Planes of maximum shear under biaxial stress: For a biaxial state with principal planes defined, what is the orientation of the planes of maximum shear stress?
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ATrue
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BFalse
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CThey are 30° to the principal planes
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DThey coincide with principal planes
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EThey are 60° to the principal planes
Answer
Correct Answer: True
Explanation
Introduction / Context:Understanding the orientation of maximum shear stresses relative to principal planes is fundamental for failure analysis and for interpreting Mohr’s circle in biaxial stress states.
Given Data / Assumptions:
- Biaxial stress condition with defined principal planes (no in-plane shear on those planes).
- Linear elasticity; standard sign conventions.
Concept / Approach:Principal planes carry only normal stress; shear is zero there. Maximum shear planes occur midway between principal planes. In physical space, this corresponds to a 45° inclination to principal planes, and two orthogonal maximum-shear planes exist, 90° apart from each other.
Step-by-Step Solution:
On Mohr’s circle, principal planes lie at the horizontal intercepts where τ = 0.Maximum shear occurs at the top and bottom of the circle (τ = R), 90° on Mohr’s circle, which translates to 45° in the physical plane.Hence, the two planes of maximum shear are orthogonal and each is 45° to the principal planes.Verification / Alternative check:Transforming stresses via the stress transformation equations and differentiating τ(θ) with respect to θ yields the same 45° condition for extremum shear.
Why Other Options Are Wrong:
- Any angle other than 45° contradicts the transformation result.
- Coincidence with principal planes is impossible because shear there is zero by definition.
Common Pitfalls:Mixing Mohr’s-circle angle (2θ) with the physical angle θ; forgetting the two orthogonal maximum-shear planes; sign convention errors.
Final Answer:True