A comprehensive peak shear strength criterion for rock jointsCritsre de resistance maximum complet pour les fractures des roches

Umfassende Spitzenfestigkeits-Kriterien fOrGesteinsfugen

T.T. PAPALIANGAS, Department of Civil Engineering, Technological Educational Institution, Thessaloniki, Greece, andDepartment of Earth Sciences, University of Leeds, UK

S. R. HENCHER & A.C. LUMSDEN, Department of Earth Sciences, University of Leeds, UK

ABSTRACT : The basic principles of a new simple peak shear strength criterion for rock joints based on well-documented theory offriction and deformation of rough surfaces are presented. According to this criterion, the peak shear strength of a rock joint at any nennalstress is interpreted as comprising two components, one purely frictional and one geometrical. The first is due to the shear strength ofrock junctions sheared under high nonnal stress and is independent of roughness and scale. The second is due to surface roughness whichcauses dilation. Under a given nonnal stress, the joint is first deformed and then sheared along an inclined plane defined by the deformedasperities. The magnitude of the frictional component is detennined from the shear strength of the rock wall material, whereas themagnitude of the geometrical component, which is normal stress dependent, is predicted from consideration of surface roughness andcontact theory. The new criterion is supported by the results of a series of direct shear tests on modelled rock joints. Changes in peakfriction angle due to nonnal stress, scale effects or roughness anisotropy are interpreted in terms of change in the geometrical componentonly.

~lThffi: Cet article presente les principes de base d'un nouveau critere, le maximum de resistance au cisaillement des joints de roche,bases sur les theories connues de frottement et de deformation des surfaces irreguliers, Selon ce critere, Ie maximum de resistance aucisaillement d'un contact rocheux sous contrainte nonnale comprend deux composantes; une purement de frottement, et une autregeometrique, La premiere est due ~ la resistance des joints rocheux soumis au cisaillement sous haute contrainte nonnale, et eUe estindependante de la rugosite et de l'~helle; la deuxieme est due ~ la rugosite de la surface, qui entraine la dilatation. Sous une contraintenormale donnee, le joint est d'abord deforme, puis cisaillee le long d'un plan incJ.iOOdefini par les asperites ainsi d~fo~. La grandeurde la composante de frottement est ~te~ ~ partir de la resistance au cisaillement du materiau de roche, alors que la grandeur de lacomposante geometrique, qui depend de la contrainte nonnale, est predite en consideriant la rugosite de la surface et les contacts. Lenouveau critere est vali~ par les resultats d'une serie de tests de cisaillements directs sur des joints de roches. Le changement de l'angledu maximum de frottement dus ~ la contrainte normale, les effets d' echelle

ZUSSAMENFASSUNG: Die Grundprinzipien fUr cine neue einfache Regel fUr sie Spitzenuerscherungskraft von GesteinsklUften,basiennd auf gut dokurnuntierter Theorie der Reibung und Deformation von unebenen ober-flllchen werden vorgestellt Die Regel besagt,daB die spitzenscherkraft einer Gesteinskluft unter Normal-druck als aus zwei komponenten bestehend auf gefasst werden kann: einerReibungs komponente und einer geometrinschen komponente. Die Reibungskomponente ist bedingt durch die scherkraft der kontaktpunkte unter hohem Nonnal-druck und ist unabhUngig von der ober-flllchenbeschaftenheit und dem maBstab. Die geometrischeKomponente ist bedingt durch die oberflllchenbeschaffenheit (Ranheit?), welche eine Mfnung der Gieitflllchen veruvsachen. Unter einemgegebenen Nonnal-druck wird die kluft zuerst deformiert und dann geschhert entlang einer Neigungsebene, die durch die deformiertenAsperitatan definiert ist. Die GrOBe der Reibungs-komponente ist bestimmt durch die scherkraft des Gesteinmaterials, wohingegen dieScherkraft des Gesteinmaterials, wohingegen die GrOBe der geometrischen komponente, welch abhungig vom Nonnaldruck ist,vorherbestimant werden kann ans Uberlegungen Uber die oberflllchenbeschaffenheit und der Kontakttheorie. Die neue Regel beruht aufden Ergebnissen einer Rihe von Scherversuchen an Modelgestuns verklUflugen. Verllnderungen des spitzenreibungswinkels auf Grunddes Nonnaldruckes, MaBstabseinflusse oder ober-flachenanistropic werden nur als Verllnderungen der geometrischen komponenteangeschen.

1. INTRODUCTION

Analysis of rock mechanics problems often requires reliableprediction of joint behaviour based on quantitative descriptions.Shear strength of rock joints is of primary importance and isassociated with dilatancy which depends on the geometricalproperties of the joint surface. These properties are random whichmakes it difficult to model shear behaviour accurately. Mostexisting theoretical models use idealised surfaces such as saw-tooth, sine-tooth etc.; the required parameters are not easilyobtained and often fail adequately to predict real behaviour. Forthis reason most rock engineers use simple empirical models,which are based on the analysis of experimental data or back-analyses. and may be of limited use or site-specific. The simpletheoretical criterion briefly described below allows accurateprediction without relying on empiricism.

2. PROPOSED CRITERION

2.1 Mechanism of shearing

The actual area of contact between two rough surfaces is a verysmall fraction of the gross area of contact and consequently thenonnal stresses at the contacts are much higher than theconventional stress calculated from the gross area (e.g. Bowden &Tabor, 1950). Terzaghi (1925) suggested that the actual contactstress is of the order of the unconfined compressive strength;Jaeger (1971) estimated that the actual area of contact Aa for arough joint is of the order of I% of the apparent area of contact A,with the normal stress at the contacts 100 times higher than theapparent nonnal stress. Barton & Choubey (1977) suggested thatreal nonnal stress at contacts may be as high as one thousandtimes the averaged stress. Logan & Teufel (1986) measured

359

fractional area of contact AeJA during sliding as 18% forlimestone at 25 MPa and about 2% for sandstone at 50 MPaaveraged nonnal stress, the approximate values of real normalstress at contacts being 200 MPa and 2,200 MPa respectively.Stesky & Hannan (1987, 1989) measured 15% area of contact at19 MPa normal stress for marble, and I % at 49 MPa for quartzite.They noticed that cataelastic flow is the main mode of defonnationunder normal stress and commented that the actual contactstresses may be close to the differential stress needed for the rockto deform by bulk cataclastic flow, i.e. close to the brittle-ductiletransition stress.

Terzaghi (1925) and Bowden & Tabor (1950) suggested that theshear strength between two surfaces arises from the shearing oflocal junctions. It is postulated that the normal stresses at junctionsbetween rock surfaces are sufficiently high locally to induce abrittle-ductile state. Depending on the apparent nonnal stress level,this state will be reached by material at contacts to a shallow deptheither side of the junction; away from that zone the rock materialwill remain essentially undefonned. The actual nonnal stress atcontacts will always be of similar magnitude, because the actualcontact area adjusts proportionally to carry the applied load. It issuggested that the average inclination of the mean planes of alljunctions will define the plane along which shearing will takeplace, as shown in Figure I. The plane may be inclined dependingupon the initial roughness of the joint and the stress level. Simplythen, and as proposed in broad terms by various authors, the peakfriction angle of a rock joint can be considered as comprising twocomponents: a) a purely frictional (independent of normal stress)component +m, attributable to shearing of rock wall materialjunctions which are under high normal stress and b) a dilationalcomponent 'If, related to the average inclination of the shear planeof all contacting asperities, or:

where 'tp is the peak shear strength, an is the averaged, effectivenormal stress, +m the friction angle of the joint wall material and'If the dilation angle at the instant of peak shear strength.Determination of the two basic parameters +m and 'If, is discussedbriefly below and in more detail by Papaliangas (1995).

2.2 Friction angle of rock wall material

Orowan (1960) suggested that the shear strength of the intact rockmaterial at the brittle-ductile transition stress is equal to thefrictional resistance of the surface along which the material fails.Therefore the fracture strength envelope of the intact rockmaterial will intersect with that for frictional sliding at the brittle-ductile transition stress. IT the transition stress is known, then thefriction angle of the material can be determined, as shown inFigure 2. According to the parabolic criterion proposed byFairhurst (1964), the shear strength 't of the intact rock material ata normal stress an is given by:

C (~-l)Rl on)'t- 0 +n-n Co

where Co is the unconfined compressive strength and n the ratio ofcompressive to tensile strength. IT the transition pressure is takenequal to the unconfined compressive strength, then the coefficientof friction will be

~-l m:J.lm----<i1+nn

For a ratio of compressive to tensile strength n = IS, which istypical for many rocks, the calculated coefficient of friction is ~m=0.8 (+m = 38.7°). This value is very close to the measured values

(lA..v :~ Rock junction

- "'

Figure 1. Contact of rock asperities

of 39° found by Mogi (1966) for a number of silicate rock typesand 40° found by Byerlee (1978) for many rock types independentof lithology. However, the assumption that the brittle/ductiletransition pressure is equal to the unconfined compressive strengthis an approximation only. Some strong rocks, such as granites,may have transition pressure five times the unconfinedcompressive strength or even higher, whereas limestones andmarbles may have a transition pressure lower than the unconfinedcompressive strength (Mogi, 1966, Paterson, 1978). An accuratedetermination of friction angle of intact rock requires knowledge

Table 1. Examples of typical brittle-ductile transition pressures(Paterson, 1978)

Rock Approximate transitionpressures (MPa)

30-100100-200 or higher

40100<20400

300-500300

200-300

(1)

Limestones and marblesDolomiteGypsumAnhydriteRocksaltTalcSerpentiniteChlorititeArgillaceous sandstone(-10% porosity)Siltstones and shales(medium to highporosity)Porous lavas 30-100

< 100

B

(2)

Brittle-ductiletransition stress

(3)

Normal stress

Figure 2. Determination offriction angle of intact rock materialfrom triaxial tests

360

of the shear strength at the brittle-ductile transition stress. Mogifound that the friction angle fOj carbonate rocks may be as highas 49°, markedly higher than that of silicate rocks. Table I,adapted from Paterson (1978), gives the brittle-ductile transitionpressures for several rock types. Papaliangas (1995) provides acompilation of data on brittle-ductile transition stresses andfriction angles predicted on this basis for a larger number of rocktypes.

2.3 Dilation angle

According to the proposed mechanism of shearing, sliding willtake place along a plane which is defined by the deformedasperities; the orientation of this plane depends on the originalroughness characteristics of the joint surface and the normal stress.A method is presented below for estimating the dilation angle atpeak shear strength, based on the principles of normal contacttheory and assuming that the surface characteristics remainessentially constant up to peak strength, which is a reasonableassumption for the small shear displacements generally involved.

Greenwood & Williamson (1966) proposed an analytical modelfor the contact between rough and flat surfaces. The basicassumptions of this model are: a) that the asperities, at least attheir summits, are spherical. b) all asperity summits have the sameradius and c) the summits deform independently. Based on theHertzian contact theory. they found that. for a surface with anexponential distribution of asperity peaks, the closure under anormal stress an is given by:

Ii-A+Blna.

where A and B are constants. This relation is independent of themode of deformation (elastic or plastic) or the shape of theasperities and results in exact proportionality between normal loadand real area of contact, so that Amonton's second law is obeyed.Brown & Scholz (1985) generalised the theory of Greenwood &Williamson and found that relation (4) holds in the case of contactbetween two rough surfaces, whether or not they are mated. Theheights of asperities on most rough surfaces are essentially ofGaussian distribution rather than exponential (Greenwood &Williamson, 1966, Swan, 1983). However. as suggested byGreenwood & Williamson, an exponential distribution is a fairapproximation to the upper quartile of a Gaussian distribution ofasperity heights. This quartile is most relevant to frictionalbehaviour (Halling. 1978) and an exponential distribution ofheights can offer a reasonable approximation for rock jointcontacts at low to medium normal stresses (Swan. 1983. Sun etal., 1985).

Consider the case of a contact between two single asperitiesinclined at an angle '1'0 to the horizontal (Figure 3a).

Under zero normal load sliding will occur without any deformationalong the plane AB defined by the angle '1'0' but under a givennormal stresses an, the asperity deforms and responds with aclosure 5. Sliding will then take place along the plane AB' definedby the deformed asperity, inclined at an angle 'I' to the horizontal(Figure 3b). The closure f> can be expressed as:

Ii- L(lan",o -Ian",)

where L is the base length over which '1'0 is calculated. For anexponential distribution of asperity heights:

L(lan",o -Ian",)- A+Blna.

When an = ano~ 0, tan'l' ~ tanwo, where '1'0 is the averageasperity angle with negligible deformation. The lower boundary fornormal stress ano can not be mathematically equal to zero. and istherefore set at a low value which causes minimal deformation, for

A

(b)

Figure 3. Simplified geometry of asperity deformation

(4)

example 1 kPa.When an=anT ' the normal stress at which all dilation is

suppressed, tan'l'=O. Using these two boundary conditionsequation (6) yields:

Ian", -Ian",o In a.T lin anTOn 0no

(7a)

or (7b)

From equations 7a and 7b. the dilation angle at any normal stressan is given as a function of the dilation angle '1'0 at very lownormal stress. and the two boundary normal stresses a no and anT.These relations suggest that for a particular joint surface, thedilation rate reduces logarithmicaly with normal stress, asconfirmed experimentally by Barton (1971) and other authors. Itmust be emphasised that anT, the transition stress from dilatant topurely frictional sliding, is generally different from the brittle-ductile transition stress, as discussed later.Assuming that anT = 10MPa, and ano = 1kPa, equations (7a)and (7b) become:

Ian", _ Ian "'0 In anT9.21 a.

(8a)

or (8b)

(5)At normal stress levels greater than anT • only non-dilationalfriction operates and the shear strength Of the joint remainsproportional to normal stress until the brittle-ductile transitionstress, beyond which the shear strength envelope of the joint is thesame as that of the intact material. Figure 4 shows the generalshape of the criterion for a friction angle +m=39°, anT = 10 MPaand three different values of '1'0 (40°, 30° and 10°).(6)

3. EXPBRIMENTAL RESULTS

A series of direct shear tests on modelled rock joints has beencarried out to test the applicability of the new criterion. A medium

361

3.0 '.= ~ ••)10 J..2.5 ; ~.

2.0 :

'" = 400

o ...,..

300

100

0.5

1.5 . ......;

1.0

0.0 1.0 1.5 2.0 2.5 3.0Nonnal stress (MPa)

Figure 4. Typical shape of predicted 't-eJ curves for cIlm=39°. anT= 10 MPa and three different '1'0 values

strength cement-based rock substitute having a density of 2.45Mg/m3, unconfined compressive strength 47 MPa, tensile strength4.5 MPa and friction angle of saw-eut surfaces 33° was used toprepare replicas of different rock joints. These joints were tested indirect shear under normal stresses in the range 0-2 MPa. in aGolder Associates shear box, using a sophisticated system forrecording loads and displacements. Detailed description of thematerial and experimental procedure is presented in Papaliangas(1995). The behaviour in triaxial compression of the modelmaterial is similar to that of most limestones. According to thecriterion proposed by Hoek & Brown (1980) the coefficient m isequal to 5.9. Stress-strain curves at different confining pressuresare shown in Figure 5. At low confining pressures, the materialbehaves in a brittle manner, which is evident from the stress dropshown by the axial stress-axial shortening (01-£1) curve; as theconfining pressure increases the curve flattens and at a confiningpressure of 20 MPa it becomes horizontal, which suggests that thebrittle-ductile transistion stress has been reached. At higherconfining pressures the behaviour becomes purely ductile which isevident from the continuing increase of 01 with e 1.

The Mohr envelope corresponding to the stress state at thebrittle-ductile transition stress is shown in Figure 6. The frictionangle of the intact rock material can be calculated from the tangentto the Mohr circle passing through the origin (Orowan, 1960),which has a slope given by:

With 03= 20 MPa and 01=123 MPa a friction angle of the rockmaterial of 46° and the normal stress at the fracture plane of 34.5MPa is derived (Figure 6). The value of 46° may seem high, but itis typical for carbonate rocks (Mogi, 1966).

Figure 7a shows the measured peak shear strength from tests ontwo different joints (designated A and B) tested in the normalstress range 0·2 MPa. When the dilational component wasseparated in the manner suggested by Hencher & Richards(1989), the best fit line for the non-dilational friction angle was46.4° as shown in Figure 7b. This value is very close to the frictionangle for intact material as determined from triaxial testing. Thissuggests that asperities are sheared under conditions which aresimilar to the brittle-ductile transition and that the frictionalcomponent of peak shear strength can be determined either fromdirect shear test results after correction for dilation or fromtriaxial testing of the rock wall material under confining pressure

150

125

.-.. 100~'-':a 75it3 50~ 0

25

25MPa

,

MPai

··-f ... ·············f

···············t··· +,

i;·········t ...!...

0.0 1.0 2.0 3.0 4.0 5.0 6.0Axial shortening (%)

Figure 5. Stress-strain curves at different confining pressuresshown on each curve

(9)

sufficient to produce a brittle-ductile transition stress. The fact thatthe value of the non-dilational component is identical for jointswith quite different roughness, emphasises its frictional origin andfully supports the shearing mechanism proposed. The non-dilational friction angle is 13.4° higher than the friction angle ofsaw-eut surfaces and somewhat higher than the residual frictionangle.The variation of dilation rate with normal stress for the two series

of tests is shown in figure 8. Dilation was calculated over a step-size of 0.2mm which is approximately 0.2% of the sample length.The angle '1'0 was calculated from direct shear tests under the selfweight of the sample (normal stress 0.6kPa for joint A and 0.8 kPafor joint B, average of 5 tests).

The rate of dilation tan'l'o reduces logarithmicaly with normalstress, as predicted from the preceding theory, over a range of fourorders of magnitude and is fully suppressed at a normal stress ofabout 3.8 MPa for both joints. This stress is only about 8% of theunconfined compressive strength and 11% of the brittle-ductiletransition stress. Assuming that at the brittle-ductile stress there is100% contact between the two surfaces and proportionalitybetween area and load exists, as it is· for the exponentialdistribution of asperity heights, these data indicate that dilation isfully suppressed when the area of contact is approximately 11%of the gross area. The experimental results presented in Figures 7and 8 fully support the proposed mechanism of shearing. Thecurves on Figure 7a were fitted using equation (1) with the valuesfor V, cIlm and anT determined from equation 7b and Figures 7band 8 respectively (i.e, cIlm = 46.4° and anT = 3.8 MPa).

100

150

50

aT 50 C 100Normal stress (MPa)

Figure 6. Stress state at the brittle-ductile transition pressure

362

2.5

"""' 2.0=~-= 1.5

ftllil 1.00'i

" B• Joint A ('10= 35.6°) , /

.11I Joint B ('10 = 23.3~)_; .... ;;."r.<Pm=46.4° / .0aT= 3.8 MPa , .//.

.. ···)··T········ __········ ....;...

.//

i/- ~.._..... .........•. . ,/'

.// :

. :AI-:·····_·······L ...._..+ ....

.4~--Dila~ion-oorrected ~eak s.s.

0.5 1.0 1.5 2.0 2.5Normal stress (1vIPa)

Figure. 7a. Measured peak shear strength

0.5 ..._....

2.5• Joint A• JointB •

2.0

1.5

1.0

0.5

0.5 1.0 1.5 2.0Normal stress (MPa)

Figure 7b. Dilation-corrected peak shear strength

2.5

4. DISCUSSION

The proposed non-linear peak criterion has the simple fonn:

(10)

where +m is the friction angle of the rock wall material and 'I' thedilation angle at peak strength which is given by:

The angle +m is the non-dilational component of peak frictionangle and can be determined either from triaxial tests at sufficientlyhigh confining pressure to produce a brittle-duetile transition stateas shown in Figure 6, or from the dilation-eorrected shearstrength-normal stress data (Hencher & Richards, 1989) as shownin Figure 7b. From an analysis of published experimental data,typical values of +m for rough textured joints are 39° for silicatesand 45° for carbonates but they may be lower for rocks like schistsand phyllites and in general for rocks rich in low friction minerals(e.g. mica or other sheet minerals). As an example, shear strengthenvelopes for 20 different natural rock contacts, including contactsbetween different rock types, published by Baldovin (1970) areshown in Figure 9a. With the exception of the envelopes No I, 3

LO ! 2 :• Joint A (r = 0.950)• Joint B (r~=0.958) .

~••0

E 0.4s::0...~

0.2

0.0

-0.2 0.001 om 0.1 1 10Normal stress (MFa)

Figure 8. Variation of dilation rate tan'l' with normal stress

(11)

and 4 corresponding to marl-shale, marl-sandstone and shale-limestone contacts respectively, the line + = 39° provides a lowerboundary of peak shear strength. Similar results are shown inFigure 9b, for 13 different natural discontinuities, tested in thenonnal stress range o-2.5MPa. Friction angles lower than 39° areshown only by phyllite and cl.l::h schist If a joint surface has beensmoothened by a natural (or artific.ial) process, or coated by lowfriction minerals, a lower value of'the non-dilational component isexpected, especially at low normal ,tresses. It is then essential thatnon-dilational friction be determined from shear tests.The angle '1'0 is the average asperity angle of the joint and can be

obtained from direct measurements of the surface roughness. Thisangle is equivalent to dilation angle of the joint when all theroughness is mobilised (negligible surface damage); for a rocksample it can alternatively be estimated by performing a shear testunder very low normal load, for example under its self weightMethods employing photogrammetric techniques, profilometry or

the plate and compass method (Peeker & Rengers 1971, Richards& Cowland, 1982) can be used to provide an appropriate value of'1'0 in the field. A base length of 0.2% of the full length of the jointappears to be appropriate for the determination of the angle '1'0'This angle can then be used directly in equations (10) and (11) togive an estimate of the shear strength.Finally, the nonnal stress anT which gives the transition fromdilational to purely frictional behaviour can be estimated from atanw vs. logan graph as shown in Figure 8. This graph will showa wide scatter at low normal stresses as different samples havedifferent roughness and exhibit different dilation angles, but as thenormal stress increases, this scatter is reduced, allowing a reliabll"estimate of anT, as shown in Figure 10, for the experimental dati.sets published by three different authors.

A better correlation between dilation angle and nonnal stress isobtained if a graph of the normalised dilation rate tanwltan'l'o andlogan is plotted. In this way individuality of joints at low normalstress is lost and anT can be estimated more confidently, as shownin Figure 11 for the data plotted in Figure 8. The value of '1'0required for each individual sample in this case can be obtained byself-weight tests prior to main testing at the appropriate normalstress.Figures 8 and 10 suggest that the transition stress anT fromdilatant to purely frictional sliding may be more than one order ofmagnitude lower than the unconfined compressive strength. Datapublished by other workers such as Goodman & Dubois (1972),Martin & Millar (1974), Schneider (1976), Bandis et al. (1981),Denby & Scobie (1984), Leichnitz (1985) and Kutter & Otto

363

1.0

0.8,-..tlS

~ 0.6'-'

~~ 0.4til

;0..c

U) 0.2

Q2 Q4 Q6 Q8Normal stress (MFa)

1.0

4

3

~ paragneiss

o clay shale+ lIlicaschisto phyllite• I1l1ltstoneX graniteo ca Ich schist••. orthognei 55

• schist '1"aphltous+ dolomit.lllll!stone*" shale~ sandstone~ serpentine

*•

2

1

(b)

123Normal stress (MFa)

Figure 9. Peak shear strength data for natural jointsa) after Baldovin (1970) (b) after Giani (1993)

(1990) support this argument The last authors suggested that thisstress is equal to the tensile strength of the rock material. For themodel material used in the present study (tensile strength 4.5 MPaand anT = 3.8 MPa) this may be a reasonable approximation. Theassumption that dilation is fully suppressed only when theaveraged normal stress is equal to the unconfined compressivestrength, as assumed in some models, is not confirmed and may beunrealistic for most natural joints.

It is not clear whether anT depends upon roughness, although itis reasonable to assume that it increases with the degree ofinterlocking. This is a matter that requires further study. Theresults shown in Figure 8 suggest that anT may be independent ofroughness; similar results were obtained by Kutter & Otto (1990).Data published by Bandis et al, (1981) and Denby & Scobie(1984) indicate that a small dependence of anT on roughness mayexist However, this small dependence does not introduce seriouserrors in the determination of peak shear strength because of thelogarithmic form of the equation (II). For example, in the case ofthe data presented in Figure 8, an increase or decrease in anT by

0.8 'Data from :1lI Gyenge & Herget (1977)......+ ..• Leichnitz (1985)~· .

, • Kutter & Otto (l ?90)............... .Ii!................. .•..........•......• ,•..•.......•...............

! GI '~ tan'l1 = 0.206-0:20310go

Iii·:········· (r2= 0.816)·'········11 ..

0.4 .... ··················· .. ·1····

. . ~

0.2 ; .

······ ..·············i··

0.0Ql 1 10

Normal stress (MPa)

Figure 10. Variation of dilation angle with normal stress forjoints in different rock types

100

L2tan'l1/~n'l1o = L~Oo-O.2791ogol1····················!····(·?~··O:955)LO

0.8

0.60.4 ···············_····························i·····

;······_···········f···-0.2

0.0 ..

-0.2 0.001 0.01 0.1 1Normal stress (MPa)

10

Figure 11. Normalised rate of dilation vs, normal stress

50% results in an overestimation or underestimation of the dilationangle 'I' by 1° approximately for a normal stress range an = 0.1-1.0 MPa, assuming '1'0=30°. The new criterion has been testedagainst well documented, published data from a variety of authorsand for a range of rock types and found to be generally applicable.A basic implication of this criterion is that any variation in shearstrength is attributed to changes in the geometrical component.Accordingly, changes due to sample size, roughness anisotropyand normal stress are explained in terms of variations only indilation as demonstrated by Papaliangas et al. (1994). Theimportance of relation (II) for determining the dilation angle atany given normal stress lies in its use of only one single surfaceparameter, the average asperity slope, which can be directlymeasured for any surface,. The strong correlation between thisparameter and shear strength has been emphasised by severalauthors, including Tabor (1975) and Koura & Omar (1982).

The form of the new criterion is similar in some respects, andprobably explains the proven usefulness of Barton's empiricalformula (Barton. 1973). However, the new criterion has a soundtheoretical base and employs physically meaningful parameterswhich can be readily established through careful testing,

364

5

4

1- Proposed criterion<p = 39° i........m ..•._ "_ .'II = 40° '

0,

0aT= 10 MPa3

2 '2 - Barton's equation: <p = 30°,r ,

..........IRC = 10L. ....JCS = 100 MPa

(. )4 5

1

1 2 3Normal stress (MFa)

100 1- Proposed criterion<p= 39°

m. °80'-"-'II~ ,= 40 .._;_. -_....-.

oaT=:' 10 ~a

60 .~ - .Barto.:"~_~~~<:l~. ·-t·<p = 30 :r, 'IRC = 10 :

40 .....JCS = 100 MFa

(b)

80 10000 oaT 20 40 60Normal stress (MFa)

Figure 12. Comparison of proposed criterion with Barton's model(a) Low normal stress (b) High normal stress

.measurement and analysis. There is no need for any index tests orpreparation of surfaces. A comparison between the two models isshown in Figure 12. The two models may give identical results atlow normal stresses, but at higher normal stresses, where Barton'sformula is known to underestimate the measured strength (Barton,1976), the new criterion predicts peak friction angle value equal tothe friction angle of the rock wall material (+p= +m)'

5. CONCLUSIONS

This new simple non-linear criterion for peak shear strength isbased on realistic mechanisms of shearing and widely acceptedtheory of asperity deformation. The basic assumption is that shearstrength is derived from shearing of rock junctions which areunder high normal stress. For quantification the criterion requiresevaluation only of the average asperity angle (function of jointroughness) and the friction angle of the rock wall material (basicrock property) together with the normal stress at which dilation isfully suppressed. The assumptions made are reasonable and therequired parameters can be easily determined through a series ofdirect shear tests. The criterion is simple, has been tested againstexperimental data and found to make accurate predictions. It is

suggested for use either to predict or extrapolate peak shearstrength data.

6. REFERENCES

Baldovin, G (1970). The shear strength of some rocks bylaboratory tests. Proc. 2nd Congr. ISRM. V5, Belgrade. pp.165-172.

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