According to the maximum strain energy theory, the yield locus is:
Try it first: pick an option, then read the explanation.
Correct answer: D. an ellipse
Asked in SSC JE 03/03/2017 (Evening)
Detailed Explanation
In two dimensions (σ₃ = 0), each theory of failure gives a closed boundary in the σ₁–σ₂ plane. Points inside are safe.
\[ \sigma_1^2 + \sigma_2^2 - 2\mu\,\sigma_1\sigma_2 = \sigma_y^2 \]
A quadratic with no linear terms and \(4\mu^2 < 4\) — an ellipse with its major axis along the 45° line.
Why “an ellipse” Is Correct
The strain energy theory gives an ellipse. (The von Mises distortion energy theory also gives an ellipse, with μ effectively 0.5.)
Why the Other Options Are Wrong
- A. a rectangle: Rankine's theory gives a square, not a general rectangle.
- B. a circle: None of the standard theories gives a circle in the σ₁–σ₂ plane.
- C. a hexagon: A hexagon is Tresca's locus.
Key Point
Yield loci: Rankine square, St. Venant rhombus, Tresca hexagon, Haigh and von Mises ellipses.
Key Concepts for Students
- Von Mises ellipse: σ₁² + σ₂² − σ₁σ₂ = σy².
- Tresca's hexagon is inscribed in the von Mises ellipse.
- Ellipses are elongated along σ₁ = σ₂.
Practice more: this is question 23 in Principal Stress and Principal Planes MCQs – Part 2. Try the whole set and check every answer instantly.