Empirical failure criterion for a jointed anisotropic rock massCritere empirique de rupture d'un massif rocheux fissure et anisotrope <

Empirisches Bruchkriterium fOrein klOftiges und anisotropisches Felsmassiv

K.THIEL, Institute of Hydroengineering ofthe Polish Academy of Sciences, Gdahsk .L.ZABUSKI, Institute of Hydroengineering of the Polish Academy of Sciences, Gdahsk

ABSTRACT: A criterion for failure of a jointed rock mass built up of rock that shows shearand tensile strength anisotropy is proposed. The criterion is based on the modifiedCoulomb condition, on the relationships describing the variation of cohesion and uniaxialtensile strength with direction, proposed by the authors, and on the shear failure criterionset out by Jaeger for a rock with a single joint. An example is given of how the criterioncan be used for describing failure of rock mass built up of metamorphic mica schists andpossessing two sets of joints. The rock mass considered was the foundation of a dam.The proposed criterion can be used in interpreting the results of strengthrtests carriedout in connection with a stability analysis of the foundation of an engineering structureor of a slope.RESUMI!: On propose un critere de rupture d'un massif rocheux constitue des roches se carac-terisant par und anisotropie de la resistance au cisaillement et a la traction. Ce critereest base sur le critere de rupture modifii de Coulomb, la proposition de la variation dela cohesion et ~e la resistance a la traction uniaxiale en fonction de la ~irection, ainsique sur le critere do rupture par cisaillement de la roche avec une fissure de Jaeger.On presente un exemple de l'application de ce critere pour decrire la rupture d'un massifrocheus compose des schistes metamorphiques a mica avec deux systemes de fissures consti-tuant la fondation d'un barrage. Le critere propose peut etre applique dans l'interpreta-tion des resultats d'essais de resistance lies avec l'analyse de la stabilite de la fonda-tion des barrages ou des talus rocheux.ZUSAMMENFASSUNG: Das Bruchkriterium des klftftigen Felsmassives welches die anisotropischeScher und Zugfestigkeit hat vorgeschlagen wurde. Dieser Vorschlag auf dem modifiziertencoulombischen Bruchkriterium, der eigenen Proposition der Beschreibung der Kohesion und dereinachsialen Zugfestigkeit als Funktionen der Richtung, und das Jaeger-Scherkriterium vonFels mit einzelner Kluft, gegrftndet wurde. Ein Beispiel der Anwendung di~ses Kriteriums fftrdie Beschreibung des Bruches des metamorphischen Glimmerschiefers mit zwei Kluftsysteme,vorgestellt wurde. Das vorgeschlagene Bruchkriterium kann bei der Interpretation von Ver-suchsergebnissen der Festigkeit mit der Stabilit!tanalyse von Dammboden und B6schungengebundenen, benfttzt werden.

1. INTRODUCTIONThe anisotropy of strength of a rock massresults from the lithological features ofrock mas~ jlithogenic anisotropy - seeMOdel I, F~.1), and from the presence ofsets of joints in it (tectonogenic aniso-tropy, Model II , Fig. 1).

When describing the lithogenic anisotropywe begin with empirical relationships deve-loped on the basis of data obtained fromstUdies on metamorphic mica shists carriedout by the authors. This relationships des-cribe how the cohesion and the uniaxial ten-sile strength vary with direction. To descri-be the tectonogenic anisotropy, on the otherhand, the known Jaeger's criterion that de-fines the shear strength of a rock dissectedby a single joint is used, in conjunctionwith the condition of limited tensilestrength.

2. FAILURE CRITERIA FOR MODELS I AND IIThe modified Coulomb criterion that takesaccount the anisotropy of the parameters ofthe shear and tensile strength has the form

a) Model I- shear

6,=[So(CX)(1+tg2~)+63'tgfi(1+)J(o<.),tgP)]'1 (1)tg~ - )J(o<.)

- tension6,+63'tg2p+To(cx)(1+tg2p1:0 (2)

where: So(cx)J)J(cx),To(o<')are the cohesion,the coefficient of internal friction, andthe uniaxial tensile strength, respectively,of the rocki the three quantities areexpressed as functions of the angle ex (for exsee Fig.1). .We can write the criteria (1) and (2) inexplicit forms by substituting empiricalformulae for So(ol) ,,u(cx), and To(Ot).Jaeger (1960) gives such a formula for thecohesion

So(cx)e S, - 52 coslex. (3)where S, and S2 are empirical constants.

McLamore and Gray (in Goodman, 19801 havealso found a formula for the variation ofthe coefficient of internal friction, the

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~LI MODELD

Fig.1. Models of an anisotropic rock massModel r - lithogenic anisotropy, Model II -tectonogenic anisotropy, Model III - combinedlithogenic and tectonogenic anisotropyA - anisotropy surface in the rock, F -failure surfac~,J.1, J;2 -sets of joints.

character of this variation being similarto that of the variation of cohesion in Eq(3)

There are no however empirical relation-ships for To(~) , and this restricts prac-tically the range of use of these criteriato compressive stresses.b~ Model II

- shear of rock61 = 2·So·tgllJ + 63(1+ 2·,IJ·tgllJ) (4)

where: IIJ '=! (45°+ ~/2)So,~ - the cohesion and the co-efficient of internal friction ofthe rock, respectively, the quan-tities which are constant andindependent of direction.

- tension of rock~ =- To,r (5)

where: To,r is the uniaxial tensilestrength of the rock, constant andindependent of direction,

- shear along the i-th set of joints

where: Sie.],i-the cohesion of a jointn'the i-th set,,.uj,i - the friction coefficientof a joint in the i-th set

'tj,i - see Flg.1,- tension along the i-th set of joints:

(7)

where: To j i-the tensile strength ofa ~olnt in the i-th set.

It follows from equations (4) and (6) I

which are the Jaeger criteria, that failuremay occur through shearing off the rock orthrough a slide along the joints. Taking inaddition equations (5) and (7) into account,we consider the possibility of extensionf'racturethrough the body of the rock oralong the joints.

3. PROPOSED CRITERION OF FAILURE FOR MODELIII.

,The general form of the criterion has beenobtained by replacing equations (4) and (5)in model II by equations (1) and (2) , whichis equivalent to replacing the rock withisotropic strength represented by model Iby a rock whose strength is anisotropic~The general form of the failure criterionthus obtained for a rock mass showing bothlithogenic- and tectonogenic strength ani-sotropy comprises equations (1) , (2) , ( 6) ,and (7), cf. Zabuski (1986).

The explicit expression for So(ot)in equa-tion (1) has been derived from the resultsof direct shear tests made on mica schistspecimens of 40x60x40 cm. The specimenswere sheared in·parallel, at an angle of 45°and perpendicularly to the foliation (i.e.ex. 0° , 45°, 90°). This expression is

So(od =Ai + Bi~ (80)

where: Ai and'Bj - empirical constants,i • 1, •••,n, where n is the number ofthe intervals into which the ex - variationrange, from 0° to 90° , is divided inorder to make the curve fit better theexperimental results, in practice n-1 or 2.For the schist ex~mined equation (8a,)has

the form (cf. Fig.2)

50,1 (ot) =QOag + O.o3Zv'(X: , MPo, oc.E <0",45°)

So,2(OC) =0.146 oU02251{cX ,MPa. exE-(45':9:J;(8b)

The assumption that 50 (ex) varies accor-ding to formula (8a) is supported by theobserved fact that, if the shear occursalong the foliation (ex. 0), the individualparticles of micaceous minerals slide oneupon another, while in shear effected in adirection inclined to the follation, even ata small angle, the particles themselves ~resheared. Therefore, since the mechanisms ofshear in two cases are different, one canexpect that the change from ct- 0 to an exonly slightly greater than zero will resultin a considerable increase of cohesion. Atgreater values ofcx the shear mechanismremains unchanged with changes of ex and,consequently, 50 varies within a narrowrange and its variations are not violent.lnFig.2, 650 for the change oi from 0° to 45°is about 0.215 MFa, whereas at greater

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cJ..=o" -m1~ cA=90"

~ --I

04 So(oC)II lJ(o()

. [MAl] IIII

0.3I

)J{ocl------...... ------ --::.-------~...-

1.0

0.6e - data from shear test

0.'

o ro w m ~ ~ 00 W 00 ~oC [degree 1

Fig.2. Shear strength parameters of therock, plotted as a function of the angle ~

values of ex and identical increase in cx,from 45" to 90", gives llSo as small asabout 0.062 MFa.

The results of the studies on shear alsoindicate that the value of the internalfriction coefficient is approximatelyconstant, irrespective of ~ (cf. Fig.2),and thus,u (ex)- con st has been taken.Wi th the parameters SQ (ex) and,u (~) sodefined, the criterion (1) can be put inthe form

, to,=[(Aj .SiVlpm-tfl(1+tg2,A.l.S3·tgpm(1 +,u.tg~m)]tg~m-,u (9)

'Where :1J3m-jl- ex, cr , Fig.1.The tensile strength of the rock has been

determined using the method of transversecOmpression at an angle of O· , 45" , 90"With respect to the foliation of the rock.To'" To(ex) has been taken to vary in the waydefined by the function (curve 1 in Fig.3):

To(ex)=To•o+ K· ex. (100)where: T 0,0 - uniaxial tensile strength in

the plane of foliation,K - empirical coefficient,For the schists examined we have obtained

Tolex) ••0.0716 + 0.0283' ex, MFa (lOb)In the studies on tests mentioned above,

it has been observed that when the load isapplied at an angle of 45" with respect tothe foliation, the rock was not fracturedbut sheared. This suggests that forex-45"the tensile strength is greater than thatfound from the studies (see points for(90·-~)_ 45"in Fig.3), and thus the theo-retical line (10b) lies above these points.To verify the hypothetical function (10a),and its special case (10b), the straightline 1 has been compared with the curvesObtained by Barla (1974) for rocks whosedegree of the tensile strength anisotropyTo 90" IT 0 O. is close to that for micascnists. 'Curves 2 and 3 in Fig.3 representthe tensile strength plotted as a functionof the angle (90" - r ).If the value of the angle (90" - t) rangesfrom 0" to about 50", then extension frac-

3 3

A: cx~Q A': cx=Q

2~ ~

2

~ ~

• - data from tests on mica schists

0 '0 20 :J) 40 so ED 70 eo 90oc (for CD), 9O·-r(for~®). [degree]

Fig.3. Uniaxial tensile strength as afunction of the angle 0(1) , or of theangle (90"- 't) (( 2 ), (3) ).1 - relationship for To(ex) , 2 - tests ongneiss, 3 - tests on serpentinite schist.

ture occurs at ~ ••(f , and not at ~.;,0"marked on the horizontal axis of the diagram.This indicates that the tensile strength at

an angle ex:l:0" must be 9reater than thatfound experimentally and represented bycurves 2 and 3. This is why the theoreticalcurve 1 lies above curves 2 and 3. If theangle (90" - t) approaches 90", then cx••(90· - r), and curves 1,2 and 3 may bepositioned close one to another.

Taking To ••To(cx) as defined by equation(10a), the criterion (2) has the form5, + 63' tg2,Bm+(To,0+K Iftm- rIH1+t91m) = 0 (11)

where: I f3m - '( I = ex: ,cf. Fig. 1•Equations (9) and (11) define the resul-

tant limit surface for a rock with lithogenicanisotropy. However, when /3m '" t , then itis not possible to determine the failuredirection and thus the strength analytically.The angle /3m should be determined numeri-cally, by seeking a minimum (equation (9).. equation (11)) of 6, forvarious values of {3 and constant values of

03 and t , or by seeking numericallysuch ~ ••13m that satisfies the conditiondo,/df3 ••0 and then sUbstituting 13m thusfound in equation (9) or (11).

Assuming in criterion (9) that f3m - 't orf3m ",1jI •• 45· + '/112 gives a simplified formof this criterion. Fig.4. shows the sectionsof the limit surface (9) by the planest= const and 0 = const. Curves I,Ia,

Ib, Ic cor.espond to shear along the folia-tion and curves II, IIa, lIb to shear at theangle 13m -ljJ • Curve III in Fig.4a representsthe approximate section of the limit surfaceby the plane r'" const for the condition

13m +\11 and f3 i: 't • It can be seenfrom the above that the surface determinedfor f3m =~ is the upper limit of the actualsurface III. Numerical calculations have

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(0) 1=0

It=const·1 /

~/

// "(-900

6'1

Fig.4. The limit surface for an anisotropic rock sUbjected to shear, referred to coordinates01' 6"3'')" ~ aj sections of the limit surface by the planes 't = cons'tj b) sections of the

limit surface by the planes 63- const. I,Ia,Ib,~c - shear along the surface of anisotropy(film. r), II, IIa, lIb - shear transversly to the surface of anisotropy, at an angle j3m=~III - shear along surfaces f>m i< rand ftm ~ 1jI. E2Z22- safe region for 63> -So (}) /)J~ safe region for 63<-So (Ol/,.u

-TO(g)") (0) - 1: (g)"-T) -T 0 61

T-Oo I - r-oo°T-90° /I In I

'I , I/

I \ II \ I l;lI I II

I \ ItoosI I '0

I \ I - 9 °-III

~

III \ ./I I //

\ ----..k:/~/ 161·const·1

/./

~ safe regon

°1/O,=6'3=-To.O 63 6'3

Fig.5. The limit surface for an anisotropic rock subjected to tension, referred to coordi-nates, 6'11 63, r ~a) sections of the limit surface by the planes '1" const , b) section ofthe limit surfaCe by the plane 6'3- const; c) section of the limit surface by the plane

f)1"" cons't , I, la, Ib - fracture along the surface of anisotropYftm ""r, II,IIa,IIb - fra-cture transversely to the surface of anisotropy, at an angle ftm" 90°, III - fracturealong surfaces jm" 90°.

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shown that the maximum difference between6"1 , found for /1m" IV , and the actual 61

does not exceed ca. 5% so that in practicalcalculations we can use the simplifiedlimit surface.Fig.5 shows the sections that the planes1''#1 const, 63= const, 6, a const cut

through the limit surface defined by (11).Also in this case curves I,Ia, Ib define thelimit sur~ace that corresponds to extensionfracture along the foliation, where ascurves II, IIa, lIb define the limit surfacecorresponding to fracture at the angle,8m = 90°. Here the surface found for.am" 90°

is the lower limit of the actual surface(i.e. for ftm •• 90° ) whose section by the

plane T= const is shown schematically bycurve III.Numerical studies for various To,o and K

have shown that the value of the angle j3min equation (11) can be found from therelationships

where: 63trr= - [To,o + K(900-'t)]

_ 1 +tg2j3mT63T - - To,o+ K9mT -r) tg2r-tg2pmT

"BmT=66,8°-0482'r , 1E<00,45°) (12c)

j3mT =0.2°+09978·y , rE(4S0,90' (12d)

Equations (12a) and (12b) have been setout based on two facts, namely, that theangle ftmTdefining the direction in whichthe rock is fractured at the point 03Tt cr, Fig. 5) is dependent only on r (equa-tions (12c) and (12dl)and that the angle~ m for a given r is a linear function of

3' that is this angle increases linearlyfrom the value )3mT to 90°, as 03 decreasesfrom 03T to 63.0The criteria ( 9) and (11) establishedabOve predict the following properties ofa lithogenetically anisotropic rock:a) for 63 ~ 0 :- the compressive strength 0, in particularthe uniaXial compressive strength 00 , isalways maximum when 't > 0° and is minimumWhen r • IV = 45°+ ¢/2,- the index of anisotropy as defined byQ'r"o/QI"1fI is maximum when 63 ~ 0 and decre-ases w th increasing 6"3: if 03 _ •••••then theindex of anisotropy tends to unity,- the direction in which the rock issheared coincides with the directio~ of thefOliation or is close to \jJ (,8m~ \jJ -10)the value of /Jm tends to '+' as 63 - • 00

bJ'for 03<0:-over a relatively wide range of r ' fail-~re may be initiated by shear and not byracture,- the direction of failure through fracturei~incides with the direction of the folia-on or is inclined to it, with 13m noteqUal to 90° , except when 6'1 •• + 00.

In considering the tectonogenic anisotropyit has been assumed that generally a rock is

,dissected by n sets of joints. The shear'strength of the i-th set of joints is givenby equation (6) where, with the referencesystem as in Fig.1 if the angle "(.i ispositioned in the 2nd quadrant, thJ'systemis represented in the 1st quadrant so thatthe angle defining its orientation is '(j.i=(180° - rj,i) , The criterion for frac'Eurealong the i-th set of joints is given by(7)

where, if rjdE( 9et,1800), thenlj,i=(laoo-ti,il

4. EXAMPLE OF USING THE CRITERION FOR ADESCRIPTION OF FAILURE OF MICA SCHISTS

We shall consider a rock mass built up ofmica schists in which the parameter So(oc)varies according to (8b), ~ ••0.9227, andthe parameter To (ex) varies according to(10b), and which is dissected by two sets ofjoints J.1 and J.2 whose strikes are appro-ximately parallel to the strike of the sur-face of the rock foliation. The joints inthe two sets have identical strength parame-~~~~ SO,j,i ••0.0, fJj,i" 0.577 and To,j,i:The angle between the J.1 and J.2 is w.670Lcf', Fig.1 l.

Individual segments of the limit surfacehave been determined from (6) , (7) , (9),and (11). By superposing these segments weobtain the resulting limit surface referredto the axes 6" 63 I r.

01 [MPa] 6',

2.0

1.0 1.0

010 20 :Il 40 50 60 70 eo0 so

[degree ] 1'= 1j,1I I I I

'00 l1b 120I I I

6770 eo SK) 1:Jl 140 1!1l1j,2=1j,I+W

G)_rOCk @--"'J.1 @--- J.2--- rock mass

Fig.6. Section of the limit surface for thestrength of jointed mica schists by theplane 6'3 = + 0.1 MPa.

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As an example, Fig.6 shows the section ofthis surface by the plane 03. + 0.1 MPa.It can be seen that the strength of therock mass is determined first of all by itstectonogenic anisotropy and that in thiscase over about 9096of the "t range (0° +

90°), failure occurs by shear along one ofthe two sets of joints. This percentagedecreases with increasing 6'3 and, for

6'3. + 00, which in practice correspondsto 63 > > So(ex) , shear along the jointsoccurs over about 7096of the t - range.

5. CONCLUSIONSThe failure criterion proposed in this pa-per for an anisotropic rock mass permits usto find the strength, the direction ofpossible failure, and also the mechnism offailure by shear or fracture along oracross the surface of joints or along oracross the surface of foliation, layering,schistosity, etc. in all possible statesof stress. To use the criterion one needsto know the strength parameters of the rockand of the ,joints, but these parameters canbe determined by relatively simple strengthtests.The proposed failure criterion for ananisotropic rock describes well the stren-gth and failure conditions of a rock in whichplanes' of weakness (e.g. foliation or schi-s1;osity planes ) are clearly marked.

The criterion presented here has beenverified by the results of investigation onmica schits. It would be useful to verifyit by tests on other rock of this type.The authors suggest that this criterion canbe applied to a wide group of lithogenti-cally and tectogenetically anisotropicrock masses.

REFERENCESBarla G. (1974) • Rock Anisotropy - Theory

and Laboratory Testing, In: L. Mnller(ed.), Rock MeOhanics, Courses andLectures, No. 165, Udine 1974, Springer-Verlag, Wien New York, pp. 35-69.

Goodman R.E. (1980). Introduction to RockMechanics, John Wiley and Sons, NewYork - Chichester-Brisbane-Toronto

Jaeger J.C. (1960). Shear Failure of Aniso-tropic Rocks, Geol. Mag. No. 97,pp.65-72. .

Jaeger J.C., Cook N.G.W. (19691.Fundamentalsof Rock Mechanics,Cbapman and Hall Ltd.,Science Paperbacks, London.Zabuski L. (1986) Failure criterion for ajointed anisotropic rock mass, Inst. ofHydroengineerin~ of the Polish Academyof Sciences Gdansk (PhD Thesis)(Kryterium zn1szczenia sp~kanego masywuskalnego 0 anizotropowych wlasnosciachwytrzym~ciowych).

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