Civil Engineering Reference
In-Depth Information
length and the spacing between discontinuities in
each set (Appendix II). The properties of discon-
tinuities typically vary over a wide range and it
is possible to describe the distribution of these
properties by means of probability distributions.
A normal distribution is applicable if a particular
property has values in which the mean value is
the most commonly occurring. This condition
would indicate that the property of each discon-
tinuity, such as its orientation, is related to the
property of the adjacent discontinuities reflect-
ing that the discontinuities were formed by stress
relief. For properties that are normally distri-
buted, the mean and standard deviation are given
by equations (1.16) and (1.17).
A negative exponential distribution is applic-
able for properties of discontinuities, such as
their length and spacing, which are randomly
distributed indicating that the discontinuities are
mutually independent. A negative exponential
distribution would show that the most com-
monly occurring discontinuities are short and
closely spaced, while persistent, widely spaced
discontinuities are less common. The general
form of a probability density function
f(x)
of
a negative exponential distribution is (Priest and
Hudson, 1981)
respectively, are
e
−
1
/
2
)
F(x)
=
(
1
−
=
40%
and
e
−
5
/
2
)
F(x)
=
(
1
−
=
92%
Equation (3.20) could be used to estimate the
probability of occurrence of discontinuities with
a specified length. This result could be used, for
example, to determine the likelihood of a plane
being continuous through a slope.
Another distribution that can often be used
to describe the dimensions of discontinuities is
the lognormal distribution which is applicable
where the variable
x
ln
y
is normally distributed
(Baecher
et al
., 1977). The probability density
function for a lognormal distribution is (Harr,
1977)
=
exp
2
ln
y
1
y
SD
x
√
2
π
1
2
−
x
f(x)
=
−
SD
x
(3.21)
where
x
is the mean value and SD is the standard
deviation.
Figure 3.11 shows the measured lengths of 122
joints in a Cambrian sandston
e
for lengths of
less than 4 m; the mean length
l
is 1.2 m (Priest
1
x
(
e
−
x/ x
)
f(x)
=
(3.19)
and the associated cumulative probability
F(x)
that a given spacing or length value will be less
than dimension
x
is given by
25
20
Exponential (
r
= 0.69)
Lognormal (
r
= 0.89)
l
15
e
−
x/ x
)
F(x)
=
(
1
−
(3.20)
10
N
=122
= 1.2 m
l
where
x
is
a measured value of length or spa-
cing and
x
is the mean value of that parameter. A
property of the negative exponential distribution
is that the standard deviation is equal to the mean
value.
From equation (3.20) for a set of discontinuities
in which the mean spacing is 2 m, the probabilities
that the spacing will be less than 1 m and 5 m,
5
0
0
1.0
2.0
3.0
4.0
Measured trace length (m)
Figure 3.11
Histogram of joint trace lengths, with
best fit exponential and lognormal curves (Priest and
Hudson, 1981).
