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Rankine Passive Earth Pressure Coefficient

Also known as Kp · passive earth pressure

Kp=tan⁡2 ⁣(45∘+ϕ2)K_p = \tan^{2}\!\left(45^\circ + \frac{\phi}{2}\right)
sig figs

Worked example: Kp = 3.6902 → φ = 35.000°. Press Try an example to run it live, then adjust anything.

Enter your known values, leave one input blank, and solves for the missing one. Tap a variable’s symbol to see what it means, with a typical value. Change any unit and the answer follows.

Where Rankine Passive Earth Pressure Coefficient fits

Your answer feeds
Plastic Zone Radius (Mohr–Coulomb)
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See also

Plastic Zone Radius (Mohr–Coulomb): Both use passive earth pressure coefficient.At-Rest Earth Pressure Coefficient (Jaky): Both use angle of internal friction.Bearing Capacity Factor Nc: Both use angle of internal friction.Bearing Capacity Factor Nq: Both use angle of internal friction.Culmann Planar Wedge Factor of Safety: Both use angle of internal friction.Infinite Slope Factor of Safety with Cohesion and Pore Pressure: Both use angle of internal friction.
4 more relatedInfinite Slope Factor of Safety with Slope-Parallel Seepage: Both use angle of internal friction.Infinite Slope Factor of Safety, Dry Cohesionless Soil: Both use angle of internal friction.Mohr–Coulomb Shear Strength: Both use angle of internal friction.Rankine Active Earth Pressure Coefficient: Both use angle of internal friction.
Learning zone

Rankine Passive Earth Pressure Coefficient explained

Kpφ

Passive pressure is the mirror image of active: instead of letting the soil spread, you shove the wall into it and ask how hard it pushes back. The soil has to lift and ride up a failure wedge, so the resistance is large: at φ = 30°, KpK_p = tan²(60°) = 3.0, exactly nine times the active coefficient. That factor of Kp/Ka=9K_p/K_a = 9 at 30° (and 13.6 at 35°) is the whole reason a sheet pile wall can be held by an embedded toe.

The trap is deformation. Full passive resistance needs wall movement of 2–4% of the wall height in dense sand and up to 10% in loose: one to two orders of magnitude more than active pressure needs. A retaining structure that is only allowed to move 5 mm has mobilised perhaps a quarter of KpK_p, which is why most codes ask you to divide passive resistance by a factor of 1.5 to 2 before relying on it, or to ignore the top metre entirely because it may be dug up for services. The second trap is wall friction: for passive pressure it helps a great deal, and Rankine's smooth-wall assumption is conservative here in a way it is not on the active side.

Rankine Passive Earth Pressure Coefficient formula

Kp=tan⁡2 ⁣(45∘+ϕ2)K_p = \tan^{2}\!\left(45^\circ + \frac{\phi}{2}\right)
Where
  • KpK_p= Passive earth pressure coefficient
  • ϕ\phi= Angle of internal friction (°)
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Rankine earth pressure

4 formulasActive, passive and at-rest earth pressure coefficients, and the thrust a retaining wall has to carry — three states set by which way the wall moves.

Revised 31 October 2026
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formula.expert. (2026, October 31). Rankine Passive Earth Pressure Coefficient Calculator. https://formula.expert/solvers/rankine-passive-coefficient

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