Mohr’s circle basics: The radius of Mohr’s circle equals which combination of the two principal stresses?
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AThe sum of the two principal stresses
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BThe difference of the two principal stresses
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CHalf the sum of the two principal stresses
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DHalf the difference of the two principal stresses
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EThe product of the two principal stresses
Answer
Correct Answer: Half the difference of the two principal stresses
Explanation
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:
Plot points A(σ1, 0) and B(σ2, 0) on the σ-axis.Center C is the midpoint of AB → (σ1 + σ2)/2.Radius R is half the length of AB → (σ1 − σ2)/2.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