Rankine Active Earth Pressure

K_{a} = tan^{2} (4 5^{\circ} - ϕ /2) = \frac{1 - sin ϕ}{1 + sin ϕ}, σ_{a} = K_{a} σ_{v}^{'} - 2 c K_{a}

Active pressure develops when a wall moves away from the backfill enough to mobilize shear strength. K_a < 1 for φ > 0. At-rest K0 ≈ 1 − sin φ is used when the wall cannot move. Passive K_p = 1/K_a is much larger.

$ϕ$ (deg): Effective friction angle
$K_{a}$: Active earth-pressure coefficient
$c$ (kPa): Cohesion

Step-by-step solved example

Smooth vertical wall, 5 m, dry sand φ = 30°, γ = 18 kN/m³. Active force per meter?

Active wedge: Pa = ½ Ka γ H² (no surcharge, c = 0).

1. Ka
Ka = tan²(30°) = 1/3.
2. Force
Pa = ½ Ka γ H² = 0.5×(1/3)×18×25 = 75 kN/m, acting at H/3 from the base.
$P_{a} = 75 kN/m$

Answer: 75 kN/m at H/3 from the base

10 practice questions

0/10 correct

1.For φ = 30°, Ka is

2.Kp for φ = 30° is

3.At-rest K0 (normally consolidated sand) ≈

4.Active resultant on a triangular diagram acts at

5.Cohesion reduces active pressure by

6.If the wall does not yield, use

7.Rankine assumes the wall is

8.Surcharge q on the backfill adds

9.Water in the backfill (no drainage) is dangerous because

10.φ = 0 clay, undrained, Ka =