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Journal of the Chinese Institute of Engineers, Vol. 26, No. 3, pp. 353-359 (2003) 353

STABILITY ANALYSIS OF ROCK SLOPES USING BLOCK

THEORY

Tien-Kuen Huang*, Jaw-Chern Chen and Chein-Chi Chang

ABSTRACT

The system of discontinuities that transverses a rock mass delimits rock blocks

of many sizes, shapes and positions in a surficial excavation. By using the block

theory, it is possible to determine the most critical of these isolated masses, which are

denoted as key blocks. After identifying the potential key blocks, their shapes and

volumes can be computed with the input of the spacing of each discontinuity. In thispaper, a sliding equilibrium stability analysis is performed with the friction of each

discontinuity, the gravity force, the sliding direction and rock bolt support for the

potential key blocks calculated from the study of some typical examples and field

cases.

Key Words: key block, kinematics, limit equilibrium method, stability analysis,

stereographic projection.

*Corresponding author. (Tel: 886-4-22872221 ext. 227; Fax:

886-4-22862857; Email: [email protected])

T. K. Huang is with the Department of Civil Engineering, Na-

tional Chung-Hsing University, Taichung, Taiwan 402, R.O.C.

J. C. Chen is with the Water Conservancy Agency, Ministry of

Economic Affairs, Taichung, Taiwan 408, R.O.C.

C. C. Chang is with the Department of Civil and Environmental

Engineering, University of Maryland, Baltimore County, MD21250, U.S.A.

I. INTRODUCTION

Discontinuities in rocks are very common in

rock slope engineering. Since discontinuities are ex-

tensive near the earth surface, they frequently inter-

sect each other and divide rock masses into jointed

rocks of various shapes and sizes. The instability

phenomena of discontinuous rock slopes may be due

to rotation, sliding and toppling. Rotational slides

mainly occur in closely jointed rock. Sliding motion

tends to follow the pre-existing discontinuity plane

(s). Toppling conditions are reached when the jointed

rock is tall and thin enough so that the gravity vector

of the jointed rock falls outside its base. Once the

blocky system has been isolated and defined, staticor dynamic methods (Lin and Fairhurst, 1988) for sta-

bility analysis can be used.

Static analysis examines the kinematic mecha-

nism of the sliding or toppling of the block which

has a face exposed on the rock slope. The acting and

resistant forces are computed and the equilibriumequations are then solved to determine whether or not

the block is stable. The analysis by using the limit

equilibrium method, considers the incipience of mo-

tion and does not examine the subsequent behavior

of the whole system of blocks. On the other hand,

dynamic analysis such as the distinct element method

(Cundall, 1988), attempts to simulate the behavior of

a blocky system in a realistic manner by taking ac-

count of discontinuity stiffness, rock block deforma-

tion, and progressive failure, etc. Since most occur-

rences of jointed rocks are near the earths surface in

rock slope engineering and if the removable blocks

around the surface are restrained, the jointed blocksbehind the surficial block will remain in a stable con-

dition and no subsequent behavior of the blocky sys-

tem needs to be examined. Therefore, the static analy-

sis based on the limit equilibrium method is still an

efficient procedure for the stability evaluation of dis-

continuous rock slopes. As for discontinuous rock

slopes, a complete stability analysis should include

three parts: (1) generate the geometry of the blocky

system isolated by discontinuities and rock slope;

(2) find the potential removable blocks and the corre-

sponding sliding modes; and (3) evaluate the stability.

Most previous research on the stability problemsof discontinuous rock slopes with static analysis was


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354 Journal of the Chinese Institute of Engineers, Vol. 26, No. 3 (2003)

only focused on some subjects, such as generation of

a blocky rock mass by Warburton (1983) and Heliot

(1988); the determination of sliding modes and safety

factors of sliding blocks by Klaus (1968), and Hoek

and Bray (1981); the distinction between single and

double plane sliding of blocks by Hocking (1976) and

Lucas (1980); the assessment of kinematic feasibility,

sliding and volumes of blocks through inclined ste-

reographic projection by Priest (1980), through vec-

tor analysis by Warburton (1981), or based on graphic

theory by Lin and Fairhurst (1988). The block theory

developed and continuously extended by Shi et al.

(Goodman and Shi, 1985; Mauldon and Goodman,

1990) can be thought of as a comprehensive proce-

dure of static analysis for stability evaluation of dis-

continuous rock slopes. The power of this block

theory lies in simplification from a fairly complicated

analysis of discontinuous rock slopes to a step-by-step analysis (Mauldon and Goodman, 1990).

This paper describes the principles of block

theory in a brief manner to evaluate the stability of

discontinuous rock slopes. Detailed fundamentals and

proofs can be referred to in the book by Goodman

and Shi (1985). Some typical examples and field

cases are illustrated to validate the theoretical results

by the block theory. It seems that very few applica-

tions are conducted for jointed rock stability in a de-

tailed manner.

II. BLOCK THEORY

1. Generation of a Block

A block is the region of intersection of half-

space formed by the discontinuities that form the

block faces. Each discontinuity is described by two

parameters: the dip angle and the dip direction .

IfD is the length perpendicular from the origin to the

discontinuity or slope plane, the equation of the dis-

continuity plane is given by:

(sinsin)X+(sincos)Y+(cos)Z=D (1)

A particular block can be created by the inter-

section of the designated upper or lower half-spaces

corresponding to each of the discontinuities. The

block corners are calculated as the intersection points

of three different planes. Only a few corners which

are real actually belong to the considered block. In

computing the volume of any type of block, it can be

subdivided into tetrahedra and then can be made of

the common formula with vector analysis.

2. Types of Blocks

There are five types of blocks in the block

theory. An infinite block (type V) is of no hazard to

an excavation. Finite blocks are divided into non-

removable and removable types. A finite block may

be non-removable because of its tapered shape (type

IV). The other three (III, II, I) are removable blocks.

Their stability depends on the orientation of the re-

sultant force, frictional resistance of discontinuities

and support implementation, etc.

3. Determination of Removable Blocks

The blocks are defined partly by discontinuity

and rock slope half-spaces. The discontinuity subset

of the half-spaces determines the joint pyramid (JP).

The set of slope half-spaces is designated as the ex-

cavation pyramid (EP). The block pyramid (BP) is

then the intersection of the JP and the EP for a par-

ticular block:

BP=JPEP (2)

If the BP is empty (), the block is infinite.

JPEP= (3)

Whether a finite block is not removable (type

IV, tapered) or removable is based on the following

conditions.

A block is removable if itsBP=and JPandbecomes non-removable if its BP=and JP=.

4. Failure Modes of Removable Blocks

Only removable blocks require for further

analysis. There are three failure modes considered.

They are lifting (or falling), sliding on a single plane,

and sliding on the intersection of two planes.

The lifting or falling mode occurs when there

are no discontinuities in contact and the sliding di-

rection is along the resultant force. In the case of

sliding in a single plane, there will be only one dis-

continuity in contact and the sliding direction is along

the orthogonal projection of the resultant force on thatcontact plane. As for sliding on the intersection of

two planes, there are two discontinuities in contact

and the sliding direction is along the intersection of

those two planes. A fully kinematic analysis used to

determine the sliding direction of the removable

blocks has been developed in the block theory.

5. Analysis of Sliding Equilibrium Stability

From the kinematic analysis of failure modes for

removable blocks, one can obtain the required infor-

mation for identifying the possible sliding conditionsof the removable blocks. If the removable blocks for


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T. K. Huang et al.: Stability Analysis of Rock Slopes Using Block Theory 355

a given rock slope do not have any failure mode, they

will be stable and safe.

An innovative approach to combining the kine-

matic conditions and stability analysis of the sliding

equilibrium for removable blocks is developed in the

block theory through the construction of sliding equi-librium regions of stereographic projection. In the

sliding equilibrium region for each of the removable

blocks, its stereographic projection is obtained from

the designated half-spaces of discontinuity planes

with the inclusion of three failure modes. Within the

region of each removable block, the contours of dif-

ferent friction angles of the discontinuities are also

constructed to provide the friction to stabilize the re-

movable block. In the projection, five symbols +, 0,

i, ij and S represent respectively the resultant force

projection, the region of lifting (or falling), the re-

gion of sliding on plane i, the region of sliding on theintersection of planes i and j and the safe region.

III. WORKED EXAMPLE

The computational procedure of the block theory

is demonstrated with a worked example. To provide

comparison with previous work, the illustration is

based on the example chosen by Priest (1980), which

involves five discontinuity planes for defining ten

tetrahedra. Two types of free surfaces are considered,

which are non-overhanging and overhanging respec-tively. The dip angles and dip directions of all the

discontinuity planes and free surfaces are listed in

Table 1. The blocks are under gravity load only.

First, the removable block with a designated JP code

number corresponding to each type of tetrahedron by

discontinuity planes and free surface can be deter-

mined through stereographic projection. Fig. 1 shows

the results of the stereographic projection of the dis-

continuity planes 1, 2, 3 and a non-overhanging face

6 with a lower focus projection. The only removable

block is JP with code number 001 inside the circle 6.

The result of sliding equilibrium regions correspond-ing to JP001 is presented in Fig. 2. It can be seen that

Table 1 Joint and slope orientations of worked example

Plane Dip, (deg) Dip direction, (deg)

1 (Joint) 65 353

2 (Joint) 42 305

3 (Joint) 50 454 (Joint) 39 153

5 (Joint) 82 179

6 (Non- overhanging slope) 80 292

7 (Overhanging slope) 30 40

011

010

110

Plane 3

111

Slope 6

011

000

001

101

Plane 2

Plane 1

13

S

23

3

0

80

70 60 50 40

2030

10

12

2

Fig. 1 Stereographic projection of plane 1, 2, 3 and 6 of worked

example

Fig. 2 Sliding equilibrium regions and contours of friction of

JP001 of worked example


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JP001 slides along plane 2 (the symbol + inside

region 2) and a minimum value of friction, over 40

degrees, is needed to stabilize the removable block.

Other computational results concerning the discon-

tinuous planes 1, 3, 4 and an overhanging face, 7, are

presented in Figs. 3 and 4. JP111 is the only remov-

able block and its sliding mode is falling. Frictiondoes not have any effect in the prevention of falling

because there are no discontinuities in contact. The

other removable tetrahedra by any three discontinu-

ity planes and the non-overhanging face 6 or over-

hanging face 7 can be obtained in a similar manner.

The results are summarized in Tables 2 and 3, in

which the corresponding sliding plane(s) and the least

friction angles to keep the removable blocks stable

are also included. When compared with the work by

Priest (1980), the sliding mode and direction corre-

sponding to each of the tetrahedra are the same as

those obtained from the block theory. But more en-gineering information such as the precise description

of removable blocks and minimum friction angles

required to prevent the movement of the potential

removable blocks are easily examined and provided

in the analysis of the block theory.

IV. FIELD CASES

In the following, two field cases are studied to

validate the practical applicability of the theoretical

results by the block theory. The in-situ discontinu-

ous rock slope is located on the side of a country road

in the central part of Taiwan. The discontinuity con-

ditions of rock mass and the surficial excavation for

the first case studied are listed in Table 4 after field

investigation. The principal joint sets are character-

ized through the measuring of many joints and then

determined from the clustering of joint orientations

in the stereographic projection. The bedding systemis laminated by sand and shale layers of different

Table 2 Summary of tetrahedral blocks under a

non-overhanging slope (=80, =292)

Planes Removable Min. frictionSliding

defining JP to stabilizeplane(s)

tetrahedra code no. (deg)123 001 2 42.0

124 101 1 and 2 29.6

125 101 1 and 2 29.6

134 010 none

135 010 1 and 5 2.9

145 010 1 and 5 2.9

234 011 2 42.0

235 011 2 42.0

245 010 2 and 5 24.4

345 101 none

Table 3 Summary of tetrahedral blocks under an

overhanging slope (=30, =40)

Planes Removable Min. frictionSliding

defining JP to stabilizeplane(s)

tetrahedra code no. (deg)

123 001 2 65.0

124 101 1 and 2 29.6

125 101 1 and 2 29.6

134 111 0

135 111 0

145 110 5 82.0

234 111 0

235 111 0

245 110 5 82.0

345 010 3 and 5 23.6

: falling mode

001

011

111

110

101

100

000

010

Slope 7

Plane 1

Plane 4

Plane 3

13

1020

3040

50

807060

12

2

S

230

1

3

Fig. 3 Stereographic projection of planes 1, 3, 4 and 7 of worked

example Fig. 4 Sliding equilibrium regions and contours of friction of

JP111 of worked example


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thickness and extends with an orientation of dip di-

rection of 145 from the north and dip angle of 40.

Two types of joint planes are developed in the bed-

ding with a spacing of 50 cm in sand layer and a spac-

ing of 10 cm in shale layers, while 10-25 cm spacing

is found in the joint planes. The other case is near

the site of the former one on the same route but the

orientation of surficial excavation is shifted for com-

plying with the local morphology. The dip angle and

dip direction for case two are 80 and 35, respec-

tively. In the former case, Figs. 5 and 6 show the

analyzed results in which JP001 is the only remov-

able block and its sliding direction is along the inter-

section of the planes 1 and 2, and a higher frictionangle of approximately 50 degree is required to con-

strain the removable block (JP 001) if no artificial

support is provided. Fig. 7 shows the wedge failure

of this case in the field. It can be clearly seen that

much more blocks with tetrahedron shapes slide in

sand layer along the intersection of the two joint

planes. Considering the maximum removable block

JP001 with the spacing of joint 1, joint 2 and bed-

ding 3 being 0.20m, 0.25m and 0.50m respectively,

the volume is computed as 0.172 m 3. The gravity

weight is about 464 kg. With the provision of rock

bolt of 10mm in diameter installed along the normaldirection of both sliding planes, the allowable shear

Table 4 Joint and slope information of field cases

Plane Dip,(deg) Dip direction, (deg) Friction angle, (deg) Spacing, m

1 (Joint) 60 300 30 0.1-0.2

2 (Joint) 70 25 30 0.1-0.25

3 (Bedding) 40 145 15-30 0.1-0.54 (Slope) 70 315

strength for each rock bolt is about 879 kg (=0.4.2800.

(1)2.3.14/4). The new orientation of the resultant can

be obtained and plotted on the stereographic projec-

tion asR1 in Fig. 6. It can be seen that the removable

block JP001 will stand safely after implementing the

rock bolt even without the provision of joint friction.

As for the latter case, the stereographic projections

of the discontinuity and surficial planes are shown in

Fig. 8. The only type of removable block is JP 101

which is inside the surficial surface 4. Following the

same procedure as in the analysis of the former case,we can construct the sliding equilibrium regions of

the block JP101 (see Fig. 9) in which the sliding mode

is along the joint plane 2, and an approximate fric-

tion angle value of 70 is needed to stabilize the re-

movable block if no artificial support is provided.

Fig. 10 shows the sliding in this case along plane 2

with a higher dip angle. In a similar manner, the maxi-

mum removable block can be obtained with the maxi-

mum spacing of discontinuities. Its volume and

weight are 0.265 m3 and 716 kg respectively. A rock

bolt of 10mm in diameter is installed along the nor-

mal direction of the sliding plane. The new orienta-tion of the resultant combined with the allowable

Plane 4

Plane 1

Plane 3

Plane 2

001

101

011

000

010100

110111

Fig. 5 Stereographic projection of field case one

23

R1

S

13

0

212

1

80

7060

50

40

30

20

3

10

Fig. 6 Sliding equilibrium regions and contours of friction of

JP001 of field case one


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shear strength can be obtained and plotted on the ste-

reographic projection asR

2 in Fig. 9. The removableblock JP101 becomes stable, as in case one, after in-

stalling the rock bolt.

V. CONCLUSIONS

In this paper the principles and procedures of

the block theory are described in a brief manner. Very

common cases of rock slope with two joint planes,

one bedding and a surficial excavation are introduced

in the example problems and field case studies. In

the block theory, the removable blocks can be pre-

cisely examined and predetermined. Only removable

blocks require further analysis. Through a kinematicanalysis the sliding mode is obtained. Then either

potential blocks with suitable frictional resistance or

unstable blocks required artificial support are calcu-

lated sequentially. The illustrations of worked ex-

ample and field cases validate the theoretical results

by the block theory. More engineering information

from the computation of block theory can provide

valuable references in the design of discontinuous

rock slopes.

NOMENCLATURE

BP block pyramid

D perpendicular length from the origin to discon-

tinuity

EP excavation pyramid

JP joint pyramid

dip angle dip direction

REFERENCES

Cundall, P. A., 1988, Formulation of a Three Di-

mensional Distinct Element Model - Part I. A

Scheme to Detect and Represent Contacts in a

System Composed of Many Polyhedral Blocks,

In ternat ional Journa l of Rock Mechanics and

Mining Science, Vol. 25, No. 3, pp. 107-116.

Goodman, R. E., and Shi, G. H., 1985, Block Theory

and Its Application to Rock Engineering, Prentice

-Hall, New Jersey, p. 338.

Heliot, D., 1988, Generating a Blocky Rock Mass,

In ternat ional Journa l of Rock Mechanics and

Mining Science, Vol. 25, No. 3, pp. 127-138.Hocking, G. A., 1976, A Method for Distinguishing

Fig. 8 Stereographic projection of field case two

R2

13

1

0

2

12

S

23

3

30

20

10

80

70 60

50 40

101

001

011

Plane 1

000

010100

110111 Plane 3

Plane 2

Slope 4

Fig. 10 Single plane sliding of field case two

Fig. 9 Sliding equilibrium regions and contours of friction ofJP101 of field case two

Fig. 7 Wedge failure of field case one


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between Single and Double Plane Sliding of

Tetrahedral Wedges, Internat ional Journal of

Rock Mechanics and Mining Science, Vol. 13, No.

7, pp. 225-226.

Hoek, E. , and Bray, J . W., 1981, Rock Slope

Engineering, Institute of Mining and Metallurgy,

London, p. 358.

Klaus, W. J., 1968, Graphical Stability Analysis of

Slope in Jointed Rock, Journal of Soil Mechan-

ics and Foundation Engineering Division, ASCE,

Vol. 94, No. 2, pp. 497-526.

Lin, D., and Fairhurst, C., 1988, Static Analysis of

the Stability of Three-Dimensional Blocky Sys-

tems around Excavations in Rock,International

Journal of Rock Mechanics and Mining Science,

Vol. 25, No. 3, pp. 139-147.

Lucas, J. M, 1980, A General Stereographic Method

for Determining the Possible Mode of Failure ofany Tetrahedral Rock Wedge, International

Journal of Rock Mechanics and Mining Science,

Vol. 17, No. 1, pp. 57-61.

Mauldon, M., and Goodman, R. E., 1990, Rotational

Kinematics and Equilibrium of Blocks in a Rock

Mass,Internat ional Journal of Rock Mechanics

and Mining Science, Vol. 27, No. 4, pp. 291-300.

Priest, S. D., 1980, The Use of Inclined Hemisphere

Projection Methods for the Determination of Ki-

nematic Feasibility, Slide Direction and Volume

of Rock Blocks, International Journal of Rock

Mechanics and Mining Science, Vol. 17, No. 1,

pp. 1-23.

Warburton, P. M., 1981, Vector Stability Analysis

of an Arbitrary Polyhedral Rock Block with any

Number of Free Faces,International Journal of

Rock Mechanics and Mining Science, Vol. 18, No.

5, pp. 415-427.

Warburton, P. M., 1983, Application of a New Com-

puter Mode for Reconstructing Blocky Rock

Geometry, Analyzing Single Block Stability and

Identifying Keystones, Proceedings of 5th In-

ternational Congress on Rock Mechanics , Mei-bourne, Balkema, Rotterdam, F225-230.

Manuscript Received: Jan. 25, 2002

Revision Received: Nov. 03, 2002

and Accepted: Dec . 22, 2002

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