Page 251 - Rock Mechanics For Underground Mining

ROOF DESIGN PROCEDURE FOR PLANE STRAIN

√
variation shown in Figure 8.8a, described by a minimum in f c at a distance s/2 2
from the midspan. For this distribution, the average longitudinal stress in the beam is
given by

1 2
f av = f c + n (8.8)
3 3
Elastic shortening of the arch accompanying vertical deflection is then given by

f av
L = L (8.9)
E
where E is the Young’s modulus of the rock mass in the direction parallel to the beam
axis.
The elastic shortening of the arch due to downward deflection of the arch at midspan
produces a new reaction moment arm given by

2
1/2
3s 8 z 0
z = − L (8.10)
8 3 s
The deflection at midspan given by equation 8.2 is determinate only if the term within
the square brackets in equation 8.10 is positive. A negative value implies that the
moment arm is notionally negative; i.e. the critical deflection of the beam has been
exceeded, and snap-though failure or buckling instability of the beam is indicated for
the specified value of arch compression thickness, nt.
In solving for the equilibrium deformation of the beam, the objective is to find the
pair of values of n and z which satisfies the set of equations 8.2–8.10. The original
analysis by Evans (1941) was based on the assumption of n = 0.5. Brady and Brown
(1985)proposedarelaxationmethodtodeterminen and z simultaneously.Duetosome
convergence problems with the relaxation method, Kaiser and Diederichs showed that
to provide an efficient and quick analysis, the value of n could best be evaluated in
a stepwise incremental fashion, to provide direct solution for the lever arm z using
equation 8.4. The equilibrium values of n and z are assumed to correspond to the
minimum in the value of f c returned in the stepwise incrementation of n, in steps of
0.01, over the range 0.0–1.0. The flow chart for the analysis is provided in Figure 8.9.
A notable conclusion of some scoping studies was that n is around 0.75 for stable
beams at equilibrium, and drops to below 0.5 as critical (unstable) beam geometry
is approached. As a point of interest, 0.75 is the equilibrium value for n returned
consistently in the analysis reported by Brady and Brown (1985). Later it is recorded
that Sofianos (1996) reported considerably lower values for the equilibrium value of
n.
Having obtained equilibrium values for n and z, the factors of safety against the
various modes of failure can be estimated from the other problem parameters and
rock mass properties. For the crushing mode of failure, which will be localized at the
lower abutments and at top midspan, the factor of safety against crushing is given by
c
FofS crush = (8.11)
f c
The value of c for the rock mass needs to be considered carefully. For isotropic
rocks, the long term strength of the rock mass is about 50% of the laboratory c
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