Page 304 - Rock Mechanics For Underground Mining

ENERGY, MINE STABILITY, MINE SEISMICITY AND ROCKBURSTS

Determination of the energy changes associated with the excavation of the open-
ing requires estimation of work done by surface forces acting through induced dis-
placements. Consider first the case of gradual excavation. At the excavation surface
S a (r = a), the radial pressure drops from p to zero, while the radial displacement is
given by

u r =−pa/4G
i.e. the induced displacement is directed radially inward.
The work W 1 done by the surface S a against the support forces within S a during
their gradual reduction is the product of the average radial force and displacement;
i.e.

2
2 3
W 1 = 4 a × p/2 × pa/4G = p a /2G (10.49)
In the far field, excavation-induced displacement is diminishingly small, but the area
affected is large, so the work done cannot be disregarded. Consider a spherical surface
S R in the medium, of radius R, and concentric with the opening. In the process of
excavating the opening, the radial stress on the spherical surface drops from p to
3
3
2
3
p[1 − (a /R )], and the corresponding radial displacement u R is −pa /4GR . The
work W 2 done on the surface S R by the exterior rock during the induced displacement
is given by the product of the average radial force and displacement; i.e.
2
3
3
2 3
3
3
3
2
W 2 = 4 R × 1/2 {p + p[1− (a /R )]} (pa /4GR ) = ( p a /G)[1− (a /2R )]
(10.50)
For the case of a remote surface, i.e. as R →∞, equation 10.50 becomes
2 3
W 2 = p a /G (10.51)
The increase in the static strain energy, W s , around the excavation is given by the
difference between the work done on the rock medium at the remote surfaces, and
the work done at the excavation surface by the medium against the support forces;
i.e.
2 3
W s = W 2 − W 1 = p a /2G (10.52)
The energy released by excavating the opening is given by the complementary work
done at the boundaries during their pseudo-static displacement; i.e.

2 3
W r = W 1 = p a /2G (10.53)
In the case of sudden generation of the spherical opening, the surface S a will do
no work during its radial displacement. Excess energy, W e , present in the excavation
peripheral rock, is expressed as a spherical stress wave which propagates away from
the opening, so that the final conditions in the medium are identical to those for the
case of gradual excavation. Thus, the increase in static strain energy W s is identical
to that given by equation 10.52. Also, the work W 2 done at the far-field surface S R

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