Mohr’s circle basics: The radius of Mohr’s circle equals which combination of the two principal stresses?

Introduction / Context:
Mohr’s circle graphically represents plane stress transformation. Knowing the center and radius allows immediate computation of principal stresses and maximum shear stress.

Given Data / Assumptions:

• Principal stresses σ1 and σ2 are real values (plane stress state).
• Standard plotting: σ-axis horizontal, τ-axis vertical.
• Positive shear convention consistent with Mohr’s construction.

Concept / Approach:
The circle’s center is at C = (σ1 + σ2)/2 on the σ-axis, and the radius is R = (σ1 − σ2)/2. The maximum in-plane shear equals this radius.

Step-by-Step Solution:

Verification / Alternative check:
General stress components (σx, σy, τxy) reduce to principal form by rotation; analytical solutions give σ1,2 = (σx + σy)/2 ± √[ ((σx − σy)/2)^2 + τxy^2 ], where the radical equals the radius R.

Why Other Options Are Wrong:
Sum or half-sum describe the center, not the radius.Full difference is twice the radius.Product has no direct geometric meaning in Mohr’s circle radius.

Common Pitfalls:
Mixing center and radius formulas; sign mistakes in τ that reflect points across the σ-axis.

Final Answer:

Half the difference of the two principal stresses