Engineering Rock Mechanics

406 Appendix A: Stress and strain analysis

The principal directions are the directions of the co-ordinate axes such
that for any given stress state the shear stresses are zero. The only stresses
acting on the elemental cube in this new direction are the principal
stresses.
In the rotated diagram above we want QY = 0, i.e.

zxtyt = zxy (cos^2 a - sin^2 a) - (ox - oy) cos a sin a = 0

:. zxy (cos^2 a - sin^2 a) = (ox - cy) cos a sin a

or

2, - - cosasina -^72 sin2a^1

(ox - ov) cos' a - sin2 a cos2a 2

• = -tan2a

inverting:

When qYr = 0, x' and y' are the principal directions and qand qare the

principal stresses.

Example. Continuing from before

TY = 10, 0, = 20, ay = 10

so

i.e.

a = 31.7".

This gives the principal directions. The principal stresses are found as

before, with 8 = 31.7".

0 = 31.7" + sin 0 = 0.526, cos 0 = 0.851

ox, = 20 x (0.851)2 + 10 (0.526)' + 2 x 10 x 0.526 x 0.851 = 26.18 MPa

or, = 20 x (0.526)' + 10 X (0.851)' - 2 x 10 x 0.526 x 0.851 = 3.82 MPa.

Check: a,, +oy, = 26.18 + 3.82 = 30.00 MPa.