Page 214 - Rock Mechanics For Underground Mining

METHODS OF STRESS ANALYSIS

The principles used in the development of a boundary element (b.e.)–distinct ele-
ment (d.e.)linkage algorithm are illustrated in Figure 6.12. The b.e. and d.e. domains
are isolated as separate problems, and during a computational cycle in the d.e. rou-
tine, continuity conditions for displacement and traction are enforced at the interface
between the domains. This is achieved in the following way. As shown by Brady and
Wassyng (1981), the boundary constraint equation developed in a direct b.e. formu-
lation can be manipulated to yield a stiffness matrix [K i ] for the interior surface of
the domain. In each computational cycle of the d.e. iteration, displacements [u i ]of
interface d.e. nodes calculated in a previous cycle are used to determine the reactions
[r i ] developed at the nodes from the expression
[r i ] = [K i ][u i ] (6.57)

Then the forces applied to the d.e. interface nodes are equal and opposite to the
reactions developed on the b.e. interface; i.e.

[q i ] =−[r i ] (6.58)
Equation 6.57 represents formal satisfaction of the requirement for continuity of
displacement at the interface, while equation 6.58 implies satisfaction of the condition
for force equilibrium.
In the d.e. routine, nodal forces determined from equation 6.58 are introduced in
the equation of motion (equation 6.47) for each block in contact with the interface.
In practice, several iterative cycles may elapse before it is necessary to update the
interface nodal forces. When equilibrium is achieved in the d.e. assembly, the interface
nodal displacements and tractions (derived from the nodal reactions) are used to
determine stresses and displacements at interior points in the b.e. domain.

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