Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 17, pp. 57 to 61 Pergamon Press Ltd 1980. Printed in Great Britain

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A General Stereographic Method for Determining the Possible Mode of Failure of Any Tetrahedral Rock Wedge J. M. LUCAS*

The paper describes a stereographic method for determining the possible mode of failure of a tetrahedral rock wedge, either in a rock slope or hounding an underground excavation. The method is more general than those pre- viously published in that all non-rotational modes of failure are considered and the method is not restricted to rock slope failures. A Table facilitates failure mode identification and worked examples are provided.

INTRODUCTION

A problem often encountered when excavating in regu- larly-jointed rock is that of identifying possible wedge or planar failure modes from a knowledge of the joint sets present and the orientation of the excavated face or faces.

Hocking [3] and Goodman [2] have presented simi-' lax methods for distinguishing between two-plane and single-plane sliding for rock slope wedges. Both methods are limited in that they ate only readily applic- able to slope wedges for which the two planes likely to be involved in sliding have been pre-determined.

Since, if the orientation of three bounding planes and an excavated face are specified, the poss~le mode of failure of a tetrahedral wedge is completely defined, it is possible to develop a method which will predict the mode of failure of any tetrahedral wedge having any~ orientation.

DISTINGUISHING BETWEEN SINGLE- AND TWO-PLANE SLIDING

As shown by Goodman [21 the condition which must be satisfied for sliding to occur through contact with one plane only is that the azimuth of the direction of maximum dip of the plane must lie between the azi- muths of the two boundary intersections. If this condi- tion, sketched in Fig. la, is not satisfied, then sliding will occur in contact with two planes as shown in Fig. lb.

Cruden [1] has shown that Goodman's condition is essentially the same as those of Panet [4] and Hocking [33.

All planes which satisfy the condition for single-plane sliding are not, however, sliding planes. One, two or three planes in a tetrahedral rock wedge may separately

* Engineering Geology Laboratories, University of Durham, South Road, Durham DH1 3LE, U.K.

satisfy this condition, so it is further necessary to determine whether the weight of the wedge will tend to act on the plane being tested. In the method presented this is determined by considering the orientation of planes and intersections relative to the intersection hav- ing the steepest downwards dip towards the excavated face.

In addition to falling by sliding in contact with one or two planes, a tetrahedral wedge with three enclosed

faces can fall verticaUy or he recumbent (i.e. unable to fail except in opposition to gravity)~

METHOD

Nomenclature of planes and intersection vectors For an underground wedge, A, B and C designate the

three enclosed planes and F the excavated face, as shown in Fig. 2a. For a slope wedge, A and B designate the two enclosed planes, C designates the upper free or excavated face and F the lower excavated or free face, as shown in Fig. 2b.

The vector normals to the planes (i.e. the poles to the planes) are designated A, B, C and F, and are all con- sidered to be positive in an upward direction.

l.b, I~c and Ib~ represent the intersection vectors (i.e. lines of intersection) of planes A and 13, A and C, and B and C, respectively. The intersection vectors are con- sidered to be positive in a direction from the wedge apex towards the excavated face. Vector azimuths are measured from 0-360 and their inclinations from -90 (vertically upwards) to +90 (vertically down- wards).

The intersection vector having the most positive dip is designated Ixy, being the line of intersection of planes X and Y. The third plane is designated plane Z.

Criteria for determining the mode of failure of a wedge The relative spatial positions of the three planes and

the excavated face determine whether or not a kinema-

57

58 J.M. Lucas

L ~ ~ ~' -

I ,)

w~

maximum 111~

0 ~ A ~ Iaei#

Fig. 1. (a) Condition requirzd for sliding on one plane and (b) condi- tion required for sliding on the intersection of two planes.

tically admissible wedge is formed. As a stereogram can only hold directional information, the kinematicaUy ad- missible combination of planes is always assumed to be present.

Recumbent wedges--If all the intersection vectors have negative clips then the wedge is recumbent.

Ag~t

Falling wedges--If the azimuths of the three intersec- tion vectors do not lie within 180 of arc, the wedge can then fall vertically.

Sliding wedges--Three comparisons of the azimuths of intersection vectors and plane vector normals are carried out in order to determine uniquely the mode of failure. These comparisons involve the azimuths of:

Ixy, X and Y (i)

Ixy. X and Lz (ii)

Ixy, Y and lyz (iii)

Within each group of three, the vector which has its azimuth intermediate between the other two is noted. From these comparisons there are three possible out- comes for (i) and two each for (ii) and (iii), giving a total of twelve possible outcomes for the three comparisons considered together. From a consideration of these out- comes the mode of failure of the wedge can be determined.

PROCEDURE

A practical stereographic method of determination based on the above criteria is presented here as a step by step procedure.

(1) Plot the poles to planes A, B, C and the excavated face F on the stereogram as upper hemisphere projec- tions. (This gives the projection of vectors A, B, C and F.)

(2) Draw the great circle described by the excavated face in upper hemisphere projection.

(3) Construct the upper hemisphere projection of the intersection vectors l.b, I~ and I.o by drawing the pole to the great circle which passes through the poles to the pair of planes defining each intersection.

N

(.I

"~~~ ~ I I I I I ~=EX \:o,'*

0 negative I vec lot Q no.thee J t~ sa~e

(b) I~litive vector Drojecting to the lower hemisl:t~-e

Fig. 2. Definition of the planes composing (a) an underground wedge Fig. 3. Stereographic projection illustrating the method for finding and (b) a surface wedge, the positive intersection vectors.

Method for Determining the Failure of any Tetrahedral Rock Wedge 59

T~st 1. Moves ov SLIDING AS DETERMINED FROM TEST 3

Plane(s) on which Result of Test 3 contact is

No. (a) Co) (c) maintained

1 X |xy Izy X Y 2 l,y lxy lay X Y 3 Y Ixy l~y XY 4 X X la) X 5 lay X I,, X 6 Y X I,y Z or none* 7 X~ I,y Y Z or none* 8 ]xy Ity Y Y 9 Y I,y Y Y

I0 X X Y Y II l,y X Y Z or none* 12 Y X Y X

Notes

Plane Y overlies Plane X

Plane X overlies Plane Y

Plane Y overlies Plane X See footnote See footnote Plane X overlies Plane Y

Plane X overlies Plane Y See footnote Plane Y overlies Plane X

*If contact is maintained on plane Z only.

(4) For non-overhan0in0 excavated faces, change to lower hemisphere projection all intersection vectors which project outside the area of the stereogram bounded by the great circle of the excavated face and containing F. In Fig. 3, I,o and Ib~ are transformed in this manner from upper to lower hemisphere projec- tion.

For overhanoino excavated faces, change to lower hemisphere projection all intersection vectors which project inside the area of the stereogram bounded by the great circle of the excavated face and containing F. (In the cases of both non-overhanging and overhanging faces the three points resulting from the transformation are the projections of the intersection vectors in a posi- tive direction.)

(5) Test I

If all the positive intersection vectors project to the upper hemisphere, contact is maintained on three planes and the wedge is recumbent.

(6) Test 2

If the azimuths of the three positive intersection vec- tors do not all fall within 180 of arc, then contact is lost on all planes and the wedge falls vertically under gra- vity loading.

(7) The positive intersection vector projecting to the lower hemisphere and having the steepest dip is determined and designated Izr For a rock slope wedge, if lay is not lab then no wedge is formed.

(8) Lines are drawn from the origin (centre) of the stereogram to the projections of l,y, Iz,. I~ X and Y so that their relative azimuths may be readily discerned.

(9) Test 3 Three comparisons are made and the result of each

noted. Test 3(a). X, Y and Izy axe considered and the vector

with azimuth intermediate between the azimuths of the other two is noted.

Test 3(b). X, Ixy and Ixz are considered and the vector with azimuth intermediate between the azimuths of the other two is noted.

the result of Test 2 indicates that contact is lost on all planes, then 'none', otherwise

Test 3(c~ Y, l~y and I~ are considered and the vector with azimuth intermediate between the azimuths of the Other two is noted .

(10) The plane or planes on which contact is main- tained are read off from Table 1, where the twelve poss- ible outcomes of 3(a), 3(b) and 3(c) are listed.

EXAMPLES OF THE METHOD The examples quoted below are taken from actual case histories.

Example I A wedge has formed in a tunnel haunch. The dips/

azimuths of the planes composing the wedge are:

Joint Plane A 500/060 Joint Plane B 40/120 Joint Plane C 10/310

Overhanging excavated Plane F 450/270

face

A stereogram of points plotted for this dete.//~ination is shown in Fig. 4. After step 4 it is found that the positive intersection vector I,o projects to the upper hemisphere and vectors lib and Ib~ project tO the lower hemisphere. l,b has the steepest dip and is hence designated lay in step 7. The following allocations are thus made:

I. -

It, - Ib~

laz -- la

X- -A

YEB

The results of Test 3 are:

Test 3(a) l,y ( - tab ) Test 3(b)lxy ( - lab )

Test 3(c)Ixy (~ lab )

From Table 1 this configuration is a wedge of type 2, with sliding occurring in contact with planes A and B.

60 J .M. Lucas

T.,3. x l r..li, T--t 3b X /m[ [,c To., 3o 6 r.,-I r,~

Fig. 4. Stereographic projection and analysis of a wedge formed in a tunnel sidewall.

Example 2

A wedge has formed in a tunnel crown. The dips/azi- muths of the planes composing the wedge are:

Joint Plane A 80/150 Joint Plane B 70/120 Joint Plane C 50/170

Tunnel roof plane Plane F 00 / -

(overhanging)

A stereogram of the points plotted for this determina- tion is shown in Fig. 5. After step 4 it is found that the positive intersection vectors lab, lbc and l~c all project to the lower hemisphere, lab has the steepest dip and is hence designated lxy in step 7. The following allocations are thus made:

Ixy = lab

Ixz = Iac

Iy z ~ Ibc

X=A

Y=B

The results of Test 3 are:

Test 3(a)Y = B

Test 3(b)X -- A

Test 3(c)Y = B

From Table 1 this configuration is a wedge of type 12 with sliding occurring in contact with plane A only.

Example 3

A wedge has formed in a sloping quarry face. The dips/azimuths of the planes composing the wedge are:

Joint Plane A 49/035 Joint Plane B 80/105

Upper free Plane C 00 / - face

Excavated Plane F 60/028 quarry face

A stereogram of the points plotted for this determina- tion is given in Fig. 6. After step 4 it is found that the

N

n

ToSt 311. ~ I ~ Teb

Test 3b Tab[ A iac Test 3c Iab j Tbc

Fig. 5. Stereographic projection and analysis of a wedge formed in a tunnel roof.

Twit 3, ~1 [ ~. I l'ab r~, 3b X T~ Tac

Te=t 3c Tab t'~

Fig. 6. Stereographic projection and analysis of a wedge formed in a quarry face.

Method for Determining the Failure of any Tetrahedral Rock Wedge 61

positive intersection vectors lac and lbc project to the horizontal and vector lab projects to the lower hemi- sphere, lab has the steepest dip and is hence designated I,y in step 7. This shows that a valid wedge is formed. The following allocations are thus made:

Ixy ~ lab

Ixz = Iac

Iy z ------- Ibc

X- -A

Y-B

CONCLUSIONS

Knowing the angular relations of the planes which compose a tetrahedral rock wedge it is possible, using the method outlined in the paper, to assess very rapidly in the field the possible non-rotational mode of wedge failure. The method is shown to be applicable to both surface slope and underground wedges.

Acknewledgemeam--The author thanks Dr P. B. Attewell for his help and encouragement in the preparation of this paper. The work on which this paper is based was funded by an award from the Natural Environment Research Council.

Received 31 July 1979.

The results of Test 3 are:

Test 3(a)X - A

Test 3(b)Ixy = lab

Test 3(c)Ixy - lab

From Table 1 this configuration is a wedge of type 1 with sliding occurring on planes A and B, plane B over- lying plane A.

REFERENCES

I. Cruden D. M. Discussion of G. Hocking's paper A method of distinguishing between single and double plane sliding of tetra- hedral wedges. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 15, 217 (1978).

"2. Goodman R. E. Methods of Geolooical E~ineering in Discomi. nuous Rocks, 472 pp. West, St. Paul (1976).

3. Hocking G. A method of distinguishing between single and double plane sliding of tetrahedral wedges. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 13, 225-226 (1976).

4. Panet M. Discussion of graphical stability analysis of slopes in jointed rock by John, K. W. J. Soil Mech. Found. Div. Proc. ASCE, 95, 685-686 (1969).