Page 279 - Rock Mechanics For Underground Mining

ROOF STABILITY ANALYSIS FOR A TETRAHEDRAL BLOCK

where
M = [(k s /k n ) cos 2 cos i + sin( − i) sin ] sin( − )/D cos i
D = cos cos k s /k n + sin sin( − i)/ cos i
2
W = R C w
2
C w = cos
(tan
+ cot ) − /2 +
The Factor of Safety against wedge failure is given by
(S + P 0 ) S
FoS = = + q (9.33)
W W
In a further development of this analysis, Nomikos et al. (2002) considered a sym-
metric roof wedge in an inclined, deviatoric stress field. Expressions more complex
than those above define the Factor of Safety against roof failure, and a reasonable
correspondence was shown between the analytical solution and some solutions using
UDEC.

9.4 Roof stability analysis for a tetrahedral block

A comprehensive relaxation analysis for a non-regular tetrahedral wedge in the crown
ofanexcavationpresents someconceptual difficulties. These arisefromtheextranum-
ber of degrees of freedom to be accommodated in the analysis. For example, on any
face of the tetrahedron it is necessary to consider two components of mutually orthog-
onal shear displacement as well as a normal displacement component. Maintenance
of statical determinacy during the relaxation process would require that the wedge
be almost isotropically deformable internally. For this reason, a thorough analysis of
the stability of a tetrahedral wedge in the crown is not handled conveniently by the
relaxation method presented earlier. A computational method which takes explicit
account of the deformation properties of the rock mass, and particular joints and joint
systems, presents the most opportune basis for a mechanically appropriate analysis.
In some circumstances, it may be necessary to assess the stability of a roof wedge
in the absence of adequate computational tools. In this case, it is possible to make
a first estimate of wedge stability from an elastic analysis of the problem and the
frictional properties of the joints. Suppose the orientation of the dip vector OA of a
joint surface, which is also a face of a tetrahedral wedge, is defined by the dip angle ,
and dip direction , measured relative to the global (x, y, z) reference axes, illustrated
in Figure 9.17a. The direction cosines of the outward normal to the plane are given
by

n = (n x , n y , n z ) = (sin cos , sin sin , − cos )

The normal component of traction at any point on the joint surface can be estimated
from the elastic stress components and the direction cosines by substitution in the
equation
2
2
2
t n = n xx + n yy + n zz + 2(n x n y xy + n y n z yz + n z n x zx )
x y z
If the normal traction t n is determined at a sufficient number of points on the joint
surface, its average value and the area of the surface can be used to estimate the total
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