Page 250 - Rock Mechanics For Underground Mining

EXCAVATION DESIGN IN STRATIFIED ROCK

force, V , be balanced by the resisting moment M R of the distributed end loads,
i.e.
1 2 1
M A = ts = M R = f c ntz (8.5)
8 2
or

1 s 2
f c = (8.6)
4 nz
where f c is the maximum compressive stress acting in the beam, operating at the
bottom edge at the abutment and at the top edge at the centre of the span.
Analysis of the three modes of roof instability involves determination of f c , n and z.
Following Sofianos (1996), Diederichs and Kaiser (1999) showed that the assumption
of Evans of n = 0.5 was inadequate, that the iterative method proposed by Brady and
Brown (1985) was appropriate for small deflections, but that convergence and stability
in the solution procedure could be improved.
The solution procedure begins with an assumption of the initial moment arm prior
to deflection, z o , which is given by

2
z o = t 1 − n (8.6)
3
The length L of the parabolic reaction arch is given by
8 z 2 o
L = s + (8.7)
3 s
To calculate the elastic shortening of the arch and the central deflection of the arch
through equation 8.2, an assumption must be made about the distribution of axial
compressive stress over the longitudinal vertical section of the beam. In their original
relaxation analysis, Brady and Brown (1985) assumed the bilinear variation shown
in Figure 8.8a. From various numerical studies, Diederichs and Kaiser proposed that
a better approximation for the simple, two-member voussoir beam is the quadratic

Figure 8.8 Assumed distributions
of axial compressive stress and
parabolic thrust line (after Diederichs
and Kaiser, 1999a).

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