Page 272 - Rock Mechanics For Underground Mining

EXCAVATION DESIGN IN BLOCKY ROCK

In analyzing the stability of the tunnel, the prospect of failure is defined by the
formation of a key block of any size within the particular cross section of interest. A
basic parameter in the analysis is the factor x, the key block size fraction of interest
in the assessment of excavation stability. It is defined by

x = minimum key block volume of interest/volume of maximum key block (9.3)

The probability of failure is given by

p f = volume of key block-forming region/volume of unit cell (9.4)

The analysis also requires the definition of the geometric parameter, C, given by

C = volume of key block-forming region/mean volume of unit cells
= 1/6 length ∗ width ∗ altitude of the maximum key block/mean
size of unit cells (9.5)

The mean size of unit cells can be calculated from the mean spacings
S1 ,
S2 ,
S3
of the three joint sets.
The unconditional probability of failure, defined as the probability that a key block
larger than size x will intersect a randomly selected tunnel cross section, is shown to
be given by

p f = C(1 − 3x 2/3 + 2x) (9.6)

To assess key block sizes, the cumulative distribution function (cdf) F X (x) is obtained
from

F X (x) = 1 − C(1 − 3x 2/3 + 2x) (9.7)

The probability density function (pdf) for key block sizes, f X (x), is given by

f X (x) = 2C(x −1/3 − 1) (0 < x ≤ 1) (9.8)

The pdf has a lumped mass at x = 0, given by p(x = 0) = F X (x = 0) = 1 − C.
An alternative to considering the probability of key block occurrence in any ran-
domly selected cross section is to assess the number E n of key blocks expected to be
present along a given length of an excavation. This is evaluated as follows.
From equation 9.8, the probability, p KB , that a particular tunnel cross section con-
tains a keyblock within a differential size range dx is given by

p KB = 2C(x −1/3 − 1) dx (9.9)

Along a particular length of tunnel, L tun , the fraction f KB of the tunnel cross sections
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