Page 276 - Rock Mechanics For Underground Mining

EXCAVATION DESIGN IN BLOCKY ROCK

distance u y . Displacements u s and u n , with the directions indicated in Figure 9.15b,
occur at the joint surface, and the normal and shear forces are incrementally perturbed,
changing to the new equilibrium values N and S. Since the prism is not deformed in
the joint relaxation, joint deformations, u s and u n , are readily related to the vertical,
rigid-body displacement, u y , of the prism. From Figure 9.15b

u s = u y cos
(9.22)
u n = u y sin

Notingthattheblockmovesawayfromthesurroundingrockduringjointrelaxation,
increments of surface force are related to displacement increments by

N 0 − N = K n u n = K n u y sin
(9.23)
S − S 0 = K s u s = K s u y cos

Also, the equation of static equilibrium of the prism for the x direction requires
H = N cos + S sin (9.24)

Substitution of N 0 , S 0 , from equation 9.21, in equation 9.23, yields

N = H 0 cos − K n u y sin
(9.25)
S = H 0 sin + K s u y cos

Since the problem being considered involves the limiting equilibrium state of the
prism, introduction of the limiting friction defined by equation 9.18 in equation 9.25
yields

H 0 sin + K s u y cos = (H 0 cos − K n u y sin ) tan

Simple trigonometric substitutions and rearrangement of this expression produce

H 0 sin( − ) = u y (K s cos cos + K n sin sin )

or
H 0 sin( − )
u y =
(K s cos cos + K n sin sin )
Inserting this expression for u y in equation 9.25 gives

K n sin sin( − )

N = H 0 cos −
K s cos cos + K n sin sin

K n cos sin( − )
S = H 0 sin −
K s cos cos + K n sin sin

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