The locus of the end point of the resultant of the normal and tangential components of stress on an inclined plane is:
Try it first: pick an option, then read the explanation.
Asked in SSC JE 02/03/2017 (Evening); also 04/03/2017 (Evening)
Detailed Explanation
On a plane with normal (cos θ, sin θ) in principal axes, the resultant stress vector is (σ₁ cos θ, σ₂ sin θ). As θ varies, this point traces a curve.
\[ p_x = \sigma_1\cos\theta,\ p_y = \sigma_2\sin\theta \Rightarrow \frac{p_x^2}{\sigma_1^2} + \frac{p_y^2}{\sigma_2^2} = 1 \]
Why “an ellipse” Is Correct
That is the equation of an ellipse (Lamé's stress ellipse), with semi-axes equal to the principal stresses. Do not confuse it with Mohr's circle, which plots (σn, τ) rather than the resultant stress vector.
Why the Other Options Are Wrong
- A. a circle: A circle results only when σ₁ = σ₂, or when plotting (σn, τ) on Mohr's diagram.
- B. a parabola: No parabola arises in stress transformation.
- D. a straight line: A straight line results only under uniaxial stress (degenerate ellipse).
Key Point
Resultant stress vector traces the ellipse of stress (semi-axes σ₁, σ₂); Mohr's circle plots σn vs τ.
Key Concepts for Students
- Lamé's stress ellipse / ellipsoid in 3-D.
- Mohr's circle is a different, more practical construction.
- Equal principal stresses make the ellipse a circle.
Practice more: this is question 16 in Principal Stress and Principal Planes MCQs – Part 2. Try the whole set and check every answer instantly.