Design against stress-controlled instability 389of zero at a strain of 0.0116, which is equivalent to a displacement of about
70 mm. The ground response curve is linear, because it has been developed
on the basis of linear elasticity theory. The complete available support
line is equivalent to the complete stress-strain for the pillar material,
under conditions of plane strain (we discussed the complete stress-strain
curve in Section 6.1, and noted the importance of the relative stiffnesses
of the loading system and the descending, post-peak, portion of the
curve),
The two curves in Fig. 20.30 now allow study of the stability of the entire
structure. The operating point, indicated by the intersection of the two
curves, represents a stage in the mechanical breakdown of the pillar that
is almost complete. The displacement at the operating point is almost the
displacement that would be reached without the pillar being present-
when the excavation would be stable anyway. The conclusion is clear: the
pillar is both ineffective and unnecessary.
There are many variations on this theme, and the way in which natural
supporting elements in mining geometries can be used optimally to ensure
stability whilst maximizing the amount of excavated material can be
studied. Our purpose here is to demonstrate one specific case where the
ground response curve analysis provides a clear conclusion, remembering
that these analyses have been in two dimensions but rock engineering is
always conducted in three dimensions.
20.2.5 Three-dimensional unulysis
An additional level of complexity is introduced by the three-dimensional
nature of the rock engineering geometry compared to studies for two-
20 -
0 2 4 68 1012
EZ io3
At the operating point
pB = 0.6 MPa eZ =I 1.3 E-3
The pillar is in an advanced stage of
breakdown and is ineffective.
Figure 2030 Ground response and available support lines for the tabular excava-
tion illustrated in Fig. 20.29.