Page 293 - Rock Mechanics For Underground Mining

MINING CONSEQUENCES OF ENERGY CHANGES

i.e. from equations 10.6 and 10.7
1
2P z u z st = P z u z st + K.E. + P z u z st
2
or
1
K.E. = P z u z st
2
The kinetic energy in the pillar (which is seen to be equal in magnitude to the
released energy defined by equation 10.2) represents excess energy U e in the system,
compared with the case of static loading. In practice, this excess energy would be
lost to the pillar during vibration, by transfer to the adjacent country rock and by
dissipation through internal frictional damping. The question that arises is – what
is the source of the excess energy, manifested as kinetic energy in the system, after
rapid excavation of adjacent rock? The answer is related to the control which is
exercised at a surface when it is created slowly. Controlled excavation implies a
gradual reduction in the support forces which operated originally on the surface of
interest. The gradual force reduction involves application of a decreasing restraining
force as the surface displaces to its final equilibrium position. In this process, the
excavated surface does work against the restraining force during the mining-induced
displacement. In the case of unrestrained displacement of the excavated surface, as in
sudden mining of the rooms in the example above, the excess energy manifested in
the system represents the work which would have been done by the mined boundary
against the support force, had the surface displacement been restrained during min-
ing. This energy is retained instantaneously, and then transmitted to the surrounding
rock.
This discussion suggests there are two factors to be considered in relation to energy
changes associated with creating excavations. First, increase of static strain energy
occurs in areas of stress concentration, equivalent (for an elastic rock mass) to the
energy released during pseudo-static displacement of excavated surfaces. Second,
sudden excavation of surfaces causes an energy imbalance in the system, and results
intransientstressesdifferentfromtheequilibriumstaticstresses.Foranelasticsystem,
the excess energy is equal in magnitude to the released energy.

10.2 Mining consequences of energy changes

Mining activity takes place in a medium subject to general triaxial stress. For pur-
poses of illustration, attention will be restricted to two-dimensional problems involv-
ing biaxial states of stress. The analyses may be simply extended to accommodate
three-dimensional problem geometries in multiaxial stress fields. The discussion is
an elaboration of ideas proposed by Cook (1967b), Salamon (1974), Bray (1977) and
Blight (1984).
The basic ideas introduced by the pillar loading problem can be extended to in-
clude energy changes at the boundaries of arbitrarily shaped excavations. Referring
to Figure 10.5a, prior to excavation a surface S is subject, at any point, to tractions t x ,
t y . In the case of gradual excavation of the material within S, the surface tractions are
gradually reduced to zero, and areas of induced stress are generated around the excava-
tion, as indicated in Figure 10.5b. The strain energy density is inferred, from equation

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