Which of the following represents the expression for the passive earth pressure coefficient?
Try it first: pick an option, then read the explanation.
Correct answer: A. tan²(45° + φ/2)
Asked in SSC JE 27/01/2018 (Morning)
Detailed Explanation
Three coefficients describe lateral earth pressure; each has a standard expression (see table).
Why “tan²(45° + φ/2)” Is Correct
The passive coefficient is the larger one, with the plus sign: Kp = tan²(45° + φ/2). For φ = 30°, tan²60° = 3.
| 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. μ/(1 − μ): μ/(1 − μ) is the at-rest coefficient K₀ from elastic theory.
- C. tan²(45° − φ/2): tan²(45° − φ/2) is the active coefficient Ka.
- D. μ/(1 + μ): μ/(1 + μ) is not a standard earth pressure coefficient.
Key Point
+ for passive, − for active; μ/(1 − μ) for at rest.
Key Concepts for Students
- Kp = 1/Ka for Rankine with level fill.
- μ = Poisson's ratio.
- Coulomb's Kp with wall friction can be much higher (and is often over-estimated).
Practice more: this is question 17 in Earth Pressure and Retaining Wall MCQs – Part 2. Try the whole set and check every answer instantly.