Rock Mechanics.pdf - Mining and Blasting

Figure 16.23 Extraction of an element
dA (after Berry, 1978).
CONTINUOUS SUBSIDENCE DUE TO THE MINING OF TABULAR OREBODIES
Profile functions may be used to describe the shape of the subsidence profile. They
take the general form
s = Smax f (B, x, c)
where Smax is the subsidence at the centre of a panel of critical width, B = h tan
is the critical radius of extraction, x is the horizontal distance of the point from the
origin of co-ordinates, and c is some function or constant.
Exponential, trigonometric, hyperbolic and error functions have been used as profile
functions. The function that appears to have given the best results is the hyperbolic
tangent function
s(x) = 1
2 Smax
bx
1 − tanh
(16.11)
h
where x is the horizontal distance measured from the point of inflection (where
s = Smax/2) in the direction of decreasing subsidence, h is the depth of the seam,
and b is a constant controlling the slope at the inflection point. For UK conditions, a
value of b = 5 is used (King et al., 1975, for example). Hood et al. (1983) found that
a value of b = 11.5 applied for a number of transverse subsidence profiles at the Old
Ben Number 24 Mine in Illinois, USA.
By differentiation of equation 16.11, the surface slope, or tilt, is given as
g = ds
dx
bSmax bx
= sech2
2h h
For b = 5, this gives the maximum slope at the point of inflection (x = 0) as
G = 2.5Smax
h
(16.12)
a similar result to that given by the NCB (1975).
The surface curvature is given by differentiation of the expression for g given in
equation 16.12 with respect to x. Methods using profile functions have been developed
for estimating subsidence profiles for subcritical widths and for making allowance
for the effects of seam inclination (Brauner, 1973).
Influence functions are used to describe the surface subsidence caused by the
extraction of an element, dA. The principle of superposition is assumed to apply, so
that the subsidence profile for the complete extraction can be found by integrating the
influence function over the complete extraction area. The use of numerical integration
permits subsidence predictions to be made for extraction areas of any shape.
The influence function p(r) gives the contribution to subsidence at a point P on
the surface due to an element of extraction dA at P ′ as a function of r, the horizontal
projection of PP ′ (Figure 16.23). If P has co-ordinates x, y referred to a set of axes in
the plane of the surface, and P ′ has co-ordinates , referred to similar axes vertically
below in the seam, the influence function takes the form
509
p(r) = w(, ) f (r) (16.13)