Page 291 - Rock Mechanics For Underground Mining

MECHANICAL RELEVANCE OF ENERGY CHANGES

Figure 10.2 (a) Static loading of a
pillar;(b)inducedload–deflectiondia-
gram for the pillar; (c) load–deflection
diagram for boundary unloading.

Work done by the applied load

u z
%
= P z du z
0
1
= P z u z (10.1)
2
= Increase in strain energy in pillar ( U st )

The complementary work (i.e. the work done by the excavation surfaces against the
reducing support force) may be identified with energy U r released by the rock mass
by the process of excavation. From Figure 10.2b, the released energy is given by
1
Work done by excavation surface loads = P z u z (10.2)
2
It is observed, from equations 10.1 and 10.2, that the released energy is equal to the
increase in pillar strain energy, consistent with energy conservation.
After the pillar load is applied, induced static stress zz and axial compression u z
are related through the elasticity of the rock; i.e.

zz = P z /A

or

P z = zz A (10.3)
and

u z = ε zz L = ( zz /E)L (10.4)

where A and L are pillar plan area and length.
From equations 10.1, 10.3, and 10.4

1 2
U st = /E × A × L (10.5)
zz
2
In the case of sudden mining of the adjacent excavations, the complete induced load
P z is rapidly applied to the pillar. Figure 10.3 shows the deformation of the pillar up
to the stage of maximum axial compression, and the corresponding load–deflection

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