36 Stress
By similar considerations in the y and t directions, we also have-+-+-- atxy aUYY atzy - ()
aY aY aY
-+-+-=O ab aryz a%,
az az az
The reason for including this question and answer is to demonstrate
that equations such as these relating to stress analysis, and which are
perhaps daunting at first sight, are not so difficult and are certainly
easier to remember once they are understood.
43.10 Given an elemental cube with a normal stress component
and two shear stress components acting on all its faces, it is always
possible to find a cube orientation such that the shear stresses dis-
appear on all faces and only normal stresses (the principal normal
stresses) remain. Is it possible to find a complementary orientation
such that the normal stresses disappear on all faces and only shear
stresses (i.e. principal shear stresses) remain? Explain the reason
for your answer.
A3.10 No, generally it is not possible. The answer is straightforward
from the mathematical point of view. The first invariant ZI explained in
A3.8 provides the direct answer to the question. As the orientation of the
cube changes, the normal and shear stresses on the cube faces change, but
ZI = a,, + ay, + azr = a1 + a2 + 03
and so, for any non-zero value of 11, it is not possible to have
q = a2 = a3 = 0.
There is the exceptional case when ZI = 0. In 2-D, for the pure shear
stress case, as defined in A3.6, where ZI = 01 + a2 = +k + (4) = 0,
there is an orientation of the elemental square with the edges having no
normal stresses, i.e. when the square is rotated by 45". Note that this can
only occur for a Mohr circle centred at the origin of the a - t axes, so
that one diameter of Mohr's circle is along the t axis where the normal
stresses are zero.
Pure shear on
this plane-a\ 45"
No normal
stresses on
the edges of
this square