Page 592 - Rock Mechanics For Underground Mining

APPENDIX B STRESSES AND DISPLACEMENTS

If the axis of rotation, A, has a high plunge, it is more convenient to rotate this axis
to the centre of the projection in the auxiliary rotation, rather than to the perimeter. As
before, during this auxiliary rotation, all other points are rotated along small circles by
the same amount and in the same direction as A. When the auxiliary rotation has been
completed, the specified rotation is achieved by moving the point through a circular
arc centred on the centre of the projection. The auxiliary rotation is then reversed and
the orientation of the rotated line read off.

Appendix B Stresses and displacements induced by
point and infinite line loads in an

infinite, isotropic, elastic continuum

B.1 A point load (the Kelvin equations)

For a point load, P, applied at the x, y, z co-ordinate origin, in the positive x direction,
and tensile normal stresses reckoned positive, etc:
P x 3x 2
xx =− 3 (1 − 2 ) + 2
8 (1 − ) R R
P x 3y 2
yy =− −(1 − 2 ) +
8 (1 − ) R 3 R 2
P x 3z 2
zz =− −(1 − 2 ) +
8 (1 − ) R 3 R 2
P y 3x 2
xy =− (1 − 2 ) +
8 (1 − ) R 3 R 2
P 3xyz
yz =−
8 (1 − ) R 5
P z 3x 2
zx =− (1 − 2 ) +
8 (1 − ) R 3 R 2
P x 2 1
u x = + (3 − 4 )
16 G(1 − ) R 3 R
P xy
u y =
16 G(1 − ) R 3
P zx
u z =
16 G(1 − ) R 3
2
2
2
2
where R = x + y + z .
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