Probabilistic excavations C.F. Leung and S.T. Quek

stability analysis in jointed rock

Introduction

Abstract: Excavations in jointed rock may liberate rock blocks that may fall by gravity or slide along the discontinuity. The orientation of discontinuities is one of the major input parameters in the conventional deterministic stability analysis of rock blocks. As the mean orientation of a discontinuity is often derived from a large number of joint-set data obtained from site investigation, Fisher's constant is commonly employed to represent the degree of dispersion of individual discontinuity orientation. However, such dispersion factors are rarely used in the conventional analysis. A probabilistic-based approach is proposed in this paper to incorporate Fisher's constant in the stability analysis of rock blocks. To account for the uncertainty reflected by the sample

- dispersion, data are generated systematically around each mean discontinuity normal, based on its Fisher's constant. The probability of rock block failure at a certain location and the largest possible block volume are determined in a logical manner. A microcomputer program has been developed to automate the analysis, and illustrative examples are shown to demonstrate the importance of incorporating the Fisher's constant of individual discontinuity in the stability analysis. In addition, risk mapping plots are presented to enable visual selection of an optimal route for excavation from one location to another.

Key words: discontinuity, factor of safety, probability of failure, rock block, rock excavation, stability analysis.

R&sum& : Des excavations dans un massif rocheux fracturC peut IibCrer des blocs de roche qui peuvent tomber par gravitC ou glisser le long de discontinuitCs. L'orientation des discontinuitCs est un des principaux paramktres qui entrent dans l'analyse des stabilitC dCterministe conventionnelle de blocs de roche. Puisque l'orientation moyenne d'une discontinuit6 est souvent dCrivCe d'un grand nombre de donnCes d'ensembles de joints obtenues par une Ctude sur le terrain, la constante de Fisher est communCment utilisCe pour representer le degrC de dispersion de l'orientation de chaque discontinuitC. Cependant, un tel facteur de dispersion est rarement utilisC dans l'analyse conventionnelle. Dans cet article, l'on propose une approche b a k e sur les probabilitCs qui incorpore la constante de Fisher dans l'analyse de stabilitC des blocs de roche. Pour tenir compte de l'incertitude rCflCtCe par la dispersion de l'Cchantillon, les donnCes sont gCnCrCes systkmatiquement autour de chaque moyenne de la distribution normale de la discontinuit6 basCe sur sa constante de Fisher. La probabilitC de rupture d'un bloc de roche 2 un certain endroit et le volume de bloc le plus grand possible sont dCterminCs de f a ~ o n logique. Un programme de micro-ordinateur a CtC mis au point pour automatiser l'analyse, et des exemples sont donnCs qui illustrent l'importance d'incorporer la constante de discontinuit6 individuelle de Fisher dans l'analyse de stabilitC. De plus, des cartes de risques sont prCsentCes pour permettre de choisir visuellement la route de optimale pour excaver d'un point 2 un autre.

Mots cle's : discontinuitC, facteur de sCcuritC, probabilitC de rupture, bloc de roche, excavation de roche, analyse de stabilitC.

[Traduit par la rCdaction]

Excavat ions in jointed rock may liberate rock blocks of various geometric configurations defined by the disconti- nuity surfaces and the excavation free face. In the event of underground excavat ions, the biggest possible block would b e a t e t rahedra l b lock that i s b o u n d e d b y th ree

C.F. Leung and S.T. Quek. Department of Civil Engineering, National University of Singapore, Singapore.

discontinuities and one free face. Figure 1 shows the pos- s ible modes of failure of the rock blocks. T h e block may fail by gravity fall o r by sliding along the discontinuity. In the even t of surface excavat ions, the block would b e bounded b y t w o discontinuities and two non-overhanging f ree faces and may fail by sliding along one discontinu- ity o r the l ine of intersection of the t w o discontinuities. A determinist ic s tabi l i ty analysis is of ten conducted t o evaluate the stability o f rock blocks based o n the g iven orientation of discontinuities and excavation free faces. In practice, the orientation of a discontinuity is usually deter- m i n e d f r o m t h e m e a n orientat ion of a l a rge n u m b e r o f

Can. Geotech. J. 32: 397-407 (1995). Printed in Canada I ImprimC au Canada

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Fig. 1. Possible modes of failure of rock blocks.

joint data obtained from site investigation. Although Fisher's (1953) constant is a commonly adopted parameter (see for example Goodman (1989) and Priest (1985)) to represent the degree of dispersion of individual discontinuity, such dis- persion data are rarely used in the stability analysis of rock blocks.

In recent years, a number of probabilistic-based approaches have been proposed to tackle stability analysis of various types of rock excavations. However, the major- ity of them only dealt with stability of rock slopes, for example, Kawamura and Honda (1985), Chowdhury (1986), Tamimi et al. (1989), Scavia et al. (1990), Genske and Walz (1991), and Carter and Lajtai (1992). Only a handful of published papers (for example Kohno et al. (1989) and Mauldon (1990)) have handled stability analysis of under- ground excavations. Moreover, none of the above-mentioned research work made use of Fisher's constant in their analy- ses. In view of this, a probabilistic-based approach is pro- posed in the present work to handle stability analysis of surface and underground excavations in jointed rocks based on the mean orientation of rock discontinuities as well as the degree of dispersion of the individual joint set.

Computer program PROCK

In the present work, only the standard case of a tetrahedral block, which is bounded by two to three natural planar discontinuities and one to two free faces, is analyzed. Sta- bility analysis involving the complex case of a polyhedral rock block with any number of discontinuities and free faces, which has been dealt with by Warburton (1981) and Goodman and Shi (1985), is outside the scope of this paper. The present work is an extension of the first author's earlier work (Leung and Kheok 1987; Leung 1990), whereby microcomputer programs have been developed to fully automate the kinematic analysis of rock slopes and limit equilibrium stability analysis of rock excavations using the vector approach.

Can. Geotech. J., Vol. 32, 1995

Fig. 2. Structural chart of computer program PROCK.

I Start I

I

Module 1:

1. Input trend and plunge of sampling line (if any) 2. Input dip directioddip angle of discontinuity data 3. Determine weight factor for individual data 4. Calculate weight at each grid point of the projection 5. Plot contour of weights 6. Obtain mean orientation and Fisher constant of individual cluster 7. Create output file for next module

Module 2: I 1. Input dip directioddip angle and Fisher constant of individual discontinuity

(or use output file from previous module) 2. Input shear strength of individual discontinuity and excavation parameters 3. Creation of combinations of three discontinuities by probability 4. For each combination, cany out stabil~ty analysis to

(a) identify mode of failure and find maximum volume of tetrahedral block (b) determine minimum and maximum safety factor against sliding failure

5. Calculate probability of failure for each case in 4 6. Create output file for risk mapping

I

Module J:

1. Select relevant output file(s) from module 2 as input files 2. Screen output or printer output on one or more of the following

(a) risk map of plane sliding failure @)risk map of wedge sliding failure (c) risk map of fallout failure (d) risk map of combined plane sliding, wedge sliding and fallout failure

I

End

The dispersion factor of an individual discontinuity, which is usually determined using the stereographic pro- jection technique, is one important input parameter in the analysis. It is thus beneficial to develop a comprehensive computer program to handle the entire analysis incorpo- rating the computer plotting and interpretation of discon- tinuity data, determination of discontinuity clusters and the associated Fisher's constant, and probabilistic stability analysis of rock blocks including identification of modes of failure and probability of rock block failure. The algo- rithm of the program, PROCK, is shown in Fig. 2. The pro- gram is purposely developed in modular form to enable the user to commence or terminate the analysis at any stage depending on the type and nature of data available and output requirement. The following sections will highlight the procedures for each module in which illustrative exam- ples are shown to describe its applications.

Stereographic projection of joint orientation data

The orientation of discontinuities can be recorded during site investigation by mapping along sampling lines set upon the rock face. Traditionally, discontinuity populations are

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Fig. 3. Determination of pole concentration on an equal- angle projection using a counting grid and a counting circle.

-Counting circle

' Percentage of weight ~ l u e s per lo/. area at grid point

identified from either equal-area or equal-angle projection of poles (normals) to the discontinuities mapped. Based on the results of a detailed comparative study, Hoek and Brown (1980) concluded that there were few differences in the results between an equal-area or equal-angle projec- tion as long as one type or the other is used consistently. The equal-angle projection, which is also termed stereo- graphic projection and commonly used by rock mechan- ics engineers, will be employed in this paper.

Priest (1985) described a graphical approach of using a counting grid and circle to determine the pole concentra- tion on an equal-angle projection. Figure 3a shows a count- ing grid that is placed on data points of discontinuity poles on an equal-angle projection. Each square within the count- ing grid has a linear dimension c, which is one tenth of the radius of the circular area of projection. A counting circle of radius c would therefore occupy exactly 1% of the area of the projection. The pole concentration of each coordinate within the counting grid can be determined by finding the total weighted subsample that appears within the counting circle whose centre is placed on the coordinate, as illustrated in Fig. 3b. These pole concentration average values can then be contoured at some appropriate interval. The mean orientation of each discontinuity cluster as well as their degree of dispersion can then be determined sta- tistically. Priest's graphical approach has been automated

Fig. 4. Cartesian coordinate system and representation of the locus of a discontinuity normal on a sphere. N , mean discontinuity normal; F, discontinuity normal at an angle $I away from N.

,Y ihorimtalmrth) /

- - X [horizontal east)

I f 'locus of F z

(vertical down)

on a microcomputer in the present work and the method is described as follows.

Based on the convention (Fig. 4) that positive x is hor- izontal to the east (azimuth 90") and positive y is hori- zontal to the north (azimuth = 0") and z is positive down- wards, the x, y Cartesian coordinates of the projection of the discontinuity normal for equal-angle projection in the lower hemisphere may be given as

[I] x = R sina,, tan 45 - 2 ( [2] y=Rcosa , , t an45 - - ( 3 where R is the radius of the projection, a,, and p,, are the trend and plunge of the discontinuity normal. The artificial sampling line will tend to intersect preferentially the larger, or more persistent, discontinuities and those discontinu- ities whose normals make a small angle to the sampling line and hence impose bias on the sample measured. Priest (1985) established a "weight factor" w to evaluate the sam- pling bias whereby

[3] w = 1

I C O S ( ~ , - a, )COS PI, cos P, + sin P, sin P, I

where a, and p, are the trend and plunge of the sampling line. In the computer program, grid coordinates are gener- ated using [ I ] and [2] and the position of all the disconti- nuity normals on the projection are checked against each of these generated coordinates. The weight assigned to this coordinate will be the sum of the weight factors of all the samples with distance less than the radius of the counting circle, c, from the coordinate. This total subsample is then expressed as a percentage of the total weighted sample size. It should be noted that when the counting circle is close to the edge of the projection, any part of the circle that extends beyond the perimeter must reenter at a dia- metrically opposite point (one projection diameter apart).

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400 Can* Geotech. J , Vol 32 , 1995

Fig. 5. Contour plot of discontinuity normal data points.

The percentage weights of all the coordinates are deter- mined and saved in a file, which is subsequently used as the input data file for the contour plot using a commercially available package program. A contour plot for a sample set of 160 discontinuity normal data points is shown in Fig. 5, which clearly shows that there are five dominant clusters. The package program allows the user to specify the range of contour percentages as well as the incremental percentage values. In addition, portion(s) of the plot can be chosen and plotted separately.

Discontinuity orientation and dispersion

Suppose the vector representing the normal to the jth dis- continuity of the selected set of data points to be nj and its trend to be anj and plunge with an associated weight factor of wj calculated using [3]. To normalize each of the weight factors wj so that the total normalized weighted sample size is N, the normalized weight factor w; is given by

N [4] w'. = w . --

I I N,v

where the total weighted size, N,, for the set is determined as

Using the same sign convention as described earlier, the unit vector for nj can be obtained by its direction cosine as follows (Fig. 4):

[6] nj = (wj' sin a,,. cos Pnj)i (wj' cos a,,. cos pllj)j + (wi sin Pnj)k

= njxi + njyj + njzk

*,

where i, j , and k are the unit vectors in the x, y, and z direc- tion, respectively. It is assumed that the representative ori- entation or mean normal for this set is given by the ori- entation of the resultant of the vectors n, where j = 1 to N. Thus data points with higher weight factors are automati- cally given greater importance. The x, y, and z Cartesian components of this resultant vector r,, are hence given by r,r, r,, and r, as

The magnitude of r,, is given by R , as follows:

The mean orientation of the discontinuity normal of a group of discontinuity data is represented by its trend a,, and plunge P,,,, and given as

The parameter q is an angle, in degrees, that ensures that a,,, lies in the correct quadrant and in the range of 0-360".

It is unlikely that structural geologic phenomena can cause discontinuities that are uniformly distributed in ori- entation across the region of interest. Instead several dom- inant discontinuity orientations are more likely to be found. Mapping data would produce joint orientations spread around these dominant directions. It is important to rec- ognize that dispersion may arise from natural variation in the populations, due to waviness for example, from regional variations due to stress differences across a structural domain, or as a result of operator error inherent in the mapping technique as described by Hoek and Brown (1980). Ignoring some of these possible influences could lead to erroneous or conservative design results.

Goodman (1989) and Priest (1985, 1993) described in some detail the use of Fisher's constant to evaluate the dispersion of joint orientations, and the technique is sum- marized as follows. Fisher (1953) assumed that the popu- lation of vectors (from which nj are samples) is randomly dispersed about some "true" orientation. This is equiva- lent to the idea of discontinuity poles being dispersed within a set. To obtain some measure of the degree of dis- persion in a given sample data set, the fundamental dis- tribution of elementary errors over the surface of a unit sphere is used, in which Fisher defined the frequency den- sity function, df, as

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Table 1. Comparison of results for stability analysis with and without Fisher's constant.

(A) Example I

Dip directionldip angleIFisher constant of discontinuity 1 Dip directionldip angle/Fisher constant of discontinuity 2 Dip directioddip angle/Fisher constant of discontinuity 3 Shear strength of discontinuities Excavation parameters

Analysis with Fisher's constant

Mode of failure: Sliding along discontinuities 1 and 3 Maximum factor of safety against sliding = 7.54 Minimum factor of safety against sliding = 2.95 Maximum block volume = 201 m3

330°/700/733 80'11 0°/67 1 30°/800/533 c = 25 kPa, 4 = 30" Roof, tunnel axis = 0°, width = 10 m

Analysis without Fisher's constant

Mode of failure: Sliding along discontinuities 1 and 3 Factor of safety against sliding = 5.32

Block volume = 23.1 m3

(B) Example 2

Dip directionldip angleIFisher constant of discontinuity 1 Dip directioddip angle/Fisher constant of discontinuity 2 Dip directionldip angle/Fisher constant of discontinuity 3 Shear strength of discontinuities Excavation parameters

Analysis with Fisher's constant

Failure mode 1: Sliding along discontinuity 2 Maximum factor of safety against sliding = 0.50 Minimum factor of safety against sliding = 0.44 Maximum block volume = 4.86 m3

Failure mode 2: Fallout Maximum block volume = 46.6 m3

23O0/5O0/2O0 330°/700/733 80°/100/67 c = 25 kPa, 4 = 30" Roof, tunnel axis = 0°, width = 10 m

Analysis without Fisher's constant

Mode of failure:

Fallout

Block volume = 24.7 m3

where 0 is the angular displacement from the "true" posi- tion where 0 = 0, and K is known as the Fisher's constant, which has a positive magnitude. It is assumed that Fisher's method (1953) is also applicable to the analysis of weighted data if these data have been normalized such that the total normalized weighted sample size is equal to N. The errors introduced by this approximating assumption are consid- erably less than those that would be introduced by sim- ply ignoring the sampling bias. Thus K can be derived based on Fisher's definition and is given by

where N , is the total weighted size of N observations [5] and R,, is the magnitude of resultant vector r,, [8]. Note that when K is large, the dispersion is limited to a small area around the "true" position and tends to a two-dimensional isotropic Gaussian distribution with variance given by 1IK. When K is small, the distribution approaches a uniform spread over the spherical surface.

The inclusion of particular data points for statistical evaluation of discontinuity orientation is primarily based on engineering judgement and experience. In general, engi- neers may select the data points by examining data within

different contouring percentages. The data points will sub- sequently be chosen to determine the mean discontinuity ori- entation if the value of the resulting K for the particular data set is above a given desirable value (say for example K > 500). The criteria for an appropriate value are (i) as many data points as possible are represented and (ii) scatter is sta- tistically representative and not too diffuse. In the com- puter program, users are given the option to carry out a parametric study on the evaluation of K using different range of percentage contours. The results are stored in a file which is subsequently used as a data file for probabilistic stability analysis.

Probabilistic stability analysis

Once the mean orientation of the discontinuities and their associated Fisher's constant, K, have been identified, analy- sis may be conducted to evaluate the stability of tetrahedral blocks that are bounded by three discontinuities and one excavation free face in the case of underground excava- tions or two discontinuities and two non-overhanging free faces in the case of surface excavations. To account for the uncertainty reflected by the sample dispersion, data are generated around each mean discontinuity normal based on the Fisher's constant of individual discontinuity in order

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402 C,an. Geotech. J., Vol. 32 , 1995

Fig. 6. Equal-angle projection of mean and possible range if K is large, 0 tends to be small and the generated data of orientation of discontinuities (example 1). would be expected to be close to the mean normal.

0 Since a, and p, are the known mean orientation of a discontinuity normal, the unknown a, and P, can be found by making use of 1131, 1141, and [15]. By statistically gen- erating many sets of discontinuities and performing sta- bility analyses on these discontinuities using an automated vector method (Leung 1990), more conclusive and mean- ingful deductions can be made since the scattering of data has been incorporated in the analysis. The proposed method is illustrated by the following examples involving under- ground excavations in jointed rock.

Example 1 Table 1A gives the orientation of three discontinuities whose K value ranges from 67 (highly scattered) to 733. Each discontinuity is assumed to possess a shear strength of cohesion c of 25 kPa and a friction angle + of 30". Sta- bility analysis is conducted on a 10 m span roof along a tunnel axis of azimuth 0". The mean and possible range of orientation of the discontinuities are represented by solid and broken lines of great circles, respectively, on an equal-angle projection shown in Fig. 6. Without consid-

Possible range of discontinuity

ering the dispersion of the discontinuities, the rock block may slide kinematically along the line of intersection of

Projection of block bounded by meon discontinuities discontinuities 1 and 3,-as represented by the cross-shaded region bounded by the-three solid lines of great circles as

Projection of block bounded by one worse combination of discontinuities shown in Fig. 6. The volume of the tetrahedral block is determined to be 23.1 m3 using the vector method pro-

deduce the probability of rock block failure at a cer- posed by Leung (1990). Based on the definition of the tain location in a meaningful manner. These generated data will cluster around the mean discontinuity normals.

Figure 4 depicts the projection of vector N, which represents the mean discontinuity normal and the corre- sponding locus of points of vector F, which represents the generated discontinuity normals at an angle 0 away from N on a sphere. The Cartesian coordinates of N having trend a , and plunge P, are given by its direction cosines as (COS P,, sin a,, cos P, cos a,, sin PI,). Similarly the coor- dinates of F having trend a , , and plunge P, are given as (COS PI sin a , , cos PI cos a , , sin PI). From geometric con- siderations, it can be shown that

2 0 2 h sin(@,, - 0) sin 0 + 8 cos p, sin - sin - [13] sin PI = 2 2

sin 0

cos 0 - sin p, sin PI [14] cos(aI, - a l ) =

cos PI, cos PI where X is an arbitrary angle that can be randomly gen- erated following a uniform distribution over the interval (0, IT), and 0 generated randomly following its probability distribution in view of [ l l ] is given by

[15] P(<0) = 1 - e-K'l-COSe'

The probability value refers to the chance that a vec- tor selected at random makes an angle less than 0 from the mean discontinuity normal. Values of P ( 4 ) are ran- domly generated following uniform distribution, and the extent of the variation of these generated data about their mean normal will depend on the value of K. For example,

factor of safety as the ratio of shear resistance versus the shear force along the sliding discontinuities, the factor of safety against wedge sliding failure is determined to be 5.32. see Table 1A.

is her's constants, K, of the three discontinuities are incorporated in the present work to systematically generate the possible combinations of discontinuity planes based on the probability function of [15]. In general, the variations in the discontinuity orientation would be large for dis- continuity having relatively small K values and vice versa, as schematically represented by the broken lines in Fig. 6. It is evident that there would be significant variations in the volume of the rock block, as illustrated by the bigger shaded region of Fig. 6 which is bounded by three dis- continuities in one worse possible combination. The factor of safety of rock block against sliding failure would hence vary due to changes in the block surface contact areas and volume. Table 1A reveals that under the best ~ossible com- bination, the factor of safety against sliding failure improves significantly to 7.54, while under the worst possible com- bination, the factor of safety decreases to 2.95. The max- imum block volume of all possible combinations is found to be 201 m3. This example truly demonstrates the impor- tance of incorporating .the dispersion factor of discontinu- ity orientation in the stability analysis.

Example 2 Table 1B shows the results of stability analysis on another three discontinuities. The rock strength and excavation parameters are essentially the same as those given in exam- ple 1. The equal-angle projection of the mean and possible range of orientation of the discontinuities are shown by

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Fig. 7. Equal-angle projection of mean and possible range of orientation of discontinuities (example 2).

Possible range of discontinuity

Mean n

Projection of block bounded by mean dircontinuilicr

Projection of block bounded by one worse combination of d i s ~ i t i e r

the solid and broken lines of great circles, respectively, in Fig. 7 . The projection reveals that the rock block will fail kinematically by falling and the volume of the falling block will vary somewhat because of the dispersion of individual discontinuity orientation. By incorporating the respective K values for each discontinuity in the analysis, Table 1B shows that the maximum volume of the falling block increases from 24.7 to 46.6 m3. In addition, the pre- sent approach determines that there is also potential slid- ing failure along one of the discontinuities, whereby such mode of failure cannot be identified if K values have not been considered in the analysis.

Probability of failure

In this paper, a simple risk model has been developed to examine the stability of rock blocks using a probabilistic- based concept. The present work assumes that once the orientations of the discontinuities have been identified for a certain location, they will represent the only discontinu- ities that can be found in the close vicinity. For example, if N "high" clusters or discontinuity normals are identified from the statistical analysis of the orientation data, the risk

Fig. 8. Equal-angle projection of mean and possible range of orientation of discontinuities (example 3).

0

Possible range of 100 discontinuities

model assumes that the characteristics of the rock mass can be well represented solely by the mean discontinuity normal of each of these N clusters. Suppose each of these discontinuity normals is denoted by vector r,,! whose mag- nitude is Ir,l,l. The probability of a rock block b,,, being formed by the excavation free face and discontinuities 1 , 2, and 3 is termed as P(b,,,) and is given by the product of the probability of three discontinuity normals, i.e.,

As mentioned earlier, data are generated around every chosen mean discontinuity normal to account for the uncer- tainty associated with K in defining these normals. The pre- vious examples illustrate that the orientation of the three discontinuity normals involved in one combination is dif- ferent from that in another combination. The probability of rock failure by a certain rock block failing in a certain mode is dependent upon the probability of occurrence of the rock block and the frequency of the rock block failing by this mode. Dershowitz and Einstein (1984) and Carter and Lajtai (1992) had employed such a definition of probability of fail- ure to assess the stability of rock slopes. The probability of rock block containing discontinuities 1, 2, and 3 sliding on any discontinuity is termed as P(b,,3),lide and is defined as

number of plane sliding failure 'I7'

P(bi23)s1ide = P(b123) number of combinations of rock block (4,) generated

The probability of failure in terms of wedge failure involving block sliding on the line of intersection of two discontinuities P(b123),ed,,, and for block fallout failure, P(b123)fallout, can be defined in a similar manner.

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404 Can. Geotech. J., Vol. 32 , 1995 .,

Table 2. Probability of failure of rock blocks.

(A) Parameters

Dip directionldip angle/Fisher constant of discontinuity 1 230°/500/200 Dip directionldip angle/Fisher constant of discontinuity 2 30°/300/1700 Dip directionldip angle/Fisher constant of discontinuity 3 330°/700/733 Dip direction/dip angleIFisher constant of discontinuity 4 80°/100/67 Dip directionldip angleIFisher constant of discontinuity 5 130°/800/533

Excavation parameters Roof, tunnel axis = 0°, width = 10 m

(B) Probability of failure (%) for different rock blocks taking rock joint strength c = 0 and + = 0

Wedge sliding Plane sliding failure along failure along

1st and 1st and 2nd and Joint 2nd 3rd 3rd 1st 2nd 3rd Fallout set Stable joints joints joints joint joint joint failure Total

1,2,3 0 0 0 0 0 0 25.2 0 25.2 1,2,4 0 0 0 0 0 0 0 4.8 4.8 1,2,5 0 0 0 0 0 0 0 6.4 6.4 1,3,4 0 0 0 0 0 0.1 0 9.1 9.2 1,3,5 0 0 0 0 0 0 0 12.2 12.2 1,4,5 0 0 0 0 0 0 2.3 0 2.3 2,3,4 0 13.0 0 0 0 0 0 0 13.0 2,3,5 0 1.7 0 15.5 0 0 0 0 17.2 2,4,5 0 0 3.3 0 0 0 0 0 3.3 3,4,5 0 0 6.3 0 0 0 0 0 6.3 Total 0 39.8 27.7 32.5 100.0

(C) Probability of failure (%) for different rock blocks taking rock joint strength c = 25 kPa and + = 30"

Wedge sliding failure along

Plane sliding failure along

1st and 1st and 2nd and Joint 2nd 3rd 3rd 1st 2nd 3rd Fallout s t Stable joints joints joints joint joint joint failure Total

1,2,3 0 1,2,4 0 1,2,5 0 1,3,4 0 1,3,5 0 1,4,5 0 2,3,4 13.0 2,3,5 17.2 2,4,5 3.3 3,4,5 6.3 Total 39.8

Example 3 of any three discontinuities. The determination of the mag- Table 2A gives the orientation of five discontinuities with nitude of the largest block volume and the range of fac- their respective K values and the excavation parameters. Ten tor of safety against sliding for the two rock blocks bounded sets of rock blocks are feasible based on the combinations by discontinuities 3, 4, 5 and 1, 3, 4, respectively, have

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Fig. 9. Probability of failure of rock blocks.

I\,\\\\\\\,\.\\\,\\\<\\V 1 Plane sliding LI Fallout

I -Total

Probability of failure (%)

been illustrated by examples 1 and 2. Figure 8 shows the equal-angle projection of the mean and possible range of orientation of the five discontinuities. Due to the dispersion in each discontinuity orientation, stability analysis would involve numerous combinations of rock blocks as illus- trated in the previous examples. This example aims to demonstrate the use of the probability of failure concept in the interpretation of stability of rock blocks involving mul- tiple discontinuities.

Without considering the shear strength of the rock joints initially, kinematic stability analysis is conducted on the 10 possible rock blocks. Figure 9 shows the probability and mode of failure for each of the 10 possible blocks. It is evident that the block bounded by discontinuities 1, 2, 3 has a probability of plane sliding failure of 25.2%, which is significantly higher than the other blocks and hence warrants more attention. Table 2B shows the detailed results identi- fying the plane(s) of sliding. For example, the probability of failure of rock block b,,, bounded by discontinuities 2, 3, 5 is about 17.2%, which comprises a probability of 15.5% that it will slide along the line of intersection of discontinuities 3 and 5 and 1.7% that it will slide along discontinuities 2 and 3. The overall probability of encoun- tering wedge sliding failure is the sum of all rock blocks failing in this particular mode and is determined to be 39.8%. Example 2 demonstrates that rock block b134 may fail by fallout or sliding along discontinuity 3. It is worth noting that Table 2B reveals that this block has a probability of fallout failure of 9.1% and a relatively small probabil- ity of plane sliding failure of 0.1%.

Stability analysis of rock blocks is then conducted by considering the shear strength of the rock joints, and the results are shown in Table 2C. It is evident that wedge sliding failure has been completely eliminated, and hence the overall probability of failure reduces from 100% to 60.2%. Options are also available to carry out parametric stability analyses on rock reinforcement, desirable factor of

safety, effects of water pressure on joints, and external forces. The respective summary table of probability of failure for different rock blocks can be readily replotted once the appropriate analyses have been carried out.

Risk mapping

To deduce the optimal excavation route from one location to another, parametric studies have to be conducted to evaluate the stability of rock blocks in different regions between the two locations. Wagner et al. (1987) produced risk maps to identify the optimal path for rock slope exca- vations between two locations. The final part of the com- puter program will yield a graphical output of risk maps whereby the preferred excavation route(s) with the least probability of failure from one location to another may be readily identified. The input data files for risk mapping plot are essentially the output files obtained from proba- bilistic stability analyses conducted for different regions between the two locations. In the present work, four dif- ferent types of risk maps are shown, namely, ( i ) plane sliding, (ii) wedge sliding, (iii) fallout, and (iv) a combi- nation of all three cases. The following example highlights the applications of risk mapping plot.

Example 4 Suppose a proposed excavation route commences at loca- tion S and terminates at location E and a route based on maximum safety is to be determined. The plan area between the two locations can be subdivided into squares by means of a pattern of regular horizontal and vertical grid lines as shown in Fig. 10. Based on the orientation and disper- sion of discontinuities identified for each square, the prob- ability of failure of rock blocks can be determined using the technique demonstrated in example 3 based on three exca- vation directions within each square. As an illustration, the probabilities of plane sliding failure for the square bounded by coordinates S, 1, 3, and 2 are shown in the

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Fig. 10. Risk mapping plot of probability of plane and wedge sliding and fallout failure (plan view).

Can. Geotech. J., Vol. 32, 1995

Fig. 11. Risk mapping plot of probability of plane sliding failure (plan view).

Starting point

loo % 90 - 99.9% 80 - 89.9% 70 - 79.9% 60 - 69.9% 50 - 59.9% 40 - 49.9% 30 - 39.9% 20 - 29.9% 10- 19.9% 0 - 9.9%

lower left comer of Fig. 11. These consist of the probability of failure for route S1 (axis = 90°), route S3 (axis = 45"), and route S2 (axis = 0"). In situations where the route is at the boundary of two squares (for example route 23, which transverses between squares S132 and 235a), the input data will be based on the results obtained from stability analysis conducted on all discontinuity data from the two adjacent squares.

The probabilities of plane sliding failure along possible routes from the starting location S to the end location E are given in Fig. 11. In the hard-copy printout version, the risk of failure is indicated by the thickness of line, as given in the lower portion of the figure. On the computer screen, the risk of failure is illustrated by lines of different colour and thickness. It is evident from Fig. 11 that route S-3-4-7-9-E is the optimal route with a probability of plane sliding fail- ure of less than 30%. Figure 10 shows another risk map where the combined probability of failure of plane sliding, wedge sliding, and fallout are shown. The most stable route can be readily established as S-2-5-8-9-E. Parametric sta- bility studies can also be conducted on a variety of influ- encing factors described in the last section, and the risk map can be readily replotted to determine the optimal route.

Conclusion A probabilistic-based approach has been proposed to carry out stability analysis of excavations in jointed rock by

Starting point

Probability of plane sliding failure

100 "I. 90 - 99.9% 80 - 89.9 % 70 - 79.9 % 60 - 69.9 % 50 - 594% 10 - 199 '1. 30 - 39.9 '1. 20 - 29.9 % 10 - 19.9% 0 - 9.9 'I.

incorporating the Fisher's constant of individual discon- tinuity in the analysis. A comprehensive microcomputer program is written in modular form for this purpose whereby the user can commence or terminate the analy- sis at any stage that they so desire depending on the type of input data and output requirement. The first few modules involve the automated procedures of contour plotting of individual clusters of discontinuity normals and the deter- mination of Fisher's constant for individual cluster. Ori- entation data are then systematically generated around the mean orientation of discontinuities based on the scatter represented by Fisher's constant. Stability analyses of the rock blocks bounded by discontinuities and excavation free face(s) are conducted to examine the mode of block failure and the largest possible block volume. In the case of sliding failure, the range of factor of safety against slid- ing failure can also be determined. Illustrative examples are shown to demonstrate the importance of incorporat- ing the dispersion factor of discontinuity in the analysis, as there are significant differences in the stability of rock block between the present proposed approach and the con- ventional deterministic stability analysis whereby Fisher's constant has not been considered. The final two modules of the computer program involve the evaluation of probabil- ity of failure against sliding and fallout failure and the risk mapping plots to enable visual selection of the optimal path for excavation from one location to another.

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Acknowledgement

The assistance of Miss Sian Hee Chng, former student of the Department of Civil Engineering, National University of Singapore, in the development of the computer pro- gram is gratefully appreciated.

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