Stress analysis 407Note: the largest of these stresses is the major principal stress, q. The
smallest is the minor principal stress, 0,.
Note that becauese (ox + oy) = (ox( + oyr) = (q + 02) which is a constant,
there can be no principal shear stresses, i.e. planes on which no normal
stresses act
Mohr’s circle of stress
This is a graphical method of transforming the stress tensor. It is easy to
use and remember, and is the best way of remembering the transformation
equations.
If we choose the global x- and y-axes to coincide with the principal
directions (and because we can choose the axes arbitrarily there is nothing
to prevent this), then the transformation equations become
a,, = q cos’ e + o2 sin’ e (4
o,,# = q sin’ e + oz cos’ e (B)
Zx‘Y‘ - - - (q - 0’) cos 8 sin (^6) (C)
where o1 and o2 are now the principal stresses, and 8 is measured
anticlockwise from the principal direction x to the local direction x’.
These new equations can be simplified still further, by making use of
trigonometric iden tities.
Let
then
sin 4 = sin (e + e) = sin COS e + COS esin e= 2 sin COS e
:. cos 8 sin 8 = Y2 sin Q
and
but
cos2 0 + sin2 e = 1
so
COS = cos2 e - (1 - cos2 e) = 2 cos2 e - 1