the effect of discontinuity - geology

Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 20, No. 5. pp. 227-236, 1983 0148-9062/8353.00 + 0.00 Printed in Great Britain. All rights reserved Copyright ( 1983 Pergamon Press Lid

The Effect of Discontinuity Persistence on Rock Slope Stability H. H. EINSTEIN* D. VENEZIANO* G. B. BAECHERt K. J. O'REILLY$

Discontinuity persistence has a major effect on rock mass resistance (strength) but, as direct mapping of discontinuities internal to a rock mass is not possible, persistence is a difficult parameter to measure. As a result, the conservative approach of assuming full persistence is often taken. In this paper a method is developed for relating rock mass stability and hence persistence to the geometry and spatial variability of discontinuities. The method is applied to slope stability calculations in which the probability of failure is related to discontinuity data, as obtained in joint surveys. The complete method makes use of dynamic programming and simulation, but a closed form e,vpression satisfactory for most purposes is also presented.

INTRODUCTION

Discontinuity (hereafter referred to as joint) persistence is among the parameters most significantly affecting rock mass strength, and is a problematic one. While relatively small bridges of intact rock between otherwise con- tinouous joints substantially increase strength, the mapping of each joint is impossible on a practical basis. An attractive alternative to separately considering specific joints is offered by statistical techniques for sampling and 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 mass strength.

A method is developed here for rock-slope reliability analysis based on a probabilistic characterization of the joint system. In doing so, it is found convenient to modify the traditional definition of rock persistence by accounting for the uncertain failure path. Preceding work is briefly described, and is followed by a descrip- tion of the present method.

T R A D I T I O N A L DEFINITION OF J O I N T

P E R S I S T E N C E AND ASSOCIATED P R O B L E M S

With reference to a joint plane (a plane through the rock mass that contains a patchwork of discontinuities and intact-rock regions), joint persistence K is usually

* Professor, ?Associate Professor and ++Formerly Research Assistant at: Department of Civil Engineering, Massachussetts Institute of Technology, Cambridge, MA 02139, U.S.A.

227

defined as the fraction of area that is actually discontinuous. One can therefore express K as the limit

Y~ aoi i

K = lim - - (i) AD'~Z~ Ao '

in which D is a region of the plane with area Ao and aol is the area of the i th joint in D (Fig. 1). The summation in equation (1) is over all joints in D. Equivalently, joint persistence can be expressed as a limit length ratio along a given line on a joint plane. In this case,

K = l i m - (2a)

in which Ls is the length of a straight line segment S and E,, is the length of the i th joint segment in S; or for a particular joint (Fig. 2),

AD

ODI - AREA OF INDIVIDUAL JOINT A D AREA OF JOINT PLANE

K lim ]~ODI ,~ Jointed Area AD-~m A D Total Area

Fig. 1. Joint persistence.

228 EINSTEIN et al.: DISCONTINUITY PERSISTENCE AND SLOPE STABILITY

/ ~ - ' f rock bridge (RBR)

~:JL+ ZRBR Fig. 2. Joint persistence as length ratio.

ZJL K - ZJL + ERBR" (2b)

Another useful index of rock mass discontinuity is joint intensity I, defined as the area of joints per unit rock volume,

a i i

I = lim - - (3) V,~ V

in which ai is the area of the i 'h joint in a 3-D region of volume V.

Joint persistence can be used to estimate the strength of a rock mass against sliding along a given plane: if the plane of sliding has area A, then shearing resistance can be adequately expressed as

Rr = (tr~ tan ~b~ + cr)A, (4)

in the case of intact rock and

Rj = (a, tan 4~j + cj)A, (5)

in the case of completely jointed region. ~br and q~i are the friction angles of intact rock and the joint respectively, c~ and cj, the intact rock- and joint-cohesion. In both cases, a, is the average normal stress across the region of sliding. If the sliding region is partitioned into an intact-rock portion of area A~ and a jointed portion of area Aj = A - A~ (Fig. 3), then following Jennings [5] one can evaluate the shear resistance to sliding, R, as a weighted combination of R~ and Rj according to the expression

A~ Aj R = ~ - Rr + ~- Rj

= (tr~ tan q~, + c.)A, (6)

f

/

];o i Aj Aj

~'Oi + ~'b i Aj4"A r A Fig. 3. Jennings" relations.

N-ECHELON FAILURE

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

where tan ~, and c,, so-called Jennings' equivalent friction and cohesion parameters, are given by

c a = ( l - K) cr + Kcj, tan Oa = (1 - K) tan 4). + K tan ~bj. (7)

The use of equations (6) and (7) for shear resistance of jointed rock masses has several shortcomings:

(1) Failure surfaces are restricted to .joint planes. En- echelon failures (Fig. 4), common in the field, are neglected.

(2) Shear failure does not typically occurjbr the usually low values ofaa. For example, for slopes of 30 m (100 ft) height, tr~ is about 0.7MPa (100psi), whereas c r is typically ten to hundred of MPa. If tr, is negligible, then the major principal stress must exceed cr for shear failure to occur (Fig. 5). This is unrealistic, as Lajtai [6] and Stimpson [9] have pointed out. Also, peak shear resistance in the intact rock and on the joint probably are not mobilized simultaneously.

(3) Small variations of persistence produce large variations of resistance. Therefore, even modest uncertainty about persistence forces the designer to the conservative assumption 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") through intact rock and joints in a rock mass (Fig. 4) constitutes a potential failure surface (failure path) with associated driving force L and resisting force R. For a given configuration of the joint system and a given set of strength parameters, there is a path of minimum safety or "critical path" (Fig. 6). The critical path for a particular joint configuration is that combination of joint and intact-rock portions having the minimum safety margin SM = R - L. If the SM for this path is negative, the rock mass fails; otherwise it resists. Thus, a critical path may or may not be a failure path. The probability of failure Pf of a randomly-jointed rock mass can be expressed as the limit of relative frequency of failure across the spectrum of joint configurations,

P f= lira Nf

in which N is the number of critical paths (failing and not failing) and Nf is the number of critical paths for

EINSTEIN e t al.: DISCONTINUITY PERSISTENCE AND SLOPE STABILITY 229

Or

1"

Fig. 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 Pf: using statistical information on joint length and spacing distributions one can simulate a number of networks of joints such as that in Fig. 7 and then determine the SM for all possible paths in each network or configuration. The critical path for a configuration of the type in Fig. 7 is obtained by identifying the path of minimum SM among all or among a reasonable number of in-plane and en-echelon paths. In some configurations the critical path will be a failure path (SM ~< l) while in others it will not be

j /

(SM > 1). Simulating many configurations (realizations) represents various ways that joint populations with the same spacing and length characterization may manifest, and at the same time produces the parametrs N and Nf for use in equation (8). In any one realization the SM for the critical path can be used to calculate an apparent persistence, and thus one obtains a relation between resistance and apparent persistence for a rock mass characterized by joint length and spacing distributions.

To date, these principles have been applied to 2-D slope stability models in which the pattern of jointing, and strength coefficients, are assumed similar for all cross-sections: 3-D extensions for slope and tunnel appli- cations have been limited [7].

Earlier probabilistic 2-D slope models

Call and Nicholas [l] and Giynn [3] have developed methods of 2-D slope stability analysis that use statistical information on jointing and that allow for both in-plane and en-echeion failures. The model of Call and Nicholas considers two random joint sets. Given distributions of joint length and separation and of spacing

Fig. 6. Critical paths for different joint configurations.

J / ~ ~ - I ~ ,,Critical oath

- S / Fig. 7. Joint configuration and its critical path in a portion of the rock

mass.

230 EINSTEIN et al.: DISCONTINUITY PERSISTENCE AND SLOPE S T A B I L I T Y

/Step path angle = a

Failure = "~- I00 % Tensile

RANDOM VARIABLES

Dip t Length Matter Joint Set Spacing Cross Joint Set

Overlap

Fig. 8. Call and Nicholas [1] model--general features.

between joint planes for each set, the procedure simu- lates critical "step-paths" as shown in Fig. 8. Specifically, for each simulated realization of the joint network, "exit points" are identified (i.e. inter- sections of the shallow joint planes with the slope face), and the critical step-path through each exit point is found. Critical paths are obtained by alternately following jointed segments, which fail in shear, and tensile fractures through the intact rock between joint planes. Shear failure of intact rock bridges between joints is considered improbable except for extremely short bridges (

EINSTEIN et al.: DISCONTINUITY PERSISTENCE AND SLOPE STABILITY 231

.:, ~,% P'~M,

I / . SMi Ri" Wiuzina

Fig. II. SLOPESIM method of slices.

pf, _ Nr~ (10) Ni,

in which N, is the total number of simulated exit joints (critical paths) in the ith elevation interval and Nf, is the number of such exit joints associated with unstable critical paths (i.e. failure paths). The identification of the critical path through each exit point is performed through a dynamic programming method: the algorithm starts with the exit points on the top of the slope and progresses backwards towards the exit points on the face of the slope. During this backwards progression, the algorithm considers all the physically realizable paths through a discrete set of points, including the end points of each joint. For more details on this procedure, see [3].

An important feature of SLOPESIM is the realistic modeling of failure mechanisms. Driving and resisting force calculations are based on the method of slices which is common to many deterministic slope stability methods (the method as applied here is simplified by neglecting interslice forces). The principle is illustrated in simplified form in Fig. 11: the slope overlying the failure path is partitioned into a series of vertical slices, bounded at their bottom end by joints or intact rock. The total driving force L and the total resistance R are calculated by summing slice contributions, i.e.

L = E W, sin ~, i

A fundamental feature of Lajtai's model is in the afiai0gy of shear resistance of intact rock bridges to resistance in direct shear tests. At least for short intact rock bridges, this analogy is justified by the assumption of rigid body motion of the overlying unstable wedge in the direction of jointing. In direct shear tests, the resistance of intact rock can be mobilized in one of two ways:

At relatively low stress levels (tra small), the applica- tion of shear stress in the direction of movement leads to a minimum principal stress (r 3 equal to the tensile strength of intact rock. Hence in this case, failure occurs by tensile fractures that develop at high angles to the direction of sliding (Fig. 12a). Simultaneously with the appearance of these fractures, peak shear resistance za in the sliding direction is attained. There- after, shearing at residual stress values takes place in the direction of sliding.

At higher normal stress levels, the minimum principal stress does not exceed the tensile strength and failure occurs when ra equals the shear resistance defined by the Coulomb failure criterion. In this case, shear fractures develop in the sliding direction at the time when the applied shear stress is maximum (see Fig. 12b).

The two modes can be visualized by use of Mohr's circle. In a direct sheart test, the center of Mohr's circle remains at all times at trJ2 as the shear stress varies from zero to the value at failure. For small (r~ (Fig. 13), Mohr's circle becomes tangent to the failure envelope at a = -Ts , ~ = 0 and thus failure occurs in tension (Mode 1). For larger O'a, the center of Mohr's circle lies more to the right and the point of first tangency is located on the linear portion of the envelope. Thus, as shown in Fig. 14, this mode of failure (mode 2) corresponds to shear failure in the traditional formulation as used by Jennings [5]. Both types of failure can occur, but Mode 2 probably only in high slopes with weak intact rock. Thus, Mode 2 is neglected in the following discussion.

Failures may be in plane or out of plane (en echelon). During in-plane failure, tension cracks develop first,

% TO..._ ~ ~- Primory tension i l l / / ] ~ frocture (high ongle) p t - - / - ~ -F - i i'--seon~..y ,o...ok.)

[ / - - - - , j ir shear frocture

o) FAILURE IN TENSION

R = ~ R~, (l I)

where ct is the angle of jointing, ~ is the weight of the i th slice, and R~ is the peak shear force mobilized by the portion of path underlying that slice. The ith portion of the path may be jointed, in which case Ri can be calculated through equation (5), or it may consist of intact rock. In the latter case R~ is best calculated using rock resistance criteria in Lajtai [6] and Einstein et al. [2].

%

~ . ~ L o w angle primary ~Nlar

I J~

b) FAILURE IN SHEAR

Fig. 12. Direct shear failure modes--after Lajtai [6].


Pointof ~ ~ $ i ~ L ~ ~, x~n' T t failur

Fig. 13. Mohr's circle--failure by tensile fracturing (Mode 1).

followed by secondary shear fractures (Fig. 15a and b). The angle of the tension cracks 0t, can be obtained.from Mohr's circle. The same mechanism applies to out-of- plane failures with low-angle transitions [with (fl - ~t) < 0,, see Fig. 16], whereas for high angle transitions a continuous tension crack occurs directly between joints, without secondary shear fractures (Fig. 17). Therefore, Mode 1 failures with initial tension fractures encompass the entire range of geometrical conditions in slopes with a single joint set, including in-plane as well as 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-plane transitions (fl < 0, + ~),

R = r,d, (12)

in which d is the "in-plane length" of the rock bridge (Fig. 16) and z, is the peak shear stress mobilized in the direction of jointing. In terms of the intact cohesion c~ and the ratio c = z,/c, the peak shear stress is

r, = ~ + I. (13)

For high angle transition (/~ >1 O, + ~),

R = T~X,

i Point of sing r O to failure

Fig. 14. Mohr's circle--failure by shear fracturing (Mode 2).

IN-PLANE FAILURE OF INTACT ROCK- PRIMARY TENSION FRACTURES @ (et) TO JOINT PLANE

/ ~ shear fracture

Fig. 15. In-plane failure of intact rock--secondary shear fractures in the joint plane.

in which X is the distance between the joint planes that define the bridge (Fig. 17) and Ts is the intact rock tensile strength. The contributions to resistance from intact rock bridges and from joint segments are added to obtain the total resisting force associated with a given path.

The driving force associated with the same path is assumed to be due solely to the overburden weight; it is therefore calculated as the sum of the driving force contributions Li from each slice above a path segment (Fig. I 1). If ~t denotes the angle of sliding (joint angle) and W~ is the weight of the i 'h slice, then

L,= W, sin ~t (15)

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

cro Tensile (]4) ~,tJ ~---'l f r o e t u r y

_ 1 . - - Y l - ~ V ~

o\ ' , , I ' ',,

S fracture

O"

Fig. 16. Failure of "'low angle" (fl

EINSTEIN et al.: DISCONTINUITY PERSISTENCE AND SLOPE STABILITY 233

c t u r e

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

analysis. Although in principle simple to do, such a feature would only be consistent if the spatial variability in pressure distributions were expressed, which has not been done so far.) SLOPESIM uses dynamic programming to scan the large number of potential failure paths and to identify, for each "exit joint", the path with minimum SM.

The program has been used to conduct a parametric study of rock slope reliability, aimed at identifying critical variables and at obtaining simple reliability formulae. Results from the numerical analyses are sum- marized in the following sections.

PARAMETRIC STUDY OF SLOPE RELIABILITY

Slope safety depends on a variety of parameters, which for the most part describe geometry and resistance. In the course of the parametric study, these parameters have been given values according to Table 1. Slope safety resulting from the parametric studies has been expressed in several terms:

(I) Failure Probability Pr(:), is defined as the ratio of

critical paths having negative SMs to all critical paths, as a function of the depth z (Fig. 18) at which the paths daylight on the slope face. If a joint plane daylights in height interval "i" on the slope face (Fig. 10), the probability that at least one wedge be- longing to this interval is unstable is Pf(zi) where ,j is the vertical distance between the mid-point of the height interval and the slope crest. At present, the only way to calculate Pf(z) is through repeated Monte Carlo simulation of the network of joints. However, it is possible to obtain analytical lower bounds to Pf(z). One such bound, in many cases close to the exact value, is derived in Appendix A.

(2) Probability Distribution of Apparent Persistence Ka. Ka is the average persistence along an existing joint plane that produces a SM equal to that of the associated critical path, the plane and its critical path daylighting at the same point on the slope face (the critical path may be the particular joint plane or it may have an en echelon shape involving other joint planes). It follows from the definition that Ka is not smaller than the actual persistence of the plane, K. The probability distribution of apparent persistence depends on depth. Of special interest is the variation with depth of the mean value mKd and the standard deviation rx~, which together with the critical persistence defined below are used to define a second moment reliability index.

(3) Critical Persistence K,.. The critical path is unstable and failure occurs if/Ca exceeds the critical value of persistence Kc which, in using the parameters in Table 1, is given by

Table 1. Parameters and their ranges used in SLOPESIM parametric study

Parameter Value or

Symbol Definition range

Geometric parameters (see Fig. 18) H Slope height Fixed 30m (100 ft) : Depth below slope apex 0-30m (0-100 ft) 0 Slope angle 50--90' JL Mean joint length (joint length 3-12 m (10--40fl)

assumed exponentially distributed) RBR Mean rock bridge length Not directly varied,

(rock bridge length assumed considered by exponentially distributed) varying persistence

__K Mean joint plane persistence 10-73/0 SP Mean joint plane spacing (joint 0.6--3 m (2-10 ft)

plane spacing assumed exponentially distributed) Mean joint intensity, a derived

K variable ) = -

SP Joint plane angle 30-80

Resistance parameters CF Intact-rock cohesion 0.30-24.0 MPa

(assumed to be twice the rock (8-500 ksf) tensile strength)

~b, Intact-rock friction angle (not Fixed 30 ~ important at low stress levels)

cj Joint cohesion Fixed 0 ~bj Joint friction angle 0-40

Other parameter 7r Unit weight Fixed 2.2 g/cm 3

( 150 Ib/ft 3)


3- z

Fig. 18. Slope geometry for parametric study.

2c(tan~ - tan qS,) (16) Kc=100 1 x / ~ + l - 2 c t a n 4 ~ , '

where W cos a

C - - - - JLcr

(4) Second-Movement Reliability Index [3 is the number of standard deviations separating the mean K. from the critical value Kc,

/3 - K~ - mx______.___~ " (17)

For Fh.(K) the cumulative distribution function of apparent persistence at a given depth, the probabil i ty of failure at that depth becomes

Pf = 1 - F~q (K~)

= 1 - F x , ( m K + / 3 a x , ) (18)

The sensitivity of these four safety measures to the parameters of Table 1 has been studied by varying one parameter at a time within the specified ranges, while holding all the other parameters fixed at given values. Of special interest is the dependence of Pr on depth z, which is shown in Fig. 19. The same figure contains a plot of the lower bound P~ (see equation A2 in the Appendix) which for slope heights up to 30m (100ft) and for typical values of c~ (c~/> 24 MPa (500 ksf)) provides a good approximation to Pf (accuracy depends also__ on other parameters, such as the mean joint length JL and the friction angle for the joints, ~j). The parametric study also revealed that dependence of Pr, mx, and 13 on intact-rock strength c~ for depths up to 30 m (100 ft), is small. This is a welcome result because it allows one to calculate the reliability index/3 in equation (17) after a single use of SLOPESIM to obtain representative values of mx~ and aKo.

Similar sensitivity analyses were made with respect to the other parameters of Table 1 (see [7]), leading to the following conclusions:

The influence of strength parameters c, and q~j usually dominates other parameters. Slopes with high values of c, and q~j (c ,>/24MPa (500ksf), % ~ a) tend to be reliable at all depths investigated (up to 30m (100 ft)), regardless of the other parameters. When c~ and q~j are small, joint and slope geometry become important. Among the joint geometry parameters (K, JL, SP), mean

persistence K has the largest influence. Changing the mean joint length JL may also substantially modify P, and/3 at any given depth, whereas mean joint spacing S-P plays a less significant role. For small K (K < 30,;) P, depends on K and S-P almost exclusively through the ratio K/SP. which to first-order accuracy equals the mean intensity 7.

Joint inclination, ~, has varying effect of reliability. Values of ~ for which reliability is smallest are typically around 45 . As ~ increases above this value, approaching the slope angle, reliability increases due to the decrease in driving force. This effect is especially significant in slopes with weak intact rock (cr < 4.8 MPa (100 kst)) and weak joints ~bj

EINSTEIN et al.: D I S C O N T I N U I T Y PERSISTENCE A N D SLOPE STABILITY 235

z c DEC.

0 Z - . . - -o -

t

0 Z -~4P-

0 Z -~- '~ , -

hc I)

o z .------~

z

0 Z -------~

\

0 Z - - -4~ 0 Z . ~ - 0 Z . - ~ -

Fig. 20. Probability of failure Pr as a function of slope depth z and of the other slope parameters.

the higher probability of a joint isolating a wedge near the crest is significant.

(2) Pf(z) can often be approximated by Pl(z) (see Appendix A), which is the probability that the joint plane at depth z is 100% persistent. This approximation is good either when Cr or q~j is high (~j ~ ~t or Cr > 24 MPa (500 ks0). Also when both cr and ~j are small (q~j


REFERENCES

1. Call R. D. and Nicholas D. E. Prediction of step path failure geometry for slope stability analysis. Proc. 19th U.S. Syrup. on Rock Mechanics (1978).

2. Einstein H. H. et al. Risk analysis for rock slopes in open pit mines, Parts I-V, USBM Technical Rept J0275015 (1980).

3. Glynn E. F. A probabilistic approach to the stability of rock slopes, Ph.D. dissertation, M.I.T. (February, 1979).

4. Hasofer A. M. and Lind N. C. Exact and invariant second moment code format. A S C E J. Engng Mech. Die. 100, 111-121, No. EMI, Proc. Paper 10376 (February, 1974).

5. Jennings J. E. A mathematical theory for the calculation of the stability of open cast mines. Proc. Symp. on the Theoretical Back- ground to the Planning o f Open Pit Mines, pp. 87-102, Johannesburg (1970).

6. Lajtai E. Z. Strength of discontinuous rocks in shear. Geotechnique 19(2), 218-233 (1969).

7. O'Reilly K. J. The effect of joint phase persistence on slope reliability, M.Sc. thesis, M.I.T., 553 pp (1980).

8. Shair A. K. The effect of two sets of joints on rock slope reliability, M.Sc. thesis, M.I.T.. 307pp (1981).

9. Stimpson D. Failure of slopes containing discontinuous planar joints. Proc. 19th U.S. Syrup. on Rock Mechanics, pp. 246-300 (1978).

A P P E N D I X A

An analytical lower bound to the probability o f slope failure

The probability of failure, Pf(z) has been defined as the fraction of unstable critical paths that daylight at depth z. Lower bounds to Pf(z) can be obtained by constraining the geometry of the critical path and the pattern of jointing that can produce failure. One such bound is obtained here under the following conditions: with reference to Fig. A l, failure of the joint plane exiting at : can occur only if:

(1) The joint plane AA' is 100% persistent, i.e. L~ >/L, and failure is by sliding along AA'.

(2) The joint plane AA' is not completely jointed; however, the next joint plane BB' is completely jointed (100% persistent) and the distance (D) between the joint planes is sufficiently small (smaller than a critical distance D,). Failure occurs by sliding along the jointed segment of AA', fracturing through intact rock to connect to BB' and sliding along BB'.

(3) Only parts of AA' and BB' are jointed but the jointed parts overlap or are equal to L (L~ + L~/> L). and the distance D is smaller

Fig. AI. Geometry for analytical lower bound to Pf(2).

than De. Failure occurs by sliding along the two jointed portions and a connecting fracture through intact rock.

Because these three failure events are mutually exclusive, the probability Pc that any one of them occurs is the sum of their individual probabilities (Pt, P2, and P3) and

Pf(z ) >1 Pc(z) = Pl(z ) + P2(z ) + P3(z ) (AI)

Let ~1: be the mean joint length, ~ the mean rock bridge length, ~" = ] 'L/(J[ + R- ]~) the mean joint plane persistence and g]5 the average spacing between joint planes. Also denote by D c the critical joint separation that corresponds to unstable wedges in cases 2 and 3 (note that D c is stress-dependent and thus dependent on its location in the slope. Since the following approximation omits /)2 and P3, no further consideration of D c is necessary). Thus, using Glynn's [3] probabilistic model of joints, one finds the following expression for P~, P2, and P3:

P~ = K e -L'E = K" e -::dL~') = K e -:'jL ~n'~

P, = (I - Pi) PI( 1 - e- Dc/sP)

Figure 19 showed Pi and Pc derived with SLOPES1M as a function of depth and of intact-rock strength c r (Pt does not depend on c,), while all other parameters are kept constant. As c r increases, Pc becomes closer to P~ because in the limit, as cr-* ~ , failure can occur only if a joint plane is 100% persistent (Mode 1). The probability PI is thus a simple and often good approximation to Pf[2,7].

the effect of discontinuity - geology

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