Page 210 - Rock Mechanics For Underground Mining

METHODS OF STRESS ANALYSIS

block centroid and the rotation of the block about the centroid; i.e. for the x direction

F x
¨ u x = (6.47)
m
where ¨ u x is the acceleration of the block centroid in the x direction, F x is the x
component of the resultant force on the block, and m is the mass of the block.
The translation of the block centroid can be determined from equation 6.47, by nu-
merical integration. Suppose a time increment t is selected, over which it is intended
to determine the block translation. Block velocity and translation are approximated
by

˙ u x (t 1 ) = ˙ u(t 0 ) + ¨ u x t
u x (t 1 ) = u x (t 0 ) + ˙ u x t
Similar expressions are readily established for block translation in the y direction,
and for block rotation.

6.7.3 Computational scheme
The distinct element method is conceptually and algorithmically the simplest of the
methods of analysis considered here. In its computational implementation, precau-
tions and some effort are required to achieve satisfactory performance. First, the time
step t in the integration of the law of motion cannot be chosen arbitrarily, and an ex-
cessively large value of t results in numerical instability. Second, for an assemblage
of blocks which is mechanically stable, the dynamic relaxation method described
above provides no mechanism for dissipation of energy in the system. Computation-
ally, this is expressed as continued oscillation of the blocks as the integration proceeds
in the time domain. It is therefore necessary to introduce a damping mechanism to
remove elastic strain energy as the blocks displace to an equilibrium position. Viscous
damping is used in practice.
The computational scheme proceeds by following the motion of blocks through a
series of increments of displacements controlled by a time-stepping iteration. Iteration
through several thousand time steps may be necessary to achieve equilibrium in the
block assemblage.

6.8 Finite difference methods for continuous rock

Dynamic relaxation, finite difference methods for continua have a long history of
application in the analysis of stress and displacement in the mechanics of deformable
bodies, traceable from the original work of Southwell (1940) and Otter et al. (1966).
Particular formulations for rock mechanics are represented by the proprietary codes
FLAC and FLAC3D (Itasca, 2003), which have gained acceptance as reference codes
for excavation engineering and support and reinforcement design. The FLAC (Fast
Langrangian Analysis of Continua) codes are intended for analysis of continuum
problems, or at most sparsely jointed media.
FLAC and FLAC3D are explicit finite difference techniques for solution of the
governing equations for a problem domain, taking account of the initial and boundary
conditions and the constitutive equations for the medium. An explicit procedure is
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