» Why Stress Transformation
So far, we have looked at stress elements (representing material points) that align with a typical x-y-z coordinate system. However, there are times when rotating or transforming the stress element to view it in another direction is necessary. Two key reasons that we may want to view an element in another orientation include:

• To determine the stress in another direction (e.g., normal and parallel to the the plane of a weld);
• To determine the Maximum Normal Stress or Maximum Shear Stress at a Point (this is useful when an element is subjected to multiple stresses);

» Rotation of a Stress Element
STRESSES of a ROTATED ELEMENT are derived using trigonometry and applying the equilibrium equations. If a Plane-Stress element is rotated by an angle of q, the Transformed Stresses are given by:

sx' = sx + sy 2 + sx sy 2 cos(2q) + txysin(2q) sy' = sx + sy 2 sx sy 2 cos(2q) txysin(2q) tx'y' = sx sy 2 sin(2q) + txycos(2q)

• Note that q is positive in the counter-clockwise direction.
• Adding the values for the two normal stresses together, we find that: s_x + s_y = s_x' + s_y'
  This is true for any q and is termed Invariance.

» Principal Stresses
PRINCIPAL STRESSES are the Maximum and Minimum Normal Stresses that occur at a point as the set of axes is rotated by angle q. The Principal Stresses are called s_I and s_II, and occur along the x_p-y_p axes. The Principal Stresses for a given stress state are given by:

sI = sx + sy 2 + sx sy 2 2 + txy2 1/2 sII = sx + sy 2 - sx sy 2 2 + txy2 1/2

The larger of the two stresses is generally termed s_I.

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

tan(2qp) = 2txy sx - sy

This equation has two results: angles, q_pI and q_pII, (90° apart). It is not always obvious which angle goes with which Principal Stress. It is often best to substitute one of principal angles into the general stress-transformation equation for s_x'(q_p). By doing so, the appropriate angle is matched with the right stress.

• KEY POINT: When the Normal Stresses are Principal Stresses the Shear Stress is Zero. This can be shown by plugging in either of the principal angles into the general equation for. t_x'y'(q_p)

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

tmax = sI sII 2 = sx sy 2 2 + txy2 1/2

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

tan(2qs) = sx sy 2txy

This equation has two solutions, q_sI and q_sII, one corresponding to a positive shear stress, the other to a negative shear stress on the new x_s-face (defined by q_s). It is often best to substitute one of angles into the general stress-transformation equation for t_x'y'(q_s). By doing so, the appropriate angle is matched with the right shear stress (positive or negative).
It should be noted that q_s =q_p±45°.

• KEY POINT: When Shear Stress is maximum, the Normal Stresses are the same and equal to their average. This can be shown by plugging in either of the Shear Stress Angles into the general equation for s_x'(q_s) or s_y'(q_s) .