Page 191 - Rock Mechanics For Underground Mining

CLOSED-FORM SOLUTIONS FOR SIMPLE EXCAVATION SHAPES

plane can be readily established from an orthogonal Cartesian system in the plane
through the transformation

z = x + iy = c cosh = c cosh( + i )

so that

x = c cosh cos , y = c sinh sin (6.17)

Equations 6.17 are also the parametric equations for an ellipse of major and minor axes
c cosh , c sinh . The analysis of the stress distribution around the opening proceeds
by expressing boundary conditions, source analytic functions and stress components
in terms of the elliptic co-ordinates. The detail of the method then matches that
described for a circular opening.
The conformal mapping method involves finding a transformation which will map
a chosen geometric shape in the z plane into a circle of unit radius in the plane. The
problem boundary conditions are simultaneously transformed to an appropriate form
for the plane. The problem is solved in the plane and the resulting expressions
for stress and displacement distributions then inverted to obtain those for the real
problem in the z plane. Problem geometries which have been analysed with this
method include a square with rounded corners, an equilateral triangle and a circular
hole with a concentric annular inclusion.

6.3 Closed-form solutions for simple excavation shapes

Theprecedingdiscussionhasestablishedtheanalyticalbasisfordeterminingthestress
and displacement distributions around openings with two-dimensional geometry. In
rock mechanics practice, there is no need for an engineer to undertake the analysis
for particular problem configurations. It has been noted already that comprehensive
collections of solutions exist for the analytically tractable problems. The collection
by Poulos and Davis (1974) is the most thorough. The practical requirement is to be
able to verify any published solution which is to be applied to a design problem. This
is achieved by systematic checking to determine if the solution satisfies the governing
equations and the specified far-field and boundary conditions. The verification tests to
be undertaken therefore match the sets of conditions employed in the development of
the solution to a problem, as defined in section 6.1, i.e. imposed boundary conditions,
differential equations of equilibrium, strain compatibility equations and constitutive
equations. The method of verification can be best demonstrated by the following
example which considers particular features of the stress distribution around a circular
opening.

6.3.1 Circular excavation
Figure6.3ashowsthecircularcrosssectionofalongexcavationinamediumsubjectto
biaxial stress, defined by p yy = p, and p xx = Kp. The stress distribution around the
opening may be readily obtained from equations 6.6, by superimposing the induced
stresses associated with each of the field stresses p and Kp. The complete solutions
for stress and displacement distributions around the circular opening, originally due
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