Civil Engineering Reference
In-Depth Information
was digitized, with 145 measurements made on
a 1 mm interval
(M
vectors
l
,
m
and
n
, where
1
)
, from
which
Z
2 was calculated from equation (3.11c)
to be 0.214 with a corresponding JRC value
of 11.4.
=
145;
x
=
⎫
⎬
l
=
sin
ψ
·
cos
α
m
=
sin
ψ
·
sin
α
(3.12)
⎭
n
=
cos
ψ
3.5 Probabilistic analysis of structural
geology
As discussed in Section 1.4.4, a measure of
the stability of a slope is the probability of
failure. Calculation of the probability of fail-
ure involves expressing the design parameters in
terms of probability distributions that give the
most likely value of each parameter (e.g. aver-
age), as well as the probability of its occurrence
within a range of possible values (e.g. standard
deviation).
This section demonstrates techniques to deter-
mine the probability distributions of structural
geology data. The distribution in orientation can
be calculated from the stereonet, while the distri-
butions of persistence and spacing are calculated
from field measurements.
For a number of poles, the direction cosines (
l
R
,
m
R
and
n
R
) of the mean orientation of the discon-
tinuity set is the sum of the individual direction
cosines, as follows:
l
i
|
m
i
|
n
i
|
l
R
=
;
m
R
=
;
n
R
=
R
|
R
|
R
|
(3.13)
where
|
R
|
is the magnitude of the resultant vector
given by
l
i
2
n
i
2
1
/
2
(3.14)
m
i
2
|
R
|=
+
+
3.5.1 Discontinuity orientation
The natural variation in orientation of discon-
tinuities results in there being scatter of the poles
when plotted on the stereonet. It can be useful
to incorporate this scatter into the stability ana-
lysis of the slope because, for example, a wedge
analysis using the mean values of pair of discon-
tinuity sets may show that the line of intersection
of the wedge does not daylight in the face and
that the slope is stable. However, an analysis
using orientations other than the mean values
may show that some unstable wedges can be
formed. The risk of occurrence of this condition
would be quantified by calculating the mean and
standard deviation of the dip and dip direction as
described next.
A measure of the dispersion, and from this the
standard deviation, of a discontinuity set can be
calculated from the direction cosines as follows
(Goodman, 1980). The direction cosines of any
plane with dip
ψ
and dip direction
α
are the unit
The dip
ψ
R
and dip direction
α
R
of the mean
orientation are
⎫
⎬
cos
−
1
(
n
R
)
α
R
=+
ψ
R
=
cos
−
1
(
l
R
/
sin
ψ
R
)
for
m
R
≥
0
⎭
cos
−
1
(
l
R
/
sin
ψ
R
)
α
R
=−
for
m
R
<
0
(3.15)
A measure of the scatter of a set of disconti-
nuities comprising
N
poles can be obtained from
the dispersion coefficient
C
d
, which is calculated
as follows:
N
C
d
=
(3.16)
(N
−|
R
|
)
