Page 212 - Rock Mechanics For Underground Mining

METHODS OF STRESS ANALYSIS

where ˚ u i is the gridpoint velocity and g i is the component of the gravitational accel-
eration in the i coordinate direction.
Introducing equation 6.48 yields

%
∂ ˚ u i /∂t = 1/m ij n j dS + g i (6.50)
S
where m = A.
If a force F i is applied at a gridpoint due, for example, to reactions mobilized by
reinforcement or contact forces between blocks, equation 6.50 becomes

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∂ ˚ u i /∂t = 1/m(F i + ij n j dS) + g i
S
= 1/mR i + g i (6.51)
where R i is the resultant (out-of-balance) force at the gridpoint.
Equation 6.51 indicates that the acceleration at a gridpoint can be calculated ex-
plicitly from the resultant force obtained by integration of the surface tractions over
the boundary contour of the region surrounding the gridpoint summed with local in-
ternally applied forces, the lumped local mass and the local gravitational acceleration.
When the acceleration of a gridpoint has been calculated, central difference equa-
tions can be used to calculate gridpoint velocities and displacements after a time
interval t:

u (t+ t/2) = u (t− t/2) + [R i /m + g i ] t (6.52)
i i
(t+ t) (t) (t− t)
x = x + x (6.53)
i i i
When a pseudo-static problem is being analysed, viscous damping terms are included
in equations 6.52 and 6.53 to increase the rate of convergence.
Calculation of changes in the state of stress proceed through calculation of strain
increments and their introduction in the constitutive equations for the medium. Strain
increments are determined directly from the velocity gradients, as follows. From the
Gauss Divergence Theorem,

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∂ ˚ u i /∂x j = 1/A ˚ u i n j dS (6.54)
s
The RHS of equation 6.54 can be evaluated as a summation over the boundary contour
of a polygon surrounding a gridpoint, and then strain increments can be determined
from the expression

1
ε ij = [∂ ˚ u i /∂x j + ∂ ˚ u j /∂x i ] t (6.55)
2
Finally, the stress increment in the time interval t is calculated directly from the
existing state of stress, the strain increments and the material constants k for the
medium, through an appropriate constitutive equation:

ij = f ( εij, ij , k ) (6.56)

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