Page 228 - Rock Mechanics For Underground Mining

EXCAVATION DESIGN IN MASSIVE ELASTIC ROCK

Figure 7.12 Ovaloidal opening in a
medium subject to biaxial stress.

in equation 7.6, as
$
!
2 × 3H
A = p 1 − 0.5 +
H/2
= 3.96p

An independent boundary element analysis of this problem yielded a sidewall bound-
ary stress of 3.60p, which is sufficiently close for practical design purposes. Although
the radius of curvature, for the ovaloid, is infinite at point B in the centre of the crown
of the excavation, it is useful to consider the state of stress at the centre of the crown
of an ellipse inscribed in the ovaloid. This predicts a value of B , from equation 7.6, of
−0.17p, while the boundary element analysis for the ovaloid produces a value of B
of −0.15p. This suggests that excavation aspect ratio (say W/H), as well as boundary
curvature, can be used to develop a reasonably accurate picture of the state of stress
around an opening.
A square hole with rounded corners, each with radius of curvature = 0.2B,is
shown in Figure 7.13a. For a hydrostatic stress field, the problem shown in Figure
7.13b is mechanically equivalent to that shown in Figure 7.13a. The inscribed ovaloid
has a width of 2B[2 1/2 − 0.4(2 1/2 − 1)], from the simple geometry. The boundary
stress at the rounded corner is estimated from equation 7.6 as
' (
2B(2 − 0.4(2 − 1))
1/2 1/2 1/2
A = p 1 − 1 +
0.2B
= 3.53p
The corresponding boundary element solution is 3.14p.
The effect of boundary curvature on boundary stress appears to be a particular
consequence of St Venant’s Principle, in that the boundary state of stress is dominated
by the local geometry, provided the excavation surface contour is relatively smooth.
An extension of this idea demonstrates an important design concept, namely that
changing the shape of an opening presents a most effective method of controlling
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