ISRM-EUROCK-1996-036_The Strength of Rock Masses With Finite Sized Joints

8/12/2019 ISRM-EUROCK-1996-036_The Strength of Rock Masses With Finite Sized Joints

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Eurock '96, Barla (ed.) 1996 Ba lkema, Rotterdam. ISBN 905410843 6

The strength of rock masses with finite sized joints

La resistance de masses rocheuses a fractures lirniteesDie Festigkeit von Gesteinsmassen mit Verbindungen begrenzter Grobe

Mohammad H. Bagheripour & Ga.ITyMostyn - The School of Civil Engineering. University of New SouthWales, Sydney. N S .w ., Australia

ABSTRACT: The strength of rock masses containing finite sized joints is discussed. Using Linear Elastic

Fracture Mechanics the authors developed a criterion to predict the brittle failure of rock specimens containing

finae sized joints. This criterion is very similar to that developed by J aeger for throughgoing joints but

unfortunately fails to predict the strengths observed in an extensive laboratory investigation. An empirical

relauonship is introduced which accurately predicts the experimental data. The paper concludes that t hecontinuity of joints has a major impact on the strength of the rock mass containing them and that the models that

have generally been applied are overly conservative. The proposed relationship provides a much better model

for uniaxial strength test results and its extension to triaxial compression is discussed.

RESUME: la resistance de massese rocheuses contenant des fractures limitees est presentee. A I aide de la LEFM,

les auteurs ont developpe un critere permeuant de c alculer Ie point de c assure netre de specimens rocheuxContenant des fractures limitees. Ce critere est tres proche de celui relatif aux fractures continues developpe par

Jaeger rnais, malheuresement, ne calcule pas les resistances observees lors d 'une recherche approfondie en

laboratoire. Une relation empirique calculant avec j usresse les donnees experimentales est i nrroduire. La

presentation conclue que la conrinuite des fractures joue un role primordial sur la resistance des masses rocheuses

qui continnent ees fractures, et que les rnodeles generalement appliques son: excessivernent prudents. La relation

proposee fournit un modele beaucoup plus approprie aux resullats des essaix uniaxes de la resistance, et son

application a la compression a trois axes est presen tee.

ZUSAMMENFASUNG : Es wird die Festigkeit von Gesteinsrnassen mit Verbindungen begrenzter Grofsebesprochen. Unter Anwendung derLinearen Elastischen Bruchrnechanik entwickelten die Autoren ein Kriteriurn

fUr die Voraussage von Sprodigkeitsbruchen von Gesteinsproben mit Verbindungen begrenzter Grobe. Dieses

Kriterium ist dem von Jaeger fur durchgehende Verbindungen entwickelren sehr ahnlich, ist jedoch leider nicht

In der Lage die in ausfuhrlichen Laborversuchen beobachteten Festigkeiten vorauszusagen. Ein empirisches

Verhiiltnis, das die experimentellen Daten genaus voraussagt wird eingefuhrt. Del' Beitrag kornmt zu dem SchluB,

daB die Durchgangigkeit der Verbindungen einen groben Einfluf auf die Festigkeit der sie eruhaltenden

Gesteinsmasse hat und d aB die allgemein angewandten Erklarungsrnodelle zu konservativ sind. Das

Vorgeschlagene Verhaltnis bietet eine viel besseres Modell fiir einachsige Versuchsergebnisse und eine

Erweiterung auf dreiachsige Kompression wird besprochen.

INTRODUCTION

In general, failure analysis of jointed rocks has been

aSSumed to occur along a throughgoing dominant

Single joint, or set of joints, such as that for a rock

slope. Further, in such jointed rocks, the designer is

essentially interested in the strength parameters of the

discontinuities. This assumption is generally quite

conservative. Stimpson (1978) discusses the

dIffiCUlties in designing ultimate open pit slopes

Where joints dip into the pit at greater than the angle of

friCtion. Most analysis would predict that such a slope

was unstable, but in fact such slopes are often observed

to be stable. Generally, this stability is attributed to the

fact that the discontinuities are not continuous but are

of finite size. Thus, as well asjoint orientation and the

confining pressure, the persistence of a joint has a

significant impact on the the strength of rocks.

Limited progress has been reponed on theoretical and

experimental approaches to strength of jointed rocks

containing finite sized joints. McClintock & Walsh(1962), Jaeger & Cook (1979) and Hoek (1968)discussed the compressive strength of rocks

containing very small finite sized joints (sometimes

called rnicrofractures). The Linear Elastic Fracture

Mechanics (LEFM) approach has been investigated

by other researchers (eg Atkinson, 1987, Erdogan

1974, Hung & Lee 1990). Others have attempted to

apply this approach to discontinuously jointed rock

slope stability problems (eg Tharp 1984, Tharp &Coffin 1985). In this approach, finite size joints are

commonly treated as friction free cracks which

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extend, or propagate, due to the stress concentration at

their tips.In this paper the concept of LEFM is reviewed and acriterion for brittle failure of rocks containing finitesized JOints subjected to compressive loads is derived.It is assumed that friction is mobilized on finite sized

joint surfaces when shear is induced under

compressive loads.Model materials were used to develop jointed rockspecimens containing finite sized joints. The modelmaterial development and characteristics aredescribed in Mostyn & Bagheripour (1995). Thespecimens were tested under uniaxial and triaxialconditions. The results were compared to the criteriondeveloped but agreement was poor. The study thenfocused on predicting the strength using empiricaln?ethods. The empirical criterion developed is verysimilar to that described for throughgoing jointedrocks (Bagheripour & Mostyn 1996). .

2 LEFM CONCEPT

The original concept of fracture mechanics is based onconsidering materials containing ideal flat, perfectlysharp cracks of negligible thickness, undergoing three

possible modes of crack tip displacement, oftenreferred to as modes of loading. Mode I is for normalloading, II for in-plane shear loading and III foranti-:-plane, sometimes referred as tearing, shearloading .. In problems concerning crack loading thesuperposiuon of these three basic modes is sufficientto describe the most general combination of crack tipdeformation and stress field.

Modes l and II are the most commonly used loadingmodes in engineenng problems. It is recognized thatmode III loading is involved in special engineering

problems which are beyond the scope of the currentstudy. Following is a brief description of mode 1andII loadings. These are then further extended to rockmechanics problems where compression and shear arethe most common loading systems.If a single isolated joint is subject to a uniaxial tensilestress, the stress intensity factor, K " is:

A list of symb?ls is given at the end of this paper. Thecondition for incipient crack propagation is obtainedby equating K, toK,c, the critical stress intensity factoror fracture toughness, and a to atj, the tensile strencihof jointed rock: '"

K,c =at/.nc)i (2)

The stress intensity factor in pure shear, Kif , is alsorelated to shear stress:

1

KII =r(.nc)2 (3)

For incipient crack propagation due to shear we have:

(4)

where rc is the critical shear stress. The solutions 10problems of linear elasticity relating the near crack tipstresses to the stress intensity factors are givenelsewhere (eg Atkinson 1987, Ingraffea 1987) and are

not presented here. Here it suffices to describe nearcrack tip stresses in mixed mode loading.The extension of LEFM principles from a solid brittlematerial problem to rock mechanics problemsdemands special consideration. Classic LEFMproblems generally involve open cracks with frictionfree surfaces under shear and/or tensile stresses.Problems in rock mechanics, however, commonlyInvolve finite Sized joints under compressive andshear loads. The most common theories of mixedmode loading which resolve the near crack tip stresses

are a((j)lIlfU' S((j)lIli" and C(j)lIlax theories. In thefollowing section, the. a((j)",[U theory is brieflydescribed first and then simplified forthecondition ofcompressive loading of a single inclined finite sizedJOInt. Thereafter, the theory is used to derive thefracture stren?th criterion of rock specimenscontaining finite sized joints in terms of jointonentuuon, con filling pressure and joint persistence.The criterion is similar to the Jaeger (1960) and Jaeger& Cook (1979) cruenon for strength variation ofthroughgoing jointed rocks.

2.1 Development of a compressive fracture strength[or rocks

If a single isolated joint under general biaxial loadingIS considered, as shown in Figure I, then the normaland shear stresses on the plane of the joint arecalculated as:

(J\+(J3 (J1-(J3

a ,,= -2---7-cos(2(J) ( 5 )

(6)

(I)

Generally for a natural joint, the surfaces will be incontact. The frictional resistance of these surfaces isassumed to be constant and equal to a" tan(1)j)'

t

0~2c"[c

(l~ 7' --:=(J1l.tan(jJ)

} ! ' O ,; 0 0

Figure 1 Cracked body subjected to biaxial

compressive loads.

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1 .2

S(())min1.0 r:

u 0.8

~'a32: O . It will be seen that this assumption resultsIn equations that are simple and comparable with

other criteria. Thus, in applying fracture mechanics to

the problem, the only stress component which will

Contribute to the behaviour of the stress state around

the joint ends is the effective shear stress defined by

the following equation:

Te==T (J-G II tan()

From equations 5 and 6, it follows that:

In the terminology of Figure I, and without loss of

generality, it will be assumed that Te >O . The solutionof the related elasticity problem gives the stress

tntensity factors and the stress Slate in the close

neighbourhood of the joint lip as:

and

The stress components near the crack tip for the

Conditions given in Equation 9 are defined by are ) II II 1X

theory as follows:

_ K" o [ o 3. ]ar-_-cos(-) 2 tan(-)--S11l(e). { i . . ; r ; - 2 2 2

au = = K" Cos(ft)[lsin(e)]. { i . . ; r ; - 2 2

K" earO = = r.:;-cos("2)[1-3 cos(()J[Zstr

( 10)

( 1 1 )

( 1 2 )

The theory assumes that crack extension occurs:

at the crack tip and in a radial direction,

when a re ) max reaches a crit ical value that isthe material constant, ie Krc. in a plane normal to the direction of greatest

tension, ie at e o that (JrfI =0,The mathematical expression equivalent for the above

conditions is:

( 13)

(14)

and if (Jr()==0 then:

( 15)

Equations 14 and 15 are the parametric equations of

a general fracture criterion locus in the K"K" plane

as shown in Figure 2. The plot in the figure shows afracture surface for mixed mode cracking of the form:

(16)

(7)

The procedure for the development of such a fracture

locus isdescribed elsewhere (eg Tharp 1984, lngraffea1987). The significance of Equation 15 is that the

direction of crack extension e o can be predicted. Thisgives:

and

( 17)

If the condition for pure mode II is considered then

K,==() which gives:

( 18)

( 9 )

This indicates that the crack, ie finite sized joint, will

nOI extend initially in its own plane. This quite

different to pure mode I loading. Combining Equation

II and 18 gives:

(Ju== ~(~) (19)

On the other hand, if (Jlj is the uniaxial tensile strength

of the rock containing a crack of the same size as that

considered in Figure I ,( iea , = = -(Jlj and (J3==0, f3=90)from the conditions of fracture assumed in a(e)IIII1X

theory, the following can be defined:

(20)

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Combining Equations 19 and 20:

Equation 21 can also be directly taken as the fracture

strength locus, shown previously in Figure 2 for t heassumed condition. Combining Equations 8,9 and 21

gives:

2 [ ~ ; r +a3 tan()]a =a +----------

1 3 [1-cot(,8) tan()J sin(2,B)

Equation 22 i s very similar to the Jaeger (1960) and

Jaeger & Cook (1979) model for the compressivestrength of rock containing through going joints.

However, it can be seen from a comparison of the two

equations that the term /3 KJC/2 k in Equation 22replaces the term C j in the Jaeger (1960) and Jaeger &Cook (1979) model.

Equation 22 i s based on mode I loading being zero.

Thus, mode II indicates the severity of applied loads

and the geometry of the cracked body.

The orientation ,8=,80, which corresponds to the

maximum value of K" for a given a,,u3 and c, andhence to t he j oint from which the fracture would

extend, may be obtained by maximizing KII. This

condition yields:

are =0 and a2re - < 0 (23)a ,8 a2 , 8

giving:

1tan(2,80)= -;--(A. ) (24)

t0, it is clear that Equations24 and 25 are subject to the following constraint:

( 0 '; 0 ' sin(2{3) )tanPJ) - < - - - -- - - (26)

(O,+o,_o,-


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jo inted sp ec im en s, th e el lip tic al jo int ca nn ot be

c ha ra cte riz ed so le ly b y a pa ra me te r suc h a s c ra ck l en gt h. T he d e sc rip ti on o f t he co rr ec ti on fa ct ors fo r ell ip tical f lat cracks is rather lengthy and is beyond thes co pe o f th is p ap er. H ow ev er , th e co rre ct io n f ac to rsh ave b een u sed to amen d the s tr ess inten si ty f ac to r o f the loaded samp les u sed in the cu rr en t s tu dy .

-x - p =1 .0

-0- p=0.2 5

-+-p=0.7 5

-p=O.O

-6- p=0.5

4

2

o

o 30 60 90

Joint orientatiomjl)

Fig ure 5 U nia xia l c om pre ss ive stre ng th v ersus

jo int or ientat io n an d pe rsi sten ce .

A p lo t o f Eq ua ti on 2 8 is s ho wn in F ig ur e 3 . T h e p lo tIS li mi te d to t he u n ia xia l c om pre ss iv e s tre ng th a ndcon siders the ro ck and jo in t s tr en gth p ar ameters to b eth ose sh ow n in Fig ure 6.

3 EXPERIMENTAL PROGRAM

An experimental program was completed onspecimen s made f ro m an art if ic ia l intac t r ock and jo in t

mater ials descr ibed in Mostyn & Bagheripour (1995).T he o bjec tive o f comp ress iv e s tr en gth tes ting o n suchspecimen s w as to s tu dy the eff ec t o f the p er si sten ce o f f rictional joints on the f racture s trength of a rock mass.T h e exp er imental p ro g ram inc lu ded b oth un iaxial andtrIa xia l c om pre ssive stre ng th te sts o n spe cim en sC on ta in in g fin ite siz ed jo in ts w ith va rio us jo in to~lentations and persistence.FIgure 4 shows schem atically the first type of cYltnd rica l specimen s tha t w er e tested . T h ese joint swere not ful ly contained within the sample. Specimensw ere cas t w i th 1 /4 , 1 /2 , 3 /4 con tinu ity r at io s, p, an dJ Oint o rien ta tion s v ar ying in 1 5 in terv als f ro m 0 to90.

3.1 Test procedure

T h e p ro ced ur e f ol lo w ed f or the comp ress io n tes ts w asIn aCco rd an ce w ith the IS RM (1 98 1, 1 98 3) sug g es tedmethOd.

- - E q. 28, p=O.75- - - - E g. 2 8, p = O.2 5

+ p=O.75 p=O.25

. - - - - . Eg . 28 . p=05-- Intact rock

6 p=O.5

1 6

1 4 " ,1 / ( ?, ' / I I" \ ,'0I

\"\, 0/,' I

, .' / t/.

\ '., .- 6./ /

\ '......... / +

" /......._--+'

+

1 2

~IO

'"0-2 8~

3 < 6b

4

2

o

Rock and joint strength data:

K /C=O.4Nmm,j, ,;, =32"

o 30 60 90

e+

~--

+I

60 90

o 30Joint or ientation (13)

Figure 6 (a) Experim ental data and plot

o f E qu ati on 28 , (b ) R es id ua ls .

3.2 Results

Th e uniaxial c om pressive stre ngth ve rsus joint

orientat ion for various joint continuity rat ios is shownin F ig ur e 5 .

3.3 Comparison with theory

E qu at io n 2 8 is o ve rl ain o n F ig ure 6 a. I t c an b e s ee nth at th e m od el pro vid es a v ery p oo r fit t o t he d ataexcept for 13=90 and 13=30. Plots of residuals result ingf ro m f it ting Equ at io n 2 8 to the ex perimen ta l d ata a res ho wn in F ig ure 6 b a nd i t is o bv io us th at t he m o de lgreatly over esurnares the s trength.It s ho ul d b e no te d t ha t t h e L EF M p ri nc ip le s u se d tod eri ve t he E qu ati on 2 8 as su me p la ne s tre ss o r pla nes tr ain co nd it io ns and th us may int ro du ce some err or sinto the results. Fracture tough ness m ay not beconstant but m ay change as the norm al stress

i nc re as es . T his e ffe ct h as n ot b e en i nc lu de d i n th isw o rk b ut has been d iscu ssed b y some inv es tiga to rs ( eg

Hung & Lee 1990).

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x p=I.O

+ p=0.75

6 p=0.5

0 p=0.25

-- Eq. 31. p=I.O

- -Eg.31,p=0.75

- Eq. 31, p=0.50

Eq 31, p=0.25

-- Eg. 31, p=O.O

1 6

]14

1 2 !- ;:; 1 0

t~c, ~6 8 f' "

~ j

b

0

2

'"c, e~en 0

'" -I:: J"0

'u;v -2~

-3

0

o.0

..6

+ x

+

~x

x,

30 60 90

+)(

l! ;

+~xo4 >

o o

x

30 60 90

Joint orientation (13)

Figure 7 (a) Experimental data and plots o f

Equation 3], (b) Residuals.

4 EMPIRICAL PREDICTION OF STRENGTH

4.1 General

Bagheripour & Mostyn (1996) modified Singh et al(1989) relationship for uniaxial compressive strength

of throughgoing joints to:

ucj=A,uci

where

A=B. cos 3/2(j3min-(1)

4.2 Unconfined compressive strength

Equation 29 can be modified to allow for joint

persistence. The following relation is proposed:

(31 )

where

B.p. cos 3/2(j3min-(1)Ap=1---------

Fp(12)

and B is a constant. It can be seen that when p-> IEquation 31 reduces to Equation 29 for throughgoing

jointed rocks. On the other hand, whenp->O, Equation

31 reduces to uei which is the strength of intact rock .

A regression analysis of the experimental data about

Equation 31 has been completed and the results are

given in Table I. It should be noted that Oei and Bvalues obtained are the same as those obtained for

testing throughgoing jointed rocks (Bagheripour &Mostyn 1996). The experimental data and Equation

31 are shown in Figure 7a. It can also be seen that the

effect of joint orientation decreases as the continuityratio decreases. Figure 7b shows a plot of the residuals

which, although showing a small trend, indicate that

the model provides a reasonable Iit ro the data. Furtherit can be seen that the empirical relation of Equation 31

provides a better estimation of strength than the

theoretical criterion of Equation 28.

A contour plot of Equation 31 is presented in Figure 8.

Table I Summary of the regression results using

Equation 31.

Parameters in Equa-tion 31

43 Triaxial strength

Hoek (1994) proposed the following modified relation

for confined compressive strength of rock:

UI-a3=(7r:i(m~3 + S ) { 1 (34)ct

The authors have fitted this relationship to our datausing:

(29)

s= 1 _ Bp(((1)Fp

where F((1) is:t c,.a,

f({J)=(cos 3/2((1l11in-(1)) +a;;- (36)

(35)

(30)Regression analysis indicated that a is very close to

unity, perhaps because the range of confining stresses

in the lest program was too small 10 demonstrate a

clearly curved failure envelope. Equation 34 has the

following properties:

when (73->0, it reduces to the already discussed

Equation 31, defined for uniaxial strength.

when p-.O, it reduces to the strength criterion forintact rock,

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Parameters in Equation 34

Table 2 Summary of the regression results using Equation 34 .

when a3--+0 and p--+O, it reduces to the uniaxial

Compressive strength of intact rock, when p-eland a3--+0, it reduces to Equation 29for uniaxial compressive strength of throughgoing

jointed rock.A regression analysis of the data about Equation 34was completed and resulted in the parameters given inTable 2.Equation 34 for a persistence equal 0.5 is plotted onFigure Figure 9a. Residuals are plotted on Figure 9b.It can be seen that the emprical relation of Equation 34reasonably defines the variation of the confinedstrength of the finite sized jointed rock samplesdeveloped and tested. The same good agreement was

observed for persistences equal to 0.25,0.75.

5 CONCLUSIONS

The strength of jointed rocks containing finite sizedJOtnts was investigated. The stress intensity factor and

a(8)max theory of mixed mode loading were used toderive an equation to predict the fracture strengthvariation of a rock containing finite sized joints intenns of persistence and confining pressure. Therelation was shown to very similar to that introducedby Jaeger (1960) and Jaeger& Cook (1979) for thestrength prediction of thoughgoing jointed rocks.Unfortunately it did not predict the results of theexperimental program at all well. An empiricalrelation, that has been used to predict the strength of

~~

'"' ~6 :~

~ :b' ' C i~.;

' i I l 'i l I l

67.5e

JOin 15.eeto . Q,,~

Denta' 'V>QtJOn 1/1)

Figure 8 Contour of Equation 31 showing the

effect of joint orientation and persistence.

throughgoing jointed rocks was modified to account

for joint persistence. The resulting relationshippredicted the results of uniaxial and triaxialcompression tests on the specimens containing singleinclined finite size joints far better than theory. The

parameters are easily understood and determined.

00.0 MPa

x 3.0 MPa

61.0 MPa 02.0 MPa

:I: 4.0 MPa

0 30 60 90

- ; : ; -1.5 T

c, I S ~ 0 x

2

O~t x e 6Vl lL-eeli e ::J

"U -0.5 II ; ; r ;'iji ~(l) _ I x

c ::: 6 X 0-1.5

0 30 60 90

Joint orientation W )

Figure 9 (a) Experimental data and plot of

Equation 34 for p=0.5, (b) Residuals.

6 NOTATION

A

B

A I '

r;pI'Il,S,n

parameter of Equat ion 30

constant of equation 31,33,36

parameter of Equation 32

persistence function

persistenceconstants of Hoek & Brown criterion

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C 1 constant of Equation 37

C half the length of joint

a , major principal stressa3 minor principal stress

e p j joint friction angle

Cj joint cohesion

aci uniaxial compressive strength of intact rockacj uniaxial compressive strength of jointed rock

f 3 joint orientation

f3 m in value of particular orientation angle, f3 ,where the strength is a minimum

K, stress intensity factor for mode I

Ki f stress intensity factor for mode 1/

e p j friction angle of joint (crack)

() the angle between the crack extension and

original crack direction.

an normal compressive stressalj tensile strength of jointed rock

T shear stress

Te critical shear stress

7 REFERENCES

Atkinson, B.K. (1987) Fracture Mechanics of Rock.

Academic Press London.Erdogan, F. (1974) Fracture problems in a

nonhomogeneous medium. Proc. NATO Advanced

Study Institute in Reykjavik, Iceland, 11-20

August, 1974.

Gdoutos, E. E., (1984) Problems of mixed mode crack

propagation. Martinus Nijhoff Publishers,

Netherlands.Hoek, E. (1968) Brittle failure of rocks. Chapter 4 of

" Rock Mechanics in Engineering Practice" Stagg,

K.G., Zienkiewicz, O.e . (Eds). John Wiley &Sons.Hung, J. J. and Lee, T. T. (1990) A study on shear

strength of rock joint of partial continuity. Int. Conf.

Rock Joints. Barton & Stephenson (eds).

Rotterdam. pp. 219-224.

Ingraffea, A. R. (1987) Theory of crack initiation and

propagation in rock. Chapter 3 of Fracture

Mechanics of Rock. (ed Atkinson B.K.) Academic

Press London.

International Society of Rock Mechanics (1981)Suggested methods for determining the strength of

rock materials in triaxial compression. in "RockCharacterization Testing & Monitoring". Brown

E.T. (ed.). Pergamon Press.

International Society of Rock Mechanics (1983).

Suggested methods for determining the strength of

rock materials in triaxial compression: Revised

Version. Int. J. Rock. Mech. Min. Sci. & Geomech.

Abstr, Vol. 20, No.6, pp. 283-290.

Jaeger, J.e. (1960) Shear fracture of anisotropic

rocks. Geol. Mag. Vol. 97, pp. 65 -72.

Jaeger, J.e . and Cook, N. G. W. (1979) Fundamentalsof Rock Mechanics. 3rd ed. Chapman & Hall.

London.

Mcf.lintock, F. A. and Walsh, J. B. (1962) Friction

on Griffith cracks in rocks under pressure. Proc.

4th US Symp. Rock Mechanics, 1962,

pp.1015-1021

Mostyn, G.R. and Bagheripour, M.H. (1995) A new

model material to simulate rock. Proc. 2nd Int.

Conf. on Mech. of Jointed and Faulted Rock

(M.lFR-l ).Vienna, Austria, 10-14 April 1995. A.A.

Balkema, Rotterdam.Bagheripour, M.H. and Mostyn G. Prediction of the

strength of jointed rock - Theory and practice. Proc.

Eurock 96. 1996 ISRM Int. Symp. on Prediction

and Performance in Rock Mechanics and Rock

Engineering. Torino, Italy, Sepl. 2-5, 1996. (In

Press)Sih, G.e . (1974) Strain energy density factor applied

to mixed crack problems, ln r .J . of Fracture, Vol. 10,

pp. 305-321.Singh, .I., Ramarnurthy, T. and Rao, G. (1989)

Strength anisotropies in rocks. Indian Geotechnical.lournal Vol. 19, No.2, pp. 147-\66.

Stimpson, B. (1978) Failure of slopes containing

discontinuous planar joints. Proc. 19th US Symp.

on Rock Mech. pp. 296-302.

Tharp. T. M. (1984) Stability of slopes in

discontinuously jointed rock. Proc. 25th US Symp.

on Rock Mechanics. pp. 891-898.Tharp, T. M. and Coffin, D. T. (1985) Field

application of fracture mechanics analysis to small

rock slopes. 26th US Symposium on Rock

Mechanics, Rapid City, SD. 26-28 .lune 1985.

pp.667-674.

Zarrabi, K. (1993) Mechanics of Fracture. Lecture

Notes. School of Mechanical Engineering, TheUniversity of New South Wales.

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ISRM-EUROCK-1996-036_The Strength of Rock Masses With Finite Sized Joints

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