The **maximum (in-plane) shear stress ** occurs when the principal stress element is rotated by 45°, and is
given by:

The angle that defines the direction nomral to the plane on which the Maximum Shear Stress occurs is:

This equation has two solutions, q_{sI}
and q_{sII}, one corresponding to a
positive shear stress on the newx_{s}-face, and 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 the angles into the general stress-transformation
equation for t_{x'y'}(q_{s}).
By doing so, the appropriate angle is matched with the correct shear stress
(positive or negative).

Note 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}) .