The coefficient of active earth pressure for a loose sand (cohesionless soil) having an angle of internal friction φ is:
Try it first: pick an option, then read the explanation.
Correct answer: B. (1 − sin φ)/(1 + sin φ)
Asked in SSC JE 2012 / SSC JE 2008
Detailed Explanation
For a cohesionless soil, Rankine's active coefficient uses the sine of φ:
\[ K_a = \frac{1-\sin\phi}{1+\sin\phi} \]
Why “(1 − sin φ)/(1 + sin φ)” Is Correct
The active coefficient must be less than 1, so the smaller term (1 − sin φ) is on top. That is option (b).
| 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
- A. (1 + sin φ)/(1 − sin φ): (1 + sin φ)/(1 − sin φ) is the passive coefficient Kp.
- C. (1 + cos φ)/(1 − cos φ): Cosine forms are not used for Rankine's coefficients.
- D. (1 − cos φ)/(1 + cos φ): Cosine forms are not used.
Key Point
Ka = (1 − sin φ)/(1 + sin φ) < 1.
Key Concepts for Students
- φ = 30° → 1/3.
- Same as tan²(45° − φ/2).
- Loose sand has a lower φ, so a higher Ka than dense sand.
Practice more: this is question 20 in Earth Pressure and Retaining Wall MCQs – Part 2. Try the whole set and check every answer instantly.