The coefficient of passive pressure Kp is ______ if the coefficient of active pressure Ka is 1/4.
Try it first: pick an option, then read the explanation.
Correct answer: A. 4
Asked in SSC JE 09/10/2023 (Morning)
Detailed Explanation
Rankine's coefficients for a level, cohesionless backfill are reciprocals of each other:
\[ K_a = \frac{1-\sin\phi}{1+\sin\phi} \qquad K_p = \frac{1+\sin\phi}{1-\sin\phi} = \frac{1}{K_a} \]
\[ K_p = \frac{1}{K_a} = \frac{1}{1/4} = 4 \]
(This corresponds to \(\sin\phi = 0.6\), i.e. \(\phi \approx 36.9^\circ\).)
Why “4” Is Correct
Because the formulas are exact inverses, Kp = 1/Ka = 4.
| 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. 0.5: 0.5 would be √Ka, not 1/Ka.
- C. 0.25: 0.25 is Ka itself.
- D. 0.33: 0.33 is Ka for φ = 30°, unrelated here.
Key Point
Rankine, level backfill: Ka × Kp = 1.
Key Concepts for Students
- This reciprocal relation does not hold for Coulomb's theory with wall friction.
- From Ka = 1/4: (1 − sin φ)/(1 + sin φ) = 1/4 → sin φ = 0.6.
- Kp is always greater than 1 and Ka always less than 1.
Practice more: this is question 7 in Earth Pressure and Retaining Wall MCQs – Part 1. Try the whole set and check every answer instantly.