Rankine's Earth Pressure Theory: Active, Passive and At-Rest Pressure with Example
Rankine's theory gives the lateral pressure that soil exerts on a retaining wall. When the wall moves slightly away from the soil, the pressure drops to the active value; when the wall is pushed into the soil, it rises to the passive value; and if the wall does not move at all, the soil stays at rest. For a dry sand backfill the active pressure at depth z is simply pa = Ka γ z.
Earth pressure coefficients
| State | Coefficient | Value for φ = 30° |
|---|---|---|
| Active (wall moves away from soil) | Ka = (1 − sin φ) / (1 + sin φ) = tan²(45° − φ/2) | 0.333 |
| Passive (wall pushed into soil) | Kp = (1 + sin φ) / (1 − sin φ) = 1 / Ka | 3.0 |
| At rest (no movement) | K0 ≈ 1 − sin φ (Jaky) | 0.5 |
So Ka < K0 < Kp. A basement wall or a bridge abutment that cannot move is designed for at-rest pressure; a free cantilever retaining wall is designed for active pressure.
Pressure and thrust for common cases
| Case | Pressure at depth z | Total thrust per metre run |
|---|---|---|
| Dry cohesionless backfill | Ka γ z | ½ Ka γ H², acting at H/3 |
| Uniform surcharge q on the backfill | Extra Ka q (constant) | Ka q H, acting at H/2 |
| Submerged backfill (water table at top) | Ka γ′ z + γw z | ½ Ka γ′ H² + ½ γw H² |
| Cohesive soil (c, φ) | Ka γ z − 2c √Ka | Negative (tension) zone up to z₀ = 2c / (γ √Ka) |
| Passive, cohesive soil | Kp γ z + 2c √Kp | Used for resistance in front of the toe |
Water roughly doubles the pressure on a wall, which is why every retaining wall needs weep holes and a filter or drainage layer behind it.
Exam tip: Remember the order Ka < K0 < Kp and the product Ka × Kp = 1. For φ = 30° the trio is 1/3, 1/2 and 3. A favourite MCQ: the depth of tension crack in a purely cohesive soil is 2c/γ, and the critical height of an unsupported vertical cut is twice that, 4c/γ.
Worked example
A retaining wall 4 m high holds back a dry sand with γ = 18 kN/m³ and φ = 30°. The backfill carries a uniform surcharge of 10 kN/m². Find the total active thrust and where it acts.
- Ka = (1 − sin 30°) / (1 + sin 30°) = 0.5 / 1.5 = 0.333.
- Pressure from soil at the base = Ka γ H = 0.333 × 18 × 4 = 24 kN/m².
- Thrust from soil, P₁ = ½ × 24 × 4 = 48 kN/m, acting at 4/3 = 1.33 m above the base.
- Pressure from surcharge = Ka q = 0.333 × 10 = 3.33 kN/m², constant over the height.
- Thrust from surcharge, P₂ = 3.33 × 4 = 13.33 kN/m, acting at 2 m above the base.
- Total thrust P = 48 + 13.33 = 61.33 kN/m.
- Overturning moment about the base = 48 × 1.333 + 13.33 × 2 = 64.0 + 26.67 = 90.67 kN·m per m.
- Height of the resultant above the base = 90.67 / 61.33 = 1.48 m.
Tension crack check (cohesive soil): for a clay with c = 10 kN/m², φ = 20° and γ = 18 kN/m³, Ka = tan²(35°) = 0.490 and √Ka = 0.700, so z₀ = 2 × 10 / (18 × 0.700) = 1.59 m. In design the tension zone is ignored and the crack is assumed to fill with water.
Rankine vs Coulomb
Rankine ignores friction between the wall and the soil, so it gives slightly conservative (higher) active pressure for a rough wall. Coulomb's wedge theory includes wall friction δ, an inclined wall back and a sloping backfill, so it is preferred for gravity walls and walls with sloping fill. For most cantilever walls with a vertical stem and level backfill, Rankine is simple and accurate enough.
Frequently asked questions
What is the coefficient of active earth pressure for φ = 30°?
Ka = (1 − sin 30°)/(1 + sin 30°) = 0.5/1.5 = 1/3. The passive coefficient Kp is 3.
Where does the active thrust act on a retaining wall?
For a dry cohesionless backfill the pressure is triangular, so the thrust ½ Ka γ H² acts at H/3 above the base. A uniform surcharge adds a rectangular block acting at H/2.
What is the depth of tension crack in clay?
z₀ = 2c / (γ √Ka). For a purely cohesive soil (φ = 0) this becomes 2c/γ.
What are the assumptions of Rankine theory?
The wall back is vertical and smooth (no wall friction), the backfill surface is horizontal, and the soil is homogeneous, semi-infinite and in a state of plastic equilibrium.
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