Page 273 - Rock Mechanics For Underground Mining

SYMMETRIC TRIANGULAR ROOF PRISM

(or the length of tunnel expected to contain blocks of this size) is

f KB = 2CL tun (x −1/3 − 1) dx (9.10)

The length (measured along the tunnel axis) of a key block of size x can be shown to
be related to the length of the maximum key block by the expression

length = length of maximum key block ∗ x 1/3 (9.11)

From equations 9.10 and 9.11, the expected number of key blocks of size x is given
by

E n = 2CL tun (x −1/3 − 1)/(length of maximum key block ∗ x 1/3 ) dx (9.12)

It is convenient to introduce a second geometric problem parameter defined by

C 2 = 6CL tun /length of maximum key block (9.13)

and E n is then given by
1
E n = C 2 (x −2/3 − x −1/3 ) dx (9.14)
3
For a finite size interval (x 1 , x 2 ), E n is given by

% x2 1
E n = C 2 (x −2/3 − x −1/3 ) dx (9.15)
x1 3

1/3 1 2/3 1/3 1 2/3
= C 2 x − x − x + x (0 ≤ x 1 ≤ x 2 ≤ 1) (9.16)
2 2 2 1 2 1
Theexpressionsinequations9.6,9.7and9.16maybeappliedinvariouswaysintunnel
design. In discussing how to apply them, Kuszmaul (1999) reiterates the importance
of bearing in mind the assumptions made in their derivation: that there are three
discontinuity sets, the sets are well defined and widely spaced (on the scale of the
excavation), the discontinuities are persistent and the rock mass characteristics remain
uniform along the planned excavation length. If these conditions are not met, such
as closely spaced discontinuities and limited persistence, the unit cell calculations
overestimate the number of key blocks in the excavation. If there are more than three
joint sets, the joint sets can be considered separately in different sets of three.
The specific value of the unit cell method is that it provides a method for making
design decisions based on likely key block sizes, rather than assuming the worst case
of taking the dimensions of the maximum key block. Alternatively, a design approach
could be based on seeking to minimize the probability of failure of an excavation
support system.

9.3 Symmetric triangular roof prism

Having identified the feasible block collapse modes associated with joint orientations
and excavation surface geometry, it is necessary to determine the potential for block
displacement under the conditions that will exist in the post-excavation state of the
255