41 4 Appendix A: Stress and strain analysis
For small rotations sin R = R, so that
du=-Rdy
dv = Rdx
or
(b) Normal strain
- Because we use compression
I
I I
I we have
- Because we use compression
(and hence contraction) positive,
I
----^__- J I
w+
from which we obtain
du
dx
& x =--
or
(c) Shear strain
This is negative shear strain: P'Q
is longer than PQ, and extension
is negative. For small angles, to a
good approximation we have
du =dysina!+dxcosa-dx
butforsmallanglessina=aandcosa=l,sothatdu= ady+dx-dx=
a dy and similarly dv = adx.
The definition of shear strain is the change in angle between two lines
originally perpendicular to each other, i.e. ywy = (p - V2).
q2 = p+2a =3 -2a= p-x/2 = r,,