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Lateral Earth Pressure (Rankine)

Active, at-rest, and passive lateral earth pressure on a vertical retaining-wall back face. Cohesionless and c-φ soils, with line surcharge. Per Rankine (1857) — assumes vertical wall, horizontal backfill, smooth wall (no wall friction).

ft
pcf
degrees
psf (use 0 for sand)
psf (uniform on backfill)
——
——
——
—lb/ft of wall
—lb/ft of wall
—ft above base

Defaults: 10-ft wall, granular backfill (γ = 120 pcf, φ = 32°, c = 0), 250 psf surcharge — typical highway right-of-way wall design assumption.

Rankine coefficients:
$$ K_a = \tan^2\!\left(45 - \tfrac{\phi}{2}\right), \qquad K_p = \tan^2\!\left(45 + \tfrac{\phi}{2}\right) $$
At-rest (Jaky, normally consolidated):
$$ K_0 = 1 - \sin\phi $$
Active thrust on wall, soil + surcharge:
$$ P_a = \tfrac{1}{2} K_a \, \gamma \, H^2 + K_a \, q \, H - 2 c \sqrt{K_a} \, H $$
Ka, Kp active and passive earth pressure coefficients · K0 at-rest coefficient · γ backfill unit weight · H wall height · q uniform surcharge · c cohesion · Pa, Pp total thrust per unit length of wall.

Active vs at-rest vs passive

The earth pressure on a wall depends on wall movement:

For a typical retaining wall, design uses Ka on the back face (driving force) and Kp at the toe (resistance), with FS > 1.5–2.0 against sliding and overturning.

Surcharge effects

Uniform surcharge q on the backfill adds Ka q H to the active thrust. The pressure distribution from surcharge is rectangular (constant with depth), unlike the triangular distribution from soil self-weight. Line surcharges (column footings, traffic loads) require Boussinesq's elastic solution and are not in this calculator.

Cohesive soil and the tension crack

For a c-φ backfill, Ka reduces the effective lateral pressure by 2c√Ka at every depth. This means the upper portion of the wall might compute negative active pressure — physically impossible because soil can't pull. A tension crack forms at depth z0 = 2c/(γ √Ka). For design, ignore the tension zone (treat the upper soil as if it weren't there) and recompute the thrust from below z0. Saturated cracks fill with water, adding hydrostatic pressure — so use Ka = 0.5 to 1.0 in the cracked zone for waterlogged conditions.

Wall friction (Coulomb)

Rankine assumes a smooth (zero friction) wall and horizontal backfill. Real walls have wall friction δ = (1/2 to 2/3) φ. Coulomb's earth pressure theory accounts for this and gives slightly lower Ka (cheaper design) and slightly higher Kp. For tall walls (H > 25 ft) or sloping backfill, use Coulomb. Rankine is conservative for active pressure.

Line of action

The triangular soil-weight pressure has its centroid at H/3 above the base. The rectangular surcharge pressure has its centroid at H/2. The combined line of action is a weighted average. This calculator reports the combined line of action above the base for the active thrust including surcharge.

Reference: Rankine, W.J.M. (1857). "On the Stability of Loose Earth." Phil. Trans. Royal Society, 147, 9-27. Das, B.M. (2014). Principles of Geotechnical Engineering, 8th ed., Cengage, ch. 13.

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