Page 314 - Rock Mechanics For Underground Mining

ENERGY, MINE STABILITY, MINE SEISMICITY AND ROCKBURSTS

The term on the left of the inequality in expression 10.82 is a quadratic form in the
¨
vector [ u], and represents the virtual work term W defined by equation 10.73. From
the theory of quadratic forms ( Jennings, 1977), the requirement that W be positive is
equivalent to the condition that the global stiffness matrix be positive definite. This is
assured if all principal minors of [K g ] are positive. The value of any principal minor
of a matrix is obtained by omitting any number of corresponding rows and columns
from the matrix, and evaluating the resultant determinant. In particular, each element
of the principal diagonal of the global stiffness matrix is, individually, a principal
minor of the matrix. Thus, a specific requirement for stability is that all elements of
the principal diagonal of [K g ] be positive.
Techniques for practical assessment of mine global stability have been proposed
by Starfield and Fairhurst (1968) and Salamon (1970). Both techniques relate to
the geometrically simple case of stope-and-pillar layouts in a stratiform orebody,
and can be shown readily to be particular forms of the criterion expressed by equa-
tion 10.82, involving the positive definiteness of [K g ]. Starfield and Fairhurst pro-
posed that the inequality 10.79 be used to establish the stability of individual pillars
in a stratiform orebody, and therefore to assess the stability of the complete mine
structure. Mine pillars are loaded by mining-induced displacement of the country
rock, which are resisted by the pillar rock. The country rock therefore represents
the spring in the loading system illustrated in Figure 10.17a, and the pillar repre-
sents the specimen. For a mining layout consisting of several stopes and pillars,
the stiffness of pillar i, i , replaces in inequality 10.79, while the correspond-
ing local stiffness, k i , replaces k. Mine global stability is then assured if, for all
pillars i

k i + i > 0 (10.83)

It is to be noted, from Figure 10.17c, that in the elastic range of pillar performance,
i is positive, and for elastic performance of the abutting country rock, k i is positive
by definition. Pillar instability is liable to occur when i is negative, in the post-peak
range, and | i | > k i .
In an alternative formulation of a procedure for mine stability analysis, Salamon
(1970) represented the deformation characteristics of the country rock enclosing a
set of pillars by a matrix [K] of stiffness coefficients. The performance of pillars
was represented by a matrix [ ], in which the leading diagonal consists of individual
pillar stiffnesses i and all other elements are zero. Following the procedure used to
develop equation 10.82, Salamon showed that the mine structure is stable if [K + ]
is positive definite. Brady and Brown (1981) showed that, for a stratiform orebody,
this condition closely approximates that given by the inequality 10.83, when this is
applied for all pillars.
In assessing the stability of a mine structure by repetitive application of the pillar
stabilitycriterion(inequality10.83),therequiredinformationconsistsofthepost-peak
stiffnesses of the pillars, and the mine local stiffnesses at the various pillar positions.
It must be emphasised that the idea that the post-peak deformation of a pillar can
be described by a characteristic stiffness, , is a gross simplification introduced for

the sake of analytical convenience. The idea is retained for the present discussion,
because it presents a practical method of making a first estimate of pillar and mine
stability, for geometrically regular mine structures.

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