
Similar to
stress analysis at a point, strain can also be
rotated to different directions and thus maximum shear and normal strains can be
found. Strain at a given point on a two dimensional object can be seen as a small
strain element with two normal strains and one shear strain, as shown in the
diagram.
The sign convention for strains at a point is similar to stresses at a point.
Normal tension strain in both the x and y direction are assumed positive. The
positive shear strain direction is shown in the diagram at
the left. Shear stresses act on four sides of the stress element, causing a pinching
or shear action. Shear strains on all four sides are the same, thus
γ_{xy} = γ_{yx}
Recall, the shear strain is actually
defined as the angle of rotation or twist
due to the shear stress. This angle is in radians and is shown at the left.

Rotated Strain Element
Rotated Strain Element


Stains at a point (strain element) can be rotated to give
a new strain state at any particular angle. The rotation angle, θ,
is assumed positive using the right hand rule (counterclockwise in the xy plane
is positive). The new coordinate system is labeled as x' and y'. The new rotated
strains are shown in the diagram at the left. The shear strains, γ_{x'y'} and γ_{y'x'}
are still equal.
As one may expect, the strain transformation equations are nearly identical
to the stress transformation angles. The only difference is a factor of 2 for
the shear strain. Other than that, the strain symbol, ε,
can be simply substituted for σ to give
the strain transformation equations (detail derivation is omitted). The final
equations are,



Generally, the largest normal strain is of most interest. This can be found
by taking a derivative of either the ε_{x´} or ε_{y´} strain
with respect to θ and equating it to zero. This will
give the principal rotation angle, θ_{p}, that
will produce the principal (maximum and minimum) strains. The resulting equations
are
These strains, ε_{1} or ε_{2} are
referred to as the principal shear strains.
