Ka, K0, Kp Calculator: Lateral Earth Pressure Coefficients

Soil pushes sideways, and how hard it pushes is one of the stranger facts in geotechnics: it depends on which way the wall moves. Let a retaining wall rotate away from the backfill by a fraction of a percent of its height and the soil mobilizes its own shear strength, dropping the push to the active minimum. Hold the wall perfectly rigid, as a braced basement wall or a heavy bridge abutment does, and the pressure stays at its geologic equilibrium, the at-rest state. Push the wall into the soil, as a foundation key or an anchor block does, and the soil resists with everything it has, the passive maximum. Three states, three coefficients, all multiplying the same vertical effective stress.

The formulas

• Active: Ka = tan^2(45 – phi/2), the Rankine minimum
• Passive: Kp = tan^2(45 + phi/2), the Rankine maximum, with the tidy identity Ka x Kp = 1
• At rest: K0 = 1 – sin(phi), the Jaky relation for normally consolidated soil
• Overconsolidated at rest: K0 x OCR^sin(phi), because soils remember past loads

Worked example

For a clean sand with phi = 30 degrees: Ka = 0.333, K0 = 0.50, Kp = 3.0. A nine-fold spread between the minimum push and maximum resistance from a single soil property. Add overconsolidation, say OCR = 4 from a preloaded clay, and K0 climbs to 1.0: the horizontal stress equals the vertical, which stiff clays routinely exhibit and heavily overconsolidated ones exceed.

Choosing the right coefficient, which is the real skill

The most common design error with these numbers is not miscalculating them but mispairing them with the structure. Active pressure requires movement, roughly 0.1 to 0.5 percent of wall height for sands, more for clays; a flexible cantilever retaining wall earns Ka, but a braced excavation strut or an integral bridge abutment that cannot yield must be designed for K0, which for phi = 30 is fifty percent more load. Passive resistance requires far more movement still, several percent of the embedment depth, before it fully mobilizes, which is why design codes divide Kp by a factor of 1.5 to 2: the wall would have to displace unacceptably far to collect the full theoretical resistance. And these Rankine values assume a vertical, frictionless wall with level backfill; wall friction and sloping backfill shift the numbers (helpfully for passive, modestly for active), which is what the Coulomb formulation and the retaining wall calculator on this site handle in full.

Related Calculators and Reference

These coefficients feed directly into the retaining wall pressure calculator, which turns them into forces and overturning checks, the friction angle input often comes from correlations with the SPT N-value calculator, and the same phi drives the bearing capacity calculator. A clear open reference is the lateral earth pressure article.