestimation. Therefore, agreement with only the stable cases was sought for in this analysis. The predicted pressures using the semi-analytical method c o m e reasonably close to the observed stable pressures, especially if we give some latitude to the insufficiency o f experimental information. The range predicted corresponds to the range of ¢0 values in Table II.
137 Barton's formula, however, does n o t give good corroboration if the aforesaid S R F values are taken. It is seen from eq.10 that the ultimate support pressure m u s t depend on the depth of cover or P0 as well as the broken zone size and the S R F values recommended do not consider this fact effectively. Cases 2 and 3 in Table II indicate t h a t in the same rock formation the support pressure increases with depth. At the same time, since eq.11 is simple to handle, it was decided to see if Barton's approach could be somewhat modified for squeezing rock. The extent o f the broken zone will depend on the cover pressure P0 vis-avis rock strength, but the support pressure will vary inversely with the broken zone radius. At the same time, the support pressure must increase with the depth of cover. Thus it is apparent t h a t the choice of S R F should consider both rock strength and cover pressure. Since the values of the rock mass strength C' were obtained by guess work in all except one case, t h e y can at best be approximate. An alternative reliable parameter, which could perhaps be accepted as a direct index of rock mass strength, is RMR. After some trial calculations it was seen that the stress reduction factor S R F could have reasonable values if it was varied against the product (RMR × P0) in the manner shown in Fig.5. It should be noted t h a t the RMR values are those from Table I. These S R F values, given in Table II, lead to ultimate (roof) support pressure estimates which are in good agreement with the observed pressures. The criterion in Fig.5 was constructed with emphasis on the stable support cases, the unstable cases giving higher estimates t h a n the 30
SRF chosen f o r case studies 20 LL n, t/1 "ID C
E o
el
10
0
200
400
600
RMR = c o v e r pressure. HPo
Fig.5. R e c o m m e n d e d values o f the stress reduction factor (SRF) for openings in squeezing rock.
138 TABLE IH R e c o m m e n d e d values of stress r e d u c t i o n factor (SRF) for squeezing rock conditions
RMR x cover pressure P0
R e c o m m e n d e d SRF
(in MPa) < 150 150--200 200--250 250--300 300--350 350--400 400--450 450--500 500--550 550--600
1--2 2--4 2.5--5 3.5--6.5 4--8 5--10 7--12 9--17 13--22 18--28
observed values. Two upper and lower curves enveloping the modified S R F values were constructed in the figure to give the S R F values of Table III recommended for squeezing conditions. Fig.6 shows a comparison between the estimates of ultimate support pressure obtained by using the modified SRF values in e q . l l , the observed pressures and those computed.from the semi-analytical approach, eq.10. 16
12
~ "~o~ $
s
\
.s
,~~
•
Measured (stable supports)
x T i r
Measured (buckled supports) Barton (modified SRF) Semi-anatytica[ method Roof pressure
s Sd i epressure_
' \T ~LT ~\
N
~eS
T,r
t
/*s-jr ~ ~-_0
r's I 1
4
,
I
I Sx~
rxe, s
rx xr xS
0
0.2
0./.
0.6
0-8
1
Jr 0~ Fig.6. C o m p a r i s o n o f u l t i m a t e s u p p o r t pressures as measured and as calculated f r o m B a r t o n ' s e q . l l ( w i t h m o d i f i e d S R F ) and f r o m the semi-analytical m e t h o d , eq.10.
139
Wall support pressure Barton et al. have suggested using t h e following substitution for Q in eq.11 for obtaining the side or wall support pressure: Q <0.1 0.1-10 >10
Replace Q b y 1.0 Q 2.5 Q 5.0 Q
In only one stable case (case 2, Table II) is the wall-support pressure distinctly different from the r o o f pressure. In the rest of the cases the t w o pressures are approximately equal. Taking S R F as 3--5 for case 2 as per Table II, Q becomes 0.083--0.05 which means that the side pressure should equal the r o o f pressure, which is contrary t o the observation. Table IV indicates that the criterion above is difficult to fit as far as Q values are concerned. The product JrQ '/3, on the other hand, offers a possible solution. It is suggested tentatively, for wall pressure in squeezing conditions, to adopt the following guideline:
JrQ '/3 <0.5 >O.5
Replace Q b y 1.0 Q 2.5 Q
More case studies, however, will be needed to corroborate this criterion. 'I'ADb 5 iV E f f e c t o f classification p a r a m e t e r s o n o b s e r v e d wall s u p p o r t pressure vis-A-vis r o o f pressure Case
Jr
SRF
Q
JrQ 1/3
Ps(wall) = P s ( r o o f ) ?
1.5 1.5 0.5 1.0
3--5 15--25 2--4 1--2
0.083--0.050 0.017-0.010 0.078-0.039 0.055-0.028
0.65-0.55 0.39---0.32 0.21-0.17 0.38-0.30
No Yes Yes Yes
no.
2 3 4 5
CONCLUSION
E q . l l proposed b y Barton et al. (1979) for ultimate r o o f support pressure does not agree with practical observations if their r e c o m m e n d e d values of the stress reduction factor S R F are used in calculating the rock quality Q. The modified values of S R F suggested in Table III and Fig.5 give support pressures in good agreement with the observed values. This modification in S R F considers the rock mass rating R M R and the cover pressure to account for the effects of depth and broken zone size on support pressure. Alterna-
140
tively, Table III can be constructed by incorporating the in-situ large-scale strength instead of R M R . The semi-analytical method proposed for stress analysis in the broken rock annulus also gives reasonable support pressure values. The method shows that Bieniawski's and Barton's rock classificationsgive good estimates of the peak and residual angles of internal friction respectively. The pre-requisite of the semi-analytical method is that the radius of the broken zone must be known from in-situ observations. If such a measurement is carried out with buckling supports, the radius obtained will be larger and the ultimate support pressure calculated will be in error. N o such measurement is required while using Barton's eq.11 with the recommended S R F values and therefore this becomes a simple formula giving reasonable estimates of the ultimate support pressure. The tentative criterion suggested for wall pressure in squeezing conditions needs further experimental verification. ACKNOWLEDGEMENTS T h a n k s are d u e t o t h e D i r e c t o r , CMRS f o r permission t o publish this p a p e r a n d t o t h e a u t h o r ' s colleagues Drs. J.L. J e t h w a and A.K. D u b e f o r their valuable d a t a o n t u n n e l s and f o r free discussions. The o p i n i o n s expressed are t h o s e o f t h e a u t h o r and n o t necessarily o f t h e CMRS.
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. Brown, E.T. et al., 1983. Ground response curves for rock tunnels. J. Geotech. Eng., 109: 15--39. Hock, 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. Sheorey, P.R., Das, M.N. and Singh, B., 1982. A numerical procedure for rock pressure problems in level seams. In: I.W. Farmer (Editor), Symposium on Strata Mechanics. Newcastle upon Tyne, pp.254--259.