Lateral Earth Pressure Calculator (Rankine & Coulomb Ka, Kp, K0)
Calculate active, passive and at-rest earth pressure coefficients (Ka, Kp, K0) with Rankine or Coulomb theory, and the thrust and pressure diagram on a wall with sloping backfill, cohesion and surcharge.
How it works
Rankine, level backfill: Ka = tan²(45° − φ/2), Kp = tan²(45° + φ/2); sloping backfill: Ka = cos β (cos β − √(cos²β − cos²φ)) / (cos β + √(cos²β − cos²φ)) · Coulomb, vertical wall: Ka = cos²φ / {cos δ [1 + √(sin(φ+δ)·sin(φ−β) / (cos δ·cos β))]²} · K0 = 1 − sin φ · σa = Ka(γz + q) − 2c√Ka, σp = Kp(γz + q) + 2c√Kp · thrust = area of the pressure diagram; Pa = ½KaγH² at H/3 for c = q = 0.
Soil pushing on a wall that moves slightly away from it reaches the active state (lowest pressure, Ka); a wall pushed into the soil mobilises the passive state (highest, Kp); a rigid wall that cannot move carries the at-rest pressure K0, estimated with Jaky’s formula 1 − sin φ for normally consolidated soil. Rankine theory assumes a smooth wall; Coulomb’s wedge theory adds wall friction δ, which lowers Ka and tilts the thrust by δ. Cohesion reduces active pressure and creates a tension crack near the surface where the calculated pressure is negative — the calculator ignores that tension, as design practice does.
Worked example
A 5 m wall retains level sand with φ = 30°, γ = 18 kN/m³, no cohesion and no surcharge (Rankine). Ka = tan²(30°) = 0.3333, so the pressure at the base is 0.3333 × 18 × 5 = 30.0 kPa and the active thrust is ½ × 0.3333 × 18 × 5² = 75.00 kN/m, acting 5/3 = 1.67 m above the base. For comparison Kp = 3.0 (passive resistance 675 kN/m) and K0 = 0.5 (at-rest thrust 112.5 kN/m).
Sources: FHWA — Soils and Foundations Reference Manual Vol. II, §9 (earth pressure theories) · Lateral earth pressure — Rankine and Coulomb theories
Frequently asked questions
What is the difference between active, passive and at-rest pressure?
At-rest (K0) is the pressure on a wall that does not move. If the wall yields outward by about 0.1–0.4% of its height, the soil relaxes to the active state (Ka, smallest). If the wall is pushed into the soil by 1–5% of its height, the soil resists with passive pressure (Kp, largest).
When should I use Coulomb instead of Rankine?
Use Coulomb when wall friction matters — rough concrete or a battered wall — because it gives a lower active thrust inclined at δ. Rankine assumes a smooth wall and is simpler and slightly conservative for active pressure. Coulomb overestimates passive pressure when δ is large, so many engineers use log-spiral values for Kp.
Which pressure should I design a basement wall for?
A basement wall braced by floors at top and bottom cannot rotate enough to reach the active state, so it is designed for at-rest pressure (K0 ≈ 0.5 for sand). Free-standing cantilever retaining walls can rotate and are designed for active pressure.
How does a surcharge change the pressure?
A uniform surcharge q adds a constant pressure Ka·q over the full height, so its thrust is Ka·q·H acting at mid-height. Traffic is often modelled as 10–12 kPa (about 250 psf) of surcharge.
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This calculator is for educational purposes. Double-check important results before acting on them.