Einstein, H.H., Veneziano, D., Baecher, G.B. and Oreilly, K.J., 1983, The Effect of Discontinuity...

8/14/2019 Einstein, H.H., Veneziano, D., Baecher, G.B. and Oreilly, K.J., 1983, The Effect of Discontinuity Persistence on Rock Slope Stability

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Int. J. Rock , \f , h . Min. Sci. G lnrech. Abslr. Vol. 20. No.5. pp. 227-236. 1983Printed in Great Brilain. All rights reserved0148-9062/83S3.00 + 0.00Copyright :D 1983 Pergamon Press Lid

The ffect iscontinuity Persistenceon Rock Slope StabilityH H. EINSTEIN*D VENEZIANO*G. B BAECHERtK J. O REILLY;

Discominuity persistence has a major effect on rock mass resistance (strength)but, as direct mapping discominuities imemal to a rock mass is not possible,persistence is a difficult parameter to measure. As a result, the conservativeapproach assuming full persistence is often taken. n this paper a methodis developed for relating rock mass stability and hence persistence to thegeometry and spatia/variability discontinuities. he method is applied toslope stability ( alculations ill which the probability offailure is related todiscominuity data, as obtained in joim surveys. The complete method makesuse dynamic programming and simulation, but a closed form expressionsatisfactory for most purposes is also presemed.

0 1 - AREA OF INDIVIDUAL JOINTAD - AREA OF JOWT PLANE

defined as the fraction of area that is actually discontinuous. One ca n therefore express as the limit

in which L s is the length of a straight line segment Sandts is the length of the i lh joint segment in S; or for ap ~ r t i u l r joint Fig. 2 ,

I

2a)

I:aDjK = lim - -

0 Din which D is a region of the plane with area Dan d ais the area of the i lh joint in Fig. I . The summationin equation l) is over all joints in D Equivalently, jointpersistence can be expressed as a limit length ratio alonga given line on a joint plane. In this case,

I: sK = lim - -

l s x L

TRADITIONAL DEFINITION OF JOINTPERSISTENCE AND ASSOCIATEDPROBLEMS

With reference to a joint plane a plane through therock mass that contains a patchwork of discontinuitiesand intact-rock regions , joint persistence K is usually

Discontinuity hereafter referred to as joint persistenceis among the parameters most significantly affecting rockmass strength, and is a problematic one. While relativelysmall bridges of intact rock between otherwise continouous joints substantially increase strength, the mapping of each joint is impossible on a practical basis. Anattractive alternative to separately considering specificjoints is offered by statistical techniques for samplingand describing the geometry of discontinuities.These techniques are at an early stage of development,but offer a significant advancement of the state of the art:they characterize persistence as a random variable and,in conjunction with a mechanical model of rock failure,produce the probability distribution of rock massstrength.

A method is developed here for rock-slope reliabilityanalysis based on a probabilistic characterization of thejoint system. In doing so, it is found convenient tomodify the traditional definition of rock persistence byaccounting for the uncertain failure path. Precedingwork is briefly described, an d is followed by a description of the present method.

INTRODUCTION

Professor, tAssociate Professor and :j:Formerly Research Assislanlat: Department of Civil Engineering, Massachusseus Institute ofTechnology. Cambridge. MA 02139. U.S.A.

K lim fODI Jointed AreaAo GO AD - Total Area

Fig. 1 Joint persistence.227


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228 EINSTEIN et at.: DISCONTINUITY PERSISTENCE AND SLOPE STABILITY

/ rock bridQ8 RBR

~ j n t ~ m n t (JL)

IJLIJL+IRBRFig. 2. Joint persistence as length ratio.

\ PL NE FAILURE

Fig. 4. En-echelon and in-plane failures.

Fig. J. Jennings' relations.

1

in which N is the number of critical paths (failing andnot failing) and Nf is the number of critical paths for

where tan cPa and ca, so-called Jennings' equivalentfriction and cohesion parameters, are given byCa = 1 c r KCjtan cPa = (I - tan cPn + K tan cPj (7)

The use of equations (6) and (7) for shear resistance ofjointed rock masses has several shortcomings:I) Failure surfaces are restricted to joint planes. Enechelon failures (Fig. 4), common in the field, areneglected.2 Shearfailure does not typically occur rthe usuallylow values Ta For example, for slopes of 30m (100 ftheight, a is about 0.7 MPa (100 psi), whereas cr istypically ten to hundred of MPa. If CT. is negligible, thenthe major principal stress must exceed Cr for shear failureto occur (Fig. 5). This is unrealistic, as Lajtai [6] andStimpson [9] have pointed out. Also, peak shear resistance in the intact rock and on the joint probably are

not mobilized simultaneously.(3) Small variations persistence produce large vari-ations resistance. Therefore, even modest uncertaintyabout persistence forces the designer to the conservativeassumption of 100 persistence.

To surmount these difficulties a new definition of persistence is required.NEW CONCEPT OF PERSISTENCE

Any planar or non-planar surface (or path ) throughintact rock and joints in a rock mass (Fig. 4) constitutesa potential failure surface (failure path) with associateddriving force L and resisting force R. For a givenconfiguration of the joint system and a given set ofstrength parameters, there is a path of minimum safetyor critical path (Fig. 6). The critical path for aparticular joint configuration is that combination ofjoint and intact-rock portions having the minimumsafety margin M = R - L. If the M for this path isnegative, the rock mass fails; otherwise it resists. Thus,a critical path mayor may not be a failure path. Theprobability of failure Pf of a randomly-jointed rock masscan be expressed as the limit of relative frequency offailure across the spectrum of joint configurations,

n I Nf~ tm (8)N N

(3)

(4)

(6)

(2b)

E ajI = lim v ~ ~ V

l;JLK = l;JL+ERBR

Another useful index of rock mass discontinuity is jointintensity I defined as the area of joints per unit rockvolume,

Rr = CTa tan c r + cr Ain the case of intact rock and

in which a is the area of the i 1h joint in a 3-D region ofvolume VJoint persistence can be used to estimate the strength

of a rock mass against sliding along a given plane: if theplane of sliding has area A then shearing resistance canbe adequately expressed as

Rj = CTa tan c j+ cj A 5in the case ofcompletelyjointed region. c r and cPj are thefriction angles of intact rock and the joint respectively,Cr and Cj, the intact rock- and joint-cohesion. In bothcases, 1a is the average normal stress across the regionof sliding. If the sliding region is partitioned into anintact-rock portion of area A r and a jointed portion ofarea Aj = A - Ar(Fig. 3 , then following Jennings [5] onecan evaluate the shear resistance to sliding, R, as aweighted combination of Rr and Rj according to theexpression


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EINSTEIN l oJ DISCONTINUITY PERSISTENCE AND SLOPE STABILITY

CT

ra- Il VnarFig. 5 Mohr s circle at failure predicted by Jennings relations at low stress levels.

which SM 0 the number of failure paths). Equation 8) suggests a way to estimate Pr: using statisticalinformation on joint length and spacing distributionso ne can si mul ate a n umber of networks of joints such ast hat in Fig. 7 and t hen d etermin e t he SM f or all possiblepaths in each network or configuration. The critical pathfor a configuration of the type in Fig. 7 is obtained byidentifying th e p at h of m in im um SM among all oramong a reasonable number of in-plane and en-echelonpaths. In some configurations the critical path will be afailure path SM I) while in others will not be

Fig. 6 Critical paths for different joint configurations.

SM I). Simulating many configurations realizations)represents various ways that joint populations with thesame spacing and length characterization may manifest.and at the same time p ro du ces the p ar amet rs Nand ff or use in equ at io n 8). In anyone realization the SM forthe critical path can be used to calculate an apparentpersistence, and thus one o bt ai ns a relation betweenresistance and apparent persistence for a rock masscharacterized by joint length and spacing distributions.To date, these principles have been applied to 2-Dslope st abi lit y models in which the p at tern of jointing,

and str en gt h coefficients, are assumed similar for allcross-sections: 3-D extensions for slope and tunnel applications have been limited [7]Earlier probabilistic D slope models

Call and Nichola s [I] a nd G ly nn [ ] have developedmethods of 2-D slope stability analysis that use statistical information on jointing and that allow for bothin-plane and en-echelon failures. The model of Call andNicholas considers two random joint sets. Given distributions of j oi nt length a nd s ep ar at ion a nd of spacing

Fig. 7 Join, configuration and its critical path in a portion of the rockmass.


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230 EINSTEIN / Ill.: DISCONTINUITY PERSISTENCE AND SLOPE STABILITY

Step path aangle

Tensile ailure 100 Fig. 9. Rock slope with single set of parallel joints.R NDOM VARIABLES

where

Fig. 8. Can and Nichol..s [1] model-gener.. ' Icatures.

J

(10)Critical pathlINr

Pr = N

Fig. 10. Elevation intervals H,.

ell prob bilistic persistence model SLOP SIMSLOPESIM is a computer code for the analysis ofrock slopes that contain one set of parallel joints (Fig.

9). SLOPESIM use Monte Carlo simulation to generatejoint patterns ( realizations ) in accordance with givenprobability distributions of joint length and plane spacing (corresponding to the statistical information on jointlength and spacing taken in surveys). For each exit pointthe algorithm finds the path of minimum SM. Criticalpaths may be planar or may involve transit ions tooverlying joint planes. The distribution of SM and inparticular the probability of unstable paths (SM < 0)depend on the elevation of the exit joint . SLOPESIMestimates the distr ibution and probabili ty by groupingexit points according to elevation intervals (Fig. 10). Forexample, the probability of unstable pa ths for the i lllelevation interval is calculated as:

Apparent persistence depends explicitly on strengthalong the path and implicitly on configuration. Repeatedsimulation of the joint pattern yields in a distribution ofKa to be used in a probabilistic version of Jennings'approach. The result is probability of slope failure.Both previous models consider stochastic jointing andfailures occuring in plane or an echelon. Results areexpressed as probability distributions. The limitations ofthese models are that they (I ) apply to a specific rockmass geometry and (2) use unsatisfactory, mechanicalmodels (i.e. shearing is ignored or unrealistically treated)[7.2).

9

Master Joint Setross Joint Set

Dip }LengthSpacingOverlap

bclween joint planes for each set. the procedure simulates critical step-paths as shown inFig. 8. Specifically. for each simulated realization of thejoint network. exit points arc identified (i.e. intersections of the shallow joint planes with the slope face).and the critical step-path through each exit point isfound. Critical paths are obtained by alternately following join ted segments. which fail in shear, and tensilefractures through the intact rock between joint planes.Shear failure of intact rock bridges between joints isconsidered improbable except for extremely shortbridges 6 cm). Faced with a choice among pathsthrough intact rock. the model chooses that with lowestangle. Through simulation of the joint ing pattern, themodel calculates the distribution of average step-pathangle and fraction of path containing jointed segments.the latter taken as a measure of persistence. Thesedistributions are conditioned on slope geometry.strength parameters, and spacing.Glynn's [2,3] JOINTSIM model generates joint networks with exponential distributions of spacing andlength. The strength of intact rock bridges. in plane oren echelon. is determined by superimposing a negativeincrement of horizontal stresses ll and a positiveincrement of shear stress AT on the initial state of stress.such that failure is caused. Using this calculated strengthof rock bridges and the resistance of the jointed segments. the critical path for a given joint pattern iscalculated. Apparent persistence Ka is defined as thevalue of along a joint plane that would have the sameresistance as the failure path,

R - RKa=R _R, J

R = resistance of failure path.RJ = resistance of joint plane if 100 persistence.R, = resistance along joint path, if intact rock only.R, = resistance along joint path, if intact rock only.


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EINSTEIN el at DISCONTINUITY PERSISTENCE AND SLOPE STABILITY 231

Fig. II. SLOPESIM method of slices.

in which is the total number of simulated exit joints critical paths) in the it h elevation interval and Nr is thenumber of such exit joints associated with u ~ s t l ecritical paths i.e. failure paths). Th e identification of thecritical path through each exit point is performedthrough a dynamic programming method: the algorithms ta rt s with the exit poi nt s on the top of the slope an dprogresses backwards towards the exit points on the faceof the slope. During this b ackw ar ds pr ogr es sion , thealgorithm considers all the physically realizable pathsthrough a discrete set of points, including the en d pointsof each joint. Fo r more details on this procedure, see [3].

An important feature o f SLOPESIM is the realisticmodeling of failure mechanisms. Driving and resistingforce calculations ar e based on t he me tho d of sliceswhich is common to many deterministic slope stabilitymethods the method as applied here is simplified byneglecting interslice forces). The principle is illustrated insimplified form in Fig. II : the slope overlying the failurepath is partitioned into a series of vertical slices,bounded at their b ot to m end by joints or intact rock.Th e t ot al dri ving force L an d the total resistance Rarecalculated by summing slice contributions, i.e.

A fundamental feature of Lajtai s model is in theanalogy of shear resistance of i ntact rock bridges toresistance in direct shear tests. A t l east for s hort i nt ac trock bridges, this analogy is justified by the assumptionof rigid body motion o f the overlying unstable wedge inthe direction o f joi nting. In direct she ar tests, the resistance o f intact rock ca n be mobilized in on e of twoways: At relatively low stress levels 0 . small), the application of shear stress in the direction of movement leads

to a min imum p rincipal s tr ess 3 equal to t he tensilestrength of intact rock. Hence in this case, failureoccurs by tensile fractures that develop at high anglesto the direction of sliding Fig. 12a). Simultaneouslywith t he a ppe ar an ce o f these fractures, peak shearresistance fa in the sliding direction is attained. Thereafter, shearing at residual stress values takes place inthe direction of sliding.

At higher normal stress levels, the minimum principalstress does no t exceed the tensile strength an d failureoccurs when fa equals the shear resistance defined byth e Coulomb failure criterion. In this case, shearfractures develop in the sliding direction at the timewhen t he appl ied shear stress is maximum see Fig.12b).

0 0 ~ ~ Primary tension. fracture high angle1 I / I Secondary low anglet shear fracture

Th e two mod es ca n be visualized by use of Mohr scircle. In a direct s hear t test, the cen ter of Mohr s circleremains at all times at O a/2 as the shear stress varies fromzero to the value at failure. Fo r small 0 . F ig. 13),Mohr s circle becomes tangent to the failure envelope at0 = - Ts f = 0 an d thus failure occurs in tension ModeI . Fo r larger 0 ., the center of Mohr s circle lies more tothe right an d the point of first tangency is located on thelinear portion of the envelope. Thus, as shown in Fig. 14this m ode of failure mod e 2) c orr es po nds to s hea rfailure in the traditional formulation as used by Jennings[5]. Both types of failure ca n occur, b ut M od e 2 probablyonly in high slopes with weak i nt act rock. T hus , Mode2 is neglected in the following discussion.

Failures may be in plane or ou t of plane en echelon).D ur in g in- plane failure, tens ion crack s develop first,

10)

7 7SM - 1: SMi-1: Rj-Wisinai i

L = Wi sin x,I 1 FAILURE IN TENSION

where ct is the angle of jointing, Wi is the weight of thei 1h slice, a nd R i is the peak s hear force mobilized by theportion of path underlying that slice. Th e i 1h portion ofthe path may be jointed, in which case Rj can becalculated thr ou gh equ atio n 5), or it may consist ofintact rock. In the l at ter case Ri is best calculated usingrock resistance criteria in Lajtai [6] an d Einstein al [ ]

R = I: Ri , 11 ) 0 0 _ 1 Low angle primary shear

b FAILURE IN SHEARFig. 12. Direct shear failure modes--after lajlai [6J.


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232 EINSTEIN al.: DISCONTINUITY PERSISTENCE N SLOPE STABILITY

Fig. 13. Mohr s circle-failure by tensile fracturing (Mode I).

(15)j= Wjsincr

IN PLANE FAILURE OF INTACT ROCK-PRIMARY TENSION FRACTURES @ lB, TO JOINT PLANE

Fig. IS. In-plane failure of intact rock-secondary shear fractures inthe joint plane.

in which is the distance between the joint planes thatdefine the bridge (Fig. 17) and T, is the intact rock tensilestrength. The contributions to resistance from intactrock bridges and from joint segments are added toobtain the total resisting force associated with a givenpath.The driving force associated with the same path isassumed to be due solely to the overburden weight; it is

therefore calculated as the sum of the driving forcecontributions L j from each slice above a path segment(Fig. II . If denotes the angle of sliding (joint angle)and Wj is the weight of the jl h slice, then

The SM of a given path can thus be calculated. (Theeffect of cleft water pressure has not been included in the

14=T X

Point oftonllney

followed by secondary shear fractures (Fig. 15a and b).The angle of the tension cracks 0. can be obtained,fromMohr s circle. The same mechanism applies to out-ofplane failures with low-angle transitions [withfJ - rx) < 0. see Fig. 16], whereas for high angle transitions a continuous tension crack occurs directly betweenjoints, without secondary shear fractures (Fig. 17).Therefore, Mode I failures with initial tension fracturesencompass the entire range of geometrical conditions inslopes with a single joint set, including in-plane as wellas low- and high-angle out-of-plane transitions.In summary, intact-rock resistance R can be calculated as follows: For in-plane or low-angle out-of-planetransitions fJ < 0. + (X),R=l ad, (12)in which is the in-plane length of the rock bridge(Fig, 6 and T. is the peak shear stress mobilized in thedirection of jointing. In terms of the intact cohesion c,and the ratio c = Talc the peak shear stress is

T a ~ (13)For high angle transition fJ + cr),

Fig. 16. Failure of low angle < ll + xl) transitions throughFig. 14. Mohr s circle-failure by shear fracturing (Mode 2). intact rock.


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EINSTEIN et al.: DISCONTINUITY PERSISTENCE AND SLOPE STABILITY 233

Fig. 17. Failure of high angle transitions through intact rock.

analysis. Although in principle simple to do, such afeature would only be consistent if the spatial variabilityin pressure distributions were expressed, which has notbeen done so far.) SLOPESIM uses dynamic programming to scan the large number of potential failurepaths and to identify, for each exitjoinC', the path withminimum SM.The program has been used to conduct a parametricstudy of rock slope reliability, aimed at identifyingcritical variables and at obtaining simple reliabilityformulae. Results from the numerical analyses are summarized in the following sections.PARAMETRIC STUDY OF SLOPERELIABILITY

Slope safety depends on a variety of parameters,which for the most part describe geometry and resistance. In the course of the parametric study, theseparameters have been given values according to Table I.Slope safety resulting from the parametric studies hasbeen expressed in several terms:I Failure Probability Pr z , is defined as the ratio of

critical paths having negative SMs toall critical paths. as a function of the depth z (Fig. 18at which the paths daylight on the slope face. a jointplane daylights in height interval i on the slope face(Fig. 10 , the probabili ty that at least one wedge belonging to this interval is unstable is Pr z, where Zi is thevertical distance between the mid-point of the heightinterval and the slope crest. At present, the only way tocalculate Pr z is through repeated Monte Carlo simulation of the network of joints. However, it is possibleto obta in analytical lower bounds to Pr z . One suchbound, in many cases close to the exact value, is derivedin Appendix A.

2 Probability Distribution of Apparent PersistenceKQ Ka is the average persistence along an existing jointplane that produces a SM equal to that of the associatedcritical path, the plane and its critical path daylightingat the same point on the slope face (the critical path maybe the particularjoint plane or it may have an en echelonshape involving other joint planes). I t follows from thedefinition that Ka is not smaller than the actual persistence of the plane, K. The probability distribution ofapparent persistence depends on depth. Of special interest is the variation with depth of the mean value mK andthe standard deviation K which together with thecritical persistence defined 'below are used to define asecond moment reliability index.

3 Critical Persistence K, The critical path is unstable and failure occurs if exceeds the critical value ofpersistence Kc which, in using the parameters in Table I,is given by

Table Parameters and their ranges used in SLOPESIM parametric studyParameter Value orSymbol Definition rangeGeometric parameters (see Fig. 18H Slope height Fixed 30 m (100ft)= Depth below slope apex 0-30 m (0-100 ft)1 Slope angle 50-90 Mean joint length (joint length 3 2m (lQ-40ft)assumed exponentially distributed)RBR Mean rock bridge length Not directly varied.(rock bridge length assumed considered by

K exponentially distributed) varying persistenceMean joint plane persistence 10-73SP Mean joint plane spacing (joint 0.6-3 m (2-10 fllplane spacing assumed exponentiallydistributed)I Mean joint intensity. a derived_ Kvariable I =-. SPI 3D-80l Jomt plane ang eResistance parametersc, Intact-rock cohesion 0.30-24.0 MPa(assumed to be twice the rock (8-500 ksf)tensile strength)

t Intact-rock friction angle (not Fixed 30important at low stress levels)c Joint cohesion Fixed 0 Joint friction angle Q-40Other parameter. Unit weight Fixed 2.2 g/cmJ (150Ib/ftJ)


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234 EINSTEIN e al.: DISCONTINUITY PERSISTENCE AND SLOPE STABILITY

JL Cr4 Second-Movement Reliability Index p is the num

ber of standard deviations separating the mean Ka fromthe critical value Ke,

For FK K) the cumulative distribution function of apparent persistence at a given depth , the probabi li ty offailure at that depth becomes

Pr= I FK Ke= I FK mK paK) 18

The sensitivity of these four safety measures to theparameters of Table I has been studied by varying oneparameter at a time within the specified ranges, whileholding all the other parameters fixed at given values. Ofspecial interest is the dependence of Pron depth z whichis shown in Fig. 19. The same figure contains a plot ofthe lower bound PI (see equation A2 in the Appendix)which for slope heights up to 30 m (100 ft) and fortypical values of c, (c, 24 MPa (500 ksf) provides agood approximation to Pr (accuracy depends also onother parameters, such as the mean joint length JL andthe friction angle for the joints, cPj) The parametricstudy also revealed that dependence of Pr, mK and ponintact-rock strength c, for depths up to 30 m (100 ft), issmall. This is a welcome result because it allows one tocalculate the reliability index { in equation (17) after asingle use of SLOPESIM to obtain representative valuesof mK and aSimilar sensitivity analyses were made with respect tothe other parameters of Table I (see [7]), leading to thefollowing conclusions:

The influence of strength parameters c, and c usuallydominates other parameters. Slopes with high values ofCr and rpj (c r 24 MPa (500 ksf), cPj ex tend to bereliable at all depths investigated (up to 30 m (100 fl,regardless of the other parameters. When Cr and c aresmall, joint and slope geometry become important.Among the joint geometry parameters l< JL, SP), mean

J

Cr =1.2-24 MPa4>i =o8 =600=40y = 2.211/cm3

X=40SP= 1.5mK=50

30

40

z_Fig. 19. Effect of intact-rock cohesion (c,) on Pr=).

0 ~ ~ 7 ~ 5 : : ~ 1 5 : : : ~ ~ 3 0m0 25 50 100 ft

50_

persistence K has the largest influence. Changing themean joint length JL may also substantially modify Prand { at any given depth, whereas mean joint spacing SPplays a less significant role. For small K K < 3 0 / ~ Prdepends on K and SP almost exclusively through theratio K/SP, which to fi rst-order accuracy equals themean intensity I

Joint inclination, ex has varying effect of reliability.Values of for which reliability is smallest are typicallyaround 45. As a increases above this value, approachingthe slope angle, reliability increases due to the decreasein driving force. This effect is especially significant inslopes with weak intact rock (cr < 4.8 MPa (100 ksf)and weak joints cPjex). Reliability increases also as decreases from 45, especially as it approaches cPjSlope depth z is a very significant parameter. For thisreason, probability of failure is presented here as Pr zcurves. Figure 20 shows a few such curves for differentslope parameter combinations and leads to two obser-vations:

I) The shape of Pr z does not vary much with anyparameter and displays a minimum at a characteristicdepth , Ze This depth is in some cases outside the rangeshown. That Pr decreases with slope height before increasing again for H > Ze seems at first glance to beincorrect. However, the result is in fact correct: i t is dueto the overriding part played by persistent portions ofjoints for low slope heights (and thus small drivingforces). For fixed mean joint length JL the probability ofa 100 persistent joint increases as z decreases at Z = 0the probability of failure equals the average persistenceK). Observations in nature are cons istent with thisresult. Natural slopes are often convex near the crest .Although weathering effects playa role in this geometry,

(17)

(16)

Ke mu _ .- .a a

Fig. 18. Slope geometry for parametric study.

w cos c

Ke= 100 _2c(tanex - tan cPi ,Jk+l 2c tan c iwhere


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EINSTEIN al : DISCONTINUITY PERSISTENCE AND SLOPE STABILITY 235

K I ~ - - - - - -

0 z 0 z zi

t\\ \ P\/PI .... 0 z 0 z z z

Fig. 20. Probability of failure Pr as a function of slope depth: and of the other slope paramctcrs.

the higher probability of a joint isolating a wedge n earthe crest is significant.2 Prj is high c/>j oc or Cr > 24 MPa 500ksf . Also when bot h Cr and c/>j are small c/>jocand Cr < 4.8 MPa 100ksf the approximation remainsgood for a) Shallow depths [z ~ 6-9 m 20-30 ft)] b) Sho rt joint length [JL ~ 3-6m 10-20 ft)]c) Large joint-plane spacing [SP > 3 m 10 ft)] d) Joint inclination angle oc close to c/>j orto 8.

At very large depths, on the order of 300 m 1000 ft), ifCr is very large) the approximation Pr z ::::::: p z losesaccuracy regardless of the other parameters.

The parametric analysis has been instrumental inassessing the influence on reliability of strength andgeometry parameters. Critical parameter combinationsthat lead to failure have been identified. These parameters e.g. JL an d K if Cr and c/>j are small) should beaccurately determined.

EXTENSION OF THE SLOPESIM APPROACHIn its present form, the model is limited to slopes witha single parallel set of joints and neglects 3-D effects.However, extensions are possible. For example, Shair [8]has developed a version of SLOPESIM for two paralleljoint sets. As expected, reliability is smaller than inotherwise comparable cases with a single joint set, theplagnitude of the safety decrease depending on theparticular p aram eter comb in ation . A first attem pt atincluding the third, along-slope dimension has beenmade by O Reilly [7], but more work is needed. Also, the

mechanical models of joint and intact-rock failures cancertainly be improved. Finally, procedures of the typeused here for slopes have potential application in rockmass stability problems in tunnelling and can be ex-tended to problems of rock-mass deformation and flow.

CONCLUSIONSJoint persistence has a major effect on rock-mass

resistance, and yet, it is difficult to define a persistenceparameter simply and directly related to resistance.First, joint geometry internal to a rock mass is notknown with certainty, a nd second, failure involves acombination of mechanisms. including shearing alongjoints and failure through intact rock, either in plane oren-echelon.The proposed approach expressed probability of rockslope failure as a function of joint geometry and intactrock and joint resistance. Spatial variabili ty of jointgeometry is taken into account by making use of statistical information obtained from standard joint surveys.Probability of failure as derived with the SLOPESIMapproach or the related expression of apparent persistence thus makes it possible to represent the effect ofjoint persistence directly.Parametric studies show the relations between rockslope reliability I Pr) and various mechanical and geometric parameters; graphs of probability of failure vsslope height are particularly illustrative. An importantresult is the indication of when strength parameters arem ore imp ortant than geometry, and vice versa.Although an initial step, the proposed approachpromises insights into a major problem in rock mechanics.

Receit ed 22 Decemher 1982; reri red 20 y 983u lOIS


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236 EINSTEIN , I al.: DISCONTINUITY PERSISTENCE AND SLOPE STABILITYREFERENCES

I. Call R. D. and Nicholas D. E. Prediction of step path failuregeometry for slope stability analysis. Proc. 19Th U.S. Symp. on Rock,\fee/ranics (1978).2. Einstein H. H. el al. Risk analysis for rock slopes in open pitmines. Parts I-V. USBM Technical Rept J0275015 1980).3. Glynn E. F. A probabilistic approach to the stability of rockslopes. Ph.D. dissertation. M.LT. February. 1979).4. Hasofer A. M. and Lind N. C. Exact and invariant second momentcode format. ASC J. KnK Mech. Dil . 100. 111-121. No. EM I.Proc. Paper 10376 February. 1974).5. Jennings J. E. A mathL'II1atical theory for the calculation of thestability of open cast mines. Proc. Symp. on Iht Theoretical Back-ground 10 Ihe Planni K Opt n Pil Afines. pp. 87-102. Johannesburg 1970).6. Lajtai E. Z. Strength of discontinuous rocks in shear. Geolee/miqlll19 2). 218-233 1969).7. O Reilly K. J. The elTL Ct of joint phase persistence on slopereliability. M.Sc. thesis. M.J.T 553 pp 1980).8. Shair A. K. The effect of two sets of joints on rock slope reliability.M.Sc. thesis. M.LT. 307pp 1981).9. Stimpson D. Failure of slopes containing discontinuous planarjoints. Pro(. 19Th U.S. Symp. II I Rock Afedwnics. pp. 246-300 1978).

APPENDIX AAn a l l a ~ r i c a l 10ll er bound 10 r prohahility (if slopt failure

The probability of failure. P, :) has been defined as the fraction ofunstable critical paths that daylight at depth : . Lower bounds to P, :)can be obtained by constraining the geometry of the critical path andthe pattern of jointing that can produce failure. One such bound isobtainL d here under the following conditions: with reference to Fig.A I. failure of th e j oi nt pl ane exiting at : can occ ur only if: I) The joint plane AN is 100 persistent. i.e. L, L. and failureis by sliding along AN(2) The joint plane AA is not completely jointed: however. the nextjoint plane BB' is completely jointed I o o ~ ~ persistent) and the distance

0 ) between thejoint planes is sulliciently small smaller than a criticaldistance D,). Failure occurs by sliding along the jointed segment ofA A . fr ac tur ing t hrou gh intact rock to connec t to BB' and slidingalong 8B .(3) Only parts of AA and BB' are j oi nt ed but the j oi nt ed p ar tsoverlap or are equal to L L I L L). and the disllmce D is smaller

Fig. A I. Geometry for analytical lower bound to Pr :) .

than Dc. Failure occurs by sliding along the two jointed portions anda connecting fracture through intact rock.Because these three failure events are mutually exclusive, the probability Pc that a ny on e o f them occurs is the sum of their individualprobabilities PI P2 and PJ) andP, :) Pc : = P, :) P2 : PJ : AI)

Let be the mean joint length. RBR the mean rock bridge length.K = [[/ JL RBR t he mean j oi nt plane persistence and S]i theaverage spacing between joint planes. Also denote by Dc the criticaljoint separation that corresponds to unstable wedges in cases 2 and 3 note thatDc is stress-dependent and thus dependent on its location inthe slope. Since the following approximation omits P2 and PJ nofurther consideration of Dc is necessary). Thus. using Glynn s [3]probabilistic model of joints. one finds the following expression for PPl and p):

P = e-L JL = e-: lJI:.. . = K e-: l L ......p = I PI) p.(1 - e-D.SP,

A2)Figure 19 showed PI and Pc derived with SLOPESIM as a function ofdepth and of intact-rock strength c, P, does not depend on c,). whileall o th er pa ra met er s a re kept co nsta nt . As cr increases. Pc becomescloser to PI because in the limit, as C,- CO, failure can occur only if ajoint plane is 100% persistent Mode I). The probability PI is thus asimple and often good approximation to Prl2.7).

II

I

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Einstein, H.H., Veneziano, D., Baecher, G.B. and Oreilly, K.J., 1983, The Effect of Discontinuity...

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