Kinematic Analysis Markland_plot1

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Kinematic analysis for sliding failure of multi-faced rock slopes

W.S. Yoon, U.J. Jeong, J.H. Kim *

School of Earth and Environmental Sciences, Seoul National University, San 56-1, Shilim-dong, Kwanak-gu, Seoul, 151-742, South Korea

Received 3 October 2001; accepted 29 April 2002

Abstract

Most kinematic analyses for rock slope stability have dealt with single-faced slopes (SFS) with a planar surface of a constant

strike. However, there are many slopes with non-planar surfaces along road cuts and in open pits, etc. Multi-faced slopes (MFS)

consist of two or more faces with different strikes. Multi-faced slopes have different sliding conditions compared to single-faced

slopes because of their geometrical characteristics, i.e. convex surface on plan view. On stereographic projection, a sliding

envelope of a multi-faced slope is a union of envelopes of individual faces formed on the slope surface. Sliding modes of multi-

faced slopes are divided into two types and they are subdivided into two modes, respectively; Type 1 single or double plane

sliding and Type 2 single or double plane sliding. Type 1 sliding failures are controlled by the same rule as the single-faced

slopes as suggested by Hocking [Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 13 (1976) 225]. Type 2 sliding failures can

occur on multi-faced slopes only. A Type 2 sliding block must have two or more adjacent slope faces. Though two joint sets

must be developed for Type 1 sliding, Type 2 single plane sliding can occur with only one joint plane in the multi-faced slopes.

A simple Type 2 single sliding block is composed of one joint plane, two slope faces and the upper natural slope surface. If twojoint planes act as sliding planes for plane failures at two adjacent faces of a multi-faced slope, they form a block of Type 2

double plane sliding in the slope. On a stereographic projection, a Type 2 sliding zone is defined as the area between the true dip

lines of two side-slope faces in the sliding envelope of a multi-faced slope. If the true dip lines of one or two joint planes plot

within a Type 2 sliding zone, Type 2 sliding failure can be possible. D 2002 Elsevier Science B.V. All rights reserved.

Keywords: Kinematic analysis; Rock slope; Single-faced slope; Multi-faced slope; Stereographic projection

1. Introduction

Kinematic analysis, which is purely geometric,

examines which modes of slope failure are possible

in a jointed rock mass. Angular relationships between

discontinuities and slope surfaces are applied to

determine the potential and modes of failures (Kliche,

1999). Numerous studies (Markland, 1972; Goodman,

1976; Hocking, 1976; Cruden, 1978; Lucas, 1980;Hoek and Bray, 1981; Matherson, 1988) have been

performed to determine failure modes utilizing stereo-

graphic projection technique since Panet (1969).

The Markland test (Markland, 1972) is one of the

kinematic analysis methods designated to evaluate the

possibility of wedge failure. A refinement to Mar-

klands test has been discussed by Hocking (1976).

He differentiated the sliding along one plane (single

plane sliding) forming the base of the wedge from the

sliding along the line of intersection of two joints

0013-7952/02/$ - see front matterD 2002 Elsevier Science B.V. All rights reserved.P I I : S 0 0 1 3 - 7 9 5 2 ( 0 2 ) 0 0 1 4 4 - 8

* Corresponding author. Tel.: +82-2-880-6733; fax: +82-2-871-

3269.

E-mail address: [email protected] (J.H. Kim).

www.elsevier.com/locate/enggeo

Engineering Geology 67 (2002) 5161


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(double plane sliding). Most methods including Mar-

klands test are restricted to rock slopes with a con-

stant strike. Even though the angle of the natural slope

(upper slope surface) has no fundamental effect on thekinematical stability of a slope, it has been used in

kinematic analysis (Ocal and Ozgenoglu, 1997).

If a slope surface has a constant strike, section view

is more important rather than the geometry on plan

view. However, many rock slopes consist of several

faces with different strikes. If upper natural slopes are

not considered, single-faced slopes (SFS) can be

defined as slopes having only one face with a constant

strike, and multi-faced slopes (MFS) can be defined as

slopes having two or more faces with different strikes.

Convex- and concave-shaped slopes on plan views are

defined as positive and negative MFSs, respectively.

This paper deals with positive MFSs.

MFSs have different sliding conditions from the

SFSs because of their geometrical characteristics. For

example, two or more joint sets are necessary for

sliding failures of SFSs. These joint sets act as sliding

planes or release surfaces of sliding blocks. Even if

only one joint plane is developed, sliding failure can

occur in MFSs because this joint can intersect two

conjoined slope faces acted as lateral boundaries of

sliding block. Therefore, upper natural slope angles on

section views can be negligible for estimating thestability of rock slopes, but geometries of slope

surfaces on plan views must be considered to deter-

mine failure modes and potential. However, many

MFSs have been analyzed as for SFSs in the processes

of stability analysis.

The aims of this paper are: (1) to describe sliding

modes of MFSs compared with SFSs, and (2) to

suggest a method of kinematic analysis for MFSs

based on the Markland test. A real example of MFS

is subjected to stability analysis.

2. Sliding failure of single-faced slopes (SFS)

The wedge failure is a common type in jointed rock

slopes. Markland (1972) proposed that wedge failures

occur along the lines of intersection of joints and

hence, these intersections must daylight in the slope

faces. In order for the line of intersection to daylight,

the plunge of the intersection should be less than the

dip angle of the slope face measured along the trend

of the intersection. In addition, the plunge of the

intersection must exceed the friction angle of the joint

planes.

In the Markland test (Markland, 1972), the greatcircle of a slope face (SL in Fig. 1) and the circle of

the friction angle, /, of the joint are plotted on a

stereographic projection. The zone (the thick-lined

envelope in Fig. 1) between the great circle (SL)

and the friction circle is called the sliding envelope.

This sliding envelope represents Marklands wedge

failure conditions, i.e. plunge of intersection of the

joints is less than slope angle and greater than friction

angle of the joint. If the intersection (L12 in Fig. 1) of

the two joints (J1 and J2, dotted great circles in Fig. 1)

is located in the sliding envelope, the wedge failure is

possible.

Hocking (1976) divided wedge failures into two

modes in term of sliding planes of wedges. One is

single plane sliding (so-called plane failure), and the

other is double plane sliding which is the general form

of wedge failure suggested by Markland (1972). If

two joint planes bound a sliding wedge, single plane

sliding (upper part of Fig. 1a) is defined as a sliding

on only one of the joints bounding the base of the

wedge, but the double plane sliding (upper part of Fig.

1b) is defined as a sliding on both joint planes parallel

to the line of intersection. If the true dip (L1) of either(J1) of the joint planes lies within the shaded area

between the line of intersection (L12) and true dip of

slope face (LSL), as shown in lower part ofFig. 1a, the

single plane sliding can occur (Hocking, 1976). Cru-

den (1978) suggested that if true dip of either or both

wedge-forming joint planes fall into the area, the

sliding mode of the wedge is a single plane sliding.

However, if the line of intersection (L12) locates in the

sliding envelope and both L1 and L2 lie outside the

area (shaded area in lower part ofFig. 1b), the double

plane sliding can occur.Hoek and Bray (1981) proposed two general con-

ditions of a plane failure by single plane sliding, (1)

the strike of sliding plane must be parallel or nearly

parallel (F 20j) to the slope face, and (2) release

surfaces must be present in the rock mass to define the

lateral boundaries of the slide. The second condition is

a critical condition of the plane failure. If the release

surface does not exist, single plane sliding cannot

occur, though the strike of sliding plane is parallel to

the slope face. These release surfaces can be divided

W.S. Yoon et al. / Engineering Geology 67 (2002) 516152


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into two types: (1) joint planes that intersect the

sliding plane and (2) free faces that intersect the slope

surface. These release surfaces lead to differences in

sliding possibilities between SFSs and MFSs. In

SFSs, joint planes act as the release surfaces, whereas

individual faces as well as joint planes can act as the

release surfaces in MFSs.

3. Sliding failures of multi-faced slopes (MFS)

3.1. Sliding modes of MFS

Various modes of failure can occur in MFSs as

shown in Fig. 2. Single plane sliding (Fig. 2a) released

by other intersected joint plane is called Type 1 single

plane sliding. This type of sliding complies with the

Hocking single plane sliding rule. Fig. 2b shows a

sliding block formed by only one joint plane without

any intersected joints. This single plane failure can

occur because two adjacent faces act as free faces for a

sliding block. Single plane sliding without the pres-

ence of any other joint plane can be defined as Type 2

single plane sliding. The sliding block of Type 2

single plane sliding must be bounded by two or more

slope faces and upper natural slope surface.Double plane sliding in MFSs has different char-

acteristics when compared with double plane sliding

in SFSs. If one joint is a potential sliding plane of

single plane sliding in one face and the other joint is a

potential sliding plane of single plane sliding in an

adjacent face, double plane sliding along the inter-

section of two joint planes can occur (Fig. 2d). This

double plane sliding is called Type 2 double plane

sliding in MFSs, and Hockings double plane sliding

for SFSs (Fig. 2c) called Type 1 double plane sliding.

Fig. 1. Hockings (1976) stereographic solution for sliding of a single-faced slope. (a) Single plane sliding. The sliding direction (thick arrow) of

the sliding block is same as L1. (b) Double plane sliding. The sliding direction (thick arrow) of the sliding block is same as L 12. SL, great circle

of slope face; J1 and J2, great circles of joint planes; LSL, true dip of slope face; L1 and L2, true dip lines of J1 and J2; L12, line of intersectionbetween J1 and J2; /= 30j, friction angle of joint planes.

W.S. Yoon et al. / Engineering Geology 67 (2002) 5161 53


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Therefore, sliding modes of MFSs can be dividedinto two types: Type 1 single and double plane sliding

and Type 2 single and double plane sliding. In case of

Type 1 sliding blocks, two wedge-forming joint planes

must be necessary, but the number of slope faces is not

important. For Type 2 sliding, at least two slope faces

are necessary. If upper slope surface (or natural slope

surface) is not considered, boundary conditions of

sliding blocks are summarized as follows:

Type 1 single or double plane sliding block: two

joint planes and one slope face (Fig. 2a and c).Type 2 single plane sliding block: one joint plane

and two slope faces (Fig. 2b).

Type 2 double plane sliding block: two joint planes

and two slope faces (Fig. 2d).

3.2. Stereographic projection technique for simple

multi-faced slopes

Fig. 3 is an example of kinematic analysis using

stereographic projection technique for two-faced

slopes with different sliding modes. A two-faced slopeis the simplest MFS. The surfaces of a two-faced slope

form a triangular shape between the two ends (S and SV

in the middle part ofFig. 3) in plan views. The straight

line connecting the two ends (S and SV) is defined as

mean slope line (MS, the dashed line in Fig. 3). RS and

LS in Fig. 3 are the side-slope faces. The side-slope

face with a positive side-slope orientation angle meas-

ured clockwise from MS is called a left side-slope face

(LS), and the face with a negative side-slope orienta-

tion angle is called a right side-slope face (RS) in this

paper. The side-slope orientation angles (a and b in themiddle part of Fig. 3) are defined as the acute angles

between the strikes of side-slope faces and MS. The

clockwise angle measured from MS has a positive

value. MS of the example in Fig. 3 trends EW, and the

latitudes of LS and RS are N45jW/60jSW and

N60jE/60jSE, respectively. The side-slope orientation

angle of LS is 45j (a), and the angle of RS is 30j

(b). Friction angle of all joints is assumed as 30j.

In the kinematic analysis, individual faces of the

MFS keep on their sliding envelopes because all faces

Fig. 2. Sliding modes of a multi-faced slope (a rectangular three-faced slope) can be divided into two types which are subdivided into two

modes, respectively. (a) Type 1 single plane sliding and (b) Type 2 single plane sliding. (c) Type 1 double plane sliding and (d) Type 2 double

plane sliding. Type 2 sliding blocks (c and d) must be bounded by two or more slope faces.



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are controlled by Hockings rule like a SFS. There-

fore, the sliding envelope of a MFS is the union of

envelopes of individual faces forming the slope sur-

face. In Fig. 3, the sliding envelope (thick-lined) iscomposed of the envelopes of LS and RS. If the

absolute values of the two side-slope orientation

angles are increased, the area of the sliding envelope

is increased.

Fig. 3a shows the Type 1 single plane sliding

condition of the two-faced slope. If one (or both) true

dip of one (or both) of the wedge-forming joints lies

within the shaded area, bounded by the trend of the line

of intersection and the true dip line of any face, the Type

1 single plane sliding is possible (see Fig. 1a).

In lower part of Fig. 3a, the intersection (L12) of

two joint planes (J1 and J2) is located within the

sliding envelope (thick-lined) of the two-faced slope,

and the true dip line (L1) of J1 joint plane plots in the

shaded area between L12 and true dip line (LRS) of RS.

This indicates that Type 1 single plane sliding on J1joint plane can occur within RS.

Even if only one joint plane is developed in a MFS,

Type 2 single plane sliding can be possible. Fig. 3b

shows Type 2 single plane sliding of a two-faced

slope. The Type 2 sliding block(Fig. 3b) is formed by

one joint plane (J1) and two side-slope faces in the

plan view. In order that a sliding block is formed byone joint plane and two or more faces, the acute angle

(h1 of Fig. 3b) between the strike of the sliding plane

(J1, dotted line) and MS (dashed line) should lie

be tw ee n th e tw o si de -s lo pe or ie nt at io n an gl es

(b< h< a). h has a positive value when measured

CW from MS. Additionally, the sliding plane should

daylight in the MFS, and its dip angle should be larger

than the friction angle.

In Fig. 3b, h1 is 15j and Type 2 single plane

sliding on J1 can occur because h1 lies between the

two side-slope orientation angles; b (

30j

) < h1(15j) < a (45j). Type 2 single plane sliding cannot

occur within the slope in Fig. 3a because the angle

(h1 = 45j) of the sliding plane (J1) for Type 1 single

plane sliding is smaller than a and b; h1 ( 45j) < b

( 30j) < a (45j).

This orientation condition for Type 2 single plane

sliding can be represented as a Type 2 sliding zone on

a stereographic projection. If h1 lies between the two

side-slope angles (b < h < a), the true dip of J1 plots

within the area between true dip lines of two side-

slope faces. This area is defined as Type 2 sliding

zone in this paper. If the true dip of a joint plane is

plotted in the Type 2 sliding zone, single plane sliding

can occur without any other joint plane.Type 1 double plane sliding within slopes is con-

trolled by Hockings double plane sliding condition

(see Fig. 1b). Fig. 3c shows the condition for Type 1

double plane sliding for two-faced slopes. The boun-

daries of the sliding block in Fig. 3c are two joint planes

(J1 and J2) and one side-slope face (RS). The inter-

section line (L12) is located within the sliding envelope

of RS, and the true dip lines (L1 and L2) of the two joint

planes (J1 and J2) lie outside the shaded area between

L12 and true dip line (LLS or LRS) of any face (LS or

RS). Type 1 double plane sliding on both joint planes

can occur within RS of the two-faced slope.

Type 2 double plane sliding has a different sliding

condition from Type 1 double plane sliding. In Fig.

3d, both joint planes for Type 2 double plane sliding

can act as sliding planes for single plane sliding. If the

two wedge-forming joint planes can respectively act

as sliding planes for Type 1 single plane sliding within

two adjacent slope faces, and the line of intersection

also satisfies Type 1 double plane sliding, the block

formed by the two joints and the two faces will slide

on both joint planes along the line of intersection. For

this Type 2 double plane sliding condition, the anglesh1 and h2 in Fig. 3d (the angles between strikes of the

two joint planes (J1 and J2) and MS) must satisfy the

following conditions: 0 < h1 < a, and b < h2 < 0. These

conditions indicate that true dip lines of both joints

will plot within the Type 2 sliding zone bounded by

the two true dip lines of side-slope faces in sliding

envelope on the stereo-net. For example, in Fig. 3d, if

h1 = 30j and h2 = 20j, then 0< h1 (30j) < a (45j)

and b ( 30j) < h2 ( 20j) < 0. The true dip lines (L1and L2) of the two wedge-forming joint planes (J1 and

J2) and line of intersection (L12) are located within theType 2 sliding zone (shaded area), and L12 plots

between L1 and L2.

In Fig. 3d, J1 and J2 joint planes can act as sliding

planes of Type 1 single plane sliding for LS and RS,

respectively. If the sliding block is formed by only or

either of two joint planes, Type 2 single plane sliding

will occur including both slope faces. However, if a

sliding block is formed by two joint planes (J1 and J2)

and two slope faces like in Fig. 3d, sliding does not

occur on either of the two planes, but occur on both



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planes along the line of intersection. This double

plane sliding is a Type 2 double plane sliding.

When two joint sets develop in an MFS rock mass,

two or more different sliding modes can be possibleand these sliding modes have different conditions to

those for SFS, because a MFS is composed of two or

more slope faces.

3.3. Stereographic projection technique for complex

multi-faced slopes (MFSs)

The stereographic overlay technique for two-facedslopes (Fig. 3) can be applied to more complicated

MFS using the same method. Fig. 4 shows examples

of three-faced (Fig. 4a) and round-faced (Fig. 4b)

Fig. 3. Sliding modes (upper), plan views (middle) and stereographic projections (lower) for a two-faced slope (SSV). (a) Type 1 single plane

sliding. The sliding block is formed by J1, J2 and RS. (b) Type 2 single plane sliding. The sliding block is formed by J1, LS and RS. (c) Type 1

double plane sliding. The sliding block is formed by J1, J2 and RS. (d) Type 2 double plane sliding. The sliding block is formed by J1, J2, LS and

RS. LS and RS, left and right side-slope faces; MS, mean slope line; a and b, side slope orientation angles of LS and RS; LLS and LRS, true dip

lines of LS and RS; h1 and h2, angles between MS and strike lines of sliding planes (J1 and J2); sliding direction, a thick arrow. For key to other

abbreviations, see Fig. 1.



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slopes. These examples have the same side-slope

orientation angles (a = 45j and b = 30j) and MS

as the two-faced slopes in Fig. 3.

Surface of a three-faced slope (Fig. 4a) consists of

LS, RS and the central slope face (CS). Sliding enve-

lope (thick-lined) of the three-faced slope is a union of

the three envelopes of the three individual slope faces,

and the Type 2 sliding zone is a shaded area between

true dip lines of RS and LS in Fig. 4a.

The surface of a round-faced slope has unlimited

number of faces (Fig. 4b). The side-slope faces of this

slope can be considered as the two tangential faces at

two ends of the slope surface as shown in Fig. 4b. The

sliding envelope (thick-lined) of a round-faced slope

is a union of unlimited number of envelopes, and the

Type 2 single plane sliding zone (shaded) is the area

between dip direction lines of two tangential faces of

the sliding envelope. If the orientations of the two

Fig. 3 (continued).



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side-slope angles (a and b) are constant, the sliding

zone increases with an increase in the number of slopefaces.

4. Case study

To evaluate the proposed method, a fracture survey

and kinematic analysis was carried out on the exposed

rock slope faces in a quarry of Jurassic granite in

Seoul, Korea, where building stones were quarried

until 1979. Recently, this site has been developed for

housing. However, sliding failures have occurred

several times during re-excavation and site prepara-tion. A rock slope in Fig. 5 is an example of these.

The rock slope has a steep-quarried surface with a

gentle upper slope surface. The quarried surface (Fig.

5) is a typical two-faced slope having LS and RS. The

latitudes of LS and RS are N12jE/75jSE and

N40jW/75jNE, respectively. The lengths of LS and

RS are about 25 and 60 m, respectively. The trend of

MS is N25jW, and side-slope orientation angles of LS

and RS are 32j and 15j, respectively. Two joint

sets are developed in the rock mass of the slope.

Fig. 4. Stereographic projections for (a) a three-faced slope and (b) a round-face d slope. CS is the central slope face, and LCS is true dip of CS.

The shaded area is a Type 2 sliding zone. For key to other abbreviations, see Fig. 3.



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Latitudes of Set 1 joints (J1) and Set 2 joints (J2) are

N14jW/43jNE and N82jW/75jSW, respectively.

A major sliding failure occurred by single planesliding within both slope faces. This sliding block is

formed by J1 (sliding plane) and two side-slope faces.

This sliding is Type 2 single plane sliding on J1 plane.

Sliding plane is partly intersected by another joint.

J1 does not satisfy the plane failure condition

(h


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the Type 2 sliding zone (shaded area) between true dip

lines, LLS and LRS, of the sliding envelope (Fig. 7).This result indicates that Type 2 single plane sliding

on J1 joint planes can occur including both slope

faces, as shown in Fig. 5. This case study shows that

the suggested kinematic analysis method is a more

effective method to determine characteristics (i.e.,

modes, locality and potential, etc.) of sliding failures

in multi-faced slopes.

5. Conclusions

A single-faced slope is straight in plan view, but a

multi-faced slope, which consists of two or more

faces, is not straight in plan view. This difference in

surface geometries in plan view can trigger different

sliding conditions related to boundary conditions of

sliding blocks. Sliding modes and the stereographic

projection technique for stability analysis of multi-

faced slopes are as follows.

(1) Sliding modes in multi-faced slopes are divided

into two types based on number of slopes involved in

Fig. 6. Kinematic analysis to estimate the sliding potential of the

rock slope shown in Fig. 5 using the stereographic projection

technique for single-faced slope. (a) Right slope (RS). Type 1 single

plane sliding can occur on joint plane J1 (N14jW/43jNE). (b) Left

slope (LS). Type 1 double plane sliding can occur along the line of

intersection between joint planes J1 and J2 (N82jW/75jSW). For

key to other abbreviations, see Fig. 3.

Fig. 7. Kinematic analysis to estimate the sliding potential of the

rock slope shown in Fig. 5 using the stereographic projection

technique for multi-faced slope. Type 2 single plane sliding on joint

plane J1 as well as Type 1 single plane sliding are possible. For key

to other abbreviations, see Fig. 3.



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sliding block formation which are further subdivided

into two modes, respectively; Type 1 single and

double plane sliding and Type 2 single and double

plane sliding.(2) Type 1 single and double plane sliding are

controlled by the same rules for single-faced slopes. A

Type 1 sliding block has at least two wedge-forming

joint planes.

(3) Type 2 single and double plane sliding are

controlled by the rules different than for single-faced

slopes. A Type 2 sliding block involved at least the

two slope faces, and two or more slope faces of the

sliding block act as release surfaces. Type 2 single

plane sliding can occur with only one joint plane and

involves two or more faces of a multi-faced slope. For

Type 2 double plane sliding, the intersection of the

two wedge-forming joints must satisfy the Type 1

double plane sliding condition, and the joints must

satisfy the Type 1 single plane sliding condition on

two adjacent slope faces, respectively.

(4) On a stereographic projection, individual faces

of a multi-faced slope have their own sliding enve-

lopes. The sliding envelope of a multi-faced slope is

defined as the union of the envelopes of individual

faces forming the slope surface. If the absolute values

of the two side-slope orientation angles and the

number of faces of the multi-faced slope increase,the area of the sliding envelope increases.

(5) On a stereographic projection, Type 2 sliding

zone is defined as an area between true dip lines of the

two side-slope faces of the sliding envelope. If one (or

two) true dip line(s) of joint(s) is (are) plotted within

the Type 2 sliding zone, Type 2 single (or double)

plane sliding is possible, involving two or more faces

of the multi-faced slope. This Type 2 sliding cannot be

predicted by kinematic analysis technique developed

for a single-faced slope.

Acknowledgements

This research was performed for the Natural

Hazards Prevention Research Project, one of theCritical Technology-21 Programs, funded by the

Ministry of Science and Technology of Korea. The

BK 21 program through SEES has supported part of

this study. We thank Dr. W.Y. Kim of KIGAM, Dr.

Y.S. Kim and S.J. Yeo of SEES, Korea, Dr. J.R.

Andrews of University of Southampton, UK and two

anonymous reviewers for helpful comments.

References

Cruden, D.M., 1978. Discussion of G. Hockings paper A method

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Goodman, R.E., 1976. Methods of Geological Engineering in Dis-

continuous Rocks. West Publishing, San Francisco.

Hocking, G., 1976. A method for distinguishing between single and

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Hoek, E., Bray, J.W., 1981. Rock Slope Engineering Institution of

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Kliche, C.A., 1999. Rock Slope Stability SME, Littleton, CO.

Lucas, J.M., 1980. A general stereographic method for determining

possible mode of failure of any tetrahedral rock wedge. Int. J.Rock Mech. Min. Sci. Geomech. Abstr. 17, 5761.

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