Engineering Rock Mechanics

408 Appendix A: Stress and strain analysis

from which

COS' e = YZ (1 + COS 4)

or

cos 4 = (1 - sin2 e) - sin' e = 1 - 2 sin2 e

from which

sin' B = YZ (1 + cos 41.

Substituting (D), (E) and (F) into equations (A), (B) and (C):

0,. = q (112(1 + cos 4)) + q(Y2(1 + cos 4))

i.e.

These two equations are simply the equations of a circle centred at

Yz (q + 02) on the o-axis in c~z space:

-7 A

+(a, + a,) +$(a, - a*)cos0

I

0

The o-axis and the

x-axis are parallel

To use Mohr's circle you must understand and remember:

1. Positive shear stresses and positive rotations have been used in
   developing the equations for a point (0, z), but the z co-ordinate is
   negative. This means the z-axis is upside down: