Principal Plane and Maximum Shear Plane Angle Explained

The question asks for the angle between a principal plane and a plane of maximum shear.

Understanding Principal Planes

In stress analysis, a principal plane is a plane within a material under stress where the shear stress is zero. Only normal stress (tensile or compressive) acts on these planes. For a 2D stress state, there are typically two principal planes, corresponding to the maximum and minimum normal stresses, often denoted as $\sigma_1$ and $\sigma_2$. These planes are perpendicular to each other.

Understanding Planes of Maximum Shear

Planes of maximum shear are planes where the shear stress acting on them reaches its maximum possible value for a given stress state. On these planes, the normal stress is intermediate between the principal stresses.

Relationship Between Principal Planes and Maximum Shear Planes

The relationship between the directions of principal planes and planes of maximum shear can be visualized using Mohr's circle:

• The principal planes correspond to points on the horizontal (normal stress) axis of Mohr's circle, where the vertical coordinate (shear stress) is zero.
• The planes of maximum shear stress correspond to the points at the very top and bottom of Mohr's circle, representing the maximum radius of the circle.
• The angle between the points representing the principal plane and the plane of maximum shear on Mohr's circle is $90^\circ$.
• Crucially, a $90^\circ$ rotation on Mohr's circle corresponds to a $45^\circ$ rotation in the actual physical element.

Therefore, the plane of maximum shear is oriented at an angle of $45^\circ$ with respect to the principal planes.

Conclusion

The angle between a principal plane and a plane of maximum shear is $45^\circ$.