Page 530 - Rock Mechanics For Underground Mining

MINING-INDUCED SURFACE SUBSIDENCE

For the two-dimensional plane strain case, the number of equations reduces to three
and the number of elastic stiffnesses to four. The solution of such problems involves
two parameters 1 , 2 which are those roots of the characteristic equation

4
2
c 11 c 44 + [c 13 (2c 44 + c 13 ) − c 11 c 33 ] + c 33 c 44 = 0
having positive real parts. Sometimes it is more convenient to use two parameters
k 1 , k 2 , which are always real and are defined by

2 1/2 2 −1/2

k 1 = 1 2 = 1 − E 1 /E 2 −
1 2
1 2 2 1 2 −1

k 2 = + = E 1 /[G 2 − 2 (1 + 1 )] E 1 /E 2 −
2 1 2 2 2
where the elastic constants are as defined in section 2.10.
Berry (1963) found that for plane strain and complete closure of an excavation
parallel to the surface, the subsidence and horizontal strain at the surface for real 1 ,
2 are given by
m −1 2 ah 1 −1 2 ah 2
s(x) = 1 tan 2 − 2 tan 2
( 1 − 2 ) x − a + h 1 x − a + h 2
2
2
2
2
" 2 2 2 2 2 2 #
2 a 1 2 m x − a − h 2 x − a − h 1
ε(x) =
( 1 − 2 ) x − a − h 2 2 + 4h x − x − a − h 2 2 + 4h x
2
2
2
2 2
2
2 2
2 2 1 1
where h 1,2 = h/ 1,2 .
Berry found that reasonable fits to some subsidence and strain data from UK
coalfields were obtained using 1 = 4.45, 2 = 0.45, or k 1 = 2, k 2 = 10. However,
these values are probably unreasonable for the coal measures rocks concerned. This
suggests that the results may be influenced by some of the assumptions made in
the analysis, or alternatively, that the observed subsidence phenomena may not be
regarded as being principally due to elastic deformation.
Salamon(1974)showedthatthevalues 1 = 9.0, 2 = 0.5givevaluesofmaximum
tilt and maximum compressive and tensile strain that are in closer agreement with
those given by the NCB observations. Avasthi and Harloff (1982) modified Berry and
Sales’ theory for the three-dimensional case, and introduced an empirical adjustment
to account for incomplete closure for those cases in which W/h < 0.6. They found
that the best fit to the NCB data base was given by 1 = 2.74, 2 = 0.65 which are
much closer to values commonly measured for UK coal measures rocks (Berry, 1978)
than the other sets of values postulated.
In a further contribution to the analysis of subsidence as elastic deformation, Sala-
mon (1991) used a linear laminated elastic model of the strata overlying the coal
seam. The strata were modelled as a series of homogeneous isotropic beds with the
interfaces between them being horizontal and free of shear stresses and cohesion. The
continuity of stresses and normal displacements across the interfaces was assured.
The lamination thickness and the elastic moduli of the laminations could vary, but
they were constant in the simplest version of the model. Salamon (1991) derived
analytical solutions for his model but used numerical methods to evaluate some of the
integrals involved. In common with some of the numerical models to be discussed
in the next section, Salamon modelled the caved material in the goaf as an isotropic
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