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Int. J. Rock Mech. Miu. S,'i. & Geomech. Abstr. Vol. 13, pp. 225--226. Pergamon Press 1976. Printed in Great Britain

Technical Note A Method for Distinguishing between Single and Double Plane Sliding of Tetrahedral Wedges G. HOCKING*

INTRODUCTION

of such a wedge is that l t2 must lie between l t t. and ltt. (or 12t~ and I,i.) (Panet[1]), and also that • ~. > ~J2 > ~i,, (where :c,_, is the dip of I~_, and a~. and ~i: are the apparent dips of the lower and upper slopes in the direction of I~2). The question now arising is, on which of the following will the wedge slide----(a) the line of intersection of planes 1 and 2; (b) plane 1; or (c) plane 2? Consider the wedge shown in Fig. 3(a). and its stereographic projection, Fig. 3(b). For the purposes of clarity the upper slope is assumed horizontal; however, the argument holds true for the general case. In the case illustrated, the wedge will tend to slide in the direction of maximum dip of plane 2. The wedge will move away from plane 1 and thus contact with plane 1 will be lost. Now examine the case illustrated in Fig. 4 in which the planes 1 and 2 are identical to those in Fig. 3, but the slope face orientation has been slightly altered. In this case, the wedge will tend to slide down the line of intersection of planes 1 and 2, and thus contact will be shared by both planes.

In assessing the stability of a tetrahedral wedge in a slope against sliding, it is important to distinguish between the cases of single and double plane sliding. If sliding down the line of intersection is always assumed, then the stability of the wedge will be overestimated for cases in which sliding down a single plane actually occurs. A simple rule to distinguish between single and double plane sliding of tetrahedral wedges, assuming only body forces and hydrostatic forces are acting, is presented. This rule differs from that given by Panet [1] who assumed that the analysis was independent of the orientation of the slope face for the plane sliding case. This assumption is shown to be incorrect. DISCUSSION Figure 1 illustrates a kinematically possible sliding wedge and Fig. 2 shows its lower hemispherical sterographic projection (Wulff net). The condition for sliding

~ L O W E R

SLOPE

"

~

O

F

Fig. 1. Rock slope containing a three-dimensional wedge.

FIGURE 3a.

N

N

UPPER SLOPE--~/I \ !

~

A

MOVEMENT

PLANE2 .OWERSLOPE

E

Fig. 2. Stereographic projection (lower hemisphere) of wedge and rock slopes illustrated in Fig. 1.

FIGURE 3~

* Imperial College of Science and Technology, London S.W.7, U.K.

Fig. 3. (a) Single plane sliding case---wedge slides on plane 2 only. (b) Stereographic projection of single plane sliding case--wedge slides on plane 2 only.

225

226

Technical Notes CONCLUSIONS

NE 1

FIGURE 4a.

N

In the case of a kinematically possible sliding wedge. one can distinguish between single and double plane sliding by applying the following rule: If the dip direction of either plane 1 or 2 lies between the dip direction of the lower slope face and the line of intersection of planes I and 2. then sliding will occur down planes I or 2 respectively (illustrated in Fig. 3). If the above condition is not satisfied then the wedge will slide down the line of intersection of planes 1 and 2 (illustrated in Fig. 4). The distinction between single and double plane sliding is important, because unsafe design would result if sliding were always assumed to occur down the line of intersection of the two planes.

Received 9 February 1976.

FIGURE 4h Fig. 4. (a) Double plane sliding case--wedge slides on line of intersection of planes 1 and 2. (b) Stereographic projection of double plane sliding case--wedges slides on line of intersection of planes 1 and 2.

REFERENCE 1. Panet M. Discussion on 'Graphical stability analysis of slopes in jointed rock' by John K. W., J. Soil Mech. and Found. Dir. Proc. A.S.C.E., Vol. 95, SM 2, pp. 685-686 (March 1969).