Plane stress relations — maximum shear stress: The maximum shear stress in plane stress equals what fraction of the algebraic difference between maximum and minimum normal stresses?

Introduction:
Mohr's circle for plane stress provides direct relations among principal stresses and maximum shear stress. This question checks your command of that geometric interpretation.

Given Data / Assumptions:

• Plane stress with principal stresses σ1 and σ2.
• Linear elasticity and small deformations.

Concept / Approach:
On Mohr's circle, the radius equals (σ1 − σ2)/2 and represents the maximum shear stress τ_max in plane stress. Hence τ_max = (σ_max − σ_min)/2.

Step-by-Step Solution:

Verification / Alternative check:
Stress transformation equations also yield the same result when maximizing shear with respect to angle, confirming the half-difference relationship.

Why Other Options Are Wrong:

• Equal to or twice: Overestimate τ_max.
• One-fourth or three-fourths: Arbitrary fractions not supported by the transformation equations.

Common Pitfalls:
Confusing average stress (σ_avg) with radius; mixing up shear in 3D with plane-stress results.

Final Answer: