The ratio of the tangential and normal components of stress on an inclined plane through θ to the direction of the force is:
Try it first: pick an option, then read the explanation.
Asked in SSC JE 02/03/2017 (Evening)
Detailed Explanation
Under an axial stress σ, the force on an inclined section resolves into components normal and tangential to the plane. Both depend on θ.
\[ \sigma_n = \sigma\cos^2\theta,\qquad \tau = \sigma\sin\theta\cos\theta,\qquad \frac{\tau}{\sigma_n} = \tan\theta \]
Why “tan θ” Is Correct
The ratio is tan θ. Strictly, this holds when θ is the angle between the force and the normal to the plane; measured to the plane itself, the ratio would be cot θ, which is not offered.
| Loading | Normal stress σn on the plane | Shear stress τ on the plane | Maximum shear |
|---|---|---|---|
| Uniaxial σ (θ = angle of the plane's normal to σ) | σ cos²θ | (σ/2) sin 2θ | σ/2 at θ = 45° |
| Biaxial σ₁, σ₂ (θ = angle of the normal to σ₁) | (σ₁ + σ₂)/2 + ((σ₁ − σ₂)/2) cos 2θ | ((σ₁ − σ₂)/2) sin 2θ | (σ₁ − σ₂)/2 at θ = 45° |
| General σx, σy, τxy | (σx + σy)/2 + ((σx − σy)/2) cos 2θ + τxy sin 2θ | ((σx − σy)/2) sin 2θ − τxy cos 2θ | √(((σx − σy)/2)² + τxy²) |
If θ is measured from the plane itself (i.e. from the direction of the stress) instead of from its normal, sin and cos swap: σn = σ sin²θ, τ = σ sinθ cosθ.
Why the Other Options Are Wrong
- A. sin θ: sin θ is a single component factor, not the ratio.
- B. cos θ: cos θ is a single component factor, not the ratio.
- D. cosec θ: cosec θ does not arise from the two components.
Key Point
τ/σn = tan θ; both components are equal at θ = 45°.
Key Concepts for Students
- Resultant stress on the plane = σ cos θ.
- Maximum τ = σ/2 at 45°.
- Angle convention changes tan ↔ cot.
Practice more: this is question 10 in Principal Stress and Principal Planes MCQs – Part 1. Try the whole set and check every answer instantly.