The total strain energy theory for the failure of a material at the elastic limit is known as:
Try it first: pick an option, then read the explanation.
Correct answer: D. Haigh's theory
Asked in SSC JE 27/01/2018 (Evening)
Detailed Explanation
Each theory of failure carries the name of the person associated with it. The total strain energy theory compares the energy stored per unit volume with that at yield in simple tension.
\[ \sigma_1^2 + \sigma_2^2 + \sigma_3^2 - 2\mu(\sigma_1\sigma_2 + \sigma_2\sigma_3 + \sigma_3\sigma_1) = \sigma_y^2 \]
Why “Haigh's theory” Is Correct
The total strain energy theory is Haigh's theory (also credited to Beltrami).
| Theory of failure | Due to | Failure when | Yield locus (2-D) | Suited to |
|---|---|---|---|---|
| Maximum principal stress | Rankine | σ₁ = σy | Square | Brittle materials |
| Maximum principal strain | St. Venant | σ₁ − μσ₂ = σy | Rhombus | Rarely used |
| Maximum shear stress | Guest / Tresca | σ₁ − σ₃ = σy | Hexagon | Ductile (conservative) |
| Total strain energy | Haigh (Beltrami) | σ₁² + σ₂² − 2μσ₁σ₂ = σy² | Ellipse | Ductile |
| Shear strain (distortion) energy | von Mises (Hencky) | σ₁² + σ₂² − σ₁σ₂ = σy² | Ellipse | Ductile (best fit) |
Why the Other Options Are Wrong
- A. Guest's or Tresca's theory: Guest/Tresca: maximum shear stress theory.
- B. St. Venant's theory: St. Venant: maximum principal strain theory.
- C. Rankine's theory: Rankine: maximum principal stress theory.
Key Point
Haigh → total strain energy; von Mises → shear strain (distortion) energy.
Key Concepts for Students
- Its 2-D yield locus is an ellipse.
- It fails to predict that hydrostatic stress does not cause yielding.
- Von Mises corrected this by using only distortion energy.
Practice more: this is question 21 in Principal Stress and Principal Planes MCQs – Part 2. Try the whole set and check every answer instantly.