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?
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Aequal to
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Bone-fourth
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Cone-half
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Dtwice
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Ethree-fourths
Answer
Correct Answer: one-half
Explanation
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:
Define principal stresses: σ_max = σ1, σ_min = σ2Mohr’s circle radius: R = (σ1 − σ2)/2Therefore τ_max = R = (σ_max − σ_min)/2Verification / 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:
one-half