A retaining wall with a vertical back retains a mass of cohesionless soil, the surface of which is level with the top of the wall. The active lateral earth pressure coefficient in terms of the angle of internal friction φ of the backfill is given by:
Try it first: pick an option, then read the explanation.
Asked in SSC JE 29/10/2020 (Morning)
Detailed Explanation
For a vertical, smooth wall with a level cohesionless backfill, Rankine's active coefficient can be written in two equivalent ways.
\[ K_a = \frac{1-\sin\phi}{1+\sin\phi} = \tan^2\!\left(45^\circ - \frac{\phi}{2}\right) \]
Check with φ = 30°: \(0.5/1.5 = 1/3\) and \(\tan^2 30^\circ = 1/3\) ✓.
Why “(1 − sin φ)/(1 + sin φ) or tan²(45° − φ/2)” Is Correct
Option (c) pairs the correct fraction with the correct tangent form. The other options either use the passive fraction or mix the active fraction with the passive tangent form.
| 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) or tan²(45° + φ/2): The fraction is right but tan²(45° + φ/2) is Kp, so the two forms contradict each other.
- B. (1 + sin φ)/(1 − sin φ) or tan²(45° + φ/2): Both forms here are Kp.
- D. (1 + sin φ)/(1 − sin φ) or tan²(45° − φ/2): (1 + sin φ)/(1 − sin φ) is Kp, while tan²(45° − φ/2) is Ka; they are not equal.
Key Point
Ka = tan²(45° − φ/2); Kp = tan²(45° + φ/2).
Key Concepts for Students
- The minus sign goes with active (smaller) pressure.
- Ka < 1 < Kp for any φ > 0.
- For an inclined backfill, Rankine's Ka includes the slope angle β.
Practice more: this is question 12 in Earth Pressure and Retaining Wall MCQs – Part 1. Try the whole set and check every answer instantly.