Rock Mechanics.pdf - Mining and Blasting

Figure 16.24 Problem definition for
elastic analysis of trough subsidence.
CONTINUOUS SUBSIDENCE DUE TO THE MINING OF TABULAR OREBODIES
If it is assumed that the thickness of the extracted seam is small compared with the
other dimensions of the excavation and with the depth, a point on the lower boundary
of the seam can be given the same co-ordinate as the nearest point on the upper boundary.
The excavation is then located by a single plane, and the convergence of opposing
points in the roof and floor can be treated as a discontinuity in displacement at a single
point. Unless the excavated width is small, the roof and floor will meet over some central
area where the displacement discontinuity has its greatest magnitude, m, the thickness
of the extraction (Figure 16.24). Where the roof and floor do not meet, the boundaries
of the excavation are traction free. A further boundary condition is given by the
fact that the upper plane surface remains traction free before, during and after mining.
Berry (1960) solved the simple two-dimensional case involving hydrostatic in situ
stress and isotropic ground, exactly for complete closure, and approximately for
less than complete closure. The calculated maximum settlements were found to be
independent of the elastic constants and were less than the values recorded in UK
coalfields. In order to give a better representation of the mechanical response of the
sedimentary strata, Berry and Sales (1961) carried out a similar analysis using the
stress–strain relations of a transversely isotropic medium with the planes of symmetry
parallel to the ground surface (Figure 2.10).
As noted in section 2.10, a transversely isotropic material has five independent
elastic constants. The stress–strain relations may be written in terms of the five elastic
stiffnesses c11, c12, c13, c33 and c44 as
511
xx = c11 εxx + c12 εyy + c13 εzz
yy = c12 εxx + c11 εyy + c13 εzz
zz = c13 εxx + c13 εyy + c33 εzz
yz = 2c44 yz
xz = 2c44 xz
xy = (c11 − c12)xy