Civil Engineering Reference
In-Depth Information
If there is little scatter in the orientation of the
discontinuities, the value of
C
d
is large, and its
value diminishes as the scatter increases.
From the dispersion coefficient, it is possible to
calculate from equation (3.17) the probability,
P
,
that a pole will make an angle
θ
◦
or less than the
mean orientation.
of the discontinuities aligned parallel to the line of
mapping are measured. This bias in the data can
be corrected by applying the Terzaghi correction
as described in Section 3.4.1.
3.5.2 Discontinuity length and spacing
The length and spacing of discontinuities deter-
mines the size of blocks that will be formed in
the slope. Designs are usually concerned with
persistent discontinuities that could form blocks
with dimensions great enough to influence overall
slope stability. However, discontinuity dimen-
sions have a range of values and it is useful to
have an understanding of the distribution of these
values in order to predict how the extreme val-
ues may be compared to values obtained from
a small sample. This section discusses proba-
bility distributions for the length and spacing of
discontinuities, and discusses the limitations of
making accurate predictions over a wide range
of dimensions.
The primary purpose of making length and
spacing measurements of discontinuities is to
estimate the dimensions of blocks of rock formed
by these surfaces (Priest and Hudson 1976;
Cruden, 1977; Kikuchi
et al
., 1987; Dershowitz
and Einstein, 1988; Kulatilake, 1988; Einstein,
1993). This information can then be used, if
necessary, to design appropriate stabilization
measures such as rock bolts and rock fall bar-
riers. Attempts have also been made to use this
data to calculate the shear strength of “stepped”
surfaces comprising joints separated by portions
of intact rock (Jennings, 1970; Einstein
et al
.,
1983). However, it has since been found that
the Hoek-Brown method of calculating the shear
strength of rock masses is more reliable (see
Section 4.5).
cos
−
1
=
[
+
−
]
θ
1
(
1
/C
d
)
ln
(
1
P)
(3.17)
For example, the angle from the mean defined by
one standard deviation occurs at a probability
P
of 0.16 (refer to Figure 1.12). If the dispersion
is 20, one standard deviation lies at 7.6
◦
from
the mean.
Equation (3.17) is applicable when the disper-
sion in the scatter is approximately uniform about
the mean orientation, which is the case in joint set
A
in Figure 2.11. However, in the case of the bed-
ding in Figure 2.11, there is less scatter in the dip
than in the dip direction. The standard deviations
in the two directions can be calculated approx-
imately from the stereonet as follows. First, two
great circles are drawn at right angles corres-
ponding to the directions of dip and dip direction
respectively. Then the angles corresponding to the
7% and 93% levels,
P
7
, and
P
93
respectively,
are determined by counting the number of poles
in the set and removing the poles outside these
percentiles. The equation for the standard devi-
ation along either of the great circles is as follows
(Morriss, 1984):
tan
−
1
SD
=
{
0.34
[
tan
(P
93
)
−
tan
(P
7
)
]}
(3.18)
More precise methods of determining the stand-
ard deviation are described by McMahon
(1982), but the approximate method given by
equation (3.18) may be sufficiently accurate con-
sidering the difficulty in obtaining a represent-
ative sample of the discontinuities in the set. An
important aspect of accurate geological invest-
igations is to account for bias when mapping a
single face or logging a single borehole when few
Probability distributions
Discontinuities are usually mapped along a scan
line, such as drill core, slope face or wall of a
tunnel. Individual measurements are made of the
properties of each fracture, including its visible
