Page 395 - Rock Mechanics For Underground Mining

ELEMENTARY ANALYSIS OF PILLAR SUPPORT

to a representative pillar is of plan dimensions (a + c), (b + c), so that satisfaction of
the equation for static equilibrium in the vertical direction requires

p ab = p zz (a + c)(b + c)

or

p = p zz (a + c)(b + c)/ab (13.3)

The area extraction ratio is defined by

r = [(a + c)(b + c) − ab]/(a + c)(b + c)

and some simple manipulation of equation 13.3 produces the expression

p = p zz [1/(1 − r)]

which is identical with equation 13.2.
For the case where square pillars, of plan dimension w p × w p , are separated by
rooms of dimension w o , equation 13.3 becomes

p = p zz [(w o + w p )/w p ] 2 (13.4)

Of course, average axial pillar stress is still related to area extraction ratio through
equation 13.2.
Equations 13.1, 13.3, and 13.4 suggest that the average state of axial stress in a
representative pillar of a prospective mining layout can be calculated directly from
the stope and pillar dimensions and the pre-mining normal stress component acting
parallel to the pillar axis. It is also observed that for any geometrically uniform mining
layout, the average axial pillar stress is directly determined by the area extraction ratio.
The relation between pillar stress level and area extraction ratio is illustrated graphi-
cally in Figure 13.9. The principal observation from the plot is the high incremental
change in pillar stress level, for small change in extraction ratio, when operating at
high extraction ratio. For example, a change in r from 0.90 to 0.91 changes the pillar
stress concentration factor from 10.00 to 11.11. This characteristic of the equation
governing stress concentration in pillars clearly has significant design and operational
Figure 13.9 Variation of pillar stress implications. It explains why extraction ratios greater than about 0.75 are rare when
concentration factor with area extrac- natural pillar support is used exclusively in a supported method of working. Below
tion ratio.
this value of r, incremental changes in p /p zz with change in r are small. For values
of r greater than 0.75, the opposite condition applies.
When calculating pillar axial stress using the tributary area method, it is appropriate
to bear in mind the implicit limitations of the procedure. First, the average axial pillar
stress is purely a convenient quantity for representing the state of loading of a pillar in a
direction parallel to the principal direction of confinement. It is not simply or readily
related to the state of stress in a pillar which could be determined by a complete
analysis of stress. Second, the tributary area analysis restricts attention to the pre-
mining normal stress component directed parallel to the principal axis of the pillar
support system. The implicit assumption that the other components of the pre-mining
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