Page 249 - Rock Mechanics For Underground Mining

ROOF DESIGN PROCEDURE FOR PLANE STRAIN

idea that the central transverse crack determines the deformational behaviour of the
discontinuous beam, the problem may be analysed in terms of the problem geometry
illustrated in Figure 8.7b. The three possible failure modes discussed earlier can be
readily appreciated from the problem configuration. These are: shear at the abutment
when the limiting shear resistance, T tan , is less than the required abutment vertical
1
reaction V (= W); crushing at the hinges formed in the beam crown and lower
2
abutment contacts; buckling of the roof beam with increasing eccentricity of lateral
thrust, and a consequent tendency to form a ‘snap-through’ mechanism. Each of these
modes is examined in the following analysis.

8.4.1 Load distribution
In analysing the performance of a roof bed in terms of a voussoir beam, it is observed
that the problem is fundamentally indeterminate. A solution is obtained in the current
analysis only by making particular assumptions concerning the load distribution in
the system and the line of action of resultant forces. In particular, the locus of the
horizontal reaction force or line of thrust in the beam is assumed to trace a parabolic
arch on the longitudinal vertical section of the jointed beam. The other assumption is
that triangular load distributions operate over the abutment surfaces of the beam and at
the central section, as shown in Figure 8.7b. These distributions are reasonable since
roof beam deformation mechanics suggest that elastic hinges form in these positions,
by opening or formation of transverse cracks.

8.4.2 Analysis and design
The problem geometry for the analysis is illustrated in Figure 8.7. The roof beam
in Figure 8.7b, of span s, thickness t and unit weight , supports its own weight
W by vertical deflection and induced lateral compression. The triangular end load
distributions each operate over a depth h = nt of the beam depth and unit thickness
is considered in the out-of-plane direction. In Figure 8.7b, the line of action of the
resultant of each distributed load acts horizontally through the centroid of each distri-
bution. In the undeflected position of the beam, the moment arm for the couple acting
at the centre and abutment of the beam is z 0 , and after displacement to the equilibrium
position, the moment arm is z. As shown in Figure 8.7b, the vertical deflection at
the beam centre is given by

= z 0 − z (8.2)

From the geometry in Figure 8.7b, the fractional loaded section n of the beam is
defined by

h
n = (8.3)
t
and the moment arm z is given by

2
z = t 1 − n (8.4)
3
Moment equilibrium of the free body illustrated in Figure 8.7b requires that the active
couple M A associated with the gravitational load and the equilibrating abutment shear
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