Engineering Rock Mechanics

334 Design of surface excavations

weight of the rock overlying the bedding plane and any water pressures
induced due to the operation of the dam to make certain of this.
In order to determine the range of x-ordinate values to the right
of this position over which the bedding plane is stable, we need to
compute the stress state induced on the plane and compare this with
its strength parameters. Again, although we could determine a solution
for the position of the specific point at which the plane is in limiting
equilibrium, it is more useful and instructive to determine the conditions
at a number of points and hence plot the variation with distance.
This first step in this analysis is to sketch the geometry of a general
point on the bedding plane, in order to determine the angles required
for computation of the radial stresses (from the equations given) and the
stress transformations:
Geometry:

From this we can see that d

or or

a,, = - (1 +cask) and r = -sin%.

2 2

A table of results can now be constructed, as shown below.

x, m -15 -10 -5 0 5 10 15 20 25 30

r, m (= dm) 16.16 11.66 7.81 6.00 7.81 11.66 16.16 20.88 25.71 30.59

a, deg (=tan-' (6/x)) -68.2 -59.0 -39.8 0.0 39.8 59.0 68.2 73.3 76.5 78.7

e, deg (= 90"-a) 158.2 149.0 129.8 90.0 50.2 31.0 21.8 16.7 13.5 11.3

o,(P), Wm2 137.6 264.0 588.6 997.4 588.6 264.0 137.6 82.4 54.3 38.4

ur(Q)r Wm2 -161.0 -206.0 -229.6 0.0 229.6 206.0 161.0 128.5 105.9 89.8

cr(wmbined)r W/m2 -23.4 58.0 359.0 997.4 818.2 470.0 298.6 210.8 160.3 128.1

a,, IdV/m2 -3.2 15.4 211.9 997.4 482.9 124.4 41.2 17.4 8.7 4.9

T, kN/m2 8.1 -25.6 -176.6 0.0 402.4 207.3 102.9 58.0 36.4 24.6

u,,, lcN/m2 (= a, + 6y) 128.8 147.4 343.9 1129.4 614.9 256.4 173.2 149.4 140.7 136.9

hquiredr deg

(=tan-' (r/un)) 3.6 -9.9 -27.2 0.0 33.2 39.0 30.7 21.2 14.5 10.2

The results for these 10 points are plotted on the following graph,

together with the curve that joins them. This curve represents the

variation in required friction angle, and from this we can see that there

is a region from about x = 4.5 m to x = 14.0 m where slip will occur

(the actual values, found algebraically, are x = 4.69 m and x = 14.38 m).

Notice that no slip occurs for negative values of x, although the change

in sign of the friction angle indicates that the sense of the shear stress

changes at x = 0.