Mohr’s circle interpretation: The endpoints of any diameter on Mohr’s circle represent stresses on which planes?
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APrincipal stresses only
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BNormal stresses on planes at 45° to principal planes
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CShear stresses on planes at 45° to principal planes
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DThe normal and shear stresses on two mutually perpendicular physical planes
Answer
Correct Answer: The normal and shear stresses on two mutually perpendicular physical planes
Explanation
Introduction / Context:Mohr’s circle is a powerful graphical tool for transforming plane stresses. Correctly reading what points and diameters represent is key to finding stresses on inclined planes without laborious algebra.
Given Data / Assumptions:
- State of plane stress is represented by a Mohr’s circle.
- A diameter is drawn through the circle with endpoints on the circumference.
- We relate these endpoints to stresses on real (physical) planes in the body.
Concept / Approach:Each point on Mohr’s circle corresponds to the pair (σ_n, τ_nt) acting on some plane through a point. Two points connected by a diameter correspond to two orthogonal (mutually perpendicular) physical planes. Principal stresses are a special case where τ = 0 (points at horizontal intercepts).
Step-by-Step Solution:
Select any reference plane; find its stress point on Mohr’s circle.Construct the diameter through the circle’s centre to the opposite point.Interpretation: the two endpoints give (σ, τ) for two perpendicular planes in the material.Recognise that rotating the physical plane by θ rotates the Mohr’s radius by 2θ.Verification / Alternative check:Transformations show that stress components on orthogonal planes are located 180° apart on Mohr’s circle, i.e., at endpoints of a diameter.
Why Other Options Are Wrong:Principal stresses only: only true at τ = 0 intercepts.Options limited to “45° planes”: too restrictive; a diameter is general.
Common Pitfalls:Mistaking principal points for arbitrary diameter endpoints; forgetting the 2θ relationship between physical and Mohr rotations.
Final Answer:
The normal and shear stresses on two mutually perpendicular physical planes