ARMA-90-0237_A Mathematical Model for the Shear Behavior of a Dilatant Rock Joint

8/12/2019 ARMA-90-0237_A Mathematical Model for the Shear Behavior of a Dilatant Rock Joint

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RockMechanicsontributionsndChallenges,usb'ulidJohnsoneds) 1990Balkema, otterdam.SBN90 6191 1230A mathematical odel or the shear ehavior f a dilatant ockjointS.SaebInternationalechnologyorporationlbuquerque,3'lex.B.AmadeiDepartmentfCivilEngineering,niversityfColorado,oulder,olo.

ABSTRACT: A mathematical model is presented o determine the effect ofboundary conditionson the shearbehaviorof a dilatant rock oint. The modelrelates the normal load-deformation esponseof a joint to its shearload-deformation and dilatant behavior.

1 INTRODUCTIONIn two recentpapers Saeband Amadei,1989and 1990), he authorspresenteda graphical method to predict the shear responseof a dilatant rockjoint under constant or variable applied stiffnessboundary conditions. Themethod used the joint shear stress-shear isplacement urvesandcorrespondingdilatancy curves or different constant normal stress evels andthe joint normal stress-normal isplacement urve. The proposedmethod wasalso verified using the results of constant stiffness ests previously reported inthe literature.

In this paper, the graphicalmethod has been shaped nto a more generalmathematical orm that can be included n the numericalmodelingof jointedrock masses.The mathematical model presentedherein makesuse of existingformulationsuchas that of Bandiset al (1983) or oint normalbehavior ndthose f Ladanyi ndArchambault1970),Goodman1976)andGoodmanand St. John 1977) or oint shearbehavior nd dilatancy.At the outset,these ormulationsare summarized.Then, they are coupled o relate thenormal load-deformation esponse f a joint to its shear oad-deformationanddilatant behavior. This coupling s used to predict the increase ndeformability of an initially mated joint as it traversesa range of unmatedconditionsduring shearing. Then the model is presented n an incrementalform that can be implementedn non-linear inite elementprograms.Finally,an example llustrateshow the model can predict the shear response f adilatant joint under boundary conditionsother than constant normal stress.

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0'n

-V m 0-closing-

O'-opening-

Figure . Normal tresss. normal isplacementurvesor a matedoint(curve ) andanunmatedoint(curve )T

Tp1s (a)Up u r u

Up u r(b)

Figure . (a) Shear tresss.shear isplacementurve,b) Dilatancyurve

residual hear isplacementsonstantconstantisplacementodel).Thevariation of the peak shear strength with the normal stresshas been modeledby manyauthors.Consider,or instance,he modified ersion f LadanyiandArchambault's1970) ailure riterion roposedy Saeb 1989)n which pand tt are related as follows:rp= r, an (q - i) (1 - a,) -Fa,ar. (4)

In eq. (4), a, is the proportion f the total oint areashearedhrough heasperitiesnd1 - ao s the proportionnwhich liding ccurs;b s the angleof friction or slidingalong he asperities; r representshe shearstrengthofthe asperitieswhich s alsoequal o the intact rockstrength.The latter canbe approximatedy the Mohr-Coulombriterionor Fairhurst's araboliccriterion ssuggestedy Ladanyi ndArchambault1970).Finally,

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i = arctan (9) where9 representshe secant ate of dilatancyat the peakshearstress.Accordingo Ladanyiand Archambault 1970),a, and/ arenormal stressdependentwithaT

- anio (6)where k and k2 are empirical constantswith suggestedvalues of 1.5 and 4,respectively. an io is the secantdilatancy rate at zero normal stressand aa' isa transitional stressbeyond which shearing through joint asperities s thedominant mechanism of shear. Thus, for an > aa,, 9 = 0 and a, = 1. Theuniaxial compressive trength of the intact rock can be used as an estimate toaa. (Goodman, 976).For the variation of the residual shear strength with the normal stress,considerhe modelof Goodman 1976)where

r, rvBo -Ban when an < or (7)aTand

rr=rp when an_>ar. (8)In eq. (7), B0(0 _


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Since q. (11) mustalso epresenthe oint behavior ndera mated ondition(u = 0) , it followshatf(on) must eequalo the ighthand ide f eq. (2).Then,eq. (11) takes he followingorm

u (10n'" On ,... tanio+ (12)When u > u, and crn< at, the joint ceaseso dilate and v is equal o its valueobtainedby substituting = u, in eq. (12). When crn at no dilation spossibleduring shearing.If the first term n eq. (12) is calledw, theneq. (12) becomes

v-to= on.V. (13)k. V. - onor

- to). n,- (14)Comparingqs. 13) and (14) to eqs. 1) and(2), it appearshat v - tohasnowbeensubstitutedor v. In otherwords, qs. (13) and (14) nowmodel henormal stress-normal isplacement ehaviorof the joint after having beensheared r mismatched y an amountequal o u. Equation 14) hasbeenplotted as curve2 in Figure 1 for an arbitrary valueof u with u _ u,. Curves1 and 2 are a distance o apart as long as crn< crTand coincidewhen crn_>If it is assumed hat the maximum closureV, is a good estimate of theapertureof the joint in its matedposition, hen the valueof to at crn 0, thatis u tan io shownas distance00 in Figure 1, represents he additionalapertureof the unmated oint created hroughdilatancy.The maximumadditionalaperture will be equal to u, tan to. Figure 1 alsoshows hat anunmated oint is more deformable han a mated one.An incrementalormulationanbe obtained y differentiatingq. (12).After rearrangement, this gives

-'2 1- . tardoEquation15) relateshe changen normal tresso the changesnnormaland sheardisplacements.ince rndepends n v and u, eq. (15) can

be rewritten in a more compact form asdan= knndv kntdu (16)

whereknn oa_ nd knt= o are stiffns ccients that depend n o du. eq. (16),k,, is now he oint gent norm stiffns(slope f curvein Figure ) when he oint hm bn sheedby ount equal o u. Theexpressionf k,, reduc to that of eq. (3) whenu = 0, that is, when he ointis in a mated pition.

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It shouldbe mentionedhere that when u > ur or o, _ oT, the joint doesnot dilate anymore nd as a resultdv - 0. Therefore,herewill be no changein normalstress do, = 0) as shearing f the oint takesplace.An equation imilar o eq. (16) can be written for the shearstress sincethe latter also dependson v and u,dr = kt,dv+ kttdu (17)

wherekt. = andku = e twoshe stiffns ccients. theliterature, it is sumed that kt = 0 d ktt= k,. This sumption is notcorrt shown y Amadei d Sb (19) since t d ktt depend n both andu. Equations16) d (17) provide n incrementalon-nearformulationof rk joint deformabiliW.As a direr application f eq. (12), the rponse of a rk joint underappliedconstantor viable normal stiffness ound conditions an bepredicted. This c be done by writing that = K dv where K is theappliedstiffns of the system,which can be constantor alsova with u and,. Substituting = K dv in eqs. 16) and 07) gives he followingrelationshi betweenchges in normal stress,normal displacementdshear displement

K.

d k general, knowing he initial normal stress ross a dilatant joint and itsshe stressvers shear displement and dilatcy cues at differentconstantnormal strs levels, he above ormulationcan be used o predictthe joint she rponse under vious boundaryconditions.For example,Figures 3a-3c show the rponse of a rk joint under vious constant normalstressesangi between0 and 20 units and for severalapplied constantnormal stiffness K of 0.1, 10 and 10 units. The units c be arbitrilychosen long they e consistent.The joint peak shear strength wsumed to follow modified dnyi nd Archambault'sriterion eq. (4))with = 20 units, o = 10 degrees, = 30 degr, and s w defined yusing Fairhurst's parabolic criterion. The ridual she strength w obtainedby takingB0 = 0.7 in eq. (7). A constant isplementmodelw selectedorthe oint she behavior nderconstant orm stresswith = 4 unitsandu = 14 units. An initial norl stressof 4 units w sumed a startingpoint for all threeconstt stiffness aths.Equation 12) w used oconstruct the normal strs-normal displement curv of Figure 3c fordifferent levels of shear displement rging between 0 d 14 units. Thenormal strs-shear displement curves and the norm displement-shedisplementcurvesn Figu 3b wereobtained singeq. (18) anddv = /K with smallsheardisplement ncrements. hen, the she

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17.5

15.012.510.0

7.5

5.02.50.0

15 20

o181614

1.o - ooo.o8 f lO.O

--o.1

5 10 25 30u

(a)

2.5

v 2.01.51.0

0.50.0

-0.5-1.0

4,f = 0.1

6-f{ = 10.0f-1000.02O

0 5 10 15 20 25 30u

(b)

22.520.017.515.012.510.0

7.55.02.50.0

-- -0.5 0 0.5 1.0 1.5 2.0 2.5

(c)

Figure3. Response urves or constantnormalstress nd stiffness aths(K = 0.1, 10, 1000) (a) Shearstress s. sheardisplacement,(b) normaldisplacements. sheardisplacement,(c) normalstress s. normaldisplacement243


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stress-sheardisplacementcurves of Figure 3a for the three constant stiffnesspathsweredeterminedsing q. (19). These igures how learly hat jointshear strength is affectedby the applied normal stiffness.4 CONCLUSIONSA mathematical model was proposed n this paper to predict the responseof adilataa-xtock joint for various boundary conditions. The model couples hejoint normal load-deformation response o its shear load-deformation aa-xddilataa-xcy ehavior. It predicts the change n normal deformability aa-xdaperture of an unmated joint as it undergoesshear displacement starting froma mated position. Finally, the model was presented n aa-xncrementalnon-linear orm that can be implemented n existing finite element progrm..Exampleof suchan implementationanbe found n Saeb 1989).5 REFERENCESAmadei, B. & Saeb, S. 1990. Effect of boundary conditionson the

constitutive modeling of a dilataa-xtock joint. Key note lecture. Proc. Int.Conf. on Rock Joints. Loen. Norway.Bandis, S.C., Luresden, A.C. & Barton, N.R. 1983. Fundamentalsof rock

joint deformation. Int. J. Rock Mech. Min. Sci., Vol 20, No. 6,pp. 249-268.

Gerrard, C. 1985. Formulations for the mechaa-xicalroperties of rock jointsProc. Int. Syrup. on Fund. of Rock Joints, pp. 405-422. 249-268.Goodman, R.E. 1976. Methods of geologicalengineering n discontinuousrocks. West Publ. Company

Goodman, R.E. & St. John, C. 1977. Finite element analysis or discontinuousrocks, in Numerical Methods in Geotech. Eng., pp. 148-175.

Ladanyi, B. & Archambault, G. 1970. Simulation of shear behavior of ajointed rock mass. Proc. 11th Syrup. on Rock Mech., pp. 105-125.

Saeb, S. 1989. Effect of boundary conditionson the behavior of a dilatantrock joint. Ph.D. Thesis, Univ. of Colorado at Boulder.

Saeb, S. & Amadei, B. 1989. Effect of boundary conditions on the shearbehavior of a dilataa-xtock joint. Proc. 30th U.S. Syrup. on Rock Mech.,pp. 107-114.

Saeb, S. & Amadei, B. 1990. Modeling joint responseunder constaa-xtrvariable normal stiffness boundary conditions. Int. J. Rock Mech.(in press).

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ARMA-90-0237_A Mathematical Model for the Shear Behavior of a Dilatant Rock Joint