Engineering Rock Mechanics

Geometrical properties of discontinuities 1 33

Proportionate error in mean discontinuity spacing, E

Figure 7.18 Determination of number of discontinuity spacings (and hence scanline
length) to estimate the mean spacing value to within a given error at given
confidence levels.

In Fig. 7.19, we show the direction cosines associated with a unit vector.
These are required for numerical calculation of the mean orientation of a
number of discontinuities, using the procedure outlined below. When
many discontinuities with different orientations are intersected by a
scanline and sampled, the mean dip direction and dip angle may be found
by using the procedure outlined in tabular form in Fig. 7.20. This procedure
corrects for orientational sampling bias through the introduction of
weighted direction cosines. The first two columns are the dip direction and
dip angle as measured, a, p, and the following two columns contain G, Pn,
the trend and plunge of the normal to each discontinuity. The direction
cosines of each of the normals are then evaluated in the next three columns
using the formulae of Fig. 7.19.

Vector geometry

m = cosol cosp

1 = sinol cosp

n = sinp

Direction cosines

1

6 vector Unit E X Angle cos6 between = 1, 1,^2 + vectors: rn, m, + n, n2

Down

2

Figure 7.19 Direction cosines of a unit vector.