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.