Page 194 - Rock Mechanics For Underground Mining

METHODS OF STRESS ANALYSIS

Figure 6.4 Problem geometry, co- i.e. the boundary stress takes the value 2p, independent of the co-ordinate angle
. This
ordinate axes and nomenclature for represents the optimum distribution of local stress, since the boundary is uniformly
specifying the stress distribution
around an elliptical excavation in a bi- compressed over the complete excavation periphery.
Equations 6.18 are considerably simplified for a hydrostatic stress field, taking the
axial stress field.
form
a
2
rr = p 1 −
r 2
a
2

= p 1 + 2 (6.20)
r
r
= 0

The independence of the stress distribution of the co-ordinate angle
, and the fact
that r
is everywhere zero, indicates that the stress distribution is axisymmetric.

6.3.2 Elliptical excavation
Solutions for the stress distribution in this case are quoted by Poulos and Davis (1974)
and Jaeger and Cook (1979). In both cases, the solutions are expressed in terms of
elliptical curvilinear co-ordinates. Their practical use is somewhat cumbersome. Bray
(1977) produced a set of formulae which results in considerable simplification of the
calculation of the state of stress at points in the medium surrounding an elliptical
opening. The problem geometry is defined in Figure 6.4a, with the global x axis
parallel to the field stress component Kp, and with an axis of the ellipse defining the
local x 1 axis for the opening. The width, W, of the ellipse is measured in the direction
of the x 1 axis, and the height, H, in the direction of the local z 1 axis. The attitude of
the ellipse in the biaxial stress field is described by the angle between the global x
and local x 1 axes. The position of any point in the medium is defined by its Cartesian
co-ordinates (x 1 , z 1 ) relative to the local x 1 , z 1 axes.
Bray’s solution specifies the state of stress at a point in the medium in terms of a set
of geometrical parameters, and relative to a set of local axes, denoted l and m, centred
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