» Strain Transformation
STRAIN TRANSFORMATION at a point (in Plane Strain) is done in a similar manner as with stresses. Simply substitute e_x for s_x and e_y for s_y. HOWEVER, because of the way the math works out, substitute g/2 for t:

ex' = ex + ey 2 + ex ey 2 cos(2q) + gxy 2 sin(2q) ey' = ex + ey 2 ex ey 2 cos(2q) gxy 2 sin(2q) gx'y' 2 = ex ey 2 sin(2q) + gxy 2 cos(2q)

• Again, the principle of Invariance still works: e_x + e_y = e_x' + e_y'

» Principal Strains
PRINCIPAL STRAINS are the Maximum and Minimum Normal Strains that occur at a point as the set of axes is rotated by a certain angle q_p. The Principal Strains are called e_I and e_II, and occur along the x_p-y_p axes. These directions are NOT necessarily the same as those for the Principal Stresses for a given stress-state.

The Principal Strains for a given strain state are given by:

eI = ex + ey 2 + ex ey 2 2 + gxy 2 2 1/2 eII = ex + ey 2 - ex ey 2 2 + gxy 2 2 1/2

• The Principal Strains occur when the element has been rotated by an angle of q_p, where:

tan(2qp) = gxy ex - ey

As with the angles for Principal Stresses, there are two solutions.

• KEY POINT: When the Normal Strains are Principal Strains the Shear Strain is Zero.

» Maximum Shear Strain
The MAXIMUM (In-Plane) SHEAR STRAIN occurs when the Principal Strain element is rotated 45° and is given by:

gmax 2 = eI + eII 2 = ex ey 2 2 + gxy 2 2 1/2

• The angle at which the Maximum Shear Strain occurs is given by:

tan(2qs) = ex ey gxy

Again, there are two solutions.

• When the shear strain is maximum, the normal strains are equal to each other and are the average of the Principal Strains.