Stress-controlled instability mechanisms 353redistribution over 10 tunnel diameters is shown in Fig. 19.13. Other
interesting statistics related to this re-distribution are that 50% of the load
is redistributed between the tunnel boundary and 0.23 tunnel diameters
into the rock, and that 95% of the load is re-distributed between the tunnel
boundary and 4.5 tunnel diameters into the rock.
Note that the curve in Fig. 19.13 applies to the load re-distribution rather
than the stress re-distribution. The curve shown is the cumulative load re-
distribution, i.e. the integration of the stress distribution represented by the
shaded area in Fig. 19.12(d).7 9.2.2 Stresses around ellipticar/ openings
The stresses around elliptical openings can be treated in an analogous way
to that just presented for circular openings. There is much greater utility
associated with the solution for elliptical openings than circular openings,
because these can provide a first approximation to a wide range of
engineering geometries, especially openings with high widthheight
ratios (e.g. mine stopes and civil engineering caverns). From a design point
of view, the effects of changing either the orientation within the stress field
or the aspect ratio of such elliptical openings can be studied to optimize
stability.
ElZipticuZ openings in isotropic rock. An elliptical opening is completely
characterized by two parameters: aspect ratio (the ratio of the major
axis to the minor axis) which is the eccentricity of the ellipse; and
orientation with respect to the principal stresses (measured, for example,
in terms of the angle between the major axis and the major principal stress).
Bray (1977) derived a suite of equations for the state of stress around
an elliptical opening in terms of these parameters and the Cartesian
co-ordinates of the location of the point in question. These equations
are given in Fig. 19.14, with reduced forms in Figs 19.15 and Fig. 19.16 for
the cases of tangential stress on the boundary of an arbitrarily orientated
excavation and the tangential stress on the boundary of an excavation
orientated with its axes aligned with the principal stress directions,
respectively. The diagram in Fig. 19.14 shows how the angle 8 defines the
orientation of the local reference axes I, m relative to the ellipse local axes
xl, zl. In Fig. 19.15, the position on the boundary, with reference to the
x-axis, is given by the angle x, and in Fig. 19.16 the ellipse is aligned by
taking p = 0.
It is instructive to consider the maximum and minimum values of the
stress concentrations around the ellipse for the geometry of the ellipse in
Fig. 19.16. It can easily be established that the extremes of stress
concentration occur at the ends of the major and minor axes-points A and
B in Fig. 19.16-and the corresponding stress magnitudes are as given by
the equations in the figure.
In a given engineering context, k cannot generally be altered and so any
design optimization must be performed through a variation in 9, which is
usually possible. An optimal design can be defined as one in which the