The coefficient of passive earth pressure for a cohesionless granular soil is represented by:
Try it first: pick an option, then read the explanation.
Correct answer: A. (1 + sin θ)/(1 − sin θ)
Asked in SSC JE 03/03/2017 (Evening)
Detailed Explanation
Here θ is used for the angle of internal friction (usually written φ).
\[ K_p = \frac{1+\sin\theta}{1-\sin\theta} = \tan^2\!\left(45^\circ + \frac{\theta}{2}\right) \]
Why “(1 + sin θ)/(1 − sin θ)” Is Correct
The passive coefficient is greater than 1, so the larger term (1 + sin θ) is in the numerator. That is option (a).
| Coefficient | Formula (cohesionless soil) | φ = 30° |
|---|---|---|
| Active, Ka | (1 − sin φ)/(1 + sin φ) = tan²(45° − φ/2) | 1/3 ≈ 0.333 |
| At rest, K0 | μ/(1 − μ) (elastic) or ≈ 1 − sin φ (Jaky) | ≈ 0.5 |
| Passive, Kp | (1 + sin φ)/(1 − sin φ) = tan²(45° + φ/2) | 3 |
For Rankine's theory with a level backfill, Ka × Kp = 1 and Ka < K0 < Kp.
Why the Other Options Are Wrong
- B. (1 + cos θ)/(1 − cos θ): Cosine forms are not used.
- C. (1 − sin θ)/(1 + sin θ): That is the active coefficient Ka.
- D. None of these: Option (a) is correct.
Key Point
Kp = (1 + sin φ)/(1 − sin φ) > 1.
Key Concepts for Students
- φ = 30° → Kp = 3.
- Kp = 1/Ka for level backfill (Rankine).
- Passive resistance is used for anchor blocks and the toe of walls.
Practice more: this is question 23 in Earth Pressure and Retaining Wall MCQs – Part 2. Try the whole set and check every answer instantly.