Page 193 - Rock Mechanics For Underground Mining

CLOSED-FORM SOLUTIONS FOR SIMPLE EXCAVATION SHAPES

so that the far-field stresses recovered from the solutions correspond to the imposed
field stresses. With regard to the equilibrium requirements, the differential equation
of equilibrium in two dimensions for the tangential direction and no body forces, is

∂ r
1 ∂

2 r
+ + = 0
∂r r ∂
r
Evaluating each of the terms of this equation from the expressions for

and r
leads to
2 4
∂ r
2a 6a
= p(1 − K) − + sin 2
∂r r 3 r 5
4
1 ∂

1 3a
=−p(1 − K) + sin 2
r ∂
r r 5
2 4
2 r
1 2a 3a
= p(1 − K) + − sin 2
r r r 3 r 5
Inspection indicates that the equilibrium equation for the tangential direction is sat-
isfied by these expressions. It is obviously an elementary exercise to confirm that the
stress components satisfy the other two-dimensional (i.e. radial) equilibrium equa-
tion. Similarly, the strain components can be determined directly, by differentiation
of the solutions for displacements, and the expressions for stress components derived
by employing the stress–strain relations. Such a test can be used to confirm the mutual
consistency of the solutions for stress and displacement components.

Boundary stresses. Equations 6.19 define the state of stress on the boundary of a
circular excavation in terms of the co-ordinate angle
. Clearly, since the surface is
traction free, the only non-zero stress component is the circumferential component

.For K < 1.0, the maximum and minimum boundary stresses occur in the side
wall (
= 0) and crown (
= /2) of the excavation. Referring to Figure 6.3b, these
stresses are defined by the following:

at point A:
= 0, (

) A = A = p(3 − K)

at point B:
= , (

) B = B = p(3K − 1)
2
These expressions indicate that, for the case when K = 0, i.e. a uniaxial field
directed parallel to the y axis, the maximum and minimum boundary stresses are

A = 3p, B =−p

These values represent upper and lower limits for stress concentration at the boundary.
That is, for any value of K > 0, the sidewall stress is less than 3p, and the crown
stress is greater than −p. The existence of tensile boundary stresses in a compressive
stress field is also noteworthy.
In the case of a hydrostatic stress field (K = 1), equation 6.19 becomes

= 2p
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